Flight and Orbital Mechanics

Transcription

1 Flight and Orbital Mechanics Lecture slides Challenge the future 1

2 Flight and Orbital Mechanics AE-104, lecture hours 1-4: Interplanetary flight Ron Noomen October 5, 01 AE104 Flight and Orbital Mechanics 1

3 Example: Galileo VEEGA trajectory [NASA, 010] Questions: what is the purpose of this mission? what propulsion technique(s) are used? why this Venus- Earth-Earth sequence?. AE104 Flight and Orbital Mechanics

4 Overview Solar System Hohmann transfer orbits Synodic period Launch, arrival dates Fast transfer orbits Round trip travel times Gravity Assists AE104 Flight and Orbital Mechanics 3

5 Learning goals The student should be able to: describe and explain the concept of an interplanetary transfer, including that of patched conics; compute the main parameters of a Hohmann transfer between arbitrary planets (including the required ΔV); compute the main parameters of a fast transfer between arbitrary planets (including the required ΔV); derive the equation for the synodic period of an arbitrary pair of planets, and compute its numerical value; derive the equations for launch and arrival epochs, for a Hohmann transfer between arbitrary planets; derive the equations for the length of the main mission phases of a round trip mission, using Hohmann transfers; and describe the mechanics of a Gravity Assist, and compute the changes in velocity and energy. Lecture material: these slides (incl. footnotes) AE104 Flight and Orbital Mechanics 4

6 Introduction The Solar System (not to scale): [Aerospace web, 010] AE104 Flight and Orbital Mechanics 5

7 Introduction (cnt d) planet mean distance [AU] eccentricity [-] inclination [ ] Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto * [Wertz, 009] AE104 Flight and Orbital Mechanics 6

8 Introduction (cnt d) Conclusions: scale of interplanetary travel >> scale of Earth-bound missions orbits of planets more-or-less circular (except Mercury and Pluto) orbits of planets more-or-less coplanar (except Pluto) -dimensional situation with circular orbits good 1 st -order model (asteroids????) AE104 Flight and Orbital Mechanics 7

9 Introduction (cnt d) [Virginia.edu, 010] AE104 Flight and Orbital Mechanics 8

10 Introduction (cnt d) how can we escape from Earth gravity? how can we travel to other planets/asteroids in the most efficient way? how can we reach beyond 10 AU? C 3 = V AE104 Flight and Orbital Mechanics 9

11 Basics Interaction between 3 bodies: definition Sphere of Influence: acc Sun,3rd acc Earth, main acc Earth,3rd acc Sun, main r SoI r3 rd M main M3rd 0.4 SoI Earth: ~930,000 km (0.006 AU; 0.6% distance Earth-Sun) at SoI Earth: acc 3rd /acc main = O(10-6 ) -body approach very good 1 st -order model AE104 Flight and Orbital Mechanics 10

12 Basics (cnt d) hyperbola (or: parabola) orbit types: circular orbit ellipse AE104 Flight and Orbital Mechanics 11

13 Basics (cnt d) main characteristics: symbol meaning ellipse hyperbola a semi-major axis > 0 < 0 e eccentricity < 1 > 1 E (specific) energy < 0 > 0 r(θ) radial distance a(1-e )/(1+e cos(θ)) r min r max minimum distance (pericenter) maximum distance (apocenter) a(1+e) a(1-e) V velocity [μ(/r 1/a)] AE104 Flight and Orbital Mechanics 1 V velocity [V esc + V ]

14 Basics (cnt d) satellite altitude [km] specific energy [km /s /kg] SpaceShipOne 100+ (culmination) ENVISAT LAGEOS 5, GEO 35, Moon 384, in parking orbit Hohmann orbit to Mars 185 (after 1 st ΔV) +4.3 AE104 Flight and Orbital Mechanics 13

15 Basics (cnt d) AE104 Flight and Orbital Mechanics 14

16 Hohmann transfer Hohmann transfer between orbits around Earth: coplanar orbits circular initial, target orbits impulsive shots transfer orbit touches tangentially minimum energy [Wertz, 009] AE104 Flight and Orbital Mechanics 15

17 Hohmann transfer (cnt d) Hohmann transfer between orbits around Earth (cnt d): 1 at ( r1 r ) 1 V1 Vper, T Vc,1 Earth( ) r a r 1 T 1 1 V V V ( ) r r a Earth c, apo, T Earth Earth T T transfer 1 T T a 3 T Earth AE104 Flight and Orbital Mechanics 16

