Astrodynamics (AERO0024)

Size: px
Start display at page:

Download "Astrodynamics (AERO0024)"

Transcription

1 Astrodynamics (AERO0024) L04: Non-Keplerian Motion Gaëtan Kerschen Space Structures & Systems Lab (S3L)

2 Non-Keplerian Motion 4 Dominant Perturbations Analytic Treatment Numerical Methods Concluding Remarks The Restricted 3-body Problem 2

3 Summary Mathematical modeling of 4 dominant perturbations. Analytic propagation: Meaning of J2 and its effect on the orbit. Periodic and secular contributions. STK: J2 and SGP4 propagators. 3

4 STK Propagators 2-body: constant orbital elements. J2: analytic propagator which accounts for secular variations in the orbit elements due to Earth oblateness. HPOP: orbit trajectories are computed using numerical integration of the equations of motion (periodic and secular effects included). 4

5 Numerical Integration The high accuracy required today for satellite orbits can only be achieved by using numerical integration. Incorporation of any arbitrary disturbing acceleration (versatile). Truncation and round-off may degrade the accuracy for lengthy propagation intervals. New data means new integration (lengthy computing times). 5

6 Numerical Integration 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? Four methods are implemented in STK. 6

7 DLR Example 7

8 DLR Example 8

9 Further Reading on the Web Site 9

10 Envisat Example 10

11

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

13 What is Numerical Integration? Given μ r = r+ a r r( t ), r ( t ) 3 perturbed n Δ t = t t i+ 1 n i Compute r( t ), r ( t ) n+ 1 n+ 1 13

14 State-Space Formulation μ r = r+ a r 3 perturbed u = f ( u, t) u r = r 6-dimensional state vector 14

15 How to Perform Numerical Integration? u( t n ) u( ) t n + 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 15

16 First-Order Taylor Approximation (Euler) along the tangent u( t +Δ t) = u( t ) +Δtu ( t ) n n n u = u +Δtf( u, t) n+ 1 n n n Euler step Exact solution x(t)=t Time t (s) The stepsize has to be extremely small for accurate predictions, and it is necessary to develop more effective algorithms. 16

17 Numerical Integration Methods m u = α u Δt β u n+ 1 j n+ 1 j j n+ 1 j j= 1 j= 0 State vector m β 0 β = Implicit, the solution method becomes iterative in the nonlinear case Explicit, u n+1 can be deduced directly from the results at the previous time steps α, β =0 j for j > 1 α, β 0 j for j > 1 j j 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. 17

18 Examples: Implicit vs. Explicit Trapezoidal rule (implicit) u n+ 1 un t ( + ) n = +Δ u u 2 n+ 1 r tn t n+ 1 Euler backward (implicit) u = u +Δt u n+ 1 n n+ 1 r tn t n+ 1 Euler forward (explicit) u = u +Δt u + n 1 n n r tn t n+ 1 18

19 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. 19

20 Runge-Kutta Family: Single-Step () t = f( u,) t with u( t 0) = u0 u Slopes at various points within the integration step u = u +Δt bk k n+ 1 n i i i= 1 = f u, t + cδt ( ) 1 n n 1 s i 1 ki = f un +Δ t aijk j, tn + ciδ t, i= 2... s j= 1 20

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

22 RK4 (Explicit) u = +Δ k k k k 6 n+ 1 un t k 1 = f u, t ( ) n n Δt Δt k2 = f un + k1, tn Δt Δt k3 = f un + k2, tn k = f u + k Δ t, t +Δt ( ) 4 n 3 n Butcher Tableau 22

23 RK4 (Explicit) k 1 = u f = +Δ k k k k 6 n+ 1 un t u, t ( ) n n Δt Δt k2 = f un + k1, tn Δt Δt k3 = f un + k2, tn k = f u + k Δ t, t +Δt ( ) 4 n 3 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 23

24 RK4 (Explicit) u k 4 k 1 k 3 Estimate at new time k 2 t n tn + Δt/2 tn + Δt 24

25 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. 25

26 RK4 in STK 26

27 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) un+ 1 = un +Δt bi ki i= 1 s ( p + 1) ( p + 1) ( p 1) n 1 n t b + + = +Δ i i i= 1 u u k 27

28 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. 28

29 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 29

30 Ode45 in Matlab / Simulink edit ode45 30

31 Ode45 in Matlab / Simulink Be very careful with the default parameters! options = odeset('reltol',1e-8,'abstol',1e-8); 31

32 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. 32

33

34 Multi-Step Methods (Predictor-Corrector) They estimate the state over time using previously determined back values of the solution. Unlike RK methods, they only perform one evaluation for each step forward, but they usually have a predictor and a corrector formula. Adams (*) Bashforth - Moulton, Gauss - Jackson. (*) The first with Le Verrier to predict the existence and position of Neptune 34

35 Multi-Step Methods: Principle u () t = f( u,) t u t u t f u t dt n+ 1 ( n+ 1) = ( ) + n (, ) tn t unknown u Four function values interpolated by a third-order polynomial Replace it by a ploynomial that interpolates the previous values t 35

36 Multi-Step Methods: Initiation () t = f( u,) t with u( t 0) = u0 u u t 36

37 Multi-Step Methods: Initiation Because these methods require back values, they are not self-starting. One may for instance use of a single-step method to compute the first four values. 37

38 Gauss-Jackson in STK One of the most recommendable fixed-stepsize multistep methods for orbit computations. 38

39 Extrapolation Methods Not discussed herein. More details in Montenbruck and Gill, Satellite orbits, Springer,

40 Integrator Selection Montenbruck and Gill, Satellite orbits, Springer,

41 Integrator Selection Multi-step Single step Pros Very fast Pros Plug and play Error control Cons Special starting procedure Fixed time steps Error control Cons Slower 41

42 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! 42

43 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). 43

44 Appropriate Step Size The problem of determining an appropriate step size is a challenge in any numerical process. Δ t = T orbit Fixed step size: (rule of thumb for standard applications). 100 But an algorithm with variable step size is really helpful. The step size is chosen in such a way that each step contributes uniformly to the total integration error. 44

45 Three Examples: XMM / OUFTI-1 / ISS Can you plot the step size vs. true anomaly? 45

46 XMM: Report in STK 46

47 XMM (e~0.8) Postprocessing in Matlab Step size (s) True anomaly (deg) 47

48 Exercise Session Reproduce this graph! 48

49 OUFTI-1 (e~0.07) Step size (s) True anomaly (deg) 49

50 ISS (e~0) Step size (s) True anomaly (deg) 50

51 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. 51

52 HPOP Propagator: ISS Example 1. J2 only 2. Drag only 3. Sun and moon only 4. SRP only 5. All together. 52

53 1. J2: Comparison of the Propagators 2-body HPOP J2 53

54 1. J2: Secular and Periodic Effects HPOP with central body (2,0 + WGS84_EGM96) (without drag/srp/sun and Moon) 54

