Re: [find_orb] (6995) Minoyama/asteroid perturbers

Bill Gray Feb 10, 2013

Hi Rob, all,

On 02/10/2013 04:35 PM, Matson, Robert D. wrote:
> As for perturbers, I checked Jim Baer's list to see if (6995)
> was expected to be perturbed by any of the larger asteroids.
> Jim *does* list this pairing: 42 (Isis) / 6995. He has the
> encounter date as 1995-08-14 at a distance of 0.013 a.u. It
> didn't look like it would yield a huge deflection, but it
> wouldn't surprise me if it was measurable after going back
> almost 40 years.

You can sort of see it, down at the milliarcsecond level:

http://www.projectpluto.com/temp/with_3.htm
http://www.projectpluto.com/temp/with_50.htm

(42) Isis is the 40th object in BC-405. 'with_3' includes
the first three asteroids in BC-405 (the "usual suspects" Ceres,
Pallas, and Vesta). 'with_50' includes the first fifty asteroids,
and therefore includes (42) Isis.

Getting a solution with fifty asteroids included took about a
minute. Doing it with all 300 would have taken about six minutes.
You can see why I'd still like to come up with some way of
speeding up the process. (*)

If you open both files in separate tabs in a browser, and
click on 'residuals', you can blink-compare them. I switched
to 0.001-arcsecond precision residuals so that the difference
would be apparent.

Again, you'll see that the fit is good, even with just
Ceres, Pallas, and Vesta... the previous trouble I saw was due
to thinking that I could get away with including asteroids only
if they were within about 0.15 AU.

-- Bill

(*) How would one do this, you ask? A couple of possibilities...
I'll go into gory details on-list; off-list replies suggest
there's a fair bit of interest in this problem. I'll go into
greater depth tomorrow, but here's a start:

Alan Harris pointed out, in a private e-mail, that the distance
within which we'd need to include an object would vary linearly with
mass. So if we have to consider Pallas from a distance of, say,
5 AU, then we'd have to consider an object 10% as massive only when
within .5 AU. I did the math and got the same answer.

Pallas has a mass about 1/5 of Ceres. But only 12 asteroids have
a mass greater than 10% that of Pallas, and only 96 have masses
greater than 1% that of Pallas. Looking at this, you'd think:
"Huh! We've got to consider the first few objects over most of the
solar system, but after a bit, we're down to that 0.15 AU limit
and can usually ignore most of the rest of the asteroids!" If only
it were so easy.

Problems crop up because of resonances. Suppose the worst-case
scenario of the perturber that stays consistently at a nearly fixed
distance r0 ahead or behind the object, accelerating it or decelerating
it constantly by an amount a=GM/r0^2. If we have an arc length t,
then the perturber will cause an offset of at^2/2 = .5 * GM(t/r0)^2.
If the maximum possible perturbation we are willing to ignore is p AU,
then we need to consider any object of mass M if it comes within

r0 = t * sqrt( GM / 2p)

For Pallas, GM=3.1E-14 AU^3/day^2. If we're willing to accept,
say, a possible perturbation of 10^-8 AU (about 1.5 km), then
r0 = t / 800. For a sixty-year integration of the sort we're
considering here, that's r0 = 27 AU.

r0 will drop for the objects lighter than Pallas, of course, but
in this resonant case, it's only dropping as the square root of the
mass, not directly with mass. For the objects with masses less than
1% of Pallas, we've dropped to 27 * sqrt(.01) = 2.7 AU. Still not
very good... we need a way to say, "These asteroids are resonant
with our object of interest, so include them for the whole
integration; the others need to be included only when they're close."
Which, I think, might be possible.

-- Bill