Astrodynamics (AERO0024)

Transcription

1 Astrodynamics (AERO0024) The Orbit in Space and Time Gaëtan Kerschen Space Structures & Systems Lab (S3L)

2 Newton s laws F = ma F g mm = G r 1 2 uˆ 2 r Relative motion μ r = r 3 r Energy conserv. 2 v μ ε = 2 r Angular mom. h= r r Azim. velocity r Kepler s 1 st law 2 h 1 = μ 1+ ecosθ The orbit equation v = h/ r Kepler s 2 nd law da / dt = h /2 We have lost track of the time variable! Kepler s 3 rd law T = 2 π a / μ

3 Motivation: Time Fundamental to astrodynamics is keeping track of when events occur. A robust time system is therefore crucial. For example: when will I see the ISS or Halley s comet in the sky? 3

4 Motivation: Space The discussion in Lecture 2 was confined to the plane of the orbit; i.e., to two dimensions. We need means of describing orbits in three-dimensional space. Example: Earth s oblateness YES! Atmosphere NO! 4

5 3-D Space: OUFTI-1 Example Two-body propagator J2 propagator 5

6 3-D Space: OUFTI-1 Example Two-body propagator HPOP propagator (two-body + drag) 6

7 STK 7

8 The Orbit in Space and Time The Orbit in Time Time Systems Concluding Remarks The Orbit in Space 8

9 Time Since Periapsis What is the time required to fly between any two true anomaly? Hint: Kepler s law 2 μ 3 h dt = dθ 1+ ecosθ ( ) 2 da 1 dθ h r dt 2 dt 2 2 = = = Kepler s second law r = 2 h 1 μ 1+ ecosθ constant 2 μ ( ) θ dθ t t 3 p = h 0 1+ ecosθ Sixth missing constant (t p =0) ( ) 2 Orbit equation 2 μ θ dθ t = 3 h 0 1+ ecosθ ( ) 2 9

10 Circular Orbits 2 μ θ dθ t = 3 h 0 1+ ecosθ ( ) 2 2 μ h t = θ 3 0 dθ 3 3/2 h r T t = θ θ θ 2 μ = μ = 2π θ = 2π t T Obvious, because the angular velocity is constant. 10

11 Elliptic Orbits Because the angular velocity of a spacecraft along an eccentric orbit is continuously varying (see constant areal velocity), the expression of the angular position versus time is no longer trivial. 2 μ h t = 3 0 0< e< 1 θ dθ ( 1+ ecosθ ) 2 Mean anomaly, M e θ e 1 e sinθ = 2 tan tan ( 2 ) 3/2 1 e 1 e ecosθ 3 a Tellip = 2π μ h= μa e 2 (1 ) 2 μ ( 2 ) 3/2 2π M = 1 e t = t = nt 3 h T 11

12 Mean Anomaly For circular orbits, the mean M and true anomalies θ are identical. For elliptic orbits, the mean anomaly represents the angular displacement of a fictitious body moving around the ellipse at the constant angular speed n. 12

13 Simplification: Eccentric Anomaly E r θ a ae 13

14 Eccentric Anomaly acos E = ae + rcosθ Graph e + cos E cos 2 1 e = r = a L2 1 + ecos 1 cosθ + e θ θ 2 2 cos E+ sin E = 1 sin E = 2 1 e sinθ 1+ ecosθ cos E sin E = e + cosθ = 1 + ecosθ 2 1 e sinθ 1+ ecosθ tan E 1 cose 1 e θ = = tan 2 1+ cose 1+ e E 1 1 e θ = 2tan tan 1+ e 2 14

15 Kepler s Equation M sin 2tan e tan e e θ θ = 1 e ecosθ M = E sin E It relates time, in terms of M=nt, to position, in terms of E, r=a(1-e.cose). 15

16 Usefulness of Kepler s Equation θ is given E 1 1 e θ = 2tan tan 1+ e 2 M = E sin E Practical application: Determine the time at which a satellite passes from sunlight into the Earth s shadow (the location of this point is known from the geometry). t = M T 2π 16

17 Example A geocentric elliptic orbit has a perigee radius of 9600 km and an apogee radius of km. Calculate the time to fly from perigee to a true anomaly of 120. L2 ra rp e = = = ra + rp e E 2 tan θ = tan = rad 1+ e 2 M = E sin E = rad 3 a rp + ra T = 2π = 2 π / μ = 18834s μ 2 M t = T s=1h07m57s 2π = 2π = 3 17

18 θ=0, t=12h θ=120, t=13h07m57s OK!

