Astrodynamics (AERO0024)

Transcription

1 Astodynamics (AERO004). The Two-Body Poblem Gaëtan Keschen Space Stuctues & Systems Lab (S3L)

2 Y. The Two-Body Poblem.1 Justification of the -body model X. Gavitational field 3.3 Relative motion.4 Resulting obits

3 Y. The Two-Body Poblem.1 Justification of the -body model X 3 3

4 N-body poblem 4

5 Two-Body Poblem: Matlab Example Two identical masses: One is at est at the oigin of the inetial fame of efeence. The othe one has a velocity diected upwad to the ight making a 45 degees angle with the X axis. Y v 0 m 1 m X.1 Justification of the two-body model 5

6 Y x 10 5 Motion elative to the inetial fame X x 10 5 Seemingly complex motion in the inetial fame m1 m cm

7 Y Y x 10 5 Motion elative to the cente of mass x 10 5 Motion of m elative to m 1 1 m1 m X x X x 10 5 Much less complex motion when viewed fom the c.o.m Much less complex motion when viewed fom m 1

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9 Y.5 x 10 7 Motion of m elative to m The initial velocity is quadupled X x 10 7

10 Thee-Body Poblem: Matlab Example Thee identical masses: Two ae at est. The thid one has a velocity diected upwad to the ight making a 45 degees angle with the X axis. Y v 0 m 1 m m 3 X.1 Justification of the two-body model 10

11 Y x 10 6 Motion elative to the inetial fame m1 m m3 cm X x 10 6

12 Y x 10 5 Motion elative to the cente of mass m1 m m X What Do You Conclude? x 10 6

13 Y x 10 5 Motion elative to the inetial fame X x 10 5 Seemingly complex motion in the inetial fame m1 m cm

14 Displacement of the fist mass (x diection) Displacement of the fist mass (x diection) Why Is the 3-Body Poblem So Difficult? 16 x 105 Compaison of time seies Nominal IC 14 Petubed IC Chaotic by natue! body poblem (initial conditions petubed by 0.1%) 14 x 106 Compaison of time seies Time x body poblem (initial conditions petubed by 0.1%) Justification of the two-body model Nominal IC Petubed IC Time x

15 Y Y Why Is the 3-Body Poblem So Difficult? x Nominal IC m1 m m3 Chaotic by natue! X x x 10 5 IC petubed by 0.1% m1 m m Justification of the two-body model X x

16 Inteest in the Two-Body Poblem? Pecise obit popagation: Elaboate models ae necessay to compute the motion of satellites to the high level of accuacy equied fo many applications today (e.g., the GPS system). The - body poblem is not helpful in that context..1 Justification of the two-body model 16

17 Inteest in the Two-Body Poblem? Qualitative undestanding: The main featues of satellite and planet obits can be descibed by a easonably simple appoximation, the two-body poblem. Mission design: Some impotant quantities (ΔV and C 3 ) can be computed faily accuately using the two-body assumption. Inteplanetay tansfe: In lectue 6, we will use a sequence of -body poblems to appoximate a complex inteplanetay mission..1 Justification of the two-body model 17

18 An analytic obit popagato, based on a closed-fom solution, also pesents inteesting featues (e.g., computation time).

19 . The Two-Body Poblem. Gavitational field:..1 Newton s law of univesal gavitation.. The Eath..3 Gavity models and geoid 3 What is the highest point on Eath? 19

20 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 ( )..1 Newton s law of univesal gavitation 0

21 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:..1 Newton s law of univesal gavitation 1

22 In Vecto Fom F mm G 1 ˆ with ˆ Newton s law of univesal gavitation

23 Gavitational Constant By measuing the mutual attaction of two bodies of known mass, the gavitational constant G can diectly be detemined fom tosion balance expeiments. Due to the small size of the gavitational foce, G is pesently only known with limited accuacy and was fist detemined many yeas afte Newton s discovey: ( ± ) m 3.kg -1.s - GmM k L ( Newton s law of univesal gavitation 3

24 Gavitational Paamete of a Celestial Body 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 -. The uncetainty is 1 to 5e8, much smalle than the uncetainties in G and M sepaately (~1 to 1e4 each)...1 Newton s law of univesal gavitation 4

