Astrodynamics (AERO0024)

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1 Astrodynamics (AERO0024) L06: Interplanetary Trajectories Gaëtan Kerschen Space Structures & Systems Lab (S3L)

2 Motivation 2

3 Problem Statement? Hint #1: design the Earth-Mars transfer using known concepts Hint #2: division into simpler problems Hint #3: patched conic method 3

4 What Transfer Orbit? Constraints? Dep. ΔV 1 Arr. Arr. ΔV 2 Dep. Dep. Arr. Motion in the heliocentric 4 reference frame

5 Planetary Departure? Constraints? Apse line v Earth / Sun v? ΔV 1 β Motion in the planetary 5 reference frame

6 Planetary Arrival? Similar Reasoning Transfer ellipse SOI SOI Arrival hyperbola Departure hyperbola 6

7 Patched Conic Method Three conics to patch: 1. Outbound hyperbola (departure) 2. The Hohmann transfer ellipse (interplanetary travel) 3. Inbound hyperbola (arrival) 7

8 Patched Conic Method Approximate method that analyzes a mission as a sequence of 2-body problems, with one body always being the spacecraft. If the spacecraft is close enough to one celestial body, the gravitational forces due to other planets can be neglected. The region inside of which the approximation is valid is called the sphere of influence (SOI) of the celestial body. If the spacecraft is not inside the SOI of a planet, it is considered to be in orbit around the sun. 8

9 Interplanetary Trajectories Patched Conic Method Gravity Assist Concluding Remarks Lambert s Problem 9

10 Interplanetary Trajectories Patched Conic Method Gravity Assist Sphere of influence Concluding Remarks Lambert s Problem 10

11 Sphere of Influence (SOI) Let s assume that a spacecraft is within the Earth s SOI if the gravitational force due to Earth is larger than the gravitational force due to the sun. Gm m r > Gm m E sat S sat 2 2 E, sat rs, sat r < 5 Esat, km The moon lies outside the SOI and is in orbit about the sun like an asteroid! 11

12 Sphere of Influence (SOI) Third body: sun or planet Spacecraft m 2 d m j r ρ μ d && r+ r = Gm j + r d ρ ρ O m 1 Central body: sun or planet Disturbing function 12

13 If the Spacecraft Orbits the Planet ( m ) G mp + v r r sv sp && rpv + r 3 pv = Gms + 3 r 3 pv r sv rsp p:planet v: vehicle s:sun && r A = P pv p s Primary gravitational acceleration due to the planet Perturbation acceleration due to the sun 13

14 If the Spacecraft Orbits the Sun && r G( m + m ) + r = Gm r + s v pv sp sv 3 sv p 3 3 r sv rpv rsp r p:planet v: vehicle s:sun && r A = P sv s p Primary gravitational acceleration due to the sun Perturbation acceleration due to the planet 14

15 SOI: Correct Definition due to Laplace It is the surface along which the magnitudes of the acceleration satisfy: P A p s = P A s p Measure of the planet s influence on the orbit of the vehicle relative to the sun Measure of the deviation of the vehicle s orbit from the Keplerian orbit arising from the planet acting by itself r SOI m m p s 2 5 r sp 15

16 SOI: Correct Definition due to Laplace P A p s > P A s If the spacecraft is inside the SOI of the planet. p A A > p The previous (incorrect) definition was 1 s 16

17 SOI Radii Planet SOI Radius (km) SOI radius (body radii) Mercury 1.13x Venus 6.17x OK! Earth 9.24x Mars 5.74x Jupiter 4.83x Neptune 8.67x

18 Validity of the Patched Conic Method The Earth s SOI is 145 Earth radii. This is extremely large compared to the size of the Earth. This is extremely small with respect to 1AU. 18

19 Extremely Large The velocity relative to the planet on an escape hyperbola is considered to be the hyperbolic excess velocity vector. vsoi v When this velocity vector is added to the planet s heliocentric velocity, the result is the spacecraft s heliocentric velocity on the interplanetary elliptic transfer orbit at the SOI in the solar system. 19

20 Extremely Small During the elliptic transfer, the spacecraft is considered to be under the influence of the Sun s gravity only. In other words, the spacecraft follows an unperturbed Keplerian orbit around the Sun. 20

21 Interplanetary Trajectories Patched Conic Method Gravity Assist Planetary departure Concluding Remarks Lambert s Problem 21

22 Outbound Hyperbola The spacecraft necessarily escapes using a hyperbolic trajectory relative to the planet. Hyperbolic excess speed Lecture 02: v = v + v v + v esc SOI esc Is v SOI the velocity on the transfer orbit? 22

