I've been trying to build a model of the solar system in a game. Thus far I've succeeded in placing each of the planets in position using Keplerian elements and formulae from https://ssd.jpl.nasa.gov/?planet_pos An example for Jupiter is:
a e I L long.peri. long.node.
AU, AU/Cy rad, rad/Cy deg, deg/Cy deg, deg/Cy deg, deg/Cy deg, deg/Cy
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Jupiter 5.20288700 0.04838624 1.30439695 34.39644051 14.72847983 100.47390909
-0.00011607 -0.00013253 -0.00183714 3034.74612775 0.21252668 0.20469106
At this point, I want to add several of the Jovian moons, however the same site doesn't seem to provide the same type of data for these moons, perhaps because the moons behave differently. Unfortunately, my math is not great; I can usually implement something that's written down, but I don't have the wherewithal to work it out myself.
For the Jovian moons, the data is listed at: https://ssd.jpl.nasa.gov/?sat_elem#jupiter Here is an example for Ganymede:
Sat. a e w M i node n P Pw Pnode R.A. Dec. Tilt
(km) (deg) (deg) (deg) (deg) (deg/day) (days) (yr) (yr) (deg) (deg) (deg)
Ganymede 1070400. 0.0013 192.417 317.540 0.177 63.552 50.3176072 7.155 63.549 132.654 268.168 64.543 0.068
Here they state that the data is "Mean orbital elements referred to the local Laplace planes", as opposed to "Keplerian Elements for Approximate Positions of the Major Planets".
Whilst the first 6 parameters appear to be the same (a, e, w, M, i, node), the Keplerian elements also have "rates", or time derivatives (according to this post: https://space.stackexchange.com/questions/8911/determining-orbital-position-at-a-future-point-in-time). The forumlae kindly provided by NASA for the Keplerian elements relies on these rates.
If I could understand how to determine the "rates" for the Jovian moons, then I might be able to use the same formula (is this the right thing to do?) It looks to me like those rates are something I should be able to compute from the other data provided for the moons, however I haven't been able to determine how.
My suspicion is that I either need a different formula for the computation using the Laplace Planes, or I need to compute those 'rate' values and use the same formula I already have.
I'm trying to generate cartesian (x, y, z) coordinates for each Moon where (0,0,0) is the centre of Jupiter itself, for a given date/time (typically "now").
I'm beginning to wonder if I'm barking up the wrong tree on this. Can anyone shed any light