Astrodynamics (AERO0024) 5. Dominant Perturbations Gaëtan Kerschen Space Structures & Systems Lab (S3L)
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
STK: Different Propagators 3
4. Non-Keplerian Motion 4.1 Dominant perturbations 4.1.2 Atmospheric drag 4.1.3 Third-body perturbations 4.1.4 Solar radiation pressure 5
Different Perturbations and Importance? In low-earth orbit (LEO)? In geostationary orbit (GEO)? 4.1 Dominant perturbations 6
Satellite dependent! Montenbruck and Gill, Satellite orbits, Springer, 2000 Fortescue et al., Spacecraft systems engineering, 2003
Orders of Magnitude 400 kms 1000 kms 36000 kms Oblateness Drag Oblateness Sun and moon Oblateness Sun and moon SRP 4.1 Dominant perturbations 8
The Earth is not a Sphere 9
Mathematical Modeling r ' P r Satellite unit mass r x y z 2 2 2 r ' 2 2 2 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 1 10
Legendre Polynomials First introduced in 1782 by Legendre as the coefficients in the expansion of the Newtonian potential x r=norm(x) r =norm(x ) x' 11
Legendre Polynomials 12
Let s Use Them 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. 1 2 13
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 14
Geometric Method: First Term U 0 G r dm r Two-body potential 15
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 16
Geometric Method: Third Term U 2 2 G 2 3cos 1 dm r 2 G 2 G 2 2 2 r ' dm 3 r ' sin dm 3 3 2r 2r G 3 3 A B C I 2r 2 2 2 2 2 2 2 2 ' r dm dm dm dm A B C 2 2 r ' sin dm I Moments of inertia Polar moment of inertia 17
Geometric Method: MacCullagh s Formula Gm G U A B C I r 2r 3... 3 Some of the simplest assumptions are - the ellipsoidal Earth (oblate spheroid) with uniform density (a=b>c). - triaxial ellipsoid (a>b>c). 18
Geometric Method: Difficult to Go Further 19
Summing up G l U Pl [cos( )] dm r body l 0 Geometric method (intuitive feel for gravity and inertia) Spherical-harmonic expansion Experiments Theory 20
Fourier Series? Complete set of orthogonal functions. Circle representation. A square wave defined using 4 Fourier terms 21
The Earth Is a Sphere Spherical Harmonics Similar in principle as Fourier series. Complete set of orthogonal functions. They can be used to represent functions defined on the surface of a sphere Our objective! 22
Spherical Harmonics If you want to model the 2-body problem or Earth s oblateness, which spherical harmonics do you need? 23
Spherical Harmonics & Legendre Polynomials l l l l l 24
Spherical Trigonometry G l U Pl [cos( )] dm r body l 0? http://mathworld.wolfram.com/ 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 25
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 26
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. 27
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 28
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)! 0.16210 5.40110 (4 0)!1(2.4 1) 5 7 (70 61)! 1.11410 1.01410 (70 61)!2(2.70 1) 116 9 29
End Result l l R U 1 Plm sinsat Cl, m cos( msat ) Sl, m sin( msat ) r l2 m0 r 30
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! 31
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. 32
Gravitational Coefficients: GRACE EGM-2008 has been publicly released: Extensive use of GRACE twin satellites. 4.6 million terms in the spherical expansion (130317 in EGM-96) Geoid with a resolution approaching 10 km (5 x5 ). 33
EGM96 (Course Web Site) 0,1? 34
EGM96 (Course Web Site) C 4,0 (4 0)! 0.16210 5.40110 (4 0)!1(2.4 1) 5 7 JGM-2 35
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 m1 36
Zonal Harmonics (m=0) Each boundary is a root of the Legendre polynomial. Vallado, Fundamental of Astrodynamics and Applications, Kluwer, 2001. 37
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 2 3 2 2,0= 0.4841 10 0.001082 38
First Zonal Harmonic: J2 Planet J 2 Mercury Venus Earth Moon Jupiter Saturn 60e-6 4.46e-6 1.08e-3 2.03e-4 1.47e-2 1.63e-2 39
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, 2001. 40
Tesseral Harmonics (lm0) Vallado, Fundamental of Astrodynamics and Applications, Kluwer, 2001. 41
Can You Recognize the Spherical Harmonics? sectorial tesseral zonal 42
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 43
Spherical Earth Gravitational force acts through the Earth s center. U r 1 1 ˆ ˆ ˆ r φ λ r r r cos 44
P Oblate Earth: J2 l l R U 1 JlPl sinsat Plm sinsat Cl, m cos( msat ) Sl, m sin( msat ) r l2 r m1 1 2 2 2[ ] 3 1 U 1 J r 2 R r 3sin sat 1 2 2 2 1 1 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 45
STK: Central Body Gravity (HPOP) 46
STK: Gravity Models (HPOP) 47
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. 4.1.2 Atmospheric drag 48
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! 4.1.2 Atmospheric drag 49
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. 4.1.2 Atmospheric drag 50
Solar Activity 4.1.2 Atmospheric drag 51
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 (90-300 km) and for re-entry research
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. 53
CHAMP Density at 410 kms 4.1.2 Atmospheric drag 54
Further Reading on the Web Site 4.1.2 Atmospheric drag 55
Harris-Priester (120-2000km) 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. 4.1.2 Atmospheric drag 56
Harris-Priester (120-2000km) 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 4.1.2 Atmospheric drag 57
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! 4.1.2 Atmospheric drag 58
Atmospheric Bulge Position Sun declination Sun right ascension Lag 4.1.2 Atmospheric drag 59
Interpolation Between Altitudes Mean solar activity 4.1.2 Atmospheric drag 60
STK Atmospheric Models (HPOP) 4.1.2 Atmospheric drag 61
STK Solar Activity (HPOP) 4.1.2 Atmospheric drag 62
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. 4.1.3 Third-body perturbations 63
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 3 3 3 3 sat sat sat r r 4.1.3 Third-body perturbations sat direct indirect G m m r Gm r r sat sat sat 3 3 3 rsat rsat r 64
STK: Third-Body Gravity (HPOP) 65
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. 4.1.4 Solar radiation pressure 66
Mathematical Modeling F p c Ae SR SR R sc / sun 2 1350 W/m psr 4.5110 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). 4.1.4 Solar radiation pressure 67
Mathematical Modeling: Eclipses Use of shadow functions: F 0 SR Dual cone 4.1.4 Solar radiation pressure Cylindrical 68
STK: Solar Radiation Pressure (HPOP) 69
STK: Shadow Models (HPOP) 70
STK: Central Body Pressure (HPOP) 71
4. Non-Keplerian Motion 4.1 DOMINANT PERTURBATIONS 4.1.2 Atmospheric drag 4.1.1 Third-body perturbations 4.1.2 Solar radiation pressure 72
Astrodynamics (AERO0024) 5. Dominant Perturbations Gaëtan Kerschen Space Structures & Systems Lab (S3L)