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My wild guess on the direction for the maths involved (probably a much better way to do this):
Establish two tables of static information when a new games starts.
Establish an Epoch reference local time for each body at position equatorial longitude 0 for the UT time of y0-d0-h0-m0-s0 when the game starts.
A table of day lengths for each body (not sure if this is pullable from the API orbit / CB info - if it is, then no lookup needed.)
Knowing what time it is at a certain location should make the calculation for any other location should just involve a greater circle calculation from the known equatorial point / starting time to the required relevant point dragged to the equator and a delta between the epoch and multiple divisions of the day length.
Sunrise
Sunset
etc
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