18 Hohmann transfer (cnt d) transfers between planets: Type II transfer unsuccessful transfer Earth Mars Hohmann transfer Type I transfer AE104 Flight and Orbital Mechanics 17

19 Hohmann transfer (cnt d) Hohmann transfer between planets around Sun: coplanar orbits circular orbits departure and target planet impulsive shots transfer orbit touches tangentially minimum energy [Wertz, 009] AE104 Flight and Orbital Mechanics 18

20 Hohmann transfer (cnt d) Hohmann transfer between planets around Sun (cnt d): 1 at ( rdep rtar ) 1 V,1 Vper, T Vdep,1 Sun( ) r a r 1 V V V ( ) Sun, tar, apo, T Sun rtar rtar at Sun dep T dep T transfer 1 T T a 3 T Sun Question: identical to transfer around Earth? AE104 Flight and Orbital Mechanics 19

21 Hohmann transfer (cnt d) Hohmann transfer between planets around Sun (cnt d): Answer: NO!!! AE104 Flight and Orbital Mechanics 0

22 Hohmann transfer (cnt d) interplanetary trajectory: succession of 3 influence areas SoI dep. planet SoI target planet planetocentric part (1) planetocentric part (3) heliocentric part () AE104 Flight and Orbital Mechanics 1

23 Hohmann transfer (cnt d) heliocentric velocities: V dep, V tar (planets) V 1, V (satellite) V (relative) AE104 Flight and Orbital Mechanics

24 Hohmann transfer (cnt d) planetocentric scale: planetocentric satellite velocities: V c (parking orbit) ΔV (maneuver) V 0 (hyperbola) V (excess velocity) parking orbit Sphere of Influence AE104 Flight and Orbital Mechanics 3

25 Hohmann transfer (cnt d) Hohmann transfer between planets around Sun (cnt d): transfer starts in parking orbit around departure planet planetocentric until leaving SoI relative velocity when crossing SoI: V V achieved by maneuver ΔV in parking orbit similarly around target planet succession of 3 -body problems patched conics AE104 Flight and Orbital Mechanics 4

26 Hohmann transfer (cnt d) Essential difference between Hohmann transfer around Earth and around Sun: Earth missions: ΔV directly changes velocity from V circ to V per (or V apo ) of Hohmann transfer orbit interplanetary missions: ΔV changes velocity from V circ to value (larger than) V esc, which results in V Q: trips to the Moon? AE104 Flight and Orbital Mechanics 5

27 Hohmann transfer (cnt d) Main elements of computation interplanetary Hohmann transfer: compute semi-major axis transfer orbit compute V at departure and target planet compute pericenter velocity of planetocentric hyperbolae compute ΔV s AE104 Flight and Orbital Mechanics 6

28 Hohmann transfer (cnt d) Recipe (1-): step parameter expression example 1 V dep (heliocentric velocity of departure planet) V tar (heliocentric velocity of target planet) 3 V c0 (circular velocity around departure planet) 4 V c3 (circular velocity around target planet) V dep = (μ Sun /r dep ) V tar = (μ Sun /r tar ) V c0 = (μ dep /r 0 ) V c3 = (μ tar /r 3 ) km/s km/s km/s km/s 5 a tr (semi-major axis of transfer orbit) a tr = (r dep + r tar ) / km 6 e tr (eccentricity of transfer orbit) e tr = r tar - r dep / (r tar + r dep ) V 1 (heliocentric velocity at departure position) 8 V (heliocentric velocity at target position) Earth(185) Mars(500) V 1 = [μ Sun (/r dep -1/a tr )] V = [μ Sun (/r tar -1/a tr )] 3.79 km/s km/s AE104 Flight and Orbital Mechanics 7

