After writing this answer to JPL Horizons - "highly accurate measurements of planetary positions" - how do they do it? which draws from Park, Folkner, Williams & Boggs 2021 The JPL Planetary and Lunar Ephemerides DE440 and DE441 I'm left wondering how the heck they actually do the numerical integration.


  1. Do they use a fixed time step for the whole solar system, or is there higher time granularity (whatever that might mean) within say the Jovian moon system than for interactions between planetary barycenters that never get near each other or have rapidly changing relative accelerations.
  2. Are two body interactions calculated hierarchically (e.g. Jovian system as a whole on Saturnian system as a whole), or are all $n(n-1)$ interactions explicitly, or is it somewhere in-between?
  3. What integrator do they use? Do they use more than one?
  4. Order of magnitude how many time steps do they use per year of simulation? Yes higher order integrators can have bigger steps and several evaluations at different times within the step, but I'm just trying to understand if it's one step per minute or millisecond.

Since computers keep getting faster, what was done for the early ephemerides several decades ago may be different than what's done today, or it may not be. I can imagine some tricks to reduce computation time are no longer needed, but there might be hesitancy to switch to simpler algorithms to simply avoid breaking things.

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    $\begingroup$ I suspect this lack of publications on the exact technique they are using is intentional. Their competition (there are only two organizations with JPL's stature with regard to solar system ephemerides, the Observatory of Paris's IMCCE (Institute of Celestial Mechanics and Ephemeris Calculations; the acronym is French), and the Institute of Applied Astronomy of the Russian Academy of Sciences) do much the same. Not releasing details on the technique that lies at the heart of the process of developing an ephemeris means that these three organizations can share observations and compare results. $\endgroup$ Jun 26, 2022 at 1:53
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    $\begingroup$ @DavidHammen yes that makes a lot of sense $\endgroup$
    – uhoh
    Jun 26, 2022 at 1:58
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    $\begingroup$ @uhoh "One minute step size"?? One might as well use RK4 for this small of a step size. Or maybe even symplectic Euler. One of the key advantages of a linear multi-step integrator (aka Adams family) is that this can enable very large step sizes. I would be very surprised if the step size for Venus orbit (for example) was much smaller than one day per step. I know of single-step / multi-stage integrators (e.g., Runge-Kutta integrators) that take steps that are larger than one orbit. They have lots of internal stages (e.g., RK4 has four stage per step). $\endgroup$ Jun 26, 2022 at 2:09
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    $\begingroup$ @uhoh The only moon addressed by the Development Ephemerides is the Earth's Moon. All other planetary systems are modeled as a point mass at the planetary system's barycenter. Modeling the Moon is critical as the Moon's mass is about 0.0123 times the Earth's mass. (That's a handy and rather precise number. There are several zeros after 0.0123 before the next non-zero number arises.) I suppose they should be modeling the Pluto system similarly, but they don't. The Pluto system has very little affect on the orbits of the planets. $\endgroup$ Jun 26, 2022 at 2:15
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    $\begingroup$ With regard to observing Juno, yes, it's important to model the behavior within the Jupiter system as well as the behavior of the Jupiter system as a whole. The problems are separable, at least with regard to impact on accuracy. $\endgroup$ Jun 26, 2022 at 3:17


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