Astrodynamics (AERO0024)

Transcription

1 Astrodynamics (AERO0024) 3B. The Orbit in Space and Time Gaëtan Kerschen Space Structures & Systems Lab (S3L)

2 Previous Lecture: The Orbit in Time 3.1 ORBITAL POSITION AS A FUNCTION OF TIME Kepler s equation Orbit propagation 3.2 TIME SYSTEMS Clocks Universal time Earth s motion Atomic time Coordinated universal time Julian date 2

3 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! 3

4 3-D Space: OUFTI-1 Example Two-body propagator J2 propagator 4

5 3-D Space: OUFTI-1 Example Two-body propagator HPOP propagator (two-body + drag) 5

6 Complexity of Coordinate Systems: STK 6

7 3. The Orbit in Space and Time? 3.3 Inertial frames 3.4 Coordinate systems ω i α δ a θ e Ω 3.5 Coordinate types 7

8 3. The Orbit in Space and Time? 3.3 Inertial frames ICRS ICRF ω i α δ a θ e Ω 8

9 Importance of Inertial Frames An inertial reference frame is defined as a system that is neither rotating nor accelerating relative to a certain reference point. Suitable inertial frames are required for orbit description (remember that Newton s second law is to be expressed in an inertial frame). An inertial frame is also an appropriate coordinate system for expressing positions and motions of celestial objects ICRS 9

10 Reference System and Reference Frame Distinction between reference system and a reference frame: 1. A reference system is the complete specification of how a celestial coordinate system is to be formed. For instance, it defines the origin and fundamental planes (or axes) of the coordinate system. 2. A reference frame consists of a set of identifiable points on the sky along with their coordinates, which serves as the practical realization of a reference system ICRS 10

11 International Celestial Reference System (ICRS) The ICRS is the reference system of the International Astronomical Union (IAU) for high-precision astronomy. Its origin is located at the barycenter of the solar system. Definition of non-rotating axes: 1. The celestial pole is the Earth s north pole (or the fundamental plane is the Earth's equatorial plane). 2. The reference direction is the vernal equinox (point at which the Sun crosses the equatorial plane moving from south to north). 3. Right-handed system ICRS 11

12 Vernal Equinox? Vallado, Kluwer, The vernal equinox is the intersection of the ecliptic and equator planes, where the sun passes from the southern to the northern hemisphere (First day of spring in the northern hemisphere). Today, the vernal equinox points in the direction of the constellation Pisces, whereas it pointed in the direction of the constellation Ram during Christ s lifetime. Why? 12

13 Need To Specify a Date Remember the complicated motion of the Earth ICRS 13

14 Need To Specify a Date Because the ecliptic and equatorial planes are moving, the coordinate system must have a corresponding date: "the pole/equator and equinox of [some date]". For ICRS, the equator and equinox are considered at the epoch J (January 1, 2000 at 11h58m56s UTC) ICRS 14

15 ICRS in Summary Quasi-equatorial coordinates at the solar system barycenter! [ in the framework of general relativity ] An object is located in the ICRS using right ascension and declination But how to realize ICRS practically? ICRS 15

16 Previous Realizations: B1950 and J2000 B1950 and J2000 were considered the best realized inertial axes until the development of ICRF. They exploit star catalogs (FK4 and FK5, respectively) which provide mean positions and proper motions for classical fundamental stars (optical measurements): FK4 was published in 1963 and contained 1535 stars in various equinoxes from 1950 to FK5 was an update of FK4 in 1988 with new positions for the 1535 stars. STK ICRF 16

17 Fifth Fundamental Catalog (FK5), available on the web site

18 Star Catalogs: Limitations and Improvement 1. The uncertainties in the star positions of the FK5 are about milliarcseconds over most of the sky. 2. A stellar reference frame is time-dependent because stars exhibit detectable motions. 1. Uncertainties of radio source positions are now typically less than one milliarcsecond, and often a factor of ten better. 2. Radio sources are not expected to show measurable intrinsic motion ICRF 18

19 Fifth Fundamental Catalog (FK5), available on the web site

20 ICRF is the Current Realization of ICRS Since 1998, IAU adopted the International Celestial Reference Frame (ICRF) as the standard reference frame: quasi-inertial reference frame with barely no time dependency. It represents an improvement upon the theory behind the J2000 frame, and it is the best realization of an inertial frame constructed to date ICRF 20

21 Very Long Baseline Interferometry STK ICRF 21

22 Further Reading on the Web Site ICRF 22

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24 Formal Definition of ICRS It is defined by the measured positions of 212 extragalactic sources (mainly quasars). 1. Its origin is located at the barycenter of the solar system through appropriate modeling of VLBI observations in the framework of general relativity. 2. Its pole is in the direction defined by the conventional IAU models for precession (Lieske et al. 1977) and nutation (Seidelmann 1982). 3. Its origin of right ascensions was implicitly defined by fixing the right ascension of the radio source 3C273B to FK5 J2000 value ICRF 24

25 3. The Orbit in Space and Time? 3.4 Coordinate systems ω i α δ a θ e Ω 25

26 Coordinate Systems Now that we have defined an inertial reference frame, other reference frames can be defined according to the needs of the considered application. Coordinate transformations between two reference frames involve rotation and translation. What are the possibilities for a satellite in Earth orbit? 3.4 Coordinate systems 26