29 Hohmann transfer (cnt d) Recipe (-): step parameter expression example 9 V,1 (excess velocity at departure planet) V,1 = V 1 - V dep.945 km/s 10 V, (excess velocity at target planet) V, = V - V tar.649 km/s 11 V 0 (velocity in pericenter of hyperbola around departure planet) 1 V 3 (velocity in pericenter of hyperbola around target planet) 13 ΔV 0 (maneuver in pericenter around departure planet) 14 ΔV 3 (maneuver in pericenter around target planet) V 0 = (μ dep /r 0 + V,1 ) V 3 = (μ tar /r 3 + V, ) ΔV 0 = V 0 V c0 ΔV 3 = V 3 V c km/s km/s km/s.070 km/s 15 ΔV tot (total velocity increase) ΔV tot = ΔV 0 + ΔV km/s AE104 Flight and Orbital Mechanics 8 16 T tr (transfer time) T tr = π (a tr3 /μ Sun ) yr

30 Hohmann transfer (cnt d) target Recipe (cnt d): ΔV dep [km/s] V,dep [km/s] C 3 [km /s ] ΔV tar [km/s] ΔV total [km/s] Mercury Venus Mars Jupiter Saturn Uranus Neptune Pluto What is possible? departure? arrival? total? AE104 Flight and Orbital Mechanics 9

31 Hohmann transfer (cnt d) Question 1: Consider a Hohmann transfer from Earth to Mercury. Begin and end of the transfer is in a parking orbit at 500 km altitude, for both cases. a) What are the semi-major axis and the eccentricity of the transfer orbit? b) What is the trip time? c) What are the excess velocities at Earth and at Mercury (i.e., heliocentric)? d) What are the circular velocities in the parking orbit around Earth and Mercury (i.e., planetocentric)? e) What are the ΔV s of the maneuvers to be executed at Earth and Mercury? What is the total ΔV? Data: μ Sun = km 3 /s ; μ Earth =398,600 km 3 /s ; μ Mercury =,034 km 3 /s ; R Earth =6378 km; R Mercury =440 km; distance Earth-Sun = 1 AU; distance Mercury- Sun = AU; 1 AU = km. Answers: see footnote below (BUT TRY YOURSELF FIRST!!) AE104 Flight and Orbital Mechanics 30

32 Hohmann transfer (cnt d) Question : Consider a Hohmann transfer from Mars to Jupiter. Begin and end of the transfer is in a parking orbit at 500 km and 50,000 km altitude, respectively. a) What are the semi-major axis and the eccentricity of the transfer orbit? b) What is the trip time? c) What are the excess velocities at Mars and at Jupiter (i.e., heliocentric)? d) What are the circular velocities in the parking orbit around Mars and Jupiter (i.e., planetocentric)? e) What are the ΔV s of the maneuvers to be executed at Mars and Jupiter? What is the total ΔV? Data: μ Sun = km 3 /s ; μ Mars =4,83 km 3 /s ; μ Jupiter = km 3 /s ; R Mars =3397 km; R Jupiter =71,49 km; distance Mars-Sun = 1.5 AU; distance Jupiter-Sun = 5.0 AU; 1 AU = km. Answers: see footnote below (BUT TRY YOURSELF FIRST!!) AE104 Flight and Orbital Mechanics 31

33 Timing Synodic period (1): planet θ 1 (t 1 ) planet 1 θ 1,0 Δθ(t 1 ) = θ (t 1 )-θ 1 (t 1 ) t = t 1 + T syn Δθ(t ) = θ (t )-θ 1 (t ) = Δθ(t 1 ) + π AE104 Flight and Orbital Mechanics 3

34 Timing (cnt d) Synodic period (): positions of planet 1and : 1( t) 1,0 n1 ( t t0 ) ( t),0 n ( t t0 ) difference: ( t) ( t) 1( t) (,0 1,0 ) ( n n1) ( t t0 ) geometry repeats after T syn : ( t) ( t1) ( n n1) ( t t1) ( n n1) Tsyn or 1 T syn 1 T 1 T1 Def: synodic period = time interval after which relative geometry repeats AE104 Flight and Orbital Mechanics 33

35 Timing (cnt d) Synodic period (3): AE104 Flight and Orbital Mechanics 34

36 Timing (cnt d) Near Earth Objects Synodic period (4): AE104 Flight and Orbital Mechanics 35

37 Timing (cnt d) Question 3: Consider a trip from Earth to Saturn. a) Compute the orbital period of the Earth around the Sun. b) Compute the orbital period of Saturn around the Sun. c) Derive a general equation for the synodic period, i.e., the period after which the relative constellation of two planets repeats. d) Compute the synodic period of the combination Earth-Saturn. e) Now consider a Near-Earth Object (NEO) with a semi-major axis of 1.05 AU (circular orbit). Compute the orbital period of this object. f) Compute the synodic period of the combination Earth-NEO. g) Discuss the physical reason for the difference between the answers for questions (d) and (f). Data: μ Sun = km 3 /s ; distance Earth-Sun = 1 AU; distance Saturn-Sun = 9.54 AU; 1 AU = km. Answers: see footnote below (BUT TRY YOURSELF FIRST!!) AE104 Flight and Orbital Mechanics 36

