From: Henry Spencer <henry@zoo.toronto.edu>
Newsgroups: sci.space.science
Subject: Re: Orbit an Asteroid?
Date: Wed, 13 Mar 1996 21:11:06 GMT

>>Can a spacecraft orbit something as small as an asteroid? ...
>	I believe that you can orbit anything that has a stronger gravity 
>well than your own.

Not quite.  There are both theoretical and practical limits.

Theoretically, to a first approximation, to be in orbit around an object
you have to be following a trajectory entirely within its sphere of
influence (which is defined with respect to whatever *it* is orbiting --
if you are far enough away from the Earth, you are in an orbit around
the Sun, not the Earth).  If you reduce the object's mass but keep its
density constant, it turns out that the size of the sphere of influence
drops off faster than the object's own size.  So eventually the sphere
of influence reaches the object's surface, and orbits around it are no
longer possible.  A quick back-of-the-envelope estimate suggests that
at 1AU from the Sun, this doesn't happen until we're talking about small
dust grains (being orbited by still smaller ones, presumably...).

More practical limits arise because orbital velocities become very slow
when the object gets small, and navigation errors and perturbations like
light pressure start to overwhelm the object's gravity.

>...the Earth moves in a very small 
>circle due to the tug of the Moon.  And if you make your refernce frame 
>the Moon then the Earth is indeed orbiting it...

Except that then the Sun's path is very peculiar indeed.  You cannot say
that the Earth is orbiting the Moon in any realistic sense of the term.
-- 
Space will not be opened by always                 |       Henry Spencer
leaving it to another generation.   --Bill Gaubatz |   henry@zoo.toronto.edu

From: Henry Spencer <henry@zoo.toronto.edu>
Newsgroups: sci.space.science
Subject: Re: Orbit an Asteroid?
Date: Wed, 13 Mar 1996 21:22:07 GMT

In article <4i51sd$j1d@athena.ulaval.ca> afour@RedStripe.gre.ulaval.ca (Alain Fournier) writes:
>> The sphere of influence is *not* where the
>> asteroid gravity exceeds other forces!  It's where the asteroid gravity
>> exceeds the *difference*, between the asteroid and NEAR, in other forces.
>
>This is quite true, an interesting fact to show this is the following:
>...the gravitational force of the Sun on the moon is about
>332 800/(390)^2 = 2.188 times the gravitational force of the Earth on the
>moon. So we could say that the moon is really orbiting the Sun, not
>Earth, Earth is only modifying the orbit of the moon around the Sun,
>but it is much more practical to consider the moon as a satellite of Earth.

It is also much more correct.  Treating the Moon as a satellite of Earth
(and ignoring the Sun) gives you a much better approximation of its path
than treating it as a satellite of the Sun and ignoring the Earth.  (This
is equivalent to saying that the Moon is well within the Earth's sphere
of influence, using the correct definition as given in >> above.)  The
Moon orbits the Earth, *not* the Sun.
-- 
Space will not be opened by always                 |       Henry Spencer
leaving it to another generation.   --Bill Gaubatz |   henry@zoo.toronto.edu

From: Henry Spencer <henry@zoo.toronto.edu>
Newsgroups: sci.space.science
Subject: Re: Orbit an Asteroid?
Date: Mon, 18 Mar 1996 16:44:28 GMT

In article <4iako1$qkr@lace.colorado.edu> fcrary@rintintin.Colorado.EDU (Frank Crary) writes:
>>...If you reduce the object's mass but keep its
>>density constant, it turns out that the size of the sphere of influence
>>drops off faster than the object's own size.  So eventually the sphere
>>of influence reaches the object's surface, and orbits around it are no
>>longer possible.
>
>I disagree. Last time I checked, the size of this sphere of 
>influence was of order the Hill radius,
>r_H = a*(m/M)^0.33

Sorry, Frank, you lose this time.  The radius of the sphere of influence
(well, it's not really exactly a sphere, but pretty close) is, to a good
approximation:

	r_SOI = a*(m/M)^0.4

Roy, Prussing&Conway, and Escobal all agree on the exponent being 0.4 for
the sphere of influence.  The Hill radius is not quite the same thing.

>...Also, if the object in question is small
>enough (say 300 km or less) then it isn't going to be 
>spherical. That introduces all sorts of perturbations...

Indeed so, and I should have mentioned that.
-- 
Americans proved to be more bureaucratic           |       Henry Spencer
than I ever thought.  --Valery Ryumin, RKK Energia |   henry@zoo.toronto.edu