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

Bill Zielenbach Feb 11, 2013

(6995) got a few other small "tugs" that I know about from my 2006 close approach database.

As I explained to Bill, it looked at interbody distances every 8 days between the first 129,000 numbered asteroids, from mid 2006 backwards to 3 years before the published discovery date of the "younger" body.
It looked for the closest approach within 0.1AU (many bodies have multiple close encounters with each other over the years) and the date and separation figures are those of the nearest 8 day integration step. (At that time I was using a fixed-step modified Cowell integrator with an 8 day step for generating this database.) The perturbers were Mercury thru Pluto, the moon, and CPV.

The E-16 numbers are the GM's for the bodies in parentheses.

08/03/95 0.060AU from (110) 3.76E-16

08/11/95 0.017 AU from (42) 2.3E-16

11/15/95 0.078AU from (96) 7.43E-16

06/19/02 0.085AU from (431) 1.32E-16


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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