38 Timing (cnt d) Interplanetary Hohmann transfer (1): when do we leave? when do we arrive? travel time? planet planet 1 AE104 Flight and Orbital Mechanics 37

39 Timing (cnt d) Interplanetary Hohmann transfer (): a transfer time: 1 TH Torbit 3 a Example: Earth Mars: a = (1+1.5)/ = 1.6 AU T H = s = 58.3 days AE104 Flight and Orbital Mechanics 38

40 Timing (cnt d) Interplanetary Hohmann transfer (3): positions at epoch 1: ( t ) ( t ) n ( t t ) ( t ) ( t ) n ( t t ) ( t ) ( t ) ( t ) n ( t t ) sat positions at epoch : so ( t ) ( t ) n T ( t ) ( t ) n T 1 ( t ) ( t ) ( t ) ( t ) sat ( t ) 1 0 H H sat ( 1 0 ) ( 0 ) ( 1 0 ) n t t t n t t n T H planet 1 at t 1 planet 1 at t planet at t or t and t 1 0 t t T 1 ( t0) 1( t0) n TH n n H 1 planet at t 1 AE104 Flight and Orbital Mechanics 39

41 Timing (cnt d) Interplanetary Hohmann transfer (4): Example: Earth Mars Data: t 0 = January 1, 000, 1 h θ Earth (t 0 ) = ; θ Mars (t 0 ) = a Earth = 1 AU; a Mars = 1.5 AU μ Sun = km 3 /s Results: a H = 1.6 AU T H = 58 days = yr n Earth = /day; n Mars = /day t 1 = 456 days after t 0 April 1, 001 t = 715 days after t 0 Dec 16, 001 Other options: modulo T synodic!! AE104 Flight and Orbital Mechanics 40

42 Timing (cnt d) Question 4: Consider a Hohmann transfer from planet 1 to planet. a) Derive the following general equations for the epoch of departure t 1 and the epoch of arrival t : ( t0) 1( t0) n TH t1 t0 n n t t T 1 1 Here, t 0 is a common reference epoch, T H is the transfer time in a Hohmann orbit, n 1 and n are the mean motion of the two planets, and θ 1 and θ are the true anomalies of the planetary positions, respectively. Assume circular orbits for both planets. b) Consider a Hohmann transfer from Earth to Neptune. What is the transfer period? c) Assuming that on January 1, 010, θ Earth =70 and θ Neptune =10, what would be the epoch of departure (expressed in days w.r.t. this January 1)? d) What would be the arrival epoch? e) Can we change the launch window? If so, how? A qualitative answer is sufficient. Data: μ Sun = km 3 /s ; distance Earth-Sun = 1 AU; distance Neptune-Sun = 30.1 AU; 1 AU = km. Answers: see footnote below (BUT TRY YOURSELF FIRST!!) H AE104 Flight and Orbital Mechanics 41

43 Timing round-trip missions after [Wertz, 003] Δθ E Δθ sat AE104 Flight and Orbital Mechanics 4

44 Timing round-trip missions (cnt d) after [Wertz, 011] angle covered by Earth : E sat N (N integer; fastest for Mars: N 1; fastest for Venus : N 1) total trip time : T T H t stay t stay E T T E ( N 1) E E E (T M E M ( N 1) M T H T H t H stay ) sat N M t stay Hohmann N AE104 Flight and Orbital Mechanics 43

45 Timing round-trip missions (cnt d) Example for round-trip travel times: Earth Jupiter Earth Verify!! N [-] stay time [yrs] round trip time [yrs] AE104 Flight and Orbital Mechanics 44

46 Timing round-trip missions (cnt d) Some examples for round-trip travel times: target planet mean orbit angular motion Hohmann transfer stay time round trip time N [-] radius [AU] [rad/s] time [yrs] [yrs] [yrs] Venus Mars Jupiter Verify!! AE104 Flight and Orbital Mechanics 45

