Orbital Mechanics

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 Mechanics
Principles of Space Systems Design

Energy in Orbit
• Kinetic Energy
1 KE . . v
KE . . = mν
2
⇒ =
2 m 2
• Potential Energy
PE . .
• Total Energy
mµ
PE . . µ
=− ⇒ =−
r m r
2
v µ µ
Const.
= − =−

Implications of Vis-Viva
• Circular orbit (r=a)
v
circular
µ
=
r
• Parabolic escape orbit (a tends to infinity)
v
escape = 2µ
r
• Relationship between circular and parabolic
orbits
vescape = 2vcircular
Orbital Mechanics
Principles of Space Systems Design

Some Useful Constants
• Gravitation constant µ = GM
– Earth: 398,604 km 3 /sec 2
– Moon: 4667.9 km 3 /sec 2
– Mars: 42,970 km 3 /sec 2
– Sun: 1.327x10 11 km 3 /sec 2
• Planetary radii
–r Earth = 6378 km
–r Moon = 1738 km
–r Mars = 3393 km
Orbital Mechanics
Principles of Space Systems Design

Classical Parameters of Elliptical Orbits
Orbital Mechanics
Principles of Space Systems Design

Basic Orbital Parameters
• Semi-latus rectum (or parameter)
2
p= a( 1−e
)
• Radial distance as function of orbital position
• Periapse and apoapse distances
• Angular momentum
r
p
=
1+ ecosθ
r = a( 1−e) r = a( + e)
1
p
r r r
h = r × v h= µ
p
a
Orbital Mechanics
Principles of Space Systems Design

The Classical Orbital Elements
Ref: J. E. Prussing and B. A. Conway, Orbital Mechanics Oxford University Press, 1993
Orbital Mechanics
Principles of Space Systems Design

The Hohmann Transfer
v1 vperigee v2 vapogee Orbital Mechanics
Principles of Space Systems Design

First Maneuver Velocities
• Initial vehicle velocity
• Needed final velocity
• Delta-V
∆v
1
=
v
µ
⎛
⎜
r ⎝
1
v
1
=
r
µ
perigee =
1
2r2
r + r
1 2
µ
r
1
⎞
−1⎟
⎠
2r2
r + r
1 2
Orbital Mechanics
Principles of Space Systems Design

Second Maneuver Velocities
• Initial vehicle velocity
• Needed final velocity
• Delta-V
∆v
2
v
v
apogee =
2
µ
⎛
= ⎜1−
r ⎝
2
=
r
µ
2
µ
r
2
2r1
r + r
1 2
⎞
⎟
⎠
2r1
r + r
1 2
Orbital Mechanics
Principles of Space Systems Design

Limitations on Launch Inclinations
Orbital Mechanics
Principles of Space Systems Design

Differences in Inclination
Orbital Mechanics
Principles of Space Systems Design

Choosing the Wrong Line of Apsides
Orbital Mechanics
Principles of Space Systems Design

Simple Plane Change
v perigee
v 1 v apogee
∆v 2
v 2
Orbital Mechanics
Principles of Space Systems Design

Optimal Plane Change
v perigee v1 v apogee
∆v 1
∆v 2
v 2
Orbital Mechanics
Principles of Space Systems Design

First Maneuver with Plane Change ∆i 1
• Initial vehicle velocity
• Needed final velocity
• Delta-V
v
v
1
=
r
µ
p =
1
µ
r
1
2r2
r + r
1 2
2 2
∆v = v + v −2vv cos( ∆i
)
1 1
p 1 p 1
Orbital Mechanics
Principles of Space Systems Design

Second Maneuver with Plane Change ∆i 2
• Initial vehicle velocity
• Needed final velocity
• Delta-V
v
v
a =
2
µ
r
2
=
r
µ
2
2r1
r + r
1 2
2 2
∆v = v + v −2v v cos( ∆i
)
2 2
a 2 a 2
Orbital Mechanics
Principles of Space Systems Design

Sample Plane Change Maneuver
Delta V (km/sec)
7
6
5
4
3
2
1
0
0 10 20 30
Initial Inclination Change (deg)
Optimum initial plane change = 2.20°
DV1
DV2
DVtot
Orbital Mechanics
Principles of Space Systems Design

Bielliptic Transfer
Orbital Mechanics
Principles of Space Systems Design

Coplanar Transfer Velocity Requirements
Ref: J. E. Prussing and B. A. Conway, Orbital Mechanics Oxford University Press, 1993
Orbital Mechanics
Principles of Space Systems Design

Noncoplanar Bielliptic Transfers
Orbital Mechanics
Principles of Space Systems Design

Calculating Time in Orbit
Orbital Mechanics
Principles of Space Systems Design

Time in Orbit
• Period of an orbit
P
• Mean motion (average angular velocity)
• Time since pericenter passage
➥M=mean anomaly
3
a
= 2π
µ
n
=
a
µ 3
M = nt = E−esin E
Orbital Mechanics
Principles of Space Systems Design

Dealing with the Eccentric Anomaly
• Relationship to orbit
r = a( 1−ecos
E)
• Relationship to true anomaly
θ 1+
e E
tan = tan
2 1−e2 • Calculating M from time interval: iterate
E + nt e E
= + 1 sin
i i
until it converges
Orbital Mechanics
Principles of Space Systems Design

Hill’s Equations (Proximity Operations)
Ref: J. E. Prussing and B. A. Conway, Orbital Mechanics
Oxford University Press, 1993
2
x˙˙ = 3n x+ 2ny˙+
adx y˙˙ =− 2nx˙+
ady 2
˙˙z=− n z+ adz Orbital Mechanics
Principles of Space Systems Design

Clohessy-Wiltshire (“CW”) Equations
x( t) = 4−3cos( nt) x
sin( nt)
2
x˙
[ 1 cos( nt) ] y˙
n n
[ ] + + −
2
y( t) = 6[
sin( nt) −nt]
x + y − 1−cos(
nt) x˙
n
zt ( ) = zcos( nt)
+
o
o o o
[ ] +
4sin( nt) − 3nt
n
o o o o
z˙
o sin( nt)
n
z˙( t) =− z nsin( nt) +
z˙ sin( nt)
o o
y˙
Orbital Mechanics
Principles of Space Systems Design

References for Lecture 3
• Wernher von Braun, The Mars Project
University of Illinois Press, 1962
• William Tyrrell Thomson, Introduction to
Space Dynamics Dover Publications, 1986
• Francis J. Hale, Introduction to Space
Flight Prentice-Hall, 1994
• William E. Wiesel, Spaceflight Dynamics
MacGraw-Hill, 1997
• J. E. Prussing and B. A. Conway, Orbital
Mechanics Oxford University Press, 1993
Orbital Mechanics
Principles of Space Systems Design