Astrodynamics (AERO0024)

Transcription

1 Astrodynamics (AERO0024) 4B. Non-Keplerian Motion Gaëtan Kerschen Space Structures & Systems Lab (S3L)

2 2. Two-body problem 4.1 Dominant perturbations Orbital elements (a,e,i,ω,ω) are constant Real satellites may undergo perturbations This lecture: 1. Effects of these perturbations on the orbital elements? 2. Computation of these effects?

3 STK: Different Propagators 3

4 Why Different Propagators? Analytic propagation: Better understanding of the perturbing forces. Useful for mission planning (fast answer): e.g., lifetime computation. Numerical propagation: The high accuracy required today for satellite orbits can only be achieved by using numerical integration. Incorporation of any arbitrary disturbing acceleration (versatile). 4

5 4. Non-Keplerian Motion a 2 (1 e ) N sin2 sin i 1 ecos 4.2 Analytic treatment r 4.3 Numerical methods tn t n Geostationary satellites 5

6 4. Non-Keplerian Motion a 2 (1 e ) N sin2 sin i 1 ecos 4.2 Analytic treatment Variation of parameters Non-spherical Earth J2 propagator in STK Atmospheric drag Third-body perturbations SGP4 propagator in STK Solar radiation pressure 6

7 Analytic Treatment: Definition Position and velocity at a requested time are computed directly from initial conditions in a single step. Analytic propagators use a closed-form solution of the time-dependent motion of a satellite. Mainly used for the two dominant perturbations, drag and earth oblateness. 4.2 Analytic treatment 7

8 Analytic Treatment: Pros and Cons Useful for mission planning and analysis (fast and insight): Though the numerical integration methods can generate more accurate ephemeris of a satellite with respect to a complex force model, the analytical solutions represent a manifold of solutions for a large domain of initial conditions and parameters. But less accurate than numerical integration. Be aware of the assumptions made! 4.2 Analytic treatment 8

9 Assumption for Analytic Developments The magnitude of the disturbing force is assumed to be much smaller than the magnitude of the attraction of the satellite for the primary. r r a r a 3 perturbed perturbed r 4.2 Analytic treatment 9

10 Variation of Parameters (VOP) Originally developed by Euler and improved by Lagrange (conservative) and Gauss (nonconservative). It is called variation of parameters, because the orbital elements (i.e., the constant parameters in the two-body equations) are changing in the presence of perturbations. The VOP equations are a system of first-order ODEs that describe the rates of change of the orbital elements. a, i, e,,, M? Variation of parameters 10

11 Disturbing Acceleration (Specific Force) 2 a F Reˆ Teˆ Neˆ perturbed R T N Rotating basis whose origin is fixed to the satellite Variation of parameters 11

12 Perturbation Equations (Gauss) Chapter 2 2 2a a 2 (1) 2a 2 h 2 (2) r sin sin r h r r e e 2 1 ecos h h (3) The generating solution is that of the 2-body problem Fr F reˆ re ˆ rr rt (4) R T Time rate-of-change of the work done by the disturbing force Variation of parameters 12

13 Perturbation Equations (Gauss) (1) (2) (3) (4) a esin h 2a h a R T esin R T h r h r 2 2a Resin T 1 ecos h h a e 2 (1 ) Chapter 2 3 a a 2 Resin T 1 ecos 2 1 e Variation of parameters 13

14 Perturbation Equations (Gauss) a 2 (1 e ) N sin2 sini 1 ecos 3 a a 2 Resin T 1 ecos 2 1 e i a 2 (1 e ) N cos2 1 ecos 2 a(1 e ) e Rsin T cos cos E 2 1 a(1 e ) Tsin 2 ecos cosi Rcos e 1 ecos M nt, with a 2 2 (1 e ) R 2e cos ecos T sin 2 ecos e 1 ecos J.E. Prussing, B.A. Conway, Orbital Mechanics, Oxford University Press Variation of parameters 14

15 Perturbation Equations (Gauss) Limited to eccentricities less than 1. Singular for e=0, sin i=0 (use of equinoctial elements). In what follows, we apply the Gauss equations to Earth oblateness and drag. Analytical expressions for third-body and solar radiation forces are far less common, because their effects are much smaller for many orbits Variation of parameters 15

16 Non-spherical Earth: J2 Focus on the oblateness through the first zonal harmonic, J2 (tesseral and sectorial coefficients ignored). The J2 effect can still be viewed a small perturbation when compared to the attraction of the spherical Earth Non-spherical Earth 16

