AERO0025 Satellite Engineering. Lecture 6. Satellite orbits

Transcription

1 AERO0025 Satellite Engineering Lecture 6 Satellite orbits 1

2 Can the Orbit Affect Mass of the satellite? Power generation? Amount of data that can be transferred to the ground? Space radiation environment? Revisit time of satellite to a point on Earth? Thermal control? Launch costs? YES! 2

3 Motivation Mission analysis Therefore A clever selection of the orbit of the satellite (and its precise knowledge) is key for the success of a mission. Astrodynamics (satellite orbits) 3

4 Satellite Orbits 1. Two-body problem 4

5 1. Gravitational Force The law of universal gravitation is an empirical law describing the gravitational attraction between bodies with mass. It was first formulated by Newton in Philosophiae Naturalis Principia Mathematica (1687). He was able to relate objects falling on the Earth to the motion of the planets. Isaac Newton ( ) 5

6 1. Gravitational Force Every point mass attracts every other point mass by a force pointing along the line intersecting both points. The force is proportional to the product of the two masses and inversely proportional to the square of the distance between the point masses: 6

7 1. In Vector Form R 1 m 1 ˆ r u r = = r R R R R m 2 R 2 F mm = G 1 2 uˆ 12 2 r r 7

8 1. Gravitational Parameter of a Body μ = GM The gravitational parameter of the Earth has been determined with considerable precision from the analysis of laser distance measurements of artificial satellites: ± km 3.s -2. The uncertainty is 1 to 5e8, much smaller than the uncertainties in G and M separately (~1 to 1e4 each). 8

9 1. Bodies with Spatial Extent An object with a spherically-symmetric distribution of mass exerts the same gravitational attraction on external bodies as if all the object's mass were concentrated at a point at its centre. = Point mass M Sphere of mass M 9

10 1. Definition of the 2-Body Problem Motion of two bodies due solely to their own mutual gravitational attraction. Also known as Kepler problem. Assumption: two point masses (or equivalently spherically symmetric objects).?? 10

11 1. Motion of the Center of Mass m m R 1 2 ˆ r R = Gm m r + Gm m = r u u 1 2 ˆ r m 1 R 1 uˆ r R 2 r = = r G R R m 2 R R m 1R1+ m2r2 = 0 R G = m + R m + m + m R Inertial frame of reference The c.o.m. of a 2-body R = R + v t system may serve as the G G0 G 11 origin of an inertial frame.

12 1. Equations of Relative Motion Gm m r R ˆ 2 = u 2 r mm Gm m r R ˆ 1 u 2 r mm = + ( + m ) G m R R = u r 1 2 ˆ r m 1 R 1 uˆ r R 2 r = = r G Inertial frame of reference R R m 2 R R μ r = r 3 r μ is the gravitational parameter The motion of m 2 as seen from m 1 is the same as the motion of m 1 as seen from m 2. 12

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

14 1. Constant Angular Momentum r = μ r 3 r r μ r r = r r = r 3 0 h= r r Specific angular momentum d / dt dh dt = r r + r r = r r dh 0 constant= dt = r r = h 14

15 1. The Motion Lies in a Fixed Plane r h ˆ = h h r r h ˆ = h h The fixed plane is the orbit plane and is normal to the angular momentum vector. r r r = constant= h 15

16 1. ISS in STK 16

17 1. STK: Satellite Tool Kit A professional software suite used in astronautic industries Orbit propagation, mission analysis and design 17

18 1. STK at ULg The Space Structures and Systems Laboratory has acquired 20 licenses of the professional edition. Thorough description and extensive use of STK in the Astrodynamics class. 18

19 1. First Integral of Motion r = μ r 3 r h μ μ r h= r h= r r r 3 3 r r ( ) d r rr rr a ( b c) = b( ac. ) c( ab. ) = 2 dt r r μ r rr d r r h= (. ) (. ) = μ = μ 3 2 r r rr r rr r r dt r r r h μ = constant= μe r e lies in the orbit plane (e.h)=0: the line defined by e is the apse line. 19 Its norm, e, is the eccentricity.