55 2. Drag: HPOP 55

56 2. Drag: Lifetime (Satellite Tools) 56

57 2. Drag HPOP with drag Harris Priester (without oblateness/srp/sun and Moon) 57

58 2. Drag: Relationship with Eclipses 58

59 3. Third-Body Gravity: HPOP 59

60 3. Third-Body Gravity HPOP with Sun and Moon (without oblateness/srp/drag) 60

61 4. SRP HPOP with SRP (without oblateness/drag/sun and Moon) 61

62 4. SRP: Relationship with Eclipses 62

63 All Perturbations Together 63

64 2-Body: ISS Example 64

65 Non-Keplerian Motion 4 Dominant Perturbations Analytic Treatment Numerical Methods Concluding Remarks The Restricted 3-body Problem 65

66 Motivation For nearly half a century, space missions planners have used tools based largely on a two-body decomposition of the solar system. This is called the patched conic approach (Lecture 06). While that approach remains very valuable for some missions, new trajectory paradigms must be developed to meet the challenges of today. 66

67 The 3-Body Problem 18 th -order system, but ten known integrals: 6 (linear momentum). 1 (energy). 3 (angular momentum). No closed-form solution. Not integrable. 67

68 Matlab Example Three identical masses: Two are at rest. The third one has a velocity directed upward to the right making a 45 degrees angle with the X axis. Y v 0 m 1 m 2 m 3 X 68

69 Why Is the 3-Body Problem So Difficult? Displacement of the first mass (x direction) 16 x 105 Comparison of time series Nominal IC Perturbed IC Time x body problem (0.1% IC perturbation) Displacement of the first mass (x direction) 14 x 106 Comparison of time series Chaotic by nature! 3-body problem (0.1% IC perturbation) Nominal IC Perturbed IC Time x 10 4

70 Why Is the 3-Body Problem So Difficult? x Nominal IC m1 m2 m3 Chaotic by nature! 2 0 Y X x x 10 5 IC perturbed by 0.1% m1 m2 m3 Y X x 10 6

71 The Planar, Circular, Restricted 3BP z m (x,y,z) m is vanishingly small compared to m 1 and m 2 (no effect on their motions) r 1 r y What is the motion of m? m 1 (x 1,0,0) CM (0,0,0) r 2 r 12 Plane of motion of m 1 and m 2. The orbit around each other is a circle of radius r 12. m 2 (x 2,0,0) x Co-moving xyz frame 71

72 Position of m 1, m 2, m (co-moving frame) mx 1 1+ mx 2 2= 0 x = x + r x x = π r = π r with m π =, π = m m1+ m2 m1+ m2 r = xˆi+ yˆj+ zkˆ & r = x+ π r ˆi+ yˆj+ zkˆ ( ) r = x π r ˆi+ yˆj+ zkˆ ( )

73 Absolute Acceleration of m dω r = acm + arel + 2Ω vrel + r+ Ω Ω r dt ( ) r R ρ Rotating body frame (angular velocity ω) Inertial frame a ω ρ ω ω ρ 2 2 d R d ρ i dρi dω i = i + i dt dt dt body body dt 73 ( )

74 Absolute Acceleration of m r = Ω ( Ω r) + 2Ω vrel + arel Inertial angular velocity ˆ μ Ω =Ω k = kˆ r 3 12 Velocity and acceleration of m in the moving frame v = x ˆi+ y ˆj+ z kˆ rel a = xˆi+ yˆj+ zkˆ rel r = x Ωy Ω x ˆi+ y + Ωx Ω y ˆj + zkˆ ( 2 2 ) ( 2 2 ) 74

75 Newton s Second Law (Secondary Body) μ m μ m m r = F + F = r r r1 r2 μ μ r = r r r r2 x Ωy Ω x ˆi+ y+ Ωx Ω y ˆj+ zkˆ = ( 2 ) ( ) μ ( ) ˆ ˆ ˆ μ x + π ( ) ˆ ˆ ˆ 212 r + y + z x π112 r + y + z 75 r i j k i j k r2

76 Overall: Three Scalar Equations μ μ x 2Ωy Ω x= x+ π r x π r r2 ( ) ( ) r1 r2 μ y y+ 2Ωx Ω y = μ z z = r μ y r1 r2 μ z 76

77 Lagrange Points The three scalar equations of motion have no closed form analytical solutions. But the equilibrium points, called the Lagrange points, can be determined. x = y = z = 0, x= y = z = 0 77

78 Lagrange Points Lie in the Orbital Plane μ μ Ω x = x+ π r + x π r ( ) ( ) r1 r2 μ y Ω y = + μ y r1 r2 μ1 μ 2 + z = r1 r2 78

79 Some Points Are Collinear μ μ Ω x = x+ π r + x π r ( ) ( ) r1 r2 μ y Ω y = + μ y r1 r2 μ1 μ 2 + z = r1 r2 79

80 Two Sets of Lagrange Points μ μ Ω x = x+ π r + x π r ( ) ( ) r1 r2 μ y Ω y = + μ y r1 r2 y 0, L4,L5 y=0, L1,L2,L3 with π = 1 π 1 2 Ω= μ 3 r 12 μ π =, π = μ μ μ

81 L4, L5: Vertices of Equilateral Triangles π x+ π r + π x+ π r r = ( )( ) ( ) ( ) r1 r2 r π + π =, if y r1 r2 r12 x = = r = r = r r r r

82 Coordinates of L4, L5 (Co-Moving Frame) π = 1 π r = x+ π r ˆi+ yˆj+ zkˆ ( ) r = x π r ˆi+ yˆj+ zkˆ 1 2 ( ) with z = 0 r1 = r2 = r12 2 ( ) = + π r x r y 2 ( ) = + π r x r r y x y r 2 12 = π 212 =± 3 2 r 12 r 82

83 L1, L2, L3: Collinear Points y = z = 0 r r = x+ π r ( ) ˆi = x+ π r r ( ) ˆi r = x+ π r r = x+ π r r with π = 1 π 1 2 Ω= μ 3 r 12 μ π =, π = μ μ μ

84 L1, L2, L3: Numerical Technique μ μ Ω x = x+ π r + x π r ( ) ( ) r1 r2 f 1 π π ξ = ξ + π ξ + π ξ = 0 ξ + π ξ + π 1 ( ) 2 ( ) 2 ( ) 2 2 μ μ π1 = 1 π 2 Ω= π =, π = 3 r 12 μ r1 = x+ π 2r12 x ξ = r = x+ π r r r μ μ

85 Earth - Moon System m m r kg = kg = km = m π = 2 2 m1+ m =

86 Earth - Moon System L3: L2:1.156 f(ksi) L1: ksi 86

87 Earth - Moon System 87

88 Lagrange Points: Stability L1,L2,L3 are unstable: the satellite drifts away from the equilibrium point for slight perturbations. L4,L5 are stable: the satellite remains around the equilibrium point for slight perturbations. 88

