# Re: Beating Ariel

• From: Erik Max Francis <max@xxxxxxxxxxx>
• Date: Thu, 07 Aug 2008 02:31:26 -0700

Michael Ash wrote:

Now, it's 5AM and so I'm not about to step in with work of my own, but I recall working out the mumble field around an infinite cylinder in college physics, and it was pretty easy. (I think it was the electrical field, but I don't recall for sure. In any case they're all inverse square.) An infinite plane gives you constant force, an infinite cylinder gives you force inversely proportional to radius, and the familiar sphere gives you inverse square.

The ever-unreliable wikipedia gives 2Gd/r as the magnitude of the force at distance r from a cylinder of length-density d. (http://en.wikipedia.org/wiki/Gauss's_law_for_gravitational_fields)

I thought of that. Trying to perform the correct integral for an
infinite cylinder is plenty ugly on its own, but by symmetry arguments
(I'm very sure) you can equivalently replace that with it an infinitely
thin line of constant mass density, in which case the integration
becomes quite trivial (I've done it a few times, I'm quite sure, in this
very newsgroups for various problems), and the formula you found on
Wikipedia happens to be correct.

The problem is one of approximations. I'm not sure how knowing this
helps you. There's no doubt that the limiting case of a sphere deformed
into a longer and longer cylinder is the infinitely-long cylindrical
(or, equivalently, line) case. But what good does this do you to answer
the question?

To put it another way, we're interested in knowing the orbital period
around a sphere that's progressively deformed into a thinner and thinner
cylinder. We know the limit of that process, which is an
infinitely-long line of constant lineic mass density, for which we have
a (very simple) formula. So ... to answer the original question, what
do you set the lineic mass density to? You can set it to a value where the far-away parts of the line will not contribute significantly to the gravitational field there, but then you're right back to the problem of choosing the right approximation.

Presuming John Park, through his comments, is pursuing this line of reasoning, it seems essentially assured that this type of approximation won't really apply to any remotely reasonable asteroid, so then the question is what kind of bound it sets.

You'll get weird orbits if you aren't circular, I think, but for circular orbits the calculation ought to be relatively straightforward.

Right, I'm assuming we only care about circular orbits here, so we only
need the simplest case possible.

--
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