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Our goal is to determine the best orbit that will allow a ground station (so this is a fixed point on earth) to communicate a maximum number of times par day with a LEO satellite.

So the question is : Is there mathematical models/equations used to determine the optimal orbit for a LEO satellite that will allow us to communicate a maximum number of times per day with a certain point on Earth ?

PS : We already used STK to test some arbitrary orbits and calculate the number of times the satellite entred in communication with the ground station (and thus the total communication time), but we want a more "scientific" method.

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It depends on what you want to do exactly, but a few rules of thumb:

  1. Typically you want the inclination to be about the latitude of where you want to have coverage.
  2. The inclination plus ground swath must cover the area you want to cover. You can never see the poles from the ISS, for instance, for that you need a polar orbit.
  3. For a near equatorial orbit, a retrograde orbit can provide more frequent repeats of passes, although each individual pass will be slower.
  4. Higher will provide more "dwell" time, but less frequent passes overall. The percentage of time in contact will go up.
  5. If you only care about coverage in one hemisphere (North or South), you can have an elliptical orbit with a higher altitude in the desired hemisphere. This will increase dwell time and reduce the repeats.

Designing orbits is more art then science, but these rules are pretty general and should get you started.

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    $\begingroup$ Higher orbits yield more dwell time, but also increase the comm range. For a fixed telecom system (fixed frequency, transmitter power, antenna gains, receiver system noise temperature, coding schemes, etc.) the maximum supportable data rate is roughly proportional to inverse range squared. A Mars Exploration Program trade study found that if the goal is to maximize the data volume (number of bits) transferred, for a fixed telecom system short passes at high data rates beat long passes at lower data rates. Some missions need contact time more than data volume, so they can use high orbits. $\endgroup$
    – Tom Spilker
    Apr 28, 2018 at 1:31

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