AERO0025 Satellite Engineering. Lecture 8. Satellite orbits

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1 AERO0025 Satellite Engineeing Lectue 8 Satellite obits 1

2 Can the Obit Affect Mass of the satellite? Powe geneation? Amount of data that can be tansfeed to the gound? Space adiation envionment? Revisit time of satellite to a point on Eath? Themal contol? Launch costs? YES! 2

3 Motivation Mission analysis Theefoe A cleve selection of the obit of the satellite (and its pecise knowledge) is key fo the success of a mission. Astodynamics (satellite obits) 3

4 Satellite Obits 1. Two-body poblem 4

5 1. Gavitational Foce The law of univesal gavitation is an empiical law descibing the gavitational attaction between bodies with mass. It was fist fomulated by Newton in Philosophiae Natualis Pincipia Mathematica (1687). He was able to elate objects falling on the Eath to the motion of the planets. Isaac Newton ( ) 5

6 1. Gavitational Foce Evey point mass attacts evey othe point mass by a foce pointing along the line intesecting both points. The foce is popotional to the poduct of the two masses and invesely popotional to the squae of the distance between the point masses: 6

7 1. In Vecto Fom R 1 m 1 ˆ u = = R R R R m 2 R 2 F m m = G 1 2 uˆ

8 1. Gavitational Paamete of a Body µ = GM The gavitational paamete of the Eath has been detemined with consideable pecision fom the analysis of lase distance measuements of atificial satellites: ± km 3.s -2. The uncetainty is 1 to 5e8, much smalle than the uncetainties in G and M sepaately (~1 to 1e4 each). 8

9 1. Bodies with Spatial Extent An object with a spheically-symmetic distibution of mass exets the same gavitational attaction on extenal bodies as if all the object's mass wee concentated at a point at its cente. = Point mass M Sphee of mass M 9

10 1. Definition of the 2-Body Poblem Motion of two bodies due solely to thei own mutual gavitational attaction. Also known as Keple poblem. Assumption: two point masses (o equivalently spheically symmetic objects).?? 10

11 1. Motion of the Cente of Mass m m R&& 1 2 ˆ R&& = Gm m + Gm m = u u 1 2 ˆ m 1 R 1 uˆ R 2 = = G R R m 2 R R m && && 1R1 + m2r 2 = 0 R G = m + R m + m R + m Inetial fame of efeence The c.o.m. of a 2-body R = R + v t system may seve as the G G0 G 11 oigin of an inetial fame.

12 1. Equations of Relative Motion Gm m R && ˆ 2 = u 2 m m Gm m R && ˆ 1 u 2 m m = + ( + m ) G m R&& R&& = u 1 2 ˆ m 1 R 1 uˆ R 2 = = G Inetial fame of efeence R R m 2 R R µ && = 3 µ is the gavitational paamete The motion of m 2 as seen fom m 1 is the same as the motion of m 1 as seen fom m 2. 12

13 1. Equations of Relative Motion && = µ 3 This is a nonlinea dynamical system. How to solve it? Find constants of the motion! How many? 13

14 1. Constant Angula Momentum && = µ 3 µ && = = 3 0 h = & Specific angula momentum d / dt dh dt = & & + && = && dh 0 constant= dt = & = h 14

15 1. The Motion Lies in a Fixed Plane & h ˆ = h h h ˆ = h h The fixed plane is the obit plane and is nomal to the angula momentum vecto. & & = constant= h 15

16 1. ISS in STK 16

17 1. STK: Satellite Tool Kit A pofessional softwae suite used in astonautic industies Obit popagation, mission analysis and design 17

18 1. STK at ULg The Space Stuctues and Systems Laboatoy has acquied 20 licenses of the pofessional edition. Thoough desciption and extensive use of STK in the Astodynamics class. 18

19 1. Fist Integal of Motion && = µ 3 h µ µ && h = h = & 3 3 ( ) d & & a ( b c) = b( a. c) c( a. b) = 2 dt µ & & d && h =.. = µ = µ 3 2 & & dt ( ) ( ) & h µ = constant= µ e e lies in the obit plane (e.h)=0: the line defined by e is the apse line. 19 Its nom, e, is the eccenticity.