20 1. Οrbit Equation ( ) r h r = + e μ r ( ) = ( ) a. b c a b. c r. r. r h rr. = + μ r ( ) ( ) 2 r. r h r r. h hh. h = = = = r + μ μ μ μ re. re. r = 2 h 1 μ 1+ ecosθ Closed form of the nonlinear equations of motion (θ is the true anomaly) 20

21 1. Conic Section r 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 circle parabola hyperbola 22

23 1. How Many Variables to Define An Orbit? r = μ r 3 r 6 3 ODEs of second-order Useful parametrization of the orbit? ISS cartesian parameters on March 4, 2009, 12:30:00 UTC (Source: Celestrak) 23

24 1. Cartesian Coordinates? r r and do not directly yield much information about the orbit. We cannot even infer from them what type of conic the orbit represents or what is the orbit altitude! Another set of six variables, which is much more descriptive of the orbit, is needed. 24

25 1. Six Orbital (Keplerian) Elements 1. e: shape of the orbit 2. a: size of the orbit 3. i: orients the orbital plane with respect to the ecliptic plane 4. Ω: longitude of the intersection of the orbital and ecliptic planes 5. ω: orients the semi-major axis with respect to the ascending node 6. θ: orients the celestial body in space definition of the ellipse definition of the orbital plane orientation of the ellipse within the orbital plane position of the satellite on the ellipse 25

26 Orbital plane orientation of the ellipse position of the satellite θ Equatorial plane

27 1. ISS in STK ISS Keplerian elements on March 4, 2009, 12:30:00 UTC (Source: Celestrak) 27

28 1. In Summary + We can calculate r for all values of the true anomaly. + The orbit equation is a mathematical statement of Kepler s first law. - We only know the relative motion (however, e.g., the motion of our sun relative to other parts of our galaxy is of little importance for missions within our solar system). - The solution of the simple problem of two bodies cannot be expressed in a closed form, explicit function of time. Do we have 6 independent constants? The two vector constants h and e provide only 5 independent constants: h.e=0 28

29 Satellite Orbits 1. Two-body problem 2. Orbit types 29

30 2.1 Circular Orbits (e=0) r 2 h = = Constant h= rv = rvcircular μ v circ = μ r μ 2π Tcirc = 2π r = r r μ 3/2 30

31 2.1 Digression: Angular Momentum v θ r r h= r r = ruˆ ( v uˆ + v uˆ ) = rv hˆ r r r v r 2 h rv r θ = = The angular momentum depends only on the azimuth component of the relative velocity [End of digression] 31

32 2.1 Orbital Speed Decreases with Altitude μ = Gm ( sat + M ) GM Orbital speed (km/s) ISS HST SPOT Altitude (km) 32

33 2.1 Orbital Period Increases With Altitude SPOT-5 Orbital period (min) ISS HST Altitude (km) 33

34 2.1 Hubble Ground Track 34

35 2.1 Hubble in Inertial Space 35

36 2.1 Two Important Cases km/s is the first cosmic velocity; i.e., the minimum velocity (theoretical velocity, r=6378km) to orbit the Earth km is the altitude of the geostationary orbit. It is the orbit at which the satellite angular velocity is equal to that of the Earth, ω=ω E = rad/s, in inertial space (*). r GEO= T μ 2π circ * A sidereal day, 23h56m4s, is the time it takes the Earth to complete one rotation relative to inertial space. A synodic day, 24h, is the time it takes the sun to apparently rotate once around the Earth. They would 36be identical if the earth stood still in space. 2/3

37 2.1 Earth s Rotation Speed The mean solar day (24h) is not the time it takes the Earth to perform one revolution in inertial space. 37

38 2.2 Elliptic Orbits (0<e<1) r = 2 h 1 μ 1+ ecosθ The relative position vector remains bounded. θ=0, minimum separation, periapse r p = 2 h μ(1 + e) θ=π, greatest separation, apoapse r a = 2 h μ(1 e) θ=π/2, semi-latus rectum p e = r r a a r + r p p 38