17 Disturbing Acceleration (Specific Force) Chapter 4A U 1 J r 2 R r 3sin sat U with ˆ ˆ ˆ F r φ λ r r r cos i 2 r T sin isin cos N sin isin cos i 3JR 2 1 3sin sin F 4 e e e r Non-spherical Earth 17

18 Physical Interpretation of the Perturbation The oblateness means that 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 to rotate in inertial space) Non-spherical Earth 18

19 Physical Interpretation of the Perturbation Non-spherical Earth 19

20 Effect of Perturbations on Orbital Elements Secular rate of change: average rate of change over many orbits. Periodic rate of change: rate of change within one orbit (J2: ~ 8-10km with a period equal to the orbital period) Non-spherical Earth 20

21 Effect of Perturbations on Orbital Elements Periodic Secular Non-spherical Earth 21

22 Secular Effects on Orbital Elements Nodal regression: regression of the nodal line: 2 1 T 3 JR 2 avg dt cosi / 2 T 2 (1 e ) a Apsidal rotation: rotation of the apse line: avg 2 1 T 3 JR 2 dt / 2 4 5sin T 0 4 (1 e ) a 2 i Mean anomaly. No secular variations for a, e, i Non-spherical Earth 22

23 Secular Effects: Node Line 2 1 T 3 JR 2 avg dt / 2 T 2 (1 e ) a cosi 0 i 90, 0 For posigrade orbits, the node line drifts westward (regression of the nodes). And conversely. i 90, 0 For polar orbits, the node line is stationary Non-spherical Earth 23

24 Vallado, Fundamental of Astrodynamics and Applications, Kluwer, 2001.

25 Exploitation: Sun-Synchronous Orbits The orbital plane makes a constant angle with the radial from the sun: Non-spherical Earth 25

26 Exploitation: Sun-Synchronous Orbits The orbital plane 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 The satellite sees any given swath of the planet under nearly the same condition of daylight or darkness day after day Non-spherical Earth 26

27 Existing Satellites SPOT-5 (820 kms, 98.7º) NOAA/POES (833 kms, 98.7º) 27

28 Secular Effects: Apse Line avg 2 1 T 3 JR 2 dt 4 5sin / 2 T 4 (1 e ) a 2 i 0 i 63.4 or i180, 0 The perigee advances in the direction of the motion of the satellite. And conversely. i 63.4 or i 116.6, 0 The apse line does not move Non-spherical Earth 28

29 Vallado, Fundamental of Astrodynamics and Applications, Kluwer, 2001.

30 Exploitation: Molniya Orbits A geostationary satellite cannot view effectively the far northern latitudes into which Russian territory extends (+ costly plane change maneuver for the launch vehicle!) Molniya telecommunications satellites are launched from Plesetsk (62.8ºN) into 63º inclination orbits having a period of 12 hours. 3 a Tellip 2 the apse line is 53000km long Non-spherical Earth 30

31 Analytic Propagators in STK: 2-body, J2 2-body: constant orbital elements. J2: accounts for secular variations in the orbit elements due to Earth oblateness; periodic variations are neglected J2 propagator in STK 31

32 J2 Propagator: Underlying Equations J2 propagator in STK 32

33 2-body and J2 Propagators Applied to ISS Two-body propagator J2 propagator J2 propagator in STK 33

34 HPOP and J2 Propagators Applied to ISS Nodal regression of the ISS 34

35 Effects of Atmospheric Drag: Semi-Major Axis a 2 Lecture 2 a a >0 Because drag causes the dissipation of mechanical energy from the system, the semimajor axis contracts. Drag paradox: the effect of atmospheric drag is to increase the satellite speed and kinetic energy! Atmospheric drag 35

36 Effects of Atmospheric Drag: Semi-Major Axis 1 A 2 1 A N R 0, T CD vr CD 2 m 2 m a 3 a a 2 Resin T 1 ecos 2 1 e A a acd m Circular orbit 0 is assumed constant CDA a a t t 2m f i f i Atmospheric drag 36

37 Effects of Atmospheric Drag: Orbit Plane a 2 (1 e ) N sin2 sini 1 ecos i a 2 (1 e ) N cos2 1 ecos The orientation of the orbit plane is not changed by drag Atmospheric drag 37

38 Effects of Atmospheric Drag: Apogee, Perigee Apogee height changes drastically, perigee height remains relatively constant. Vallado, Fundamental of Astrodynamics and Applications, Kluwer, Atmospheric drag 38

39 Effects of Atmospheric Drag: Eccentricity Vallado, Fundamental of Astrodynamics and Applications, Kluwer, Atmospheric drag 39

