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. = = <VisViva Equation r a
3 Implications of VisViva 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 Semilatus 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 DeltaV 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 DeltaV 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 DeltaV 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 DeltaV 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 ClohessyWiltshire ( 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 PrenticeHall, 1994 William E. Wiesel, Spaceflight Dynamics MacGrawHill, 1997 J. E. Prussing and B. A. Conway, Orbital Mechanics Oxford University Press, 1993
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