Spacecraft TrajectoriesSpacecraft Trajectories

You Can Get There from Here!You Can Get There from Here!

John F SantariusJohn F Santarius

Lecture 9Lecture 9

Resources from SpaceResources from Space

NEEP 533/ Geology 533 / Astronomy 533 / EMA 601NEEP 533/ Geology 533 / Astronomy 533 / EMA 601

University of WisconsinUniversity of Wisconsin

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Newton’s Laws of MotionNewton’s Laws of Motion

• The fundamental laws of mechanical motion were first formulated by Sir Isaac Newton (1643-1727), and were published in his Philosophia Naturalis Principia Mathematica.

• Calculus, invented independently by Newton and Gottfried Leibniz (1646-1716), plus Newton's laws of motion are the mathematical tools needed to understand rocket motion.

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Newton’s Laws of MotionNewton’s Laws of Motion

• Every body continues in its state of rest or of Every body continues in its state of rest or of uniform motion in a straight line except insofar as it uniform motion in a straight line except insofar as it is compelled to change that state by an external is compelled to change that state by an external impressed force.impressed force.

• To every action there is an equal and opposite To every action there is an equal and opposite reaction.reaction.

dt

dpF

• The rate of change of The rate of change of momentummomentum of of the body is proportional to the the body is proportional to the impressed force and takes place in impressed force and takes place in the direction in which the force acts. the direction in which the force acts.

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Newton’s Law of GravitationNewton’s Law of Gravitation

• Every particle of matter attracts every other particle Every particle of matter attracts every other particle of matter with a force directly proportional to the of matter with a force directly proportional to the product of the masses and inversely proportional to product of the masses and inversely proportional to the square of the distance between them. the square of the distance between them.

rF ˆ2

12

21

r

mGm

G=6.67×10-11 m3 s-2 kg-1 is the gravitational constant.

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Kepler’s Laws of Planetary MotionKepler’s Laws of Planetary Motion

• The planets move in ellipses The planets move in ellipses with the sun at one focus.with the sun at one focus.

• Areas swept out by the radius Areas swept out by the radius vector from the sun to a planet vector from the sun to a planet in equal times are equal.in equal times are equal.

• The square of the period of The square of the period of revolution is proportional to revolution is proportional to the cube of the semimajor axis. the cube of the semimajor axis. TT22 aa33

Conic sections

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Essential Orbital DynamicsEssential Orbital Dynamics

• Circular velocityCircular velocity

• Escape velocityEscape velocity

• Energy of a vehicle Energy of a vehicle following a conic section following a conic section ((aasemimajor axis)semimajor axis) a

GMmE

r

GMv

r

GMv

conic

esc

cir

2

22/1

2/1

G=6.67×10-11 m3 s-2 kg-1

Hohmann’s Minimum-Energy Hohmann’s Minimum-Energy Interplanetary TransferInterplanetary Transfer

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• The minimum-energy transfer between circular orbits is an elliptical trajectory called the Hohmann trajectory. It is shown at right for the Earth-Mars case, where the minimum total delta-v expended is 5.6 km/s.

• The values of the energy per unit mass on the circular orbit and Hohmann trajectory are shown, along with the velocities at perihelion (closest to Sun) and aphelion (farthest from Sun) on the Hohmann trajectory and the circular velocity in Earth or Mars orbit.

• The differences between these velocities are the required delta-v values in the rocket equation.

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Calculating Hohmann TransfersCalculating Hohmann Transfers

• Kepler's third law, Kepler's third law, TT aa3/23/2, can be used to calculate , can be used to calculate the time required to traverse a Hohmann trajectory by the time required to traverse a Hohmann trajectory by raising to the 3/2 power the ratio of the semimajor raising to the 3/2 power the ratio of the semimajor axis of the elliptical Hohmann orbit to the circular axis of the elliptical Hohmann orbit to the circular radius of the Earth's orbit and dividing by two (for radius of the Earth's orbit and dividing by two (for one-way travel). one-way travel).

• For example, call the travel time for an Earth-Mars For example, call the travel time for an Earth-Mars trip trip TT and the semimajor axis of the Hohmann ellipse and the semimajor axis of the Hohmann ellipse a.a.

aa = (1 AU + 1.5 AU)/2 = 1.25 AU = (1 AU + 1.5 AU)/2 = 1.25 AU TT = 0.5 ( = 0.5 (aa / 1 AU) / 1 AU)3/23/2 years years = ~0.7 years = ~8.4 months= ~0.7 years = ~8.4 months

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Rocket EquationRocket Equation

• Conservation of momentum leads to the so-called rocket Conservation of momentum leads to the so-called rocket equation, which trades off exhaust velocity with payload equation, which trades off exhaust velocity with payload fraction. Based on the assumption of short impulses with fraction. Based on the assumption of short impulses with coast phases between them, it applies to chemical and coast phases between them, it applies to chemical and nuclear-thermal rockets. First derived by Konstantin nuclear-thermal rockets. First derived by Konstantin Tsiolkowsky in 1895 for straight-line rocket motion with Tsiolkowsky in 1895 for straight-line rocket motion with constant exhaust velocity, it is also valid for elliptical constant exhaust velocity, it is also valid for elliptical trajectories with only initial and final impulses. trajectories with only initial and final impulses.

