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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
Orbital MechanicsPrinciples of Space Systems Design
U N I V E R S I T Y O F
MARYLAND
Energy in Orbit
• Kinetic Energy
• Potential Energy
• Total Energy
K.E. =12
mν 2 ⇒K.E.m
=v2
2
P.E. =−mμr
⇒P.E.m
=−μr
Const. =v2
2−
μr
=−μ2a
<--Vis-Viva Equation
Orbital MechanicsPrinciples of Space Systems Design
U N I V E R S I T Y O F
MARYLAND
Implications of Vis-Viva
• Circular orbit (r=a)
• Parabolic escape orbit (a tends to infinity)
• Relationship between circular and parabolic orbits
vcircular =μr
vescape =2μr
vescape = 2vcircular
Orbital MechanicsPrinciples of Space Systems Design
U N I V E R S I T Y O F
MARYLAND
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
U N I V E R S I T Y O F
MARYLAND
Classical Parameters of Elliptical Orbits
Orbital MechanicsPrinciples of Space Systems Design
U N I V E R S I T Y O F
MARYLAND
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+ecosθ
rp =a(1−e) ra =a(1+e)
r h = rr × rv h = μp
Orbital MechanicsPrinciples of Space Systems Design
U N I V E R S I T Y O F
MARYLAND
The Classical Orbital Elements
Ref: J. E. Prussing and B. A. Conway, Orbital Mechanics Oxford University Press, 1993
Orbital MechanicsPrinciples of Space Systems Design
U N I V E R S I T Y O F
MARYLAND
The Hohmann Transfer
vperigee
v1
vapogee
v2
Orbital MechanicsPrinciples of Space Systems Design
U N I V E R S I T Y O F
MARYLAND
First Maneuver Velocities
• Initial vehicle velocity
• Needed final velocity
• Delta-V
v1 =μr1
vperigee =μr1
2r2r1 + r2
Δv1 =μr1
2r2
r1 + r2
−1 ⎛
⎝ ⎜
⎞
⎠ ⎟
Orbital MechanicsPrinciples of Space Systems Design
U N I V E R S I T Y O F
MARYLAND
Second Maneuver Velocities
• Initial vehicle velocity
• Needed final velocity
• Delta-V
v2 =μr2
vapogee =μr2
2r1r1 + r2
Δv2 =μr2
1−2r1
r1 + r2
⎛
⎝ ⎜
⎞
⎠ ⎟
Orbital MechanicsPrinciples of Space Systems Design
U N I V E R S I T Y O F
MARYLAND
Limitations on Launch Inclinations
Orbital MechanicsPrinciples of Space Systems Design
U N I V E R S I T Y O F
MARYLAND
Differences in Inclination
Orbital MechanicsPrinciples of Space Systems Design
U N I V E R S I T Y O F
MARYLAND
Choosing the Wrong Line of Apsides
Orbital MechanicsPrinciples of Space Systems Design
U N I V E R S I T Y O F
MARYLAND
Simple Plane Change
vperigee
v1 vapogee
v2
Δv2
Orbital MechanicsPrinciples of Space Systems Design
U N I V E R S I T Y O F
MARYLAND
Optimal Plane Change
vperigee v1 vapogee
v2
Δv2Δv1
Orbital MechanicsPrinciples of Space Systems Design
U N I V E R S I T Y O F
MARYLAND
First Maneuver with Plane Change Δi1
• Initial vehicle velocity
• Needed final velocity
• Delta-V
v1 =μr1
vp =μr1
2r2r1 + r2
Δv1 = v12 + vp
2 − 2v1vp cos(Δi1)
Orbital MechanicsPrinciples of Space Systems Design
U N I V E R S I T Y O F
MARYLAND
Second Maneuver with Plane Change Δi2
• Initial vehicle velocity
• Needed final velocity
• Delta-V
v2 =μr2
va =μr2
2r1r1 + r2
Δv2 = v22 + va
2 − 2v2va cos(Δi2 )
Orbital MechanicsPrinciples of Space Systems Design
U N I V E R S I T Y O F
MARYLAND
Sample Plane Change Maneuver
01234567
0 10 20 30
Initial Inclination Change (deg)
Delta V (km/sec)
DV1DV2DVtot
Optimum initial plane change = 2.20°
Orbital MechanicsPrinciples of Space Systems Design
U N I V E R S I T Y O F
MARYLAND
Bielliptic Transfer
Orbital MechanicsPrinciples of Space Systems Design
U N I V E R S I T Y O F
MARYLAND
Coplanar Transfer Velocity Requirements
Ref: J. E. Prussing and B. A. Conway, Orbital Mechanics Oxford University Press, 1993
Orbital MechanicsPrinciples of Space Systems Design
U N I V E R S I T Y O F
MARYLAND