39 2.2 Geometry of the Elliptic Orbit a apse line θ b p r a r p 39

40 2.2 Digression: Angular Momentum r = 2 h 1 μ 1+ ecosθ r 2 (1 e ) a = 1 + ecosθ Orbit equation Polar equation of an ellipse (a, semimajor axis) h= μa e 2 (1 ) [End of digression] 40

41 2.2 Kepler s Second Law da = r r dt = h dt = hdt r() t r( t+ dt) r dt da h 1 dθ r dt 2 2 dt 2 = = = constant The line from the sun to a planet sweeps out equal areas inside the ellipse in equal lengths of time. 1 Area = AB AC 2 reminder 41

42 2.2 Kepler s Third Law T enclosed area = = da/ dt 2π ab h h a e 2 2 = μ (1 ) b= a 1 e 3 a Tellip = 2π μ The elliptic orbit period depends only on the semimajor axis and is independent of the eccentrivity. T T a = a2 The squares of the orbital periods of the planets are proportional to the cubes of their mean distances 42 from the sun.

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

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

45 2.2 OUFTI-1 in Inertial Space 45

46 2.2 OUFTI-1 Ground Track 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) True anomaly θ (deg) 48

49 2.2 HEO in Inertial Space Perigee = 500 km Apogee = km 49

50 2.3 Parabolic Orbits (e=1) r = 2 h 1 μ 1+ cosθ θ π, r v parab = 2μ r The satellite will coast to infinity, arriving there with zero velocity relative to the central body. 50

51 2.3 Escape Velocity, V esc 11.2 km/s is the second cosmic velocity; i.e., the minimum velocity (theoretical velocity, r=6378km) to orbit the Earth. v circ = μ r v parab = 2μ r 11.2 km/s = km/s 51

52 2.4 Hyperbolic Orbits (e>1) r = 2 h 1 μ 1+ ecosθ r p 2a r a v μ a v = v + v = C + v = esc 3 esc Hyperbolic excess speed C 3 is a measure of the energy for an interplanetary mission: 16.6 km 2 /s 2 (Cassini-Huygens) 8.9 km 2 /s 2 (Solar Orbiter, phase A) 52

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

54 2.4 Proton 54

55 What Do you Think? Assume we have a circular or elliptic orbit for our satellite. Will it stay there??? 55

56 Satellite Orbits 1. Two-body problem 2. Orbit types 3. Orbit perturbations 56

57 3.1 Non-Keplerian Motion In many practical situations, a satellite experiences significant perturbations (accelerations). These perturbations are sufficient to cause predictions of the position of the satellite based on a Keplerian approach to be in significant error in a brief time. 57

58 3.1 STK: Different Propagators 58

59 Different Perturbations? LEO? GEO? Satellite dependent! Montenbruck and Gill, Satellite orbits, Springer, 2000

60 3.1 Respective Importance 400 kms 1000 kms kms Oblateness Oblateness Oblateness Drag Sun and moon Sun and moon SRP 60

61 3.2 The Earth is not a Sphere 61

62 3.2 Physical Interpretation The force of gravity is no longer within the orbital plane: non-planar motion will result. The equatorial bulge exerts a force that pulls the satellite back to the equatorial plane and thus tries to align the orbital plane with the equator. Due to its angular momentum, the orbit behaves like a spinning top and reacts with a precessional motion of the orbital plane (the orbital plane of the satellite rotates in inertial space). 62

63 3.2 What Do You See? OUFTI-1 in September 2009 OUFTI-1 in March

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

65 3.2 Secular Effects Nodal regression: regression of the nodal line. 2 1 T 3 μ JR 2 Ω avg = Ω dt = /2 T 2(1 e ) a cosi Apsidal rotation: rotation of the apse line. ω avg 2 1 T 3 μ JR 2 = ωdt = /2 T 4(1 e ) a ( 2 4 5sin i) No secular variations for a, e, i. 65

66 3.2 Exploitation: Sun-Synchronous Orbit The orbital plane makes a constant angle with the radial from the sun. It must rotate in inertial space with the angular velocity of the Earth in its orbit around the Sun: 360º per days or º per day. 66

