Orbital Mechanics. Orbital Mechanics. Principles of Space Systems Design David L. Akin - All rights reserved

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1 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 ) 001 David L. Akin - All rights reserved

2 Energy in Orbit Kinetic Energy KE.. Potential Energy PE.. Total Energy 1 KE.. v = mν = m mµ PE.. µ = = r m r v µ µ Const. = = <--Vis-Viva Equation r a

3 Implications of Vis-Viva Circular orbit (r=a) v = µ circular r Parabolic escape orbit (a tends to infinity) vescape = µ r Relationship between circular and parabolic orbits v escape = v circular

4 Some Useful Constants Gravitation constant µ = GM Earth: 398,604 km 3 /sec Moon: km 3 /sec Mars: 4,970 km 3 /sec Sun: 1.37x10 11 km 3 /sec Planetary radii r Earth = 6378 km r Moon = 1738 km r Mars = 3393 km

5 Classical Parameters of Elliptical Orbits

6 Basic Orbital Parameters Semi-latus rectum (or parameter) p= a( 1 e ) Radial distance as function of orbital position Periapse and apoapse distances Angular momentum r p = 1+ ecosθ r = p a ( 1 e ) r = a( 1+ e) a r r h = r v h= µ p

7 The Classical Orbital Elements Ref: J. E. Prussing and B. A. Conway, Oxford University Press, 1993

8 The Hohmann Transfer v v apogee v 1 v perigee

9 First Maneuver Velocities Initial vehicle velocity Needed final velocity Delta-V v 1 = v1 = µ r1 v = µ r perigee r r + r µ r 1 r r + r

10 Second Maneuver Velocities Initial vehicle velocity Needed final velocity Delta-V v v = µ r1 apogee r r + r v µ = 1 r = µ r r1 r + r 1 1

11 Limitations on Launch Inclinations

12 Differences in Inclination

13 Choosing the Wrong Line of Apsides

14 Simple Plane Change v v perigee v 1 v apogee v

15 Optimal Plane Change v perigee v 1 v apogee v 1 v v

16 First Maneuver with Plane Change i 1 Initial vehicle velocity Needed final velocity v1 = µ r1 v = µ r p r r + r 1 1 Delta-V v = v + v vv cos( i ) 1 1 p 1 p 1

17 Second Maneuver with Plane Change i Initial vehicle velocity Needed final velocity v = µ r1 a r r + r v = µ r 1 Delta-V v = v + v v v cos( i ) a a

18 Sample Plane Change Maneuver Delta V (km/sec) DV1 DV DVtot Initial Inclination Change (deg) Optimum initial plane change =.0

19 Bielliptic Transfer

20 Coplanar Transfer Velocity Requirements Ref: J. E. Prussing and B. A. Conway, Oxford University Press, 1993

21 Noncoplanar Bielliptic Transfers

22 Calculating Time in Orbit

23 Time in Orbit Period of an orbit P 3 a = π µ Mean motion (average angular velocity) n = Time since pericenter passage M=mean anomaly µ 3 a M = nt = E esin E

24 Dealing with the Eccentric Anomaly Relationship to orbit r = a( 1 ecos E) Relationship to true anomaly θ tan = Calculating M from time interval: iterate until it converges 1+ e E tan 1 e E = nt + i+ 1 esin E i

25 Hill s Equations (Proximity Operations) x = 3n x+ ny + a dx y = nx + a dy ż= n z+ a dz Ref: J. E. Prussing and B. A. Conway, Oxford University Press, 1993

26 Clohessy-Wiltshire ( CW ) Equations x( t) = 4 3cos( nt) x sin( nt) x [ 1 cos( nt) ] y n n [ ] + + o o o 4sin( nt) 3nt y( t) = 6[ sin( nt) nt] x + y [ 1 cos( nt) ] x + n n zt ( ) = zcos( nt) + o o o o o z o sin( nt ) n z ( t) = z nsin( nt) + z sin( nt) o o y

27 References for Lecture 3 Wernher von Braun, The Mars Project University of Illinois Press, 196 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