• Conservation of momentum for a rocket and its exhaust leads Conservation of momentum for a rocket and its exhaust leads to to

exi

fex v

v

m

m

dt

dv

m

dtdmv

dt

dmv

dt

dvm

dt

mvd

dt

dpexp

/0

)(

High Exhaust Velocity GivesHigh Exhaust Velocity GivesLarge Payloads or Fast TravelLarge Payloads or Fast Travel

• The rocket equation shows why high exhaust The rocket equation shows why high exhaust velocity has historically been a driving force for velocity has historically been a driving force for rocket design: payload fractions depend strongly rocket design: payload fractions depend strongly upon the exhaust velocity.upon the exhaust velocity.

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Chemicalrocket

Gravity AssistsGravity AssistsEnable or Facilitate Many MissionsEnable or Facilitate Many Missions

• A spacecraft arrives within the sphere of A spacecraft arrives within the sphere of influence of a body with a so-called influence of a body with a so-called hyperbolic excess velocity equal to the hyperbolic excess velocity equal to the vector sum of its incoming velocity and vector sum of its incoming velocity and the planet's velocity.the planet's velocity.

• In the planet's frame of reference, the In the planet's frame of reference, the direction of the spacecraft's velocity direction of the spacecraft's velocity changes, but not its magnitude. In the changes, but not its magnitude. In the spacecraft's frame of reference, the net spacecraft's frame of reference, the net result of this trade-off of momentum is a result of this trade-off of momentum is a small change in the planet's velocity and small change in the planet's velocity and a very large delta-v for the spacecraft.a very large delta-v for the spacecraft.

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• Starting from an Earth-Jupiter Hohmann trajectory and performing a Jupiter flyby at one Jovian radius, as Starting from an Earth-Jupiter Hohmann trajectory and performing a Jupiter flyby at one Jovian radius, as shown above, the hyperbolic excess velocity vshown above, the hyperbolic excess velocity vhh is approximately 5.6 km/s and the angular change in is approximately 5.6 km/s and the angular change in

direction is about 160direction is about 160oo..

Vh

Vh

Motion in planet’sframe of reference

Efficient Solar-System Travel RequiresEfficient Solar-System Travel RequiresHigh-Exhaust-Velocity, Low-Thrust PropulsionHigh-Exhaust-Velocity, Low-Thrust Propulsion

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• Electric power can be used to drive high-exhaust-velocity Electric power can be used to drive high-exhaust-velocity plasma or ion thrusters, or fusion plasmas can be directly plasma or ion thrusters, or fusion plasmas can be directly exhausted.exhausted.

– Allows fast trip times Allows fast trip times or large payload or large payload fractions for long-fractions for long-range missions.range missions.

• Uses relatively small Uses relatively small amounts of propellant, amounts of propellant, reducing total mass.reducing total mass.

Fusion rocketFusion rocket(( specific powerspecific power))

How Do Separately Powered Systems How Do Separately Powered Systems Differ from Chemical Rockets?Differ from Chemical Rockets?

• Propellant not the power source.Propellant not the power source.

• High exhaust velocity (High exhaust velocity (101055 m/s). m/s).

• Low thrust (Low thrust (1010-2-2 m/s} m/s}1010-3-3 Earth gravity). Earth gravity).

• Thrusters typically operate for a large fraction of Thrusters typically operate for a large fraction of the mission duration.the mission duration.

• High-exhaust-velocity trajectories are High-exhaust-velocity trajectories are fundamentally differentfundamentally different from chemical-rocket from chemical-rocket trajectories.trajectories.

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Chemical rocket trajectoryChemical rocket trajectory(minimum energy)(minimum energy)

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Taking Full Advantage of High Exhaust Taking Full Advantage of High Exhaust Velocity Requires Optimizing TrajectoriesVelocity Requires Optimizing Trajectories

Earth

Mars

Sun

Fusion rocket trajectoryFusion rocket trajectory(variable acceleration)(variable acceleration)

Earth

MarsSun

Note: Trajectories are schematic, not calculated.Note: Trajectories are schematic, not calculated.

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mission power-on timemission power-on time

MMw w power plant masspower plant mass

MMl l payload masspayload mass

MMp p propellant masspropellant mass

MM0 0 total masstotal mass = M = Mww + M + Mll + M + Mpp

M M propellant flow ratepropellant flow rate = M = Mpp// F F thrustthrust = M v = M vex ex

PPww thrust powerthrust power = ½ M v = ½ M vexex22

[kW/kg][kW/kg] specific power specific power = P = Pww / M / Mww

vvch ch characteristic velocity = (2 characteristic velocity = (2))½½

Useful Propulsion DefinitionsUseful Propulsion Definitions

-1.5 -1 -0.5 0 0.5 1

EXHAUST VELOCITY / CHARACTERISTIC VELOCITY

0

0.2

0.4

0.6

0.8

1

NOI

TC

AR

F D

AO

LY

AP

Rocket Equation forRocket Equation forSeparately Powered SystemsSeparately Powered Systems

• Explicitly including the power-plant mass through the Explicitly including the power-plant mass through the characteristic velocity modifies the rocket equation:characteristic velocity modifies the rocket equation:

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exch

ex

ex

l

v

u

v

v

v

u

M

Mexp1exp

2

2

0

U/Vch=0.1

0.3

0.5

0.7

U mission energy requirement

vch characteristic velocity

= (2)½

Fusion Propulsion Would Enable Fusion Propulsion Would Enable Efficient Solar-System TravelEfficient Solar-System Travel

Comparison of trip times and payload Comparison of trip times and payload fractions for chemical and fusion rocketsfractions for chemical and fusion rockets

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00.10.20.30.40.50.60.70.80.91

Payl

oad

frac

tion

Earth-Mars(260 days)

Earth-Jupiter(1000 days)

Chemical

Fusion (1 kW/ kg)

Fusion (10 kW/ kg)