23 Magnitude of V SOI The velocity v D of the spacecraft relative to the sun is imposed by the Hohmann transfer (i.e., velocity on the transfer orbit). H. Curtis, Orbital Mechanics for Engineering Students, Elsevier. 23

24 Magnitude of V SOI By subtracting the known value of the velocity v 1 of the planet relative to the sun, one obtains the hyperbolic excess speed on the Earth escape hyperbola. = = μ 2R sun 2 vsoi vd v1 1 v ( ) R 1 R1+ R 2 Imposed Known Lecture05 24

25 Direction of V SOI What should be the direction of v SOI? For a Hohmann transfer, it should be parallel to v 1. 25

26 Parking Orbit A spacecraft is ordinary launched into an interplanetary trajectory from a circular parking orbit. Its radius equals the periapse radius r p of the departure hyperbola. H. Curtis, Orbital Mechanics for Engineering Students, Elsevier. 26

27 Delta-V Magnitude and Location a Lecture 02: known v = = 2 h μ e μ a r p = known μ h = 2 h μ(1 + e) e v 2 1 e = 1+ h= r v + p rv p μ 2 μ 2 2 rp Δ h v = v p v = μ c r 1 β = cos 1 p rp e 27

28 Planetary Departure: Graphically Departure to outer or inner planet? H. Curtis, Orbital Mechanics for Engineering Students, Elsevier. 28

29 Sensitivity Analysis The maneuver occurs well within the SOI, which is just a point on the scale of the solar system. One may therefore ask what effects small errors in position and velocity (r p and v p ) at the maneuver point have on the trajectory (target radius R 2 of the heliocentric Hohmann transfer ellipse). 29

30 Sensitivity Analysis 2μ1 v + R2 2 δr 1 p rp δv δ μ p = + 2 R2 Rv 1 D vdv rp rp vd vp 1 2 μ sun H. Curtis, Orbital Mechanics for Engineering Students, Elsevier. 30

31 Earth-Mars (300km Parking Orbit) μ R r sun = km, R = km p = km / s, μ = km / s = 6678 km v = km / s, v = km / s D δ R δ r 2 p δ v = R r v 2 p p p 31

32 Earth-Mars (300km Parking Orbit) A 0.01% variation in the burnout speed v p changes the target radius by 0.067% or km. A 0.01% variation in burnout radius r p (670 m!) produces an error over km. Midcourse maneuvers during the coasting flight along the elliptical transfer trajectory are mandatory. 32

33 Midcourse Maneuvers: Cassini-Huygens TCM-1: correct errors due to launch TCM-2: very small corrections before Venus encounter 33

34 Interplanetary Trajectories Patched Conic Method Gravity Assist Hohmann transfer Concluding Remarks Lambert s Problem 34

35 Planetary Orbits Lie Close to the Ecliptic Planet Inclination of the orbit to the ecliptic plane Sun --- Mercury 7.00º Venus 3.39º Earth 0.00º Mars 1.85º Jupiter 1.30º Saturn 2.48º Uranus 0.77º Neptune 1.77º Pluto 17.16º 35

36 Planetary Orbits Have Small Eccentricities Planet Inclination of the orbit to the ecliptic plane Eccentricity Sun Mercury 7.00º Venus 3.39º Earth 0.00º Mars 1.85º Jupiter 1.30º Saturn 2.48º Uranus 0.77º Neptune 1.77º Pluto 17.16º Assumption of circular, coplanar orbits is valid 36

37 Transfer to Outer Planet H. Curtis, Orbital Mechanics for Engineering Students, Elsevier. 37

38 Velocities v v D 2 μ 2R sun 2 v1 = R ( ) 1 R1+ R2 μ sun 2R1 va = 1 R R + R ( ) Signs v 2 -v A, v D -v 1 >0 for transfer to an outer planet v 2 -v A, v D -v 1 <0 for transfer to an inner planet 38

39 Transfer to Inner Planet H. Curtis, Orbital Mechanics for Engineering Students, Elsevier. 39

40 Timing is Crucial Phasing maneuvers are not practical due to the large periods of the heliocentric orbits. The planet should arrive at the apse line of the transfer ellipse at the same time the spacecraft does. Launch window. 40

41 Rendez-vous Opportunities θ = θ + θ = θ φ = θ θ 2 1 nt nt φ = φ + n n t ( ) φ π = φ + 2 ( n n ) Tsyn T syn = 2π n n 1 2 T syn = TT 1 2 T T 1 2 Synodic period 41

42 Transfer Time t 12 π 1 2 = μ sun R + R 2 3/2 φ = π nt

Earth-Mars Example T syn 365.26 687.99 = = 365.26 687.99 777.