40 Early Reentry of Skylab (1979) Increased solar activity, which increased drag on Skylab, led to an early reentry. Earth reentry footprint could not be accurately predicted (due to tumbling and other parameters). Debris was found around Esperance (31 34 S, E). The Shire of Esperance fined the United States $400 for littering, a fine which, to this day, remains unpaid Atmospheric drag 40

41 Effects of Third-Body Perturbations The only secular perturbations are in the node and in the perigee. For near-earth orbits, the dominance of the oblateness dictates that the orbital plane regresses about the polar axis. For higher orbits, the regression will be about some mean pole lying between the Earth s pole and the ecliptic pole. Many geosynchronous satellites launched 30 years ago now have inclinations of up to ±15º collision avoidance as the satellites drift back through the GEO belt Third-body perturbations 41

42 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, Third-body perturbations 42

43 STK: Analytic Propagator (SGP4) The J2 propagator does not include drag. SGP4, which stands for Simplified General Perturbations Satellite Orbit Model 4, is a NASA/NORAD algorithm SGP4 propagator in STK 43

44 STK: Analytic Propagator (SGP4) Several assumptions; propagation valid for short durations (3-10 days). TLE data should be used as the input (see Lecture 03). It considers secular and periodic variations due to Earth oblateness, solar and lunar gravitational effects, and orbital decay using a drag model SGP4 propagator in STK 44

45 SGP4 Applied to ISS: RAAN SGP4 propagator in STK 45

46 Further Reading on the Web Site SGP4 propagator in STK 46

47 Secular Effects: Orders of Magnitude Vallado, Fundamental of Astrodynamics and Applications, Kluwer,

48 Periodic Effects: Orders of Magnitude Vallado, Fundamental of Astrodynamics and Applications, Kluwer,

49 4. Non-Keplerian Motion r a 2 (1 e ) N sin2 sin i 1 ecos tn t n Numerical methods Orbit prediction Numerical integration Single-step methods: Runge-Kutta Multi-step methods Integrator and step size selection ISS example 49

50 STK Propagators 2-body: analytic propagator (constant orbital elements). J2: analytic propagator (secular variations in the orbit elements due to Earth oblateness. HPOP: numerical integration of the equations of motion (periodic and secular effects included). Accurate Versatile Errors accumulation for long intervals Computationally intensive Orbit prediction 50

51 Real-Life Example: German Aerospace Agency Orbit prediction 51

52 Real-Life Example: German Aerospace Agency 52

53 Further Reading on the Web Site Orbit prediction 53

54 Real-Life Example: Envisat ENVpred.html Orbit prediction 54

55 Why do the predictions degrade for lower altitudes?

56 Did you Know? NASA began the first complex numerical integrations during the late 1960s and early 1970s Orbit prediction 56

57 What is Numerical Integration? Given r r a r r( t ), r( t ) 3 perturbed n t t t n1 n n Compute r( t ), r( t ) n1 n Numerical integration 57

58 State-Space Formulation r r a r 3 perturbed u f ( u, t) u r r 6-dimensional state vector Numerical integration 58

59 How to Perform Numerical Integration? u( t n ) u( ) tn 1 h h f t h f t hf t f t f t R 2 s! 2 s ( s) ( n ) ( n) '( n) ''( n)... ( n) s Taylor series expansion Numerical integration 59

60 First-Order Taylor Approximation (Euler) along the tangent u( t t) u( t ) t u( t ) n n n u u t f ( u, t ) n1 n n n Euler step Exact solution x(t)=t Time t (s) Numerical integration The stepsize has to be extremely small for accurate predictions, and it is necessary to develop more effective algorithms. 60

61 Numerical Integration Methods m u u t u n1 j n1 j j n1 j j1 j0 State vector m 0 0 Implicit, the solution method becomes iterative in the nonlinear case 0 0, =0 j for j 1, 0 j for j 1 j j Explicit, u n+1 can be deduced directly from the results at the previous time steps Single-step, the system at time t n+1 only depends on the previous state t n Multi-step, the system at time t n+1 depends several previous states t n,t n-1,etc Numerical integration 61

62 Examples: Implicit vs. Explicit Trapezoidal rule (implicit) u n u u 2 n1 un t Euler backward (implicit) u u t u n1 n n1 n1 r r tn t n 1 tn t n 1 Euler forward (explicit) r u u t u n 1 n n Numerical integration tn t n 1 62

63 Why Different Methods? A variety of methods has been applied in astrodynamics. Each of these methods has its own advantages and drawbacks: Accuracy: what is the order of the integration scheme? Efficiency: how many function calls? Versatility: can it be applied to a wide range of problems? Complexity: is it easy to implement and use? Step size: automatic step size control? Orbit prediction 63

