Surfaces Prepaper

Both icy and rocky bodies show a transition in shape with size. Large bodies 
have oblate ellipsoidal shapes, whereas small bodies have irregular shapes. 
Ellipsoidal shapes are in hydrostatic equilibirum and are controlled by self-
gravity, whereas irregular shapes are controlled by material strength. The 
transition occurs at around 200 km radius for icy bodies and between 300-500 km 
radius for rocky bodies, and is accompanied by changes in roughness and in the 
relation between maximum topographic height and radius [Slyuta and Voropaev, 
1997; Thomas, 1989; Croft, 1992].

Shape models for well-imaged asteroids and satellites in the strength regime 
[Thomas et al, 1994; Thomas et al, 1996; Thomas et al, 1998; note that NEAR 
didn't image Mathilde very well and hasn't published much on the Eros flyby] 
suggest that slopes on these irregular bodies are always less than 35', a 
typical angle of repose [Jaeger and Nagel, 1992]. Using this as a constraint on 
the maximum slope on a small body gives us a technique for calculating the 
maximum possible departure from sphericity.

First order techniques, as applied by myself in the first homework, are 
unsatisfactory and do not give reliable results. However, they can be extended 
to give self-consistent, reliable results, as I propose to do in this project.

The aim is to obtain a shape for an axisymmetric, non-rotating, homogeneous body 
such that the surface normal is everywhere at a fixed angle from the local 
gravity vector and see how this shape varies with size and density.

The procedure will be as follows:

1: Begin with a massive sphere as the small body with specified radius and          
density.
2: Overlay a massless, axisymmetric shell, touching the small body at one pole,     
such that its surface normal is everywhere 35` from the local gravity vector.
3: Fill the shell with mass to make a homogeneous body
4: Calculate the new gravitational potential
5: Using filled body, repeat from step 2.
6: Stop when change between iterations is sufficiently small.

The final shape can then be compared with those of real small bodies (whose 
shapes are tabulated in almost all the references and in many not listed in this 
short prepaper). It will then be possible to discuss whether these real bodies 
have shapes which as far from sphericity as allowed by this model, or whether 
some other mechanism must be controlling their departures from sphericity. 
Density, size, and angle of repose can be varied to investigate how changes in 
these properties affect the results.

Possible problems:

The method may not converge. My intuition does not give me any ideas as to 
whether it will or not. If it doesn't, and if minor alterations won't make it 
converge, a backup plan is needed. The backup plan is a shotgun approach to 
finding a solution. A whole host of possible limb profiles for an axisymmetric, 
non-rotating, homogeneous body will be generated. Their perpendicular distance 
from the symmetry axis will be constrained to increase monotonically from zero 
to a maximum then decrease monotonically back to zero. Size and density can then 
be varied for each shape. Some results will exceed the angle of repose, some be 
nearly spherical, and (hopefully) some will be close to the angle of repose over 
much of their surface. The latter can be compared with real small bodies as 
discussed above. Note that using a fixed set of shapes makes it easy to add in 
rotation, even about an axis that is not the symmetry axis.

Possible extensions for further study, if results are interesting:
[Aimed at primary plan, some applicable to backup plan]

Compare limb of model to best-fit ellipse, calculate surface roughness following 
Thomas [1989, especially figure 4] and compare to his results for real small 
bodies.

The homogeneity assumption should be straightforward to relax. A high density 
region within the asteroid overlain by a lower density mantle could represent 
differentiation or the effects of gardening to yield a low density regolith, 
depending on the magnitude and extent of the prescribed density difference.
[I need to do some reading on internal structure of asteroids to do this]

Rotation about the symmetry axis should be straightforward to add, being merely 
an additional term in the gravitational potential. Unfortunately, this will be 
rotation around the long axis of the body, whereas asteroids tend to rotate 
around their short axis [minimization of rotational kinetic energy for a given 
angular momentum]. Nevertheless, it should be interesting to see how large the 
effects of rotation are.

It will be harder to have rotation about a more physically realistic axis. One 
possible approach is take the axisymmetric, non-rotating, homogeneous final body 
and add a rotation axis through the centre of mass perpendicular to the symmetry 
axis. Calculate the new gravitational potential, including rotation, in three 
dimensions, and repeat the shell overlaying procedure described above. There are 
potential problems in the centre of mass shifting off the rotation axis and in 
where to have each axisymmetric shell touch the "core". This is not a trivial 
problem and I do not think I have thought it through completely. I don't expect 
to ever do this.

Investigate shapes of equipotential surfaces.

Things I need to sort out to do this:

How to do an axisymmetric problem - eg do I have to have different densities 
closer to the core to accounts for changes in r? - ask Gareth, Betty.
How to do this kind of coding, checking for artifacts, etc.
More reading on angle of repose.

References:

Croft, S. K. 1992, Proteus: Geology, Shape, and Catastrophic Destruction. Icarus 
99, 402-419
Hartmann, W. K. 1983, Moons and Planets, 2nd Ed., publisher
Jaeger, H. M., and S. R. Nagel 1992, Physics of the Granular State. Science 255, 
1523-1531
Johnson, T. V., and T. R. McGetchin 1973, Topography on Satellite Surfaces and 
the Shape of Asteroids. Icarus 18, 612-620
Slyuta, E. N., and S. A. Voropaev 1997, Gravitational Deformation in Shaping 
Asteroids and Small Satellites. Icarus 129, 401-414
Thomas, P. C. 1989, The Shapes of Small Satellites. Icarus 77, 248-274
Thomas, P. C. 1993, Gravity, Tides, and Topography on Small Satellites and 
Asteroids: Application to Surface Features of the Martian Satellites. Icarus 
105, 326-344
Thomas, P. C. et al 1994, The Shape of Gaspra. Icarus 107, 23-36
Thomas, P. C. et al 1996, The Shape of Ida. Icarus 120, 20-32
Thomas, P. C. et al 1998, The Small Inner Satellites of Jupiter. Icarus 135, 
360-371
Thomas, P. C. et al 1999, Mathilde: Size, Shape, and Geology. Icarus 140, 17-27