27 Geocentric Inertial (ECI) A geocentric-equatorial system is clearly convenient. The geocentric celestial reference frame (GCRF) is the geocentric counterpart of the ICRF and is the standard inertial coordinate system for the Earth. Within AGI products, the term ICRF coordinate system is not restricted to the system whose origin is at the solar system barycenter--- rather, the term describes a coordinate system whose origin is determined from context whose axes are aligned with the axes of the BCRF. 3.4 Coordinate systems 27

28 ICRF in STK 3.4 Coordinate systems 28

29 B1950, J2000 and ICRF in STK 29

30 Geocentric Fixed (ECEF) 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. For ground tracks and force computation. 3.4 Coordinate systems 30

31 OUFTI-1 in ECEF 3.4 Coordinate systems 31

32 ECEF-ECI Transformation It includes precession, nutation, and rotation effects, as well as pole wander and frame corrections. ECEF e.g., ITRF PM ST N P ECI e.g., FK5 3.4 Coordinate systems 32

33 ECEF-ECI Transformation Vallado, Fundamental of Astrodynamics and Applications, Kluwer,

34 STK Illustration Period of the Earth s precession = years 34

35 ECEF-ECI Transformation Simplified transformation Precession, nutation, polar motion ignored 3.4 Coordinate systems 35

36 Topocentric On the Earth s Surface SEZ or NEZ For satellite tracking 3.4 Coordinate systems 36

37 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 3.4 Coordinate systems 37

38 3. The Orbit in Space and Time? ω i α δ a θ e Ω 3.5 Coordinate types 38

39 39

40 Cartesian and Spherical 1. Cartesian: for computations 2. Spherical: azimuth and elevation (for ground station) right ascension and declination (for astronomers) ˆK Ĵ Î 3.5 Coordinate types 40

41 Cartesian Spherical r XIˆ YJˆ ZKˆ ruˆ r uˆ cos cos Iˆ cos sin Jˆ sin Kˆ r 3.5 Coordinate types 41

42 Orbitron 42

43 Orbitron: Close-Up 3.5 Coordinate types 43

44 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. 3.5 Coordinate types 44

45 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 3.5 Coordinate types 45

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

47 Orbital Elements From State Vector rv, a,, i e,, 3.5 Coordinate types 47

48 Orbital Elements From State Vector ˆK Ĵ Î 3.5 Coordinate types 48

49 Hints Order: e,a / i,ω,,ω,θ e,a: L2 i: where is h? Ω: use the line of nodes ω: use e θ: use ω 3.5 Coordinate types 49

50 Eccentricity (First Integral of Motion) r r r e v h v r v r e v r v r r / 3.5 Coordinate types 50

51 Semi-Major Axis (Vis-Viva Equation) v ellip 2 1 r a r r, v v a 2 r 2 rv 3.5 Coordinate types 51

52 Inclination (Geometrical Arguments) Normal to the orbit plane h rv h hk. ˆ z Normal to the equatorial plane i ˆ 1h 1 ( ). cos z cos r v K h r v 3.5 Coordinate types 52

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

54 Argument of Perigee (Geom. Arguments) e vr v r r Its direction corresponds to the apse line If cos ek. ˆ 0 ˆ r v vr v r K. ne. r cos r v ne ˆ r v v r v r K r r v

55 True Anomaly (Geometrical Arguments) cos If rv. 0 vr v r r. re. r cos re vr v r r r Coordinate types 55

56 Remarks If If nj. ˆ 0 ek. ˆ 0 1.ˆ 2 cos ni n 1 ne. 2 cos ne If rv. 0 1 re. 2 cos re 3.5 Coordinate types 56

57 Matlab and STK Examples 57

58 State Vector From Orbital Elements 1. Solve for r r 2 a(1 e ) 1 ecos 2. Solve for v v 2 1 r a 3. Solve for γ h rv cos a 1 e 2 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. 3.5 Coordinate types 58

59 State Vector From Orbital Elements 5. Define a vector normal to the orbit plane c rnˆ r(cos Iˆ sin Jˆ) 6. Solve the system v v v v x y z rv. rvsin r v r v r v x x y y z z cv. 0 c v c v c v x x y y z z 3.5 Coordinate types 59

60 Matlab Example 3.5 Coordinate types 60

61 Two-Line Elements (TLE) For monitoring by Norad * * North American Aerospace Defense Command 61

62 STK: Generate and Propagate TLE U

63 Celestrak: Update TLE 63

64 Celestrak: ISS, February 24, Coordinate types 64

65 Celestrak: IRIDIUM 33, February 24,

66 3. The Orbit in Space and Time 3.1 ORBITAL POSITION AS A FUNCTION OF TIME Kepler s equation Orbit propagation 3.2 TIME SYSTEMS Clocks Universal time Earth s motion Atomic time Coordinated universal time Julian date 66

67 3. The Orbit in Space and Time 3.3 INERTIAL FRAMES ICRS ICRF 3.4 COORDINATE SYSTEMS 3.5 COORDINATE TYPES 67

68 Lecture 2 Lecture 3 68

69 Astrodynamics (AERO0024) 3B. The Orbit in Space and Time Gaëtan Kerschen Space Structures & Systems Lab (S3L) 69