Are Retrograde Satellites the Key to Stable and Optimal Orbits?

[mollwollfumble]

TL;DR Summary

Information wanted on three-body solvers, and on high inclination coorbitals

I was working on a proposal for a spacecraft , and suddenly realized that the ideal orbit may be a high inclination type of near-Earth coorbital called a "retrograde satellite" or RS orbit. Do you know of:

* A person who can compute 100 years of coorbital stability using three body (sun, earth, satellite) or four body (with moon) kinematics?

* A simple-to-use approximate 3-D calculator for the three body problem, simplified so the third body has negligible mass.

* Where I can find as much information as possible on RS orbits in general. eg. ellipticity vs inclination.

* Any information at all on periodic RS orbits with 1:1 resonance.

So far, all I know about RS orbits comes from "Coorbital Dynamics at Large Eccentricity and Inclination ", https://journals.aps.org/prl/pdf/10.1103/PhysRevLett.83.2506
The inclination range I'm most interested in is 15 to 45 degrees.

 

[mollwollfumble]

These are the sort of orbits I'm hoping for. These are heliocentric orbits, but appear to circle the Earth. The Earth is at (0, 0). The L4 and L5 Lagrangian points are at +-60 degrees on the horizontal axis. In the background, contours of the intensity of the inner zodiacal light are drawn. These orbits see through the fringes of the inner zodiacal light at all points on the orbit and so, with luck and planning, could see faint galaxies and transient events on the far side of the Sun. They are closer than orbits around L4 or L5 and more stable than Earth- leading and Earth-trailing trajectories.

 

[mollwollfumble]

> A simple-to-use approximate 3-D calculator for the three body problem, simplified so the third body has negligible mass.

I think I've found one. http://www.fisica.edu.uy/~gallardo/atlas/
Program name atlas2bgeneral.f

It needs at least a slight modification in that it calculates number of resonances at assumed eccentricity and inclination. Whereas I want best (ie periodic orbit) eccentricity and inclination for a specific (Earth 1:1) resonance. I'm intrigued to know if its calculation of semimajor axis is correct, I don't think it can be because the semimajor axis has to depend on orbital inclination. The closer the inclination brings us to Earth, the larger the semimajor axis has to be, not by much, but by enough to make a difference as to whether the orbit is exactly 1:1 resonant.

 

[stefan r]

...
So far, all I know about RS orbits comes from "Coorbital Dynamics at Large Eccentricity and Inclination ", https://journals.aps.org/prl/pdf/10.1103/PhysRevLett.83.2506
The inclination range I'm most interested in is 15 to 45 degrees.

Retrograde is an inclination greater than 90 degrees. Did you mean 135 to 165 degrees?

 

[mollwollfumble]

Retrograde is an inclination greater than 90 degrees. Did you mean 135 to 165 degrees?

Thank you for the reply. No. It's prograde around the Sun but appears retrograde around the Earth. The orbit is around the Sun, slightly modified (approximately by one part in 50) by the Earth's gravity. The inclination 15 to 45 degrees is for the prograde orbit around the Sun. It's annoying that the literature refers to it as a "retrograde satellite" orbit, but that's what it's called. Like the "tadpole" and "horseshoe" orbits, the name refers to its appearance from Earth. I've now written a little program in Fortran 77 to approximate the coorbital orbit (lumping the Moon's gravity in with the Earth's and approximating the Earth's orbit as circular). The program still isn't fully debugged, and for some reason can't I get orbit closure closer than 12,000 km.

 

[mollwollfumble]

Results are coming out better than i could have possibly hoped. I speculate that there may even be a class of equilibrium periodic 1:1 resonant coorbitals of the "retrograde satellite" type. Instead of closure errors of 12,000 km or so, the periodic orbits have minuscule closure errors of order 1 km per year or less, and closure errors of 1 m/s per year or less. This chart gives error in periodic orbit closure, as a function of inclination and eccentricity for orbit inclinations of 15 to 45 degrees. Error in periodic orbit closure. This chart gives the locus of minimal closure error Best eccentricity vs inclination for exact 1:1 resonance. Orbits look like this when seen from above. Lagrangian points are at (-0.5, +-sqrt(3)/2) on this chart.

and like this when viewed along the Earth-Sun axis.

Weird or what? But then coorbital orbits when viewed from Earth are always weird. The minimum-closure-error coorbital with the minimum maximum distance from Earth (ie. best communications) has an inclination near 19.5 degrees and eccentricity near 0.22. And is also a really good orbit for seeing past the inner zodiacal light.