MARYLAND U N I V E R S I T Y O F. Orbital Mechanics. Principles of Space Systems Design

Transcription

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 ) 2003 David L. Akin - All rights reserved

2 Energy in Orbit Kinetic Energy K.E. = 1 2 mn 2 fi K.E. m Potential Energy P.E. = - mm P.E. fi r m Total Energy Const. = v2 2 - m r = - m 2a = v2 2 = - m r <--Vis-Viva Equation

3 Implications of Vis-Viva Circular orbit (r=a) v circular = m r Parabolic escape orbit (a tends to infinity) v escape = 2m r Relationship between circular and parabolic orbits v escape = 2v circular

4 Some Useful Constants Gravitation constant m = GM Earth: 398,604 km 3 /sec 2 Moon: 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

5 Classical Parameters of Elliptical Orbits

6 Basic Orbital Parameters Semi-latus rectum (or parameter) p = a(1 - e 2 ) Radial distance as function of orbital position r = Periapse and apoapse distances p 1+ ecosq r p = a(1 - e) r a = a(1 + e) Angular momentum r h = r v r h = mp

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

8 The Hohmann Transfer v 2 v apogee v 1 v perigee

9 First Maneuver Velocities Initial vehicle velocity v 1 = m r 1 Needed final velocity Delta-V v perigee = m 2r 2 r 1 r 1 + r 2 Dv 1 = m Ê 2r 2 ˆ Á -1 r 1 Ë r 1 + r 2

10 Second Maneuver Velocities Initial vehicle velocity Needed final velocity v apogee = m r 2 2r 1 r 1 + r 2 v 2 = m r 2 Delta-V Dv 2 = m Ê Á 1- r 2 Ë 2r 1 ˆ r 1 + r 2

11 Limitations on Launch Inclinations

12 Differences in Inclination

13 Choosing the Wrong Line of Apsides

14 Simple Plane Change Dv 2 v perigee v 1 v apogee v 2

15 Optimal Plane Change v perigee v 1 v apogee Dv 1 Dv 2 v 2

16 First Maneuver with Plane Change Di 1 Initial vehicle velocity v 1 = m r 1 Needed final velocity v p = m r 1 2r 2 r 1 + r 2 Delta-V Dv 1 = v v p 2-2v 1 v p cos(di 1 )

17 Second Maneuver with Plane Change Di 2 Initial vehicle velocity Needed final velocity v a = m r 2 2r 1 r 1 + r 2 v 2 = m r 2 Delta-V Dv 2 = v v a 2-2v 2 v a cos(di 2 )

18 Sample Plane Change Maneuver Delta V (km/sec) DV1 DV2 DVtot Initial Inclination Change (deg) Optimum initial plane change = 2.20

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 = 2p a3 Mean motion (average angular velocity) m n = m a 3 Time since pericenter passage M = nt = E - esin E ÂM=mean anomaly

24 Dealing with the Eccentric Anomaly Relationship to orbit r = a(1 - e cose) Relationship to true anomaly tan q 2 = Calculating M from time interval: iterate until it converges 1+ e 1- e tan E 2 E i+1 = nt + esin E i

25 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.

26 Example: Lunar Orbit Insertion v 2 is velocity of moon around Earth Moon overtakes spacecraft with velocity of (v 2 -v apogee ) 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 v 2 v apogee

27 Planetary Approach Analysis Spacecraft has v h hyperbolic excess velocity, which fixes total energy of approach orbit Vis-viva provides velocity of approach v = 2 v 2 h 2 + m r Choose transfer orbit such that approach is tangent to desired final orbit at periapse v m r = - m 2a = v h 2 2 Dv = 2 v 2 h 2 + m - r orbit m r orbit

28 DV Requirements for Lunar Missions Project Diana From: To: Low Earth Orbit Lunar Transfer Orbit Low Lunar Orbit Lunar Descent Orbit Low Earth Orbit km/sec Lunar Transfer Orbit km/sec km/sec Low Lunar Orbit km/sec km/sec Lunar Descent Orbit km/sec Lunar Landing km/sec km/sec Lunar Landing km/sec km/sec Space Systems Laboratory University of Maryland

29 Hill s Equations (Proximity Operations) x = 3n 2 x + 2n y + a dx y = -2n x + a dy z = -n 2 z + a dz Ref: J. E. Prussing and B. A. Conway, Oxford University Press, 1993

30 Clohessy-Wiltshire ( CW ) Equations x(t) = [ 4-3cos(nt) ]x o + sin(nt) n x o + 2 n y(t) = 6[ sin(nt) - nt]x o + y o - 2 n [ 1-cos(nt) ] x o + z(t) = z o cos(nt) + z o n sin(nt) z (t) = -z o nsin(nt) + z o sin(nt) [ 1- cos(nt) ] y o 4sin(nt) - 3nt n y o

31 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

32 R-Bar Approach Approach from along the radius vector ( Rbar ) 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

33 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