Dimensional Analysis for the Design of Satellites in...

Lehrstuhl fur Raumfahrttechnik

Prof. Dr. rer. nat.

Ulrich Walter

Diplomarbeit

RT-DA 2010/14

Dimensional Analysis

for the Design of Satellites

in LEO

Author: Tanja Nemetzade

Primary Advisor: Dipl.-Ing. Andreas Hein

Lehrstuhl fur Raumfahrttechnik / Institute of Aeronautics

Technische Universitat Munchen

Local Advisors: Prof. Olivier de Weck and Dr. Afreen Siddiqi

Engineering Systems Division

Massachusetts Institute of Technology

Bestatigung der eigenstandigen Arbeit

Ich erklare hiermit, dass ich diese Arbeit ohne fremde Hilfe angefertigt und nur die in dem

Literaturverzeichnis angefuhrten Quellen und Hilfsmittel benutzt habe.

Garching, den

Name: Tanja Nemetzade

Matrikelnummer 2723592

Page III

Danksagung

Ich danke meinen Professoren und Betreuern, die mir wahrend meines kompletten Studiums

ermoglicht haben, den Weg zu dieser Arbeit zu ebnen. Insbesondere danke ich Herrn Professor

Ulrich Walter und Herrn Dipl.-Ing. Andreas Hein von der TU Munchen sowie Herrn Professor

Olivier de Weck und Frau Dr. Afreen Siddiqi vom Massachusetts Institute of Technology

fur die Moglichkeit am MIT forschen zu durfen wie auch fur die fachliche und personliche

Unterstutzung.

Ein ganz besonderer Dank gilt meiner Familie und meinen Freunden fur ihre bedingungslose

Unterstutzung durch mein Studium hindurch und daruber hinaus. Vor allem danke ich meiner

Mutter von ganzem Herzen. Diese Arbeit sei ihr gewidmet.

Page IV

Zusammenfassung

In dieser Arbeit wird die Dimensionsanalyse als leistungsstarke Methode fur den Entwurf

von Satelliten vorgestellt. Diese Anwendung basiert auf der Annahme, dass mit Hilfe der

Ahnlichkeitstheorie, die sich dimensionsloser Kennzahlen aus der Dimensionsanalyse bedient,

Satelliten schneller und damit kostengunstiger entworfen werden konnen als es die gebrauch-

lichen Methoden erlauben. Ein weiterer Vorteil der Konzeption von Satelliten mit dimensions-

losen Großen ist die mogliche Uberprufung, ob ein Design sich in den richtigen Großenordnun-

gen bewegt. Außerdem wird gezeigt, dass die Dimensionsanalyse auch verwendet werden

kann, um Satellitenstandards in Ihrer Missionsperformance zu untersuchen und damit auch

zu uberdenken indem in einem neuartigen Ansatz physikalische Charakeristiken des Satelliten

mit der erreichbaren Missionsperformance des Raumflugkorpers in Beziehung gesetzt werden.

Ein besonderer Schwerpunkt der Arbeit liegt auf der Anwendung der theoretischen Ergebnisse

auf den CubeSat-Standard da diese Satellitenklasse an Bedeutung gewinnt, jedoch nur wenige

systemspezifische Ergebnisse bis jetzt vorhanden sind.

Nach einer Einfuhrung in die theoretischen Grundbegriffe dieser Arbeit, die CubeSat-spezifische

Definitionen wie auch Begriffsdefinitionen aus der Dimensionsanalyse aufgreift, folgt die Vorstel-

lung des Buckingham-Π-Theorems. Diese vor allem in der Aerodynamik verwendete Methode

aus der Dimensionsanalyse wird in dieser Arbeit genutzt um den Satelliten mit dimensionslosen

Kennzahlen zu modellieren. Die Anwendung der theoretischen Grundlagen auf den Satellite-

nentwurf erfolgt mit zwei verschiedenen Ansatzen. Der Top-Down-Ansatz bedient sich als

Ausgangspunkt nutzlastspezifischer Parameter, wohingegen der Bottom-Up-Ansatz Charak-

teristiken des Satelliten als Eingangsgroßen fur den Satellitenentwurf verwendet. Als Einstieg

in die Dimensionsanalyse werden zunachst im Satellitenentwurf gebrauchliche Verhaltnisse an-

hand von Daten von nicht-geosynchronen Kommunikationssatelliten validiert. Anschließend

werden dimensionslose Kennzahlen fur einen Top-Level Ansatz und den Entwurf von drei ex-

emplarischen Subsystemen und einer optischen Nutzlast hergeleitet. Die Analysen munden

in der Erstellung eines sogenannten Missionsperformanceindex, welcher die Quantifizierung

des Missionserfolgs eines gegebenen Satelliten ermoglicht. Eine Anwendung der theoretis-

chen Ergebnisse anhand von drei verschiedenen hypothetischen Satelliten veranschaulicht ab-

schließend das praktische Vorgehen mit dimensionslosen Kennzahlen. Insbesondere der dritte

Beispielfall, ein 2U CubeSat, verdeutlicht das Potential der Ergebnisse dieser Arbeit, namlich

die Moglichkeit Satellitenstandards auch in Ihrer Missionsleistung zu untersuchen und sie da-

raufhin gegebenenfalls zu andern.

Dass die Dimensionsanalyse vielseitig einsetzbar ist, zeigt sich anschließend in einer kurzen

Einfuhrung in die potentielle Anwendung der Ergebnisse auf Satellitencluster und der damit

verbundenen Moglichkeit anhand der Dimensionsanalyse die Missionsperformance von Clus-

Page V

Zusammenfassung

terarchitekturen mit jenen von einzelnen Satelliten zu vergleichen. Der Appendix mit einer

Detaillierung der Berechnung der Eklipsenzeit veranschaulicht schließlich beispielhaft, dass die

Modellierung des Satelliten in dieser Arbeit nicht die vollstandige Komplexitat der Wirkungszu-

sammenhange der einzelnen Parameter in der Realitat darstellt und folglich weiterentwickelt

werden muss. Eine kurze Beschreibung einer im Rahmen dieser Arbeit erstellten Datenbank,

die die momentan auf dem Markt erhaltlichen CubeSat-spezifischen Standardkomponenten

zusammenfasst, schließt die Diplomarbeit ab.

Page VI

Abstract

In this work the powerful technique of dimensional analysis is used to benefit from the simi-

larity of spacecrafts and missions to facilitate and accelerate the design of newly developed

satellites. By means of the Buckingham-Π-Theorem, a widely employed method in dimen-

sional analysis, ratios and non-dimensional similarity parameters have been identified in this

work which formalize and facilitate the comparison between the characteristics of satellites of

different sizes. As a consequence new satellite designs can be based on results from former

missions and existing designs can be verified. Furthermore it will be shown that dimensional

analysis can be used in a newly developed approach to examine the mission performance of a

satellite and therefore provide a method for the verification of satellite standards by relating

physical characteristics of the spacecraft to the achievable mission performance of the satellite.

A special emphasis during the work is put on the application of the theoretical results on the

CubeSat standard as this satellite class is gaining more and more significance but only few

systems engineering guidelines are developed for them.

After the presentation of the theoretical basis of this work, including an introduction to the

CubeSat standard, a presentation of the most common notions in dimensional analysis and

an overview of the history and applications of dimensional analysis is given. Subsequently, an

introduction to the Buckingham-Π-Theorem with an exemplary application in aerodynamics

is presented. The theoretical basis will be applied to two different design approaches: in the

Top-Down approach payload specific quantities will be the input parameters for the satellite

design whereas the Bottom-Up approach uses spacecraft specific variables as point of depar-

ture for the satellite design. The analysis firstly consists of the validation of ratios common in

satellite design by means of data from non-geosynchronous communication satellites. After-

wards, non-dimensional parameters for a top-level approach, three exemplary subsystems and

an optical payload are derived. These results are then used to create the Mission Performance

Index which enables the quantification of the mission accomplishment of a satellite. The the-

oretical results are finally applied on three different hypothetical satellites. Especially the third

example, a 2U CubeSat, shows the potential of the results of this work, namely the possibility

to quantify the mission performance of satellite standards in order to optimize them.

A short introduction to the application of dimensional analysis on cluster architectures is given

afterwards, proving that dimensional analysis can be used to develop a possibility to compare

single satellite missions with cluster architectures. Finally, the appendix deals with an accu-

rate calculation of the eclipse time, showing indirectly the complexity of the interdependencies

between the various quantities in the system and the need for further detailed development

of the theoretical results. The work finishes with a brief presentation of a CubeSat specific

COTS component database, which was also created during this work.

Page VII

Contents

1. Introduction 1

2. Theoretical Basis 4

2.1. CubeSat-Specific Definitions and Standards . . . . . . . . . . . . . . . . . . . 4

2.2. International Vocabulary in Metrology . . . . . . . . . . . . . . . . . . . . . . 6

2.3. Standardized Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.4. Dimensional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4.1. Buckingham-Π-Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4.2. Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.4.3. Advantage of the Use of Dimensional Analysis during the Design Phase

of Satellites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4.4. Limits of the Buckingham-Π-Theorem and Dimensional Analysis . . . . 17

3. Dimensional Analysis of a Single Satellite 19

3.1. Design Approaches: Top-Down and Bottom-Up . . . . . . . . . . . . . . . . . 19

3.2. Proceeding of the Validation of the Non-Dimensional Parameters . . . . . . . 21

3.3. Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.3.1. Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.3.2. Volume: Packing Factor . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.3.3. Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.3.4. Mass and Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.4. Non-Dimensional Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.4.1. A Top-Level Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.4.2. Subsystem Power: Battery and Solar Array . . . . . . . . . . . . . . . 63

3.4.3. Subsystem AOCS: Reaction Wheel . . . . . . . . . . . . . . . . . . . 70

3.4.4. Subsystem Communication . . . . . . . . . . . . . . . . . . . . . . . 75

3.4.5. Payload . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

3.5. Mission Performance Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

3.6. Application of the Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

3.6.1. The CubeSat - a Bottom-Up Approach . . . . . . . . . . . . . . . . . 90

3.6.2. The NGSO-Satellite - a Top-Down Approach . . . . . . . . . . . . . . 96

3.6.3. A New CubeSat Standard? - a Bottom-Up Approach . . . . . . . . . . 103

4. Dimensional Analysis of Clusters 112

4.1. The F6-Project and the Concept of Fractionated Spacecrafts . . . . . . . . . . 112

Page IX

Contents

4.2. Ratios and Non-Dimensional Parameters . . . . . . . . . . . . . . . . . . . . 113

4.3. Mission Performance Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

5. Conclusion and Future Prospects 117

A. Calculation of the Eclipse Time tEclipse 118

B. CubeSat-Specific Commercial off-the-Shelf Component Database 121

Page X

List of Figures

1.1. Thesis roadmap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1. Artistic view of a 1U CubeSat in orbit . . . . . . . . . . . . . . . . . . . . . . 4

2.2. 3U P-POD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3.1. Top-Down design approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2. Bottom-Up design approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.3. Mass ratios of NGSO-satellites . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.4. Mass ratios of NGSO-satellites plotted against mS/Cdry. . . . . . . . . . . . . 26

3.5. Mass ratios of NGSO-satellites plotted against mS/Cdryexcluding

mS/Cwet

mS/Cdry

. . . 27

3.6. Mass ratios of NGSO-satellites with mS/Cdryabove 1000 kg . . . . . . . . . . 27

3.7. Mass ratios of NGSO-satellites with mS/Cdryunder 300 kg . . . . . . . . . . . 28

3.8. Histogram formS/Cwet

mS/Cdry

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.9. Histogram formP/L

mS/Cdry

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.10. Histogram for mAOCS

mS/Cdry

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.11. Histogram for mStructure

mS/Cdry

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.12. Histogram for mPower

mS/Cdry

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.13. Histogram for mThermal

mS/Cdry

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.14. Histogram for mC&DH+TT&C

mS/Cdry

. . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.15. Histogram formPropulsion

mS/Cdry

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.16. 1U CubeSat MOVE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.17. Exploded view of the CubeSat MOVE . . . . . . . . . . . . . . . . . . . . . . 32

3.18. mStructure

mS/Cdry

for hypothetical CubeSats, calculated based on data from COTS-

components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.19. mStructure

mS/Cdry

for hypothetical CubeSats, calculated based on Aluminium wall as-

sumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.20. VStructure

VS/Cfor hypothetical CubeSats, calculated based on Aluminium wall as-

sumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34


mP/L. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.22. Histogram formS/Cdry

mP/L. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.23. Histogram for mAOCS

mP/L. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.24. Histogram for mStructure

mP/L. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.25. Histogram for mPower

mP/L. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Page XI

List of Figures

3.26. Histogram for mThermal

mP/L. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.27. Histogram for mC&DH+TT&C

mP/L. . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.28. Histogram formPropulsion

mP/L. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.29.PP/L

PSAeol

and PBus

PSAeol

plotted against PSAeol. . . . . . . . . . . . . . . . . . . . . 43

3.30.PP/L

PSAeol

and PBus

PSAeol

plotted against PSAeolwithout outlier . . . . . . . . . . . . . 43

3.31.PP/L

PSAeol

and PBus

PSAeol

plotted against mS/Cdry. . . . . . . . . . . . . . . . . . . . 44

3.32.PSAeol

PP/Land PBus

PP/Lplotted against mS/Cdry

without outliers . . . . . . . . . . . . 44

3.33. Histogram forPP/L

PSAeol

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.34. Histogram for PBus

PSAeol

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.35. Histogram forPSAeol

PP/L. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.36. Histogram for PBus

PP/L. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.37.mS/Cdry

mP/L· PSAeol


. . . . . . . . . . . . . . . . . . . . . 48

3.38.mP/L

mS/Cdry

· PP/L

PSAeol

plotted against mS/Cdry. . . . . . . . . . . . . . . . . . . . . 49

3.39.mS/Cdry

mP/L· PP/L

PSAeol


3.40.mP/L

mS/Cdry

· PSAeol


. . . . . . . . . . . . . . . . . . . . . 49


mP/L· PSAeol

PP/L. . . . . . . . . . . . . . . . . . . . . . . . . . 50


mS/Cdry

· PP/L

PSAeol

. . . . . . . . . . . . . . . . . . . . . . . . . . 50


mP/L· PP/L

PSAeol

. . . . . . . . . . . . . . . . . . . . . . . . . . 50


mS/Cdry

· PSAeol

PP/L. . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.45.mS/Cwet

mP/L· PS/Ceol

PP/Lplotted against mS/Cwet . . . . . . . . . . . . . . . . . . . . 51

3.46.mP/L

mS/Cwet

· PP/L

PS/Ceol


3.47.mS/Cwet

mP/L· PP/L

PS/Ceol

plotted against mS/Cwet . . . . . . . . . . . . . . . . . . . . 51

3.48.mP/L

mS/Cwet

· PS/Ceol

PP/Lplotted against mS/Cwet . . . . . . . . . . . . . . . . . . . . 52


mP/L· PS/Ceol

PP/L. . . . . . . . . . . . . . . . . . . . . . . . . . 52


mS/Cwet

· PP/L

PS/Ceol

. . . . . . . . . . . . . . . . . . . . . . . . . . 52


mP/L· PP/L

PS/Ceol

. . . . . . . . . . . . . . . . . . . . . . . . . . 52


mS/Cwet

· PS/Ceol

PP/L. . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.53. mS/Cwet plotted against tOrbit . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.54. PSAeolplotted against tOrbit . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.55. ρS/C plotted against tOrbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.56. Histogram for Π1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.57. Histogram for Π1 for the NGSO-satellites in LEO . . . . . . . . . . . . . . . . 58

3.58. Π∗3 plotted against mS/Cwet . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.59. Zoom of Π∗3 plotted against mS/Cwet . . . . . . . . . . . . . . . . . . . . . . 59

Page XII

List of Figures

3.60. Π∗3 plotted against tOrbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.61. Zoom of Π∗3 plotted against tOrbit . . . . . . . . . . . . . . . . . . . . . . . . 59

3.62. Histogram for Π∗3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.63. Histogram for Π∗3 for the NGSO-satellites in LEO . . . . . . . . . . . . . . . . 60


3.65. Zoom of Π∗4 plotted against mS/Cwet . . . . . . . . . . . . . . . . . . . . . . 62


3.67. Histogram for Π∗4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.68. Histogram for Π∗5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.69. Top-Down network flow diagram for an Earth Observation mission . . . . . . . 110

3.70. Bottom-Up network flow diagram for an Earth Observation mission . . . . . . 111

B.1. Screenshot of a part of the AOCS-sheet of the CubeSat COTS component

database . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

Page XIII

List of Tables

2.1. The characteristics of the most common CubeSats . . . . . . . . . . . . . . . 5

2.2. SI base quantities, units and dimensions . . . . . . . . . . . . . . . . . . . . . 9

3.1. xav, s and sxav

for Bottom-Up mass ratios - Part 1 . . . . . . . . . . . . . . . 25


for Bottom-Up mass ratios - Part 2 . . . . . . . . . . . . . . . 26

3.3. xav for mass ratios for smaller NGSO-satellites, a hypothetical 1U CubeSat and

real flown/planned CubeSats - Part 1 . . . . . . . . . . . . . . . . . . . . . . 30

3.4. xav for mass ratios for smaller NGSO-satellites, a hypothetical 1U CubeSat and

real flown/planned CubeSats - Part 2 . . . . . . . . . . . . . . . . . . . . . . 31


for Bottom-Up mass ratios with mS/Cwet - Part 1 . . . . . . . . 35


for Bottom-Up mass ratios with mS/Cwet - Part 2 . . . . . . . . 35


for Top-Down mass ratios - Part 1 . . . . . . . . . . . . . . . 36


for Top-Down mass ratios - Part 2 . . . . . . . . . . . . . . . 36

3.9. xeq for standard CubeSats . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40


for power ratios . . . . . . . . . . . . . . . . . . . . . . . . . 41


for power ratios excluding outliers . . . . . . . . . . . . . . . . 42


for rMass Power 1 to rMass Power 4 with mS/Cdry. . . . . . . . . 47


for rMass Power 1 to rMass Power 4 with mS/Cdrywithout outlier . 47


for rMass Power 1 to rMass Power 4 with mS/Cwet . . . . . . . . . 47


for rMass Power 1 to rMass Power 4 with mS/Cwet without outlier . 47


for Π1 for all NGSO-satellites and those in LEO . . . . . . . . 57


forPS/C

VS/C23mS/Cwet

for all NGSO satellites and those in LEO . . . 57


for Π∗4, Π∗5 and Π∗6 . . . . . . . . . . . . . . . . . . . . . . . . 62

3.19. Top-level Mission Performance Parameters . . . . . . . . . . . . . . . . . . . 82

3.20. Ranges for the subsystem exponents α, β, χ, . . . for single satellites . . . . . . 88

3.21. Mass ratios for the hypothetical 1U CubeSat - Part 1 . . . . . . . . . . . . . 91


3.23. Mass ratios for the hypothetical NGSO-satellite - Part 1 . . . . . . . . . . . . 97

3.24. Mass ratios for the hypothetical NGSO-satellite - Part 2 . . . . . . . . . . . . 97



4.1. Ranges for the subsystem exponents α, β, χ, . . . for cluster architectures, in-

cluding tendency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

Page XV

Symbols

AS/C m2 spacecraft surface area

b bitspixel

number of bits to encode each pixel

BER − bit error rate

c ms

velocity of sound

CBattery Ws capacity of the battery

cSolarWm2 solar constant

d 1yr

degradation per year

D m aperture diameter of payload instrument

DAntenna m antenna diameter

DTransmitter m transmitter antenna diameter

DReceiver m receiver antenna diameter

e ◦ pointing loss

ECA ◦ Earth Central Angle

f 1s

frequency of communication signal

GAntenna − antenna gain

GParaboleAntenna − gain of parabole antenna

hOrbit m orbital altitude

HRWkgm2

sangular momentum of reaction wheel

i − number of non-dimensional products in a system

i ◦ orbit inclination

IRW kgm2 reaction wheel moment of inertia

IS/C kgm2 spacecraft moment of inertia

k − number of quantities in the k-set of a system

ki variable part of a non-dimensional parameter

l m distance between two points

Page XVII

Symbols

Li − communication losses

Ma − Mach number

mi kg mass

n − number of quantities in a system

p % packing factor

Pi W power consumption or production

PTransmitter W transmitter output power

Qi variable quantity

r − ratio

R bitss

data rate

REarth m radius of the Earth

Re − Reynolds number

s − standard deviation

SCom m communication path length

SSolarWm2 product of cSolar and cos(θ)

SW m swath width

tOrbit s orbital period

tSatellite s satellite time

tEclipse s eclipse time

tDaylight s daylight time

TAero Nm aerodynamic torque

TSolar Nm solar radiation torque

TGravity Nm gravity torque

TOperating K operating temperature

Vi m3 volume

vims

velocity

X mpixel

cross-track ground pixel resolution

x mm dimension of a CubeSat along the x-axis

xav variable arithmetical average

xdeployed m equivalent edge length of spacecraft with deployed devices

xeq m equivalent edge length

Y mpixel

along-track ground pixel resolution

y mm dimension of a CubeSat along the y-axisPage XVIII

Symbols

z mm dimension of a CubeSat along the z-axis

βs◦ incident sun angle on the orbital plane

ε ◦ elevation angle

η − antenna efficiency

λi m wavelength

µ m3

s2Earth’s gravity constant

ωRW1s

angular velocity of reaction wheel

Φ ◦ angular of the eclipse from the orbit

Πi - non-dimensional parameter

Ψi variable component of the Mission Performance Index

ρ ◦ angular radius of the Earth

ρikgm3 mass density

θ ◦ average incident sun angle on the spacecraft surface

Page XIX

Abbreviations

AOCS Attitude and Orbit Control System

BER Bit Error Rate

BOL Begin of Life

C&DH Command and Data Handling

Cal Poly California Politechnic State University

CDS CubeSat Design Specification

CGPM Conference General de Poids et Mesures (French for General Confer-

ence on Weights and Measures

Com Communication

COTS Commercial Off-The-Shelf

DARPA Defense Advanced Research Projects Agency

DOD Depth of Discharge

ECA Earth Central Angle

EOL End of Life

EOM Earth Observation mission

ESA European Space Agency

Fig Figure

GEO Geostationary Orbit

IEC International Electrotechnical Commission

ISO International Organisation for Standardization

ISQ International Systems of Quantities

LEO Low Earth Orbit

LRT Lehrstuhl fur Raumfahrttechnik

MIT Massachusetts Institute of Technology

MPI Mission Performance Index

Page XXI

Abbreviations

MPP Mission Performance Parameter

NASA National Aeronautics and Space Administration

NGSO Non-Geosynchronous Orbit

P/L Payload

POD Picosatellite Orbital Deployer

Prop Propulsion

SA Solar Array

S/C Spacecraft

SI Systeme International d’unites (French for International System of

Units)

SSM Space Science mission

SSDL Space Systems Development Laboratory

Tab Table

TDM Technology Demonstration mission

TT&C Telemetry, Tracking and Command

TUM Technische Universitat Munchen

VIM Vocabulaire international des termes fondamentaux et generaux de

metrologie (French for International vocabulary of basic and general

terms in metrology)

U Unit

Page XXII

1. Introduction

In this work the powerful technique of dimensional analysis is used to benefit from the sim-

ilarity of spacecrafts and missions to facilitate and accelerate the design of newly developed

satellites. By means of the Buckingham-Π-Theorem, a widely employed method in dimen-

sional analysis, ratios and non-dimensional similarity parameters have been identified in this

work which formalize and facilitate the comparison between the characteristics of satellites of

different sizes. As a consequence new satellite designs can be based on results from former

missions and existing designs can be verified. It will be shown that dimensional analysis can

be used to examine the mission performance of a satellite and therefore provide a method

for the verification of satellite standards by relating physical characteristics of the spacecraft,

like its size, to the achievable mission performance of the satellite - this being a connection

which have been not implemented yet in current design approaches. Two approaches will be

in the focus of this work: the results of the payload-centric Top-Down approach will enable

the engineer to predict the design characteristics of a satellite correspondingly to a required

Mission Performance Index. In the spacecraft-centric Bottom-Up approach the mission ac-

complishment of a satellite with set design characteristics will be predicted by means of an

achievable Mission Performance Index. Consequently, the Mission Performance Index will also

be an assessment of the efficiency of satellite classes.

A special emphasis during the work is put on the application of the theoretical results on the

CubeSat standard as this satellite class is gaining more and more significance but only few

systems engineering guidelines are developed for them. Especially the efficiency of the CubeSat

standard will be in the focus of this work as the efficiency of this satellite concept is yet not

investigated.

Within the last 20 years CubeSats have established themselves in the educational program

of many universities worldwide, in order to provide its students hands-on-experience and to

offer universities and industries a low-cost and quickly designed solution to test their new

technologies and hardware components under real conditions. Also companies are showing

increasing interest in picosatellites, trying to exploit their enormous potential (e.g. QuakeSat

of Quakefinder launched in 2003, AeroCube 2 of The Aerospace Corporation, CSTB from

Boeing and MAST of Tethers Unlimited Inc., all last three launched in 2007). Conferences

and workshops around the world are hold to debate over the use of CubeSats and small satel-

lites (in 2010 i.a.: the 7th Annual CubeSat Developers’ Workshop by Cal Poly within the

24th Small Satellite Conference in Utah, USA, the 3rd European CubeSat Workshop initiated

by ESA in Noordwijk, Netherlands) and even the market is reacting to the increasing need

of CubeSat specified commercial off-the-shelf components, accelerating the development of

CubeSats even more by creating new businesses which offer only CubeSat related products.

Page 1

Currently 38 CubeSats were successfully launched of which 19 are still operational (two of them

since 2003) and two other launches are upcoming: Taurus XL is planned to be launched in

February 2011, bringing three more CubeSats in position and ESA’s new launch rocket VEGA

is supposed to carry eleven CubeSats into orbit on its maiden flight mid 2011. Furthermore,

NASA recently announced the second round of its CubeSat launch initiative, enabling more

CubeSat missions in 2011 and 2012 [NASA, 2010]. Another approximately 70-100 missions

are planned throughout the institutions of the world in the next years, above all the campaign

QB50 by the Von Karman Institute, ESA and NASA [Institute, 2010].

However, the mission efficiency of the CubeSat standard was not yet sufficiently questioned.

Consequently, this work tries to analyze the efficiency of the CubeSat standard with the help

of the Mission Performance Index and gives guidelines to a possible new CubeSat standard

which combines all the advantages of the current CubeSats as their cost-, size- and mass-

effectiveness but also a higher Mission Performance Index.

Furthermore, the work can be also used to create a design tool based on dimensional anayl-

sis which accelerates the design of CubeSats. Some related work has been already done by

[Aas et al., 2009a], [Aas et al., 2009b], however, not by means of dimensional analysis and not

providing a possibility for the quantification of the mission performance of a satellite either.

A method for the comparison of single satellite missions and cluster architectures in terms of

their mission performance is also believed to be not yet developed. First answers to this open

research questions are given in this work.

After the introduction, chapter 2 provides the theoretical basis of this work, including an

introduction to the CubeSat standard in 2.1, a presentation of the most common notions

in dimensional analysis in 2.2 and 2.3 and an overview of the history and applications of

dimensional analysis in 2.4. An introduction to the Buckingham-Π-Theorem in 2.4.1 with

an exemplary application in aerodynamics is given in 2.4.2, as well as an assessment of the

theorem’s assets and drawbacks in satellite design in 2.4.3 and 2.4.4. Chapter 3 presents

the results of the dimensional analysis for satellite design by means of ratios in 3.3, non-

dimensional parameters for the spacecraft and its subsystems in 3.4 as well as the Mission

Performance Indices in 3.5. Afterwards, the theoretical results are applied on three different

hypothetical satellites in 3.6. A short introduction to the application of dimensional analysis

on cluster architectures is given in chapter 4, proving the power of dimensional anaylsis based

on its various application areas. A conclusion of the work, based especially on the results in

chapter 3, is given in chapter 5. Finally, Appendix A deals with an accurate calculation of the

eclipse time, showing indirectly the complexity of the interdependencies between the various

quantities in a system. The work finishes with Appendix B which provides a brief presentation

of a CubeSat specific COTS component database, which was also created during this work.

Page 2

Introduction

Figure 1.1.: Thesis roadmap

Page 3

2. Theoretical Basis

2.1. CubeSat-Specific Definitions and Standards

The CubeSat standard originates from the efforts of the California Polytechnic State Univer-

sity, San Luis Obispo (Cal Poly) and the Space Systems Development Laboratory (SSDL) at

Stanford University in 1999 within the CubeSat program in order to develop a new class of

satellites [Heidt et al., 2001], [Toorian et al., 2005].

In its standard configuration the satellite measures 100.00 mm x 100.00 mm x 113.50 mm

with a weight of up to 1.33 kg and notably uses commercial off-the-shelf (COTS) electrical

components. This configuration is known as 1 unit CubeSat or 1U CubeSat. Other built

and launched standardized configurations are 2U (100.00 mm x 100.00 mm x 227.00 mm,

mass up to 2.66 kg) and 3U (100.00 mm x 100.00 mm x 340.50 mm, mass up to 4.00 kg)

CubeSats. Smaller formats like 0.5U CubeSats and larger ones (4U, 5U and 6U) are also

possible. The CubeSat Design Specification (CDS) [CalPoly, 2009] defines all these standards

by stating the nominal dimensions of the standard 1U CubeSat, dimension tolerances, the

reference coordinate system, acceptable materials and other information.

Please note that the presented density of the CubeSats in table 2.1 represents the theoretical

upper bound density. It is calculated assuming homogeneous mass distribution in the CubeSat,

thus

ρCubeSat =mCubeSat

VCubeSat(2.1)

with

VCubeSat = x · y · z (2.2)

where x, y and z represent the dimensions of the CubeSat.

Figure 2.1.: Artistic view of a 1U CubeSat in orbit [NASA, 2010]

Page 4

Theoretical Basis

Table 2.1.: The characteristics of the most common CubeSats

System Dimensions Mass Max. Density

mm mm mm kg kgm3

1U 100.00 100.00 113.50 1.33 1171.81

2U 100.00 100.00 227.00 2.66 1171.81

3U 100.00 100.00 340.50 4.00 1171.81

In practice, this density is never achieved and the packing factor p is of more significance for

assembly and integration of the CubeSat. A more detailed discussion about the differentiation

between ρS/C and p is given in section 3.3.2.

Figure 2.2.: 3U P-POD [Cal Poly, 2010]

A unique feature of the CubeSat program is the use of a standard deployment system: the

CubeSats are launched in standardized Picosatellite Orbital Deployers (PODs) as piggybacks

of large spacecrafts and hence achieve their orbit at greatly reduced costs. The POD is

the interface between the launch vehicle and the CubeSats. Its most important objective is

to guarantee the protection of the launch vehicle and other payloads from any mechanical,

electrical or electromagnetic interference from the CubeSats in the event of a catastrophic

CubeSat failure. Beyond that, the POD enables a simple implementation of the CubeSats

with most launch vehicles. The most popular version is the P-POD, the Poly-POD, designed

and built by Cal Poly [CalPoly, 2007]. It is a rectangular container of anodized aluminium.

