Orbital Mechanics - University Of Maryland · 2003-09-08 · Orbital Mechanics Principles of Space Systems Design U N I V E R S I T Y O F MARYLAND Patched Conics •Simple approximation

Orbital MechanicsPrinciples of Space Systems Design

U N I V E R S I T Y O F

MARYLAND

Orbital Mechanics

• Energy and velocity in orbit• Elliptical orbit parameters• Orbital elements• Coplanar orbital transfers• Noncoplanar transfers• Time and flight path angle as a function of

orbital position• Relative orbital motion (“proximity operations”)

© 2003 David L. Akin - All rights reservedhttp://spacecraft.ssl.umd.edu



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Energy in Orbit

• Kinetic Energy

• Potential Energy

• Total Energy

K.E. =12

mn 2 fiK.E.

m=

v2

2

P.E. = -mmr

fiP.E.m

= -mr

Const. =v2

2-

mr

= -m

2a<--Vis-Viva Equation



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Implications of Vis-Viva

• Circular orbit (r=a)

• Parabolic escape orbit (a tends to infinity)

• Relationship between circular and parabolicorbits

vcircular =mr

vescape =2mr

vescape = 2vcircular



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Some Useful Constants

• Gravitation constant m = GM– Earth: 398,604 km3/sec2

– Moon: 4667.9 km3/sec2

– Mars: 42,970 km3/sec2

– Sun: 1.327x1011 km3/sec2

• Planetary radii– rEarth = 6378 km– rMoon = 1738 km– rMars = 3393 km



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Classical Parameters of Elliptical Orbits



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Basic Orbital Parameters• Semi-latus rectum (or parameter)

• Radial distance as function of orbital position

• Periapse and apoapse distances

• Angular momentum

p = a(1- e2 )

r =p

1+ ecosq

rp = a(1 - e) ra = a(1+ e)

r h = r r ¥ r v h = mp



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The Classical Orbital Elements

Ref: J. E. Prussing and B. A. Conway, Orbital Mechanics Oxford University Press, 1993



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The Hohmann Transfer

vperigee

v1

vapogee

v2



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First Maneuver Velocities

• Initial vehicle velocity

• Needed final velocity

• Delta-V

v1 =mr1

vperigee =mr1

2r2

r1 + r2

Dv1 =mr1

2r2

r1 + r2-1

Ê

Ë Á

ˆ

¯ ˜



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Second Maneuver Velocities



• Delta-V

v2 =mr2

vapogee =mr2

2r1

r1 + r2

Dv2 =mr2

1-2r1

r1 + r2

Ê

Ë Á

ˆ

¯ ˜



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Limitations on Launch Inclinations



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Differences in Inclination



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Choosing the Wrong Line of Apsides



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Simple Plane Change

vperigee

v1 vapogee

v2

Dv2



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Optimal Plane Change

vperigee v1 vapogee

v2

Dv2Dv1



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First Maneuver with Plane Change Di1



• Delta-V

v1 =mr1

vp =mr1

2r2

r1 + r2

Dv1 = v12 + vp

2 - 2v1vp cos(Di1)



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Second Maneuver with Plane Change Di2



• Delta-V

v2 =mr2

va =mr2

2r1r1 + r2

Dv2 = v22 + va

2 - 2v2va cos(Di2 )



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Sample Plane Change Maneuver

01234567

0 10 20 30Initial Inclination Change (deg)

Delta

V (k

m/s

ec)

DV1DV2DVtot

Optimum initial plane change = 2.20°



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Bielliptic Transfer



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Coplanar Transfer Velocity Requirements

Ref: J. E. Prussing and B. A. Conway, Orbital Mechanics Oxford University Press, 1993



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Noncoplanar Bielliptic Transfers



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Calculating Time in Orbit



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Time in Orbit

• Period of an orbit

• Mean motion (average angular velocity)

• Time since pericenter passage

ÂM=mean anomaly

P = 2pa3

m

n =ma3

M = nt = E - esin E



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Dealing with the Eccentric Anomaly

• Relationship to orbit

• Relationship to true anomaly

• Calculating M from time interval: iterate

until it converges

r = a(1- e cosE)

tanq2

=1+ e1- e

tan E2

Ei+1 = nt + esin Ei



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Patched Conics

• Simple approximation to multi-body motion (e.g.,traveling from Earth orbit through solar orbit intoMartian orbit)

• Treats multibody problem as “hand-offs” betweengravitating bodies --> reduces analysis tosequential two-body problems

• Caveat Emptor: There are a number of formal methods toperform patched conic analysis. The approach presented here is avery simple, convenient, and not altogether accurate method forperforming this calculation. Results will be accurate to a fewpercent, which is adequate at this level of design analysis.



