1. University of Colorado Boulder ASEN 5070 Statistical Orbit determination I Fall 2012 Professor George H. Born Professor Jeffrey S. Parker Lecture 3: Astro and Coding 1

2. University of Colorado Boulder  Homework 1  Another change in office hours  Yep, D2L  HW 2 will be posted after this class 2

3. University of Colorado Boulder 3 79% of responses were correct. Answer in mid-lecture. 79% of responses were correct. Answer in mid-lecture.

4. University of Colorado Boulder 4 98% of responses were correct.

5. University of Colorado Boulder 5 70% of responses were correct. Answer in mid-lecture. 70% of responses were correct. Answer in mid-lecture.

6. University of Colorado Boulder 6 Answer in mid-lecture.

7. University of Colorado Boulder 7 95% of responses were correct. Answer in mid-lecture. 95% of responses were correct. Answer in mid-lecture.

8. University of Colorado Boulder  Review solutions for HW1  Show HW2 8

9. University of Colorado Boulder  Coordinate Frames and Time Systems  Homework details ◦ Cartesian to Keplerian conversions ◦ When elements aren’t well-defined.  Integrators  Coding hints and tricks ◦ LaTex: intro ◦ MATLAB: ways to speed up your code ◦ Python: intro 9

10. University of Colorado Boulder  Inertial: fixed orientation in space ◦ Inertial coordinate frames are typically tied to hundreds of observations of quasars and other very distant near-fixed objects in the sky.  Rotating ◦ Constant angular velocity: mean spin motion of a planet ◦ Osculating angular velocity: accurate spin motion of a planet 10

11. University of Colorado Boulder  Coordinate Systems = Frame + Origin ◦ Inertial coordinate systems require that the system be non-accelerating.  Inertial frame + non-accelerating origin ◦ “Inertial” coordinate systems are usually just non- rotating coordinate systems.  Is the Earth-centered J2000 coordinate system inertial? 11

12. University of Colorado Boulder  ICRF  International Celestial Reference Frame, a realization of the ICR System.  Defined by IAU (International Astronomical Union)  Tied to the observations of a selection of 212 well-known quasars and other distant bright radio objects. ◦ Each is known to within 0.5 milliarcsec  Fixed as well as possible to the observable universe.  Motion of quasars is averaged out. ◦ Coordinate axes known to within 0.02 milliarcsec  Quasi-inertial reference frame (rotates a little)  Center: Barycenter of the Solar System 12

13. University of Colorado Boulder  ICRF2  Second International Celestial Reference Frame, consistent with the first but with better observational data.  Defined by IAU in 2009.  Tied to the observations of a selection of 295 well-known quasars and other distant bright radio objects (97 of which are in ICRF1). ◦ Each is known to within 0.1 milliarcsec  Fixed as well as possible to the observable universe.  Motion of quasars is averaged out. ◦ Coordinate axes known to within 0.01 milliarcsec  Quasi-inertial reference frame (rotates a little)  Center: Barycenter of the Solar System 13

14. University of Colorado Boulder  EME2000 / J2000 / ECI  Earth-centered Mean Equator and Equinox of J2000 ◦ Center = Earth ◦ Frame = Inertial (very similar to ICRF)  X = Vernal Equinox at 1/1/2000 12:00:00 TT (Terrestrial Time)  Z = Spin axis of Earth at same time  Y = Completes right-handed coordinate frame 14

15. University of Colorado Boulder  EMO2000  Earth-centered Mean Orbit and Equinox of J2000 ◦ Center = Earth ◦ Frame = Inertial  X = Vernal Equinox at 1/1/2000 12:00:00 TT (Terrestrial Time)  Z = Orbit normal vector at same time  Y = Completes right-handed coordinate frame ◦ This differs from EME2000 by ~23.4393 degrees. 15

16. University of Colorado Boulder  Note that J2000 is very similar to ICRF and ICRF2 ◦ The pole of the J2000 frame differs from the ICRF pole by ~18 milliarcsec ◦ The right ascension of the J2000 x-axis differs from the ICRF by 78 milliarcsec  JPL’s DE405 / DE421 ephemerides are defined to be consistent with the ICRF, but are usually referred to as “EME2000.” They are very similar, but not actually the same. 16