After a signal is sent from the launch vehicle to the release mechanism, a spring-loaded door

opens and the CubeSats are deployed by a spring and glide along smooth flat rails into their

orbits. The most common configuration is to launch three 1U CubeSats in one P-POD.

However, also CubeSats of different lengths can be accommodated in the same P-POD (e.g.

one 1U and one 2U CubeSat) which gives the launch provider some flexibility.

Page 5

International Vocabulary in Metrology

2.2. International Vocabulary in Metrology

Before explaining the theoretical foundation of this work, namely the Buckingham-Π-Theorem,

it is important for the later comprehension to clarify some notions and expressions which will be

used in this work. Most of them are essential and widely used without further thoughts about

their definitions but are mentioned here for the sake of completeness. All citations in this sec-

tion are extracted from the International Vocabulary in Metrology (VIM) [JCGM/WG2, 2008],

the standard work for terms in metrology. One of the basic terms frequently used in this work

is quantity. According to the VIM, it is the ”property of a phenomenon, body, or substance,

to which a magnitude can be assigned”. Hence, quantities are e.g. the mass and the length of

a body. In this context a so called system of quantities is a ”set of quantities together with

a set of non-contradictory equations relating those quantities”. A sizable amount of different

systems is possible as many different combinations of quantities are thinkable. Therefore it is

always important to indicate the system of quantity which is used as it can differ from case to

case and especially from scientist to scientist. An important example for a system of quantities

is the International System of Quantities (ISQ), which will be explained in the next section.

So called base and derived quantities are part of a system of quantities. The definition of a

base quantity states that it is ”chosen by convention, used in a system of quantities to define

other quantities”. Hence, the characteristic of a base quantity is the fact that it cannot be

derived from other quantities with the help of an equation. The base quantities are therefore

independent by definition. We will see that seven independent base quantities are used in the

today’s most common unit system, the International System of Units [BIPM, 2006], from

which all other quantities can be derived with the help of the equations within the respec-

tive system of quantities, namely the ISQ. The choice of the independent base quantities is

made more or less arbitrarily with regard to the actual scientific theories in order to choose

the quantities which are the most convenient. In the end, it is up to every scientist to fix

his/her base quantities as it is possible to establish a sizable number of system of quanti-

ties. Consequently to the definition of a base quantity, a derived quantity is a ”quantity, in a

system of quantities, defined as a function of base quantities”. Velocity is an example for a de-

rived quantity in the ISQ, as it is composed of length and time, both base quantities in the ISQ.

Physical quantities are organized in a system of dimensions. A dimension is the ”dependence

of a given quantity on the base quantities of a system of quantities, represented by the product

of powers of factors corresponding to the base quantities”. The symbolic representation of

the dimension of a base quantity (in the following referred to as base dimension) is a single

upper case letter in roman (upright) sans-serif type and its denotation equates the name of

the respective base quantity (e.g. M for mass or L for length). The symbolic representation

of the dimension of a derived quantity (in the following referred to as derived dimension) is

the product of powers of the dimensions of the base quantities using the equations that relate

Page 6

Theoretical Basis

the derived quantities to the base quantities. Consequently, the dimension of any quantity Q

can be derived from the base dimensions in the following form

dimQ = D1αD2

βD3γD4

δ . . .Dnν (2.3)

with D1, . . . ,Dn as the n base dimensions in the chosen system of quantities and the expo-

nents α, β, γ, δ, . . . , ν as the n dimensional exponents. The dimensional symbols and expo-

nents are manipulated using the usual rules of algebra. This notation and the fact that every

dimension can be expressed in terms of base dimensions is a fundamental requirement for the

Buckingham-Π-Theorem. It is important to notice that quantities having the same dimension

are not necessarily quantities of the same kind. An example is given by the quantities torque

and work/heat in the ISQ where all three quantities have the same dimension ML2T−2 with

the length L, the mass M and the time T as base dimensions. As we will see later on, so called

quantities of dimension one or dimensionless quantities play a very essential role in the

Buckingham-Π-Theorem. Therefore we quote its definition here: a dimensionless quantity is

a ”quantity for which all the exponents of the factors corresponding to the base quantities in

the representation of its dimension are zero”. However, the term ’quantity of dimension one’

reflects the convention that the symbolic representation of the dimension for such quantities

is the symbol 1. Examples of dimensionless quantities are the Reynolds number Re and the

Mach number Ma, both fundamental quantities in aerodynamics which can be derived by

using the Buckingham-Π-Theorem. The derivation of the former one will be shown in section

2.4.2. In everyday life the wordings ’unit’ and ’dimension’ can be misunderstood as synonyms.

Buckingham himself did not separate the terms sharply in his statements to dimensional anal-

ysis [Buckingham, 1914], [Buckingham, 1915a], [Buckingham, 1915b]. In order to clarify the

difference between them, the definition of the term unit shall be given: a unit is a ”scalar

quantity, defined and adopted by convention, with which other quantities of the same kind are

compared in order to express their magnitudes”. Hence when talking about masses, for exam-

ple, we indirectly compare the mass of the considered object to the world’s standard kilogram

in Paris. Several units can be chosen for a quantity with a dimension (e.g. ft, inch, mile, mm,

cm, m, and km for the dimension length), but only one dimension can clearly refer to a unit.

So, when building dimensionless expressions, non-dimensionality can be also achieved when

one quantity in the equation is given in inch and the other in cm. Only a conversion factor

has to be considered in that case. However, for the sake of coherence, it is paid attention

in this work to always use the same units in numerator and denominator before cancelling

down. Another important point to notice is that conformable to the fact that the symbol 1

denotes the dimension of a dimensionless quantity, the units of quantities of dimension one

are simply numbers, e.g. radian and steradian. Pursuant to the existence of base dimensions

for base quantities, there also exist base units which are defined to be ”conventionally and

uniquely adopted for a base quantity in a given system of quantities”. We will see that all the

base quantities in the ISQ have corresponding base units which are designated in a well-known

system of units, namely the International System of Units (SI). Conformable to the definition

of a base unit, a derived unit is a ”unit for a derived quantity”. Thus the meter per second

Page 7

Standardized Systems

is a derived unit of velocity in the SI as meter and second are base units in the SI for the base

quantities length and time in the ISQ.

As we already mentioned the International System of Units as a system of units, it may be

useful to state its definition: it is a ”conventionally selected set of base units and derived units,

and also their multiples and submultiples, together with a set of rules for their use”. Multiples

and submultiples of units are simply units ”formed from a given unit by multiplying/dividing

by an integer greater than one” e.g. the kilometer is a multiple and the millimeter a submultiple

of meter.

The following definitions shall complete the list of definitions: A coherent derived unit is

a ”derived unit that, for a given system of quantities and for a chosen set of base units, is

a product of powers of base units with the proportionality factor one”. A coherent system

of units is a ”system of units, based on a given system of quantities, in which the unit for

each derived quantity is a coherent derived unit”. The value of a quantity or the value is

the ”magnitude of a quantity represented by a number and a reference”. A quantity value

can be expressed inter alia as a product of a number and a unit, or a number for a quantity

of dimension one (the unit one is generally not written out). A quantity equation is an

”equation relating quantities”. Correspondingly, an unit equation is an ”equation relating

units” and a numerical value equation or a numerical quantity value equation is an

”equation relating numerical quantity values”. In order to convert values of quantities of the

same kind but with different units conversion factors between units are necessary which

are defined as ”ratio of two units for quantities of the same kind”, e.g. ft/cm = 30,48 and

thus 1 ft = 30,48 cm. Please note finally that in literature the term ”base” is often used

synonymously with ”fundamental”. Thus fundamental units/dimensions/quantities are base

units/dimensions/quantities.

2.3. Standardized Systems

One of the fundamental requirements of our everyday communication is the fact that we express

our scientific knowledge in standardized systems of base and derived quantities, units and

dimensions. Without a standardization, not only our everyday life would be more complicated

(conversions between the European and Anglo-American unit system still produce confusion

and errors), but also the scientific communication would be harder (e.g. imagine when symbols

for quantities would not be the same at all in the different countries).

Today’s most common standardized quantity system is The International System of Quan-

tities (ISQ), the ”system of quantities, together with the equations relating the quanti-

ties, on which the SI is based”. Since November 2009 the ISQ is published in the In-

ternational Standard ISO 80000 or IEC 80000 - formerly ISO 31 and partially IEC 60027

- [ISO-TC12 and IEC-TC25, 2009], depending on which of the two international standards

Page 8

Theoretical Basis

bodies, either the International Organization for Standardization or the International Elec-

trotechnical Commission, is responsible for each respective part. The base quantities used

in the ISQ are length, mass, time, electric current, thermodynamic temperature, amount of

substance, and luminous intensity.

Today’s most common unit system is The International System of Units (or SI from its

French name Systeme International d’unites) which was already mentioned in the definition of

the ISQ above. It is a ”coherent system of units based on the ISQ, their names and symbols,

and a series of prefixes and their names and symbols, together with rules for their use, adopted

by the General Conference on Weights and Measures (CGPM)”. The system is not static but

evolves in terms of technological advances in scientific measurement [BIPM, 2006]. Therefore

additions and changes are done regularly. With regard to the base quantities in the ISQ, the

CGPM chose the base units of the SI to be the meter, the kilogram, the second, the ampere,

the kelvin, the mole, and the candela. Finally, each of the seven base quantities chosen for the

ISQ has its own dimension. The symbols used to signify the dimensions are L, M, T, I, θ,

N and J. Table (2.2) summarizes the seven base quantities with their respective base units

and base dimensions [BIPM, 2006].

Table 2.2.: The seven base quantities as defined in the SI [BIPM, 2006] with their respective

base units and base dimensions

Base Quantity Base Unit Base Dimension

Name Symbol Name Symbol Name Symbol

length l meter m length L

mass m kilogram kg mass M

time t second s time T

electric current I, i ampere A electric current I

thermodyn. temp. T kelvin K thermodyn. temp. θ

amount of substance n mole mol amount of substances N

luminous intensity IV candela cd luminous intensity J

Please note that although the seven base quantities are by convention seen as independent,

their respective base units are in a number of instances interdependent. Thus the definition

of the meter incorporates the second; the definition of the ampere incorporates the meter,

kilogram, and second; the definition of the mole incorporates the kilogram; and the definition

of the candela incorporates the meter, kilogram, and second. In the following the analyses

will be done essentially within the ISQ and the SI. However, it is also possible to carry out the

investigations in this work with a different system of quantities and units. Correspondingly the

choice of base units is also dependent on each engineer or scientist: it is also true to choose

Page 9


e.g. gram instead of kilogram as a base unit for the base quantity mass. Thus, whenever

another system as the SI and ISQ is used in this work, it will be explicitly mentioned.

2.4. Dimensional Analysis

The basic ideas of the dimensional analysis, as they are used today, date back to the 19th

century: the French mathematician Joseph Fourier spread the idea that physical equations

should be dimensional homogeneous equations in their different units of measurement in his

work Theorie Analytique de la Chaleur of 1822 [Buckingham, 1914], [Macagno, 1971]. Hence,

the dimensional analysis is still one important and very simple method to check the plausibility

of physical equations: the two terms of an equation have to have the same dimension to be

valid in terms of a first plausibility check. Or, as expressed in [Langhaar, 1951] ”an equation

will be said to be dimensionally homogeneous if the form of the equation does not depend on

the fundamental units of measurement”.

However, Fourier failed to see the connection between dimensional analysis and similarity. It

was up to the English physicist and mathematician Lord Rayleigh to develop a method to

the principle of Fourier in his work Theory of Sound from 1877/8 [Strutt and Rayleigh, 1894].

Even dimensionless groups were found by Rayleigh, like the Reynolds number Re in its inverse

form.

In continental Europe, the French physicist and mathematician Aime Vaschy achieved a more

general formulation of the theorem for the method of dimensions and came closer to a justifica-

tion of the theorem in mathematical form [Vaschy, 1892]. The Russian Dimitri Riabouchinsky

is said to have used dimensional analysis independently from the results of Vaschy and Rayleigh

[Riabouchinsky, 1911]. He also initiated a discussion about the temperature as fundamental

unit with Lord Rayleigh and Edgar Buckingham after the publication of an article by Lord

Rayleigh which was intended to stimulate the use of his method of dimensions among engi-

neers [Buckingham, 1915b], [Rayleigh, 1915].

Buckingham himself is said to have adopted the idea of the method of dimensional analysis

from his antecessors, so the theorem should not only be cited after him. Correspondingly, the

theorem is often cited as Vaschy-Buckingham-Theorem in France. However, Buckingham is

the only one, who very clearly pointed out that dimensional analysis has two further interesting

results besides the homogeneity of equations which are explained in the following.

A very important application of dimensional analysis can be often found in scientific situations

where characteristics of nearly unknown phenomena are sought. Here dimensional analysis

helps to state hypotheses about the influence of quantities of a complex system on the at-

titude and performance of the system, without knowing the exact relation between these

quantities. On top of that, even exact equations, namely scaling laws, can be found by di-

mensional analysis with some contribution of experimental results or physical reasoning.

One of the most famous examples of dimensional reasoning was in real not due to the method

of dimensional analysis. Geoffrey Taylor is to be said to have used dimensional analysis in

1950 to easily determine the discharged energy by stating a formula relating the radius r

Page 10

Theoretical Basis

of a spherical blast wave produced by the release of a quantity of energy E, at a point

in the air of density ρAir 0 and polytropic index γ to the two-fifths power of the time t

[Taylor, 1950a], [Taylor, 1950b]. In reality, fastidious calculations with the equations of motion

and the Rankine-Hugoniot equations as starting point guided him to the following equation:

r = t25

(E

ρAir 0

) 15

f(γ) (2.4)

Nevertheless, this equation can easily be reproduced by the method of dimensional anal-

ysis.Genuine examples of dimensional analysis include the most common ones in aerody-

namics/fluid dynamics [Riabouchinsky, 1911], [Anderson, 2007] and heat and mass trans-

fer [Rayleigh, 1915], [Richardson, 1919], electrotechnology [Vaschy, 1892], chemical process-

ing [Zlokarnik, 2006], mechanics [Vaschy, 1892], [Sedov, 1993] and biology [Stahl, 1961a],

[Stahl, 1961b], [Tennekes, 1996], [McMahon and Bonner, 1983]. But also not so evident sci-

entific areas as economy [de Jong, 1967] and psychology [Lehman and Craig, 1963] use di-

mensional analysis.

References for literature that deals with dimensional analysis in general are [Bridgman, 1948],

[Langhaar, 1951], [de St. Q. Isaacson and de St. Q. Isaacson, 1975], [Taylor, 1974] and also

[Sedov, 1993]. They give various examples from biology over cosmology to the classical fields

of fluid mechanics and heat transfer. The latter reference [Bridgman, 1948] is often referred

to as the first work which outlines the Π-theorem in a book with various examples.

Finally, Buckingham also stated clearly in his explanations the significant role of dimensional

analysis in the similitude theory. It is in fact especially frequently used in aerodynamics and

thermodynamics: instead of taking measurements of real sized objects, the similitude theory

states that results of a real sized object and a smaller corresponding model are the same under

the condition that the so called similarity parameters of the investigated system are the same

in reality and in the modeled situation. Similarity parameters, in turn, can be found with

dimensional analysis. Hence, this approach makes wind tunnels for real sized airplanes, for

example, nearly dispensable for most of the investigated problems.

In this context a very useful side effect of the similarity parameters and the dimensional anal-

ysis has to be mentioned which is especially useful when series of experiments are done: the

number of the influential quantities in the investigated system is reduced so that time and

effort for the series as well as the amount of data are significantly reduced since the number

of quantities which have to vary during the series, is reduced.

Another purpose of dimensional analysis which was not mentioned yet, is to quickly check the

order of magnitude of quantities and thus to see if a design is plausible or not. This is mainly

done by comparing the numerical value of non-dimensional parameters of similar designs. The

values are supposed to be in the same range for plausibility. The theory of the non-dimensional

parameters will be presented in detail in the next section 2.4.1.

Page 11


2.4.1. Buckingham-Π-Theorem

The Buckingham-Π-Theorem is a fundamental theorem in dimensional analysis named after

the American physicist Edgar Buckingham. It states that an equation written in the general

form

f(Q1, Q2, . . . , Qn) = 0 (2.5)

where the n symbols Q1, . . . , Qn denote physical quantities of n different kinds with k inde-

pendent dimensions can be rewritten as

F (Π1,Π2, . . . ,Πi) = 0 (2.6)

where the Πs represent i = n− k independent dimensionless products of the form

Πi = Qa11 Q

a22 . . . Qan

n (2.7)

The use of Π as a dimensionless product was introduced by Edgar Buckingham in his original

paper on the subject in 1914 [Buckingham, 1914] from which the theorem draws its name.

However, in order to do justice to the historical facts/origins, the theorem should instead be

called Vaschy- or Rayleigh-Theorem than Buckingham-Theorem as already seen in Chapter

2.4.

Because of the principle of dimensional homogeneity, equation (2.7) can be rewritten as

[Πi] = [Qa11 Q

a22 . . . Qan

n ] = [1] (2.8)

Since only k out of n dimensions are independent these equations can be further developed to

[Π1] = [Qα11 Q

β12 . . . Qκ1

k P1] = [1]

[Π2] = [Qα21 Q

β22 . . . Qκ2

k P2] = [1]

.................................................

[Πi] = [Qαi1 Q

βi2 . . . Qκi

k Pi] = [1]

(2.9)

with i = n − k in which the P ’s represent Qk+1, . . . , Qn, thus the quantities which are

”derived” from the base quantities Q1, . . . , Qk, also called the k-set. It is important to notice

that the base quantities with their respective base units and dimensions, as defined in the

SI-system, are not forcingly to be used as the fundamental units for the theorem. Buckingham

himself states in his explanations [Buckingham, 1914] that the dimension F, Newton as the

unit of force, could be used as a base dimension.

Hence the n− k dimensional products Π can be written as

Π1 = f1(Q1, Q2, . . . , Qk, Pk+1)

Π2 = f2(Q1, Q2, . . . , Qk, Pk+2)

................................................

Πi = fi(Q1, Q2, . . . , Qk, Pn)

(2.10)

Page 12

Theoretical Basis

thus reducing the variables of the problem from n to i. It is very important to choose the k

repeating quantities, the k-set, in a way that the quantities include all the k base dimensions

used in the problem [Anderson, 2007], [Buckingham, 1914]. The derived quantity P , however,

appears only in one Π-product.

A ”scaling law” results when the function F of equation (2.6) is determined in its nature. A

usual approach for that is a product approach following

(Π1)a1(Π2)a2 . . . (Πi)ai = c (2.11)

with a1, . . . , ai being the exponents in the function and c being a constant.

As it can occur that quantities of the same kind play a role in the system, briefly shall it be

discussed how they are to be treated during the analysis. Two different masses can for example

be considered to be significant to model a system. As Buckingham states, however, that

Q1, . . . , Qn in equation (2.5) are physical quantities of different kinds, he proposes to include

further quantities of a kind as ratios to the one which is included in the list of Q1, . . . , Qn

[Buckingham, 1914, p.345]. Equation (2.5) is thus only a reduction of the more general

equation

f ∗(Q1, Q2, . . . , Qn, r′, r′′, . . . ) = 0 (2.12)

In this equation the r’s represent ”all the independent ratios of quantities of the same kind”

[Buckingham, 1915a, p. 291] and each Π can be determined as discussed above. Ratios are

strictly speaking not considered as non-dimensional products in terms of the Buckingham-

Π-Theorem. Nevertheless, they can have an influence on the system and thus, have to be

considered as further parameters of the system, and so as further arguments in the function

F of equation (2.6). Therefore equation (2.6) can be rewritten as

F ∗(Π1,Π2, . . . ,Πn−k, r′, r′′, . . . ) = 0 (2.13)

Only when the ratios are considered to be constants of the investigated systems, can they

be neglected during the determination of the scaling laws. In Buckingham’s own expla-

nations the ratios are usually geometrical ones and considered to be of fixed value dur-

ing the analysis (e.g. dimensional analysis of the resistance of immersed bodies at moder-

ate speeds [Buckingham, 1915a, p. 271 ff.]). But as they can potentially play an essen-

tial role, they should be taken into consideration when scaling laws have to be established

[Buckingham, 1915a, p.291]. Thus equation (2.11) can be rewritten to

(Π1)a1(Π2)a2 . . . (Πi)ai(r′)b1(r′′)b2 · · · = c (2.14)

with a1, . . . , ai and b1, b2, . . . being the exponents in the function and c being again a constant.

Please note as an anticipation, that we will use another method in this work to show the

influence of the ratios on the system as we will not be able to determine scaling laws in this

work because of the lack of data: the non-dimensional parameters Πi will be directly expanded

Page 13


with the system influencing ratios. Examples of that proceeding will be shown in sections 3.4.2

to 3.4.5.

To finish the theoretical part of this work, it can be important to know that equation (2.6)

can be rewritten as

Π1 = G(Π2, . . . ,Πi). (2.15)

thus allowing to express the derived quantity P1 in Π1 as function of Π2, . . . ,Πi and the

residual quantities in Π1.

2.4.2. Example

One of the most well known applications of the Buckingham-Π-Theorem can be found in

the aerodynamics of a body (e.g. an airfoil) [Anderson, 2007]. Dimensional analysis is used

in order to determine the so called similarity parameters to compare the performances of the

model in a wind tunnel with the original object under real conditions. We consider a body with

a given shape and a given angle of attack and try to find out what physical quantities influence

the resulting aerodynamic force R being applied on the body. Thus, in a first step, we are

looking for the quantities Q1, . . . , Qn which are driving the performance of the investigated

system. It is up to the physical understanding about the system to identify these quantities.

We suppose R to be dependent on

• the freestream velocity V∞ [ms

]

• the freestream density ρ∞ [ kgm3 ]

• the viscosity of the fluid, represented by the dynamic viscosity coefficient µ∞ [ kgm s

]

• the size of the body, represented by the chord length c [m]

• the compressibility of the fluid, represented by the freestream speed of sound a∞ [ms

]

Thus the number of the independent quantities is n = 6 and we can write

R = g(ρ∞, V∞, c, µ∞, a∞) (2.16)

Equation (2.16) can be rewritten as follows

f(R, ρ∞, V∞, c, µ∞, a∞) = 0 (2.17)

Page 14

Theoretical Basis

which is equal to equation (2.5). In a next step, we consider the dimensions of the quantities.

Being in the SI-system, the fundamental dimensions of the problem are to be chosen as defined

in the standardized system:[R] = MLT−2

[ρ∞] = ML−3

[V∞] = LT−1

[c] = L

[µ∞] = ML−1T−1

[a∞] = LT−1

(2.18)

therefore the fundamental dimensions of this problem are mass M, length L and time T, thus

k = 3. As seen in section 2.4.1 the number of dimensionless products Πi can be determined

as i = n − k = 6 − 3 = 3. We choose the repeating quantities Q1, Q2 and Q3 to be ρ∞,

V∞ and c. Another choice of parameters is possible as long as the rules as discussed in 2.4.1

are followed. Note that the resulting aerodynamic force R is not a repeating quantity as we

defined it to be one of the dependent variables for which we want to find a relation. Hence,

resulting from equation (2.10) we can write

ΠEx 1 = f1(ρ∞, V∞, c, R)

ΠEx 2 = f2(ρ∞, V∞, c, µ∞)

ΠEx 3 = f3(ρ∞, V∞, c, a∞)

(2.19)

Then, with (2.18)

ΠEx 1 = ρ∞α1V∞

β1cγ1Rδ1


β2cγ2µ∞δ2


β3cγ3a∞δ3

(2.20)

or in dimensional terms

[ΠEx 1] = [ρ∞]α1 [V∞]β1 [c]γ1 [R]δ1 = [1]

[ΠEx 2] = [ρ∞]α2 [V∞]β2 [c]γ2 [µ∞]δ2 = [1]

[ΠEx 3] = [ρ∞]α3 [V∞]β3 [c]γ3 [a∞]δ3 = [1]

(2.21)

Thus[ΠEx 1] = (ML−3)α1(LT−1)β1(L)γ1(MLT−2)δ1 = 1

[ΠEx 2] = (ML−3)α2(LT−1)β2(L)γ2(ML−1T−1)δ2 = 1

[ΠEx 3] = (ML−3)α3(LT−1)β3(L)γ3(LT−1)δ3 = 1

(2.22)

For demonstration purposes we concentrate on the determination of Π1. In order to fulfill

the above equations, the exponents of each dimension must be cancelled. Thus we get k

equations by equating coefficients, one for each fundamental dimension we use:

For M: α1 + δ1 = 0 (2.23)

For L: −3α1 + β1 + γ1 + δ1 = 0 (2.24)

For T: −β1 − 2δ1 = 0 (2.25)

Page 15


So we find α1 = −1, β1 = −2, γ1 = −2 and δ1 = 1. Thus equation (2.22) can be rewritten

as follows

ΠEx 1 =R

ρ∞V∞2c2

(2.26)

where c2 has the dimension of an area. So it can be replaced by any reference area (e.g. the

planform area S of a wing) and ΠEx 1 will still be dimensionless. We can also multiply Π1

with a pure number and still, it will remain dimensionless [Buckingham, 1914]. Thus we can

rewrite equation (2.26) and get

ΠEx 1 =R

12ρ∞V∞

2S(2.27)

We know the term 12ρ∞V∞

2 as dynamic pressure q∞, so we can rewrite equation (2.27) to

ΠEx 1 =R

12ρ∞V∞

2S=

R

q∞S(2.28)

and can identify ΠEx 1 to be the force coefficient CR. The same analysis as for ΠEx 1 is to

be done for ΠEx 2 and ΠEx 3 and we get

ΠEx 2 =ρ∞V∞c

µ∞

ΠEx 3 =V∞a∞

(2.29)

Hence we can identify ΠEx 2 as the freestream Reynolds number Re and ΠEx 3 as the

freestream Mach number Ma. As a result of our analysis we can rewrite equation (2.17)

F

(R

12ρ∞V∞

2S,ρ∞V∞c

µ∞,V∞a∞

)= 0 (2.30)

or

F (CR, Re,Ma) = 0 (2.31)

Thus

CR = G(Re,Ma) (2.32)

So, one clear advantage of the Buckingham-Π-Theorem is the reduction of the number of

independent variables from which the resulting aerodynamic force R is dependent, from five

to two. This is especially important for investigative tests in a wind tunnel. Instead of varying

five different parameters, we only have to vary two, namely Re and Ma to get a formulation

for R. But the most important power of our results lies in the fact that we can use these

parameters to define dynamic similarity of two different flows: we are allowed to compare

the results of a wind tunnel test with the original conditions for a object when Re and Ma

are the same in both cases and when geometric similarity (i.e. the bodies and any other solid

boundaries are geometrically similar for both cases) is given. Re and Ma are called similarity

parameters.

Page 16

Theoretical Basis

2.4.3. Advantage of the Use of Dimensional Analysis during the

Design Phase of Satellites

One main purpose of this work is to find non-dimensional similarity parameters, similar to Re

and Ma in section 2.4.2, to formalize and facilitate the comparison between the performances

and characteristics of satellites of different sizes. In comparison to the aerodynamic example,

the CubeSat will be ideally what is the the model in the wind tunnel, and the bigger satellite

will be what is the real object like the plane. In addition, comparisons of satellites within whole

satellite classes will be possible as it is expected that the non-dimensional products will take at

least constant numerical values within a satellite class and hopefully also over a wider range.

Thus a very important advantage of using dimensional analysis during the spacecraft design

compared to usual calculation formulas presented in literature (i.e. [Larson and Wertz, 1999]),

is the fact that comparisons of the new design to already existing satellites in different missions

are possible. This enables the engineer ideally to avoid the whole set of calculations which is

normally necessary during the early development of a new satellite and just fall back on the

design of a ”similar” spacecraft, with the same non-dimensional parameters and hence adjust

its design right in the beginning, if necessary. Furthermore, existing designs can be checked

for their plausibility by investigating if their non-dimensional parameters can be found in the

usual value range of similar spacecrafts.

As this work will culminate in the creation of Mission Performance Indices, scalars which will

combine several non-dimensional parameters and ratios - thus characterizing a mission as a

whole - the engineer will quickly be able to get an idea of the performance that the designed

spacecraft is able to achieve. Thus, the user of the results of this work will be able to forecast

the ability of the so designed spacecraft already in the beginning of the design process and will

be, when required, easily and quickly redirected to other designs with the required performance.

Concludingly dimensional analysis helps to enable and formalize the potential hidden in the

similarity of spacecrafts and missions in order to facilitate and accelerate the design phase of

a satellite.

2.4.4. Limits of the Buckingham-Π-Theorem and Dimensional

Analysis

The quality of the results of the Buckingham-Π-Theorem is highly dependent on the choice

of the quantities Q1, . . . , Qn which drive the performance of the investigated system as they

are not fixed by the theorem itself. It is up to the physical understanding of the user of the

theorem to identify them. So, when a chosen quantity does not appear in a later dimensionless

product Π, it could mean that this quantity is redundant to the other chosen quantities of the

system, but it could also give the hint that another quantity is still missing in the modell of

the system.

Dimensional analysis can help to find non-dimensional parameters describing a problem but

does not determine the exact relation between them as it became clear in 2.4.2: equation (2.32)

Page 17


only states a dependence between CR, Re and Ma but does not give more information about

the function G. Physical comprehension and an extensive database of already flown satellites

are necessary to exactly determine the relation between the non-dimensional parameters and

the ratios and thus to determine the exponents a1, . . . , ai and b1, b2, . . . in equation (2.13).

Furthermore, the non-dimensional parameters and the Mission Performance Indices will provide

only a first estimation for the spacecraft design. The use of the parameters cannot substitute

detailed calculations that are definitely indispensable for a more sophisticated design.

Finally, as with every usual calculation formula, the non-dimensional parameters and also the

Mission Performance Indices will only be a model of reality and thus a simplification of all the

interdependencies which determine the characteristics of a system in the real world.

Page 18

3. Dimensional Analysis of a Single

Satellite

The aim of this chapter is to provide the engineer with theoretical scaling laws based on di-

mensional analysis which facilitate and accelerate the design phase of a satellite in general

and especially the design of a CubeSat by using the potential lying in the concept of simi-

larity. Based on the Buckingham-Π-Theorem presented in 2.4.1, non-dimensional parameters

will be presented in this chapter which will interrelate the characteristics of a satellite and its

subsystems to each other. This will establish correlations between the design parameters of

the spacecraft such as the power consumed by the spacecraft, the orbital period or simply

the volume and the size of the satellite. Based on these investigations, Mission Performance

Indices which characterize the performance of a mission as a whole by connecting the design

parameters, will be presented for three different mission types, namely Earth Observation,

Space Science and Technology Demonstration, these being analyzed as representing the most

frequent ones in recent CubeSat missions [Thomsen, 2010].

Additionally, using data from communication satellites in non-geosynchronous orbits (NGSO)

[Springmann and de Weck, 2004], some of the results of the dimensional analysis are empiri-

cally validated. Data from previously developed CubeSats was intended to be used instead but

the current available information is not extensive enough to be exclusively used for the valida-

tion of the products. Some estimations, however, are given based on this information and that

on the CubeSat-specific commercial off-the-shelf components (see Appendix B). Ideally, the

scaling laws would thus be a combination of both physics based as well as empirically derived

parameterized relationships.

