# How does JPL define the exact position of Lagrange points in the real solar system where (I thought that) they can't be defined?

My answer to Does the distance to L2 vary? is

The correct answer is "No, they are only mathematically defined when the orbit is circular".

but @DavidHammen's answer explains that they can be defined in the elliptical restricted three body problem and @PM2Ring's answer shows that JLP's Horizons provides eight specifically defined Lagrange points from which you can calculate distances to other things; L1, L2, L4 and L5 for the Earth/Moon system and for the Sun/Earth-Moon-Barycenter system.

Question: How (the heck) does JPL define the exact position of Lagrange points in the real solar system where (I thought that) they can't be defined?

There must be some kind of compromise here as Horizons reflects a realistic n-body simulation without any "restricted three body" conditions that I know of. They must use some equation or rule to calculate the position of these spots that I don't believe can be defined. What is it?

• I suppose you could ask Horizons to compute a trajectory from TLE data for a body at a Lagrange point, and see if it drifts. It's a pity that the Lagrange point body data files don't link to the relevant docs... Feb 17, 2022 at 23:15
• My guess is pretty much the same as that of @PM2Ring : Calculate the osculating elements of the Earth-Moon barycenter with respect to the Sun and apply the very ugly but well-known algorithms for the five ER3PB Lagrange points. But that's just a guess. Feb 18, 2022 at 9:07
• @DavidHammen Sure, it uses pre-computed trajectories for major bodies (including spacecraft), but Horizons does compute trajectories of small bodies. From ssd.jpl.nasa.gov/horizons/manual.html#defs "the trajectory is numerically integrated on demand from an initial set of previously statistically estimated orbital elements in the JPL database." Also see ssd.jpl.nasa.gov/horizons/manual.html#user & ssd.jpl.nasa.gov/horizons/manual.html#tles Feb 18, 2022 at 10:49
• The JPL Planetary and Lunar Ephemerides DE440 and DE441 doesn't mention "Lagrange". Here's a tantalizing snippet "These four Lagrange Point SPK files were generated this date by Min-Kun Chung of the Navigation and Mission Design Section at JPL, using a program named LTOOL and based on the just released de431 planetary ephemeris." Feb 18, 2022 at 12:28
• @PM2Ring A google search for "LTOOL site:nasa.gov" yields lots of hits. One of the earliest says "This software is available for commercial licensing." JPL oftentimes does not quite grok the concept of open source. That applies to NASA as a whole. There is a lot of software I've written for NASA that an ordinary person cannot get their hands on. Feb 18, 2022 at 15:19