17. University of Colorado Boulder  ECF / ECEF / Earth Fixed / International Terrestrial Reference Frame (ITRF)  Earth-centered Earth Fixed ◦ Center = Earth ◦ Frame = Rotating and osculating (including precession, nutation, etc)  X = Osculating vector from center of Earth toward the equator along the Prime Meridian  Z = Osculating spin-axis vector  Y = Completes right-handed coordinate frame 17

18. University of Colorado Boulder  The angular velocity vector ω E is not constant in direction or magnitude ◦ Direction: polar motion  Chandler period: 430 days  Solar period: 365 days ◦ Magnitude: related to length of day (LOD)  Components of ω E depend on observations; difficult to predict over long periods  Earth Rotation 18

19. University of Colorado Boulder  Principal Axis Frames  Planet-centered Rotating System ◦ Center = Planet ◦ Frame:  X = Points in the direction of the minimum moment of inertia, i.e., the prime meridian principal axis.  Z = Points in the direction of maximum moment of inertia (for Earth and Moon, this is the North Pole principal axis).  Y = Completes right-handed coordinate frame 19

20. University of Colorado Boulder  IAU Systems  Center: Planet  Frame: Either inertial or fixed  Z = Points in the direction of the spin axis of the body.  Note: by convention, all z-axes point in the solar system North direction (same hemisphere as Earth’s North).  Low-degree polynomial approximations are used to compute the pole vector for most planets wrt ICRF.  Longitude defined relative to a fixed surface feature for rigid bodies. 20

21. University of Colorado Boulder  Example: ◦ Lat and Lon of Greenwich, England, shown in EME2000. ◦ Greenwich defined in IAU Earth frame to be at a constant lat and lon at the J2000 epoch. 21

22. University of Colorado Boulder  Synodic Coordinate Systems  Earth-Moon, Sun-Earth/Moon, Jupiter- Europa, etc ◦ Center = Barycenter of two masses ◦ Frame:  X = Points from larger mass to the smaller mass.  Z = Points in the direction of angular momentum.  Y = Completes right-handed coordinate frame 22

23. University of Colorado Boulder  Converting from ECI to ECF 23  P is the precession matrix (~50 arcsec/yr)  N is the nutation matrix (main term is 9 arcsec with 18.6 yr period)  S’ is sidereal rotation (depends on changes in angular velocity magnitude; UT1)  W is polar motion ◦ Earth Orientation Parameters  Caution: small effects may be important in particular application

24. University of Colorado Boulder  Question: How do you quantify the passage of time? 24

25. University of Colorado Boulder  Question: How do you quantify the passage of time?  Year  Month  Day  Second  Pendulums  Atoms 25

26. University of Colorado Boulder  Question: How do you quantify the passage of time?  Year  Month  Day  Second  Pendulums  Atoms 26 What are some issues with each of these? Gravity Earthquakes Snooze alarms

27. University of Colorado Boulder  Countless systems exist to measure the passage of time. To varying degrees, each of the following types is important to the mission analyst: ◦ Atomic Time  Unit of duration is defined based on an atomic clock. ◦ Universal Time  Unit of duration is designed to represent a mean solar day as uniformly as possible. ◦ Sidereal Time  Unit of duration is defined based on Earth’s rotation relative to distant stars. ◦ Dynamical Time  Unit of duration is defined based on the orbital motion of the Solar System. 27

28. University of Colorado Boulder  The duration of time required to traverse from one perihelion to the next.  The duration of time it takes for the Sun to occult a very distant object twice. 28 (exaggerated) These vary from year to year. Why? These vary from year to year. Why?

29. University of Colorado Boulder  Definitions of a Year ◦ Julian Year: 365.25 days, where an SI “day” = 86400 SI “seconds”. ◦ Sidereal Year: 365.256 363 004 mean solar days  Duration of time required for Earth to traverse one revolution about the sun, measured via distant star. ◦ Tropical Year: 365.242 19 days  Duration of time for Sun’s ecliptic longitude to advance 360 deg. Shorter on account of Earth’s axial precession. ◦ Anomalistic Year: 365.259 636 days  Perihelion to perihelion. ◦ Draconic Year: 365.620 075 883 days  One ascending lunar node to the next (two lunar eclipse seasons) ◦ Full Moon Cycle, Lunar Year, Vague Year, Heliacal Year, Sothic Year, Gaussian Year, Besselian Year 29