3.1. Design Approaches: Top-Down and Bottom-Up

In general, two different problem assignments are possible for the design of a satellite: either

the parameters of the mission are fixed (i.e. the type of the mission, the payload to be used

on the spacecraft) and the ideal spacecraft fulfilling the requirements of the mission has to be

found - thus the bottom limits of the performance of the spacecraft are set; or the parameters

of a possible spacecraft are fixed (i.e. the size of the spacecraft, the power which can be

provided; thus the upper performance limits of the spacecraft are fixed) as well as the orbital

parameters and a mission fitting to the spacecraft and orbit has to be determined. In this

work, the first payload-centric approach is called ”Top-Down” design approach while the

Page 19

Design Approaches: Top-Down and Bottom-Up

latter spacecraft-centric approach is called ”Bottom-Up” design approach. These notations

will be used equally in this work. Figures 3.1 and 3.2 display the two approaches graphically.

Figure 3.1.: Top-Down design approach

Figure 3.2.: Bottom-Up design approach

Most of the usual industrial and scientific missions are done in a Top-Down design approach

as weight and space issues of the spacecraft play often only a secondary role as long as the

main function of the mission (i.e. scientific research with special payload) is accomplished.

However, this is one important difference to the CubeSat-missions: as most of the CubeSats

do obey a standard, size and mass limitations are the consequence and hence limitations in

power and reduced room for payloads are the result. Furthermore, as CubeSats are mostly

launched in a piggyback launch, the orbit of the CubeSat cannot be chosen independently but

is fixed by the primary payload on-board of the launcher. Thus CubeSats are definitely more

Page 20

Dimensional Analysis of a Single Satellite

often build in a Bottom-Up approach than in a Top-Down approach. However, in order to

keep the results of this chapter as general as possible, both approaches will be investigated

and are distinguished since according to the chosen one, the results are going to be different

in most of the analyses.

In order to have applicable results, the non-dimensional products Π as result of the dimensional

analysis are composed of quantities in a way, that they include ideally only one parameter

which is unknown in the chosen design approach and which can be determined by employing

the equation for the non-dimensional product with its numerical value. In the Top-Down

approach the known quantities will be characteristics of the payload (i.e. size, mass, power

consumption) whereas the unknown parameter is a characteristic of the spacecraft (i.e. size,

mass, power production). In the Bottom-Up approach it will be exactly vice-versa with known

spacecraft characteristics and unknown payload parameters. In coherence to the Buckingham-

Π-Theorem the unknown parameter will be the P and the known parameters will be equal to

the Qk’s in equation (2.10).

Please note as a final remark that in practice the orbital altitude, an approximate satellite

lifetime and the mission type are usually also given as input parameters in the Bottom-Up

approach.

3.2. Proceeding of the Validation of the

Non-Dimensional Parameters

The results of this chapter, the non-dimensional parameters, will be as general as possible and

thus will be applicable to any class of satellite. Only the empirical validation of them will

lead to different results depending on the various satellite classes as the numerical values of

the non-dimensional products Π can be different for each satellite class. Hence, some of the

theoretical results of this chapter will be confirmed by use of data from non-geosynchronous

satellites [Springmann and de Weck, 2004] as far as the data is available. Data for the flown

CubeSats was tried to be gathered together but the current available information is not exten-

sive enough to be used for the validation of the non-dimensional parameters and the Mission

Performance Indices. Some numerical estimations for the expected mass ratios of the picosatel-

lites, however, are given based on some real flown satellite data and the characteristics of the

commercial-off-the-shelf components which are gathered together in a component database

that was created during this work (see Appendix B). Thus, the main emphasis during the em-

pirical validation is put on the condition in similarity (see section 2.4) if the non-dimensional

products Π do have a fixed value for a given satellite class and thus can be applied as design

tool, and, in a first order approximation, not which value they will take.

In order to quantify the quality of the results of the application of the non-dimensional pa-

rameters on the NGSO-data, different statistical parameters were used. In a first step, the

Page 21

Proceeding of the Validation of the Non-Dimensional Parameters

arithmetical average of the results over all satellites was built. Secondly, the standard deviation

was calculated based on the following equation:

s =

√√√√ 1

N − 1

N∑i=1

(xi − xav)2 (3.1)

with N as the total number of the samples, xi the result for the satellite i and xav the arith-

metical average of the results over all satellites.

And finally, in order to be able to compare the results, the percentage of the deviation from

the arithmetical average, the coefficient of variation sxav

was calculated. The smaller this

characteristical value, the higher is in general the quality of the non-dimensional parameter,

as the deviation from the arithmetical average is as small as possible.

The validation of the non-dimensional parameters for the subsystems is more complex than for

the ratios as not only more quantities play a role but also complete ratios based on equation

(2.13). Their influence on the subsystem and correspondingly on the numerical value of the

non-dimensional parameter is expressed by exponents a, b, c, . . . that have to be determined

during the validation process. The influence of the non-ratio part nr of a non-dimensional

parameter is expressed by an exponent 1. Thus every non-dimensional parameter can be

rewritten to

1 = k · nr1r1ar2

br3c . . . (3.2)

k is a constant and the reciprocal of the latter numerical value of the non-dimensional param-

eter Π. a, b, c, . . . are the exponents of the ratios ri, expressing the level of influence of the

different ratios within the system which is described by the non-dimensional parameter. Before

computing k, nr and a, b, c, . . . , however, data of the investigated satellite class is required so

that the numerical values for nr and ri can be calculated. After that, the exponents a, b, c, . . .

and the constant k have to be determined in a way that they take ideally the same value for

every satellite in the given database, so that equation (3.2) is satisfied for every element of the

database, thus for every satellite in it. As this is very difficult and in general all but impossible

the larger the number of samples in the database gets, the left term of the equation (3.2) has

to be allowed to be slightly variable, for example variations of +/- 0.1 has to be possible in

order to get a solution. The solution which allows the slightest variations in the left term shall

be then the solution (3.2) which gives us a scaling law for the investigated satellite class.

In our case, however, this validation needs more detailed information on the satellites than it

is available in the NGSO-database. Therefore this explanations shall only be theoretical and

provide the user with the necessary information for the practical validation. An example of the

notions and terms will be given in section 3.4.2 by means of ΠBat 3 and ΠBat 4.

Theoretically, the results, especially the non-dimensional parameters, have to be checked in

their physical sense. This is formalized in an approach by [Bhaskar and Nigam, 1990] and

Page 22


is called intraregime-approach. Basically a regime is a physical aspect of a system. Thus a

non-dimensional parameter Π describes a regime. The intraregime approach consists of the

creation of partial differentials out of the quantities generating the non-dimensional parameters

in order to see how the quantities within the regime, thus the Π, are related to each other. In

the end this is a formalized expression of what will be described later for every non-dimensional

parameter. We will check for each of them if they are ”physically” consistent, thus we will

investigate, for example, if a raise in the quantities of the denominator which leads to an

increase in the quantities of the numerator for a constant Π, is logical in terms of the usual

relationships between the quantities. For example a raise in the mass of the spacecraft usually

entails also a raise in the power consumption of the spacecraft. Thus we will find the termPS/C

mS/C

in the top-level approach. Correspondingly, the intraregime approach indicates us a differential∂PS/C

∂mS/C> 0. For further interest in that topic, please refer to [Bhaskar and Nigam, 1990].

3.3. Ratios

3.3.1. Mass

A commonly used first approach in the sizing of satellites is to estimate the mass of the subsys-

tems of a satellite as percentage of the dry mass mS/Cdryor wet mass mS/Cwet of the spacecraft

[Larson and Wertz, 1999], [Pritchard, 1984], [Kiesling, 197172], [Saleh et al., 2002]. This point

of departure is definitely not in the classical sense of the dimensional analysis and the Buckingham-

Π-Theorem, as ratios of quantities are not seen as non-dimensional parameters. However, as

approximate mass ratios can be often found in literature, especially for communication satel-

lites or satellites in general, they appear to offer a quick and reliable possibility to get a first

idea of the preliminary mass budget of a satellite. Therefore, mass ratios were considered to

be worth further investigation.

In the Top-Down approach, the mass of the payload mP/L is considered to be given, thus

all the subsystems’ masses are divided by it. The Bottom-Up approach requires a mass of

the spacecraft, either the dry mass mS/Cdryor the wet mass mS/Cwet , to be fixed. Hence the

following ratios were investigated with the data of non-geosynchronous satellites as already

mentioned in section 3.2:

Page 23

Ratios

For the Top-Down approach we will investigate:

rMass TD1 =mS/C

mP/L

rMass TD2 =mAOCS

mP/L

rMass TD3 =mPower

mP/L

rMass TD4 =mStructure

mP/L

rMass TD5 =mThermal

mP/L

rMass TD6 =mC&DH+TT&C

mP/L

rMass TD7 =mPropulsion

mP/L

(3.3)

and for the Bottom-Up approach we will have a closer look at:

rMass BU1 =mP/L

mS/C

rMass BU2 =mAOCS

mS/C

rMass BU3 =mPower

mS/C

rMass BU4 =mStructure

mS/C

rMass BU5 =mThermal

mS/C

rMass BU6 =mC&DH+TT&C

mS/C

rMass BU7 =mPropulsion

mS/C

(3.4)

with mS/C being either mS/Cdryor mS/Cwet .

The data for the C&DH and TT&C subsystem was not differentiated in the given database of

the NGSO-satellites so in the following investigations they will be considered together as one

subsystem. Generally speaking, but especially in case of the CubeSats, it is recommended to

consider these subsystems separately.

Propulsion is normally not provided on CubeSats as it is a very heavy subsystem. Accord-

ingly the propulsion ratio in both approaches is not of importance for CubeSats. However, in

order to demonstrate a non-dimensional approach for satellites of all different classes which

do include propulsion, the mass of the propulsion system is considered in this investigation.

Furthermore, intensified research is ongoing in this area, also at MIT in the Space Propulsion

Laboratory under Professor Lozano who tries to find a new technology to provide propulsion

especially for small satellites [Courtney and Lozano, 2010]. Thus, this issue is assumed to be

one of the main picosatellite research topics in the near future and this work can help to model

Page 24


the resource allocation, including the propulsion subsystem.

In the following, the demonstration of the analysis is focused on the Bottom-Up approach with

the use of the dry spacecraft mass mS/Cdry, as this will be the principal case for designing the

CubeSats. The investigation of the Bottom-Up approach with use of the wet spacecraft mass

mS/Cwet and the Top-Down approach were done in the same way. Results of these two cases

will be discussed shortly afterwards.

Figure 3.3 shows the values of the ratios of equation (3.4) with the different satellites repre-

sented on the abscissa. Tables 3.1 and 3.2 represent the arithmetical average xav as well as

the standard deviation s and the coefficient of variation sxav

for the available data points.

At first glance the ratios shown in figure 3.3, seem to be similar for the different satellites

with some outliers. Tables 3.1 and 3.2 confirm this impression by showing percentages of

deviation of only 30% for mPL

mS/Cdry

, 39% for mPower

mS/Cdry

and 43% for mThermal

mS/Cdry

. Considering the

large number of satellites in the database of different masses, sizes and orbits, these deviations

were expected and are quite small.

Figure 3.3.: Mass ratios of NGSO-satellites plotted against the various satellites, represented

by a number

Table 3.1.: xav, s and sxav

of the mass ratiosmS/Cwet

mS/Cdry

,mP/L

mS/Cdry

and of the subsystems AOCS,

Power and Structure over mS/Cdry

mS/Cwet

mS/Cdry

mP/L

mS/Cdry

mAOCS

mS/Cdry

mPower

mS/Cdry

mStructure

mS/Cdry

Arithm. Average xav 1.217 0.363 0.072 0.297 0.195

Std. Deviation s 0.255 0.110 0.049 0.115 0.103

Coeff. of Variation sxav

[%] 21.00 30.43 68.73 38.74 52.55

Page 25

Ratios


for the mass ratios of the subsystems Thermal, C&DH+TT&C and

Propulsion over mS/Cdry

mThermal

mS/Cdry

mC&DH+TT&C

mS/Cdry

mPropulsion

mS/Cdry

Arithm. Average xav 0.071 0.060 0.090

Std. Deviation s 0.030 0.060 0.070


[%] 42.76 99.91 77.77

The high deviations occur mainly in the subsystems where the subsystem contributes to a very

small amount of the total mass of the satellite, namely in C&DH, AOCS and Propulsion (the

Structure subsystem is an exception as it does contribute to almost 20% of the total mass

but is still very dispersed). Small changes in the design of the subsystem can easily lead to

high percentage changes in the mass contribution. Those systems are hence more sensitive to

changes in their design, which, in turn, leads to higher coefficients of variation.

As a next step, it was assumed that the mass ratios would show more constancy in different

mass ranges. Figure 3.4 shows this to be a reasonable assumption. In the figure the mass ratios

are plotted against the dry mass mS/Cdryand here areas of constant ratios can be observed,

with the outliers inmS/Cwet

mS/Cdry

becoming more important with rising mS/Cdry:

Figure 3.4.: Mass ratios of NGSO-satellites plotted against mS/Cdry

Figure 3.4 also shows thatmS/Cwet

mS/Cdry

ranges between 1 and 1.5 for nearly all of the satellites with

an increase in wet mass for heavier satellites. This increase, however, could not be explained.

It is assumed that it can be due to the further propulsion mass heavier satellites require. In

case of most of the CubeSats this value will be equal to 1 anyway, as no propulsion is used on

CubeSats and thus mS/Cwet = mS/Cdry, the graphical presentation of

mS/Cwet

mS/Cdry

was eliminated

from the next figure to focus on the other mass ratios (see figure 3.5).

Page 26


Figure 3.5.: Mass ratios of NGSO-satellites plotted against mS/Cdryexcluding

mS/Cwet

mS/Cdry

A slight difference in constant values is visible between the satellites of masses above 1000

kg and under 300 kg. So, the data was divided into these two areas and the satellites were

investigated in their different mass classes.

In the dry mass range of 1000 kg and more (see figure 3.6), it becomes clear at first glance

that the payload and the Power subsystem are responsible for a good portion of the com-

plete satellite mass as it is also modelled in the known rules of thumb for first mass budget

estimations [Larson and Wertz, 1999, p.316]. Their values are concentrated around 40% formP/L

mS/Cdry

and around 30% for mPower

mS/Cdry

. The very few available ratios for the subsystems AOCS,

Thermal, C&DH + TT&C and Propulsion are around 10% and the ratio for the Structure

subsystem is around 15%.

Figure 3.6.: Mass ratios of NGSO-satellites with mS/Cdryabove 1000 kg

In the range of dry masses under 300 kg (see figure 3.7) some more data is available, thus the

conclusion of these results are based on a slightly wider foundation as for satellites over 1000

kg. The unfortunately few payload ratios are gathered around 25%, thus a small difference

Page 27

Ratios

to the satellite class with mS/Cdryabove 1000 kg can be observed. A correlation between the

two mass ranges can be detected for the power ratio which is again around 30%, as well as

for the subsystems AOCS, Thermal, C&DH + TT&C and Propulsion which are again in the

range of 10%. Only the subsystem Structure seems to be relatively heavier for the smaller

satellites as its ratio can be found between 10 to 35% with a rise to smaller satellite masses.

Figure 3.7.: Mass ratios of NGSO-satellites with mS/Cdryunder 300 kg

So, with the exception of the payload and structure ratio where small differences between

the mass ranges can be detected, all the other ratios were roughly in the same area for a

representative number of all 38 satellites.

The numerical results read from the figures are confirmed by the use of histograms (see figures

3.8 to 3.15) which plot the frequency of the different ratios against different value ranges, so

called classes. Important peaks can be seen for nearly every ratio except for the Structure

subsystem where the values range almost equally from 10 to 40%. (Please note that the values

are rounded up to the nearest class. Thus, all the ratios from 0.31 till 0.4 are allocated to 0.4

for example).

To summarize, it can be stated that the results of the analysis for a representative number of

satellites which ranges from 9 to up to 33, clearly show that the mass ratios can be mostly

found in a fixed value range. Deviations from the arithmetical average are of 30% for the

payload ratio up to nearly 100% for the C&DH+TT&C subsystem. Outliers do exist and

are included in the deviations, since the database is composed of satellites of very different

sizes and masses, thus with very different densities (between 4.44 and 698.6 kgm3 ). But as

the CubeSats all have the same idealized maximum density of 1171.81 kgm3 (see table 2.1),

one can assume that the mass ratios for this class of satellites will be much less dispersed

than it is the case for the NGSO-satellites. However, there is also a likely tendency that

the value for the payload mass ratio will decrease relative to the structure ratio as this trend

to higher structure ratios and lower payload ratios for smaller satellites was already seen for

the small NGSO-satellites. And in fact, a rough estimation based on the characteristics of

Page 28


Figure 3.8.: Histogram formS/Cwet

mS/Cdry

Figure 3.9.: Histogram formP/L

mS/Cdry

Figure 3.10.: Histogram for mAOCS

mS/Cdry

Figure 3.11.: Histogram for mStructure

mS/Cdry

the CubeSat-specific COTS components and some data on flown CubeSats retrieved only by

personal communication, confirms this assumption (see tables 3.3 and 3.4).

Page 29

Ratios

Figure 3.12.: Histogram for mPower

mS/Cdry

Figure 3.13.: Histogram for mThermal

mS/Cdry

Figure 3.14.: Histogram for mC&DH+TT&C

mS/Cdry

Figure 3.15.: Histogram formPropulsion

mS/Cdry

Table 3.3.: xav [%] for the Bottom-Up approach of the smaller NGSO-satellites in comparison

to an estimation for 1U CubeSat based on COTS components and real flown/-

planned CubeSats for the mass ratiosmP/L

mS/Cdry

and of the subsystems AOCS, Power

and Structure over mS/Cdry

mP/L

mS/Cdry

mAOCS

mS/Cdry

mPower

mS/Cdry

mStructure

mS/Cdry

NGSO 25 7.2 29.7 27

1U CubeSat 0-20 10-25 30-40 30-40

CP3 29.4 11.8 11.8 20

CP4 40.0 10.0 10.0 17

Cute1.7+APD 12.0 7.0 8 57

Exoplanet 13.5 30.8 28.6 18.2

SMAD 15-50 - - 15-25

Page 30


Table 3.4.: xav [%] for the Bottom-Up approach of the smaller NGSO-satellites in comparison

to an estimation for 1U CubeSat based on COTS components and real flown/-

planned CubeSats for the mass ratios of the subsystems Thermal, C&DH+TT&C,

C&DH and Communication over mS/Cdry

mThermal

mS/Cdry

mC&DH+TT&C

mS/Cdry

mC&DH

mS/Cdry

mCom

mS/Cdry

NGSO 7.1 6.0 - -

1U CubeSat 1-5 - 5-10 15-20

CP3 0 - 4 23.5

CP4 0 - 3 20.0

Cute1.7+APD 0 - 3 13

Exoplanet 0 - 2.8 6.1

SMAD 2-5 - - -

In this context, it should be noticed that in general the mass of the AOCS system highly

depends on the chosen components. The mass ratio for the 1U CubeSat is expected to around

10% when solar panels are used which already have sun sensors, magnetorquers, gyroscopes

and thermal sensors embedded. Only reaction wheels and star trackers are then potentially

added. 15 to 20% mass ratio are expected when the solar panels do not include AOCS

components and no star tracker is used. About 25% mass ratio is assumed when additionally

a star tracker is included in the design. This assumption can also be confirmed by the data on

the planned and flown CubeSats. 10% AOCS mass seems to be usual except for Exoplanet, a

3U CubeSat from MIT students. The high AOCS mass is due to the use of reaction wheels.

The assumed high power ratios for a hypothetical 1U CubeSat are due to the COTS-batteries

on-board (over 60 g for each battery with a capacity of 1200 mAh) and especially the solar

panels. The flown satellites possess lower ratios but it could not be confirmed why. We assume

that smaller or fewer batteries have been used.

Figure 3.16.: 1U CubeSat MOVE [MOVE, 2010a]

Page 31

Ratios

Figure 3.17.: Exploded view of the CubeSat MOVE [MOVE, 2010b]

The structure ratio is assumed to be higher for CubeSats than for the NGSO-satellites, taking

not only the outer walls of the spacecraft into account but also two Printed Circuit Boards

with a weight of 150g where especially the AOCS sensors are mounted on. However, two

1U CubeSats, CP3 and CP4 of Cal Poly, show smaller structure ratios than even the smaller

NGSO satellites. It is assumed that this is due to a special own build light weight structure.

Cute1.7+APD, in turn, is flying with a very high mass ratio of 57%, although the satellite is

Page 32


in the class of a 3U CubeSat. It seems that the structure ratio for CubeSats is as dispersed as

for the NGSO-satellites. One clear trend, however, can be seen by means of the rule of thumb

out of [Larson and Wertz, 1999, Tab.10-10]: larger satellites are assumed to have smaller

structure ratios of 15 to 25% of the dry spacecraft mass than nano- and picosatellites. The

tendency that the structure ratio gets larger for smaller CubeSats can be also confirmed when

plotting the mass structure ratio for several CubeSats, calculated with the potential CubeSat-

COTS components (see figure 3.18). Figures 3.19 and 3.20 show the mass and volume ratio

evolution when assuming a cube with edge length xeq and a outer structure consisting of a

5mm 5052-H32 Aluminum wall with a mass density of 2.68 gcm3 . This metal was chosen as it is

also used on the CubeSatKit from Pumpkin Inc. that builds the structure COTS components

on which the ratios from figure 3.18 are based.

Figure 3.18.: mStructure

mS/Cdry

for hypothetical CubeSats, calculated based on data from COTS-

components

Figure 3.19.: mStructure

mS/Cdry

for hypothetical CubeSats, calculated based on Aluminium wall

assumption

Page 33

Ratios

Figure 3.20.: VStructure

VS/Cfor hypothetical CubeSats, calculated based on Aluminium wall

assumption

The differences in the structure ratios in figure 3.18 and 3.19 are due to the different assump-

tions. In the calculation which is based on the COTS-components (please note the difference

when calculating with the skeletonized or solid walls provided by Pumpkin Inc.), the CubeSat

is assumed with its standard values as defined in table 2.1, thus with dimensions of a quarder.

In case of the calculations with the aluminium wall, the CubeSat is a cube with an edge length

which equals the so called equivalent edge length, a quantity which will be explained in detail

in the next section 3.3.2. For the moment it is enough to know that we assume cubes. The

main result is the fact that the mass structure ratio decreases with increasing dimensions of

the CubeSat. This can be advantageous for the payload mass.

Back to tables 3.3 and 3.4 we can see that the estimated CubeSat’s Thermal mass ratio is

smaller than for the NGSO-satellites as usually only thermal sensors are integrated in a CubeSat

and no complicated thermal control system. The real flown CubeSats even declared a thermal

mass ratio of zero percent. The C&DH subsystem of a CubeSat consists principally of a

motherboard including the microprocessor and the required memory storage causing relatively

low mass contribution. The hypothetical assumption is not far from the real flown CubeSat

data.

The communication subsystem is basically, depending on the chosen design, composed of

transceivers or transmitters and receivers and antennas. According to the chosen redundancy

a greater or fewer number of components are integrated into the satellite. The numerical

value for the theoretical 1U CubeSat in table 3.4 is based on a redundancy which necessitates

two transceivers or two transmitter/receiver pairs for the CubeSat. The real flown CubeSat

data confirms this assumed range with a tendency to smaller Communication ratios for bigger

satellites as it can be seen for Exoplanet and Cute1.7+APD. This tendency can not be as easily

developed with further evaluations as for the structure mass ratio. However, it is assumed

that the communication mass ratio decreases further for bigger satellites which would be again

Page 34


advantageous for the payload mass.

So, in the end, the residual payload mass is approximated to be between 10 and a maximum

of 20%. Surprisingly, the two bigger CubeSats Exoplanet and Cute1.7+APD show smaller

payload ratios than CP3 and CP4, both 1U CubeSats. This was not expected and is due to

the heavy subsystem of the satellites, the structure subsytem for Cute1.7+APD and Power

and AOCS for Exoplanet. CP4, on the other hand is the forth developed CubeSat by CalPoly,

thus clearly showing the gain in experience in design of CubeSats by the high payload ratio.

However, still a lot of flown CubeSats are platforms for Technology Demonstration missions.

In this respect, the payload is part of the subsystems, for example as in the case of the

CubeSat BeeSat of the Technische Universitat Berlin, Germany, which tested newly developed

reaction wheels, and is not considered as payload anymore. By that the mass restrictions for

the payload due to the other subsystems are avoided.

Despite the very rough estimations of the numerical values of the mass ratios for CubeSats,

the most important point from this section, is to see that the mass of the subsystems of a

satellite can definitely be estimated as percentage of the spacecraft’s dry mass for a whole

satellite class as a starting point in the design. This result was proven for the NGSO-satellites

and is also very likely to be valid for the CubeSats.

For the sake of completeness, the NGSO-results for the Bottom-Up approach with considera-

tion of the wet spacecraft mass mS/Cwet can be found in tables 3.5 and 3.6.


for the mass ratiosmS/Cdry

mS/Cwet

,mP/L

mS/Cwet

and of the subsystems AOCS,

Power and Structure over mS/Cwet

mS/Cdry

mS/Cwet

mP/L

mS/Cwet

mAOCS

mS/Cwet

mPower

mS/Cwet

mStructure

mS/Cwet


Std. Deviation s 0.133 0.087 0.048 0.105 0.106


[%] 15.62 29.44 72.59 40.04 59.11



Propulsion over mS/Cwet

mThermal

mS/Cwet

mC&DH+TT&C

mS/Cwet

mPropulsion

mS/Cwet


Std. Deviation s 0.028 0.062 0.063


[%] 44.26 109.22 83.87

Page 35

Ratios

In comparison to the results of the ratios which used mS/Cdryas main spacecraft mass (see ta-

ble 3.1 and 3.2) the percentage of the standard deviation to the arithmetical average is higher

here for the Bottom-Up design approach with mS/Cwet in all ratios except for the one which

relates the spacecraft’s dry mass and wet mass with each other and the payload ratio. But as

the first is not important in case of the CubeSats as they do not have a propulsion subsystem,

and as the latter only varies with marginal 0.99% difference and the other ratios do show

better values in the range of an delta of 1.3% (Power) to up to 9.31% (C&DH+TT&C), it

can be clearly concluded that the Bottom-Up approach with the spacecraft’s dry mass mS/Cdry

is more promising for the CubeSats.

To complete the analysis, the discussion of the results of the Top-Down design approach

follows. Tables 3.7 and 3.8 present the results of the investigation and one can clearly see that

the values for the Top-Down ratios are more dispersed than for the Bottom-Up approaches:

the values sxav

have a delta of plus 2.8% (Power) to 20.8% (Structure) for the subsystems’

ratios of the Top-Down approach in comparison to the Bottom-Up wet mass approach and

a delta plus of 4.1% (Power) to 38.47% (C&DH+TT&C) for the subsystems’ ratios of the

Bottom-Up approach with dry mass.


for the mass ratios of wet mass over payload mass, dry mass over

payload mass and of the subsystems AOCS, Power and Structure over payload

mass

mS/Cwet

mP/L

mS/Cdry

mP/L

mAOCS

mP/L

mPower

mP/L

mStructure

mP/L


Std. Deviation s 1.259 1.080 0.233 0.388 0.494


[%] 33.84 35.41 88.60 42.84 79.81



Propulsion over the payload mass

mThermal

mP/L

mC&DH+TT&C

mP/L

mPropulsion

mP/L


Std. Deviation s 0.107 0.201 0.261


[%] 47.65 137.38 91.84

The histograms 3.21 to 3.28 confirm graphically the higher distribution of the ratios. All

of them represent more values with lower frequencies than the histograms 3.8 to 3.15 of the

Page 36


Bottom-Up approach. One clear example is the power ratio: figure 3.12 shows only four values

with a high peak in 0.3 whereas figure 3.25 shows a distribution of ten values from 0.3 to 1.5

without a clear peak.

The reason for the higher distribution here is definitely the smaller divisor (mP/L is only 44%

of mS/Cdryand even 34% of mS/Cwet). However, a clearer picture was expected as the payload

mass coefficient of variation sxav

is only 67.9% in comparison to 80.3% for mS/Cdryand 86.0%

for mS/Cwet . Hence, the reason for the higher distribution seems to be a lack in a continuous

interdependency between the payload mass and the other subsystems’ masses for the NGSO

satellites. However, it is again expected for the CubeSats that they will show a higher coherence

between the payload mass and the subsystem masses as they are more standardized in their

mass distributions because of the mass limitations due to the standard they obey.

But in the end, a non-dimensional first mass estimation will be best when using the Bottom-Up

dry mass approach which is fortunately also the most important design approach in terms of

the CubeSats.


mP/LFigure 3.22.: Histogram for

mS/Cdry

mP/L

Figure 3.23.: Histogram for mAOCS

mP/LFigure 3.24.: Histogram for mStructure

mP/L

Page 37

Ratios

Figure 3.25.: Histogram for mPower

mP/LFigure 3.26.: Histogram for mThermal

mP/L

Figure 3.27.: Histogram for mC&DH+TT&C

mP/LFigure 3.28.: Histogram for

mPropulsion

mP/L

3.3.2. Volume: Packing Factor

In order to get design information about the dimensions of the considered spacecraft, the next

step in the development of the non-dimensional parameters is to put the volume and thus

the edge length of the spacecraft into consideration. Especially important for CubeSats is a

packing factor as CubeSats are subject not only to mass but also to size restrictions. One

possible definition of this parameter is

p :=VusedVS/C

(3.5)

with

Vused = VP/L + VBus = ΣVi = Σmi

ρi(3.6)

Page 38


i being the representative for every different subsystem, thus AOCS, Power, Structure, Ther-

mal, C&DH, Communication and Propulsion so that we can write

p =VBusVS/C

+VP/LVS/C

(3.7)

Thus this parameter is again a ratio and not a non-dimensional product in terms of the

Buckingham-Π-Theorem. In almost the same manner as the packing factor p was defined

in equation (3.5) and which is especially of importance in the Bottom-Up design approach,

other ratios are possible to be created and investigated. Ratios which can be of great interest

for the Top-Down approach are VBus

VP/Land

VS/CVP/L

, both giving the engineer a first estimation of

how much space is occupied by a payload in comparison to the spacecraft bus or the whole

spacecraft.

Since no data was available for the volume of the payload or the subsystems, the usefulness

of those ratios can not be investigated with NGSO-data. However, as the volume ratios are

supposed to give information about the dimensions of the satellite which is very important

with regard to the volume restrictions of CubeSats, the alluded ratios are assumed to be of

high interest during the design of picosatellites.

Generally speaking, the higher the packing factor, the better is the conception and design

of the spacecraft. However, the value of the above limit is constrained and can practically

never be 100%, not only because of unadept ways of putting the spacecraft together but also

because of the dimensions of the different components which can be unpropitious, too. After

discussion with some student teams involved in CubeSat projects (CP3 and CP4 from CalPoly,

Exoplanet at MIT and MOVE at TUM), typical values for CubeSats are assumed to be in the

range of 60% and ideally up to 80% for the smaller ones. Thus an increase is expected for

smaller CubeSat units and not a constant value over the whole range of CubeSats which is

important to notice in terms of normally desired constancy over a whole satellite class for the

non-dimensional parameters.