25 Satellite Lase Ranging TIGO (Concepcion, Chile) LAGEOS-1 Lases measue anges fom gound stations to satellite bone eto-eflectos. Because the events of sending and eceiving a pulse can be egisteed within a few picoseconds, the distance between the gound station and the satellite is detemined within a few millimetes...1 Newton s law of univesal gavitation 5

26 Acceleation of Gavity GM g eath, SL m/s g g g eath, aicaft eath, SL g g eath, ISS eath, SL % % We sense ou own weight by feeling contact foces acting on us in opposition to the foce of gavity: W=mg. If planetay gavity is the only foce acting on a body, then the body is said to be in fee fall. Thee ae, by definition, no contact foces, so thee can be no sense of weight. A peson in fee fall expeiences weightlessness: gavity is still thee, but he cannot feel it...1 Newton s law of univesal gavitation 6

27 Bodies with Spatial Extent Up to now, point masses wee consideed. But 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..1 Newton s law of univesal gavitation 7

28 Spheically Symmetic Mass Distibution ' M dv( ',, ) m V M Gm v dv dv ' sin v dv d d d ' R ' ' Rcos d d ' Rsin..1 Newton s law of univesal gavitation 8

29 Spheically Symmetic Mass Distibution R R d sind ' d ' 4 ' d ' M 0 0 R sind V Gm ' d ' R 1 R ' 0 R ' R' 4 Gm R 0 GMm ' d ' R 0 R 0 Gm d ' d ' OK!..1 Newton s law of univesal gavitation 9

30 What is the Highest Point on Eath? Mount Chimboazo (6310 m), located in Ecuado, may be consideed as the highest point on Eath. It is the spot on the suface fathest fom the Eath s cente km (Chimboazo) vs km (Eveest).. The Eath 30

31 The Eath is not a Sphee.. The Eath 31

32 1 st Ode Effect: Equatoial Bulge Because ou planet otates, the centifugal foce tends to pull mateial outwads aound the Equato whee the velocity of otation is at its highest: The Eath s adius is 1km geate at the Equato compaed to the poles. The foce of gavity is weake at the Equato (g=9.78 m/s ) than it is at the poles (g=9.83 m/s )... The Eath 3

33 nd Ode Effect: Mountains and Oceans Rathe than being smooth, the suface of the Eath is elatively lumpy : Thee is about a 0 km diffeence in height between the highest mountain and the deepest pat of the ocean floo... The Eath 33

34 3 d Ode Effect: Intenal Mass Distibution The diffeent mateials that make up the layes of the Eath s cust and mantle ae fa fom homogeneously distibuted: Fo instance, the cust beneath the oceans is a lot thinne and dense than the continental cust... The Eath 34

35 The Idealized Geometical Figue of the Eath Because of its elative simplicity, a flattened ellipsoid, called the efeence ellipsoid, is typically used as the idealized Eath: Ellipsoid of evolution. The size is epesented by the adius at the equato, a. The shape of the ellipsoid is given by the flattening, f, which indicates how much the ellipsoid depats fom spheical. f=(a-b)/a, whee b is the pola adius...3 Gavity models and geoid 35

36 Most Common Refeence Ellipsoid WGS84 (Wold Geodetic System 1984, evised in 004): Oigin at the cente of mass of Eath. a= km, b= km, f=0.335 %. Refeence system used by the GPS. Official document on the couse web site (inteesting to ead!). WGS84 fou defining paametes..3 Gavity models and geoid 36

37 WGS84 Coodinate System..3 Gavity models and geoid 37

38 Longitude Point coodinates such as latitude, longitude and elevation ae defined fom the efeence ellipsoid. The meidian of zeo longitude is the IERS Refeence Meidian, which lies 5.31 east of the Geenwich Meidian...3 Gavity models and geoid 38

39 GPS Receive at the Geenwich Meidian 5.31/3600= OK!..3 Gavity models and geoid 39

40 Latitude Also called geodetic latitude..3 Gavity models and geoid 40

41 The Tue Figue of the Eath The geoid is that equipotential suface which would coincide exactly with the mean ocean suface of the Eath, if the oceans wee in equilibium, at est, and extended though the continents: It is by definition a suface to which the foce of gavity is eveywhee pependicula. It is an iegula suface but consideably smoothe than Eath's physical suface. While the latte has excusions of almost 0 km, the total vaiation in the geoid is less than 00 m...3 Gavity models and geoid 41

