Orbital parameters of Asteroids using analytical Propagation

Guided by

Dr. R.V Ramanan

Team Members:Chetana D.Lakshmi Narsimhan Lokeswara Rao.NRamiz AhmadRanupriya Didwania

Plan of the talk:

1. Objective2. Introduction to various coordinate system3. Problems/complexities associated with the parameter calculations.4. Equations of motion (for two-body motion).5. Ephemeris generation (related formulas and codes).6. Conclusions.

Objective

To obtain the orbital parameters of the celestial objects ( asteroids ) at any time with respect to its reference parameters.

To study the time evolution of asteroids

Introduction to the coordinate system

Various coordinate systems:1. Inertial coordinate system (commonly used) Origin - Centre of Earth Principal axis (x-axis) - towards the vernal equinox ( intersection of the Earth Equator and ecliptic plane) from the origin Fundamental Plane - Earth equator

2. The right ascension –declination coordinate system Origin - Centre of Earth Principal axis (x-axis) - towards the vernal equinox ( intersection of the Earth Equator and ecliptic plane) from the origin Fundamental Plane - Earth equator

3. The latitude – longitude coordinate system Origin - Centre of Earth Principal axis (x-axis) - towards the Greenwich meridian from the origin Fundamental Plane - Earth equator

Spacecraft / Celestial body is acted upon by multiple gravity fields (For e.g.. Earth , Sun and Mars for an Earth –Mars Transfer) - 4-body equations of motion must be solved - No closed form solution

OTHERSM

M

M

MM

E

E

E

EES R

R

R

r

r

R

R

r

r

R

R

dt

Rd

333332

2

SRPDragNSGPlanetsOthers RRRRR

- To be solved numerically - Ephemeris (solution) accuracy depends on the Force Model

Complexity in determination of the motion of the body

Two body Motion and ConicAssumptions

The motion of a body is governed by attraction due to a single central body. The mass of the body is negligible compared to that of the central body The bodies are spherically symmetric with the masses concentrated at the center. No forces act on the bodies except for gravitational and centrifugal forces acting along the line of centers

If these assumptions hold, it can be shown that conic sections are the only possible paths for orbiting bodies and that the central body must be at a focus of the conic

32

2

r

r

dt

rd

Fundamental equations of motion that describe two-body motion under the assumptions

)( 21 mmG where

cos

)(

cos e

ea

e

pr

1

21

1

Closed form Solution

- Conic Equation

Relative Form

a - semi major axisb - semi minor axisr - radial distanceν - true anomaly

Size and Shape of a Conic

Representation of a point (spacecraft / body) in motion

True anomaly

Position and velocity vectors represent a point in motion in space uniquely

Vernalequinox Node

i

wW

n

Z

0

Satelliteperigee

Equator

i Inclination ; W

True anomalyw Argument of perigee;

n

a semi major axis ; e Eccentricity

Right ascension of ascending nodeW

Two-Body Motion : General Description

1tatVr

,2tatVr

,

Given full characteristics of spacecraft in a conic at time t1

- either state vector (both position and velocity vectors together) or orbital elements Find the characteristics of the spacecraft in the conic at time t2

Ephemeris Generation

Ephemeris generation using analytical techniques(Time evolution)

Algorithm:1. Input the present value of the true anomaly (nu0).2. Using the following equations, calculate the value of E0

3. And then calculate the value of M from the following equationM0=E0-e*sinE0

4. The value of ΔM is then calculated using the following equation.ΔM =n(t-t0)

5. Using the calculated value of ΔM we find the value of E as per the equationM0+ ΔM = E-e*sinE

6. And finally the value of nu at any time is found from the equation

sinE

Calculating the transformation equationsAlgorithm:1. From the calculated value of nu we get the value of parameters using the transformation equations

http://ssd.jpl.nasa.gov/horizons.cgi#top

T0= 2004-Oct-01T= 2005-Oct-01

Pallas From To

2004-Oct-01 2004-Oct-02

Parameters JPL calculated Error

rx (km) -204349767.594200000 -205016214.226665000 666446.632464975

ry (km) 242717229.395200000 242888592.882376000 -171363.487175971

rz (km) -34002778.020140000 -33939477.594100100 -63300.426039897

vx (km/sec) -17.500293070 -17.504129882 0.003836812

vy (km/sec) -13.720366896 -13.730516703 0.010149807

vz (km/sec) 3.945413510 3.940343099 0.005070411

v (km/sec) 22.584840337 22.593095308 -0.008254971

r (km) 319102914.236415000 319653569.959065000 -550655.722650230

alpha (degree) 130.094900000 130.166879000 -0.071979000

delta (degree) -6.117083078 -6.095094145 -0.021988932

Ephemeris (successive years)

Source of error

Conclusion:Based on the two body model ephemeris generation was carried out.Even though the % error is of the order of 10-1 -10-2 , , their value in absolute is very high This stress the fact that we need to have a detailed model and do the calculation using them, even though carrying them out is very tedious and takes a lot of time.Precision takes precedence over time!

Referenceshttp://ssd.jpl.nasa.gov/horizons.cgi#top

Lecture notes on Orbital Dynamics, Dr Ramanan M V, IIST

“Orbital mechanics for Engineering Students”, Howard Curtis, Elsevier Aerospace Engineering Series

Thank You