Please note that it is very important to differentiate between the density ρi of the whole space-

craft or a subsystem and the packing factor p which are physically and also from a practical

point of view not the same. The definition of the density assumes that the mass is homoge-

neously distributed over the complete volume of the spacecraft, a fact that is not realized in

praxis. Thus the definition of the packing factor which takes account of this fact. Finally, the

packing factor is of higher and more practical importance for the design of the spacecraft than

its density.

As it was already mentioned in table 2.1, it shall be kept in mind that CubeSats in their

standard configuration all have the same maximum density of ρCubeSat = 1171.81 kgm3 . This

allows to simplify the mass and volume relationship

mS/C = ρS/CVS/C (3.8)

Page 39

Ratios

to

mS/C ∼ VS/C (3.9)

To get a relation between the mass and the edge length of the spacecraft, it is necessary to

establish a relationship between the volume and the edge length of the spacecraft. In order to

facilitate the further investigations, an equivalent edge length xeq is introduced in this work

which is defined as

xeq := VS/C13 (3.10)

so that equation (3.9) can be rewritten as

mS/C ∼ xeq3 (3.11)

For the standard CubeSats the values of the equivalent edge length are presented in table 3.9.

Table 3.9.: Equivalent Edge Length xeq for standard CubeSats

System Volume Equivalent Edge Length xeq

units mm3 mm

1U 1.135 · 106 104.31

2U 2.270 · 106 131.42

3U 3.405 · 106 150.44

Please notice that it is assumed that the concept of the equivalent edge length is still ”phys-

ical” for the whole spacecraft but not for every single subsystem as they consist of different

components and are often not integrated closely together.

3.3.3. Power

In the same way as the mass of the spacecraft is influenced by the mass of the payload, the

power consumption or requirement of the spacecraft is driven by the power requirement of the

payload. Thus there is an interdependency between payload and spacecraft power and thus

an interest rises in power ratios just in the same manner as the interest in the mass ratios.

The equations which are going to be investigated in this section are as follows:

For the Top-Down approach:

rPower TD1 =PS/CPP/L

rPower TD2 =PBusPP/L

(3.12)

Page 40


and for the Bottom-Up approach:

rPower BU1 =PP/LPS/C

rPower BU2 =PBusPS/C

(3.13)

with PS/C being the power consumption of the spacecraft and PP/L and PBus being the

normal (and not the peak) power consumption of the payload and the spacecraft bus. It is

also desirable to analyze the power consumption of every single subsystem in comparison to

the payload or spacecraft power consumption as in the case of the mass ratios. However, as

those values are not presented in the data base, only the bus power consumption was used

here. Nevertheless, further investigations should definitely take the subsystems separately into

consideration as demonstrated with the mass ratios.

Please note that the total spacecraft consumption PS/C was not given in the NGSO-database.

Instead, we use PSAeol. Both values should, however, be in the same order of magnitude and

differ from each other only by a margin and the influence of the power need during eclipse

PEclipse so that PSAeol is slightly higher than PS/C . This is typically determined during the

design process for the solar arrays where a closer investigation of the different power quantities

is done. Results of the interdependencies can be find in section 3.3.3. Thus, in the following

investigations PSAeol is used instead of PS/C .

Table 3.10 shows the results of the investigation for equations (3.12) and (3.13).


forPSAeol

PP/L, PBus

PP/L,

PP/L

PSAeol

and PBus

PSAeol

PSAeol

PP/L

PBus

PP/L

PP/L

PSAeol

PBus

PSAeol

Arithm. Average xav 4.351 0.488 0.715 0.201

Std. Deviation s 14.792 0.900 0.239 0.186


[%] 339.95 184.49 33.42 92.78

It is easy to see directly the high percentage of deviation forPSAeol

PP/Land PBus

PP/L. A closer look

at the data base shows the responsible outlier to be the satellites of Pentriad, a constellation

with very low normal payload and bus power consumption of 100 W and 448 W but an end

of life power of 7684 W and a begin of life power of 10247 W. Those high differences can

only be possible when the power budget was not well calculated or, more likely, the peak

power consumptions are very high. Unfortunately, the data base does not contain the peak

power consumptions of the Pentriad satellites. Thus, these explanations are only specula-

tions. Excluding Pentriad of the investigations gives the results shown in table 3.11, thus an

improvement in every ratio, from 1.89% for PBus

PSAeol

to up to 308.68% forPSAeol

PP/L.

Page 41

Ratios


forPSAeol

PP/L, PBus

PP/L,

PP/L

PSAeol

and PBus

PSAeol

excluding Pentriad

PSAeol

PP/L

PBus

PP/L

PP/L

PSAeol

PBus

PSAeol


Std. Deviation s 0.454 0.426 0.195 0.188


[%] 31.27 127.24 26.27 90.87

Especially the results for the ratiosPSAeol

PP/Land

PP/L

PSAeol

are promising. Furthermore, they do con-

firm the results of [Springmann and de Weck, 2004] where a correlation between the spacecraft

power consumption PSAeoland the payload power consumption PP/L was discovered.

The results for PBus

PP/Land PBus

PSAeol

are less promising. On the other hand, a look at their dis-

tributions plotted against the corresponding divisor (see 3.29 and 3.30, showing a clearer

distribution of the ratios than when plotted against the dry mass mS/Cdryin figures 3.31 and

3.32) states that the ratios are not much varying for higher abscissa values, thus especially for

end of life powers and payload power consumption bigger than 2500 W. The high deviations

in table 3.11 are thus especially due to the satellites with low payload power consumption

and/or low end of life power. A glance at the corresponding histograms 3.34 and 3.36 fortifies

this statement by showing clear peaks in the distribution for these ratios which are even more

developed as forPSAeol

PP/Land

PP/L

PSAeol

. (Please note that the outlier Pentriad is still present in the

histograms and can be detected as the single ratio 0.1 in figure 3.33, as the single ratio 77 in

figure 3.35 and as the single ratio 5 in figure 3.36. It is not visible in figure 3.34 as its value

is of 0.05 and counts to the 0.1 bar).

Thus a conclusion only out of the arithmetical average, standard deviation and the coefficient

of variation is not sufficient here and one needs to be careful with those analyses.

What becomes also obvious in figures 3.29 and 3.30 is the fact, that the values for the bus

power ratios are much less distributed graphically than those of the payload or the spacecraft.

The reason for the high percentage of deviation for the bus ratios in table 3.11 becomes here

clear, too: the values for the bus power ratios are so dispersed that they partially vary with

a factor of 20 (see values at 0.1 and then one value at 2 in figure 3.30). The factor for the

payload and spacecraft power values is far less important and can be analyzed to be about 3

in figure 3.29.

In the end, for the purpose of this section to detect constant ratios and/or constant non-

dimensional parameters, all four ratios show potential to be similarity parameters for the

design of the spacecraft and mission but are not as satisfying as expected. Furthermore as

the outliers are more present in the areas of smaller payload or spacecraft power as the divisor

becomes smaller and thus the ratios more sensitive to deviations, high deviations are also

Page 42


expected for the CubeSats as they in general have power consumptions of only several watts.

Nevertheless, constant values for the power ratios of CubeSats are assumed to be possible as

they obey a standard in mass and size which could possibly also lead to a restricted range in

power requirement and power production.

Figure 3.29.: Power ratiosPP/L

PSAeol

and PBus

PSAeol

plotted against PSAeol

Figure 3.30.: Power ratiosPSAeol

PP/Land PBus

PP/Lplotted against PP/L without outliers

Page 43

Ratios

Figure 3.31.: Power ratiosPP/L

PSAeol

and PBus

PSAeol

plotted against mS/Cdry

Figure 3.32.: Power ratiosPSAeol

PP/Land PBus


without outliers

Figure 3.33.: Histogram forPP/L

PSAeol

Figure 3.34.: Histogram for PBus

PSAeol

Page 44


Figure 3.35.: Histogram forPSAeol

PP/LFigure 3.36.: Histogram for PBus

PP/L

Finally, which parameter is best to describe the power interactions, cannot be definitely decided.

Hence the idea for the next step arose in which it was tried to find an interdependency between

mass and power.

3.3.4. Mass and Power

In the classical Top-Down approach, the dry mass of a satellite mS/Cdryis primarily influenced

by the mass of its payload mP/L which in turn is driven by the payload power PP/L. On the

other hand the spacecrafts’ power consumption PS/C is influenced by the power requirement

of the payload, too. These are only two examples of the interactions between mass and

power which play an important role during the spacecraft design and which can be also found

in literature [Larson and Wertz, 1999, p.334],[Springmann and de Weck, 2004]. Because of

these interdependencies it was assumed that the combination of power and mass ratios would

lead to more satisfying, thus less dispersed results than the mass and the power ratios alone.

That is why it was tried in a next step to connect power and mass in ratios which are still

not non-dimensional products in terms of the Buckingham-Π-Theorem but considered to be

worth the investigation. As the payload and the complete spacecraft are the most important

players in the design process, the further investigation will be focused on their characteristics.

For the Top-Down approach it is again important to chose the characteristics of the payload

to be given, thus the payload power consumption PP/L and the payload mass mP/L. For

the Bottom-Up approach it is again vice versa: the mass of the spacecraft (either mS/Cdryor

mS/Cwet) is supposed to be known as well as the power consumption of the spacecraft PS/C .

As the NGSO-data set is the most complete for PSAeol, its value was used for the investigation

instead of PS/C as in the case of the power ratios.

Page 45

Ratios

Four different combinations of the four quantities are possible for the Top-Down and the

Bottom-Up approach likewise and shown in the following:

rMass Power 1 =mS/C

mP/L

·PS/CPP/L

(3.14)

its reciprocal

rMass Power 2 =mP/L

mS/C

·PP/LPS/C

(3.15)

then

rMass Power 3 =mS/C

mP/L

·PP/LPS/C

(3.16)

and its reciprocal

rMass Power 4 =mP/L

mS/C

·PS/CPP/L

(3.17)

Tables 3.12 to 3.15 show the arithmetical average, the standard deviation and the coefficient

of variation for all the four above combinations but with the small difference whether the

dry mass mS/Cdryor the wet mass mS/Cwet of the satellite is used in the ratios and, further,

whether or not the satellites of the Pentriad constellation are taken into account. Generally

speaking, the results are again better without Pentriad. So, the further analysis is based

on the values built without Pentriad which are shown in tables 3.13 and 3.15. First of all,

the percentage of deviation for all the four combination is very similar and also relatively

low considering again all the different satellites in the data base. Thus, it can be generally

concluded that this approach is definitely not on the wrong track to useful results. There are

also no significant differences between the ratios when using the spacecraft’s dry or wet mass.

However, a remarkable difference is the fact that the deviation is smaller for rMass Power 1 and

rMass Power 2 when using mS/Cwet instead of mS/Cdryand, in contrast, for rMass Power 3 and

rMass Power 4 smaller deviations are achieved when using mS/Cdryinstead of mS/Cwet . The

first observation can be explained by the higher deviation in the mass ratios formS/C

mP/Land

mP/L

mS/Cwhen using the dry spacecraft mass mS/Cdry

in comparison to the values when using the

spacecraft’s wet mass mS/Cwet .

Surprising, however, are the coefficients of variation for rMass Power 3 and rMass Power 4. These

are smaller when using the spacecrafts dry mass, whereasmS/C

mP/Land

mP/L

mS/Cshowed higher

deviations in section 3.3.1 when using the spacecraft’s dry mass mS/Cdry. This cannot be

easily explained and is surely associated with the interdependency between mass and power:

the power ratios seem to compensate the deviation in the mass ratios. This effect is apparently

so complimentary that it even produces the lowest percentage of deviation of 33.30% for

rMass Power 3 with use of dry spacecraft mass in comparison to all other combinations. The

lowest deviation of 36.62% for rMass Power 2 by using mS/Cwet was, however, expected as this

ratio combines the ratios with the lowest single deviation, namely 29.3% for the mass ratio

and 26.27% for the power ratio.

Page 46



for rMass Power 1 to rMass Power 4 with mS/Cdry

mS/C

mP/L· PSAeol

PP/L

mP/L

mS/C· PP/L

PSAeol

mS/C

mP/L· PP/L

PSAeol

mP/L

mS/C· PSAeol

PP/L


Std. Deviation s 43.471 0.131 0.847 7.247


[%] 289.48 51.93 42.09 323.82


for rMass Power 1 to rMass Power 4 with mS/Cdrywithout Pentriad

mS/C

mP/L· PSAeol

PP/L

mP/L

mS/C· PP/L

PSAeol

mS/C

mP/L· PP/L

PSAeol

mP/L

mS/C· PSAeol

PP/L


Std. Deviation s 2.826 0.120 0.709 0.214


[%] 58.99 44.66 33.30 40.26


for rMass Power 1 to rMass Power 4 with mS/Cwet

mS/C

mP/L· PSAeol

PP/L

mP/L

mS/C· PP/L

PSAeol

mS/C

mP/L· PP/L

PS/C

mP/L

mS/C· PSAeol

PP/L


Std. Deviation s 60.886 0.089 1.258 4.658


[%] 316.71 44.04 50.69 312.32


for rMass Power 1 to rMass Power 4 with mS/Cwet without Pentriad

mS/C

mP/L· PSAeol

PP/L

mP/L

mS/C· PP/L

PSAeol

mS/C

mP/L· PP/L

PSAeol

mP/L

mS/C· PSAeol

PP/L


Std. Deviation s 2.823 0.078 1.150 0.1932


[%] 50.20 36.62 44.09 42.87

So, in the end, what becomes clear from this short analysis is that single ratios are not neces-

sarily showing smaller deviations than combinations of ratios. In the case ofmS/Cdry

mP/L· PP/L

PS/Ceol

smaller deviations are possible than taking only its mass ratio alone (35.41%). Hence, for

an exhaustive analysis all possible combinations rMass Power 1 to rMass Power 4 should be in-

vestigated. On the other hand, the clear advantage of combining ratios and the desired

Page 47

Ratios

compensating effects for a less dispersed distribution, could not be confirmed for the NGSO-

satellite database for the whole range of possible parameters. Further validations are therefore

recommended in order to decide if the single ratios or the combinations are best to describe

the relations between mass and power for a given database.

The numerical results are confirmed by the graphical presentation of the ratios, firstly for the

ratios created with the the dry spacecraft mass mS/Cdry. Figure 3.37 and the corresponding

histogram 3.41 show how dispersed the results for the ratio rMass Power 1 are, ranging from 2

to 12, although a small peak in frequency can be spotted around 3 and 4, reflecting the ratios

behavior to be concentrated more densely around this values for increasing mS/Cdry. But this

trend is not interesting in terms of the CubeSats whose mS/Cdrydecreases in comparison to the

NGSO-satellites. The reciprocal of rMass Power 1, rMass Power 2, reflects the slightly smaller

deviation of 44.66% in figures 3.38 and 3.42 where the results are distributed over a smaller

range, namely from 0.1 to about 0.6. The graphical representations of rMass Power 4 are very

similar to those of rMass Power 2. However, the smaller occupied range became also obvious,

as most of the data points can be found between 0.4 and 0.6. The smaller percentage of

deviation is provided by rMass Power 3 which can be especially seen in its histogram 3.43 where

the most significant frequency peak of all the four investigated ratios with dry spacecraft mass

can be found. Thus, keeping in mind that the main CubeSat design approach is a Bottom-Up

approach with use of the dry spacecraft mass, ratio rMass Power 3 seems to be the most useful

for that purpose.

But one should also note that all four ratios had most of their outliers in the small dry mass

range, thus the potential mass range of the picosatellites. Intensive investigations of the ratios

(3.14) to (3.17) for the application on CubeSats is thus highly recommended.

Figure 3.37.:mS/Cdry

mP/L· PSAeol


Page 48


Figure 3.38.:mP/L

mS/Cdry

· PP/L

PSAeol


Figure 3.39.:mS/Cdry

mP/L· PP/L

PSAeol


Figure 3.40.:mP/L

mS/Cdry

· PSAeol


Page 49

Ratios

Figure 3.41.: Histogram formS/Cdry

mP/L

PSAeol

PP/LFigure 3.42.: Histogram for

mP/L

mS/Cdry

PP/L

PSAeol

Figure 3.43.: Histogram formS/Cdry

mP/L

PP/L

PS/Ceol


mS/Cdry

PSAeol

PP/L

For the sake of completeness, a brief analysis of the results of the ratios (3.14) to (3.17)

follows which focuses on the comparison of the results when using the mS/Cwet compared to

those using mS/Cdry. A comparison of the corresponding figures and histograms shows that

the value ranges for the ratios (3.14) to (3.17) using the wet mass are smaller for rMass Power 1

and rMass Power 2 (see figures 3.45, 3.46, 3.49 and 3.50) than when using the spacecraft’s dry

mass. That is why the coefficient of variation for rMass Power 1 and rMass Power 2 is smaller

when using mS/Cwet than when applying mS/Cdry. And similarly, the figures and histograms

for rMass Power 3 and rMass Power 4 (see figures 3.47, 3.48, 3.51 and 3.52) show bigger value

ranges for the corresponding ratios when using mS/Cwet in comparison to the results when

using the dry mass mS/Cdry, and hence the bigger the coefficient of variation.

Page 50


Figure 3.45.:mS/Cwet

mP/L· PS/Ceol

PP/Lplotted against mS/Cwet

Figure 3.46.:mP/L

mS/Cwet

· PP/L

PS/Ceol


Figure 3.47.:mS/Cwet

mP/L· PP/L

PS/Ceol

plotted against mS/Cwet

Page 51

Ratios

Figure 3.48.:mP/L

mS/Cwet

· PS/Ceol

PP/Lplotted against mS/Cwet


mP/L

PS/Ceol

PP/LFigure 3.50.: Histogram for

mP/L

mS/Cwet

PP/L

PS/Ceol


mP/L

PP/L

PS/Ceol


mS/Cwet

PS/Ceol

PP/L

Page 52


3.4. Non-Dimensional Parameters

3.4.1. A Top-Level Approach

In the previous sections, interdependencies between different masses, powers and the volume

and edge length became obvious. Apparently, these characteristics are highly interconnected

as the following examples summarizes the results of the previous sections and also give further

relations:

• The spacecraft mass is influenced by the power requirement because according to the

power requirement the corresponding components are heavier or lighter.

• The spacecraft mass is limited by the volume of the spacecraft as no infinite density is

possible.

• The volume is driven by the mass (as no infinite density is possible: usually, the higher

the mass, the bigger the volume) and power (i.e. the higher the power requirement, the

more complex is the power requirement, thus the bigger and numerous the components).

• Volume and mass are both driven by the orbital period (or, even more precisely, the

eclipse time) and power requirement: the longer the eclipse time or the power require-

ment, the bigger and heavier are the batteries and thus the bigger are the solar panels.

Thus, the orbital period is supposed to interplay in the system, too, as it can also be seen

in the figures 3.53 and 3.54 for the NGSO-satellites: the higher the orbit, the bigger is the

orbital period, thus in general the higher is the spacecraft mass and hence also the spacecraft

power requirement.

Figure 3.53.: mS/Cwet plotted against tOrbit Figure 3.54.: PSAeolplotted against tOrbit

Page 53


Figure 3.55.: ρS/C plotted against tOrbit

Because of the high interdependencies between those four parameters mass, power, vol-

ume/length and orbital period, it was assumed that a non-dimensional parameter build with

these four quantities could be very useful for design purposes. Hence, taking these four main

parameters of a spacecraft into consideration, one can write:

0 = f(mS/C , PS/C , VS/C , tOrbit) (3.18)

With the number of system influencing quantities n to be four and with L, M and T being

the base dimensions of the quantities, thus k = 3, one non-dimensional product has to be

determined. The k-set is chosen to contain mS/C although its importance is minimal here

as only one non-dimensional parameter is expected. Dimensional analysis according to the

Buckingham-Π-Theorem described in section 2.4.1 directs us to the following non-dimensional

product

Π1 =PS/C · t3OrbitVS/C

23 ·mS/C

(3.19)

As the edge length of a spacecraft is more practical for the design of a spacecraft than its

volume, the non-dimensional product can also be expressed with the equivalent edge length

xeq by applying equation (3.10):

Π2 =PS/C · t3Orbitx2eq ·mS/C

(3.20)

In terms of the intraregime approach [Bhaskar and Nigam, 1990], these two parameters can

also be physically confirmed. A higher spacecraft mass mS/C leads in general to higher power

consumption of the spacecraft PS/C . The same argumentation is valid for bigger satellites since

a raise in the dimensions of a satellite, here presented by the equivalent edge length x2eq, usually

leads to a raise in the mass of the spacecraft and thus again in the power consumption. The

position of the orbital period tOrbit in the parameter can be explained by the fact that usually

heavier and bigger satellites can be found in higher orbital altitudes, thus with higher orbital

Page 54


period. And those heavier and bigger satellites, in turn, have again higher power requirements.

Thus all these relationships are perfectly expressed by Π1 and Π2.

In the following, we will also deal with dimensional parameters derived from the non-dimensional

parameters Π1 and Π2 in order to reduce the number of influencing quantities in the similarity

parameters. These dimensional parameters are assumed to be useful for the design of satellites

with similar quantities. For example, as mS/C scales with x3eq (see equation (3.11)) Π2 can be

rewritten to

Π∗1 =PS/C · t3Orbit

x5eq

(3.21)

for satellites with equal mass densities. This replacement changes the numerical value of

Π2 by the reciprocal of the spacecraft’s density ρS/C and makes Π2 dimensional. In order to

distinguish between non-dimensional and dimensional quantities the dimensional Πs are marked

with a ”*”. The approximation of constant mass densities is especially true for CubeSats as

they have constant maximum densities over the whole CubeSat range. This assumption is

unfortunately not true for the NGSO-satellites as it can be seen in figure 3.55. The lack

in density constancy of the NGSO-satellites will be one of the main reasons for the high

distribution of the parameters when applied for NGSO-satellites later on in this section.

Supposing equal orbital periods for the investigated satellites, the orbital period tOrbit can be

integrated in Π∗1, leading to

Π∗2 = Π∗1 · t3Orbit =PS/Cx5eq

(3.22)

This equation is especially interesting as it reflects a result of [Springmann and de Weck, 2004]:

with increasing edge length the power of the spacecraft has also to increase in order to main-

tain a constant non-dimensional parameter in terms of the similitude theory. For example,

doubling the edge length of the satellite necessitates to take the power times 25. Thus an

increase in the specific power Peol

xeqwith increasing equivalent edge length xeq can be observed,

or in other words, some economies of scale can be detected.

Integrating tOrbit in Π1 which is again valid for satellites with same orbit periods, leads us to

Π∗3 = Π1 · t3Orbit =PS/C

VS/C23 ·mS/C

(3.23)

Some remarks shall follow these equations before the analysis of the results by the NGSO-data

is presented:

• The above equations can be applied to each different subsystems by using the corre-

sponding quantities. As the volume and also the equivalent edge length of a subsystem

is difficult to be determined, it should to be reasonable to substitute the volume in equa-

tion (3.19) by the mass of the subsystem which is more easily to be identified. With

(3.9) Π1 can be rewritten to

Π∗4 = Π1 · ρS/C =PS/C · t3Orbit

VS/C53

(3.24)

Page 55


• In case of the subsystem Power, one possibility is to replace PS/C by PBattery and to

substitute tOrbit by the eclipse time tEclipse. mPower includes, among other things, the

mass of the batteries, the solar panels and also the mass of the power distribution unit.

Thus equation (3.19) becomes

Π∗Power =PBattery · t3Eclipse

mPower53

(3.25)

Other replacements with tchargingtime instead of tEclipse or PSolarPanel instead of PBatteryare also reasonable and should be investigated in future work.

• As the payload is in the focus during the design of the spacecraft, its consideration in

above equations should not be ignored. Possible replacements for the quantities in Π1

are simply mP/L, PP/L, VP/L and Tdutycycle.

ΠP/L 1 =PS/C · t3dutycycleVP/L

23 ·mP/L

(3.26)

A ”hybrid” approach in terms of relating spacecraft specific quantities and payload

characteristics simultaneously in equation (3.19) may also be possible. One example

could be

ΠP/L 2 =PP/L · t3dutycycleVP/L

23 ·mP/L

(3.27)

• Discussing the time choice, it was considered to use the energy over an orbit or the

whole lifetime instead of the power requirement and the orbital period, in order to get

a more general equation. However, as most of the satellites do have similar lifetimes

and orbits in their own class (i.e. communication satellites are normally in GEO with a

lifetime of about 15 years; CubeSats are in LEO with a design lifetime of about a year)

this idea was considered to account for nothing more than the original equation with the

orbital period and the power requirement. Furthermore, as the same orbital period can

lead to different eclipse and sunlight times, it is supposed that it is more useful to take

the eclipse time tEclipse instead of tOrbit as time quantity. The calculation of the eclipse

time is presented in Appendix A. Especially for the minimal eclipse time and also for

sun-synchronous missions in general the knowledge of the inclination i of the orbit plane

is essential. For the sun-synchronous orbits the Right Ascension of Ascending Node also

needs to be known for the computation of the eclipse time. In order to keep the design

problem as simple as possible, however, the consideration of the orbital period tOrbit is

assumed to be sufficient enough for our estimations at this point.

The non-dimensional and dimensional parameters of this section were investigated for the

available data of the NGSO-satellites. Tables 3.16 and 3.17 summarize the results for Π1 =PS/Ct

3Orbit

VS/C23mS/Cwet

and Π∗3 =PS/C

VS/C23mS/Cwet

:

Page 56



for Π1 for all NGSO-satellites and those in LEO

Π1 Π1 for satellites in LEO

Arithm. Average xav 5.74082E+12 4.44645E+11

Std. Deviation s 1.61752E+13 6.51342E+11


[%] 281.76 146.49


forPS/C

VS/C23mS/Cwet

for all NGSO satellites and those in LEO

Π∗3 Π∗3 for satellites in LEO

Arithm. Average xav 1.360 1.696

Std. Deviation s 2.074 2.663


[%] 152.50 157.02

First of all, it shall be mentioned that the available data for those products was extremely rare,

so that only 10 data points out of 38 possible satellites were available for the investigation

of Π1. What becomes obvious, however, is the relatively large deviation of nearly 282%.

Consequently, the next step was again to eliminate the orbital period tOrbit and investigate

Π∗3. The results are definitely better then, showing a coefficient of variation of only 152.5%

which, in comparison to the result for Π1, is an improvement of nearly 130%.

To confirm the assumption that the orbital period is not a suitable time choice, the numerical

values for Π1 and Π∗3 were computed only for those satellites in LEO, all with a similar orbital

period of about 100 minutes, of which data points were available for both parameters Π1

and Π∗3. But, surprisingly, the percentage of deviation is smaller for the LEO satellites when

taking the orbital period into account, namely 146.5% in comparison to 157.0%. One can

assume that this is due to computational mistakes in MS Excel since the numerical values for

Π1 get very high because of the contribution of the orbital period, but also when introducing

tOrbit in minutes and not seconds the coefficient of variation becomes even better for Π1,

namely 142%, thus an improvement of 10% in comparison to the parameter Π∗3. Thus, for

the CubeSats which are all suited in orbits with relatively similar parameters (mostly LEO and

sun-synchronous), it seems reasonable to use Π1 and take account of the orbital period tOrbit.

When the orbital parameters are more scattered for the class of the investigated satellites, Π∗3will be better suited. However, given the very rare datapoints, these are only speculations and

need to be verified with more empirical data.

As the available data points for Π1 are so random, only the histograms of the results are

presented (see figures 3.56 and 3.57), showing the dispersed characteristic of the results but

also the improvement when eliminating the non-LEO satellites.

Page 57


Figure 3.56.: Histogram for Π1 Figure 3.57.: Histogram for Π1 for the NGSO-

satellites in LEO

Figure 3.58 shows the data points of Π∗3 plotted against mS/Cwet . The outliers in those figures

are detected to be the satellites of the constellation Ellipso and Orbcomm, both having very

small volumes: 0.6 m3 for Orbcomm and 0.09 m3 for Ellipso. Pentriad is not an outlier

in that case, although it has a very large end of life power production, since its volume of

approximately 5 m3 together with its mass of about 2000 kg compensate the high power.

A closer examination of the graph 3.58 leads to the figure 3.59 that highlights the range of

most of the data points to be between 0.2 and 1.5.

A presentation of the results plotted against tOrbit (see figure 3.60 and 3.61) does not help

much further at first glance as the number of available data points is reduced. What becomes

clear, however, is that the outliers are LEO satellites and not satellites with complete different

orbital parameters. This means that their deviation from the other satellites is not due to

a difference in the orbital periods but really in the spacecraft’s characteristics. Thus the

neglection of tOrbit in Π∗3 does not cause any outliers. Besides, the only two data points

which are not representing satellites in LEO are located in the range between 0.2 and 1.5, as

well, thus reflecting the fact that the consideration of the orbital period with power 3 in the

non-dimensional product would definitely let disperse the data points.

A look at the histograms 3.62 and 3.63 possibly reveals why the percentage of deviation for

Π∗3 is slightly worse when taking only the satellites in LEO into consideration whose values are

available for both Π1 and Π∗3: on the one hand, this eliminates three data points on the bar

with value 2 (these being unfortunately the data points where no information on the orbital

parameters was available), on the other hand, this also deletes three satellites on the bar with

value 1 which reflects definitely the majority of the data points and the significant outliers on

4 and 9 are not touched at all by this selection.

It can be summarized so far that the inclusion of the orbital period tOrbit reduces the disper-

sion for satellites with similar orbital parameters. Thus the conclusion from the theoretical

presentation of the findings that for satellites with similar orbital period Π∗3 instead of Π1 is

Page 58


suited, is not generally true and proves the importance of numerical validation of the theoreti-

cal findings. On the other hand, when considering satellites with different orbits, taking tOrbitinto account leads only to more dispersion. This result is not satisfying and has to be further

investigated with a more extensive database than given, hoping that the high dispersion is due

to the lack datapoints.

Figure 3.58.: Π∗3 plotted against mS/Cwet Figure 3.59.: Zoom of Π∗3 plotted against

mS/Cwet

Figure 3.60.: Π∗3 plotted against tOrbit Figure 3.61.: Zoom of Π∗3 plotted against tOrbit

Page 59


Figure 3.62.: Histogram for Π∗3 Figure 3.63.: Histogram for Π∗3 for the NGSO-

satellites in LEO

In a further step, the substitution of the mass by the volume (instead of the equivalent edge

length) was investigated which leads us to

Π∗4 = Π∗3 · ρS/C =PS/C

V53

(3.28)

as well as the elimination of the orbital period tOrbit and the mass of the spacecraft mS/C

from Π1 supposing similar masses and orbits

Π∗5 = Π∗3 ·mS/C =PS/C

V23

(3.29)

Also of interest is to eliminate the spacecraft’s volume VS/C and the orbital period tOrbit.

Π∗6 = Π∗3 · V23 =

PS/CmS/C

(3.30)

with mS/C = mS/Cwet .

