Page:SATCON2 Algorithms Report.pdf/24

For each satellite, we calculate the position at a series of time steps specified by the user. For each step, we perform the following steps:

1. Calculate the geocentric satellite position and uncertainty (interpolating state and covariance from the ephemeris or propagating TLE and using a fixed uncertainty estimate). Note that covariance interpolation can lead to unphysical results (results which are non-positive-definite or have negative variance). See the astroplan package (https://astronplan.readthedocs.io/en/latest/) for approaches to handling this.
2. Calculate the difference vector between the observatory location and the satellite position.
3. Transform this to a unit vector (the “line of sight”) in the observatory topocentric frame.
4. Determine the telescope field of view at this time in the topocentric frame.
   1. There remain detailed issues to resolve: how to handle aberration and refraction; whether we convert the FOV from apparent to true RA/Dec, or conversely convert the satellite vectors from true to apparent. An observing schedule is normally given in true RA/Dec rather than apparent, so the former approach seems better.
   2. If the line of sight is within the field of view, the satellite is geometrically observable (Alfano, Negron & Moore, 1992).
5. Calculate the satellite topocentric angular velocity in the instrument frame.
   1. This is not necessarily tracking at the sidereal rate; however, this is unlikely to be a significant issue.
6. Calculate the geocenter-Sun vector from, e.g., the JPL ephemeris.
7. Subtract the satellite-geocenter vector to obtain the satellite-Sun unit vector.
8. Compare the geocenter-Sun vector and Earth radius to see if the Sun is above the horizon as seen by satellite — if so, it is illuminated.

In simple mode:

1. Output the result line if the satellite is geometrically observable.
2. Keep track of whether the satellite was observable at the previous time step; if so, it’s the same streak and has the same streak ID (identification) number. If not, increment the streak ID number.
3. The result line indicates: satellite catalog number; streak ID number; predicted topocentric right ascension and declination at the time step; a flag to say whether the satellite is illuminated or not; the location of the streak in the field of view.

In advanced mode:

1. Use the BRDF (and attitude model if available) to determine the predicted magnitude in specified bands. Add to the output line.
2. Use the angular velocity and pixel size to predict the surface brightness of the streak per pixel. Add to the output line.