Re: Hill Sphere versus Laplace Sphere

From: pervect (pervect_at_invalid.invalid)
Date: 08/02/04 

Date: Mon, 02 Aug 2004 14:35:45 -0700

On Mon, 02 Aug 2004 04:37:58 -0700, Jake <jakmal@NOSPAMhobonet.com>
wrote:

>Please tell me which one is the real SOI (sphere of influence).
>
>Apparently there are two contenders for the title "sphere of influence"
>: that region of space around an astronomical body within which it can
>hold satellites in stable orbits around itself. Because the body in
>question is always(?) itself a satellite of a bigger body, the SOI is
>calculated by considering the two-body system of body #1 and its
>primary.
>
>
>Contender # 1 :
>
>The Hill sphere is named for George William Hill (1838-1914). It is
>determined by the Hill radius :
>
>r_H = D_sp·(M_s / (3·M_p))^(1/3),
>
>where r_H is measured from the center of the body (satellite), D_sp is
>the center-to-center distance between the satellite and the primary,
>M_s is the mass of the satellite and M_p is the mass of the primary.
>
>http://en.wikipedia.org/wiki/Hill_sphere
>
>Apparently it also is determined by L1 and L2, the Lagrange points
>closest to the satellite.
>
>http://stars5.netfirms.com/lagrangian.htm
>
>
>Contender # 2 :
>
>Don't bother looking up "Laplace sphere", it's my own coinage.
>Apparently the first person to calculate a SOI was French astronomer
>Pierre Simon de Laplace (1749-1827). It is determined by the following
>radius :
>
>r_L = D_sp·(M_s / M_p)^(2/5),
>
>where r_L is measured from the center of the satellite, and D_sp, M_s
>and M_p are defined as above.
>
>http://www.go.ednet.ns.ca/~larry/orbits/gravasst/gravasst.html
>
>Sincere insights are greatly appreciated.
>
>Jake

For some insight, I'd try

http://www.ima.umn.edu/talks/workshops/10-29-11-2.2001/koon/tour.pdf

Hill's formulation of the problem sounds to me more like it addresses
the issue at hand - can the body escape - than does the simplistic
quesiton of "where are the forces equal". I'd like to particularly
draw your attention to the plots of the "allowed region"
on page 12 of the pdf document I mentioned. The figure labelled case
2 should be the case where the body is no longer considered
"bound" to the moon. While one can see that that the orbit of the
body reaches the L1 point in order to escape (this is the point where
the forces balance), one can also see that the orbit isn't exactly
circular at this point anymore. So I think the answer to your
question may simply lie in what is being measured. Do we measure the
distance fromt the moon to the L1 point to determine the "region of
influence", or do we more conservatively draw a sphere around the moon
and insist that the whole sphere remain inside the "white"
energetically allowed region of figure 2, when the body has "just
enough" energy to escape the moon?

I unfortunately haven't gone through Hill's analysis thoroughly enough
to be positive of exactly how he's actually defined the Hill sphere,
though I suspect he probably did what I described above. Anyway, I
thought that this post might help you out more than it hurt,
apologizies in advance if I turn out to be totally wrong :-).

In addition

http://scienceworld.wolfram.com/physics/JacobiIntegral.html

might be helpful as another source / name for the conserved quantity
that is being plotted and used to determine the "allowed regions" in
the figures I mentioned.

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