The numerical results for these investigations can be found in table 3.18. Π∗4 and Π∗5 were

especially investigated to see how significant the influence of the diverse densities of the

satellites is. First of all, the figures 3.64, 3.65 and 3.67 for Π∗4 prove the fact that for the

given NGSO-database mass and volume of the spacecrafts do not relate all with the same

proportionality factor, which is the density, since the results are more scattered for Π∗4 than for

Π∗3. Thus the substitution of the mass by the volume is not possible. The higher dispersion

not only becomes clear in the higher coefficient of variation of 272.5% in comparison to

152.5% for Π∗3 but also in the histogram 3.67: here, in comparison to figure 3.62 a third,

very significant outlier can be determined on 6000. Furthermore the peak on the first class is

not as significant as in case of Π∗3, the second class, however, increases in frequency. For Π∗4,

Ellipso is the significant outlier at 6000 again because of its very small volume which is now

even more emphasized because of its higher exponent 53

in equationPS/C

VS/C53

. The outlier on 900

Page 60


is Orbcomm, again because of its small volume, and the new outlier on 500 is Pentriad where

the relatively small volume alone is appareantly no longer able to compensate the immense

power production at the end of life.

Thus, the mass and volume of the presented spacecrafts cannot be substituted by each other

without changing the numerical value of the non-dimensional product. In the NGSO-case,

Π∗4 showed significant differences and even higher dispersion than the results for Π∗3. For the

CubeSats, however, no important difference in the dispersion is expected as they are expected

to show a constant interdependency between mS/C and VS/C with the proportionality factor

ρS/C .

Additionally, investigating Π∗5 confirms that mass and volume are definitely not playing hand

in hand for the satellites in the given database, as the coefficient of variation gets even smaller

when ignoring the mass at all in the non-dimensional product. This can not only be seen

by a smaller coefficient of variation (120.9% in comparison to 152.5% when taking also the

mass into consideration in Π∗4) but also in the histogramm 3.68 where the only bars can be

found on 1000, 2000 and 3000 in comparison to 3.62 where four different bars were visible

and the first bar’s frequency was smaller than here. One can argument that the value range

is now multiplied by 103 but this is only a constant which can be eliminated by inserting the

numerical value of PS/C in kW and not in W. The presentation in figure 3.66 lets assume

the same concern but a closer look at the y-axis confirms the smaller value range (from 0 to

2600, or clearer, from 0 to 2.6 · 103) in comparison to figure 3.66 where the numerical values

ranged from 0 to over 8. Outliers, thus satellites with values above 1000, in figure 3.66 are

Pentriad, Virgo, Orblink and Teledisc (enumerated with anticlimatic deviation from 1000), all

satelittes with very high powers and/or small volumes. Comparatively, even better results can

be obtained with Π∗6 , thus completely eliminating the volume of the spacecraft.

However, the non-dimensional product and its derivatives are expected to show better results

when validated for CubeSat use. Π∗3 and Π∗4 are expected to provide promising results, as well

as the parameters Π∗1 and Π∗2 with the equivalent edge length xeq as quantity. Π∗5 and Π∗6were only investigated to show the discrepancy of volume and mass of the NGSO-satellites,

but could possibly be useful for other satellite classes again. Π1 and Π2, however, have to be

further investigated especially because of the influence of the orbital period tOrbit.

Nevertheless, especially the two parameters Π1 and Π2 are assumed to be very useful for quick

design estimations as they combine four main quantities of a satellite design: mass, power,

length/volume and orbital period. Three parameters of them, mass, power and length/volume,

are both for the Top-Down and the Bottom-Up approach input quantities for the design

process, for the Top-Down approach with payload specific parameters and for the Bottom-Up

approach with spacecraft specific quantities. We will see later on in figures 3.69 and 3.70

that Π1 and Π2 are not directly integrated in the flow of the design calculations based on

dimensional analysis. However, they are especially advantageous when one of the four design

quantities is missing as input parameters. It can then easily be calculated with the knowledge

of the numerical value of the non-dimensional parameter and three given quantities.

Page 61



for Π∗4, Π∗5 and Π∗6

Π∗4 Π∗5 Π∗6


Std. Deviation s 1477.130 748.720 1.638


[%] 272.50 120.89 69.24

Figure 3.64.: Π∗4 plotted against mS/Cwet Figure 3.65.: Zoom of Π∗4 plotted against

mS/Cwet

Figure 3.66.: Π∗5 plotted against mS/Cwet

Page 62


Figure 3.67.: Histogram for Π∗4 Figure 3.68.: Histogram for Π∗5

3.4.2. Subsystem Power: Battery and Solar Array

One of the most important subsystems of a satellite is its power system as its enables the

satellite to be operational. The payload, the on-board computer, the communication system

none of these could work without power. On the other hand, the power subsystem is also

one of the most significant mass contributors. Thus, because of its high importance for the

success of the mission and its significance in the design, the two most typical design solutions

shall be discussed in the following.

BATTERY

The basic power source used on satellites is a battery. If no other power source is present

on-board, it provides the satellite with the required power over its whole lifetime. More

sophisticated solutions use solar arrays in addition to batteries for storing and providing energy

when the satellite is in eclipse and during the start phase. Thus a closer look on the design of

these devices in this work is considered to be significant.

The main characteristic performance parameter of a battery is its capacity CBattery[Ws]. On

one hand, the capacity of a battery is driven by the power required of the satellite during

eclipse PEclipse and the eclipse duration tEclipse. On the other hand, the satellite lifetime

tSatellite determines the number of charges/discharges, thus the number of duty cycles of the

battery and thus the Depth of Discharge (DOD). The DOD, in turn, contributes to the choice

of the battery’s capacity: the higher the Depth of Discharge, the lower is usually the required

capacity for a fixed power requirement. Hence four parameters play a role in the battery

system:

0 = f(CBattery, PEclipse, tEclipse, tSatellite) (3.31)

with

• the capacity of the battery CBattery [Ws] = [kgm2

s2]

Page 63


• the power required of the spacecraft during eclipse PEclipse [W ] = [kgm2

s3]

• the eclipse duration tEclipse [s]

• and the satellite lifetime tSatellite [s]

So, with the definition of a combined non-fundamental SI-dimension [ML2], it is n = 4 and

k = 2, thus the number of non-dimensional parameters is i = 2. With a k− set consisting of

PEclipse and CBattery, equation (3.31) can be rewritten to

ΠBat1 = f1(PEclipse, CBattery, tEclipse)

ΠBat2 = f2(PEclipse, CBattery, tSatellite)

}(3.32)

and the following non-dimensional parameters can be found:

ΠBat1 =CBattery

PEclipse · tEclipse

ΠBat2 =CBattery

PEclipse · tSatellite

(3.33)

The two non-dimensional parameters can also be confirmed physically in terms of the intra-

regime approach [Bhaskar and Nigam, 1990]: for both of them, an increase in the power

requirement of the spacecraft PEclipse during eclipse goes along with an increase in the nec-

essary capacity of the battery CBattery for a constant Π. Furthermore, supposing ΠBat1 to be

constant over the satellite class, an increase in the eclipse time tEclipse leads to an increase

in the necessary capacity of the battery CBattery for constant PEclipse. Considering ΠBat2 , it

is known that an increase in the lifetime of the satellite causes an increase in the number

of charges and discharges of the battery, thus the cycle life of the battery. This, in turn,

reduces the depth of discharge (see [Larson and Wertz, 1999, p.421]) which then necessitates

an increase in the capacity of the battery for a constant energy requirement PEclipse · tEclipseof the spacecraft during eclipse. These interdependencies are hence appropriately expressed

by ΠBat2 . The non-dimensional products ΠBat1 and ΠBat2 could be now further developed to

a scaling law according to (2.11). As we will not be able to determine scaling laws in this

work because of the lack of data, another method of expansion is chosen to directly show the

influence of ratios on subsystems: the non-dimensional parameters are directly expanded with

the subsystem influencing ratios. This proceeding will be carried out for this and the following

subsystems.

A further expansion of the non-dimensional products ΠBat1 and ΠBat2 is possible with the

mass and power consumption of the spacecraft since these quantities are also the main input

during the Bottom-Up design approach. We will focus on ΠBat1 to show the expansion. Hence

we can write

ΠBat3 =CBattery

PEclipse · tEclipse·(

PS/CPEclipse

)a·(mS/C

mPower

)b(3.34)

This is also physically consistent and a constant Π can be achieved as an increase in the

power requirement of the spacecraft PS/C normally leads to a higher power requirement in

Page 64


eclipse PEclipse (it can be assumed that PS/C is equal to PEclipse although this is not generally

true as the power consumption is usually held lower in eclipse by turning off the payload

for example) which in turn necessitates a higher battery capacity CBattery. A higher battery

capacity usually causes a higher power mass mPower - easy to understand when imagining that

the requirement for higher capacity is satisfied by an additional battery - which, in turn, entails

a higher spacecraft mass mS/C .

From the payload-centric point of view which necessitates the use of the payload mass and

power consumption, ΠBat3 can be re-expressed with PP/L instead of PS/C as a change in the

payload power involves the same reasoning and changes as a change in the spacecraft power

requirement. In terms of the payload mass, it is assumed that a higher payload mass implies a

higher payload power requirement [Springmann and de Weck, 2004] and thus a higher battery

capacity which in turn causes a higher power mass. All these interdependencies are implied in

the following representation of ΠBat4 :

ΠBat4 =CBattery


PP/LPEclipse

)a·(mP/L

mPower

)b(3.35)

Interdependencies with the other subsystems are certainly possible as well since the power

subsystem is highly interconnected with every subsystem of the satellite that consumes power.

Thus it would be physically consistent to express ΠBat3 with for example PCom and mCom in-

stead of PS/C and mS/C . However, those interdependencies are already expressed by the power

and mass ratios in sections 3.3.1 and 3.3.3 and are thus implied when using the spacecraft

mass and the spacecraft power.

To complete this section, it shall be mentioned that is was attempted to express the eclipse

time tEclipse by the orbital period tOrbit in order to standardize the quantities in the non-

dimensional products used in accordance to the use of tOrbit in section 3.4.1. However, as

shown in the Appendix A of this work, tEclipse is not simply proportional to the orbital period

tOrbit but a complex function of the orbital altitude hOrbit. In case of the CubeSats which are

mostly launched in sun-synchronous orbits, tEclipse is additionally dependent on the inclination

i of the orbital plane and the Right Ascension of the Ascending Node RAAN . In order to

keep the level of complexity in this work adequate to a first order model, the approach with

tEclipse as a quantity of the battery system is considered to be sufficient.

Another remark has to be done regarding the efficiencies of the battery. Efficiencies are not

directly considered in the non-dimensional products above but can be retrieved in the numerical

value of the non-dimensional products. For a satellite class and the same battery type (i.e.

Lithium-Ion), the non-dimensional products describing the battery should have the same value

for every kind of satellite. The last remark in this section refers to the number of batteries

used on the satellite. The amount of batteries is not explicitly listed but can be retrieved in

CBattery which shall be the representative for the accumulated capacity of all batteries used

on the spacecraft.

As a conclusion, it can be stated that the design of the battery is influenced by several

quantities. However, in a first order model, the capacity of the battery CBattery, the power

requirement during eclipse PEclipse, the eclipse time tEclipse, the satellite lifetime tSatellite, and,

Page 65


depending on the design approach, the mass of the spacecraft mS/C and the power requirement

of the spacecraft PS/C or the mass of the payload mP/L and the power requirement of the

payload PP/L are the most significant quantities for the battery-system.

To finish this section, some notions and expressions of 3.2 shall be exemplary shown on the

non-dimensional parameters of the battery in terms of the practical validation. The same

explanations will be valid for the following sections. Equations (3.34) and (3.35) show the

nature of (3.2). Thus nr can be identified to beCBattery

PEclipse·tEclipsefor both parameters.

PS/C

PEclipse,

mS/C

mPower,

PP/L

PEclipseand

mP/L

mPowerare ratios ri in terms of equation (3.2) and k is equal to 1

ΠBat3

for equation (3.34) and equal to 1ΠBat4

for equation (3.35). Hence the validation of the

non-dimensional parameter would include the determination of the numerical value of every

quantity as described in 3.2. As the currently available information is not sufficient for this

kind of detailed validation, the results in this and the following sections will remain theoretical.

A first order numerical application will be given in section 3.6.

SOLAR ARRAY

Solar arrays are used on spacecrafts in order to provide the satellite with power over a longer

lifetime than it is possible with primary batteries. They not only provide the satellite with power

during daylight but also recharge the secondary batteries on-board which are used during an

eclipse. Solar arrays have been widely adopted, ranging from the International Space Station

to CubeSats. Its investigation in terms of the Buckingham-Π-Theorem shall be done in this

section.

In a first order approximation, the power amount produced by a solar array is mainly dependent

on the type of the cells, the solar array area, the operating temperature, the sun angle and

the solar constant:

0 = f(PSAbol, type of cells, ASolarArray, θ, Toperating, cSolar) (3.36)

with

• the begin of life power production by the solar arrays PSAbol[kgm

2

s3]

• the type of the solar array cells (i.e. Silicium, Gallium Arsenide)

• the surface of the solar array ASolarArray [m2]

• the average sun angle over the whole mission θ [deg]

• the operating temperature Toperating [K]

• the solar constant cSolar which is equal to 1367 kgs3

The characteristic ”type of the cells” is ideally an efficiency which is neglected in the following

analysis as it will emerge again in the numerical value of the non-dimensional product. Typ-

ically, the non-dimensional product should take a constant value for solar arrays of the same

type over a satellite range.

Page 66


The influence of the orbit on the performance of the solar array is represented by θ, the aver-

age incident sun angle over the whole mission, and the solar constant cSolar, which normally

changes in conjunction with the season but is supposed to be constant in our investigations.

Both quantities are considered together in the quantity SSolar which equals cosθ · cSolar. A

remark in terms of θ should be stated before continuing the analysis: as in case of the batteries

and the eclipse time, the average solar angle is also dependent on the orientation of the orbital

plane in the ecliptic as well as the satellite itself (in case of body mounted solar panels) or

the orientation of the deployable panels. As this work is a first attempt to analyze satellites

with dimensional analysis, dependencies between θ and the orbital parameters as well as the

orientation of the satellite are not taken into consideration in order to reduce the complexity

of the analysis. Please note, however, that θ equals the solar angle βs (see Appendix A) for

satellites which are launched in sunsynchronous orbits and whose orientation to the Sun is

supposed to be constant.

Continuing now on the analysis, it can be stated that the operating temperature Toperating of

the solar array affects the performance of the array and is driven by the orbital parameters. The

highest power is produced when the satellites comes out of eclipse, thus when the temperature

of the array is as low as possible. But since its influence is also complex, it was neglected in

the following first order analysis.

In general, the cells do not cover the whole outside area of the spacecraft as some space is

also required for the Communication subsystem (antenna) and for the payload (i.e. camera).

However, for the sake of simplification, it is assumed that the whole outside area of the satellite

is used for the solar cells, thus ASolarArray ∼ x2eq (an equalization is intentionally not made

as the amount of illuminated area changes during the orbit. In the case of a cube, though, a

maximum of three sides is simultaneously illuminated). In the case of deployable solar arrays,

another equivalent length quantity than xeq must be used, which represents not only the

body-mounted solar cells but also the deployable solar panels. Furthermore, a more detailed

approach should also consider areas of the satellite’s surface which are not covered by solar

cells or obscured by the deployables. In this work, however, we will continue with the body

mounted solar panels since this is usually adopted for CubeSats. Taking into account these

remarks, equation (3.36) can be finally simplified to

0 = f(PSAbol, xeq, SSolar) (3.37)

The base dimensions of the quantities describing the system are the length L and the combined

dimension watt W. Thus with the number of quantities describing the system n to be three

and the number of the base dimensions to be two, one non-dimensional product can be found:

ΠSA1 =PSAbol

x2eq · SSolar

(3.38)

In terms of the intraregime approach [Bhaskar and Nigam, 1990] ΠSA1 perfectly expresses

that the power produced by the solar array PSAbolincreases with increasing solar array area,

represented by x2eq, and/or increasing quantity SSolar which increases with higher average sun

angle θ assuming cSolar to take the constant average value of 1367 Wm2 . A more sophisticated

Page 67


approach would take into account that cSolar actually changes during the seasons (i.e. maximal

in winter).

In order to get information about the design of the spacecraft over the whole mission length, it

is necessary to incorporate also the lifetime degradation of the solar cells into the considerations

which can be expressed by Ld = (1− d)tSatellite [Larson and Wertz, 1999, p.417] with

• the yearly degradation d = degradationyear

[ 1yr

] which takes a typical value for every kind of

solar cell (i.e. satellites in LEO with cells out of Gallium Arsenide are subject to a yearly

degradation of 2.75% [Larson and Wertz, 1999, p.417])

• and the satellite lifetime tSatellite [yr]

Thus equation (3.37) can be expanded to

0 = f(PSAeol, xeq, SSolar, tSatellite, d) (3.39)

and hence

ΠSA2 =PSAeol

· d · tSatellitex2eq · SSolar

(3.40)

which can be also confirmed physically in terms of the intraregime approach again: assuming

ΠSA2 to be constant over the satellite class, an increase in the yearly degradation d or in the

satellite lifetime tSatellite will cause a decrease in PSAeolfor a constant PSAbol

.

On the other hand, an expression with PSAbolis also possible:

ΠSA3 =PSAbol

x2eq · SSolar · d · tSatellite

(3.41)

thus an increase in the yearly degradation d or in the satellite lifetime tSatellite will lead to a

necessary increase in PSAbolin order to achieve a constant PSAeol

.

Taking both these approaches into consideration, we can write

ΠSA4 =PSAbol


·(PSAbol

PSAeol

)a(3.42)

which expresses the same interdependencies as the non-dimensional products ΠSA2 and ΠSA3

before but in a combined equation.

As the overall aim of this work is to determine key design characteristics of a spacecraft

and/or a mission with few known characteristics of the payload (Top-Down approach) or of

the spacecraft (Bottom-Up approach), it is of special interest now to include not only the

equivalent edge length of the satellite xeq into the non-dimensional products but also its mass

and power. In terms of the mass of the solar arrays, this weight is already included in mPower

and its relation to mS/C and mP/L was investigated in section 3.3.1. As for the power, the

power productions PSAboland PSAeol

are a function of PS/C , tEclipse and tOrbit (representative

for tDaylight) since

PSolarArray · tDaylight = PS/C · tDaylight + PEclipse · tEclipse

PSolarArray = PS/C + PEclipse ·tEclipsetDaylight

(3.43)

Page 68


with

tDaylight = tOrbit − tEclipse (3.44)

Furthermore, in a simplified model, PEclipse can be supposed to be equal to PS/C (which is

normally not true as the operations on board are reduced in eclipse). Hence, equation (3.43)

can be rewritten to

PSolarArray = PS/C ·1

1− tEclipse

tOrbit

(3.45)

and finally

ΠSA5 =PSAbol


·(PSAbol

PSAeol

)a·(mPower

mS/C

)b·(PSAbol

PS/C

)c·(tOrbittEclipse

)d(3.46)

Again, this can be physically confirmed: ΠSA5 appropriately expresses that for a satellite class

an increase in the satellite mass mS/C usually entails an increase in the spacecraft’s power

requirement PS/C , and thus an increase in PSAboland also in the mass of the power system

mPower. Furthermore, equation (3.45) states that with increasing eclipse time tEclipse the

power produced by the solar array has also to be increased for a constant power requirement

PS/C . This interdependency is perfectly expressed by ΠSA5 as well.

While ΠSA5 is again a spacecraft-centric approach, it can be rewritten with mP/L and PP/Linstead of mS/C and PS/C as in the case of the battery and we get

ΠSA6 =PSAbol


·(PSAbol

PSAeol

)a·(mPower

mP/L

)b·(PSAbol

PP/L

)c·(tOrbittEclipse

)d(3.47)

The analysis for the Top-Down approach will again be the same as for the Bottom-Up approach.

Interdependencies to other subsystems are again possible as in the case of the batteries.

Please notice as a final remark of this section that the non-dimensional product ΠSA2 can be

expanded in the same way as ΠSA3 leading to

ΠSA7 =PSAeol


·

·(PSAeol

PSAbol

)a·(mPower

mS/C

)b·(PSAeol

PS/C

)c·(tOrbittEclipse

)d (3.48)

for the Bottom-Up approach and for the Top-Down approach:

ΠSA8 =PSAeol


·

·(PSAeol

PSAbol

)a·(mPower

mP/L

)b·(PSAeol

PP/L

)c·(tOrbittEclipse

)d (3.49)

In conclusion it can be stated that the solar array system is at least as complex as the battery

system since again a significant number of quantities play an important role in their design,

beginning with the powers PSAboland PSAeol

produced by the solar array, over spacecraft and

payload parameters such as mS/C , PP/L or tSatellite up to orbital quantities such as the eclipse

time PEclipse.

Page 69


3.4.3. Subsystem AOCS: Reaction Wheel

The Attitude and Orbit Control System (AOCS) of a spacecraft keeps the spacecraft on the

planned orbit and enables the accuracy for the pointing of instruments and/or the antenna(s)

on-board. In order to fulfill these functions, propulsion systems are often used which allow

significant attitude corrections. A propulsion system also increases the lifetime of the satellite

as it is able to increase the orbital altitude of the satellite which decreases in time due to

atmospheric drag and other disturbances. However, propulsion systems are not yet mature

enough for use on CubeSats (see Appendix B). Another common approach to the AOCS of

spacecrafts, however, is to use reaction wheels, mechanical devices which provide the satellite

with angular momentum in order to compensate for disturbance torques by aerodynamic drag,

gravity and solar radiation. The first two disturbance torques are especially significant in Low

Earth Orbits where the CubeSat are deployed. It should be noted that a reaction wheel can not

compensate for altitude loss due to these disturbances, it will only control the orientation of

the spacecraft. Propulsion systems, in turn, are a possibility for restoring the orbital altitude.

Reaction wheels are relatively heavy devices and hence not very often used on CubeSats be-

cause of their weight limitations. However, specially designed reactions wheels for nano- and

picosatellite applications have risen in interest in the industry. For example the CubeSat BeeSat

from the Technical University of Berlin, launched in September 2009, tested successfully reac-

tions wheels from the company Astro- und Feinwerktechnik Adlershof GmbH for picosatellite

use [TUBerlin, 2010]. Further designs from other companies are also expected. A brief ap-

plication of the Buckingham-Π-Theorem for dimensioning reactions wheels shall therefore be

presented in this section.

As already mentioned, a reaction wheel is supposed to compensate the disturbance torques

which act on a spacecraft. So its most important performance characteristics are its angular

momentum HRW [kgm2

s] and its inertial momentum IRW [kgm2]. The disturbance torques, on

the other hand, are dependent on a number of factors [Larson and Wertz, 1999, p.366]:

• Aerodynamic Torque: TAero = f(ρAir, vS/C , AS/C)

• Gravity Gradient: TGravity = f(IS/C , hOrbit)

• Solar Radiation Torque: TSolar = f(AS/C , surface properties)

with

• the air density ρAir [ kgm3 ]

• the velocity of the spacecraft vS/C [ms

]

• the surface of the spacecraft AS/C [m2] normal to the spacecraft velocity vector

• the inertial momentum of the spacecraft IS/C [kgm2]

• and the orbital altitude hOrbit [m]

Page 70


In section 3.3.2 the concept of the equivalent edge length was introduced, making the simpli-

fication that the spacecraft is designed as a cube. Furthermore, supposing that the mass is

homogeneously distributed inside the spacecraft, the principal moment of inertia of the space-

craft around its principal axes can be easily expressed with the mass of the spacecraft mS/C

and its equivalent edge length xeq [Gross et al., 2010]:

IS/C =1

12·mS/C · (x2

eq + x2eq)

=1

6·mS/C · x2

eq

(3.50)

This representation of the spacecraft does not take into account deployables by using xeqsince this quantity represents the dimensions of the stowed spacecraft. However, the inertial

momentum of a body is always a function of its mass and representative lengths. So, in case of

deployables, another length representing the complete deployed dimensions of the spacecraft

has to be added to the list of parameters which will cause a supplementary ratio in the non-

dimensional representation, namely for example xeqxdeployed

. A detailed explanation of how to

calculate the inertial momentum of bodies with extensions can be found in [Gross et al., 2010].

In the following, however, the representation without deployables will be considered since

CubeSats do have deployables only in a minority of cases. Thus the surface of the spacecraft

AS/C on which the disturbances act, can also be modelled with the equivalent edge length

xeq:

AS/C ∼ x2eq (3.51)

Here again, deployables are not taken into consideration. For a more sophisticated approach, a

representative length should be taken into consideration which takes account of the spacecraft

extensions.

Assuming circular orbits around the Earth, the velocity of the spacecraft vS/C can be repre-

sented by the orbital altitude hOrbit as v =√

µREarth+hOrbit

. The density of the air ρAir is also a

function of the orbital altitude hOrbit, decreasing with increasing orbital altitude [Hedin, 1991].

Developing this further, the orbital period is also a function of the orbital altitude with

tOrbit = 2π

√(REarth + hOrbit)

3

µ(3.52)

Instead of expressing the torques as functions of orbital altitude hOrbit, they can be presented

as functions of the orbital period tOrbit:

• Aerodynamic Torque: TAero = f(tOrbit, xeq)

• Gravity Gradient: TGravity = f(mS/C , xeq, tOrbit)

• Solar Radiation Torque: TSolar = f(xeq, surfaceproperties)

Thus the system of the reaction wheel can be described as following

0 = f(HRW , IRW , tOrbit, xeq,mS/C) (3.53)

Page 71


with tOrbit, xeq and mS/C being the parameters of the disturbance torques. Please note that

the surface properties of the spacecraft which have an influence on the solar radiation torque

are neglected in the following for the sake of simplification. Furthermore the accuracy range

is assumed to be 100% (this is to say we assume that the disturbance torque is completely

compensated by the torque produced by the reaction wheels) which is also a simplification. In

practice an accuracy of 100% is not feasible and precision near 100% is very expensive in terms

of mass, power and complexity. An accuracy less than 100% would imply another parameter

in equation (3.53) which can be an angle representing the accuracy range. The numerical

value of the non-dimensional parameter Π would change according to that range but finally

not the relations between the above parameters describing the system.

For the sake of simplification the reaction time of the satellite to a change in the angular

momentum by the reaction wheel is not considered, either. However, a more detailed model

of the system should take its influence into account as well.

It is also important to mention that in practice disturbances of different magnitudes act on

different satellite axes. In addition, the satellites are mostly not designed as cubes and have

extensions which also influences the degree of disturbance. Thus different sized reaction wheels

can be needed for the satellite. Hence, in this approach the non-dimensional parameter will

only provide a first approximation of the dimensions of the reaction wheels.

Given these caveats, the non-dimensional analysis can now be discussed. In accordance to

relation (3.53), five parameters describe this subsystem with three base dimensions namely

mass M, length L and time T. Thus with n = 5 and k = 3, i will be equal to two. The k-set

is chosen to contain mS/C , xeq and tOrbit. For ΠRW1 we can rewrite

0 = f1(mS/C , xeq, tOrbit, HRW ) (3.54)

and for ΠRW2

0 = f2(mS/C , xeq, tOrbit, IRW ) (3.55)

After the usual proceeding as described in section 2.4.1, it can be found that

ΠRW1 =HRW · tOrbitmS/C · x2

eq

ΠRW2 =IRW

mS/C · x2eq

(3.56)

which can be confirmed in terms of the intraregime approach [Bhaskar and Nigam, 1990]:

supposing ΠRW1 to be constant over the satellite class, an increase in spacecraft mass or

spacecraft dimensions increases the exposure to the disturbance torques. That is why more

angular momentum HRW is required. However, increasing the orbital period assuming circular

orbits means increasing the orbital altitude. As aerodynamic torque and disturbances due to

gravity decrease with increasing orbital altitude, less angular momentum has to be provided

by the reactions wheels as expressed by ΠRW1 .

Considering ΠRW2 , an increase in spacecraft mass or the equivalent edge length of the space-

craft causes an increased requirement of the reaction wheel’s inertial momentum IRW . An

Page 72


increase in mS/C and/or xeq also causes an increase in HRW as seen in ΠRW1 . Thus in terms

of the inter-regime approach, the partial derivative ∂HRW

∂IRWis positive which is not surprising as

it is also known that

HRW = IRW · ωRW (3.57)

thus an increase or decrease in IRW leads to an increase or decrease in HRW with constant

angular velocity of the wheel ωRW and vice versa.

In order to get clearer understanding of the dimensions of the reaction wheel, it was decided

to represent the principal moment of inertia of the reaction wheel IRW with the mass of the

reaction wheel mRW and its radius rRW . It is known according to [Gross et al., 2010] that

IRW =1

2·mRW · r2

RW (3.58)

Thus the system can be described with the following parameters

0 = f(HRW ,mRW , rRW ,mS/C , xeq, tOrbit) (3.59)

with n = 6 and k = 3. Thus i = 3, however, taking into account that two of those assumed

non-dimensional parameters are simply ratios. Hence after applying the process of dimensional

analysis, we get for the spacecraft-centric Bottom-Up approach


eq

·(mRW

mS/C

)a·(rRWxeq

)b·(mAOCS

mS/C

)c(3.60)

including an additional expansion by(mAOCS

mS/C

)c.

As the power consumption and the power production of the spacecraft also play an important

role in the design of the spacecraft, the dimensioning of the reaction wheel shall also be

considered from this point of view as its operation also consumes power. The system will then

be best described by

0 = f(HRW ,mRW , rRW , tOrbit, PS/C ,mS/C , xeq) (3.61)

Thus with the spacecraft-centric chosen k-set PS/C ,mS/C , xeq, the following results can be

found

ΠRW4 =H3RW

PS/C ·m2S/C · x4

eq

ΠRW5 =mS/C · x2

eq

PS/C · t3OrbitΠRW6 =

mRW

mS/C

ΠRW7 =rRWxeq

(3.62)

ΠRW4 expresses in terms of the intraregime-approach [Bhaskar and Nigam, 1990] that an in-

crease in power PS/C leads to an increase in the performance of the reaction wheel namely the

Page 73


angular momentum HRW . This is true as with more power, more operation of the reaction

wheel is possible. The second parameter ΠRW5 is already well-known and was first derived in

section 3.4.1. Here, however, the orbital period tOrbit represents the orbital altitude and thus

the magnitude of disturbances which act on the spacecraft. With increasing orbital period

tOrbit the orbital altitude hOrbit also rises which means that the disturbances get less signif-

icant. This, on the other hand, means that the need for compensation of the disturbance

torques by the reactions wheels is reduced. Consequently, the operation time of the reaction

wheels is lower and hence the power consumption by the wheels is less which is perfectly

expressed by ΠRW5 . On the other hand, an increase in spacecraft mass or edge length means

a more significant requirement for disturbance torque compensation. The operation time of

the reaction wheels is higher and hence also the power consumption by them, which is also

represented by ΠRW5 .

ΠRW6 expresses that an increase in spacecraft mass leads to an increase in the mass of the

reaction wheels which is also comprehensible as higher spacecraft masses raise the requirement

for disturbance torque compensation which can be realized by heavier reaction wheels. The

same reasoning is valid for ΠRW7 as bigger spacecraft dimensions raise the disturbance torques

which act on the spacecraft and thus cause a higher requirement in compensating angular

momentum HRW which can be fulfilled by bigger wheels.

It shall mentioned that the inclusion of the angular velocity ωRW was attempted in the analysis

but only limited satisfying results were found. However, one approach shall be presented here

which includes ωRW as parameter. The system is described with

0 = f(HRW , ωRW , PS/C) (3.63)

With n = 3 and k = 2 (assuming the combination of dimensions ML2 to be one base

dimension) i equals one. Through dimensional analysis one gets

ΠRW8 =HRW · ω2

RW

PS/C(3.64)

This expresses that a rise in the power of the spacecraft leads to an increase in HRWω2RW .