19 Usefulness of Kepler s Equation t is given M = 2π t T M = E sin E Practical application: Perform a rendez-vous with the ISS (ATV, STS, Soyuz, Progress). Transcendental equation!!! (with a unique solution) 19

20 Numerical Solution: Newton-Raphson Algorithm for finding approximations to the zeros of a nonlinear function. Recursive application of Taylor series truncated after the first derivative. The initial guess should be close enough to the actual solution. 20

21 Numerical Solution: Newton-Raphson Example: find the zero of f(x)=0.5(x-1) f(x) x 21

22 Analytic Solution: Lagrange E M a e = + n= 1 n n floor( n /2) 1 1 n 1 2 k = 0 n k! k! k n 1 ( 1) ( 2 ) sin ( 2 ) ( ) an = n k n k M Convergence if e< For small values of the eccentricity a good agreement with the exact solution is obtained using a few terms (e.g., 3). 22

23 Analytic Solution: Bessel Functions 2 E = M + Jn ( ne) sin nm n J n ( x) n= 1 ( 1) ( ) k n+ 2k x = k = 0 n+ k! k! 2 Convergence for all values of the eccentricity less than 1. 23

24 Parabolic Orbits 2 μ θ dθ 1 θ 1 3 θ t = = tan + tan h ( 1+ cosθ ) M = t = tan + tan 3 h μ θ θ 24

25 Hyperbolic Orbits F e+ 1 e 1tan( θ /2) = ln e+ 1 e 1tan( θ /2) M = esinh F F 25

26 Prediction of the Position and Velocity If the position and velocity r 0 and v 0 of an orbiting body are known at a given instant t 0, how can we compute the position and velocity r and v at any later time t? Concept of f and g function and series: r() t = f(, t t, r, v ) r + g(, t t, r, v ) v a f = 1 1 cos( E E0) r 0 (*) [ ] (*) 3 a g = ( t t0) ( E E0) sin( E E0) μ (*) J.E. Prussing, B.A. Conway, Orbital Mechanics, Oxford University Press 26

27 Prediction of the Position and Velocity Some form of Kepler s equation must still be solved by iteration. However, Gauss developed a series expansion in the elapsed time parameter t-t 0, and there is no longer the need to solve Kepler s equation: r() t = f(, t t, r, v ) r + g(, t t, r, v ) v (*) f μ ( 0) 5 ( 0)... 2 t t μ r v r0 2 r t t = (*) g ( t t μ ) 3 ( 0) 5 ( 0)... 6 t t μ r v r0 4 r t t = (*) J.E. Prussing, B.A. Conway, Orbital Mechanics, Oxford University Press 27

28 Matlab Example 28

29 Verification in STK t r 0 and v 0 29

30 OK!

31 The Orbit in Space and Time The Orbit in Time Time Systems Concluding Remarks The Orbit in Space 31

32 Did you adjust your watch last New Year s Eve?

33 Everyday Time Systems Conventional local time systems are not robust enough for orbital mechanics: They depend on the user s position on Earth. They are in a format (Y/M/D/H/M/S) that does not lend itself to use in a computer-implemented algorithm. For instance, what is the difference between any two dates? What would be a meaningful time system? 33

34 The Ingredients of a Time System 1. The interval. 2. The epoch. 34

35 International Atomic Time (TAI) Standard for the SI unit of duration: second. Duration of cycles of radiation corresponding to the transition between two hyperfine levels of the ground state of 133 Cs. Weighted average of the time kept by about 300 atomic clocks in over 50 national laboratories worldwide. What is the inherent problem of TAI? No connection with the motion of the sun across the sky! 35

36 Solar Time The time used in everyday life. Apparent solar day is the time required for the sun to lie on the same meridian. The length varies throughout the year (due to the eccentricity of Earth s orbit). 36

37 Solar Time The real sun is not well suited for time reckoning purposes (seasonal variations of the sun s apparent motion). A mean solar day comprises 24 hours. It is the time interval between successive transits of a fictitious mean sun over a given meridian. A constant velocity in the motion about the sun is therefore assumed. Concept of universal time (UT), based on the Earth s rotation on its axis. 37