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31. University of Colorado Boulder  Same variations in definitions exist for the month, but the variations are more significant. 31

32. University of Colorado Boulder  Civil day: 86400 SI seconds (+/- 1 for leap second on UTC time system)  Mean Solar Day: 86400 mean solar seconds ◦ Average time it takes for the Sun-Earth line to rotate 360 degrees ◦ True Solar Days vary by up to 30 seconds, depending on where the Earth is in its orbit.  Sidereal Day: 86164.1 SI seconds ◦ Time it takes the Earth to rotate 360 degrees relative to the (precessing) Vernal Equinox  Stellar Day: 0.008 seconds longer than the Sidereal Day ◦ Time it takes the Earth to rotate 360 degrees relative to distant stars 32

33. University of Colorado Boulder  From 1000 AD to 1960 AD, the “second” was defined to be 1/86400 of a mean solar day.  Now it is defined using atomic transitions – some of the most consistent measurable durations of time available. ◦ One SI second = the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the Cesium 133 atom. ◦ The atom should be at rest at 0K. 33

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35. University of Colorado Boulder  TAI = The Temps Atomique International ◦ International Atomic Time  Continuous time scale resulting from the statistical analysis of a large number of atomic clocks operating around the world. ◦ Performed by the Bureau International des Poids et Mesures (BIPM) 35 TAI

36. University of Colorado Boulder  UT1 = Universal Time  Represents the daily rotation of the Earth  Independent of the observing site (its longitude, etc)  Continuous time scale, but unpredictable  Computed using a combination of VLBI, quasars, lunar laser ranging, satellite laser ranging, GPS, others 36 UT1

37. University of Colorado Boulder  UTC = Coordinated Universal Time  Civil timekeeping, available from radio broadcast signals.  Equal to TAI in 1958, reset in 1972 such that TAI-UTC=10 sec  Since 1972, leap seconds keep |UT1-UTC| < 0.9 sec  In June, 2012, the 25 th leap second was added such that TAI- UTC=35 sec 37 UTC

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39. University of Colorado Boulder 39 What causes these variations?

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42. University of Colorado Boulder  TT = Terrestrial Time  Described as the proper time of a clock located on the geoid.  Actually defined as a coordinate time scale.  In effect, TT describes the geoid (mean sea level) in terms of a particular level of gravitational time dilation relative to a notional observer located at infinitely high altitude. 42 TT  TT-TAI= ~32.184 sec

43. University of Colorado Boulder  TDB = Barycentric Dynamical Time  JPL’s “ET” = TDB. Also known as T eph. There are other definitions of “Ephemeris Time” (complicated history)  Independent variable in the equations of motion governing the motion of bodies in the solar system. 43 TDB  TDB-TAI= ~32.184 sec+ relativistic

44. University of Colorado Boulder  Long story short  In astrodynamics, when we integrate the equations of motion of a satellite, we’re using the time system “TDB” or ~”ET”.  Clocks run at different rates, based on relativity.  The civil system is not a continuous time system.  We won’t worry about the fine details in this class, but in reality spacecraft navigators do need to worry about the details. ◦ Fortunately, most navigators don’t; rather, they permit one or two specialists to worry about the details. ◦ Whew. 44

45. University of Colorado Boulder  Questions on Coordinate or Time Systems?  Quick Break  Next topics: ◦ Cartesian to Keplerian conversions. ◦ Integration ◦ Coding Tips and Tricks 45

46. University of Colorado Boulder  Shape: ◦ a = semi-major axis ◦ e = eccentricity  Orientation: ◦ i = inclination ◦ Ω = right ascension of ascending node ◦ ω = argument of periapse  Position: ◦ ν = true anomaly 46

47. University of Colorado Boulder  Shape: ◦ a = semi-major axis ◦ e = eccentricity  Orientation: ◦ i = inclination ◦ Ω = right ascension of ascending node ◦ ω = argument of periapse  Position: ◦ ν = true anomaly  What if i=0? 47

48. University of Colorado Boulder  Shape: ◦ a = semi-major axis ◦ e = eccentricity  Orientation: ◦ i = inclination ◦ Ω = right ascension of ascending node ◦ ω = argument of periapse  Position: ◦ ν = true anomaly  What if i=0? ◦ If orbit is equatorial, i = 0 and Ω is undefined.  In that case we can use the “True Longitude of Periapsis” 48