The influence of PS/C is intentionally not considered in terms of the dividend separately, as

a rise in ωRW implies a rise in HRW and vice versa. So one term of the dividend can not be

held constant while the other changes. That would mean the inertial momentum of the wheel

IRW decreases at the same time which is not the intention.

In conclusion, it can be stated that the dimensioning of the reaction wheel is not only dependent

on the orbital parameters but also on the characteristics of the spacecraft namely its mass

mS/C , its equivalent edge length xeq (and another length representing the completely deployed

spacecraft) and the power provided by the spacecraft PS/C . The above presented approaches

are completely spacecraft-centric (Bottom-Up) which is also the most important design case for

CubeSats. In order to have a payload-centric (Top-Down) design one can include the accuracy

range of the payload into the list of system describing quantities. Other than that, the same

parameters as for the spacecraft-centric approach can be used as still spacecraft mass and its

Page 74


external dimensions are the most important quantities determining the disturbance torques.

The mass and the power consumption of the payload do play an indirect role in designing the

reaction wheels as they are related to the spacecraft mass and spacecraft power consumption.

However, this is enough to rewrite the equations with payload-centric ratios, as for example

ΠRW3 . So we get


eq

·(mRW

mP/L

)a·(

rRWxeq P/L

)b·(mAOCS

mP/L

)c(3.65)

The same porcedure can be done for the remaining parameters.

3.4.4. Subsystem Communication

The communication subsystem is one of the most significant subsystems as it enables the

communication between the ground station and the satellite. Without it, no data transmission

and health monitoring would be possible.

The path of the signal from the satellite to the ground station is characterized by a number

of influences. The satellite itself carries a communication subsystem which consists of at least

an antenna and a transmitter and receiver or a transceiver, a device combining the functions

of the latter two. The performance of the transmitter is characterized by its output power

PTransmitter. The performance of the antenna is characterized by its peak gain GAntenna which

is in general a function of the dimensions of the antenna and the wavelength λCom of the

transmitted signal. As an example, the gain for a parabolic antenna can be calculated by

Gparabolic antenna = η ·(π ·DAntenna

λCom

)2

(3.66)

The gain of the antenna is therefore a function of the antenna diameter DAntenna and the

signal wavelength λCom. η describes the efficiency of the antenna and takes a value between 0

and 1. In the case of the parabolic antenna, η is equal to 0.55. In the following we will assume

this type of antenna but other antenna types are also possible. The exact calculation of their

gain can be found in common literature, for example [Larson and Wertz, 1999, p. 571].

The important aspect for the further calculation is, however, that the gain of an antenna can

be calculated based on its dimensions and the signal wavelength.

It is needless to say that not only the satellite has to provide an antenna for the communication

but also the ground station. The calculation of its gain is performed using the same equation

as for the satellite antenna.

A number of losses occurs in the transmission. The space loss Ls is by far the most important

loss. It is calculated by(

λCom

4ΠSCom

)2

, with SCom being the communication path length, thus

the distance between the spacecraft and the ground station, and λCom expressing the signal

wavelength. So the space loss becomes more important with higher distance SCom.

The transmission path loss La is another loss, taking into account atmospheric and rain

absorption. It is often far less significant than the space loss, but still recommended to be

Page 75


always considered in the calculations. It is highly dependent on the chosen frequency and on

the elevation angle ε [Larson and Wertz, 1999, p.564] and can become significant when ε and

λCom are not chosen with caution.

The transmitter line loss Ll occurs between the antenna and the transmitter. It is a loss due

to the hardware and not the environment or the orbital parameters as Ls and La.

The antenna pointing losses Lp Transmitter and Lp Receiver are based on a imperfect orientation

of the on-board antenna towards the one on the ground station and vice versa. They can be

expressed as a function of the pointing error e and the half-power beamwidth θ which in turn

is a function of the signal wavelength λCom and the geometrical dimensions of the antenna.

We can sum up the losses and their interdependencies to

• space loss Ls = f(λCom, SCom) = g(λCom, hOrbit, ε)

• transmission path loss La = f(λCom)

• line loss Ll

• and antenna pointing loss Lp = f(e, θ)

As the line loss Ll is only dependent on the quality of the hardware, it is a characteristic of

its own, without any interrelations with other parameters.

The most important quantity of the system, however, is the data rate R [ bits

], the amount of

information that is transmitted by the communication subsystem.

As important as the communication subsystem is, as heavy and power intensive is it very often.

The high mass is usually due to the mass of the antenna mAntennaT which in turn is driven by

the antenna’s dimensions (see [Saleh et al., 2002]). Thus it is recommended to take account

of the antenna mass when listing the quantities influencing the communication subsystem.

The high power requirement of the subsystem is due to the transmitter or transceiver power

requirement which in turn is a high contributor to the power budget of the spacecraft in

general.

Taking all the above discussions into account, the communication subsystem can be expressed

by

0 = f(PTransmitter, PS/C ,mAntennaT , DTransmitter, DReceiver, λCom, SCom, R, e) (3.67)

with

• the power output of the transmitter PTransmitter [kgm2

s3]

• the power consumption of the spacecraft PS/C [kgm2

s3]

• the mass of the transmitter antenna mAntennaT [kg]

• the diameter of the transmitter antenna DTransmitter [m]

• the diameter of the receiver antenna DReceiver [m]

• the signal wavelength λCom [m]

• the communication path length SCom [m]

Page 76


• the data bit rate R [ bits

]

• and the pointing error e [deg]

Please note that SCom is a function of the orbital altitude hOrbit, the position of the ground

station to the subsatellite point and also of the elevation angle ε (see [Larson and Wertz, 1999,

p. 113]). Assuming the best possible orientation of satellite and ground station, namely the

satellite in the zenith of the ground station, SCom is then equal to hOrbit. The worst position

is given when satellite and ground station are just in optical range (assuming that the minimal

ε required for communication is 0◦ which is practically not true as a minimum angle of 5◦ is

usually needed). This is when the ground station is in the true outer horizon of the satellite.

SCom can then be calculated with

SCom =

√(REarth + hOrbit)2 −REarth

2 (3.68)

In both extreme cases which are important when designing a spacecraft, S is only a function of

the orbital altitude hOrbit. Furthermore, no pointing losses will be considered in the following,

assuming perfect alignment between the antenna of the spacecraft and the ground station.

Thus the number of the system’s quantities can be reduced and changed to

0 = f(PTransmitter, PS/C ,mAntennaT , DTransmitter, DReceiver, λCom, hOrbit, R) (3.69)

After dimensional analysis and in consideration of the ratios, equation (3.69) can be rewritten

as

ΠCom1 =mAntennaT · h2

Orbit ·R3

PTransmitter·(

λComDTransmitter

)a·(

λComDReceiver

)b·(

PS/CPTransmitter

)c(3.70)

Please note that the bit error rate (BER) is included in the numerical value of ΠCom1 as BER

is already a non-dimensional quantity. However, in order to display the influence of the bit

error rate more clearly, it can be simply added to equation (3.70) and we get


Orbit ·R3

PTransmitter·

·(

λComDTransmitter

)a·(

λComDReceiver

)b·(

PS/CPTransmitter

)c·(mCom

mS/C

)d·BERe

(3.71)

The signal modulation is taken into consideration indirectly by adding the bit error rate BER

into the list of the influencing quantities. It can be confirmed that this non-dimensional

product is also physically consistent: a higher transmitted data rate R generally requires a

higher transmitter output power PTransmitter; an increase in hOrbit leads to an increase in the

communication path length S. This in turn implies an increased space loss which necessitates

higher transmitter output power in order to transmit the fixed data rate R; a larger antenna

diameter, be it the one of the transmitter DTransmitter or of the receiver DReceiver, leads to a

decrease in the transmitter output power PTransmitter for a constant data rate R; an increase

in the antenna mass usually comes along with an increase in its diameter and thus leads

Page 77


to a decrease in required transmitter output power PTransmitter for a constant data rate R;

and finally, a raise in the transmitter output power PTransmitter is usually possible by heavier

devices, thus increasing the mass of the communication subsystem mCom which leads to an

increase in the spacecraft mass mS/C . All these interdependencies are appropriately expressed

by ΠCom2 .

To conclude it can be stated that the dimensioning of the communication subsystem is de-

pendent on a number of factors, taking not only spacecraft specific parameters into account

but also the characteristics of the ground station(s) and the orbital parameters as well as the

environment.

This approach is nearly completely spacecraft-centric (Bottom-Up design) except for the data

rate R which is not only influenced by the number of ground stations the satellite can have

contact with to transmit the data but also the performance of the payload. Depending on

the data rate and data volume produced by the payload, the data rate R to be transmitted

to Earth is determined. It is expected that there will be a direct interdependency between the

communication subsystem and the payload, expressed by a common quantity of the systems,

namely the data rate R. This assumption will indeed be confirmed in the next section.

Other than that, a payload-centric Top-Down approach uses nearly the same quantities as a

spacecraft-centric Bottom-Up approach, since the antenna diameters are payload-independent

and the mass and the power consumption of the payload do only play an indirect role in

designing the communication subsystem. But as in case of the battery and the solar arrays

the mass and power ratios can be adapted to the Top-Down approach so that we can re-express

equation (3.71) to


Orbit ·R3

PTransmitter·

·(

λComDTransmitter

)a·(

λComDReceiver

)b·(

PP/LPTransmitter

)c·(mCom

mP/L

)d·BERe

(3.72)

.

3.4.5. Payload

The payload is the most important part of the satellite as it gives the satellite its significance

and value. An immense range of payloads is possible as there are many possible mission pro-

files. Three kinds of missions will be in focus in this work, namely Earth Observation, Space

Science and Technology Demonstration. The payload will be different, depending on the type

of the mission.

Page 78


When taking a closer look at the payload of the satellite fulfilling an Earth Observation mission,

the payload will most likely be a sensor measuring the incoming radiation from the Earth. Based

on the characteristic wavelength of the radiation which will be measured during the mission,

the type of the sensor is then chosen. However, as the type of the design of the optical

sensors is independent from the radiation wavelength their performance is described by the

same characteristics.

First of all, there is the ground pixel resolution of the instrument (often indicated at nadir)

X [ mpixel

] and Y [ mpixel

], the first one representing the cross-track ground pixel resolution, the

latter one the along-track pixel resolution. The higher the resolution is, the smaller is X

and/or Y and the better is the quality of the image taken. The ground resolutions have direct

interdependencies with the instrument on-board, especially the diameter of the aperture of the

sensor D [m], and the orbital altitude hOrbit. Their interrelations can be expressed by

X = Y = 2.44 ·hOrbit · λP/L

D(3.73)

from [Larson and Wertz, 1999, p.264].

Thus with increasing orbital altitude hOrbit which implies an increase in tEclipse the resolution

gets worse, thus bigger, but with a bigger aperture D the resolution gets better. The data

rate R, which is produced by the instrument, depends in first order on the ground resolution

and the coverage area of the instrument: with the swath width SW of the instrument and

the cross-track ground pixel resolution X, SW

Xpixels per swath line has to be taken by the

instrument. vGround

Yswath lines, in turn, are scanned in a second as vGround expresses the

spacecraft ground velocity and Y the along-track ground pixel resolution. Hence the data rate

R produced by the instrument can be calculated with

R =SWX· vGround

Y· b (3.74)

[Larson and Wertz, 1999, p.287] with

• the swath width SW [m]

• the cross-track ground pixel resolution at nadir X [ mpixel

]

• the ground-track velocity of the spacecraft vGround [ms

]

• the along-track ground pixel resolution at nadir Y [ mpixel

]

• and the number of bits used to encode each pixel b [ bitspixel

].

The term SW

X· vGround

Ycan be thus understood as the total number of pixels taken by the instru-

ment during one second. As the swath width SW represents the coverage of the instrument,

it is assumed that it is a function of the orbital altitude hOrbit, so

SW = 2 · ECAmax (3.75)

with the Earth Central Angle ECA whose absolute maximum can be calculated by means of

equation (A.3) since ECA = 90◦ − ρ. In practice the finally chosen ECA for the design of

Page 79


the payload is less than the theoretical maximum in order to prevent distortions in the pixels

on the limits of the coverage area due to the curvature of the Earth and the position of the

instrument to the scanned area. Thus, as in case of the communication path length SCom in

section 3.4.4, ECA is a function of the orbital altitude hOrbit and the elevation angle ε. For the

sake of simplification, however, it is assumed in the following, that the width of the coverage

area is only a function of the orbital altitude (or, also possible, a function of the orbital period

tOrbit as with tOrbit = 2π√

(REarth+hOrbit)3

µ) like in the case of ECAmaxabs).

Furthermore, the ground velocity of the spacecraft is also a function of the orbital period tOrbitas

vground =2 · π ·REarth

tOrbit(3.76)

Thus, taking all these explanations into consideration, the payload for a Earth Observation

mission can be described by

0 = f(R, tOrbit, b, λP/L, D) (3.77)

and it is

ΠP/LEarth Observation 1 =R · tOrbit

b·(λP/LD

)a(3.78)

With the help of equations (3.73) and (3.74), this non-dimensional product can also be con-

firmed physically in terms of the intraregime approach [Bhaskar and Nigam, 1990]: supposing

a constant ΠP/LEOover a satellite class, an increase in b leads to an increase in the data rate

R. An increase in the aperture length D causes a decrease in the ground resolutions X and

Y , thus causes a better resolution and hence also a higher data rate R. An increase in tOrbitthus in hOrbit causes higher ground track resolutions, thus worse resolution, and decreases the

coverage area by decreasing SW and the ground velocity vground, both effects leading to lower

data rate R.

As mass and power of payload and spacecraft play an important role in the dimensioning of

the payload, these quantities shall also be taken into account:

ΠP/LEarthObservation 2 =R · tOrbit

b·(λP/LD

)a·(mS/C

mP/L

)b·(PS/CPP/L

)c(3.79)

ΠP/LEarthObservation 2 can be equally used for the Top-Down as well as for the Bottom-Up design

approach.

Space Science missions are often missions where the payload measures special wavelengths

emitted by a certain object. So the design of an space science instrument is in general not

very different to an Earth Observation instrument: an object with distance l to the satellite

emits a signal with a special wavelength λP/L which, in turn, is detected by an instrument

on-board of the satellite with a lens diameter D. The data rate R is consequently produced.

So based on ΠP/LEO 3 the following non-dimensional product can be derived

ΠP/LSpaceScience 1 =R · l

c

b·(λP/LD

)a·(mS/C

mP/L

)b·(PS/CPP/L

)c(3.80)

Page 80


with l as the distance to the measured object and c the velocity of sound and thus lc

is the time

the signal needs to get to the satellite. Another difference between the Earth Observation and

Space Science mission will be seen in the following section 3.5 where the Mission Performance

Index will be presented.

In terms of the Technology Demonstration, the payload can be any kind of component from

any kind of subsystem thus no further non-dimensional parameter is required to express it. It

is perfectly described by the ratios and already created parameters for the subsystems and the

spacecraft. The only aspect added is pointed out in section 3.5, where the mission-driving

parameters are presented. Those will be different from mission to mission and thus those

of a Technology Demonstration mission are different from those of the Earth Observation

mission and Space Science mission. To already take it upfront: important parameters for

the Technology Demonstration mission are the mass of the payload mP/L, the volume of

the payload VP/L, the satellite lifetime tSatellite and also the data rate R which results from

observing the performance of the tested component from the ground station.

3.5. Mission Performance Index

An all-embracing scaling law shall be found in this section which expresses the Mission Perfor-

mance Index, including implicitly all the relevant non-dimensional products and ratios which

are assumed to be significant for influencing the performance of a mission. The MPI shall be

designed to take values in a given range, for example between 0 and 100, in order to be able to

compare different missions simply by their numerical values of their MPI. The higher the MPI,

the ”better performant” is the mission. In order to get a relation between the MPI and the

non-dimensional parameters and ratios of section 3.3 and 3.4, it is not possible to simply insert

the non-dimensional parameters in the MPI as they are as they do not express compellingly a

better performance with higher numerical value. The non-dimensional parameters are build to

save their constant value within a satellite class when the quantities generating them change.

First of all, it shall be defined what a ”performant mission” is. In order to do so, it is

important to define the Mission Performance Parameter (MPP) for every type of mission,

thus the quantities whose magnitude shall be as large as possible for a successful mission. We

differentiate between primary and secondary top-level MPPs. The primary MPPs express the

most important quantities of the mission, the latter describing quantities which are significant

for the mission accomplishment, too, but which are not considered to be the main parameters

of the mission. This differentiation is used for the order of maximization: a primary MPP

has to be maximized for a ”performant” mission. The simultaneous maximization of the

secondary parameters, however, is desirable, but not critical. ”Top-Level” means that only the

overall characteristics are considered. As we will see later on, a satellite can be also ”of high

performance” for a maximum or minimum of subsystems’ characteristics.

With regard to the already completed, planned and current CubeSat missions [Thomsen, 2010],

we identified three types of missions on which we will focus on, namely Earth Observation mis-

Page 81

Mission Performance Index

sions (EOM), Space Science missions (SSM) and Technology Demonstration missions (TDM).

Their Mission Performance Parameters can be found in table 3.19.

Table 3.19.: Top-level Mission Performance Parameters

Mission type Primary MPPs Secondary MPPs

EOM R mP/L, VP/L, tSatellite

SSM R mP/L, VP/L, tSatellite

TDM R, mP/L, VP/L, tSatellite -

For an Earth Observation mission, for example, a maximization of the data rate R is de-

sirable: the higher the data rate, the more performant is the mission. A good performant

mission would make a simultaneous maximization of the secondary parameters, the mass of

the payload mP/L, its volume VP/L and the satellite lifetime tSatellite, also possible - these

maximizations, however, are considered to be secondary. The numerical value for the MPI

of an Earth Observation mission should therefore express the achievable maximum datarate

ideally under the condition of maximum possible payload mass and volume as well as satellite

lifetime. As already mentioned above, a satellite can be also considered to be performant in

terms of its subsystems characteristics. It is, for example performant when as much power PSAas possible is produced by the solar cells of the satellite. The ratios and non-dimensional pa-

rameters presented in sections 3.3 and 3.4 were thus investigated for their Mission Performance

Parameters. In terms of subsystems’ characteristics, we define a satellite to be performant

when CBattery is minimized (this is not valid for payloads which are only active in eclipse; here a

performant mission would have a maximized CBattery) and PSAbol/eoland HRW are maximized.

The definitions are based on the argumentation that it is possible to influence the choice of

an orbit in LEO for a mission so that no eclipse occurs. Hence a mission that needs as less

battery power as possible is a performant mission. However, disturbations cannot be avoided

for LEO satellites. Therefore, a mission that is prepared to compensate as much disturbances

as possible, is a performant mission, thus HRW needs to be maximized.

Please note for the understanding of further explanations, that the consequence of a maximum

mP/L is that for the Top-Down approach

{mS/C

mP/L

,mAOCS

mP/L

,mPower

mP/L

,mStructure

mP/L

,mThermal

mP/L

,mC&DH

mP/L

,mCom

mP/L

,mProp

mP/L

}are minimized

(3.81)

and for the Bottom-Up approach{mAOCS

mS/C

,mPower

mS/C

,mStructure

mS/C

,mThermal

mS/C

,mC&DH

mS/C

,mCom

mS/C

,mProp

mS/C

}are minimized (3.82)

Page 82


andmP/L

mS/C

are maximized. (3.83)

mCom

mS/C, for example, shall be minimized, expressing that as much mass as possible shall be

assigned to the payload. The remaining subsystems shall be as light weighted as possible.

Therefore the interest in a low mCom. This remark is especially important for the arrangement

of the ratios in the non-dimensional parameters and thus in the MPI later on.

In order to fall back on the results expressed in the non-dimensional parameters, we rearrange

the quantities in them for use in the MPI. To understand the following explanations better,

the ideas and resulting proceeding are shown exemplarily by means of the Communication

subsystem and its non-dimensional parameter ΠCom 2. The key driving parameter for the

mission success which can be found in ΠCom 2 is the data rate R that shall be maximized. For

the sake of simplification ΠCom 2 is re-expressed to

ΠCom 2 =R3

kCom(3.84)

with

kCom =PTransmitter

mAntennaT · h2Orbit

·(DTransmitter

λCom

)a·

·(DReceiver

λCom

)b·(PTransmitter

PS/C

)c·(mS/C

mCom

)d·(

1

BER

)e (3.85)

R, in turn, is maximal, for a maximal kCom - for ΠCom 2 being constant which is the condition

for applying the similitude theory within a satellite class. Please note that kCom is arranged

in a way that the conditions of the mass ratios explained above are satisfied within the kCom.mCom

mS/C, for example, shall be minimized. This is equivalent to the condition that its reciprocal

mS/C

mComshall be maximized. That is the reason why the latter ratio is presented in kCom as kCom

can be only maximized when its single factors are maximized.

Concludingly, the MPI will include a part (R3 ·kCom)χ which expresses a good performant mis-

sion when maximized. The same proceeding can be done with the remaining non-dimensional

parameters so that the MPI can be defined for both the Bottom-Up and the Top-Down ap-

proach asMPI = (ΨP/L

α) · (ΨAOCS)β · (ΨPower)χ · (ΨStructure)

δ·· (ΨThermal)

ε · (ΨC&DH)φ · (ΨCom)ϕ · (ΨProp)γ

(3.86)

with ΨCom = R3 ·k. The exponents α, β, χ, . . . express a weighting in terms of the importance

of every subsystem to the performance of the mission. Thus, the higher the exponent will be,

the higher is the significance of the corresponding subsystem. A possible value range of them

for the three chosen mission types is presented in table 3.20.

Please notice that Ψi is a dimensional quantity. However, as long as one is aware of this fact

and uses the same dimensions for the quantities for the MPIs of the missions one wants to

Page 83


compare, only the numerical values of the Φs are important for the numerical value of the

MPI. This idea is leaned from aeronautics where the performance indices are also dimensional.

For example the specific fuel consumption (SFC) is defined as

SFC =mFuel

F(3.87)

thus being the mass of fuel mFuel needed to provide the net thrust F for a given period and

thus often having the unit gkN ·s .

However, a normalization of the MPI avoids the dimensionality of MPI. We will see at the end

of this section which different kinds of normalization are possible.

The other dimensional Ψi based on the results in section 3.4 for the Bottom-Up approach are

for the payload

ΨP/L BU = R · kP/L = R · b

tOrbit·(

D

λP/L

)a·(mP/L

mS/C

)b·(PP/LPS/C

)c(3.88)

For the AOCS we can write

ΨAOCS BU = HRW · kRW

= HRW ·mS/C · x2

eq

tOrbit·(mS/C

mRW

)a·(xeqrRW

)b·(mS/C

mAOCS

)c (3.89)

The Power subsystem is described by batteries and solar arrays. Therefore ΨPower BU consists

of a part related to the batteries and one referring to the solar arrays:

ΨPower BU =1

ΨBattery BU

·ΨSolar Array BU (3.90)

with

ΨBattery BU = CBattery · kBattery

= CBattery · PEclipse · tEclipse ·(

PS/CPEclipse

)a·(mS/C

mPower

)b (3.91)

andΨSolar Array BU = P 1+a+c

SolarArraybol· kSolar Array

= P 1+a+cSolarArraybol

· x2eq · SSolar · d · tSatellite·

· P aSolarArrayeol

·(mS/C

mPower

)b· P c

S/C ·(tEclipsetOrbit

)d (3.92)

Please note that CBattery shall be minimized for a performant mission, thus CBattery · kBatteryshall be minimized. This is equivalent to a maximization of 1

CBattery ·kBatterywhich is demanded

for a maximum MPI. That is why the reciprocal of ΨBattery is taken into account in ΨPower.

The subsystems C&DH, Structure, Thermal and Propulsion where not further investigated in

the sections 3.3 and 3.4. For the latter two this is especially because of their lack of importance

Page 84


for the CubeSats. The Structure and C&DH subsystem where not further investigated because

of their simplicity in CubeSats. They are most easily described by a simple mass ratio. Further

development of the ratios and non-dimensional parameters is surely possible and desirable

but as this work is a first order modeling of a satellite by dimensional analysis, the following

definitions of the Ψs are considered to be sufficient for our purposes. For the Structure

subsystem

ΨStructure BU =mS/C

mStructure

(3.93)

and for the C&DH subsystem we can write

ΨC&DH BU =mS/C

mC&DH

(3.94)

The Thermal subsystem can be represented by

ΨThermal BU =mS/C

mThermal

(3.95)

and the Propulsion subsystem by

ΨPropulsion BU =mS/C

mPropulsion

(3.96)

The above explanations refer to the Bottom-Up approach. As the ratios and non-dimensional

parameters change for the Top-Down approach, the Ψs shall be presented for that approach

in the following. It is for the Earth Observation payload

ΨP/L TD = R · kP/L = R · b

tOrbit·(

D

λP/L

)a·(mP/L

mS/C

)b·(PP/LPS/C

)c(3.97)

and for the AOCS subsystem

ΨAOCS TD = HRW · kRW = HRW ·mS/C · x2

eq

tOrbit·(mP/L

mRW

)a·(xeq/P/LrRW

)b·(mP/L

mAOCS

)c(3.98)

The Power subsystem is as in the case of the Bottom-Up approach represented by a battery

related and a solar array related part, leading to

ΨPower TD =1

ΨBattery TD

·ΨSolar Array TD (3.99)

with

ΨBattery TD = CBattery · kBattery

= CBattery · PEclipse · tEclipse ·(

PP/LPEclipse

)a·(mP/L

mPower

)b (3.100)

Page 85


andΨSolar Array TD = P 1+a+c

SolarArraybol· kSolar Array

= P 1+a+cSolarArraybol

· x2eq · SSolar · d · tSatellite·

· P aSolarArrayeol

·(mP/L

mPower

)b· P c

P/L ·(tEclipsetOrbit

)d (3.101)

A simple mass ratio represents as in the case of the Bottom-Up approach the subsystems

Structure, C&DH, Thermal and Propulsion. So

ΨStructure TD =mP/L

mStructure

(3.102)

ΨC&DH TD =mP/L

mC&DH

(3.103)

ΨThermal TD =mP/L

mThermal

(3.104)

and

ΨPropulsion TD =mP/L

mPropulsion

(3.105)

For the Communication subsystem we can write

ΨCom TD = R3 · kCom

=PTransmitter

mAntennaThOrbit2 ·(DTransmitter

λCom

)a·

·(DReceiver

λCom

)b·(PTransmitter

PP/L

)c·(mP/L

mCom

)d· 1

BER

e

(3.106)

Please note that equations (3.88) to (3.105) include a choice of the non-dimensional param-

eters and ratios presented in sections 3.3 and 3.4. Not every parameter of sections 3.3 and

3.4 is integrated into the MPI but the parameters which are taking the most quantities into

consideration. Furthermore the parameters of section 3.4.1 provide a top-level approach to

the design. They can thus, as mentioned in that section, be used for quick design estimations

or provide a possibility to compute a missing input quantity for the Top-Down and Bottom-

Up design approach, but are not integrated into the MPI. However, the MPI can be further

developed as it was the case for the non-dimensional parameters, or also newly composed.

Nonetheless, a standardized definition of the MPI is considered to be essential and desirable

in order to be able to compare and exchange results through the satellite world.

A further important remark is that equation (3.86) is valid for any kind of mission or satellite.

The numerical values of the Φi, ki and the exponents change, however, depending on the mis-

sion and the satellite. Table 3.20 gives an overview of the estimated ranges for the exponents

α, β, χ, . . . for the three mission types we identify to be the most frequent and important

ones. As already mentioned, the higher the exponent, the higher is the functional importance

Page 86


of the subsystem for the mission accomplishment. The Earth Observation and the Space Sci-

ence missions can be considered together, as the only difference lies in the exponents for the

Thermal subsystem. This will be explained later on. The exponent ranges for the Technology

Demonstration mission are bigger as various payloads are possible so that the subsystems are

differently engaged.

The payload as most important part of the spacecraft has the highest exponent. The Power

and Communication subsystems have the second biggest exponent, as without power the

satellite and most importantly the payload cannot work. Without a working Communication

subsystem a satellite is operational but the value of the mission, the collected data, cannot

be transmitted to the Earth. In case of operational problems the mission is often lost because

of the lack of command and health monitoring links. The C&DH subsystem is responsible

for the data processing and distribution. Its significance for the mission success is almost

as important as the Communication subsystem. Once in orbit, the AOCS and Propulsion

subsystem basically carry out the same function, namely the compensation of disturbances.

Their significance is determined by the required payload accuracy. Here, among others con-

sidering CubeSat missions where a minimum of accuracy is required, a medium importance

is chosen for the AOCS and Propulsion subsystem to take CubeSats and more sophisticated

missions into consideration. A low exponent range is assigned to the Structural subsystem

assuming that the design principally gets on without mechanisms. Every mechanism, however,

adds complexity and thus the exponents should be adjusted for systems where critical parts

needs deployment, for example an antenna. The Thermal subsystem concludes the list of

subsystems, being generally the less important subsystem in most of the CubeSat missions.

However, in special cases, it is especially this subsystem which can get really critical and decide

on the mission success. It is partly taking account of this fact by widening the range for the

Space Science missions to higher exponents and thus to higher importance of the Thermal

subsystem. As satellites of Space Science missions can be placed literally everywhere in the

universe, thermal aspects can get significant when being far in the outer solar system or in

the near of the Sun. The Thermal subsystem can also be of high importance for sensitive

components and the payload on-board. However, in the case of CubeSats, Space Science and

Earth Observation missions are both executed in Earth orbits where most satellites need a

passive thermal subsystem at the most.

Page 87


Table 3.20.: Ranges for the subsystem exponents α, β, χ, . . . for single satellites

Subsystems EOM SSM TDM

Payload 0.8-1.0 0.8-1.0 0.8-1.0

Power 0.7-0.9 0.7-0.9 0.6-0.9

Communication 0.7-0.9 0.7-0.9 0.6-0.9

Structure 0.2-0.5 0.2-0.5 0.2-0.5

AOCS 0.5-0.7 0.5-0.7 0.4-0.8

Thermal 0.1-0.3 0.1-0.6 0.1-0.6

C&DH 0.6-0.8 0.6-0.8 0.5-0.8

Propulsion 0.5-0.7 0.5-0.7 0.4-0.8

The product approach for the composition of the MPI entails the fact that a non-dimensional

parameter which takes the value zero, leads to a Mission Performance Index of zero. In order

to avoid this result, non-dimensional parameters which are equal to zero shall be considered

during the calculation of the MPI with a very low value for Ψ close to zero, for example 10−20,

assuming that a complete lack in a subsystem is not conducive for the mission performance.

This can for example be the case for the Propulsion subsystem on CubeSats. This assumption

makes it still possible to compare satellites which are not equally equipped with subsystems.

Otherwise, only ignoring the subsystem in the MPI could potentially lead to better results for

satellites with a lacking subsystem.

As the numerical values of the non-dimensional parameters are theoretically not limited, so it

is with the numerical value of the MPI. In order to avoid unwieldy large values to compare the

MPIs of different satellites more easily, it is therefore recommended to normalize the numerical

values of the MPI so that it can only take values in a range, for example 0 < MPI ≤ 100. The

normalization can be classically done by means of a reference mission. As a reference mission is

always difficult to determine, another possibility shall be presented here on which we will focus.

