1. University of Colorado Boulder ASEN 5070: Statistical Orbit Determination I Fall 2014 Professor Brandon A. Jones Lecture 3: Basics of Orbit Propagation

2. University of Colorado Boulder  Monday is Labor Day!  Homework 0 & 1 Due September 5  I am out of town Sept. 9-12 ◦ Would anyone be interested in attending the recording of a lecture? 2

3. University of Colorado Boulder  Orbital elements – Notes on Implementation  Perturbing Forces – Wrap-up  Coordinate and Time Systems  Flat Earth Problem 3

4. University of Colorado Boulder 4 Orbit Elements – Review and Implementation

5. University of Colorado Boulder  The six orbit elements (or Kepler elements) are constant in the problem of two bodies (two gravitationally attracting spheres, or point masses) ◦ Define shape of the orbit  a: semimajor axis  e: eccentricity ◦ Define the orientation of the orbit in space  i: inclination  Ω: angle defining location of ascending node (AN)   : angle from AN to perifocus; argument of perifocus ◦ Reference time/angle:  t p : time of perifocus (or mean anomaly at specified time)  v,M: True or mean anomaly 5

6. University of Colorado Boulder  a – Size  e – Shape  v – Position 6

7. University of Colorado Boulder  i - Inclination  Ω - RAAN  ω – Arg. of Perigee 7

8. University of Colorado Boulder  Will get an imaginary number from cos -1 (a) if a=1+1e-16 (for example)  The 1e-16 is a result of finite point arithmetic  You may need to use something akin to the pseudocode: 8

9. University of Colorado Boulder  Inverse tangent has an angle ambiguity  Better to use atan2() when possible: 9

10. University of Colorado Boulder 10 Perturbing Forces – Wrap-up

11. University of Colorado Boulder  “Potential Energy is energy associated with the relative positions of two or more interacting particles.”  It is a function of the relative position ◦ Should it be positive or negative? 11

12. University of Colorado Boulder  For a conservative system: 12

13. University of Colorado Boulder 13 Coordinate and Time Frames

14. University of Colorado Boulder 14  Define xyz reference frame (Earth centered, Earth fixed; ECEF or ECF), fixed in the solid (and rigid) Earth and rotates with it  Longitude λ measured from Greenwich Meridian 0≤ λ < 360° E; or measure λ East (+) or West (-)  Latitude (geocentric latitude) measured from equator (φ is North (+) or South (-)) ◦ At the poles, φ = + 90° N or φ = -90° S

15. University of Colorado Boulder 15  The transformation between ECI and ECF is required in the equations of motion ◦ Depends on the current time! ◦ Thanks to Einstein, we know that time is not simple…

16. 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. 16

17. University of Colorado Boulder 17

18. University of Colorado Boulder  Question: How do you quantify the passage of time?  Year  Month  Day  Second  Pendulums  Atoms 18 What are some issues with each of these? Gravity Earthquakes Errant elbows

19. 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 19

20. University of Colorado Boulder  Equinox location is function of time ◦ Sun and Moon interact with Earth J2 to produce  Precession of equinox (ψ)  Nutation (ε)  Newtonian time (independent variable of equations of motion) is represented by atomic time scales (dependent on Cesium Clock)

21. 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 21

22. 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. 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  We did not spend a lot of time on this subject, but it is very, very important to orbit determination!  What impact can the coordinates and time have on propagation and observing a spacecraft? 24

25. University of Colorado Boulder 25 Flat Earth Problem

26. University of Colorado Boulder 26

27. University of Colorado Boulder  Assume linear motion: 27

28. University of Colorado Boulder  Given an error-free state at a time t, we can solve for the state at t 0  What about when we have a different observation type? 28

29. University of Colorado Boulder  Relationship between the estimated state and the observations is no longer linear  For our purposes, let’s assume the station coordinates are known.  You will solve one case of this problem for HW 1, Prob. 6 29