Astrodynamics (AERO0024)

Transcription

1 Astrodynamics (AERO0024) 5. Dominant Perturbations Gaëtan Kerschen Space Structures & Systems Lab (S3L)

2 Motivation Assumption of a two-body system in which the central body acts gravitationally as a point mass. 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. 2

3 STK: Different Propagators 3

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5 4. Non-Keplerian Motion 4.1 Dominant perturbations Earth s gravity field Atmospheric drag Third-body perturbations Solar radiation pressure 5

6 Different Perturbations and Importance? In low-earth orbit (LEO)? In geostationary orbit (GEO)? 4.1 Dominant perturbations 6

7 Satellite dependent! Montenbruck and Gill, Satellite orbits, Springer, 2000 Fortescue et al., Spacecraft systems engineering, 2003

8 Orders of Magnitude 400 kms 1000 kms kms Oblateness Drag Oblateness Sun and moon Oblateness Sun and moon SRP 4.1 Dominant perturbations 8

9 The Earth is not a Sphere Earth s gravity field 9

10 Mathematical Modeling r ' P r Satellite unit mass r x y z r ' U G M body dm 2 r r ' 2 2 r ' r cos U G body U=-V, potential, V, potential energy r U dm 2 r 1 2cos. ' cos rr rr. ' r ' r Earth s gravity field 10

11 Binomial Theorem 1 1 l l P [cos ] 2 1 2cos 1 x l0 P[ ] 1 d 2 l! l l l l 2 d 1 l Legendre polynomials P P P 2 0[ ] 1, 1[ ], 2[ ] 3 1, etc Earth s gravity field 11

12 Summing up G l U Pl [cos( )] dm r body l 0 Geometric method (intuitive feel for gravity and inertia) U U0 U1 U2... Theory Spherical-harmonic expansion Experiments Earth s gravity field 12

13 Geometric Method: First Term U 0 G r dm r Two-body potential Earth s gravity field 13

14 Geometric Method: Second Term G G x y z U 1 cos dm dm 2 r r r G x dm y dm z dm 3 r 0 Center of mass at the origin of the coordinate frame Earth s gravity field 14

15 Geometric Method: Third Term U 2 2 G 2 3cos 1 dm r 2 G 2 G r ' dm 3 r ' sin dm 3 3 2r 2r G 3 3 A B C I 2r ' r dm dm dm dm A B C 2 2 r ' sin dm I Moments of inertia Polar moment of inertia Earth s gravity field 15

16 Geometric Method: MacCullagh s Formula Gm G U A B C I r 2r Some of the simplest assumptions are - the ellipsoidal Earth (oblate spheroid) with uniform density (a=b>c). - triaxial ellipsoid (a>b>c) Earth s gravity field 16

17 Spherical Trigonometry G l U Pl [cos( )] dm r body l 0? SphericalTrigonometry.html, geocentric latitude, longitude cos cos(90 )cos(90 )... P sin(90 )sin(90 ) cos( ) sat P sat P sat cos sin( )sin( ) cos( )cos( )cos( ) P sat P sat P sat Earth s gravity field 17

18 Addition of Spherical Harmonics l l m! P cos P sin P sin 2 ( A A' B B' ) l l P l sat l, m l, m l, m l, m m1 l m! with A P sin cos( m ), A' P sin cos( m ) l, m l, m P P l, m l, m sat sat B P sin sin( m ), B' P sin sin( m ) l, m l, m P P l, m l, m sat sat l, degree m, order Earth s gravity field 18

19 Associated Legendre Polynomials P m lm 2 m/ 2 d P[ ] 1 2 m/ 2 l d [ ] 1 1 d 2 l! d lm, m l lm 2 1 l P 1, P sin, P cos, etc. 0,0 1,0 sat 1,1 sat For m=0, the associated Legendre functions are the conventional Legendre polynomials Earth s gravity field 19

20 Gravitational Coefficients ( l m)! C ' r ' 2 P sin cos( m ) dm l l, m l, m P P ( l m)! body ( l m)! S ' r ' 2 P sin sin( m ) dm l l, m l, m P P ( l m)! body (independent of satellite) C ' C R m, S ' S R m (nondimensionalization) l l l, m l, m l, m l, m Earth s gravity field 20

