ASEN 5070: Statistical Orbit Determination I Fall 2014 Professor Brandon A. Jones

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

2. Announcements • 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?

3. Today’s Lecture • Orbital elements – Notes on Implementation • Perturbing Forces – Wrap-up • Coordinate and Time Systems • Flat Earth Problem

4. Orbit Elements – Review and Implementation

5. Six Orbit Elements • 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: semimajoraxis • 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: • tp: time of perifocus (or mean anomaly at specified time) • v,M: True or mean anomaly

6. Orbit Size and Position • a – Size • e – Shape • v – Position

7. Orbit Orientation • i - Inclination • Ω - RAAN • ω – Arg. of Perigee

8. Numeric Issues • 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:

9. atan() versus atan2() • Inverse tangent has an angle ambiguity • Better to use atan2() when possible:

10. Perturbing Forces – Wrap-up

11. Potential Energy • “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?

12. Potential Energy • For a conservative system:

13. Coordinate and Time Frames

14. Coordinate Frames • 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. Coordinate Systems and Time • 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. 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.

17. Time Systems: Time Scales

18. 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 • Errant elbows

19. 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

20. Coordinate Systems and Time • 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. 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

22. 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.

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 and Coordinates • 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?

25. Flat Earth Problem

26. Flat Earth Problem

27. Flat Earth Problem • Assume linear motion:

28. Flat Earth Problem • Given an error-free state at a time t, we can solve for the state at t0 • What about when we have a different observation type?

29. Flat-Earth Problem • 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