20 1. Οbit Equation & ( ) h = + e µ ( ) = ( ) a. b c a b. c.. & h. = + µ ( ) ( ) 2. & h &. h h. h h = = = = + µ µ µ µ. e. e = 2 h 1 µ 1+ ecosθ Closed fom of the nonlinea equations of motion (θ is the tue anomaly) 20

21 1. Conic Section p = 1 + e cos θ e=1 e=0 0<e<1 e>1 21

22 1. Possible Motions in the 2-Body System ellipse cicle paabola hypebola 22

23 1. How Many Vaiables to Define An Obit? && = µ ODEs of second-ode Useful paametization of the obit? ISS catesian paametes on Mach 4, 2009, 12:30:00 UTC (Souce: Celestak) 23

24 1. Catesian Coodinates? & and do not diectly yield much infomation about the obit. We cannot even infe fom them what type of conic the obit epesents o what is the obit altitude! Anothe set of six vaiables, which is much moe desciptive of the obit, is needed. 24

25 1. Six Obital (Kepleian) Elements 1. e: shape of the obit 2. a: size of the obit 3. i: oients the obital plane with espect to the ecliptic plane 4. Ω: longitude of the intesection of the obital and ecliptic planes 5. ω: oients the semi-majo axis with espect to the ascending node 6. θ: oients the celestial body in space definition of the ellipse definition of the obital plane oientation of the ellipse within the obital plane position of the satellite on the ellipse 25

26 Obital plane oientation of the ellipse position of the satellite θ Equatoial plane

27 1. ISS in STK ISS Kepleian elements on Mach 4, 2009, 12:30:00 UTC (Souce: Celestak) 27

28 1. In Summay + We can calculate fo all values of the tue anomaly. + The obit equation is a mathematical statement of Keple s fist law. - We only know the elative motion (howeve, e.g., the motion of ou sun elative to othe pats of ou galaxy is of little impotance fo missions within ou sola system). - The solution of the simple poblem of two bodies cannot be expessed in a closed fom, explicit function of time. Do we have 6 independent constants? The two vecto constants h and e povide only 5 independent constants: h.e=0 28

29 Satellite Obits 1. Two-body poblem 2. Obit types 29

30 2.1 Cicula Obits (e=0) 2 h = = Constant h = v = vcicula µ v cic = µ µ 2π Tcic = 2π = µ 3/ 2 30

31 2.1 Digession: Angula Momentum v θ & h = & = uˆ ( v uˆ + v uˆ ) = v hˆ v 2 h v θ = = & The angula momentum depends only on the azimuth component of the elative velocity [End of digession] 31

32 2.1 Obital Speed Deceases with Altitude = G( m + M ) GM µ sat Obital speed (km/s) ISS HST SPOT Altitude (km) 32

33 2.1 Obital Peiod Inceases With Altitude SPOT-5 Obital peiod (min) ISS HST Altitude (km) 33

34 2.1 Hubble Gound Tack 34

35 2.1 Hubble in Inetial Space 35

36 2.1 Two Impotant Cases km/s is the fist cosmic velocity; i.e., the minimum velocity (theoetical velocity, =6378km) to obit the Eath km is the altitude of the geostationay obit. It is the obit at which the satellite angula velocity is equal to that of the Eath, ω=ω E = ad/s, in inetial space (*). GEO= T µ 2π cic * A sideeal day, 23h56m4s, is the time it takes the Eath to complete one otation elative to inetial space. A synodic day, 24h, is the time it takes the sun to appaently otate once aound the Eath. They would 36 be identical if the eath stood still in space. 2/3

37 2.1 Eath s Rotation Speed The mean sola day (24h) is not the time it takes the Eath to pefom one evolution in inetial space. 37

38 2.2 Elliptic Obits (0<e<1) = 2 h 1 µ 1+ ecosθ The elative position vecto emains bounded. θ=0, minimum sepaation, peiapse p = 2 h µ (1 + e) θ=π, geatest sepaation, apoapse a = 2 h µ (1 e) θ=π/2, semi-latus ectum p e = a a + p p 38

39 2.2 Geomety of the Elliptic Obit a apse line θ b p a p 39

40 2.2 Digession: Angula Momentum = 2 h 1 µ 1+ ecosθ 2 a(1 e ) = 1 + ecosθ Obit equation Pola equation of an ellipse (a, semimajo axis) h = µ a e 2 (1 ) [End of digession] 40

2.2 Keple s Second Law 1 1 1 da = & dt = h dt = hdt 2 2 2 ( t) ( t + dt) &dt da h 1 dθ dt 2 2 dt 2 = = = constant The line

41 2.2 Keple s Second Law da = & dt = h dt = hdt ( t) ( t + dt) &dt da h 1 dθ dt 2 2 dt 2 = = = constant The line fom the sun to a planet sweeps out equal aeas inside the ellipse in equal lengths of time. 1uuu uuu Aea = AB AC 2 eminde 41

42 2.2 Keple s Thid Law T enclosed aea = = da / dt 2π ab h h a e 2 2 = µ (1 ) b = a 1 e 3 a Tellip = 2π µ The elliptic obit peiod depends only on the semimajo axis and is independent of the eccentivity. T T a = a2 The squaes of the obital peiods of the planets ae popotional to the cubes of thei mean distances 42 fom the sun.

