Orbital MechanicsPrinciples of Space Systems Design

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”)

© 2001 David L. Akin - All rights reserved

Orbital MechanicsPrinciples of Space Systems Design

Energy in Orbit

• Kinetic Energy

• Potential Energy

• Total Energy

K E mK E

mv

. .. .

= ⇒ =12 2

22

ν

P Emr

P Em r

. .. .

= − ⇒ = −µ µ

Constv

r a. = − = −

2

2 2µ µ

<--Vis-Viva Equation

Orbital MechanicsPrinciples of Space Systems Design

Implications of Vis-Viva

• Circular orbit (r=a)

• Parabolic escape orbit (a tends to infinity)

• Relationship between circular and parabolicorbits

vrcircular =µ

vrescape =

2µ

v vescape circular= 2

Orbital MechanicsPrinciples of Space Systems Design

Some Useful Constants

• Gravitation constant µ = 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

Orbital MechanicsPrinciples of Space Systems Design

Classical Parameters of Elliptical Orbits

Orbital MechanicsPrinciples of Space Systems Design

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

• Radial distance as function of orbital position

• Periapse and apoapse distances

• Angular momentum

p a e= −( )1 2

rp

e=

+1 cosθ

r a ep = −( )1 r a ea = +( )1

r r rh r v= × h p= µ

Orbital MechanicsPrinciples of Space Systems Design

The Classical Orbital Elements

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

Orbital MechanicsPrinciples of Space Systems Design

The Hohmann Transfer

vperigee

v1

vapogee

v2

Orbital MechanicsPrinciples of Space Systems Design

First Maneuver Velocities

• Initial vehicle velocity

• Needed final velocity

• Delta-V

vr11

=µ

vr

rr rperigee =

+µ

1

2

1 2

2

∆vr

rr r1

1

2

1 2

21=

+−

µ

Orbital MechanicsPrinciples of Space Systems Design

Second Maneuver Velocities

• Initial vehicle velocity

• Needed final velocity

• Delta-V

vr22

=µ

vr

rr rapogee =

+µ

2

1

1 2

2

∆vr

rr r2

2

1

1 2

12

= −+

µ

Orbital MechanicsPrinciples of Space Systems Design

Limitations on Launch Inclinations

Orbital MechanicsPrinciples of Space Systems Design

Differences in Inclination

Orbital MechanicsPrinciples of Space Systems Design

Choosing the Wrong Line of Apsides

Orbital MechanicsPrinciples of Space Systems Design

Simple Plane Change

vperigee

v1 vapogee

v2

∆v2

Orbital MechanicsPrinciples of Space Systems Design

Optimal Plane Change

vperigee v1 vapogee

v2

∆v2∆v1

Orbital MechanicsPrinciples of Space Systems Design

First Maneuver with Plane Change ∆∆∆∆i1

• Initial vehicle velocity

• Needed final velocity

• Delta-V

vr11

=µ

vr

rr rp =

+µ

1

2

1 2

2

∆ ∆v v v v v ip p1 12 2

1 12= + − cos( )

Orbital MechanicsPrinciples of Space Systems Design

Second Maneuver with Plane Change ∆∆∆∆i2

• Initial vehicle velocity

• Needed final velocity

• Delta-V

vr22

=µ

vr

rr ra =

+µ

2

1

1 2

2

∆ ∆v v v v v ia a2 22 2

2 22= + − cos( )

Orbital MechanicsPrinciples of Space Systems Design

Sample Plane Change Maneuver

01234567

0 10 20 30

Initial Inclination Change (deg)

Del

ta V

(km

/sec

)

DV1DV2DVtot

Optimum initial plane change = 2.20°

Orbital MechanicsPrinciples of Space Systems Design

Bielliptic Transfer

Orbital MechanicsPrinciples of Space Systems Design

Coplanar Transfer Velocity Requirements

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

Orbital MechanicsPrinciples of Space Systems Design

Noncoplanar Bielliptic Transfers

Orbital MechanicsPrinciples of Space Systems Design

Calculating Time in Orbit

Orbital MechanicsPrinciples of Space Systems Design

Time in Orbit

• Period of an orbit

• Mean motion (average angular velocity)

• Time since pericenter passage

➥M=mean anomaly

Pa

= 23

πµ

na

=µ

3

M nt E e E= = − sin

Orbital MechanicsPrinciples of Space Systems Design

Dealing with the Eccentric Anomaly

• Relationship to orbit

• Relationship to true anomaly

• Calculating M from time interval: iterate

until it converges

r a e E= −( cos )1

tan tanθ2

11 2

=+−

ee

E

E nt e Ei i+ = +1 sin

Orbital MechanicsPrinciples of Space Systems Design

Hill’s Equations (Proximity Operations)

˙̇ ˙x n x ny adx= + +3 22

˙̇ ˙y nx ady= − +2

˙̇z n z adz= − +2

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

Orbital MechanicsPrinciples of Space Systems Design

Clohessy-Wiltshire (“CW”) Equations

x t nt xnt

nx

nnt yo o o( ) cos( )

sin( )˙ cos( ) ˙= −[ ] + + −[ ]4 3

21

y t nt nt x yn

nt xnt nt

nyo o o o( ) sin( ) cos( ) ˙

sin( )˙= −[ ] + − −[ ] +

−6

21

4 3

z t z ntz

nnto

o( ) cos( )˙

sin( )= +

˙( ) sin( ) ˙ sin( )z t z n nt z nto o= − +

Orbital MechanicsPrinciples of Space Systems Design

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