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Example: Lunar Orbit Insertion• v2 is velocity of moon

around Earth• Moon overtakes

spacecraft with velocityof (v2-vapogee)

• This is the velocity of thespacecraft relative to themoon while it iseffectively “infinitely”far away (before lunargravity accelerates it) =“hyperbolic excessvelocity”

vapogee

v2



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Planetary Approach Analysis

• Spacecraft has vh hyperbolic excessvelocity, which fixes total energy ofapproach orbit

• Vis-viva provides velocity of approach

• Choose transfer orbit such that approachis tangent to desired final orbit at periapse

†

v 2

2-

mr

= -m2a

=vh

2

2

†

v = 2 vh2

2+

mr

†

Dv = 2 vh2

2+

mrorbit

-m

rorbit

Space Systems Laboratory – University of Maryland

Pr

oje

ct

Dia

na

DV Requirements for Lunar MissionsTo:

From:

Low EarthOrbit

LunarTransferOrbit

Low LunarOrbit

LunarDescentOrbit

LunarLanding

Low EarthOrbit

3.107km/sec

LunarTransferOrbit

3.107km/sec

0.837km/sec

3.140km/sec

Low LunarOrbit

0.837km/sec

0.022km/sec

LunarDescentOrbit

0.022km/sec

2.684km/sec

LunarLanding

2.890km/sec

2.312km/sec



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Hill’s Equations (Proximity Operations)

†

˙ ̇ x = 3n2x + 2n˙ y + adx

†

˙ ̇ y = -2n˙ x + ady

†

˙ ̇ z = -n 2z + adz

Ref: J. E. Prussing and B. A. Conway, Orbital MechanicsOxford University Press, 1993



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Clohessy-Wiltshire (“CW”) Equations

†

x(t) = 4 - 3cos(nt)[ ]xo +sin(nt)

n˙ x o +

2n

1- cos(nt)[ ] ˙ y o

†

y(t) = 6 sin(nt) - nt[ ]xo + yo -2n

1-cos(nt)[ ] ˙ x o +4sin(nt) - 3nt

n˙ y o

†

z(t) = zo cos(nt) +˙ z on

sin(nt)

†

˙ z (t) = -zonsin(nt) + ˙ z o sin(nt)



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“V-Bar” Approach

Ref: Collins, Meissinger, and Bell, Small Orbit Transfer Vehicle (OTV) for On-Orbit SatelliteServicing and Resupply, 15th USU Small Satellite Conference, 2001



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“R-Bar” Approach

• Approach from alongthe radius vector (“R-bar”)

• Gravity gradientsdecelerate spacecraftapproach velocity - lowcontaminationapproach

• Used for Mir, ISSdocking approaches Ref: Collins, Meissinger, and Bell, Small Orbit Transfer

Vehicle (OTV) for On-Orbit Satellite Servicing andResupply, 15th USU Small Satellite Conference, 2001



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References for Lecture 3

• Wernher von Braun, The Mars ProjectUniversity of Illinois Press, 1962

• William Tyrrell Thomson, Introduction toSpace Dynamics Dover Publications, 1986

• Francis J. Hale, Introduction to SpaceFlight Prentice-Hall, 1994

• William E. Wiesel, Spaceflight DynamicsMacGraw-Hill, 1997

• J. E. Prussing and B. A. Conway, OrbitalMechanics Oxford University Press, 1993

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Orbital Mechanics - University Of Maryland · 2003-09-08 · Orbital Mechanics Principles of Space Systems Design U N I V E R S I T Y O F MARYLAND Patched Conics •Simple approximation