89 Lagrange Points: Physical Interpretation Are the Lagrange points such that the gravitational pulls of the Sun and of the Earth are equal? L1 The forces in equilibrium are the gravitational attraction of the massive bodies and the centrifugal force. 89

90 L1, Physical Interpretation 2-body reasoning: any object going around the Sun in an orbit smaller than Earth's will not keep a fixed station relative to Earth. 3-body reasoning: Earth's gravity cancels some of the gravitational pull of the Sun. With a weaker pull towards the Sun, the spacecraft then needs less speed to maintain its orbit. At L1, the spacecraft will need just one year to go around the Sun. 90

91 L2, Physical Interpretation Same reasoning! 91

92 Sun - Earth System 92

93 Sun - Earth System L1, a solar observatory: ISSE-3, SOHO, ACE L2, a stable environment: Herschel, Planck, JWST, Gaia 93

94 Sun - Earth System Assuming that the required stationkeeping is performed (unstable equilibrium), is it meaningful to maintain at spacecraft at L1? The tracking antennas are also aimed at the Sun, a source of interfering radio waves! Use of a Halo orbit. 94

95 Sun Jupiter System: Trojan Asteroids 95

96 Look for A Constant of the Motion μ μ x 2Ωy Ω x= x+ π r x π r r2 ( ) ( ) r1 r2 μ y y+ 2Ωx Ω y = z μ z = r μ y r1 r2 μ z x y z Σ d ( 2 2) μ1 μ 2 v Ω x + y = dt 2 2 r1 r2 0 96

97 Jacobi Constant ( ) μ r μ r v 2 2 x 2 y Ω + = C 1 2 Kinetic energy + rotation of the reference frame + gravitational pull of the two primary masses = C v is the speed of the secondary mass relative to the rotating frame 97

98 Allowed Regions of Motion ( ) 2μ 2μ 0 Ω r r v x y C 1 2 ( ) 2μ 2μ Ω x + y C = r r Boundaries: Zero velocity curves for a given Jacobi constant. 98

99 Earth - Moon System 99

100

101 Exercise A spacecraft has a burnout velocity at a point on the Earth- Moon line with an altitude of 200 km. Find the value of this velocity for the different scenarios discussed. m m r = kg = kg = km m π = 1 1 m1+ m = x1 = π1r12 = km x = = km y = 0 Coordinates of the burnout point 101

102 Solution ( ) 2μ 2μ r r v= Ω x + y C 1 2 C C C C C C : v = km/s bo : v = km/s bo : v = km/s bo : v = km/s bo : v = km/s bo : v = km/s bo A few m/s can have a significant influence on the regions of Earth- Moon space accessible to the spacecraft! 102

103 Comparison with Escape Velocity v esc,200 km 2μ = = = 11.01km/s r 6578 C C C C C C : v = km/s bo : v = km/s bo : v = km/s bo : v = km/s bo : v = km/s bo : v = km/s bo 103

104 Further Reading on the Course Web Site 104

105 Non-Keplerian Motion 4 Dominant Perturbations Analytic Treatment Numerical Methods Concluding Remarks The Restricted 3-body Problem 105

106 Perturbations Perturbations are small with respect to the spherical central body attraction. But they must be taken into account for accurate orbit propagation. Realistic models for the 4 dominant perturbations and understanding of their effects. 106

107 Orbit Propagation Analytical methods Exact solutions to simple approximating problems. Approximate solutions to approximating problems. Semi-analytical methods Better approximate solutions to realistic problems. Numerical methods Best solutions to most realistic problems. 107

108 STK Propagators 108

109 STK Integrators and Step Size Importance 109

110 Practical Example: GEO Satellites Nice illustration of: 1. Perturbations of the 2-body problem. 2. Secular and periodic contributions. 3. Accuracy requires by practical applications. 4. The need for orbit correction and thrust forces. And it is a real-life example (telecommunications, meteorology)! 110

111 Practical Example: GEO Satellites 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 111

112 Practical Example: GEO Satellites A GEO satellite orbit changes over time due to perturbations: 1. Inclination modified by the lunisolar attraction: N/S. 2. The longitude is modified by the tesseral terms of the geopotential: E/W. 112

113 GEO: Inclination Vector Drift Period HPOP with Sun and Moon (without oblateness/srp/drag) 113

114 GEO: Inclination Vector Drift Period HPOP with Sun and Moon (without oblateness/srp/drag) 114

115 GEO: Longitude Drift C 2,2 corresponds to the equatorial ellipticity. 115

116 GEO: Longitude Drift HPOP with 2,2 (without Sun and moon/srp/drag) 116

117 GEO: Longitude Drift HPOP with 2,2 (without Sun and moon/srp/drag) 117

118 GEO: Longitude Drift HPOP with 2,2 (without Sun and moon/srp/drag) 118

119 Thrust Forces GEO spacecraft require continual stationkeeping to stay within the authorized box using an onboard thruster system. Next lecture: basic orbital maneuvers. 119

120 Astrodynamics (AERO0024) L04: Non-Keplerian Motion Gaëtan Kerschen Space Structures & Systems Lab (S3L)

Astrodynamics (AERO0024)

Astrodynamics (AERO0024) Astrodynamics (AERO0024) 5. Numerical Methods Gaëtan Kerschen Space Structures & Systems Lab (S3L) Why Different Propagators? Analytic propagation: Better understanding of the perturbing forces. Useful

More information

Astrodynamics (AERO0024)

Astrodynamics (AERO0024) Astrodynamics (AERO0024) 4B. Non-Keplerian Motion Gaëtan Kerschen Space Structures & Systems Lab (S3L) 2. Two-body problem 4.1 Dominant perturbations Orbital elements (a,e,i,ω,ω) are constant Real satellites

More information

Astrodynamics (AERO0024)

Astrodynamics (AERO0024) 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

More information

Astrodynamics (AERO0024)

Astrodynamics (AERO0024) 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

More information

Section 13. Orbit Perturbation. Orbit Perturbation. Atmospheric Drag. Orbit Lifetime

Section 13. Orbit Perturbation. Orbit Perturbation. Atmospheric Drag. Orbit Lifetime Section 13 Orbit Perturbation Orbit Perturbation A satellite s orbit around the Earth is affected by o Asphericity of the Earth s gravitational potential : Most significant o Atmospheric drag : Orbital

More information

Astrodynamics (AERO0024)

Astrodynamics (AERO0024) Astrodynamics (AERO0024) 5B. Orbital Maneuvers Gaëtan Kerschen Space Structures & Systems Lab (S3L) Previous Lecture: Coplanar Maneuvers 5.1 INTRODUCTION 5.1.1 Why? 5.1.2 How? 5.1.3 How much? 5.1.4 When?

