ASEN 5070 Statistical Orbit determination I Fall 2012 Professor George H. Born Professor Jeffrey S. Parker Lecture 3:

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

2. Announcements • Homework 1 • Another change in office hours • Yep, D2L • HW 2 will be posted after this class

3. Quiz Results 79% of responses were correct. Answer in mid-lecture.

4. Quiz Results 98% of responses were correct.

5. Quiz Results 70% of responses were correct. Answer in mid-lecture.

6. Quiz Results Answer in mid-lecture.

7. Quiz Results 95% of responses were correct. Answer in mid-lecture.

8. Homeworks • Review solutions for HW1 • Show HW2

9. Today’s Lecture • 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

10. Coordinate Frames • 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

11. Coordinate Systems • 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?

12. Useful Coordinate Systems • 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

13. Useful Coordinate Systems • 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

14. Useful Coordinate Systems • 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

15. Useful Coordinate Systems • 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.

16. Useful Coordinate Systems • 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.

17. Useful Coordinate Systems • 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

18. Useful Coordinate Systems • Earth Rotation • The angular velocity vector ωE 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 ωE depend on observations; difficult to predict over long periods

19. Useful Coordinate Systems • 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

20. Useful Coordinate Systems • 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.

21. Useful Coordinate Systems • 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.

22. Useful Coordinate Systems • 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

23. Coordinate System Transformations • Converting from ECI to ECF • 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. Time Systems • Question: How do you quantify the passage of time?

25. Time Systems • Question: How do you quantify the passage of time? • Year • Month • Day • Second • Pendulums • Atoms

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

27. Time Systems • 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.

28. Time Systems: The Year • 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. (exaggerated) These vary from year to year. Why?

29. Time Systems: The Year • 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

30. Quiz Results

31. Time Systems: The Month • Same variations in definitions exist for the month, but the variations are more significant.

32. Time Systems: The Day • 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

33. Time Systems: The Second • 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.

34. Time Systems: Time Scales

35. Time Systems: TAI • 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) TAI

36. Time Systems: UT1 • 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 UT1

37. Time Systems: UTC • 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 25th leap second was added such that TAI-UTC=35 sec UTC

38. Time Systems: UTC

39. Time Systems: UTC What causes these variations?

40. Quiz Results

41. Quiz Results

42. Time Systems: TT • 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. • TT-TAI=~32.184 sec TT

43. Time Systems: TDB • TDB = Barycentric Dynamical Time • JPL’s “ET” = TDB. Also known as Teph. 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. • TDB-TAI=~32.184 sec+relativistic TDB

44. Time Systems: Summary • 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.

45. Questions • Questions on Coordinate or Time Systems? • Quick Break • Next topics: • Cartesian to Keplerian conversions. • Integration • Coding Tips and Tricks

46. Keplerian Orbital Elements • Shape: • a = semi-major axis • e = eccentricity • Orientation: • i = inclination • Ω = right ascension of ascending node • ω = argument of periapse • Position: • ν = true anomaly

47. Keplerian Orbital Elements • 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?

48. Keplerian Orbital Elements • 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”

49. Keplerian Orbital Elements • 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?

50. Keplerian Orbital Elements • 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”