64 Runge-Kutta Family: Single-Step Perhaps the most well-known numerical integrator. Difference with traditional Taylor series integrators: the RK family only requires the first derivative, but several evaluations are needed to move forward one step in time. Different variants: explicit, embedded, etc Single-step methods: Runge-Kutta 64

65 Runge-Kutta Family: Single-Step u( t ) u u( t) f ( u, t) with 0 0 Slopes at various points within the integration step u u tbk k n1 n i i i1 f u, t c t 1 n n 1 s i1 ki f un taijk j, tn cit, i 2... s j Single-step methods: Runge-Kutta 65

66 Runge-Kutta Family: Single-Step The Runge-Kutta methods are fully described by the coefficients: c 1 c 2 a 21 s i1 c 1 b i 0 1 c s a s1 b 1 a s2 b 2 a s,s-1 b s-1 b s c i i1 j1 a ij Butcher Tableau Single-step methods: Runge-Kutta 66

67 RK4 (Explicit) u 2 2 k k k k 6 n1 un t k 1 f u n, t n t t k 2 f un k1, tn 2 2 t t k 3 f un k 2, tn 2 2 k f u k t, t t 4 n 3 n Butcher Tableau Single-step methods: Runge-Kutta 67

68 k RK4 (Explicit) 1 u f 2 2 k k k k 6 n1 un t u n, t n t t k 2 f un k1, tn 2 2 t t k 3 f un k 2, tn 2 2 k f u k t, t t 4 n Single-step methods: Runge-Kutta n Estimated slope (weighted average) Slope at the beginning Slope at the midpoint (k 1 is used to determine the value of u Euler) Slope at the midpoint (k 2 is now used) Slope at the end 68

69 RK4 (Explicit) u k 4 k 1 k 3 Estimate at new time k 2 t t t/2 n n tn t Single-step methods: Runge-Kutta 69

70 RK4 (Explicit) The local truncation error for a 4 th order RK is O(h 5 ). The accuracy is comparable to that of a 4 th order Taylor series, but the Runge-Kutta method avoids the calculation of higher-order derivatives. Easy to use and implement. The step size is fixed Single-step methods: Runge-Kutta 70

71 RK4 in STK Single-step methods: Runge-Kutta 71

72 Embedded Methods They produce an estimate of the local truncation error: adjust the step size to keep local truncation errors within some tolerances. This is done by having two methods in the tableau, one with order p and one with order p+1, with the same set of function evaluations: s ( p) ( p) ( p) n1 n tbi i i1 s ( p 1) ( p 1) ( p 1) n 1 n t b i i i1 u u k Single-step methods: Runge-Kutta u u k 72

73 Embedded Methods The two different approximations for the solution at each step are compared: If the two answers are in close agreement, the approximation is accepted. If the two answers do not agree to a specified accuracy, the step size is reduced. If the answers agree to more significant digits than required, the step size is increased Single-step methods: Runge-Kutta 73

74 Ode45 in Matlab / Simulink Runge-Kutta (4,5) pair of Dormand and Prince: Variable step size. Matlab help: This should be the first solver you try Single-step methods: Runge-Kutta 74

75 Ode45 in Matlab / Simulink edit ode Single-step methods: Runge-Kutta 75

76 Ode45 in Matlab / Simulink Be very careful with the default parameters! options = odeset('reltol',1e-8,'abstol',1e-8); Single-step methods: Runge-Kutta 76

77 RKF 7(8): Default Method in STK Runge-Kutta-Fehlberg integration method of 7th order with 8th order error control for the integration step size. 77

78 4.3.3 Single-step methods: Runge-Kutta

79 Integrator Selection Montenbruck and Gill, Satellite orbits, Springer, Integrator and step size selection 79

80 Why is the Step Size So Critical? Theoretical arguments: 1. The accuracy and the stability of the algorithm are directly related to the step size. 2. Nonlinear equations of motion. Data for Landsat 4 and 6 in circular orbits around 800km indicates that a one-minute step size yields about 47m error. A three-minute step size produces about a 900m error! Integrator and step size selection 80

81 Why is the Step Size So Critical? More practical arguments: 1. The computation time is directly related to the step size. 2. The particular choice of step size depends on the most rapidly varying component in the disturbing functions (e.g., 50 x 50 gravity field) Integrator and step size selection 81

82 6000 XMM (e~0.8) 5000 Step size (s) True anomaly (deg) Integrator and step size selection 82