Noncoplanar Bielliptic Transfers
Orbital MechanicsPrinciples of Space Systems Design
U N I V E R S I T Y O F
MARYLAND
Calculating Time in Orbit
Orbital MechanicsPrinciples of Space Systems Design
U N I V E R S I T Y O F
MARYLAND
Time in Orbit
• Period of an orbit
• Mean motion (average angular velocity)
• Time since pericenter passage
M=mean anomaly
P =2π a3
μ
n =μa3
M =nt=E −esinE
Orbital MechanicsPrinciples of Space Systems Design
U N I V E R S I T Y O F
MARYLAND
Dealing with the Eccentric Anomaly
• Relationship to orbit
• Relationship to true anomaly
• Calculating M from time interval: iterate
until it converges
r =a(1−ecosE)
tanθ2
=1+e1−e
tanE2
Ei+1 =nt+esinEi
Orbital MechanicsPrinciples of Space Systems Design
U N I V E R S I T Y O F
MARYLAND
Patched Conics
• Simple approximation to multi-body motion (e.g., traveling from Earth orbit through solar orbit into Martian orbit)
• Treats multibody problem as “hand-offs” between gravitating bodies --> reduces analysis to sequential two-body problems
• Caveat Emptor: There are a number of formal methods to perform patched conic analysis. The approach presented here
is a very simple, convenient, and not altogether accurate method for performing this calculation. Results will be accurate to a few percent, which is adequate at this level of design analysis.
Orbital MechanicsPrinciples of Space Systems Design
U N I V E R S I T Y O F
MARYLAND
Example: Lunar Orbit Insertion
• v2 is velocity of moon around Earth
• Moon overtakes spacecraft with velocity of (v2-vapogee)
• This is the velocity of the spacecraft relative to the moon while it is effectively “infinitely” far away (before lunar gravity accelerates it) = “hyperbolic excess velocity”
vapogee
v2
Orbital MechanicsPrinciples of Space Systems Design
U N I V E R S I T Y O F
MARYLAND
Planetary Approach Analysis
• Spacecraft has vh hyperbolic excess velocity, which fixes total energy of approach orbit
• Vis-viva provides velocity of approach
• Choose transfer orbit such that approach is tangent to desired final orbit at periapse
€
v2
2−
μr
=−μ2a
=vh
2
2
€
v =2vh
2
2+
μr
€
Δv =2vh
2
2+
μrorbit
−μ
rorbit
Space Systems Laboratory – University of Maryland
Pro
ject
Dia
na
ΔV Requirements for Lunar Missions
To:
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
Orbital MechanicsPrinciples of Space Systems Design
U N I V E R S I T Y O F
MARYLAND
Hill’s Equations (Proximity Operations)
€
˙ ̇ x =3n2x+2n˙ y +adx
€
˙ ̇ y =−2n˙ x +ady
€
˙ ̇ z =−n2z+adz
Ref: J. E. Prussing and B. A. Conway, Orbital Mechanics Oxford University Press, 1993
Orbital MechanicsPrinciples of Space Systems Design
U N I V E R S I T Y O F
MARYLAND
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)
Orbital MechanicsPrinciples of Space Systems Design
U N I V E R S I T Y O F
MARYLAND
“V-Bar” Approach
Ref: Collins, Meissinger, and Bell, Small Orbit Transfer Vehicle (OTV) for On-Orbit Satellite Servicing and Resupply, 15th USU Small Satellite Conference, 2001
Orbital MechanicsPrinciples of Space Systems Design
U N I V E R S I T Y O F
MARYLAND
“R-Bar” Approach
• Approach from along the radius vector (“R-bar”)
• Gravity gradients decelerate spacecraft approach velocity - low contamination approach
• Used for Mir, ISS docking approaches
Ref: Collins, Meissinger, and Bell, Small Orbit Transfer Vehicle (OTV) for On-Orbit Satellite
Servicing and Resupply, 15th USU Small Satellite Conference, 2001
Orbital MechanicsPrinciples of Space Systems Design
U N I V E R S I T Y O F
MARYLAND
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
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