More information

APPENDIX B SUMMARY OF ORBITAL MECHANICS RELEVANT TO REMOTE SENSING

APPENDIX B SUMMARY OF ORBITAL MECHANICS RELEVANT TO REMOTE SENSING APPENDIX B SUMMARY OF ORBITAL MECHANICS RELEVANT TO REMOTE SENSING Orbit selection and sensor characteristics are closely related to the strategy required to achieve the desired results. Different types

More information

An Analysis of N-Body Trajectory Propagation. Senior Project. In Partial Fulfillment. of the Requirements for the Degree

An Analysis of N-Body Trajectory Propagation. Senior Project. In Partial Fulfillment. of the Requirements for the Degree An Analysis of N-Body Trajectory Propagation Senior Project In Partial Fulfillment of the Requirements for the Degree Bachelor of Science in Aerospace Engineering by Emerson Frees June, 2011 An Analysis

More information

Astrodynamics (AERO0024)

Astrodynamics (AERO0024) Astrodynamics (AERO0024) 5B. Orbital Maneuvers Gaëtan Kerschen Space Structures & Systems Lab (S3L) Previous Lecture: Coplanar Maneuvers 5.1 INTRODUCTION 5.1.1 Why? 5.1.2 How? 5.1.3 How much? 5.1.4 When?

More information

CS 450 Numerical Analysis. Chapter 9: Initial Value Problems for Ordinary Differential Equations

CS 450 Numerical Analysis. Chapter 9: Initial Value Problems for Ordinary Differential Equations Lecture slides based on the textbook Scientific Computing: An Introductory Survey by Michael T. Heath, copyright c 2018 by the Society for Industrial and Applied Mathematics. http://www.siam.org/books/cl80

More information

Lecture V: The game-engine loop & Time Integration

Lecture V: The game-engine loop & Time Integration Lecture V: The game-engine loop & Time Integration The Basic Game-Engine Loop Previous state: " #, %(#) ( #, )(#) Forces -(#) Integrate velocities and positions Resolve Interpenetrations Per-body change

More information

Lecture IV: Time Discretization

Lecture IV: Time Discretization Lecture IV: Time Discretization Motivation Kinematics: continuous motion in continuous time Computer simulation: Discrete time steps t Discrete Space (mesh particles) Updating Position Force induces acceleration.

More information

Applied Math for Engineers

Applied Math for Engineers Applied Math for Engineers Ming Zhong Lecture 15 March 28, 2018 Ming Zhong (JHU) AMS Spring 2018 1 / 28 Recap Table of Contents 1 Recap 2 Numerical ODEs: Single Step Methods 3 Multistep Methods 4 Method

More information

Astrodynamics (AERO0024)

Astrodynamics (AERO0024) Astrodynamics (AERO0024) 3B. The Orbit in Space and Time Gaëtan Kerschen Space Structures & Systems Lab (S3L) Previous Lecture: The Orbit in Time 3.1 ORBITAL POSITION AS A FUNCTION OF TIME 3.1.1 Kepler

More information

Astrodynamics (AERO0024)

Astrodynamics (AERO0024) Astrodynamics (AERO0024) L06: Interplanetary Trajectories Gaëtan Kerschen Space Structures & Systems Lab (S3L) Motivation 2 Problem Statement? Hint #1: design the Earth-Mars transfer using known concepts

More information

Third Body Perturbation

Third Body Perturbation Third Body Perturbation p. 1/30 Third Body Perturbation Modeling the Space Environment Manuel Ruiz Delgado European Masters in Aeronautics and Space E.T.S.I. Aeronáuticos Universidad Politécnica de Madrid

More information

Astrodynamics (AERO0024)

Astrodynamics (AERO0024) Astrodynamics (AERO0024) 3. The Orbit in Space Gaëtan Kerschen Space Structures & Systems Lab (S3L) Motivation: Space We need means of describing orbits in three-dimensional space. Example: Earth s oblateness

More information

Numerical Methods for Differential Equations

Numerical Methods for Differential Equations Numerical Methods for Differential Equations Chapter 2: Runge Kutta and Multistep Methods Gustaf Söderlind Numerical Analysis, Lund University Contents V4.16 1. Runge Kutta methods 2. Embedded RK methods

More information

AS3010: Introduction to Space Technology

AS3010: Introduction to Space Technology AS3010: Introduction to Space Technology L E C T U R E S 8-9 Part B, Lectures 8-9 23 March, 2017 C O N T E N T S In this lecture, we will look at factors that cause an orbit to change over time orbital

More information

Astrodynamics (AERO0024)

Astrodynamics (AERO0024) Astrodynamics (AERO0024) 10. Interplanetary Trajectories Gaëtan Kerschen Space Structures & Systems Lab (S3L) Motivation 2 6. Interplanetary Trajectories 6.1 Patched conic method 6.2 Lambert s problem

More information

Scientific Computing: An Introductory Survey

Scientific Computing: An Introductory Survey Scientific Computing: An Introductory Survey Chapter 9 Initial Value Problems for Ordinary Differential Equations Prof. Michael T. Heath Department of Computer Science University of Illinois at Urbana-Champaign

More information

ANNEX 1. DEFINITION OF ORBITAL PARAMETERS AND IMPORTANT CONCEPTS OF CELESTIAL MECHANICS

ANNEX 1. DEFINITION OF ORBITAL PARAMETERS AND IMPORTANT CONCEPTS OF CELESTIAL MECHANICS ANNEX 1. DEFINITION OF ORBITAL PARAMETERS AND IMPORTANT CONCEPTS OF CELESTIAL MECHANICS A1.1. Kepler s laws Johannes Kepler (1571-1630) discovered the laws of orbital motion, now called Kepler's laws.

More information

Lecture Notes to Accompany. Scientific Computing An Introductory Survey. by Michael T. Heath. Chapter 9

Lecture Notes to Accompany. Scientific Computing An Introductory Survey. by Michael T. Heath. Chapter 9 Lecture Notes to Accompany Scientific Computing An Introductory Survey Second Edition by Michael T. Heath Chapter 9 Initial Value Problems for Ordinary Differential Equations Copyright c 2001. Reproduction

More information

Experimental Analysis of Low Earth Orbit Satellites due to Atmospheric Perturbations

Experimental Analysis of Low Earth Orbit Satellites due to Atmospheric Perturbations Experimental Analysis of Low Earth Orbit Satellites due to Atmospheric Perturbations Aman Saluja #1, Manish Bansal #2, M Raja #3, Mohd Maaz #4 #Aerospace Department, University of Petroleum and Energy

More information

A SEMI-ANALYTICAL ORBIT PROPAGATOR PROGRAM FOR HIGHLY ELLIPTICAL ORBITS

A SEMI-ANALYTICAL ORBIT PROPAGATOR PROGRAM FOR HIGHLY ELLIPTICAL ORBITS A SEMI-ANALYTICAL ORBIT PROPAGATOR PROGRAM FOR HIGHLY ELLIPTICAL ORBITS M. Lara, J. F. San Juan and D. Hautesserres Scientific Computing Group and Centre National d Études Spatiales 6th International Conference