47 Timing round-trip missions (cnt d) AE104 Flight and Orbital Mechanics 46

48 Timing round-trip missions (cnt d) Question 6: Consider a round trip mission to Saturn. a) Determine the Hohmann transfer time for a trip to Saturn. b) Derive the relation for round trip travel time, for a mission to Saturn. c) Derive the relation for the stay time at Saturn. d) What would be the minimum stay time? Data: μ Sun = km 3 /s ; distance Earth-Sun = 1.0 AU; distance Saturn-Sun = AU; 1 AU = km. Answers: see footnotes below (BUT TRY YOURSELF FIRST!!) AE104 Flight and Orbital Mechanics 47

49 Timing round-trip missions (cnt d) Question 7: Consider a round trip mission to Mercury. a) Determine the Hohmann transfer time for a trip to Mercury. b) Derive the relation for round trip travel time, for a mission to Mercury. c) Derive the relation for the stay time at Mercury. d) What would be the minimum stay time? Data: μ Sun = km 3 /s ; distance Earth-Sun = 1.0 AU; distance Mercury-Sun = AU; 1 AU = km. Answers: see footnotes below (BUT TRY YOURSELF FIRST!!) AE104 Flight and Orbital Mechanics 48

50 Gravity assist C 3 = V Can we reach Saturn? Pluto? AE104 Flight and Orbital Mechanics 49

51 50 AE104 Flight and Orbital Mechanics Gravity assist (cnt d) V V V V e e a e e a r 1 lim ) cos( 1 0 cos 1 ) (1 r θ hyperbola:

52 Gravity assist (cnt d) V + [ ( lim )] lim ΔΨ/ π-θ lim V V r a a so a V always: at infinite distance, so V V r p so: a (1 e) e 1 rp a V - a( V ), e( V, r p ), lim ( V, r p ), ( V, r p ) AE104 Flight and Orbital Mechanics 51

53 Gravity assist (cnt d) bending: increases for heavier planets increases for smaller pericenter distances decreases with increasing excess velocity AE104 Flight and Orbital Mechanics 5

54 Gravity assist (cnt d) assumption (!!): hyperbola symmetric w.r.t. V planet V V + ΔΨ V - V 1 V 1 V!! V planet AE104 Flight and Orbital Mechanics 53

55 Gravity assist (cnt d) heliocentric V 1 V with cos( ) velocities satellite V planet V planet V Vplanet V cos( ) V Vplanet V cos( ) sin( ) and assumption (!!): hyperbola symmetric w.r.t. V planet (before,after encounter): cos( ) sin( ) energy gain : E V V 1 V planet V sin( ) AE104 Flight and Orbital Mechanics 54

56 Gravity assist (cnt d) assumption (!!): hyperbola symmetric w.r.t. V planet energy gain: not proportional to mass of planets increases with decreasing pericenter distances strong dependence on excess velocity AE104 Flight and Orbital Mechanics 55

57 Gravity assist (cnt d) Example: Consider a Gravity Assist along Mars (μ Sun = km 3 /s ; μ Mars =4,83 km 3 /s ; R Mars = 3397 km; distance Mars-Sun = 1.5 AU; 1 AU = km). Assume a relative velocity when entering the Sphere of Influence of 4 km/s, and a pericenter distance of 1.1 * R Mars. a=-μ Mars /(V ) a = km r p =a(1-e) e =.396 r = a(1-e )/(1+e*cos(θ)) θ lim = ΔΨ=*θ lim -π ΔΨ=49.34 Mars at 1.5 AU V Mars = km/s (heliocentric) velocity triangle before GA: V sat,1 =.780 km/s velocity triangle after GA: V sat, = 6.08 km/s assumption (!!): hyperbola symmetric w.r.t. V planet ΔE=V /-V 1 / ΔE = km /s AE104 Flight and Orbital Mechanics 56

