03 orb mech - UMD

transcript

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”)

Energy in Orbit

• Kinetic Energy

• Potential Energy

• Total Energy

K E mK E

. .. .

= ⇒ =12 2

P Em r

. .. .

= − ⇒ = −µ µ

Constv

r a. = − = −

2 2µ µ

<--Vis-Viva Equation

Implications of Vis-Viva

• Circular orbit (r=a)

• Parabolic escape orbit (a tends to infinity)

• Relationship between circular and parabolicorbits

vrcircular =µ

vrescape =

v vescape circular= 2

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

Classical Parameters of Elliptical Orbits

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

+1 cosθ

r a ep = −( )1 r a ea = +( )1

r r rh r v= × h p= µ

The Classical Orbital Elements

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

The Hohmann Transfer

vperigee

vapogee

First Maneuver Velocities

• Initial vehicle velocity

• Needed final velocity

• Delta-V

rr rperigee =

Second Maneuver Velocities

• Delta-V

rr rapogee =

= −+

Limitations on Launch Inclinations

Differences in Inclination

Choosing the Wrong Line of Apsides

Simple Plane Change

vperigee

v1 vapogee

Optimal Plane Change

vperigee v1 vapogee

∆v2∆v1

First Maneuver with Plane Change ∆∆∆∆i1

• Delta-V

rr rp =

∆ ∆v v v v v ip p1 12 2

1 12= + − cos( )

Second Maneuver with Plane Change ∆∆∆∆i2

• Delta-V

rr ra =

∆ ∆v v v v v ia a2 22 2

2 22= + − cos( )

Sample Plane Change Maneuver

01234567

0 10 20 30

Initial Inclination Change (deg)

DV1DV2DVtot

Optimum initial plane change = 2.20°

Bielliptic Transfer

Coplanar Transfer Velocity Requirements

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

Noncoplanar Bielliptic Transfers

Calculating Time in Orbit

Time in Orbit

• Period of an orbit

• Mean motion (average angular velocity)

• Time since pericenter passage

➥M=mean anomaly

M nt E e E= = − sin

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

E nt e Ei i+ = +1 sin

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

Clohessy-Wiltshire (“CW”) Equations

x t nt xnt

nnt yo o o( ) cos( )

sin( )˙ cos( ) ˙= −[ ] + + −[ ]4 3

y t nt nt x yn

nt xnt nt

nyo o o o( ) sin( ) cos( ) ˙

sin( )˙= −[ ] + − −[ ] +

z t z ntz

o( ) cos( )˙

sin( )= +

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

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

03 orb mech - UMD

Documents