It is the normalization by means of a numerical value based on the highest MPI value within a

satellite group which shall be investigated. It can for example simply be a hundredth of the MPI

of the mission with the highest MPI. Another possibility is to take for example 110% of this

value as normalization figure, leaving still some space to the upper MPI border in case further

satellites shall be included into the group with higher MPIs than the actual highest performant

one. The normalization can also be already done for the single Ψs, thus for each subsystem

separately by dividing the Ψis with the highest numerical value of a Ψi within the group of

satellites one wants to compare. This leads to 0 < Ψi ≤ 1 and thus, because of the exponents

α, β, γ, · · · ≤ 1 to a MPI with a value range between zero and one. The quality of the results

is in any case the same and a useful consequence of the normalization is also the regain of a

non-dimensional MPI or Ψi. Important for the normalization as well as general remark is the

Page 88


fact that the numerical value of the MPI when using the Top-Down related Ψs is different

from the MPI when using the Bottom-Up related Ψs. Thus the same mission with exactly the

same parameters leads to two different numerical values for the MPI. This difference is based

on the nature of the Ψs as they are differently composed for the two approaches. However, to

avoid as much misunderstandings as possible, it is recommended to always indicate if a MPI

is calculated with the Top-Down or the Bottom-Up approach. Furthermore, normalizations

are recommended to be seperately done for Top-Down and Bottom-Up MPIs. Other than

this, the normalization can be done within a group of satellites one wants to compare, or in a

wider database, taking ideally all ever flown satellites into account. A normalization within a

mission type, however, is recommended, taking only the satellites with same mission type into

account to compare the numerical results more easily. The only disadvantage of that kind of

normalization is the fact that all the normalized MPIs have to be recalculated when a new

mission happens within a group whose MPI or Ψi is bigger than the recent ones. In order

to avoid the recalculation as often as possible, it is recommended to normalize the MPIs not

with the highest MPI in the group but with a figure which is bigger than even the highest MPI

as already suggested above. A wide application of the MPI will help to crystallize a suited

normalization figure.

In the end, when enough data on flown satellites is available so that a whole database of MPIs

can be created, it is assumed that it will be possible to assign MPI ranges to satellite classes.

Two further applications of the MPI than simply qualitative comparisons of the missions will

be possible. The assignment of MPI ranges to satellite classes will firstly enable the user

of the database to see if its finished satellite design with the corresponding calculated MPI

fits the ranges of the original satellite class or if its design is actually in another satellite

class. Secondly, being redirected to another satellite class with the intended MPI, means to

be redirected within the non-dimensional parameters which are also different from satellite

class to satellite class. With a definition of ranges for the input parameters of the Top-Down

and Bottom-Up approach for the different satellite classes, the user can than recalculate its

satellite design by means of the new non-dimensional parameters.

3.6. Application of the Results

It is now important to see how the ratios, non-dimensional parameters and the Mission Per-

formance Index can be practically used. First of all, it is important to determine numerical

values for the ratios and non-dimensional parameters of a satellite class. This procedure neces-

sitates the validation of the parameters with data from real flown satellites as already shown

for the ratios and the top-level approach for the NGSO-satellites. Once the non-dimensional

parameters are determined for a satellite class, they can be used for designing a spacecraft

with as little as four main input parameters and some further secondary design inputs such as

the type of the solar cells. Figures 3.69 and 3.70 show the interdependencies of the quantities

in the system which are expressed in the results of sections 3.3 and 3.4. The figures shall

help calculating the unknown quantities of a given system by indicating the equation number

Page 89

Application of the Results

and the input and output quantities of the equations. The Mission Performance Index of the

satellite can finally be calculated, based on the results of the previous steps and indicating the

mission accomplishment which is achievable with the satellite. In this section, it will be shown

how the calculations are to be done by means of two different satellite types, the CubeSat

and the NGSO satellite. Hereby, hypothetical input values will be used, modeling the ”most

likely” CubeSat and NGSO-satellite. Most of the assumptions for the CubeSat architectures

will be based on the few data which was gathered together in an Excel-Sheet during this

work. The information originates from personal communication with the different student

CubeSat team around the world, the component database described in Appendix B as well as

two main websites on CubeSats [Thomsen, 2010] and [Amsat, 2010]. The calculation of the

non-dimensional parameters as well as the computation of the unknown quantities by means

of the non-dimensional parameters will be done together as the determination of most of the

non-dimensional parameters requires a number of assumptions. A third example, namely the

application of the equations for a larger 2U CubeSat, will be done to conclude this section in

order to show the sole calculation of the unknown quantities and the MPI for a satellite with

a few inputs.

3.6.1. The CubeSat - a Bottom-Up Approach

In order to make all the exemplary calculations for a 1U CubeSat, it is important to determine

the most important input quantities of the system in the beginning. According to figures 3.2

and 3.70 the main input quantities are mS/C , xeq or VS/C and PS/C . Based on the 1U CubeSat

standard and the available information on the flown CubeSats, we assume for the calculations

• mS/C = 1.33kg

• xeq = 104.31mm

• PS/C = 1W .

Furthermore, we will already assume an Earth Observation mission for the satellite, an esti-

mated satellite lifetime tSatellite of 1 year and an orbital altitude hOrbit of 700 km although

this is strictly not a pure Bottom-Up approach as presented in figures 3.2 and 3.70. However,

for a first order approximation as this is here the case, this estimations shall be allowed.

The determination of the Πs and the calculation of the unknown quantities

1. The mass ratios can be determined by the information about the COTS components

(see Appendix B) and the flown CubeSats in table 3.3 and 3.4. We chose the numer-

ical values of the mass ratios as presented in table 3.21 and 3.22 and calculate the

corresponding subsystem masses, also presented in the tables.

Page 90


Table 3.21.: Mass ratios of the payload and the subsystems AOCS, Power and Structure build

with mS/C CubeSat and the corresponding subsystem masses for the given mass

ratios for the hypothetical 1U CubeSat

P/L AOCS Power Structure

Mass ratio [%] 25 15 15 25

Subsystem mass [g] 332.5 199.5 199.5 332.5

Table 3.22.: Mass ratios of the subsystems Thermal, C&DH and Communication build with

mS/C CubeSat and the corresponding subsystem masses for the given mass ratios

for the hypothetical 1U CubeSat

Thermal C&DH Com Prop

Mass ratio [%] 0 3 17 0

Subsystem mass [g] 0 39.9 226.1 0

2. The volume ratios or packing factor are defined according to (3.5) and (3.7). The

collected data of the already flown CubeSats and by own estimations, a packing factor

p = VusedVS/C

of 80% can be expected for the 1U CubeSats. The ratios VBus

VS/Cand

VP/L

VS/C

cannot fall back on given data, so estimations are necessary. We assume that(VBusVS/C

)CubeSat

= 0.6− 0.7(VP/LVS/C

)CubeSat

= 0.1− 0.2

(3.107)

With VS/C = 1.135 · 10−3 m3 this leads us to

VBus CubeSat = 6.81 · 10−4 − 7.945 · 10−4 m3

VP/L CubeSat = 1.135 · 10−4 − 2.27 · 10−4 m3(3.108)

which equals equivalent edge lengths of

xeq Bus CubeSat = 8.80 · 10−2 − 9.26 · 10−2 m

xeq P/L CubeSat = 4.84 · 10−2 − 6.1 · 10−2 m(3.109)

Page 91


3. The power ratios can be determined with (3.13). After the consultation of the Exo-

planet student team from MIT and the MOVE team from TUM and by own estimations,

it can be expected that (PP/LPS/C

)CubeSat

= 0.3(PBusPS/C

)CubeSat

= 0.7

(3.110)

thus with PS/C = 1W leading to PP/L = 0.3W and PBus = 0.7W .

4. The power and mass ratios (3.14) to (3.17) can be thus calculated to

rMass Power 1 CubeSat =mS/C

mP/L

·PS/CPP/L

= 4 · 1

0.3= 13.333

rMass Power 2 CubeSat =mP/L

mS/C

·PP/LPS/C

= 0.25 · 0.3 = 0.075

rMass Power 3 CubeSat =mS/C

mP/L

·PP/LPS/C

= 4 · 0.3 = 1.2

rMass Power 4 CubeSat =mP/L

mS/C

·PS/CPP/L

= 0.25 · 1

0.3= 0.833

(3.111)

5. The top-level approach with the non-dimensional parameters (3.19) and (3.20) can

be calculated for the hypothetical 1U CubeSat to be

Π1 CubeSat = Π2 CubeSat =PS/C · t3Orbitx2eq ·mS/C

= 308.260 (3.112)

with tOrbit calculated to be 1.64hours. The orbit time is chosen to be inserted in hours

in order to keep the numerical value of Π1 CubeSat as small as possible for manipulation

reasons.

6. The Power subsystem is expressed with the non-dimensional parameters (3.34) for

the battery and (3.46) for the solar array. Both equations as well as the upcoming

parameters of the remaining subsystems possess still unknown exponents expressing the

importance of the corresponding ratios for the non-dimensional parameter. However, as

this is only a first order approximation, we assume all unknown exponents to be equal

to one. An example of this simplification shall be given in terms of ΠBat3 . It is generally

determined by

ΠBat3 =CBattery


PS/CPEclipse

)a·(mS/C

mPower

)b(3.113)

Page 92


The exponents a and b being simplified to one, we get

ΠBat3 CubeSat =CBattery

PEclipse · tEclipse·PS/CPEclipse

·mS/C

mPower

=36000Ws

1W · 600s· 1W

1W· 1

0.15

= 400

(3.114)

assumingPEclipse

PS/C= 1 and CBattery = 36000s based on technical data on the corre-

sponding COTS products.

For the solar array (body-mounted) equation (3.46) leads us to

ΠSA5 CubeSat =PSAbol


·

· PSAbol

PSAeol

· mPower

mS/C

· PSAbol

PS/C· tOrbittEclipse

=1.2W

(0.10431m)2 · 1367 Wm2 · 0.0275 · 1year

·

· 1.2W

1.1W· 0.15 · 1.2W

1W· 98.77min

10min

= 5.690

(3.115)

Please note that the maximum eclipse time in an orbital altitude of 700km would be

35 min. We assume 10 min eclipse time for unpredicted conditions as CubeSats are

generally launched in sun-synchronous orbits where no eclipse is expected. We also as-

sume a maximum incident sun angle of 90◦ and a solar constant of 1367 Wm2 . Another

assumption is that Gallium Arsenide cells are used with a yearly degradation d of 2.75%.

Furthermore, with an assumed eclipse time of 10 min and equation (3.45), we can cal-

culate PSAeol= 1.1W . With d = 0.0275 and and estimated satellite lifetime tSatellite

of one year, PSAbolcan be computed to 1.13W with [Larson and Wertz, 1999, p.417],

rounded up to 1.2W.

7. The AOCS subsystem is represented by the calculation of a reaction wheel by means

of equation (3.60). Assuming the reaction wheel RW1 Type B of the company Astro-

und Feinwerktechnik GmbH to be used on the hypothetical CubeSat as tested on BeeSat

we get

ΠRW3 CubeSat =HRW · tOrbitmS/C · x2

eq

· mRW

mS/C

· rRWxeq· mAOCS

mS/C

=0.0001Nms · 5926.2s

1.33kg · (0.10431m)2· 0.012kg

1.33kg· 21mm

104.31mm· 0.15

= 0.011

(3.116)

Page 93


8. The Communication subsystem is represented by equation (3.71). With the collected

data on the flown CubeSats, is it assumed that mAntennaT = 0.01kg, R = 9.6kbps,

PTransmitter = 0.5W , the frequency of the signal f = 437Mhz which leads to λCom =

0.6858m and BER = 10−4. This leads to

ΠCom2 CubeSat =mAntennaT · h2

Orbit ·R3

PTransmitter· λComDTransmitter

·

· λComDReceiver

·PS/C

PTransmitter· mCom

mS/C

·BER

=0.01kg · (700km)2 · (9.6kbps)3

0.5W· 0.6858m

0.3m·

· 0.6858m

1m· 1W

0.5W· 0.17 · 10−4

= 462.160

(3.117)

9. The Payload subsystem of an Earth Observation mission can be represented by equa-

tion (3.79). Based on explanations in [Larson and Wertz, 1999, Chapter 9] and data of

real flown CubeSats, we assume b = 8 bitpixel

, λP/L = 7 · 10−7m and D = 0.02m and a

ground-pixel resolution X of the instrument of 60 m. This leads to

ΠP/LEarthObservation CubeSat =R · tOrbit

b·λP/LD·mS/C

mP/L

·PS/CPP/L

=9.6kbps · 5926.2s

8 bitpixel

· 7 · 10−7m

0.02m· 4 · 1W

0.3W

= 3.319

(3.118)

The Mission Performance Index

With equations (3.86) and (3.88) to (3.96) and table 3.20 we can now calculate the Mission

Performance Index of the 1U CubeSat.

It is for the payload

(ΨP/L CubeSat)α =

(R · kP/L

)α= (9.6 · 2.893)1

= 27.773

(3.119)

The AOCS system is represented by

(ΨAOCS CubeSat)β = (HRW · kRW )β

=(0.0001 · 8.96 · 10−3

)0.5

= 9.47 · 10−4

(3.120)

Page 94


As already mentioned in section 3.5, the modeled Power subsystem consists of batteries and

solar arrays, thus

(ΨPower CubeSat)χ =

(1

ΨBattery

·ΨSolar Array

)χ=

(1

CBattery · kBattery· P 3

SAbol· kSolarArray

)χ=

(1

36000 · 4000· 1.23 · 0.304

)0.8

= 1.777 · 10−7

(3.121)

The Structure subsystem as well as the C&DH subsystem are presented simply by their mass

ratios.

(ΨStructure CubeSat)δ =

(mS/C

mStructure

)δ=

(1

0.25

)0.3

= 1.516 (3.122)

(ΨC&DH CubeSat)φ =

(mS/C

mC&DH

)φ=

(1

0.03

)0.6

= 8.198 (3.123)

We assume that the hypothetical CubeSat does not have a Thermal subsystem. Thus the rule

applies, that its Ψ is equal to 10−20.

(ΨThermal CubeSat)ε =

(mS/C

mThermal

)ε= (10−20)0.2 = 10−4 (3.124)

The Communication subsystem is calculated with

(ΨCom CubeSat)ϕ =

(R3 · kCom

)ϕ=(9.63 · 1.914

)0.8

= 382.885

(3.125)

And finally, as in case of the Thermal subsystem, the Propulsion subsystem is represented by

a Ψ of 10−20.

(ΨPropulsion CubeSat)γ =

(mS/C

mPropulsion

)γ= (10−20)0.5 = 10−10 (3.126)

Thus, with a choice of the exponent α, β, χ, . . . from table 3.20, we can write

MPICubeSat = (ΨP/L CubeSat)α · (ΨAOCS CubeSat)

β · (ΨPower CubeSat)χ·

· (ΨStructure CubeSat)δ · (ΨC&DH CubeSat)

φ · (ΨThermal CubeSat)ε·

· (ΨCom CubeSat)ϕ · (ΨPropulsion CubeSat)

γ

= 27.773 · (9.47 · 10−4) · (1.777 · 10−7) · 1.516 · 8.198 · 10−4 · 382.885 · 10−10

= 2.224 · 10−5 · 10−4 · 10−10

(3.127)

Page 95


and separately showing the factors 10−4 and 10−10 due to the lack of the Propulsion and

Thermal subsystem.

Please note that a Mission Performance Index only makes sense in comparison to the MPI

of other satellites. This is the reason why a normalization is recommended for the MPI. The

normalization can be only done when another satellite can be used for comparison.

3.6.2. The NGSO-Satellite - a Top-Down Approach

In a Top-Down-approach for a Technology Demonstration mission the payload parameters

mP/L, xeqP/L or VP/L and PP/L and tSatellite are the input variables of the design approach.

As figure 3.69 deals with an Earth Observation mission, the parameters R, X and Y are

shown as further input variables. However, the payload of an hypothetical NGSO-satellite is

in general its Communication subsystem, especially the antenna. Thus the Communication

subsystem is used in two ways, as provider of the communication link to the ground station

and other satellites of the constellation but also as payload, providing different services for the

user. This double function will be reflected in the exponent γ when calculating the MPI. In

terms of the input variables this means that X and Y are unnecessary parameters, only the

datarate R counts. However, R and hOrbit are connected by the relatively complex equations

of the link budget [Larson and Wertz, 1999, p.550 ff.]. In order to avoid those equations as

this is only a first order approximation, we assume that hOrbit and R are given simultaneously.

Based on the available information about NGSO-satellites, we assume for the calculations

• mP/L = 100kg

• xeq P/L = 0.5m

• PP/L = 500W

• R = 0.65Mbps

• hOrbit = 1000km

• tSatellite = 5years

The determination of the Πs and the calculation of the unknown quantities

1. The mass ratios can be determined by means of the results of the empirical validation

of the ratios in section 3.3.1. The ratios are set equal to the arithmetical averages of

the mass ratios of the database satellites presented in tables 3.7 and 3.8.

Page 96



with mP/L and the corresponding subsystem masses for the given mass ratios for

the hypothetical NGSO-satellite

S/Cwet S/Cdry AOCS Power Structure

Mass ratio [%] 371.9 304.9 26.2 90.6 61.9

Subsystem mass [kg] 371.9 304.9 26.2 90.6 61.9

Table 3.24.: Mass ratios of the subsystems Thermal, C&DH, Communication and Propulsion

build with mP/L and the corresponding subsystem masses for the given mass

ratios for the hypothetical NGSO-satellite

Thermal C&DH+TT&C Com Propulsion

Mass ratio [%] 22.5 14.7 114.7 28.4

Subsystem mass [kg] 22.5 14.7 114.7 28.4

Please note that not the whole Communication subsystem is considered as payload.

That is the reason why the C&DH+TT&C-ratio is not zero. However, the communica-

tion subsystem is hereby defined to consist of the payload and the C&DH and TT&C

subsystem. That is the reason for the high mass ratio of 114.7 % for Communication,

emerged simply because an addition of the payload and the C&DH+TT&C percentage.

This figure will especially be necessary when designing the Communication subsystem.

2. The packing factor or volume ratios are defined according to (3.7) and (3.5). As

no data is available on the packing factor for the NGSO-satellites, we assume that

pNGSO is about 60% to 70% as the satellites are bigger than CubeSats so that the size

restrictions are less critical. The ratios VBus

VS/Cand

VP/L

VS/Ccannot fall back on given data,

either, so estimations are necessary. We assume that(VBusVS/C

)NGSO

= 0.3− 0.5(VP/LVS/C

)NGSO

= 0.2− 0.3

(3.128)

With VP/L = 0.125 m3 this leads us to

VS/C NGSO = 0.417− 0.625 m3

VBus NGSO = 0.156− 0.260 m3(3.129)

Page 97



xeq Bus NGSO = 0.539− 0.639 m

xeq S/C NGSO = 0.747− 0.855 m(3.130)

Please note that the Bus volume VBus NGSO was build with VS/C NGSO = 0.521m3

which is the arithmetical average of the two results in equation (3.129).

3. The power ratios can be determined with (3.13) and the results of the empirical

validation of the power ratios by means of the NGSO-satellite database in subsection

3.3.3. The ratios are set equal to the arithmetical averages of the power ratios presented

in table 3.11. Please remember that the power ratio results in section 3.3.3 were based

on data for PSAeoland not as intended in equations (3.13) for PS/C .

(PSAeol

PP/L

)NGSO

= 1.451(PBusPP/L

)NGSO

= 0.335

(3.131)

thus with PP/L = 500W leading to PSAeol= 725.5W and PBus = 167.5W and thus

PS/C = PP/L + PBus = 667.5W . This numerical result can be confirmed with equation

(3.45) since

PS/C = PSAeol· 1

1 +tEclipse

tDaylight

PS/C = 725.5W · 1

1 + 8.73min105.12−8.73min

= 665.25W

(3.132)

which is remarkable as it proves the simplicity and the validity of working with non-

dimensional relationships.

4. The power and mass ratios (3.14) to (3.17) can be thus either read in table 3.15

what leads to

rMass Power 1 NGSO =mS/C

mP/L

· PSAeol

PP/L= 5.624

rMass Power 2 NGSO =mP/L

mS/C

·PP/LPSAeol

= 0.212


mP/L

·PP/LPSAeol

= 2.609


mS/C

· PSAeol

PP/L= 0.451

(3.133)

Page 98


or be calculated with the results of tables 3.23 and 3.24 and equation (3.131) to


mP/L

· PSAeol

PP/L= 3.719 · 1.451 = 5.396


mS/C

·PP/LPSAeol

= 0.269 · 0.689 = 0.185


mP/L

·PP/LPSAeol

= 3.719 · 0.689 = 2.562


mS/C

· PSAeol

PP/L= 0.269 · 1.451 = 0.390

(3.134)

5. The top-level approach with the non-dimensional parameters (3.26) can be calculated

for the hypothetical NGSO-satellite with tOrbit instead of tdutycycle to be

ΠP/L 1 NGSO =PP/L · t3Orbitx2eq P/L ·mP/L

=500W · (1.752hr)3

(0.5m)2 · 100kg

= 61.39

(3.135)

with tOrbit calculated to be 105.12min = 1.752hours. The orbit time is chosen to

be inserted in hours in order to keep the numerical value of ΠP/L 1 NGSO as small as

possible for manipulation reasons.

6. The Power subsystem is expressed with the non-dimensional parameters (3.35) for

the battery and (3.47) for the solar array. Both equations as well as the upcoming

parameters of the remaining subsystems possess as in the case of the CubeSat still

unknown exponents expressing the importance of the corresponding ratios for the non-

dimensional parameter. However, as this is only a first order approximation, we assume

all unknown exponents again to be equal to one.

The battery subsystem can be calculated with equation (3.35) to

ΠBat4 NGSO =CBattery

PEclipse · tEclipse·PP/LPEclipse

·mP/L

mPower

=5.4 · 106Ws

667.5W · 523.8s· 500W

667.5W· 1

0.906

= 12.769

(3.136)

with PEclipse = PS/C = 667.5W and tEclipse for hOrbit = 1000km and an inclination of

i = 50◦ is calculated with Appendix A to 8.73min. CBattery was calculated based on

the assumption that 15kg of Lithium-ion batteries with a specific energy of 100Whkg

are

Page 99


used on-board. Thus the accumulated capacity of the batteries CBattery is 1500Wh,

thus 5.4 · 106Ws.

The solar array is described by equation (3.47) and (3.49). As PSAeolis given by equation

(3.131), we use (3.49):

ΠSA8 NGSO =PSAeol

· d · tSatellitex2eq · S

·

· PSAeol

PSAbol

· mPower

mP/L

· PSAeol

PP/L· tOrbittEclipse

=667.5W · 0.0275 · 5years

(0.8m)2 · 1367 Wm2

· 667.5W

767.4W· 0.906 · 667.5W

500W· 105.12min

8.73min

= 1.330(3.137)

We chose an average equivalent spacecraft edge length of xeq S/C NGSO = 0.8m based

on the results (3.130). We assume a maximum incident sun angle of 90◦ and a solar

constant of 1367 Wm2 . Another assumption is that Gallium Arsenide cells are used with

a yearly degradation d of 2.75%. With d = 0.0275 and and estimated satellite lifetime

tSatellite of 5 years, PSAbolcan be computed to 767.4W with [Larson and Wertz, 1999,

p.417].

7. The AOCS subsystem of an NGSO-satellite in LEO does imply reaction wheels very

rarely. However, the calculations shall be done here for demonstration purposes. The

reaction wheels of the satellite can be designed by means of equation (3.65) in the

Top-Down approach. This leads to

ΠRW4 NGSO =HRW · tOrbitmS/C · x2

eq

· mRW

mP/L

· rRWxeq P/L

· mAOCS

mS/C

=0.7Nms · 6307.2s

371.9kg · (0.8m)2· 2kg

100kg· 0.1m

0.5m· 0.262

= 0.0194

(3.138)

HRW , mRW and rRW are all calculated, based on the assumptions that the overall

disturbance torque is in the order of 10−5. Even the small picosatellite reaction wheel

tested on Beesat can compensate this torque. What is important, however, is the mo-

mentum storage of the reaction wheel. Taking all the disturbances into consideration

for the given hypothetical NGSO-satellite, a reaction wheel with HRW = 0.7Nms is

best suited for our needs. The mass mRW for such a reaction wheel is estimated to be

about 2kg. The radius of the reaction wheel rRW will be about 100mm.

Page 100


8. The Communication subsystem can be best described with equation (3.72), leading

to

ΠCom3 NGSO =mAntennaT · h2

Orbit ·R3

PTransmitter·

· λComDTransmitter

· λComDReceiver

·PP/L

PTransmitter· mCom

mP/L

·BER

=40kg · (1000km)2 · (0.65Mbps)3

50W· 0.1875m

1.5m·

· 0.1875m

5m· 500W

50W· 1.147 · 10−5

= 0.116

(3.139)

As already mentioned in the mass ratio section above, mCom consists of the payload

and the C&DH and TT&C subsystem, that is the reason why mCom

mP/L> 1. The unknown

quantities are estimated based on the NGSO-database and for the antenna mass also

with [Richharia, 1999] and an estimation for DTransmitter.

9. The Payload is the Communication subsystem. Its influence as payload will become

clearer when determining the MPI and especially the exponent ϕ in the next section.


With equation (3.86), (3.97) to (3.105) and table 3.20 we can now calculate the Mission

Performance Index of the NGSO-satellite. It is

(ΨAOCS NGSO)β = (HRW · kRW )β

= (0.7 · 36.01)0.7

= 9.573

(3.140)

Please note that β has risen from 0.5 to 0.7 in comparison to the CubeSat. This is due to the

higher importance of pointing accuracy for communication satellites.

The Power subsystem is again represented by batteries and solar arrays, leading to

(ΨPower NGSO)χ =

(1

ΨBattery NGSO

·ΨSolar Array NGSO

)χ=

(1


SAeol· kSolarArray

)χ=

(1

5.4 · 106 · 4.229 · 105· 667.53 · 2.238 · 108

)0.8

= 3729.48

(3.141)

Page 101


The Structure subsystem is represented simply by its mass ratio

(ΨStructure NGSO)δ =

(mP/L

mStructure

)δ=

(1

0.619

)0.3

= 1.155 (3.142)

Communication satellites are provided with Thermal Control Systems. This is the reason why

ΨThermal is not 10−20 in comparison to the 1U CubeSat investigation.

(ΨThermal NGSO)ε =

(mP/L

mThermal

)ε=

(1

0.225

)0.2

= 1.348

(3.143)

The Communication subsystem can be calculated to

(ΨCom NGSO)ϕ =(R3 · kCom

)ϕ=(0.653 · 2.37

)1+0.8

= 0.461

(3.144)

As NGSO-satellites are usually equipped with a Propulsion subsystem, ΨPropulsion has to be

considered. Its exponent γ is chosen to be equal to β since AOCS and Propulsion subsystem

fulfill similar functions once the satellite is in orbit.

(ΨPropulsion NGSO)γ =

(mP/L

mPropulsion

)γ=

(1

0.284

)0.7

= 2.414

(3.145)

Please note that the C&DH subsystem is already considered in the Communication subsystem

by addition of its mass ratio to the payload ratio which leads to the communication ratio.

Therefore the C&DH subsystem is not considered separately in the Ψs.

Finally we can write with (3.86) and with the choice of the exponents α, β, χ, . . . from table

3.20

MPINGSO = (ΨAOCS NGSO)β · (ΨPower NGSO)χ · (ΨStructure NGSO)δ · (ΨThermal NGSO)ε·· (ΨCom NGSO)ϕ · (ΨProp NGSO)γ

= 27.773 · (9.47 · 10−4) · 1.155 · 1.348 · 0.461 · 2.414

= 4.559 · 10−2

(3.146)

As in the case of the CubeSats the MPI only makes sense in comparison to further satellites

with MPIs as it is a quantity expressing the value of a mission in relation to other missions.

Page 102


3.6.3. A New CubeSat Standard? - a Bottom-Up Approach

As it was partly seen in section 3.3.1 with the mass ratios of the structure and Communication

subsystem, the 1U CubeSat is assumed to be too small to be advantageous. A closer look shall

therefore be given here in this section to a 2U CubeSat in order to quantify its performance

by means of the Mission Performance Index. Again it is important to determine the most

important input quantities of the system in the beginning. According to figures 3.2 and 3.70

the main input quantities are mS/C , xeq or VS/C and PS/C . Based on the 2U CubeSat standard

and the available information on the flown CubeSats, we assume for the calculations

• mS/C = 2.66kg

• xeq = 131.42mm

• PS/C = 2W

Furthermore, as in the case of the 1U CubeSat, we will already assume an Earth Observation

mission for the satellite, an estimated satellite lifetime tSatellite of 1 year and an orbital altitude

hOrbit of 700 km. This is the same orbital altitude as it was supposed for the 1U CubeSat in

order to change as less quantities as possible in comparison to the 1U CubeSat. By this, the

comparison between the two different sized satellites is simplified. With the same argumenta-

tion, we have also chosen tSatellite.

The calculation of the unknown quantities

With the knowledge of the numerical values of the non-dimensional parameters for the CubeSat

satellite class, the computation will focus on the determination of the unknown quantities of

the design.

1. The mass ratios can be determined by the results presented in tables 3.21 and 3.22.

Although those values are supposed to be valid for a whole satellite class, it is assumed

that the mass ratios will change for bigger CubeSats as it was already seen for the mass

ratios in section 3.3.1. However, we will continue with those values and show nonetheless

that the 2U CubeSat is more performant than the 1U CubeSat.

Thus with mS/C = 2.66kg we find(mP/L

)CubeSat

= 0.25 · 2.66kg = 0.665kg

(mAOCS)CubeSat = 0.15 · 2.66kg = 0.399kg

(mPower)CubeSat = 0.15 · 2.66kg = 0.399kg

(mStructure)CubeSat = 0.25 · 2.66kg = 0.665kg

(mThermal)CubeSat = 0 · 2.66kg = 0kg

(mC&DH)CubeSat = 0.03 · 2.66kg = 0.0798kg

(mCom)CubeSat = 0.17 · 2.66kg = 0.4522kg

(mPropulsion)CubeSat = 0 · 2.66kg = 0kg

(3.147)

Page 103


The tables 3.25 and 3.26 summarize the results.


with mS/C CubeSat and the corresponding subsystem masses for the given mass

ratios for the hypothetical 2U CubeSat

P/L AOCS Power Structure

Mass ratio [%] 25 15 15 25

Subsystem mass [g] 665 399 399 665

Table 3.26.: Mass ratios of the subsystems Thermal, C&DH and Communication build with

mS/C CubeSat and the corresponding subsystem masses for the given mass ratios

for the hypothetical 2U CubeSat

Thermal C&DH Com Prop

Mass ratio [%] 0 3 17 0

Subsystem mass [g] 0 79.8 452.2 0

Please remember that we expect lower mass ratios for the subsystems in real - except

for the payload, where we expect a higher ratio. However, we will continue with the cal-

culation with the above values in order to fulfill the condition that the non-dimensional

parameter is constant for a satellite class.

