1. ASEN 5050 SPACEFLIGHT DYNAMICS Coordinate, Time, Conversions Prof. Jeffrey S. Parker University of Colorado – Boulder Lecture 7: Coordinate, Time, Conversions 1

2. Announcements Office hours today, cancelled (PhD prelim exam). Let me know if you need to chat and can’t make it to any other office hours. Homework #3 is due Friday 9/19 at 9:00 am Concept Quiz #6 will be available at 10:00 am, due Wednesday morning at 8:00 am. Reading: Chapter 3 Lecture 7: Coordinate, Time, Conversions 2

3. Quiz 5 Lecture 7: Coordinate, Time, Conversions 3 Nobody selected these. Good!

4. Quiz 5 Lecture 7: Coordinate, Time, Conversions 4 Only ½ of the class got the right answer. Please convince your neighbor that you know the correct answer! Only ½ of the class got the right answer. Please convince your neighbor that you know the correct answer!

5. Quiz 5 Lecture 7: Coordinate, Time, Conversions 5

6. Quiz 5 Lecture 7: Coordinate, Time, Conversions 6 Z ambiguity!

7. Quiz 5 Lecture 7: Coordinate, Time, Conversions 7

8. Quiz 5 Lecture 7: Coordinate, Time, Conversions 8

9. Challenge #3 We examined Pluto’s and Neptune’s orbits last time. Question: since Pluto sometimes travels interior to Neptune’s orbit, could they ever collide? –If so, what sort of order of duration do we need to wait until it may statistically happen? Years? Millennia? Eons? Lecture 7: Coordinate, Time, Conversions 9

10. Challenge #3 They are statistically never going to collide! (unless something crazy happens, like we encounter another star) Pluto and Neptune are quite far non coplanar –Pluto’s inclination is ~17 deg –Neptune’s inclination is ~2 deg –Pluto’s Longitude of Ascending Node is ~110 deg –Neptune’s Longitude of Ascending Node is ~131 deg Pluto and Neptune are in resonance –Neptune orbits the Sun 3x when Pluto orbits 2x. Lecture 7: Coordinate, Time, Conversions 10 8 people got a point!

11. Do they ever get close to colliding? Lecture 7: Coordinate, Time, Conversions 11

12. Do they ever get close to colliding? Lecture 7: Coordinate, Time, Conversions 12

13. Neptune’s and Pluto’s Orbit Do the orbits intersect? Neptune’s Orbit Pluto’s Orbit Lecture 7: Coordinate, Time, Conversions 13

14. Neptune and Pluto’s Closest Approach Lecture 7: Coordinate, Time, Conversions 14

15. ASEN 5050 SPACEFLIGHT DYNAMICS Coordinate and Time Systems Prof. Jeffrey S. Parker University of Colorado - Boulder Lecture 7: Coordinate, Time, Conversions 15

16. Coordinate Systems Given a full state, with position and velocity known. Or, given the full set of coordinate elements. What coordinate system is this state represented in? Could be any non- rotating coordinate system! Earth J2000 or ecliptic J2000 or Mars, etc. Lecture 7: Coordinate, Time, Conversions 16

17. Coordinate Systems Celestial Sphere –Celestial poles intersect Earth’s rotation axis. –Celestial equator extends Earth equator. –Direction of objects measured with right ascension (  ) and declination (  ). Lecture 7: Coordinate, Time, Conversions 17

18. Coordinate Systems The Vernal Equinox defines the reference direction. A.k.a. The Line of Aries The ecliptic is defined as the mean plane of the Earth’s orbit about the Sun. The angle between the Earth’s mean equator and the ecliptic is called the obliquity of the ecliptic,  ~23.5 . Lecture 7: Coordinate, Time, Conversions 18

19. 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 Coordinate Frames Lecture 7: Coordinate, Time, Conversions 19

20. 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? Coordinate Systems Lecture 7: Coordinate, Time, Conversions 20

21. 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 Useful Coordinate Systems Lecture 7: Coordinate, Time, Conversions 21

22. 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 Useful Coordinate Systems Lecture 7: Coordinate, Time, Conversions 22

23. 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 Useful Coordinate Systems Lecture 7: Coordinate, Time, Conversions 23

24. 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. Useful Coordinate Systems Lecture 7: Coordinate, Time, Conversions 24

25. 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. Useful Coordinate Systems Lecture 7: Coordinate, Time, Conversions 25

26. 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 Useful Coordinate Systems Lecture 7: Coordinate, Time, Conversions 26

27.  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 Useful Coordinate Systems Lecture 7: Coordinate, Time, Conversions 27

28. 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 Useful Coordinate Systems Lecture 7: Coordinate, Time, Conversions 28

29. 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. Useful Coordinate Systems Lecture 7: Coordinate, Time, Conversions 29

30. 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. Useful Coordinate Systems Lecture 7: Coordinate, Time, Conversions 30

31. 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 Useful Coordinate Systems Lecture 7: Coordinate, Time, Conversions 31

32. Converting from ECI to ECF Coordinate System Transformations  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 Lecture 7: Coordinate, Time, Conversions 32

33. Question: How do you quantify the passage of time? Time Systems Lecture 7: Coordinate, Time, Conversions 33

34. Question: How do you quantify the passage of time? Year Month Day Second Pendulums Atoms Time Systems Lecture 7: Coordinate, Time, Conversions 34

35. Question: How do you quantify the passage of time? Year Month Day Second Pendulums Atoms Time Systems What are some issues with each of these? Gravity Earthquakes Snooze alarms Lecture 7: Coordinate, Time, Conversions 35

36. 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. Time Systems Lecture 7: Coordinate, Time, Conversions 36

37. 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. Time Systems: The Year (exaggerated) These vary from year to year. Why? These vary from year to year. Why? Lecture 7: Coordinate, Time, Conversions 37

38. 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 Time Systems: The Year Lecture 7: Coordinate, Time, Conversions 38

39. Same variations in definitions exist for the month, but the variations are more significant. Time Systems: The Month Lecture 7: Coordinate, Time, Conversions 39

40. 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 Time Systems: The Day Lecture 7: Coordinate, Time, Conversions 40

41. 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. Time Systems: The Second Lecture 7: Coordinate, Time, Conversions 41

42. Time Systems: Time Scales Lecture 7: Coordinate, Time, Conversions 42

43. 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) Time Systems: TAI TAI Lecture 7: Coordinate, Time, Conversions 43

44. 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 Time Systems: UT1 UT1 Lecture 7: Coordinate, Time, Conversions 44

45. 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 Time Systems: UTC UTC Lecture 7: Coordinate, Time, Conversions 45

46. Time Systems: UTC Lecture 7: Coordinate, Time, Conversions 46

47. Time Systems: UTC What causes these variations? Lecture 7: Coordinate, Time, Conversions 47

48. 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. Time Systems: TT TT  TT-TAI= ~32.184 sec Lecture 7: Coordinate, Time, Conversions 48

49. 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. Time Systems: TDB TDB  TDB-TAI= ~32.184 sec+ relativistic Lecture 7: Coordinate, Time, Conversions 49

50. 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. Time Systems: Summary Lecture 7: Coordinate, Time, Conversions 50

51. Announcements Office hours today, cancelled (PhD prelim exam). Let me know if you need to chat and can’t make it to any other office hours. Homework #3 is due Friday 9/19 at 9:00 am Concept Quiz #6 will be available at 10:00 am, due Wednesday morning at 8:00 am. Reading: Chapter 3 Lecture 7: Coordinate, Time, Conversions 51