[meteorite-list] Spherical Shapes of Planets, Asteroids.

From: almitt <almitt_at_meteoritecentral.com>
Date: Thu Apr 22 09:48:14 2004
Message-ID: <3BDB1321.8B2F9226_at_kconline.com>

Dear list,

Here is a discussion on shapes of the minor planets and so forth. Thought it might be
relevant to this group. Comes from THE MINOR PLANET MAILING LIST. I have included
credits and subscription information to those interested. Best!

--AL

>During some discussion on 2001 KX76, the question arose as to what
size an object would have to be before you could reasonable assume that
its gravity was sufficient to assure that it would have a spherical shape due
to gravity.>

>Can anyone assist on this question?>

Dear Larry,

My mentor and thesis advisor, William Kaula, put the answer to this in
terms of gravitational harmonics, in what has sometimes been referred
to as Kaula's rule. The basis of his rule is simply the following. On
Earth, we see deviations from the equilibrium figure of the planet of the
order of one part in a thousand. That is, the Earth has a radius of 6378
km and we see mountains and ocean trenches around 6 km from equil-
ibrium "sea level".

Since the force of gravity is proportional to the dimension of the
body, you can stack rocks twice as high on a body half as big with
the same stress (pressure) at the base. So assuming rocks on other
bodies (Mars, moon, asteroids) have the same strength, the maximum
height or depth of deviations from figure should be inversely proportional
to the dimension of the body, or expressed as a proportion, inverse
with the square of the dimension. For the Earth, deviations are one
part in a thousand. For a body sqrt(1000), or about 30 times, smaller,
deviations reach the level of order unity, thus the body is no longer
"spherical" in any sense. So at r = 6000/30 = 200 km, or diameter
around 400 km, rocky bodies should take on close to arbitrary shapes
rather than spherical.

However, as deviations from "equilibrium" shape become very large,
the"equilibrium" gravity field itself distorts in the direction of the
deviation, so the non-equilibrium stresses are not so great as you
would estimate by not accounting for this effect. Allowing for this,
one would expect asteroids as large as 500 or so km to be able to
sustain fairly major deviations from equilibrium. Observation is the
final arbitor of nature, of course. Looking at actual lightcurve
amplitudes, we see that the very largest asteroids (Ceres, Pallas,
Vesta) have fairly modest lightcurve amplitudes. About the largest
large-amplitude asteroids are 87 Sylvia and 15 Eunomia at around
250 km diameter. But these are fairly fast rotators so their irregular
shapes are partially equilibrated by centrifugal force. You have to go
down to around 200 km before you find really irregular shapes with
no significant compensation by rapid rotation. This would suggest
that asteroids are somewhat weaker than terrestrial (and lunar and
Martian) rocks and have a bit less ability to sustain non-equilibrium
shapes, but only by a factor of 2 or so.

Regards,

Alan

****************************************************
Alan Harris
Senior Research Scientist
MS 183-501 Phone: 818-354-6741
Jet Propulsion Laboratory Fax: 818-354-0966
Pasadena, CA 91109 email: Alan.W.Harris_at_jpl.nasa.gov
****************************************************

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Received on Sat 27 Oct 2001 04:03:45 PM PDT