1. ASEN 5050 SPACEFLIGHT DYNAMICS Time Systems, Conversions, f & g Prof. Jeffrey S. Parker University of Colorado – Boulder Lecture 8: Time, Conversions 1

2. Announcements Homework #3 is due Friday 9/19 at 9:00 am –You must write your own code. –For this HW, please turn in your code (preferably in one text/Word/PDF document) –After this assignment, you may use Vallado’s code, but if you do you must give him credit for work done using his code. If you don’t, it’s plagiarism. Concept Quiz 7 active and due Friday at 8:00 am. I’ll be at the career fair Monday, so I’m delaying Monday’s office hours to 2:00. Reading: Chapter 3 Lecture 8: Time, Conversions 2

3. Concept Quiz 6 Lecture 8: Time, Conversions 3

4. Concept Quiz 6 Lecture 8: Time, Conversions 4 Scheduling spacecraft observations requires complete knowledge of time! UT1 and UTC are unpredictable.

5. Concept Quiz 6 Lecture 8: Time, Conversions 5 x y

6. Space News NASA just announced which companies will be used to launch our astronauts into orbit! Boeing –CST-100 –$4.2 Billion SpaceX –Dragon –$2.6 Billion Lecture 8: Time, Conversions 6

7. Final Project Reminder to think about your final project, even now. Objective: go beyond the scope of this class in some way. Build an informative website describing your project. Gloat to your friends. I have an opportunity for several people to work on the mission design for a mission to Mars. If you’re interested, email me or come by office hours. –Today and any Wednesday 2-4 –Next Monday at 2:00 (future Mondays at 11) Lecture 8: Time, Conversions 7

8. ASEN 5050 SPACEFLIGHT DYNAMICS Time Systems Prof. Jeffrey S. Parker University of Colorado - Boulder Lecture 8: Time, Conversions 8

9. Time Systems Time is important Signal travel time of electromagnetic waves –Altimetry, GPS, SLR, VLBI For positioning –Orbit determination –One nanosecond (10 –9 second) is 30 cm of distance –Relative motion of celestial bodies Scheduling maneuvers Lecture 8: Time, Conversions 9

10. 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 8: Time, Conversions 10

11. Time Systems: Time Scales Lecture 8: Time, Conversions 11

12. 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) Atomic clocks drift 1 second in about 20 million years. Time Systems: TAI TAI Lecture 8: Time, Conversions 12

13. 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 8: Time, Conversions 13

14. 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 8: Time, Conversions 14

15. 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 8: Time, Conversions 15

16. 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 8: Time, Conversions 16

17. Present time differences As of 17 Sept 2014, –TAI is ahead of UTC by 35 seconds. –TAI is ahead of GPS by 19 seconds. –GPS is ahead of UTC by 16 seconds. The Global Positioning System (GPS) epoch is January 6, 1980 and is synchronized to UTC. Lecture 8: Time, Conversions 17

18. Fundamentals of Time Julian Date (JD) – defines the number of mean solar days since 4713 B.C., January 1, 0.5 (noon). Modified Julian Date (MJD) – obtained by subtracting 2400000.5 days from JD. Thus, MJD commences at midnight instead of noon. Civilian Date JD 1980 Jan 6 midnight 2444244.5GPS Standard Epoch 2000 Jan 1 noon 2451545.0J2000 Epoch Algorithm 14 in book. Lecture 8: Time, Conversions 18

19. 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 8: Time, Conversions 19

20. ASEN 5050 SPACEFLIGHT DYNAMICS Coordinate Systems Prof. Jeffrey S. Parker University of Colorado - Boulder Lecture 8: Time, Conversions 20

21. Coordinate Systems An interesting scenario that involves two coordinate frames playing together: Lecture 8: Time, Conversions 21

22. The Moon’s Librations The librations can be explained via three facts: 1.The Moon spins about its axis at a very consistent rate And it is tidally locked to the Earth 2.The Moon’s orbit is not circular. 3.The Moon’s spin axis is not aligned with its orbital axis Lecture 8: Time, Conversions 22

23. The Moon’s Librations The librations can be explained via three facts: 1.The Moon spins about its axis at a very consistent rate And it is tidally locked to the Earth 2.The Moon’s orbit is not circular. Moon’s orbit (exaggerated) Periapse M = 0° Apoapse M = 180° M = 90° M = 270° Lon = 0° Lecture 8: Time, Conversions 23

24. Coordinate Systems An interesting scenario that involves two coordinate frames playing together: So this image may be interpreted as being a view of the Moon in the Earth-Moon rotating frame, where the Moon’s surface rotates according to the “Moon Fixed” coordinate system. Lecture 8: Time, Conversions 24