21 Normalization C S ( l m)! k(2l 1) C, S, P P, ( l m)! l, m l, m l, m l, m l, m l, m l, m l, m l, m l, m k 1 if m 0, k 2 if m 0 C C 4,0 70,61 (4 0)! (4 0)!1(2.4 1) 5 7 (70 61)! (70 61)!2(2.70 1) Earth s gravity field 21

22 End Result l l R U 1 Plm sinsat Cl, m cos( msat ) Sl, m sin( msat ) r l2 m0 r Earth s gravity field 22

23 Very Important Remark l l R U 1 Plm sin sat Cl, m cos( m sat ) Sl, m sin( m sat ) r l 2 m 0 r Many different expressions exist in the literature: V=±V P lm =(-1) m P lm Normalized or non-normalized coefficients Latitude or colatitude (sin or cos) Be always aware of the conventions/definitions used! Earth s gravity field 23

24 Determination of Gravitational Coefficients Because the internal distribution of the Earth is not known, the coefficients cannot be calculated from their definition. They are determined experimentally; e.g, using satellite tracking. Satellite-to-satellite tracking: GRACE employs microwave ranging system to measure changes in the distance between two identical satellites as they circle Earth. The ranging system detects changes as small as 10 microns over a distance of 220 km Earth s gravity field 24

25 Gravitational Coefficients: GRACE EGM-2008 has been publicly released: Extensive use of GRACE twin satellites. 4.6 million terms in the spherical expansion ( in EGM-96) Geoid with a resolution approaching 10 km (5 x5 ) Earth s gravity field 25

26 EGM96 (Course Web Site) 0,1? 26

27 EGM96 (Course Web Site) C 4,0 (4 0)! (4 0)!1(2.4 1) 5 7 JGM Earth s gravity field 27

28 Zonal Harmonics (m=0) The zonal coefficients are independent of longitude (symmetry with respect to the rotation axis). l l R U 1 Plm sinsat Cl, m cos( msat ) Sl, m sin( msat ) r l2 m0 r J l C l,0 Sl,0 0 (definition) l l R U 1 JlPl sinsat Plm sinsat Cl, m cos( msat ) Sl, m sin( msat ) r l2 r m Earth s gravity field 28

29 Zonal Harmonics (m=0) Each boundary is a root of the Legendre polynomial. Vallado, Fundamental of Astrodynamics and Applications, Kluwer, Earth s gravity field 29

30 First Zonal Harmonic: J2 It represents the Earth s equatorial bulge and quantifies the major effects of oblateness on orbits. It is almost a thousand times as large as any of the other coefficients. J 2.1.(2.2+1) C ,0= Earth s gravity field 30

31 First Zonal Harmonic: J2 Planet J 2 Mercury Venus Earth Moon Jupiter Saturn 60e e e e e e Earth s gravity field 31

32 Sectorial Harmonics (l=m) The sectorial coefficients represent bands of longitude. The polynomials P l,l are zero only at the poles. Vallado, Fundamental of Astrodynamics and Applications, Kluwer, Earth s gravity field 32

33 Tesseral Harmonics (lm0) Vallado, Fundamental of Astrodynamics and Applications, Kluwer, Earth s gravity field 33

34 Can You Recognize the Spherical Harmonics? sectorial tesseral zonal Earth s gravity field 34

35 Resulting Force 1 1 U with ˆ ˆ ˆ F r φ λ r r r cos 1 1 ˆ ˆ ˆ r φ θ r r r sin Earth s gravity field 35

36 Spherical Earth Gravitational force acts through the Earth s center. U r 1 1 ˆ ˆ ˆ r φ λ r r r cos Earth s gravity field 36

37 P Oblate Earth: J2 l l R U 1 JlPl sinsat Plm sinsat Cl, m cos( msat ) Sl, m sin( msat ) r l2 r m [ ] 3 1 U 1 J r 2 R r 3sin sat ˆ ˆ ˆ r φ λ r r r cos Perturbation of the radial acceleration Longitudinal acceleration that can be decomposed into an azimuth and normal accelerations Earth s gravity field 37

38 STK: Central Body Gravity (HPOP) Earth s gravity field 38

39 STK: Gravity Models (HPOP) Earth s gravity field 39

40 Atmospheric Drag Atmospheric forces represent the largest nonconservative perturbations acting on low-altitude satellites. The drag is directly opposite to the velocity of the satellite, hence decelerating the satellite. The lift force can be neglected in most cases Atmospheric drag 40

