Bill Zielenbach Feb 11, 2013
On Feb 10, 2013, at 9:05 PM, Bill Gray wrote:
> 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
>
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