42 The Tue Figue of the Eath..3 Gavity models and geoid 4

43 Gavitational Modeling Spheical hamonics ae used to model the Eath gavitational model: Gavitational potential function The cuent set is EGM008 (Eath Gavity Model 008). The model compises 4.6 million tems in the spheical expansion (ode and degee 159). Geoid with a esolution appoaching 10 km (5 x5 ). Moe details in Chapte 4 (Non-Kepleian motion)...3 Gavity models and geoid 43

44 EGM008 is Available in STK Gavity models and geoid 44

45 EGM008 Made Use of Gace Satellites GRACE employs micowave anging system to measue changes in the distance between two identical satellites as they cicle Eath. The anging system detects changes as small as 10 micons ove a distance of 0 km...3 Gavity models and geoid 45

46 Geoid Definition EGM008 contains no explicit infomation about which level suface, out of the infinitely many that may be geneated fom the potential coefficients, is "the" geoid. EGM008 model is theefoe used to compute geoid undulations with espect to WGS84 ellipsoid. The esult is efeed to as WGS84-EGM08 geoid. Geoid calculato fo EGM96: Gavity models and geoid 46

47 1: ocean : efeence ellipsoid 3: local plumb 4: continent 5: geoid

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49 GPS Receives You ae on a boat in the middle of the Atlantic ocean, what will be the height indicated by you GPS? You ae at sea level, but the height will be diffeent fom 0. The GPS satellites can only measue heights elative to the WGS84 model, which is an idealized figue of the Eath...3 Gavity models and geoid 49

50 GPS Receives Some GPS eceives can obtain the geoid height ove the WGS ellipsoid fom the cuent position. They ae then able to coect the height above WGS ellipsoid to the height above the geoid. You ae on a boat in the middle of the Atlantic ocean, will the height indicated by this GPS be equal to zeo? Not necessaily, because thee ae tides..3 Gavity models and geoid 50

51 Residual Sea Suface Slopes (EGM-96)..3 Gavity models and geoid 51

52 Residual Sea Suface Slopes (EGM-008)..3 Gavity models and geoid 5

53 Futue Impovements: GOCE, 009 (EGM96: ~0.5 m) (1mGal = 10-5 m/s )..3 Gavity models and geoid 53

54 Why So Many Effots??? 1. GPS and an advanced map of the geoid can eplace time-consuming leveling pocedues.. Physics of the Eath s inteio (gavity is diectly linked to the distibution of mass). 3. Undestanding of ocean ciculation, which plays a key ole in enegy exchanges aound the globe. 4. Computation of the motion of satellites to the level of accuacy equied today...3 Gavity models and geoid 54

55 Futhe Reading on the Couse Web Site..3 Gavity models and geoid 55

56 Digession: Geneal Relativity Einstein's theoy is the cuent desciption of gavity in moden physics. This couse will not cove the theoy of geneal elativity, but Newton's law is still an excellent appoximation of the effects of gavity if: GM v 1, and 1 c c c..3 Gavity models and geoid 56

57 Geneal Relativity: Eath-Sun Example 8 v obit 8 GM sun ~ 10, and ~ 10 c obit c c 1 yea. c OK! G= m 3.kg -1.s - obit = m (1 AU) M sun = kg c=3e8 m.s Gavity models and geoid 57

58 The Quest of a Unifying Theoy What is the elationship between the gavitational foce and othe known fundamental foces? That one body may act upon anothe at a distance though a vacuum without the mediation of anything else, by and though which thei action and foce may be conveyed fom one anothe, is to me so geat an absudity that, I believe, no man who has in philosophic mattes a competent faculty of thinking could eve fall into it. (Newton, 169) The question is not yet fully esolved today!..3 Gavity models and geoid 58

59 The Quest of a Unifying Theoy [End of digession]..3 Gavity models and geoid 59

60 10-min. beak

61 . The Two-Body Poblem 3.3 Relative motion:.3.1 Equations of motion.3. Closed-fom solution 61

62 Definition of the -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).??.3.1 Equations of motion 6

63 Motion of the Cente of Mass m m R 1 ˆ 1 1 R Gm m + Gm m u u 1 ˆ m 1 R 1 uˆ R G R R m R R 1 1 m 1R1 mr 0 R G m + R m m R m Inetial fame of efeence The c.o.m. of a -body R R v t system may seve as the G G0 G oigin of an inetial fame..3.1 Equations of motion 63