43 Earth-Mars Example T syn = = days It takes 2.13 years for a given configuration of Mars relative to the Earth to occur again. t 12 φ 0 = = = s days 44 o The total time for a manned Mars mission is = days = 2.66 years 43

44 Interplanetary Trajectories Patched Conic Method Gravity Assist Planetary arrival Concluding Remarks Lambert s Problem 44

45 Arrival at an Outer Planet For an outer planet, the spacecraft s heliocentric approach velocity v A is smaller in magnitude than that of the planet v 2. v v v = 2 A 45

46 Arrival at an Outer Planet v 2 v Because and have opposite signs, the spacecraft crosses the forward portion of the SOI. 46

47 Enter into an Elliptic Orbit If the intent is to go into orbit around the planet, then Δ must be chosen so that the Δv burn at periapse will occur at the correct altitude above the planet. Δ= r 1+ p 2μ rv p 2 h μ(1 + e) 2 μ μ(1 + e) Δ v= vp, hyp vp, capture = = v + r r r r 2 p p p p 47

48 Planetary Flyby Otherwise, the specacraft will simply continue past periapse on a flyby trajectory exiting the SOI with the same relative speed v it entered but with the velocity vector rotated through the turn angle δ. rv p e = 1+ μ 2 δ = 2sin 1 1 e 48

49 Interplanetary Trajectories Patched Conic Method Gravity Assist Example Concluding Remarks Lambert s Problem 49

51 Earth-Jupiter Example: Hohmann Velocity when leaving Earth's SOI: v v v D μ 2R E sun 2 1 = = = R ( ) 1 R1+ R km/s Velocity relative to Jupiter at Jupiter's SOI: J μ sun 2R v2 va = v = = R ( ) 2 R1 R km/s Transfer time: years 51

52 Earth-Jupiter Example: Departure Velocity on a circular parking orbit (300km): v c μ R + h E = = E 7.726km/s 2 2μ Δ v= v km/s = 6.298km/s r p e 2 rv p = 1+ = μ 52

53 Earth-Jupiter Example: Arrival Final orbit is circular with radius=6r 2 2 μ μ(1 + e) Δ v= v + = = 7.77km/s r r p p J e=

54 Interplanetary Trajectories Patched Conic Method Gravity Assist Concluding Remarks Lambert s Problem 54

55 Definition Also known as planetary flyby trajectory and swingby trajectory. Useful in interplanetary missions to obtain a velocity change without expending propellant. This free velocity change is provided by the gravitational field of the flyby planet and can be used to lower the delta-v cost of a mission. 55

56 Basic Principle Spacecraft outbound velocity SOI V out = V in Spacecraft inbound velocity What did we gain? 56

57 Basic Principle Resultant V out SOI Resultant V in Planet s sun relative velocity Δv = v v, out, in 57

58 Basic Principle Inertial frame Frame attached to the train Frame attached to the train Inertial frame 58

59 Leading-Side or Trailing-Side Flyby? A leading-side flyby results in a decrease in the spacecraft s heliocentric speed. On the other hand, a trailing-side flyby increases that speed. 59

60 Leading-Side Planetary Flyby H. Curtis, Orbital Mechanics for Engineering Students, Elsevier. 60

61 Trailing-Side Planetary Flyby H. Curtis, Orbital Mechanics for Engineering Students, Elsevier. 61

63 VEEGA

64 VEEGA 64

65 Launch Window 65

66 V V E J G A 66

67 Swingby Effects V V E J S 67

68 Cassini Follow-On: Itinerary Comparison 68

69 Interplanetary Trajectories Patched Conic Method Gravity Assist Concluding Remarks Lambert s Problem 69

70 Non-Hohmann Transfers Send a spacecraft from planet 1 to planet 2 in a specified time t 12. Why (2 reasons)? Find the departure and arrival velocities. Solution: use Lambert s theorem. 70

71 Non-Hohmann Transfers 71

73 Porkchop plot

77 Mars Express 207 days 77

78 Interplanetary Trajectories Patched Conic Method Gravity Assist Concluding Remarks Lambert s Problem 78

79 Interplanetary Trajectories Patched conic method. Hohmann and non-hohmann transfers. Gravity assist. 79

80 More Complex Trajectories 80

81 Astrodynamics (AERO0024) L06: Interplanetary Trajectories Gaëtan Kerschen Space Structures & Systems Lab (S3L)

Astrodynamics (AERO0024)

Astrodynamics (AERO0024) 10. Interplanetary Trajectories Gaëtan Kerschen Space Structures & Systems Lab (S3L) Motivation 2 6. Interplanetary Trajectories 6.1 Patched conic method 6.2 Lambert s problem