67 3.2 SPOT-5 Orbit is a SSO The satellite sees any given swath of the planet under nearly the same condition of daylight or darkness day after day. 820 kms, 98.7º 67

68 68

69 3.2 Exploitation: Molniya Orbits A GEO satellite cannot view effectively the far northern latitudes into which Russian territory extends (+ costly plane change maneuver for the launch vehicle!) Molniya telecommunications satellites have 63º inclination orbits with a period 12 hours. Why 63º inclination? 69

70 3.2 Exploitation: Molniya Orbits 70

71 3.3 The Earth Has an Atmosphere Atmospheric forces represent the largest nonconservative perturbations acting on low-altitude satellites. The drag is directly opposite to the velocity of the satellite. The lift force can be neglected in most cases. 71

72 3.3 Effects of Atmospheric Drag 72

73 3.4 Third-Body Perturbations For an Earth-orbiting satellite, the Sun and the Moon should be modeled for accurate predictions. Their effects become noticeable when the effects of drag begin to diminish. 73

74 3.4 Effects of Third-Body Perturbations The Sun s attraction tends to turn the satellite ring into the ecliptic. The orbit precesses about the pole of the ecliptic. Vallado, Fundamental of Astrodynamics and Applications, Kluwer,

75 3.5 Solar Radiation Pressure Solar radiation (photons) Solar wind (particles) It produces a nonconservative perturbation on the spacecraft, which depends upon the distance from the sun. It is usually very difficult to determine precisely, but the effects are usually small for most satellites. 800km is regarded as a transition altitude between drag and SRP. 75

76 3.6 In Summary (STK) 76

77 What Do you Think? We have an orbit. Do we have a launch vehicle??? 77

78 Satellite Orbits 1. Two-body problem 2. Orbit types 3. Orbit perturbations 4. Orbit transfer 78

79 4. Tsiolkovsky s Rocket Equation (1903) Δ ( mv) = 0 m i m i Δ v= vej ln = Ispg0 ln m f m f 79

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

81 4. Three-Stage Rocket (Similar Stages) [ ] Δ v= Δ vi = 3vej ln km/s i The payload is 0.1% of the initial mass! 81

82 4. Specific Impulse 82

83 4. But Coquilhat ( , Belgian) Established the rocket equation in 1873! Trajectoires des fusées volantes dans le vide, Mémoires de la Société Royale des Sciences de Liège. Recent discovery. 83

84 4. Motivation Without maneuvers, satellites could not go beyond the close vicinity of Earth. For instance, a GEO spacecraft is usually placed on a transfer orbit (LEO or GTO). 84

85 4. From LEO to GEO: STK Video 85

86 4. From GTO to GEO: Ariane V Ariane V is able to place heavy GEO satellites in GTO: perigee: km and apogee: ~35786 km. GTO GEO 86

87 4. Delta-V: Order of Magnitudes (1000,0.288) Δ m/m Δ v (m/s) Isp=300s 87

88 4. Delta-V Budget: GEO 88

89 4. How to Go to Saturn? V V E J G A 89

90 4. Gravity Assist Also known as planetary flyby trajectory, slingshot maneuver 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. 90

91 4. Basic Principle Resultant V out SOI Resultant V in Planet s sun relative velocity Δv = v v, out, in 91

92 4. Basic Principle Inertial frame Frame attached to the train Frame attached to the train Inertial frame A gravity assists looks like an elastic collision, although there is no physical contact with the planet. 92

93 4. Cassini: Swingby Effects V V E J S 93

94 Satellite Orbits 1. Two-body problem 2. Orbit types 3. Orbit perturbations 5. Conclusions 4. Orbit transfer 94

95 Satellite Orbits Gentle introduction to satellite orbits; more details in the astrodynamics course. Closed-form solution of the 2-body problem from which we deduced Kepler s laws. Orbit perturbations cannot be ignored for accurate orbit propagation and for mission design. Orbit transfers are commonly encountered. Satellite must often have their own propulsion. 95