38 Apparent and Mean Solar Days 38

39 Equation of Time Approximation where E is in minutes, sin and cos in degrees, and N is the day number: E = 9.87sin 2B 7.53cos B 1.5sin B with B = 360 ( N 81)

40 Universal Time: Different Versions UT: At noon the fictitious sun lies on the Greenwich meridian. UT0: Determined at an observatory by observing the diurnal motion of stars (more precise than sun-based observations). UT1: UT0 with a correction for polar motion. UT1 is the same everywhere on Earth, and is proportional to the true rotation angle of the Earth with respect to a fixed frame of reference. But the rotational speed of the Earth is not uniform! UT2: rarely used. 40

41 Digression: Polar Motion Movement of Earth s rotation axis across its surface. Difference between the instantaneous rotational axis and the conventional international origin (CIO a conventionally defined reference axis of the pole's average location over the year 1900). The drift, about 20 m since 1900, is partly due to motions in the Earth's core and mantle, and partly to the redistribution of water mass as the Greenland ice sheet melts and to isostatic rebound. 41

42 Polar Motion CIO: fixed with respect to the surface of the Earth CEP: periodic motion (celestial ephemeris pole) 42 [End of digression]

43 Physical and Astronomical Times Astronomical clocks: Related to everyday life. Not consistent; the length of one second of UT is not constant. Typical accuracies ~10-8. Atomic clocks: Consistent. Typical accuracies ~ Not related to everyday life. If no adjustment is made, then within a millennium, local noon (i.e., the local time associated with the Sun s zenith position) would occur at 13h00 and not 12h00. 43

44 Coordinated Universal Time (UTC) A good practical compromise? It is the international standard on which civil time is based. Its time interval corresponds to atomic time TAI: It is accurate. Its epoch differs by no more than 0.9 sec from UT1: The mean sun is overhead on the Greenwich meridian at noon. But leap seconds are introduced to account for the fact that the Earth currently runs slow at 2 milliseconds per day (tidal friction in the Earth-Moon system). 44

45 DUT1 = UT1-UTC <0.9s 45

46 UTC, UT1 & TAI 46

47 Le Soir Article 47

48 STK 48

49 Atomic times TAI Earth rotation times Sidereal time GMST in UT1 = T T T 3 with T=(JD )/36525 Solar time UT0: determined from the motion of the stars Polar motion UT1= UT0 - tan(lat)[x.sin(long)+y.cos(long)] UTC = TAI - # of leap seconds = UT1 - DUT1

50 Leap Seconds: Pros and Cons This is currently the subject of intense debate (UT vs. TAI; i.e., UK vs. France). + Abandoning leap seconds would break sundials. In thousands of years, 16h00 would occur at 03h00. The British who have to wake up early in the morning to have tea Astronomers - Leap seconds are a worry with safety-critical real-time systems (e.g., air-traffic control GPS' internal atomic clocks race ahead of UTC). 50

51 Earth s Rotation Speed The mean solar day (24h) is not the time it takes the Earth to perform one revolution in inertial space. 51

52 Another Time System: Sidereal Time The time used in astronomy. A sidereal day is the time it takes for a distant star to lie on the same meridian. One sidereal day is 23h56m4s. There is one more day in a sidereal year than in a solar year. The Greenwich sidereal day begins when the vernal equinox is on the Greenwich Meridian. Greenwich mean sidereal time (GMST) is the hour angle of the average position of the vernal equinox. 52

53 Vernal Equinox: Reference Direction The vernal equinox is the intersection of the ecliptic and equator planes, where the sun passes from the southern to the northern hemisphere. The Sun is located vertically above a point on the equator. First day of spring in the northern hemisphere. The number of hours of daylight and darkness is equal. 53

54 Vernal Equinox: Lunisolar Precession Because of the gravitational tidal forces of the Moon and Sun The Earth s spin axis precesses westward around the normal to the ecliptic at a rate of 1.4 /century. ω E F: dominant force on the spherical mass. ecliptic f 2 f 1 F f 1, f 2 : forces due to the bulging sides; f 1 > f 2, which implies a net clockwise moment. 54

55 Vernal Equinox: Lunisolar Precession Competition between two effects: 1. Gyroscopic stiffness of the spinning Earth (maintain orientation in inertial space). T=26000 years 2. Gravity gradient torque (pull the equatorial bulge into the plane of the ecliptic). 55

56 Vernal Equinox: Nutation The obliquity of the Earth varies with a maximum amplitude of over a period of 18.6 years. This nutation is caused by the motion of Lunar orbital nodes. They complete a revolution in 18.6 years. 56