25. Coordinate Systems Geocentric Coordinate System (IJK) - aka: Earth Centered Inertial (ECI), or the Conventional Inertial System (CIS) - J2000 – Vernal equinox on Jan 1, 2000 at noon - non-rotating Intersection of ecliptic and celestial eq Lecture 8: Time, Conversions 25

26. Coordinate Systems Earth-Centered Earth-Fixed Coordinates (ECEF) Topocentric Horizon Coordinate System (SEZ) Lecture 8: Time, Conversions 26

27. Coordinate Systems Perifocal Coordinate System (PQW) Lecture 8: Time, Conversions 27

28. Coordinate Systems Satellite Coordinate Systems: RSW – Radial-Transverse-Normal NTW – Normal-Tangent-Normal; VNC is a rotated version Lecture 8: Time, Conversions 28

29. Coordinate Systems Satellite Coordinate Systems: RSW – Radial-Transverse-Normal NTW – Normal-Tangent-Normal; VNC is a rotated version C V R S Lecture 8: Time, Conversions 29

30. Coordinate Transformations Coordinate rotations can be accomplished through rotations about the principal axes. Lecture 8: Time, Conversions 30

31. Coordinate Transformations To convert from the ECI (IJK) system to ECEF, we simply rotate around Z by the GHA: ignoring precession, nutation, polar motion, motion of equinoxes. Lecture 8: Time, Conversions 31

32. Coordinate Transformations To convert from ECEF to SEZ: Lecture 8: Time, Conversions 32

33. Coordinate Transformations One of the coolest shortcuts for building transformations from one system to any other, without building tons of rotation matrices: The unit vector in the S-direction, expressed in I,J,K coordinates (sometimes this is easier, sometimes not) Lecture 8: Time, Conversions 33

34. Coordinate Transformations You can check Vallado, or some of the appendix slides of this presentation for additional transformations. I’d like to provide some conceptual purpose for considering different coordinate systems! Lecture 8: Time, Conversions 34

35. Scenario: Tracking Stations Consider a satellite in orbit. How long is the satellite overhead, as viewed by a ground station in Goldstone, California? –What’s the elevation/azimuth time profile of the pass? Need: elevation (and azimuth) angles of satellite as viewed by station. –Need: satellite’s states represented in SEZ coordinates Transform satellite from IJK to ECEF Transform satellite from ECEF to SEZ Compute elevation and azimuth angles Lecture 8: Time, Conversions 35

36. Scenario: Solar Power A satellite is nadir-pointed with body-fixed solar panels pointed 90 deg away from nadir. How should the satellite rotate to maximize the energy output of the panels? What is the incidence angle of the Sun over time? Need: satellite state represented as RSW Compute angles to the Sun in that frame Lecture 8: Time, Conversions 36

37. Brainteaser If you were to plot the position and velocity of a satellite over time using RSW coordinates, what would you find? –Say, an elliptical orbit Lecture 8: Time, Conversions 37 R S

38. Challenge #4 If you were to plot the position and velocity of a satellite over time using VNC (Velocity-Normal- Conormal) coordinates, what would you find? –Say, an elliptical orbit Lecture 8: Time, Conversions 38 C V

39. (Vallado, 1997) Latitude/Longitude Geocentric latitude Lecture 8: Time, Conversions 39

40. (Vallado, 1997) Latitude/Longitude For geodetic latitude use: where e  =0.081819221456 Lecture 8: Time, Conversions 40

41. Announcements Homework #3 is due Friday 9/19 at 9:00 am –You must write your own code. –For this HW, please turn in your code (preferably in one text/Word/PDF document) –After this assignment, you may use Vallado’s code, but if you do you must give him credit for work done using his code. If you don’t, it’s plagiarism. Concept Quiz 7 due Friday at 8:00 am. I’ll be at the career fair Monday, so I’m delaying Monday’s office hours to 2:00. Reading: Chapter 3 Lecture 8: Time, Conversions 41

42. Coordinate Transformations To convert between IJK and PQW: To convert between PQW and RSW: Thus, RSW  IJK is: R S P Lecture 8: Time, Conversions 42

43. Latitude/Longitude Rotate into ECEF Lecture 8: Time, Conversions 43

44. Right Ascension/Declination thus, Lecture 8: Time, Conversions 44

45. Azimuth-Elevation Compute slant-range vector from site to satellite: Rotate into SEZ Lecture 8: Time, Conversions 45

46. Topocentric Horizon System (SEZ) Lecture 8: Time, Conversions 46

47. Azimuth-Elevation Alternatively: Lecture 8: Time, Conversions 47