64 Equations of Relative Motion mm Gm m 1 1 R ˆ u mm Gm m 1 1 R ˆ 1 u + G m m R R u 1 ˆ 1 m 1 R 1 uˆ R G Inetial fame of efeence R R m R R μ is the gavitational paamete The motion of m as seen fom m 1 is the same as the motion of m 1 as seen fom m..3.1 Equations of motion 64

65 Equations of Relative Motion 3 This is a nonlinea dynamical system. How to solve it? Find constants of the motion! How many?.3. Closed-fom solution 65

66 Constant Angula Momentum h Specific angula momentum d / dt dh dt dh 0 constant= dt h.3. Closed-fom solution 66

67 The Motion Lies in a Fixed Plane hˆ h h hˆ h h The fixed plane is the obit plane and is nomal to the angula momentum vecto. constant= h.3. Closed-fom solution 67

68 Azimuth Component of the Velocity v h uˆ ( v uˆ v uˆ ) v hˆ v h v The angula momentum depends only on the azimuth component of the elative velocity.3. Closed-fom solution 68

69 Fist Integal of Motion h h h d ab c b a. c ca. b dt d h.. 3 dt h constant= e e lies in the obit plane (e.h)=0: the line defined by e is the apse line. Its nom, e, is the eccenticity..3. Closed-fom solution 69

70 Οbit Equation h e.. h. e. a. b c a b. c. h. h hh. h e. h 1 1 ecos Closed fom of the nonlinea equations of motion.3. Closed-fom solution 70

71 Conic Section in Pola Coodinates Constant: angula momentum Semi-latus ectum h 1 p 1ecos 1ecos Constant: gavitational paamete Independent vaiable: tue anomaly (=0 at the peiapsis) Constant: eccenticity.3. Closed-fom solution 71

72 Conic Section e=1 e=0 0<e<1 e>1.3. Closed-fom solution 7

73 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=0.3. Closed-fom solution 73

74 . The Two-Body Poblem 3.4 Resulting obits:.4.1 Cicula obits.4. Elliptic obits.4.3 Paabolic obits.4.4 Hypebolic obits 74

75 Thee Objectives 1. Peiod. Velocity 3. Enegy 75

76 Digession: Enegy of the Obit T mv V m The gavitational foce is consevative Specific enegy v constant 76

77 Enegy of the Obit at Peiapsis vp v h p p p p p p h 1 h 1ecos (1 e) 0 1 h 1 e [End of digession] 77

78 Possible Motions in the -Body System ellipse cicle paabola hypebola 78

79 Cicula Obits (e=0) h Constant h v vcicula v cic Tcic 3/ cic Cicula obits 79

80 Obital speed (km/s) Obital Speed G( msat M ) GM ISS 7.6 HST 7.5 SPOT Altitude (km).4.1 Cicula obits 80

81 Obital peiod (min) Obital Peiod SPOT HST 95 ISS Altitude (km).4.1 Cicula obits 81

82 Hubble Space Telescope.4.1 Cicula obits 8

83 Hubble Space Telescope.4.1 Cicula obits 83

84 Two Paticula Cases km/s is the fist cosmic velocity; i.e., the minimum velocity (theoetical velocity, = 6378 km) 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 cic * A sideeal day, 3h56m4s, is the time it takes the Eath to complete one otation elative to inetial space. A synodic day, 4h, is the time it takes the sun to appaently otate once aound the eath. They would be identical if the eath stood still in space. 84 /3

85 Elliptic Obits (0<e<1) h 1 1 ecos The elative position vecto emains bounded. θ=0, minimum sepaation, peiapse p h (1 e) θ=π, geatest sepaation, apoapse a h (1 e) θ=π/, semi-latus ectum p e a a p p.4. Elliptic obits 85

86 Geomety of the Elliptic Obit a apse line θ b p a p.4. Elliptic obits 86

87 Angula Momentum and Enegy h 1 1 ecos a(1 e ) 1 ecos Obit equation Pola equation of an ellipse (a, semimajo axis) h a e (1 ) 1 h 1 e ellip a 0 Independent of eccenticity!.4. Elliptic obits 87