43 2.2 Vis-Viva Equation v ellip 2 1 = µ a 43

44 2.2 OUFTI-1 Example = = 6732 km = = 7825 km p a e a p a + p = = 0.075, a = = km + 2 a p 3 a T = 2π = s = 103min µ v 2 1 = µ a v p = v a = 7.98 km/s 6.86 km/s 44

45 2.2 OUFTI-1 in Inetial Space 45

46 2.2 OUFTI-1 Gound Tack 46

47 2.2 OUFTI-1 Velocity 8 Absolute velocity (km/s) Time (h) 47

48 2.2 OUFTI-1 Velocity 8 Absolute velocity (km/s) Tue anomaly θ (deg) 48

49 2.2 HEO in Inetial Space Peigee = 500 km Apogee = km 49

50 2.3 Paabolic Obits (e=1) = 2 h 1 µ 1+ cosθ θ π, v paab = 2µ The satellite will coast to infinity, aiving thee with zeo velocity elative to the cental body. 50

51 2.3 Escape Velocity, V esc 11.2 km/s is the second cosmic velocity; i.e., the minimum velocity (theoetical velocity, =6378km) to escape the gavitational attaction of the Eath. v cic = µ v paab = 2µ 11.2 km/s = km/s 51

52 2.4 Hypebolic Obits (e>1) = 2 h 1 µ 1+ ecosθ p 2a a v µ a v = v + v = C + v = esc 3 esc Hypebolic excess speed C 3 is a measue of the enegy fo an inteplanetay mission: 16.6 km 2 /s 2 (Cassini-Huygens) 8.9 km 2 /s 2 (Sola Obite, phase A) 52

53 2.4 Soyuz ST v2-1b (Kouou Launch) 53

54 2.4 Poton 54

55 What Do you Think? Assume we have a cicula o elliptic obit fo ou satellite. Will it stay thee??? 55

56 Satellite Obits 1. Two-body poblem 2. Obit types 3. Obit petubations 56

57 3.1 Non-Kepleian Motion In many pactical situations, a satellite expeiences significant petubations (acceleations). These petubations ae sufficient to cause pedictions of the position of the satellite based on a Kepleian appoach to be in significant eo in a bief time. 57

58 3.1 STK: Diffeent Popagatos 58

59 Diffeent Petubations? LEO? GEO? Satellite dependent! Montenbuck and Gill, Satellite obits, Spinge, 2000

60 3.1 Respective Impotance 400 kms 1000 kms kms Oblateness Dag Oblateness Sun and moon Oblateness Sun and moon SRP 60

61 3.2 The Eath is not a Sphee 61

62 3.2 Physical Intepetation The foce of gavity is no longe within the obital plane: non-plana motion will esult. The equatoial bulge exets a foce that pulls the satellite back to the equatoial plane and thus ties to align the obital plane with the equato. Due to its angula momentum, the obit behaves like a spinning top and eacts with a pecessional motion of the obital plane (the obital plane of the satellite otates in inetial space). 62

63 3.2 What Do You See? OUFTI-1 in Septembe 2009 OUFTI-1 in Mach

64 3.2 What Do You See? OUFTI-1: Two-body popagato OUFTI-1: J2 popagato 64

65 3.2 Secula Effects Nodal egession: egession of the nodal line. 2 1 T 3 µ J 2R Ω & avg = Ω & dt = / 2 T 2 (1 e ) a cosi Apsidal otation: otation of the apse line. & ω avg 2 1 T 3 µ J 2R = & ω dt = / 2 T 4 (1 e ) a ( 2 4 5sin i) No secula vaiations fo a, e, i. 65

66 3.3 The Eath Has an Atmosphee Atmospheic foces epesent the lagest nonconsevative petubations acting on low-altitude satellites. The dag is diectly opposite to the velocity of the satellite. The lift foce can be neglected in most cases. 66

67 3.3 Effects of Atmospheic Dag 67

68 3.4 Thid-Body Petubations Fo an Eath-obiting satellite, the Sun and the Moon should be modeled fo accuate pedictions. Thei effects become noticeable when the effects of dag begin to diminish. 68

