ASEN 5050 SPACEFLIGHT DYNAMICS Time Systems, Conversions, f & g

1. ASEN 5050SPACEFLIGHT DYNAMICSTime Systems, Conversions, f & g Prof. Jeffrey S. Parker University of Colorado – Boulder

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

3. Concept Quiz 6

4. Concept Quiz 6 Scheduling spacecraft observations requires complete knowledge of time! UT1 and UTC are unpredictable.

5. Concept Quiz 6 y x

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

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)

8. ASEN 5050SPACEFLIGHT DYNAMICSTime Systems Prof. Jeffrey S. Parker University of Colorado - Boulder

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

10. 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.

11. Time Systems: Time Scales

12. 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) • Atomic clocks drift 1 second in about 20 million years. TAI

13. 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

14. 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

15. 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

16. 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

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.

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 DateJD 1980 Jan 6 midnight 2444244.5 GPS Standard Epoch 2000 Jan 1 noon 2451545.0 J2000 Epoch Algorithm 14 in book.

19. Time Systems: Summary • 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.

20. ASEN 5050SPACEFLIGHT DYNAMICSCoordinate Systems Prof. Jeffrey S. Parker University of Colorado - Boulder

21. Coordinate Systems • An interesting scenario that involves two coordinate frames playing together:

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

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

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.

25. Intersection of ecliptic and celestial eq 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

26. Coordinate Systems Earth-Centered Earth-Fixed Coordinates (ECEF) Topocentric Horizon Coordinate System (SEZ)

27. Coordinate Systems Perifocal Coordinate System (PQW)

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

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

30. Coordinate Transformations Coordinate rotations can be accomplished through rotations about the principal axes.

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.

32. Coordinate Transformations To convert from ECEF to SEZ:

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)

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!

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

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

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 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 V C

39. (Vallado, 1997) Latitude/Longitude Geocentric latitude

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

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

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

43. Latitude/Longitude Rotate into ECEF

44. Right Ascension/Declination thus,

45. Azimuth-Elevation Compute slant-range vector from site to satellite: Rotate into SEZ

46. Topocentric Horizon System (SEZ)

47. Azimuth-Elevation Alternatively: