Could Earth Capture a SECOND Satellite the size of the Moon?

[tony873004]

You can't use RK4 and expect anything remotely resembling reality after even one orbit. It is not a stable integrator.

I know in my previous post I'm essentially disagreeing with you when you say "You can't use RK4 and expect anything remotely resembling reality after even one orbit. It is not a stable integrator. "
But with all due respect, you helped me get my degree in Physics and Astronomy. Google "tony87004 thanks DH". I used to be an electrical contractor. Now I'm a high school physics teacher :)

 

[|Glitch|]

The Moon orbits the Earth WAY slower than that. It's more like 1 km/s. Unless you're talking about the Moon's velocity with respect to the Sun. In that case, it spends about half its time orbiting the Sun up to about 1 km/s faster than the Earth, and the other half of the time orbiting the Sun up to about 1 km/s slower than than the Earth. I think you know what you're talking about, but we're just having some trouble communicating.

I think we are talking about the same thing. Only I am referring to the total orbital velocity, rather than the relative orbital velocity. I completely agree that the Moon only orbits the Earth at some thing just under 1 km/s. Furthermore, Ceres would orbit the Moon at less than 1 km/s as well. However, their total velocity includes Earth's velocity around the Sun since it was a four-body problem. If I worked out the simulation as just a three-body problem (Earth, Moon, and Ceres) then the Moon would have an orbital velocity around Earth of 996 m/s and Ceres would have an orbital velocity around the Moon of 860 m/s (at a distance of 70,000 km).

 

[D H]

I worked it out as a four-body problem, using the Sun, Earth, Moon, and Ceres. In order for the Earth to maintain a solar orbit at 1 AU with an eccentricity of 0.0167, it needs to travel at 29.8 km/s. The Moon orbits the Earth slightly faster than that, and in order for Ceres to orbit the Moon it needs to travel slightly faster than the Moon's velocity.

There's your problem.

If you are using native floating point (specifically, double precision; if you are using floats you are completely lost) and if want any hope of capturing the dynamics you need to set up a hierarchy of reference frames. If you want to do all of your computations in a common solar system barycenter frame, you *must* use some kind of extended precision arithmetic to avoid the huge truncation error problems in your integrators that would otherwise result with using double precision numbers.

For example, it's best to represent an object orbiting the Earth using an "Earth-centered inertial" (aka ECI) frame. I put that in quotes because (a) it's not an inertial frame and (b) it is a very commonly used term. There's even a wikipedia page on Earth-centered inertial: http://en.wikipedia.org/wiki/Earth-centered_inertial.

This ECI frame is not inertial. It is an accelerating frame. The astronomical / aerospace engineering term for the fictitious forces that results from using this accelerating frame is "third body force". (Physicists would call these apparent forces "tidal forces", but that term is used to denote something else in modeling solar system dynamics.) You need to model these apparent forces or you need to use extended precision arithmetic. If you use the latter, your integration will proceed extremely slowly.

I know in my previous post I'm essentially disagreeing with you when you say "You can't use RK4 and expect anything remotely resembling reality after even one orbit. It is not a stable integrator. "

These selenocentric distant retrograde orbits (SDROs) are in the class of "exotic orbits", particularly when the orbital distance is greater than the distance to the Earth-Moon L1 point (for example, a 70,000 km orbit about the Moon). These very distant SDROs don't look anything close to elliptical when viewed from the perspective of a Moon-centered inertial frame. They look more like a square with rounded corners. They do look roughly elliptical when viewed from the perspective of an Earth-centered inertial frame, but the Earth is not at one of the foci of the ellipse.

These very distant SDROs are of increasing interest to NASA, other space agencies, and also to the (not anywhere close to ready for prime time) asteroid mining community. So we've studied them. You need a very good integrator to model the behaviors. RK4 just doesn't cut it. Things get wonky after just an orbit or so when one uses RK4 against these exotic orbits.

But with all due respect, you helped me get my degree in Physics and Astronomy. Google "tony87004 thanks DH". I used to be an electrical contractor. Now I'm a high school physics teacher :)

Thanks! I'm blushing.