57 Vernal Equinox: Overall Polar motion 57

58 Yet More Time Systems! GPS time: running ahead of UTC but behind TAI (it was set in 1980 based on UTC, but leap seconds were ignored since then). Time standards for planetary motion calculations: Ephemeris time (obsolete, fraction of the tropical year). Terrestrial dynamic time: tied to TAI but with an offset of s to provide continuity with ephemeris time. Barycentric dynamic time: similar to TDT but includes relativistic corrections that move the origin of the solar system barycenter. 58

59 Greenwich Mean Time (GMT) Obsolete time systems referring to mean solar time. Superseded by UT1. Control room Mercury program 59

60 Further Reading on the Course Web Site 60

61 Julian Date The Julian day number is the number of days since noon January 1, 4713 BC Continuous time scale and no negative dates. For historical reasons, the Julian day begins at noon, and not midnight, so that astronomers observing the heavens at night do not have to deal with a change of date. The number of days between two events is found by subtracting the Julian day of one from that of the other. 61

62 Forward Computation Conversion from conventional time (YYYY, MM, DD, hh:mm:ss.ss) to Julian Date: 7 MM +9 JD = 367. YYYY floor floor 4 YYYY M 1 1 ss floor + DD mm + hh Valid for the period 1 st March 1900 and 28 th February

63 Backward Computation 63

64 Matlab and STK Examples 12 May 2004 at 14:45:30 UTC: days 64

65 Computation of Elapsed Time Find the elapsed time between 4 October 1957 at 19:26:24 UTC and 12 May 2004 at 14:45:30 UTC 4 October 1957 at 19:26:24 UTC: days 12 May 2004 at 14:45:30 UTC: days The elapsed time is days 65

66 Standard Epoch Used Today: J2000 To lessen the magnitude of the Julian date, a constant offset can be introduced. A different reference epoch 1 st January 2000 at noon is used: J2000 = JD

67 Remarks Typically, a 64-bit floating point (double precision) variable can represent an epoch expressed as a Julian Date to about 1 millisecond precision. Leap seconds and Julian day? Leap seconds are not taken into account. Therefore calculations made for Julian Dates where a leap second is inserted will be inaccurate by up to one second. Since leap seconds are only inserted at midnight UTC, they will always occur at the middle of a Julian Day. Julian Date calculator: 67

68 The Orbit in Space and Time The Orbit in Time Time Systems Concluding Remarks The Orbit in Space 68

69 Reference Frames & Coordinate Systems A single coordinate system will not suffice: 1. Geocentric right ascension-declination frame (spherical) 2. Geocentric equatorial frame (cartesian, inertial ) 3. Geocentric equatorial frame (cartesian, Earth-fixed) 4. Topocentric horizon coordinate system (polar) 5. Orbital elements (classical) 69

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71 Geocentric RA-Declination Frame For astronomers Celestial coordinate system (spherical coordinates): ˆK The N and S poles correspond to those of the Earth. Right ascension α (longitude) is measured along the celestial equator in degrees east from the vernal equinox. Latitude is the declination δ (in degrees, positive to the North). Current reference epoch: J2000. Î Ĵ Celestial sphere: imaginary sphere with a gigantic radius 71

72 A View of the Sky View above the eastern horizon from 0 longitude on the equator at 9am local time, 20 March,

73 Geocentric Equatorial Frame (ECI) For computation Same as the celestial coordinate system but Cartesian coordinates Origin at the Earth s center. z-axis is parallel to Earth s rotation vector. x-axis points in the vernal equinox direction. y-axis: right-handed set. Does not rotate with the Earth; assumed to be inertial. 73

74 ISS in ECI 74

75 Cartesian Spherical r = XIˆ+ YJˆ + ZKˆ = ruˆ r uˆ cos cos ˆ cos sin ˆ sin ˆ δ α δ α δ r = I+ J+ K 75

76 Geocentric Equatorial Frame (ECEF) For ground tracks Earth-fixed cartesian coordinate system. Origin at the Earth s center. z-axis is parallel to Earth s rotation vector. x-axis passes through the Greenwich meridian. y-axis: right-handed set. 76

