Orion computes FINCH’s data linking and imaging encounters by taking advantage of the satellite’s regular ground track shift from one orbit to the next. A ground track is the projection of a spacecraft’s path on the Earth given in terms of latitude and longitude coordinates. The idea of a ground track “shift” value is illustrated in the example below:
In the first figure is FINCH’s ground track over one orbit, and in the second is FINCH’s ground track over ten orbits. By comparing the first orbit (in red) to the remaining nine, we observe that they each follow the same exact path over the Earth; however, with each orbit, the longitude values get translated westward by an almost perfectly constant shift value.
Finding this ground track shift value allows pass calculations to be significantly simplified. Rather than using an orbit propagator to propagate days’ worth of position data, we now only need to propagate over one orbital period. With this single orbit’s data, we can very efficiently determine the data for subsequent orbits by adding the shift value.
The first couple versions of Orion relied solely on a numerical method to determine the ground track shift value. This method takes in accurate position data as returned from Poliastro’s propagator and splits it up into orbit segments with equal time lengths. If the latitude values from two points in adjacent orbit segments are approximately equal, the difference between their associated longitude values is recorded. After iterating through every pair of adjacent orbits, the function will return the average of all these longitude differences. While this method provides an accurate shift value, its major limitation is that it is extremely inefficient.
Orion v4 also includes an analytical method for determining the ground track shift. This method makes use of the following parameters:
The regression rate of the ascending node due to J2 perturbations is calculated using the following equation (Curtis pg 226):
Using this value, as well as the angular velocity of the earth and the orbital period, the closed-form expression for the ground track shift between orbits can be given as:
This method is many orders of magnitude faster than the numerical method; however, when comparing the shift values from both methods, they are generally equivalent only up to one decimal placeㅡ the numerical output remains the more accurate value. Another drawback of this method is that it does not include the effects of atmospheric drag due to the difficulty of analytically modelling them.
We can procure data for any desired number of orbits as long as we have the shift value and the propagated data for only one orbit. Each shifted orbit will have the same number of points and the same latitude coordinates as the reference orbit, however the longitude coordinates between adjacent orbits will differ by the predetermined shift value. This solution is significantly faster than propagating forwards for the same number of orbits.