More information

Celestial Mechanics and Satellite Orbits

Celestial Mechanics and Satellite Orbits Celestial Mechanics and Satellite Orbits Introduction to Space 2017 Slides: Jaan Praks, Hannu Koskinen, Zainab Saleem Lecture: Jaan Praks Assignment Draw Earth, and a satellite orbiting the Earth. Draw

More information

The Three Body Problem

The Three Body Problem The Three Body Problem Joakim Hirvonen Grützelius Karlstad University December 26, 2004 Department of Engineeringsciences, Physics and Mathematics 5p Examinator: Prof Jürgen Füchs Abstract The main topic

More information

AM205: Assignment 3 (due 5 PM, October 20)

AM205: Assignment 3 (due 5 PM, October 20) AM25: Assignment 3 (due 5 PM, October 2) For this assignment, first complete problems 1, 2, 3, and 4, and then complete either problem 5 (on theory) or problem 6 (on an application). If you submit answers

More information

Identifying Safe Zones for Planetary Satellite Orbiters

Identifying Safe Zones for Planetary Satellite Orbiters AIAA/AAS Astrodynamics Specialist Conference and Exhibit 16-19 August 2004, Providence, Rhode Island AIAA 2004-4862 Identifying Safe Zones for Planetary Satellite Orbiters M.E. Paskowitz and D.J. Scheeres

More information

Fundamentals of Astrodynamics and Applications

Fundamentals of Astrodynamics and Applications Fundamentals of Astrodynamics and Applications Third Edition David A. Vallado with technical contributions by Wayne D. McClain Space Technology Library Published Jointly by Microcosm Press Hawthorne, CA

More information

Tides and Lagrange Points

Tides and Lagrange Points Ast111, Lecture 3a Tides and Lagrange Points Arial view of Tidal surge in Alaska (S. Sharnoff) Tides Tidal disruption Lagrange Points Tadpole Orbits and Trojans Tidal Bulges Tides Tidal Force The force

More information

Review Higher Order methods Multistep methods Summary HIGHER ORDER METHODS. P.V. Johnson. School of Mathematics. Semester

Review Higher Order methods Multistep methods Summary HIGHER ORDER METHODS. P.V. Johnson. School of Mathematics. Semester HIGHER ORDER METHODS School of Mathematics Semester 1 2008 OUTLINE 1 REVIEW 2 HIGHER ORDER METHODS 3 MULTISTEP METHODS 4 SUMMARY OUTLINE 1 REVIEW 2 HIGHER ORDER METHODS 3 MULTISTEP METHODS 4 SUMMARY OUTLINE

More information

Bindel, Fall 2011 Intro to Scientific Computing (CS 3220) Week 12: Monday, Apr 18. HW 7 is posted, and will be due in class on 4/25.

Bindel, Fall 2011 Intro to Scientific Computing (CS 3220) Week 12: Monday, Apr 18. HW 7 is posted, and will be due in class on 4/25. Logistics Week 12: Monday, Apr 18 HW 6 is due at 11:59 tonight. HW 7 is posted, and will be due in class on 4/25. The prelim is graded. An analysis and rubric are on CMS. Problem du jour For implicit methods

More information

Extending the Patched-Conic Approximation to the Restricted Four-Body Problem

Extending the Patched-Conic Approximation to the Restricted Four-Body Problem Monografías de la Real Academia de Ciencias de Zaragoza 3, 133 146, (6). Extending the Patched-Conic Approximation to the Restricted Four-Body Problem Thomas R. Reppert Department of Aerospace and Ocean

More information

Chapter 5 - Part 1. Orbit Perturbations. D.Mortari - AERO-423

Chapter 5 - Part 1. Orbit Perturbations. D.Mortari - AERO-423 Chapter 5 - Part 1 Orbit Perturbations D.Mortari - AERO-43 Orbital Elements Orbit normal i North Orbit plane Equatorial plane ϕ P O ω Ω i Vernal equinox Ascending node D. Mortari - AERO-43 Introduction

More information

ECE257 Numerical Methods and Scientific Computing. Ordinary Differential Equations

ECE257 Numerical Methods and Scientific Computing. Ordinary Differential Equations ECE257 Numerical Methods and Scientific Computing Ordinary Differential Equations Today s s class: Stiffness Multistep Methods Stiff Equations Stiffness occurs in a problem where two or more independent

More information

Session 6: Analytical Approximations for Low Thrust Maneuvers

Session 6: Analytical Approximations for Low Thrust Maneuvers Session 6: Analytical Approximations for Low Thrust Maneuvers As mentioned in the previous lecture, solving non-keplerian problems in general requires the use of perturbation methods and many are only

More information

Solving PDEs with PGI CUDA Fortran Part 4: Initial value problems for ordinary differential equations

Solving PDEs with PGI CUDA Fortran Part 4: Initial value problems for ordinary differential equations Solving PDEs with PGI CUDA Fortran Part 4: Initial value problems for ordinary differential equations Outline ODEs and initial conditions. Explicit and implicit Euler methods. Runge-Kutta methods. Multistep

More information

Chapter 6 - Ordinary Differential Equations

Chapter 6 - Ordinary Differential Equations Chapter 6 - Ordinary Differential Equations 7.1 Solving Initial-Value Problems In this chapter, we will be interested in the solution of ordinary differential equations. Ordinary differential equations

More information

Mathematics for chemical engineers. Numerical solution of ordinary differential equations

Mathematics for chemical engineers. Numerical solution of ordinary differential equations Mathematics for chemical engineers Drahoslava Janovská Numerical solution of ordinary differential equations Initial value problem Winter Semester 2015-2016 Outline 1 Introduction 2 One step methods Euler

More information

Dynamics and Control of Lunisolar Perturbations for. Highly-Eccentric Earth-Orbiting Satellites

Dynamics and Control of Lunisolar Perturbations for. Highly-Eccentric Earth-Orbiting Satellites Dynamics and Control of Lunisolar Perturbations for Highly-Eccentric Earth-Orbiting Satellites by Matthew Bourassa A thesis submitted to the Faculty of Graduate and Postdoctoral Affairs in partial fulfilment

More information

CS520: numerical ODEs (Ch.2)

CS520: numerical ODEs (Ch.2) .. CS520: numerical ODEs (Ch.2) Uri Ascher Department of Computer Science University of British Columbia ascher@cs.ubc.ca people.cs.ubc.ca/ ascher/520.html Uri Ascher (UBC) CPSC 520: ODEs (Ch. 2) Fall

More information

Numerical Methods - Initial Value Problems for ODEs

Numerical Methods - Initial Value Problems for ODEs Numerical Methods - Initial Value Problems for ODEs Y. K. Goh Universiti Tunku Abdul Rahman 2013 Y. K. Goh (UTAR) Numerical Methods - Initial Value Problems for ODEs 2013 1 / 43 Outline 1 Initial Value

More information

Solving Ordinary Differential equations

Solving Ordinary Differential equations Solving Ordinary Differential equations Taylor methods can be used to build explicit methods with higher order of convergence than Euler s method. The main difficult of these methods is the computation