77 OUFTI-1 in ECEF 77

78 ECI ECEF Precession, nutation, polar motion ignored 78

79 Further Reading on the Course Web Site 79

80 Topocentric Horizon Coordinate System SEZ or NEZ For satellite tracking 80

81 Orbitron 81

82 Orbitron: Close-Up 82

83 Orbital (Keplerian) Elements For interpretation r and v do not directly yield much information about the orbit. We cannot even infer from them what type of conic the orbit represents! Another set of six variables, which is much more descriptive of the orbit, is needed. 83

84 6 Orbital (Keplerian) Elements 1. e: shape of the orbit 2. a: size of the orbit 3. i: orients the orbital plane with respect to the ecliptic plane 4. Ω: longitude of the intersection of the orbital and ecliptic planes 5. ω: orients the semi-major axis with respect to the ascending node 6. ν: orients the celestial body in space definition of the ellipse definition of the orbital plane orientation of the ellipse within the orbital plane position of the satellite on the ellipse 84

85 Orbital plane orientation of the ellipse position of the satellite Ecliptic plane

86 Orbital Elements From State Vector rv, aie,, Ω ω θ,, 86

87 Orbital Elements From State Vector ˆK Ĵ Î 87

88 Hints Where is h? Where is e? Use the line of nodes Start with i and Ω 88

89 Inclination (Geometrical Arguments) Normal to the orbit plane h= r v h = hk. ˆ z Normal to the equatorial plane ˆ 1 h 1 ( ). i cos z cos r v K = = h r v 89

90 Longitude Ω (Geometrical Arguments) n = Kˆ h h Nodal vector Where is n? In the orbital and equatorial planes, and, hence, along the line of nodes ˆ r v K. Iˆ 1 ni.ˆ 1 Ω= cos = cos r v n ˆ r v K If nj. ˆ 0 r v 90

91 Eccentricity (First Integral of Motion) μ μ r = = ( ) r μ r e v h v r v r e = v r v r r ( ) μ / μ 91

92 Argument of Perigee (Geom. Arguments) e v ( r v) r = μ r Its direction corresponds to the apse line ˆ r v v ( r v) r K. 1. μ r ne 1 ω = cos = cos r v ne ˆ ( ) r v v r v r K μ r If ek. ˆ 0 92 r v

93 True Anomaly (Geometrical Arguments) v ( r v) r r. 1 re. 1 μ r θ = cos = cos re v ( r v) r r μ r If rv. 0 93

94 Semi-Major Axis (Vis-Viva Equation) v ellip 2 1 = μ r a r = r, v= v a = 2 r 2 rv μ 94

95 Remarks If nj. ˆ < 0 1.ˆ 2π cos ni Ω= n If ek<. ˆ 0 If rv<. 0 ne ω = 2π cos ne 1. re θ = 2π cos re 1. 95

96 Matlab and STK Examples 96

97 State Vector From Orbital Elements 1. Solve for r r 2 a(1 e ) = 1 + ecosθ 2. Solve for v v 2 1 = μ r a 3. Solve for γ ( 2 ) h= rvcosγ = μa 1 e 4. Find the unit vector rhat in the direction of r by rotating the unit vector Ihat (direction of first point in Aries) through the Euler angles Ω,I and θ. 97

98 State Vector From Orbital Elements 5. Define a vector normal to the orbit plane c= r nˆ = r (cos Ω Iˆ+ sin ΩJˆ) 6. Solve the system v = v + v + v x y z rv. = rvsinγ = r v + r v + r v x x y y z z cv. = 0= cv + cv + cv x x y y z z 98

99 Matlab Example 99

100 Two-Line Elements (TLE) For monitoring by Norad * * North American Aerospace Defense Command 100

101 STK: Generate and Propagate TLE U

102 Orbitron: Load and Propagate TLE 102

103 Celestrak: Update TLE 103

104 Celestrak: ISS, February 24,

105 Celestrak: IRIDIUM 33, February 24,

106 Yet More Coordinate Systems! Satellite coordinate system For ADCS Perifocal coordinate system Heliocentric coordinate system Natural frame for an orbit (z is zero) For interplanetary missions Non-singular elements For particular orbits 106

107 The Orbit in Space and Time The Orbit in Time Time Systems Concluding Remarks The Orbit in Space 107

108 Concluding Remarks We have recovered the time variable, and we can compute the spacecraft position versus time. We have a 3D view of the orbit. Time and coordinate systems are complex! 108

109 Astrodynamics (AERO0024) The Orbit in Space and Time Gaëtan Kerschen Space Structures & Systems Lab (S3L)