69 3.4 Effects of Thid-Body Petubations The Sun s attaction tends to tun the satellite ing into the ecliptic. The obit pecesses about the pole of the ecliptic. Vallado, Fundamental of Astodynamics and Applications, Kluwe,

70 3.5 Sola Radiation Pessue Sola adiation (photons) Sola wind (paticles) It poduces a nonconsevative petubation on the spacecaft, which depends upon the distance fom the sun. It is usually vey difficult to detemine pecisely, but the effects ae usually small fo most satellites. 800km is egaded as a tansition altitude between dag and SRP. 70

71 3.6 In Summay (STK) 71

72 What Do you Think? We have an obit. Do we have a launch vehicle??? 72

73 Satellite Obits 1. Two-body poblem 2. Obit types 3. Obit petubations 4. Obit tansfe 73

74 4. Tsiolkovsky s Rocket Equation (1903) ( mv) = 0 m i v = vej ln = Ispg m f 0 m i ln m f 74

75 4. Single-Stage Rocket 100 v = v ln = v ln 5 = 1.61v ej ej ej [ ] 5 7 km/s 80%: fuel 10%: dy mass 10%: payload V ej = [3-4] km/s What to conclude? 75

76 4. Thee-Stage Rocket (Simila Stages) [ ] v = vi = 3vej ln km/s i The payload is 0.1% of the initial mass! 76

77 4. Specific Impulse 77

4. But Coquilhat (1811-1890, Belgian) Established the ocket equation in 1873!

78 4. But Coquilhat ( , Belgian) Established the ocket equation in 1873! Tajectoies des fusées volantes dans le vide, Mémoies de la Société Royale des Sciences de Liège. Recent discovey. 78

79 4. Motivation Without maneuves, satellites could not go beyond the close vicinity of Eath. Fo instance, a GEO spacecaft is usually placed on a tansfe obit (LEO o GTO). 79

80 4. Fom LEO to GEO: STK Video 80

81 4. Fom GTO to GEO: Aiane V Aiane V is able to place heavy GEO satellites in GTO: peigee: km and apogee: ~35786 km. GTO GEO 81

82 4. Delta-V: Ode of Magnitudes (1000,0.288) m/m v (m/s) Isp=300s 82

83 4. Delta-V Budget: GEO 83

84 4. How to Go to Satun? V V E J G A 84

85 4. Gavity Assist Also known as planetay flyby tajectoy, slingshot maneuve and swingby tajectoy. Useful in inteplanetay missions to obtain a velocity change without expending popellant. This fee velocity change is povided by the gavitational field of the flyby planet and can be used to lowe the delta-v cost of a mission. 85

86 4. Basic Pinciple Resultant V out SOI Resultant V in Planet s sun elative velocity v = v v, out, in 86

gavity assists looks like an elastic collision,

87 4. Basic Pinciple Inetial fame Fame attached to the tain Fame attached to the tain Inetial fame A gavity assists looks like an elastic collision, although thee is no physical contact with the planet. 87

88 4. Cassini: Swingby Effects V V E J S 88

89 Satellite Obits 1. Two-body poblem 2. Obit types 3. Obit petubations 5. Conclusions 4. Obit tansfe 89

90 Satellite Obits Gentle intoduction to satellite obits; moe details in the astodynamics couse. Closed-fom solution of the 2-body poblem fom which we deduced Keple s laws. Obit petubations cannot be ignoed fo accuate obit popagation and fo mission design. Obit tansfes ae commonly encounteed. Satellite must often have thei own populsion. 90

91 METEOSAT 6-7, HST, OUFTI-1, SPOT-5, MOLNIYA GEO LEO LEO (LEO) SSO Molniya i=0 i=28.5 i=71 i=98.7 i=63.4

92 STK Simulations METEOSAT 6-7, HST, OUFTI-1, SPOT-5, MOLNIYA GEO LEO LEO (LEO) SSO Molniya i=0 i=28.5 i=71 i=98.7 i=

93 AERO0025 Satellite Engineeing Lectue 8 Satellite obits 93

AERO0025 Satellite Engineering. Lecture 2. Satellite orbits

AERO0025 Satellite Engineering. Lecture 2. Satellite orbits AERO0025 Satellite Engineeing Lectue 2 Satellite obits 1 Can the Obit Affect Mass of the satellite? Powe geneation? Amount of data that can be tansfeed to the gound? Space adiation envionment? Revisit