More information

Fourth Order RK-Method

Fourth Order RK-Method Fourth Order RK-Method The most commonly used method is Runge-Kutta fourth order method. The fourth order RK-method is y i+1 = y i + 1 6 (k 1 + 2k 2 + 2k 3 + k 4 ), Ordinary Differential Equations (ODE)

More information

Ordinary differential equations - Initial value problems

Ordinary differential equations - Initial value problems Education has produced a vast population able to read but unable to distinguish what is worth reading. G.M. TREVELYAN Chapter 6 Ordinary differential equations - Initial value problems In this chapter

More information

CHAPTER 5: Linear Multistep Methods

CHAPTER 5: Linear Multistep Methods CHAPTER 5: Linear Multistep Methods Multistep: use information from many steps Higher order possible with fewer function evaluations than with RK. Convenient error estimates. Changing stepsize or order

More information

Searching for less perturbed elliptical orbits around Europa

Searching for less perturbed elliptical orbits around Europa Journal of Physics: Conference Series PAPER OPEN ACCESS Searching for less perturbed elliptical orbits around Europa To cite this article: J Cardoso dos Santos et al 2015 J. Phys.: Conf. Ser. 641 012011

More information

Satellite Orbital Maneuvers and Transfers. Dr Ugur GUVEN

Satellite Orbital Maneuvers and Transfers. Dr Ugur GUVEN Satellite Orbital Maneuvers and Transfers Dr Ugur GUVEN Orbit Maneuvers At some point during the lifetime of most space vehicles or satellites, we must change one or more of the orbital elements. For example,

More information

Initial value problems for ordinary differential equations

Initial value problems for ordinary differential equations AMSC/CMSC 660 Scientific Computing I Fall 2008 UNIT 5: Numerical Solution of Ordinary Differential Equations Part 1 Dianne P. O Leary c 2008 The Plan Initial value problems (ivps) for ordinary differential

More information

Chapter 13: universal gravitation

Chapter 13: universal gravitation Chapter 13: universal gravitation Newton s Law of Gravitation Weight Gravitational Potential Energy The Motion of Satellites Kepler s Laws and the Motion of Planets Spherical Mass Distributions Apparent

More information

8.1 Introduction. Consider the initial value problem (IVP):

8.1 Introduction. Consider the initial value problem (IVP): 8.1 Introduction Consider the initial value problem (IVP): y dy dt = f(t, y), y(t 0)=y 0, t 0 t T. Geometrically: solutions are a one parameter family of curves y = y(t) in(t, y)-plane. Assume solution

More information

Scientific Computing with Case Studies SIAM Press, Lecture Notes for Unit V Solution of

Scientific Computing with Case Studies SIAM Press, Lecture Notes for Unit V Solution of Scientific Computing with Case Studies SIAM Press, 2009 http://www.cs.umd.edu/users/oleary/sccswebpage Lecture Notes for Unit V Solution of Differential Equations Part 1 Dianne P. O Leary c 2008 1 The

More information

Jim Lambers MAT 772 Fall Semester Lecture 21 Notes

Jim Lambers MAT 772 Fall Semester Lecture 21 Notes Jim Lambers MAT 772 Fall Semester 21-11 Lecture 21 Notes These notes correspond to Sections 12.6, 12.7 and 12.8 in the text. Multistep Methods All of the numerical methods that we have developed for solving

More information

Math 411 Preliminaries

Math 411 Preliminaries Math 411 Preliminaries Provide a list of preliminary vocabulary and concepts Preliminary Basic Netwon's method, Taylor series expansion (for single and multiple variables), Eigenvalue, Eigenvector, Vector

More information

Numerical Methods for the Solution of Differential Equations

Numerical Methods for the Solution of Differential Equations Numerical Methods for the Solution of Differential Equations Markus Grasmair Vienna, winter term 2011 2012 Analytical Solutions of Ordinary Differential Equations 1. Find the general solution of the differential

More information

ACCURACY ASSESSMENT OF GEOSTATIONARY-EARTH-ORBIT WITH SIMPLIFIED PERTURBATIONS MODELS

ACCURACY ASSESSMENT OF GEOSTATIONARY-EARTH-ORBIT WITH SIMPLIFIED PERTURBATIONS MODELS ARTIFICIAL SATELLITES, Vol. 51, No. 2 2016 DOI: 10.1515/arsa-2016-0005 ACCURACY ASSESSMENT OF GEOSTATIONARY-EARTH-ORBIT WITH SIMPLIFIED PERTURBATIONS MODELS Lihua Ma, Xiaojun Xu, Feng Pang National Astronomical

More information

9.6 Predictor-Corrector Methods

9.6 Predictor-Corrector Methods SEC. 9.6 PREDICTOR-CORRECTOR METHODS 505 Adams-Bashforth-Moulton Method 9.6 Predictor-Corrector Methods The methods of Euler, Heun, Taylor, and Runge-Kutta are called single-step methods because they use

More information

AN ANALYTICAL SOLUTION TO QUICK-RESPONSE COLLISION AVOIDANCE MANEUVERS IN LOW EARTH ORBIT

AN ANALYTICAL SOLUTION TO QUICK-RESPONSE COLLISION AVOIDANCE MANEUVERS IN LOW EARTH ORBIT AAS 16-366 AN ANALYTICAL SOLUTION TO QUICK-RESPONSE COLLISION AVOIDANCE MANEUVERS IN LOW EARTH ORBIT Jason A. Reiter * and David B. Spencer INTRODUCTION Collision avoidance maneuvers to prevent orbital

More information

Space Travel on a Shoestring: CubeSat Beyond LEO

Space Travel on a Shoestring: CubeSat Beyond LEO Space Travel on a Shoestring: CubeSat Beyond LEO Massimiliano Vasile, Willem van der Weg, Marilena Di Carlo Department of Mechanical and Aerospace Engineering University of Strathclyde, Glasgow 5th Interplanetary

More information

MODELLING OF PERTURBATIONS FOR PRECISE ORBIT DETERMINATION

MODELLING OF PERTURBATIONS FOR PRECISE ORBIT DETERMINATION MODELLING OF PERTURBATIONS FOR PRECISE ORBIT DETERMINATION 1 SHEN YU JUN, 2 TAN YAN QUAN, 3 TAN GUOXIAN 1,2,3 Raffles Science Institute, Raffles Institution, 1 Raffles Institution Lane, Singapore E-mail:

More information

Circular Motion and Gravitation. Centripetal Acceleration

Circular Motion and Gravitation. Centripetal Acceleration Circular Motion and Gravitation Centripetal Acceleration Recall linear acceleration 3. Going around a curve, at constant speed 1. Speeding up vi vi Δv a ac ac vi ac 2. Slowing down v velocity and acceleration

More information

Angle recap. Angular position: Angular displacement: s. Angular velocity: Angular Acceleration:

Angle recap. Angular position: Angular displacement: s. Angular velocity: Angular Acceleration: Angle recap Angular position: Angular displacement: s Angular velocity: Angular Acceleration: Every point on a rotating rigid object has the same angular, but not the same linear motion! Today s lecture

More information

Linear Multistep Methods I: Adams and BDF Methods

Linear Multistep Methods I: Adams and BDF Methods Linear Multistep Methods I: Adams and BDF Methods Varun Shankar January 1, 016 1 Introduction In our review of 5610 material, we have discussed polynomial interpolation and its application to generating

More information

On Sun-Synchronous Orbits and Associated Constellations

On Sun-Synchronous Orbits and Associated Constellations On Sun-Synchronous Orbits and Associated Constellations Daniele Mortari, Matthew P. Wilkins, and Christian Bruccoleri Department of Aerospace Engineering, Texas A&M University, College Station, TX 77843,

More information

Solving Ordinary Differential Equations

Solving Ordinary Differential Equations Solving Ordinary Differential Equations Sanzheng Qiao Department of Computing and Software McMaster University March, 2014 Outline 1 Initial Value Problem Euler s Method Runge-Kutta Methods Multistep Methods

More information

Astrodynamics (AERO0024)

Astrodynamics (AERO0024) Astrodynamics (AERO0024) 5A. Orbital Maneuvers Gaëtan Kerschen Space Structures & Systems Lab (S3L) Course Outline THEMATIC UNIT 1: ORBITAL DYNAMICS Lecture 02: The Two-Body Problem Lecture 03: The Orbit

More information

Study of the Fuel Consumption for Station-Keeping Maneuvers for GEO satellites based on the Integral of the Perturbing Forces over Time

Study of the Fuel Consumption for Station-Keeping Maneuvers for GEO satellites based on the Integral of the Perturbing Forces over Time Study of the Fuel Consumption for Station-Keeping Maneuvers for GEO satellites based on the Integral of the Perturbing Forces over Time THAIS CARNEIRO OLIVEIRA 1 ; ANTONIO FERNANDO BERTACHINI DE ALMEIDA

More information

Radial Acceleration. recall, the direction of the instantaneous velocity vector is tangential to the trajectory

Radial Acceleration. recall, the direction of the instantaneous velocity vector is tangential to the trajectory Radial Acceleration recall, the direction of the instantaneous velocity vector is tangential to the trajectory 1 Radial Acceleration recall, the direction of the instantaneous velocity vector is tangential

More information

Numerical solution of ODEs

Numerical solution of ODEs Numerical solution of ODEs Arne Morten Kvarving Department of Mathematical Sciences Norwegian University of Science and Technology November 5 2007 Problem and solution strategy We want to find an approximation

More information

ABSTRACT. Manifolds and optimal control were used to better understand trajectories in the

ABSTRACT. Manifolds and optimal control were used to better understand trajectories in the ABSTRACT Title of Thesis: INVESTIGATION OF MANIFOLDS AND OPTIMIZED TRAJECTORIES IN THE THREE-BODY PROBLEM Raquel L. Jarabek, M.S., 004 Thesis Directed By: Assistant Professor Benjamin Shapiro Department

More information

Speed and Accuracy Tests of the Variable-Step Störmer-Cowell Integrator

Speed and Accuracy Tests of the Variable-Step Störmer-Cowell Integrator Speed and Accuracy Tests of the Variable-Step Störmer-Cowell Integrator Matt Berry Analytical Graphics, Inc. Liam Healy Naval Research Laboratory 1 Overview Background Integrators Orbit Propagation Tests

More information

A Variable-Step Double-Integration Multi-Step Integrator

A Variable-Step Double-Integration Multi-Step Integrator A Variable-Step Double-Integration Multi-Step Integrator Matt Berry Virginia Tech Liam Healy Naval Research Laboratory 1 Overview Background Motivation Derivation Preliminary Results Future Work 2 Background

More information

Orbital and Celestial Mechanics

Orbital and Celestial Mechanics Orbital and Celestial Mechanics John P. Vinti Edited by Gim J. Der TRW Los Angeles, California Nino L. Bonavito NASA Goddard Space Flight Center Greenbelt, Maryland Volume 177 PROGRESS IN ASTRONAUTICS

More information

Satellite meteorology

Satellite meteorology GPHS 422 Satellite meteorology GPHS 422 Satellite meteorology Lecture 1 6 July 2012 Course outline 2012 2 Course outline 2012 - continued 10:00 to 12:00 3 Course outline 2012 - continued 4 Some reading

More information

Numerical Methods for Differential Equations

Numerical Methods for Differential Equations Numerical Methods for Differential Equations Chapter 2: Runge Kutta and Linear Multistep methods Gustaf Söderlind and Carmen Arévalo Numerical Analysis, Lund University Textbooks: A First Course in the

More information

Optimal Gravity Assisted Orbit Insertion for Europa Orbiter Mission

Optimal Gravity Assisted Orbit Insertion for Europa Orbiter Mission Optimal Gravity Assisted Orbit Insertion for Europa Orbiter Mission Deepak Gaur 1, M. S. Prasad 2 1 M. Tech. (Avionics), Amity Institute of Space Science and Technology, Amity University, Noida, U.P.,

More information

Gravitation. Luis Anchordoqui

Gravitation. Luis Anchordoqui Gravitation Kepler's law and Newton's Synthesis The nighttime sky with its myriad stars and shinning planets has always fascinated people on Earth. Towards the end of the XVI century the astronomer Tycho

More information

Long-Term Evolution of High Earth Orbits: Effects of Direct Solar Radiation Pressure and Comparison of Trajectory Propagators

Long-Term Evolution of High Earth Orbits: Effects of Direct Solar Radiation Pressure and Comparison of Trajectory Propagators Long-Term Evolution of High Earth Orbits: Effects of Direct Solar Radiation Pressure and Comparison of Trajectory Propagators by L. Anselmo and C. Pardini (Luciano.Anselmo@isti.cnr.it & Carmen.Pardini@isti.cnr.it)

More information

PowerPoints organized by Dr. Michael R. Gustafson II, Duke University

PowerPoints organized by Dr. Michael R. Gustafson II, Duke University Part 6 Chapter 20 Initial-Value Problems PowerPoints organized by Dr. Michael R. Gustafson II, Duke University All images copyright The McGraw-Hill Companies, Inc. Permission required for reproduction

More information

MTH 452/552 Homework 3

MTH 452/552 Homework 3 MTH 452/552 Homework 3 Do either 1 or 2. 1. (40 points) [Use of ode113 and ode45] This problem can be solved by a modifying the m-files odesample.m and odesampletest.m available from the author s webpage.