2. The volume ratios are defined according to (3.107). With VS/C = 2.27 · 10−3 m3 this

leads us to

VBus CubeSat = 0.6− 0.7 · VS/C = 1.362 · 10−3 − 1.589 · 10−3 m3

VP/L CubeSat = 0.1− 0.2 · VS/C = 2.27 · 10−4 − 4.54 · 10−4 m3(3.148)


xeq Bus CubeSat = 1.108 · 10−1 − 1.167 · 10−1 m

xeq P/L CubeSat = 6.10 · 10−2 − 7.69 · 10−2 m(3.149)

3. The power ratios can be determined with (3.110).

PP/L CubeSat = 0.3 · PS/C = 0.3 · 2W = 0.6W

PBus CubeSat = 0.7 · PS/C = 0.7 · 2W = 1.4W(3.150)

Page 104


4. The power and mass ratios (3.111) can be used as an alternative for the calculation

of the power quantities. For example, taking rMass Power 3 as this ratio showed the least

deviation for the NGSO-satellites, and assuming that we are looking for the numerical

value of PP/L we can compute

PP/L = rMass Power 3 CubeSat ·mP/L

mS/C

· PS/C

= 1.2 · 0.25 · 2W = 0.6W

(3.151)

thus the same result as in (3.150).

5. The Power subsystem is defined by (3.114) for the battery and (3.115) for the solar

array. Thus with similar assumptions as for the 1U CubeSat, namely tEclipse = 10min

andPEclipse

PS/C= 1, we can determine CBattery by

CBattery = PEclipse · tEclipse · ΠBat3 CubeSat ·PEclipsePS/C

· mPower

mS/C

= 2W · 600s · 400 · 2W

2W· 0.15

= 72000Ws

(3.152)

Concludingly, the battery system of the 2U CubeSat will lead to a lower MPI in com-

parison to the 1U CubeSat.

For the solar array (body-mounted) equation (3.115) with PSAeol= 2.2W calculated by

(3.45) with GalliumArsenide cells and a assumed satellite lifetime of 1 year, leads us to

P 3SolarArraybol

= ΠSA5 CubeSat · x2eq · S · d · tSatellite·

· PSAeol·mS/C

mPower

· PS/C ·tEclipsetOrbit

= 5.690 · (0.13142m)2 · 1367W

m2· 0.0275 · 1year·

· 2.2W · 1

0.15· 2W · 10min

98.77min

= (2.25W )3

(3.153)

So, in terms of the solar array, however, the 2U CubeSat seems to be more performant

than in case of the 1U CubeSat.

Page 105


6. The AOCS subsystem can be calculated by means of (3.116). As there are three

unknowns which cannot be as easily calculated as PSAeol, we firstly determine the product

of unknowns to

HRW ·mRW · rRW =ΠRW3 CubeSat

tOrbit· (mS/C)2 · x3

eq ·mS/C

mAOCS

=0.011

5926.2s· (2.66kg)2 · (0.13142m)3 · 1

0.15

= 1.987 · 10−7Nms kg m

(3.154)

giving HRW in Nms, mRW in kg and rRW in m. Assuming the mass and the radius of

the reaction wheel to be 0.03kg and 0.004m based on the data for the Beesat reaction

wheel, we can detemine HRW to be 1.656 · 10−3. Concludingly, the AOCS system of

the 2U CubeSat will lead to a higher MPI in comparison to the 1U CubeSat.

7. The Communication subsystem can be designed by means of equation (3.117).

Assuming mAntennaT = 0.02kg, PTransmitter = 1W , the same frequency of the sig-

nal f = 437Mhz as for the 1U CubeSat, which leads to λCom = 0.6858m, and

BER = 10−4, we can determine the data rate R with

R3 =ΠCom2 CubeSat

mAntennaT · h2Orbit

· PTransmitter ·DTransmitter

λCom·

· DReceiver

λCom· PTransmitter

PS/C·mS/C

mCom

· 1

BER

=462.160

0.02kg · (700km)2· 1W · 0.5m

0.6858m· 1m

0.6858m· 1W

2W· 1

0.17· 1

10−4

= (11.38kbps)3

(3.155)

with DTransmitter = 0.5m and DReceiver = 1m. This is an increase in comparison to the

1U CubeSat of 1.78kbps or 18.5%, thus leading to a higher MPI.

8. The Payload subsystem of an Earth Observation mission can be represented by equa-

tion (3.118). With the assumptions as for the 1U CubeSat b = 8 bitpixel

and λP/L =

7 · 10−7m, we can determine firstly D and then a ground-pixel resolution X. First of

all, D can be determined with the data rate calculated in (3.155)

D =R tOrbit

ΠP/LEarthObservation CubeSat · b· λP/L ·

mS/C

mP/L

·PS/CPP/L

=11.38kbps · 5926.2s

3.319 · 8 bitpixel

· 7 · 10−7m · 4 · 2W

0.6W

= 0.023m

(3.156)

Page 106


With equation (3.73) we can compute the ground-pixel resolution X to

X = 2.44 ·hOrbit · λP/L

D

= 2.44 · 700 · 1000m · 7 · 10−7m

0.023m

= 51.98m

(3.157)

thus leading to a better resolution than for the 1U CubeSat. The result is probably

too overoptimistic but this can be due to previous assumptions which were too high.

However, the tendency to better scientific results with larger spacecrafts shall be in the

focus here and not the numerical results which should only show how the equations can

be applied.


The results for the unknown quantities of the design let suppose that the MPI for the 2U

CubeSat will be higher than for the 1U CubeSat. With equation (3.86) we can finally calculate

the Mission Performance Index of the 2U CubeSat. It is for the Earth Observation payload

(ΨP/L CubeSat2)α =(R · kP/L

)α= (11.38 · 3.43)1

= 39.02

(3.158)

The AOCS subsystem is represented by a reaction wheel. Its Ψ can be calculated by

(ΨAOCS CubeSat2)β = (HRW · kRW )β

=(1.656 · 10−3 · 0.151

)0.5

= 0.016

(3.159)

The power subsystem consists again of batteries and solar arrays:

(ΨPower CubeSat2)χ =

(1

ΨBattery

·ΨSolar Array

)χ=

(1


SAbol· kSolarArray

)χ=

(1

72000 · 90· 2.253 · 2.00

)0.8

= 4.337 · 10−5

(3.160)

As the same mass ratios are used for the 2U CubeSat as for the 1U CubeSat since they are in

one satellite class, (ΨStructure)δ and (ΨC&DH)ε have the same values as for the 1U CubeSat.

(ΨStructure CubeSat2)δ =

(mS/C

mStructure

)δ=

(1

0.25

)0.3

= 1.516 (3.161)

Page 107


(ΨC&DH CubeSat2)φ =

(mS/C

mC&DH

)φ=

(1

0.03

)0.6

= 8.198 (3.162)

The 2U CubeSat is designed without Thermal subsystem as in the case of the hypothetical

1U CubeSat.

(ΨThermal CubeSat2)ε =

(mS/C

mThermal

)ε= (10−20)0.2 = 10−4 (3.163)

The Communication subsystem is described by

(ΨCom CubeSat2)ϕ =(R3 · kCom

)ϕ=(11.383 · 3.189

)0.8

= 866.26

(3.164)

The 2U CubeSat is designed without Propulsion subsystem as in the case of the hypothetical

1U CubeSat.

(ΨPropulsion CubeSat2)γ =

(mS/C

mPropulsion

)γ= (10−20)0.5 = 10−10 (3.165)

Thus we can write

MPICubeSat2 = (ΨP/L CubeSat2)α · (ΨAOCS CubeSat2)β · (ΨPower CubeSat2)χ·· (ΨStructure CubeSat2)δ · (ΨC&DH CubeSat2)φ · (ΨThermal CubeSat2)ε·· (ΨCom CubeSat2)ϕ · (ΨPropulsion CubeSat2)γ

= 39.02 · (0.016) · (4.337 · 10−5) · 1.516 · 8.198 · 10−4 · 866.26 · 10−10

= 0.292 · 10−4 · 10−10

(3.166)

This is a performance gain in comparison to the 1U CubeSat which has a MPI of 4.627 ·10−4 ·10−4 · 10−10. A comparison in percentages is intentionally not given as both calculations, for

the 1U and 2U CubeSat, were based on assumptions and not on a wider database. Thus a

certain inaccuracy is the consequence due the modelling nature of the assumptions. However,

a clear tendency to a higher performance for larger satellites can be noticed. This is a very

important result as it can provide guidelines to a small satellite which is still in the range of

CubeSats but with higher performance, thus combining the advantages of the CubeSat with

a higher mission performance.

As we have now two different MPIs for different satellites in one satellite class, we can exemplary

normalize the results. Our group of missions we want to compare consists of two Earth

Page 108


Observation missions. The MPI of the 2U CubeSat being the larger numerical value, we

define the 2U CubeSat mission to be the mission with a MPI of 100. This means that the 1U

CubeSat mission has a normalized MPI of

MPICubeSat normalized =MPICubeSatMPICubeSat2

· 100

=4.627 · 10−4 · 10−4 · 10−10

0.292 · 10−4 · 10−10· 100

= 0.16

(3.167)

thus showing clearly a significant difference between the 1U and the 2U CubeSat mission.

Page 109


Figure 3.69.: Top-Down network flow diagram for an Earth Observation mission

Page 110


Figure 3.70.: Bottom-Up network flow diagram for an Earth Observation mission

Page 111

4. Dimensional Analysis of Clusters

As dimensional analysis can be in general applied to systems of any possible kind, also satellite

clusters can be investigated with it. The intention of this chapter is therefore an introduction

to the analysis of satellite clusters with the Buckingham-Π-Theorem similar to the investigation

of the single satellites in Chapter 3. The ratios, non-dimensional parameters and especially the

Mission Performance Indices will make it possible to easily compare the performance of the

cluster with the performance of an equivalent monolithic satellite and facilitate the decision

whether a cluster or a single satellite will best suit the mission requirements.

4.1. The F6-Project and the Concept of Fractionated

Spacecrafts

As this work was intended to be embedded in the current F6-program of the Defense Advanced

Research Projects Agency (DARPA) in collaboration with Orbital Sciences Corporation, the

program shall be described here in some sentences.

F6 is the abbreviation for ”Future, Fast, Flexible, Fractionated, Free-Flying Spacecraft United

Information Exchange”, a program that intends to demonstrate a new approach in space

architecture design: instead of using one sizable monolithic satellite for a mission, several

smaller satellites are set in orbit, each fulfilling one functionality of the subsystems of the

monolith satellite, organized in a cluster, sharing resources and communicating via a wireless-

network. Advantages of this new approach are among other benefits essentially a gain in

flexibility and robustness: the new architecture is more flexible than the traditional one as

elements of the cluster can be easily replaced when they become obsolete. Furthermore new

elements can be integrated into the cluster when necessary. The cluster’s robustness is based

on the fact that a mission is, contrary to an architecture with one monolith satellite, not lost

when the launch fails or a subsystem of the satellite does not work properly since the elements

of the system are replaceable. However there are also potential downsides to a fractionated

cluster of small satellites such as the duplication of subsystems across satellites and an increased

complexity of the inter-spacecraft communication system. Dimensional analysis can help to

facilitate the decision between a cluster architecture and a monolithic satellite for a mission.

The project was in Phase 2 before the collaboration between Orbital Sciences Corporation and

MIT was paused in August 2010. Phase 2 means that DARPA’s industrial partner for the

project, Orbital Sciences Corporation, was in the phase of detailed design. MIT supported

their activity by further investigations. Therefore the MIT Space Systems Laboratory planned

Page 112

Dimensional Analysis of Clusters

to launch a prototype of an F6-CubeSat cluster in 2012/13, which would been designed and

built by MIT students in the coming years.

Although the collaboration is set on pause, cluster architectures have risen the attention of

scientists and industries worldwide and will definitely be a challenge in the coming years.

Therefore first investigations to find scaling laws and Mission Performance Indices for whole

clusters are believed to be foresighted.

4.2. Ratios and Non-Dimensional Parameters

Similar to a single spacecraft which can be investigated on a top-level approach, considering

only the spacecraft as a whole, or on a more detailed approach on a subsystem level, a cluster

can be investigated on two different levels, too. The top level is the cluster itself whereas the

cluster’s ”subsystems” are the different single satellites which collectively build the cluster.

For the single satellites of the cluster, no further investigation in terms of dimensional analysis

is necessary as the same non-dimensional parameters and ratios that are presented in chapter

3 can be applied to them. A further step, however, is required to characterize the cluster as

a whole. One approach to do so is to express the non-dimensional parameters and ratios of

chapter 3 with quantities describing the whole cluster. Those quantities can simply be the

sums of the corresponding quantities of the single satellites. Thus, for example, the total mass

of the cluster mCluster can be used which is defined as the sum of the masses of all the single

satellites of the cluster together. Similarly, further possible quantities are the total mass of

the power subsystem of the cluster mClusterPower, the total volume of the cluster VCluster, the

total power need of the cluster PCluster or the total data rate of the cluster transmitted to the

Earth RCluster.

The two different design approaches, the payload-centric Top-Down and the spacecraft-centric

Bottom-Up approach as presented in section 3.1, are both also valid for the design of the

clusters.

Hence, similar to the ratios (3.4) for a single satellite, we can rewrite the mass ratios in a

Bottom-Up approach for a cluster

rCluster Mass BU1 =mClusterP/L

mCluster

rCluster Mass BU2 =mClusterAOCS

mCluster

rCluster Mass BU3 =mClusterPower

mCluster

rCluster Mass BU4 =mClusterStructure

mCluster

rCluster Mass BU5 =mClusterThermal

mCluster

rCluster Mass BU6 =mClusterC&DH+TT&C

mCluster

rCluster Mass BU7 =mClusterPropulsion

mCluster

(4.1)

Page 113

Ratios and Non-Dimensional Parameters

with mClusteri expressing the sum of the i subsystem masses of all the satellites generating

the cluster.

The same proceeding as for the above mass ratios can be done for the Top-Down mass ratios

(3.3), the power ratios (3.12) and (3.13), the mass and power combining ratios (3.14) to

(3.17) and the top-level non-dimensional parameters (3.19) to (3.24) and (3.28) and (3.29).

In terms of the non-dimensional parameters for the subsystems, the results of section 3.4 can

be changed to be applicable to clusters, however, the physical information of the results is

reduced as the subsystems are especially designed to fulfill the requirements of a single satellite.

This means, for example, that we can build a non-dimensional parameter for the design of a

reaction wheel for the whole cluster based on the results in section 3.4.3 but this information

makes no physical sense as every single satellite requires a reaction wheel which is especially

customized for it. Nevertheless, when dealing with the cluster as if it is a monolithic satellite,

also on subsystem level, comparisons to real monolithic satellites are possible and this can

help to decide whether a cluster architecture or a single satellite is best suited for the mission

requirements. Subsystems which can be treated as one representative subsystem for the whole

cluster and still make physical sense, are the subsystems Power, Structure, Communication and

C&DH. For the power subsystem the results of section 3.4 - (3.33) to (3.35) for the battery

and (3.38), (3.40), (3.41), (3.42), (3.46) and (3.47) for the solar array - can be conveyed for

the cluster by using the corresponding quantities describing the cluster as a whole, such as

the total power need of the cluster during eclipse PClusterEclipseor the capacity of the batteries

on-board CClusterBattery. For the Structure subsystem no further analyses are necessary as

this subsystem is already covered by the corresponding mass and volume ratios. In terms of

the Communication subsystem, additional intersatellite links has to be taken into account as

they play a significant role in the functionality of a cluster. The Communication subsystem

does not only enable the communication with the ground stations but also the data transfer

between the satellites of the cluster so that all resources of the cluster can be used for data

processing. Thus the Communication subsystem becomes more important and complex but

can be still expressed by the non-dimensional parameters (3.70), (3.71) and (3.72). However,

the numerical values of the non-dimensional parameter ΠCom 1, ΠCom 2 and ΠCom 3 will be

different according to the kind of link they will represent: an intersatellite link will show

different characteristics than a link to the ground. Furthermore, only an overall representation

of the satellites’ Communication subsystems for a ground link, and not for a intersatellite link,

will make physical sense. This will be done with equivalent antenna diameters which are not

equal to the real antenna diameters. However, this investigation is especially interesting to

get information about the cluster’s transmitter output power PClusterTransmitterand the data

rate to the ground RCluster. In terms of the C&DH subsystem, one can surely state that this

subsystem will increase in complexity in comparison to a single satellite. Also more power and

more storage will be necessary to process the data. However, its non-dimensional parameter

is supposed to combine the same quantities as for a single satellite.

The subsystems AOCS and Thermal as well as the payload are believed to be too spacecraft

specific to be investigated for the cluster as a whole. Therefore no cluster-specific non-

dimensional parameters except for the corresponding mass and power ratios are supposed for

Page 114

Dimensional Analysis of Clusters

this three subsystems. However, their weighting in the Mission Performance Index will change

as, for example, the AOCS system in a cluster is more critical for the mission success as for a

single satellite since the intersatellite links require a certain station keeping and accuracy. The

different payloads in a cluster can also be interrelated to each other so that their performance is

also more critical in a cluster than when separately used on single satellites. All these changes

and more differences between clusters and single satellites will be issue of the next section

when dealing with a cluster’s Mission Performance Index.

4.3. Mission Performance Index

When investigating the advantages and disadvantages of a cluster in comparison to a single

satellite, the Mission Performance Index is one method which facilitates the decision between

these two mission architectures. The Mission Performance Index of a cluster architecture can

be calculated in two steps. In a first order approach only the characteristics of the cluster

are considered, thus MPICluster only combines ΠP/L Cluster, ΠAOCS Cluster, ΠPower Cluster,

ΠStructure Cluster, . . . . It is calculated as shown in section 3.5 and also normalized to become

a figure between 0 and 100. In a second step also the performance of the single satellites

generating the cluster is taken into account. For that, the MPISingle Sat for every single

satellite is calculated as shown in section 3.5. Afterwards, an arithmetical average is build

over all the satellites which gives us the quantity MPIall/Sats. A weighing can come in here,

too, in case some satellites are considered to be more critical for the mission accomplishment

than others. MPIall Sat is then normalized to a number between 0 and 1 and then multiplied

with MPICluster to create MPISat+Cluster, again a number between 0 and 100. As the

importance of the different subsystems for the mission success changes now in comparison

to the ones for a monolithic satellite, a closer look at the exponents for MPICluster follows

in form of the table 4.1. An increase in the numerical values of the ranges can be seen for

the payload and the subsystems Communication, C&DH, AOCS and Propulsion in comparison

the the exponents for the monolithic satellites. The increase for the latter two subsystems

is due to the more important requirement in satellite station keeping within the cluster to

enable the flawless communication and data transmission between the satellites. An increase

for the Communication and C&DH exponents is caused by the higher complexity of the two

systems. Intersatellite links has to be made possible as well as data processing on-board and

resource sharing. Without all that, the advantages of the cluster architecture are massively

diminished. Finally, the payload increases in importance as the interdependency between the

different payloads generally increases in cluster architectures. The same exponent ranges are

allocated to the subsystems Power, Structure and Thermal here for the cluster architectures

as for the monolithic satellites presented in 3.20 since their importance is believed to stay in

a similar range.

Page 115


Table 4.1.: Ranges for the subsystem exponents α, β, χ, . . . for cluster architectures, including

the tendency of the change in the ranges compared to the single satellite exponents

Subsystems EOM SSM TDM Tendancy

Payload 0.8-1.0 0.8-1.0 0.8-1.0 ↑

Power 0.7-0.9 0.7-0.9 0.6-0.9 const

Communication 0.8-0.9 0.8-0.9 0.7-0.9 ↑

Structure 0.2-0.5 0.2-0.5 0.2-0.5 const

AOCS 0.7-0.9 0.7-0.9 0.6-0.9 ↑

Thermal 0.1-0.3 0.1-0.6 0.1-0.6 const

C&DH 0.7-0.9 0.7-0.9 0.6-0.9 ↑

Propulsion 0.7-0.9 0.7-0.9 0.6-0.9 ↑

Please note that the current form of the Mission Performance Index only implies physical

quantities which directly influence the design of a spacecraft such as the mass and volume of the

spacecraft or the orbit altitude. But a mission consists of more influencing parameters than only

those describing the physics of a spacecraft. Further parameters such as launcher availability,

costs, development time of the design or robustness and flexibility of the architecture are not yet

implied in the Mission Performance Index. Especially the latter two quantities, however, express

the main advantage of cluster architectures in comparison to monolithic spacecraft mission

architectures. Thus, a further development of the MPI shall include also those quantities

expressing the feasibility, usability and reliability of the mission design by taking all stakeholders

of a mission into account. That means a turn from a systems engineering to a engineering

systems approach. This development is necessary in order to do justice to the mission as a

whole. With that alteration, a change in the numerical values of the MPIs is expected with a

tendency to higher MPIs for the clusters.

Page 116

5. Conclusion and Future Prospects

The application of dimensional analysis by means of the Buckingham-Π-Theorem on satellite

design was shown in this work. Ratios and non-dimensional parameters were developed as

well as a possibility for the quantification of mission accomplishment by means of the Mission

Performance Index. The validation of several ratios has been done with data from non-

geosynchronous communication satellites, showing promising results, limited by the nature of

the given data. Numerical applications of the theoretical results for three different hypothetical

satellites have proven that the ratios, non-dimensional parameters and the Mission Performance

Index can be used in practice. Taking especially the application on a hypothetical 1U and

2U CubeSat into consideration, dimensional analysis is a possibility to quickly design a new

spacecraft based on former designs and to examine the mission performance of a satellite in

comparison to other designs. Therefore, the results of this work can be used to optimize given

satellite standards in terms of their mission performance. In this context, it is believed that

the 1U CubeSat standard is too small to be efficient in terms of mission performance. A

further development of the MPI in terms of parameters i.e. costs and launcher availability is

supposed to prove this assumption and to provide the foundation for the definition of a new,

high-performant CubeSat standard.

Future work based on this thesis might include the further improvement and development

of the non-dimensional parameters for every subsystem as well as the validation of them by

means of extensive databases for several satellite classes. Beside the implementation of further

stakeholders into the MPI, the development of an ”universal MPI” which combines all mission

types as well as the two design approaches Top-Down and Bottom-Up, is also desirable.

Furthermore, as clusters gain increasing importance in the space sector, the detailed application

of the theoretical results of the dimensional analysis on cluster architectures is also supposed

to be worth the investigation.

Finally, the combination of the component database with the theoretical results into a design

tool for CubeSats is also desirable. The design tool will ideally enable the quick design of a

spacecraft based on the theoretical results of this work and indicate the COTS-components

which are best suited for the design. A continuously updated database is therefore necessary.

Page 117

A. Calculation of the Eclipse Time

tEclipse

For circular orbits the eclipse time tEclipse for a specific position of the Earth in the ecliptic

can be calculated with the following equation [Larson and Wertz, 1999, p. 107 f]

tEclipse = tOrbit ·φ

360◦(A.1)

thus with a direct proportionality between tEclipse and tOrbit. However, a change in tOrbit means

that the orbital altitude of the spacecraft hOrbit has changed because of equation (3.52). A

change in orbital altitude hOrbit, in turn, also influences the quantity φ in equation (A.1) which

can be calculated by

cos

(φ

2

)=

cos(ρ)

cos(βs)(A.2)

with ρ as the angular radius of the Earth, calculated by

sin(ρ) =REarth

REarth + hOrbit(A.3)

Thus an increase in the orbital altitude hOrbit leads to an decrease in φ which reacts in

direct opposition to the development of tOrbit with increasing orbital altitude. The maximum

eclipse time for different orbital altitude can be found listed in common literature (see Errata

of [Larson and Wertz, 1999]): the maximum eclipse time decreases with increasing orbital

altitude only until about 1500 km. Increasing the altitude further causes the eclipse time to

rise again. That is why it was decided to leave the eclipse time tEclipse and not the orbital

period tOrbit as influencing parameter of the battery system in section 3.4.2.

Please notice that CubeSats often play a special role in terms of these calculations. They

are often launched in sun-synchronous orbits which means their orientation to the sun is the

same over the whole revolution of the Earth around the Sun. This in turn leads to constant

eclipse times during the whole year. Furthermore, as sun-synchronous orbits in LEO have an

inclination of 96.0 to 98.0◦, orbits can be achieved which do not have any eclipses at all. This,

however, also depends on the choice of the Right Ascension of the Ascending Node (RAAN).

Thus for the sake of completeness of this topic, the following further explanations are given.

The only quantity in equation (A.2) not yet explained is βs as it adds a further complexity to

the calculations. βs, also called the β-angle, represents the position of the orbital plane to

the Sun: it is the angle of the Sun above the orbital plane, thus the angle between the orbital

plane and the incident sun. Its calculation is more complex than in case of the angular radius

Page 118

Calculation of the Eclipse Time tEclipse

of the Earth ρ as it generally changes during the revolution of the Earth around the Sun.

During a year, βs ranges from ±(23.5◦ + i). From the design point of view, important values

are the maximum and minimum eclipse time: tEclipse reaches its maximum for βs = 0◦,

i.e. when the sun vector is in the orbital plane, and its minimum when βs takes its extreme

values of ±(23.5◦ + i). Hence, the calculation for the extremes of the eclipse time tEclipsedoes not seem complicated. However, as already stated, CubeSats are often launched in sun-

synchronous orbits where the β-angle is fixed during the year but dependent on the orientation

of the orbit to the sun which is not necessarily βs = 0◦ or βs = i+ 23.5◦. Thus a closer look

at the calculation of βs is recommended.

βs can be calculated in general as∣∣∣90◦ − βs

′∣∣∣ with βs

′being the angle between the normal of

the orbital plane ~nOrbit and the sun vector ~s.

In a fixed heliocentric coordinate system with the vernal equinox as the x-axis, ~nOrbit can be

calculated by

~nOrbit =

sin(RAAN) · sin(α)

−cos(RAAN) · sin(α)

cos(α)

(A.4)

with RAAN being the Right Ascension of the Ascending Node and α being i + 23.5◦. The

sun vector ~s can be expressed by

~s =

cos(δ)

sin(δ)

0

(A.5)

assuming a perfectly circular Earth orbit around the Sun, with δ being 180◦ + δ′. The 180◦

enables the sun vector to ”start” at spring and is necessary because of the definition of the

x-axis (vernal equinox). δ′

represents the actual position of the Earth relatively to the vernal

equinox, thus δ′

= 360◦

365,25TPosition with TPosition being the day for which the eclipse time shall

be calculated, counting from the vernal equinox on.

Thus with the scalar product for ~nOrbit and ~s, βs′

can be calculated as

cos(βs′) = sin(RAAN) · sin(α) · cos(δ)− cos(RAAN) · sin(α) · sin(δ) (A.6)

and with βs =∣∣∣90◦ − βs

′∣∣∣, one can calculate the beta angle βs which in turn determines the

eclipse time with equations (A.2) and (A.1).

For the CubeSats only one calculation is actually necessary - for example in autumn because of

the simplicity of the sun vector ~s - and βs is constant over the whole mission lifetime assuming

a constant orbital altitude over the mission duration.

Page 119

To sum up, the complexity of the calculation of the eclipse time tEclipse has been highlighted.

It is dependent on a number of factors and especially for the CubeSats the following function

applies

tEclipse = f(hOrbit, i, RAAN) (A.7)

Thus for a more detailed investigation, i and RAAN should be taken into consideration for

the calculation of the eclipse time.

Page 120

B. CubeSat-Specific Commercial

off-the-Shelf Component Database

Commercial off-the-shelf (COTS) components are already very often used in satellite engineer-

ing as COTS components are normally less expensive than in-house solutions and usually faster

available. So, the development time of a satellite can be effectively shortened and thus the all

in all costs decreased. However, as CubeSats have strong weight and size limitations, many

”normal” sized components are not suitable for the use on picosatellites. Thus, with the rising

of the importance of CubeSats and new minituarizing technologies, a market has established

itself which is specialized on CubeSats specific off-the-shelf components. Most companies like

Pumpkin Inc., Clyde Space Ltd. and ISIS - Innovative Solutions in Space are only specialized

in components for pico- and nanosatellites, few others like Astro- und Feinwerktechnik Adler-

shof GmbH offer also products for other satellite classes. In order to facilitate and speed up

the choice for a COTS component, a database was created during this work which lists many

of the currently available solutions with their costs, availabilities, technical characteristics and

also the internet links where the product can be find online. The list is created under Microsoft

Office Excel so that it is guaranteed that its use is familiar to an extensive number of satellite

developers. It is organized in several sheets, including CubeSat kits, Energy Power System

solutions, batteries, solar panels, CubeSat platforms, components for the AOCS, structures,

transceivers, receivers, transmitters, antennas, on-board computers, software, harnesses, elec-

trical and mechanical ground support equipment, ground station related products, possible

payloads, the latest research results in propulsion, GPS receivers, various miscellaneous prod-

ucts and also a short list of forthcoming products. Available data sheets of the listed products

are provided on the CD attached to this work and can be consulted for further information. It

is also intended to use the list in a future MATLAB program which shall, based on inputs from

the engineer like the mission performance of the satellite, recommend COTS components for

the mission.

Some universities already showed their interest in such a database and mentioned existing

plans to create one by themselves. The realization though has never been accomplished so

far, in most cases due to a lack of time.

Currently many subsystems can fall back on a wide range of specialized CubeSat COTS compo-

nents which are highly matured. Especially the Attitude and Orbit Control subsystem seems to

have been in the focus of many companies so that the customer can choose between an exten-

sive number of AOCS components. Other subsystems like power, communication and structure

Page 121

are in the product range of a smaller number of companies but interested customers can still

choose between specific flight-proven components. Beside the usual satellite components such

as magnetorquers, batteries and antennas, it is worth mentioning that whole CubeSat kits can

be purchased which promise shorter development times and affordable projects, thus are espe-

cially suited for student missions. The kits of the company Pumpkin Inc., for example, consist

of a complete lightweight structure for the CubeSat as well as a flight module, power supplies

for the use on Earth, programming adapters, cables and tools and a software. In the easiest

case, the customer has only to add a power supply (i.e. batteries, solar panel) and a payload

and the satellite is ready to be launched. Depending on the requirements of the mission, a

communication system consisting of a transceiver and an antenna as well as an attitude and

orbital determination system can also be added. However, what the users of the kits complain

about is the loss of the flexibility in design. More sophisticated missions for the CubeSats re-

quire higher accuracies and longer satellite lifetimes. Thus, picosatellite propulsion is a current

research area in many institutions and companies worldwide. However, many approaches still

need time to mature and existing technologies and components require a decrease in costs to

be used by universities with a restricted mission budget. It also became obvious during the

research, that many universities passed on COTS components in their design as they are still

regarded to be too expensive and especially not suitable for the intended purposes. Higher

development times are therefore required but still preferred to the COTS components. On

the other hand, also not flight-approven components are often used in the satellite designs.

Hence, extensive testings are necessary in this cases but still preferred to much more expensive

COTS components. In the end, it will be one of the future main tasks of a successful CubeSat

component industry to develop components which are much less expensive and meet the needs

of the customer more precisely.

Page 122

CubeSat-Specific Commercial off-the-Shelf Component Database

Figure B.1.: Screenshot of a part of the AOCS-sheet of the CubeSat COTS component

database

Page 123

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