1. University of Colorado Boulder ASEN 5070: Statistical Orbit Determination I Fall 2015 Professor Brandon A. Jones Lecture 3: Time and Coordinate Systems

2. University of Colorado Boulder  Homework 0 - Not required  Homework 1 Due September 4  I am out of town Sept. 15-18 2

3. University of Colorado Boulder  Provide enough detail to answer the problem and allow for grading ◦ For a figure, we need labeled axes, fonts big enough to read, etc. ◦ We do not need a detailed description unless it is requested in the problem set  As stated in the syllabus, legible hand-generated derivations are okay as an image in the PDF  If you are spending more than 20 minutes on the write-up, you may be including too much detail ◦ This will not be true for the project write-up! 3

4. University of Colorado Boulder  Orbital elements – Notes on Implementation  Perturbing Forces – Wrap-up  Coordinate and Time Systems 4

5. University of Colorado Boulder 5 Orbit Elements – Notes on Implementation

6. 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 6

7. University of Colorado Boulder  You 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 this pseudocode: 7

8. University of Colorado Boulder  Inverse tangent has an angle ambiguity  Better to use atan2() when possible: 8 (1,1) (-1,-1) Same value for atan

9. University of Colorado Boulder 9 Perturbed Satellite Motion

10. University of Colorado Boulder  The 2-body problem provides us with a foundation of orbital motion  In reality, other forces exist which arise from gravitational and nongravitational sources  In the general equation of satellite motion, a is the perturbing force (causes the actual motion to deviate from exact 2-body) 10

11. University of Colorado Boulder  Sphere of constant mass density is not an accurate representation for planets  Define gravitational potential function such that the gravitational acceleration is: 11

12. University of Colorado Boulder 12  The commonly used expression for the gravitational potential is given in terms of mass distribution coefficients J n, C nm, S nm  n is degree, m is order  Coordinates of evaluation point are given in spherical coordinates: r, geocentric latitude φ, longitude

13. University of Colorado Boulder  U.S. Vanguard satellite launched in 1958, used to determine J 2 and J 3  J 2 represents most of the oblateness; J 3 represents a pear shape  J 2 ≈ 1.08264 x 10 -3  J 3 ≈ - 2.5324 x 10 -6  You will only need to implement a J 2 model for this class (HW2)

14. University of Colorado Boulder  Our new orbit energy is  Is this constant over time? Why or why not?  What if we only include J 2 in U’ and pure rotation about Z-axis for Earth? 14

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16. University of Colorado Boulder 16  Atmospheric drag is the dominant nongravitational force at low altitudes if the celestial body has an atmosphere  Drag removes energy from the orbit and results in da/dt < 0, de/dt < 0  Orbital lifetime of satellite strongly influenced by drag  You will use a simple exponential model for the atmospheric density (HW2) From D. King-Hele, 1964, Theory of Satellite Orbits in an Atmosphere

17. University of Colorado Boulder  What are the other forces that can perturb a satellite’s motion? ◦ Solar Radiation Pressure (SRP) ◦ Thrusters ◦ N-body gravitation (Sun, Moon, etc.) ◦ Electromagnetic ◦ Solid and liquid body tides ◦ Relativistic Effects ◦ Reflected radiation (e.g., ERP) ◦ Coordinate system errors ◦ Spacecraft radiation 17

18. University of Colorado Boulder 18 Coordinate and Time Frames

19. University of Colorado Boulder  Countless systems exist to measure the passage of time. To varying degrees, each of the following types is important in astrodynamics: ◦ Atomic Time  Unit of duration is defined based on an atomic clock. ◦ Sidereal Time  Unit of duration is defined based on Earth’s rotation relative to distant stars. ◦ Universal Time  Unit of duration is designed to represent a mean solar day as uniformly as possible. ◦ Dynamical Time  Unit of duration is defined based on the orbital motion of the Solar System. 19

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21. University of Colorado Boulder  Question: How do you quantify the passage of time?  Year  Month  Day  Second  Pendulums  Atoms  Sundial 21 What are some issues with each of these? Gravity Earthquakes Errant elbows

22. 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)  Main idea: When we talk about time, we need to be precise with our statements! 22

23. University of Colorado Boulder  Rotating: changing orientation in space ◦ The Earth is a rotating bodyThink about the motion of a top. The Earth has similar changes in the rotation axis  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. ◦ Underlying problem: How do we estimate the inertial frame from a rotating one? 23

24. University of Colorado Boulder 24  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

25. University of Colorado Boulder 25  In this class, we will keep the transformation simple:  In reality, this is a poor model!

26. University of Colorado Boulder 26  The transformation between ECI and ECF is required in the equations of motion ◦ Why? ◦ Depends on the current time! ◦ Thanks to Einstein, we know that time is not simple…

27. University of Colorado Boulder  Accurate representations must account for precession, nutation, and other effects  Classic definition of ECF and ECI transformation based on an `Equinox’  Modern definitions instead use the “Celestial Intermediate Origin” (CIO) Animations/Images courtesy of WikiCommons

28. 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.  Why is a frame at the center of the Earth not a true inertial frame? 28

29. University of Colorado Boulder  Converting from ECR to ECI 29  BPN accounts for nutation, precision and a bias term  R is the Earth’s rotation, which is not constant! (In this class, we only include this component)  W is polar motion ◦ Earth Orientation Parameters  Caution: small effects may be important in particular application

30. 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? 30

31. University of Colorado Boulder  Propagate a spacecraft where the model includes the two-body, J 2, and drag forces  Observe the change in the orbital elements over time as a result of these forces ◦ Why would they change?  Make sure your propagator is working by looking at the constants of motion ◦ Specific energy ◦ Specific angular momentum  Once you complete HW 1, you are ready to start HW 2! 31