More information

Lecture D30 - Orbit Transfers

Lecture D30 - Orbit Transfers J. Peraire 16.07 Dynamics Fall 004 Version 1.1 Lecture D30 - Orbit Transfers In this lecture, we will consider how to transfer from one orbit, or trajectory, to another. One of the assumptions that we

More information

Lecture 4: Numerical solution of ordinary differential equations

Lecture 4: Numerical solution of ordinary differential equations Lecture 4: Numerical solution of ordinary differential equations Department of Mathematics, ETH Zürich General explicit one-step method: Consistency; Stability; Convergence. High-order methods: Taylor

More information

RAPID GEOSYNCHRONOUS TRANSFER ORBIT ASCENT PLAN GENERATION. Daniel X. Junker (1) Phone: ,

RAPID GEOSYNCHRONOUS TRANSFER ORBIT ASCENT PLAN GENERATION. Daniel X. Junker (1) Phone: , RAPID GEOSYNCHRONOUS TRANSFER ORBIT ASCENT PLAN GENERATION Daniel X. Junker (1) (1) LSE Space GmbH, Argelsrieder Feld 22, 82234 Wessling, Germany, Phone: +49 160 9111 6696, daniel.junker@lsespace.com Abstract:

More information

orbits Moon, Planets Spacecrafts Calculating the and by Dr. Shiu-Sing TONG

orbits Moon, Planets Spacecrafts Calculating the and by Dr. Shiu-Sing TONG A Science Enrichment Programme for Secondary 3-4 Students : Teaching and Learning Resources the and Spacecrafts orbits Moon, Planets Calculating the 171 of by Dr. Shiu-Sing TONG 172 Calculating the orbits

More information

Ulrich Walter. Astronautics. The Physics of Space Flight. 2nd, Enlarged and Improved Edition

Ulrich Walter. Astronautics. The Physics of Space Flight. 2nd, Enlarged and Improved Edition Ulrich Walter Astronautics The Physics of Space Flight 2nd, Enlarged and Improved Edition Preface to Second Edition Preface XVII Acknowledgments XIX List of Symbols XXI XV 1 Rocket Fundamentals 1 1.1 Rocket

More information

Interplanetary Trajectory Design using Dynamical Systems Theory

Interplanetary Trajectory Design using Dynamical Systems Theory Interplanetary Trajectory Design using Dynamical Systems Theory THESIS REPORT by Linda van der Ham 8 February 2012 The image on the front is an artist impression of the Interplanetary Superhighway [NASA,

More information

AST111, Lecture 1b. Measurements of bodies in the solar system (overview continued) Orbital elements

AST111, Lecture 1b. Measurements of bodies in the solar system (overview continued) Orbital elements AST111, Lecture 1b Measurements of bodies in the solar system (overview continued) Orbital elements Planetary properties (continued): Measuring Mass The orbital period of a moon about a planet depends

More information

Computational Techniques Prof. Dr. Niket Kaisare Department of Chemical Engineering Indian Institute of Technology, Madras

Computational Techniques Prof. Dr. Niket Kaisare Department of Chemical Engineering Indian Institute of Technology, Madras Computational Techniques Prof. Dr. Niket Kaisare Department of Chemical Engineering Indian Institute of Technology, Madras Module No. # 07 Lecture No. # 05 Ordinary Differential Equations (Refer Slide

More information

Astrodynamics (AERO0024)

Astrodynamics (AERO0024) Astrodynamics (AERO0024) 5A. Orbital Maneuvers Gaëtan Kerschen Space Structures & Systems Lab (S3L) GEO drift: period The orbit inclination will increase to a maximum of 15 deg after 27.5 years and return

More information

AP Physics 1 Chapter 7 Circular Motion and Gravitation

AP Physics 1 Chapter 7 Circular Motion and Gravitation AP Physics 1 Chapter 7 Circular Motion and Gravitation Chapter 7: Circular Motion and Angular Measure Gravitation Angular Speed and Velocity Uniform Circular Motion and Centripetal Acceleration Angular

More information

ESMO Mission Analysis

ESMO Mission Analysis Changing the economics of space ESMO Mission Analysis SRR Workshop Alison Gibbings 22 nd 26 th March 2010 Review of the existing baseline Sensitivity analysis Contents At lunar Injection Along the WSB-Moon

More information

Orbital Mechanics! Space System Design, MAE 342, Princeton University! Robert Stengel

Orbital Mechanics! Space System Design, MAE 342, Princeton University! Robert Stengel Orbital Mechanics Space System Design, MAE 342, Princeton University Robert Stengel Conic section orbits Equations of motion Momentum and energy Kepler s Equation Position and velocity in orbit Copyright

More information

Deorbiting Upper-Stages in LEO at EOM using Solar Sails

Deorbiting Upper-Stages in LEO at EOM using Solar Sails Deorbiting Upper-Stages in LEO at EOM using Solar Sails Alexandru IONEL* *Corresponding author INCAS National Institute for Aerospace Research Elie Carafoli, B-dul Iuliu Maniu 220, Bucharest 061126, Romania,

More information

Keplerian Elements Tutorial

Keplerian Elements Tutorial Keplerian Elements Tutorial This tutorial is based on the documentation provided with InstantTrack, written by Franklin Antonio, N6NKF. Satellite Orbital Elements are numbers that tell us the orbit of

More information

Satellite Geodesy and Navigation Present and Future

Satellite Geodesy and Navigation Present and Future Satellite Geodesy and Navigation Present and Future Drazen Svehla Institute of Astronomical and Physical Geodesy Technical University of Munich, Germany Content Clocks for navigation Relativistic geodesy

More information

Astrodynamics (AERO0024)

Astrodynamics (AERO0024) Astrodynamics (AERO0024) L05: Orbital Maneuvers Gaëtan Kerschen Space Structures & Systems Lab (S3L) North Korea Launch Vehicle WorldView1 satellite (Google Earth). 0.5m resolution. 2 Course Outline THEMATIC

More information

MULTI PURPOSE MISSION ANALYSIS DEVELOPMENT FRAMEWORK MUPUMA

MULTI PURPOSE MISSION ANALYSIS DEVELOPMENT FRAMEWORK MUPUMA MULTI PURPOSE MISSION ANALYSIS DEVELOPMENT FRAMEWORK MUPUMA Felipe Jiménez (1), Francisco Javier Atapuerca (2), José María de Juana (3) (1) GMV AD., Isaac Newton 11, 28760 Tres Cantos, Spain, e-mail: fjimenez@gmv.com

More information

5.12 The Aerodynamic Assist Trajectories of Vehicles Propelled by Solar Radiation Pressure References...

5.12 The Aerodynamic Assist Trajectories of Vehicles Propelled by Solar Radiation Pressure References... 1 The Two-Body Problem... 1 1.1 Position of the Problem... 1 1.2 The Conic Sections and Their Geometrical Properties... 12 1.3 The Elliptic Orbits... 20 1.4 The Hyperbolic and Parabolic Trajectories...

More information

Celestial Mechanics Lecture 10

Celestial Mechanics Lecture 10 Celestial Mechanics Lecture 10 ˆ This is the first of two topics which I have added to the curriculum for this term. ˆ We have a surprizing amount of firepower at our disposal to analyze some basic problems

More information