49. University of Colorado Boulder  Shape: ◦ a = semi-major axis ◦ e = eccentricity  Orientation: ◦ i = inclination ◦ Ω = right ascension of ascending node ◦ ω = argument of periapse  Position: ◦ ν = true anomaly  What if e=0? 49

50. University of Colorado Boulder  Shape: ◦ a = semi-major axis ◦ e = eccentricity  Orientation: ◦ i = inclination ◦ Ω = right ascension of ascending node ◦ ω = argument of periapse  Position: ◦ ν = true anomaly  What if e=0? ◦ If orbit is circular, e = 0 and ω is undefined.  In that case we can use the “Argument of Latitude” 50

51. University of Colorado Boulder  Shape: ◦ a = semi-major axis ◦ e = eccentricity  Orientation: ◦ i = inclination ◦ Ω = right ascension of ascending node ◦ ω = argument of periapse  Position: ◦ ν = true anomaly  What if i=0 and e=0? 51

52. University of Colorado Boulder  Shape: ◦ a = semi-major axis ◦ e = eccentricity  Orientation: ◦ i = inclination ◦ Ω = right ascension of ascending node ◦ ω = argument of periapse  Position: ◦ ν = true anomaly  What if i=0 and e=0? ◦ If orbit is circular and equatorial, neither ω nor Ω are defined  In that case we can use the “True Longitude” 52

53. University of Colorado Boulder  Shape: ◦ a = semi-major axis ◦ e = eccentricity  Orientation: ◦ i = inclination ◦ Ω = right ascension of ascending node ◦ ω = argument of periapse  Position: ◦ ν = true anomaly  Special Cases: ◦ If orbit is circular, e = 0 and ω is undefined.  In that case we can use the “Argument of Latitude” ( u = ω+ν ) ◦ If orbit is equatorial, i = 0 and Ω is undefined.  In that case we can use the “True Longitude of Periapsis” ◦ If orbit is circular and equatorial, neither ω nor Ω are defined  In that case we can use the “True Longitude” 53

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56. University of Colorado Boulder  Handout offers one conversion.  I’ve coded up Vallado’s conversions ◦ ASEN 5050 implements these ◦ Check out the code RVtoKepler.m  Check errors and/or special cases when i or e are very small. Also good to check the angular momentum vector. 56

57. University of Colorado Boulder  An “Integrator” in the context of astrodynamics is something that propagates the state of an object forward (or backward) in time.  Say we have a satellite in a realistic force model (not just 2-body) at some state “X”.  We have models to describe the accelerations on it.  Where will the satellite go in the future? 57 Earth

58. University of Colorado Boulder  We know the current state and time.  We know the derivative of the state. 58 Earth Accelerations are the equations of motion (two-body, J2, Drag, etc) 2nd order ODE may be integrated as system of 1 st order ODEs.

59. University of Colorado Boulder  Many techniques to predict the motion of an object.  Simplest: 1 st order Taylor expansion: Euler’s Method 59 X(0) X(10) X(20)

60. University of Colorado Boulder  Many techniques to predict the motion of an object.  Euler’s Method with predictor/corrector using trapezoidal rule 60 X(0) X(10) X(20)

61. University of Colorado Boulder  Many techniques to predict the motion of an object.  Runge-Kutta (RK4) 61

62. University of Colorado Boulder  Many techniques to predict the motion of an object.  Varying time-step / adaptive time-step methods ◦ Perform a 4 th and a 5 th -order approximation. Check the difference. ◦ If smaller than tolerance, keep the 5 th order state and move on. ◦ If not smaller than tolerance, reduce the time-step and repeat.  Small time-steps when trajectory and/or force model changes rapidly. 62

63. University of Colorado Boulder  Many techniques to predict the motion of an object.  Symplectic integrators ◦ Rather than focusing on maintaining position accuracy, these integrators focus on conserving energy. ◦ Tend to drift along-track ◦ In a conservative force-field, a satellite’s specific energy should be constant. 63

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65. University of Colorado Boulder  (Show course website’s MATLAB integrator handout) 65

66. University of Colorado Boulder  Questions on Integrators?  Next topics: ◦ Coding Tips and Tricks 66

67. University of Colorado Boulder  MATLAB! 67