41 Mathematical Modeling Knowledge of attitude Velocity with respect to the atmosphere The atmosphere co-rotates with the Earth. vr vω r r 1 C 2 A v m 2 sat D r v v r r Inertial velocity Earth s angular velocity [1.5-3] Atmospheric density All these parameters are difficult to estimate! Atmospheric drag 41

42 Atmospheric Density The gross behavior of the atmospheric density is well established, but it is still this factor which makes the determination of satellite lifetimes so uncertain. There exist several models (e.g., Jacchia-Roberts, Harris- Priester). Dependence on temperature, molecular weight, altitude, solar activity, etc Atmospheric drag 42

43 Solar Activity Atmospheric drag 43

44 von Karman Institute for Fluid Dynamics An international network of 50 double CubeSats for multipoint, in-situ, long-duration (3 months) measurements in the lower thermosphere ( km) and for re-entry research

45 Atmospheric Density using CHAMP An accelerometer measures the non-gravitational accelerations in three components, of which the along-track component mainly represents the atmospheric drag. By subtracting modeled accelerations for SRP and Earth Albedo, the drag acceleration is isolated and is proportional to the atmospheric density. 45

46 CHAMP Density at 410 kms Atmospheric drag 46

47 Further Reading on the Web Site Atmospheric drag 47

48 Harris-Priester ( km) Static model (e.g., no variation with the 27-day solar rotation). Interpolation determines the density at a particular time. Simple, computationally efficient and fairly accurate Atmospheric drag 48

49 Harris-Priester ( km) Account for diurnal density bulge due to solar radiation Height above the Earth s reference ellipsoid Angle between satellite position vector and the apex of the diurnal bulge Atmospheric drag 49

50 Atmospheric Bulge The high atmosphere bulges toward a point in the sky some 15º to 30º east of the sun (density peak at 2pm local solar time). The observed accelerations of Vanguard satellite (1958) indicated that the air density at 665 km is about 10 times as great when perigee passage occur one hour after noon as when it occurs during the night! Atmospheric drag 50

51 Atmospheric Bulge Position Sun declination Sun right ascension Lag Atmospheric drag 51

52 Interpolation Between Altitudes Mean solar activity Atmospheric drag 52

53 STK Atmospheric Models (HPOP) Atmospheric drag 53

54 STK Solar Activity (HPOP) Atmospheric drag 54

55 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 Third-body perturbations 55

56 Mathematical Modeling (Sun Example) r r r sat r sat r Sat sat m r sat sat Gm m Relative motion is of interest r Gm m sat sat sat 3 3 r sat r sat r sat Inertial frame of reference r sat Gm r Gm r Gm r Gm r r r r sat sat sat sat sat sat sat r r Third-body perturbations sat direct indirect G m m r Gm r r sat sat sat rsat rsat r 56

57 STK: Third-Body Gravity (HPOP) 57

58 Solar Radiation Pressure It produces a nonconservative perturbation on the spacecraft, which depends upon the distance from the sun. It is usually very difficult to determine precisely. It is NOT related to solar wind, which is a continuous stream of particles emanating from the sun. Solar radiation (photons) Solar wind (particles) 800km is regarded as a transition altitude between drag and SRP Solar radiation pressure 58

59 Mathematical Modeling F p c Ae SR SR R sc / sun W/m psr N/m 3e8 m/s 6 2 The reflectivity c R is a value between 0 and 2: - 0: translucent to incoming radiation. - 1: all radiation is absorbed (black body). - 2: all radiation is reflected. The incident area exposed to the sun must be known. The normals to the surfaces are assumed to point in the direction of the sun (e.g., solar arrays) Solar radiation pressure 59

60 Mathematical Modeling: Eclipses Use of shadow functions: F 0 SR Dual cone Solar radiation pressure Cylindrical 60

61 STK: Solar Radiation Pressure (HPOP) 61

62 STK: Shadow Models (HPOP) 62

63 STK: Central Body Pressure (HPOP) 63

64 4. Non-Keplerian Motion 4.1 DOMINANT PERTURBATIONS Earth s gravity field Atmospheric drag Third-body perturbations Solar radiation pressure 64

65 Astrodynamics (AERO0024) 5. Dominant Perturbations Gaëtan Kerschen Space Structures & Systems Lab (S3L)