88 Vis-Viva Equation v a v ellip 1 a.4. Elliptic obits 88

89 Keple s Second Law da dt h dt hdt () t ( t dt) dt da h 1 d dt dt constant The line fom the sun to a planet sweeps out equal aeas inside the ellipse in equal lengths of time. eminde 1 Aea ABAC.4. Elliptic obits 89

90 Keple s Thid Law enclosed aea T da/ dt ab h h a e (1 ) b a 1e 3 a Tellip The elliptic obit peiod depends only on the semimajo axis and is independent of the eccentivity. T T a a The squaes of the obital peiods of the planets ae popotional to the cubes of thei mean distances fom the sun..4. Elliptic obits 90

91 STK Simulation The elliptic obit peiod depends only on the semimajo axis and is independent of the eccentivity. 91

92 OUFTI km km p a e a p a p 0.075, a 778.5km a p 3 a T s 103min v 1 a vp va 7.98km/s 6.86 km/s.4. Elliptic obits 9

93 OUFTI-1.4. Elliptic obits 93

94 OUFTI-1.4. Elliptic obits 94

95 GTO and GEO Fo an obit with a peigee at 30 km and an apogee at km, what is the velocity incement equied to each the geostationay obit? v 1 a.4. Elliptic obits 95

96 GTO and GEO Fo an obit with a peigee at 30 km and an apogee at km, what is the velocity incement equied to each the geostationay obit? a vp va a GTO p 10.13km/s 1.61km/s 4430 km vcic Answe: 1.46 km/s (apogee moto) GEO km/s.4. Elliptic obits 96

97 m/m GTO and GEO (1000,0.88) (1460,0.391) v (m/s) Isp=300s 97

98 Paabolic Obits (e=1) h 1 1 cos, 1 paab h 1 e 0 The satellite has just enough enegy to escape fom the attacting body. v v paab The satellite will coast to infinity, aiving thee with zeo velocity elative to the cental body..4.3 Paabolic obits 98

99 Escape Velocity, V esc 11. km/s is the second cosmic velocity; i.e., the minimum velocity (theoetical velocity, = 6378km) to obit the Eath. v cic v paab 11. km/s 7.9 km/s.4.3 Paabolic obits 99

100 Hypebolic Obits (e>1) h 1 1 ecos p a a a p h (1 e) h 0 (1 e) a a p a h e 1 h e hype a Hypebolic obits 100

101 C 3 Velocity v v a a Hypebolic excess speed v v v v vesc C3 v esc C 3 is a measue of the enegy fo an inteplanetay mission: 16.6 km /s (Cassini-Huygens) 8.9 km /s (Sola Obite, phase A).4.4 Hypebolic obits 101

102 Soyuz ST v-1b (Kouou Launch).4.4 Hypebolic obits 10

103 Delta II, Delta III and Atlas IIIA.4.4 Hypebolic obits 103

104 Poton.4.4 Hypebolic obits 104

105 Falcon Hypebolic obits 105

106 The Two-Body Poblem.1 JUSTIFICATION OF THE -BODY MODEL. GRAVITATIONAL FIELD..1 Newton s law of univesal gavitation.. The Eath..3 Gavity models and geoid.3 RELATIVE MOTION.3.1 Equations of motion.3. Closed-fom solution.4 RESULTING ORBITS.4.1 Cicula obits.4. Elliptic obits.4.3 Paabolic obits.4.4. Hypebolic obits 106

107 Newton s laws F ma F g mm G 1 uˆ Relative motion 3 Enegy consev. v Angula mom. h Azim. velocity Keple s 1 st law h 1 1 ecos The obit equation v h / Keple s nd law da/ dt h/ Keple s 3 d law T a /

108 Concluding Remaks Closed-fom solution fom which we deduced Keple s laws. Analytic fomulas fo obital enegy, velocity and peiod. Two-body popagato available in STK. Often used in ealy studies to pefom tending analysis. But We have lost tack of the time vaiable! 108

109 Did you Know? Compactness of the sola system: measued by the atio of the distance a of a planet fom the Sun to the adius R of the Sun. a R 00 Compactness of the hydogen atom: measued by the atio of the distance a of an electon fom the nucleus to the adius R of the nucleus. a R 54 e 109

110 Astodynamics (AERO004). The Two-Body Poblem Gaëtan Keschen Space Stuctues & Systems Lab (S3L)