83 ISS(e~0) Step size (s) True anomaly (deg) Integrator and step size selection 83

84 Difficult Orbits Automatic time step is especially nice on highly eccentric orbits (Molniya, XMM). These orbits are best computed using variable step sizes to maintain some given level of accuracy: Without this variable step size, we waste a lot of time near apoapsis, when the integration is taking too small a step. Likewise, the integrator may not be using a small enough step size at periapsis, where the satellite is traveling fast Integrator and step size selection 84

85 HPOP Propagator: ISS Example 1. Earth s oblateness only 2. Drag only 3. Sun and moon only 4. SRP only 5. All together ISS example 85

86 Earth s Oblateness Only: Ω 2-body HPOP J ISS example 86

87 Earth s Oblateness Only: i, Ω, a HPOP with central body (2,0 + WGS84_EGM96) (without drag/srp/sun and Moon) ISS example 87

88 Drag Only: i, Ω, a HPOP with drag Harris Priester (without oblateness/srp/sun and Moon) ISS example 88

89 Drag: Relationship with Eclipses 89

90 SRP Only: i, Ω, a HPOP with SRP (without oblateness/drag/sun and Moon) ISS example 90

91 SRP: Relationship with Eclipses 91

92 All Perturbations Together ISS example 92

93 4. Non-Keplerian Motion a 2 (1 e ) N sin2 sin i 1 ecos 4.4 Geostationary satellites 93

94 Practical Example: GEO Satellites Nice illustration of: 1. Perturbations of the 2-body problem. 2. Secular and periodic contributions. 3. Accuracy required by practical applications. 4. The need for orbit correction and thrust forces. And it is a real-life example (telecommunications, meteorology)! 4.4 GEO satellites 94

95 Three Main Perturbations for GEO Satellites 1. Non-spherical Earth 2. SRP 3. Sun and Moon 4.4 GEO satellites 95

96 Station Keeping of GEO Satellites The effect of the perturbations is to cause the spacecraft to drift away from its nominal station. If the drift was allowed to build up unchecked, the spacecraft could become useless. A station-keeping box is defined by a longitude and a maximum authorized distance for satellite excursions in longitude and latitude. For instance, TC2: -8º ± 0.07º E/W ± 0.05º N/S 4.4 GEO satellites 96

97 East-West and North-South Drift What are the perturbations generating these drifts? N/S drift E/W drift 4.4 GEO satellites 97

98 East-West Drift A GEO satellite drifts in longitude due to the influence of two main perturbations: 1. The elliptic nature of the Earth s equatorial crosssection: J22 (and not from the N/S oblateness J2). 2. v sat ΔV SRP v sat ΔV 4.4 GEO satellites 98

99 East-West Drift due to Equatorial Ellipticity 4.4 GEO satellites 99

100 East-West Drift due to Equatorial Ellipticity 4.4 GEO satellites 100

101 East-West Drift: HPOP (2,0) vs. HPOP (2,2) 4.4 GEO satellites 101

102 East-West Drift: Stable Equilibirum HPOP with 2,2 (without Sun and moon/srp/drag) 102

103 East-West Drift: Stable Equilibirum HPOP with 2,2 (without Sun and moon/srp/drag) 103

104 East-West Drift: Stable Equilibirum HPOP with 2,2 (without Sun and moon/srp/drag) 104

105 North-South Drift The perturbations caused by the Sun and the Moon are predominantly out-of-plane effects causing a change in the inclination and in the right ascension of the orbit ascending node. Similar effects on the orbit to those of the Earth s oblateness (but here with respect to the ecliptic) A GEO satellite therefore drifts in latitude with a fundamental period equal to the orbit period. 105

106 North-South Drift Period? HPOP with Sun and Moon (without oblateness/srp/drag) 106

107 North-South Drift Period? HPOP with Sun and Moon (without oblateness/srp/drag) 107

108 Thrust Forces for Stationkeeping GEO spacecraft require continual stationkeeping to stay within the authorized box using onboard thrusters. 4.4 GEO satellites 108

109 4. Non-Keplerian Motion 4.2 ANALYTIC TREATMENT Variation of parameters Non-spherical Earth J2 propagator in STK Atmospheric drag Third-body perturbations SGP4 propagator in STK 4.3 NUMERICAL METHODS Orbit prediction Numerical integration Single-step methods: Runge-Kutta Multi-step methods Integrator and step size selection ISS example Solar radiation pressure 4.4 GEOSTATIONARY SATELLITES 109

110 Astrodynamics (AERO0024) 4B. Non-Keplerian Motion Gaëtan Kerschen Space Structures & Systems Lab (S3L)