58 Gravity assist (cnt d) Question 8: Consider a Gravity Assist along Jupiter. a) Assuming a relative velocity when entering the Sphere of Influence of 10 km/s, and a nearest passing distance w.r.t. the center of Jupiter of 00,000 km, compute the value of the semi-major axis and the eccentricity of this orbit (use the vis-viva equation V /-μ/r=-μ/(a), amongst others). b) Compute the heliocentric velocity of Jupiter. c) Derive the following relation, which indicates the maximum value for the direction of motion: cos(θ lim ) = -1/e. d) Derive the following relation for the bending angle around the central body: ΔΨ=*θ lim -π. e) Using the velocity diagrams, compute the heliocentric velocity of the satellite before and after the encounter, respectively. f) Compute the gain in energy caused by the Gravity Assist. assumption (!!): hyperbola symmetric w.r.t. V planet Data: μ Sun = km 3 /s ; μ Jupiter = km 3 /s ; distance Jupiter-Sun = 5.0 AU; 1 AU = km. Answers: see footnote below (BUT TRY YOURSELF FIRST!!) AE104 Flight and Orbital Mechanics 57

59 Gravity assist (cnt d) example 1: Voyager-1, - [NASA, 010] [NASA, 010] AE104 Flight and Orbital Mechanics 58

60 Gravity assist (cnt d) the Grand Tour AE104 Flight and Orbital Mechanics 59

61 Gravity assist (cnt d) Voyager-1 Voyager- launch September 5, 1977 August 0, 1977 Jupiter flyby March 5, 1979 July 9, 1979 Saturn flyby November 1, 1980 August 5, 1981 Uranus flyby January 4, 1986 Neptune flyby August 5, 1989 [ AE104 Flight and Orbital Mechanics 60

62 Gravity assist (cnt d) reconstructed reality velocity [km/s] date AE104 Flight and Orbital Mechanics 61

63 Gravity assist (cnt d) example : Ulysses [ESA, 010] [Uni-Bonn, 010] AE104 Flight and Orbital Mechanics 6

64 Gravity assist (cnt d) [Fortescue, Stark & Swinerd, 003] Solution: 3-dimensional Gravity Assist at Jupiter heliocentric velocity effectively in ecliptic (small component perpendicular to ecliptic) swing-by changes heliocentric inclination to 80. AE104 Flight and Orbital Mechanics 63

65 Gravity assist (cnt d) Some characteristics: reality reconstruction distance Earth-Sun [AU] 1.0 distance Jupiter-Sun [AU] heliocentric velocity Jupiter [km/s] heliocentric velocity Ulysses near Earth [km/s] 41. heliocentric velocity Ulysses near Jupiter [km/s]?? 16.4 excess velocity Ulysses w.r.t. Jupiter [km/s] required deflection angle [ ]?? 73.1 minimum distance Ulysses w.r.t. center Jupiter 6.3 R J 5. R J minimal distance Ulysses w.r.t. Sun [AU] distance Ulysses over poles Sun [AU] travel time to closest approach Sun [yr] AE104 Flight and Orbital Mechanics 64 time δ S > 70 [yr]

66 Miscellaneous Topics not treated here (1): 3D ephemerides : How can we model the elements in the real solar system? Consequences for mission design? Lambert targeting * : How can we obtain parameters of trip between arbitrary positions (and epochs) in space? Low-thrust propulsion * : How can we compute orbits with (semi)continuous low-thrust propulsion? Howe to optimize them? AE104 Flight and Orbital Mechanics 65

67 Miscellaneous (cnt d) Topics not treated here (): Aero Gravity Assist : Can we improve the efficiency of a planetary flyby by using the atmosphere of the flyby planet? Local geometry * : What are the geometrical conditions when departing from and arriving at a planet? Optimization * : How can we find the most attractive trajectory (ΔV, time-of-flight, geometry, ) AE104 Flight and Orbital Mechanics 66

68 Appendix: eqs. for Kepler orbits Appendix: elementary equations for Kepler orbits AE104 Flight and Orbital Mechanics 67

69 68 AE104 Flight and Orbital Mechanics ) (1 ; ) (1 ; cos 1 cos 1 ) 1 ( e a r e a r e p e e a r a p a r V E tot r V a r V a r V esc circ ; ; 1 3 a T Appendix: eqs. for Kepler orbits (cnt d)

70 Appendix: eqs. for Kepler orbits (cnt d) ellipse (0 e 1) : n a 3 1 e E tan tan 1 e M E esin E M n( t t0) M E i esin E i Ei1 Ei 1ecosE i r a(1 ecose) hyperbola (e 1) : n tan M M r V ( a) 3 n ( t t0 ) a (1 e coshf ) Vesc e e e sinh F 1 1 V F tanh F V r AE104 Flight and Orbital Mechanics 69