Re: {MPML} Orbit determination exercise

Bill J Gray Jan 29, 2011

Hi Alessandro,

(Side note: this discussion appears to be continuing on both the
Find_Orb and MPML lists. Fortunately, both are open to viewing by
non-members.)

With the short arc you have here (two hours long), you can get
an astonishing variety of orbits. That's basically standard
procedure for very short arcs, except in some unusual situations
(mostly involving objects that are really close to us). We can
say, though, with confidence, that the object isn't more than
.447 AU away. If it were, the orbit would have to be hyperbolic,
and nobody has seen that happen (yet). That upper distance limit
is just about the only constraint on the orbit right now.

Within .447 AU, there is a huge range of orbits. Find_Orb
looks through a broad sample of them and picks one that looks most
"reasonable", given what we know about the distribution of orbits
among observed objects. Usually, that leads to a main-belt
solution. (This is the approach that says, "When you hear
hoofbeats, expect horses, not zebras".)

In this case, Find_Orb's default orbit has i=27, q=.13, Q=1.53.
That's a pretty weird orbit. Find_Orb should have looked further
(i.e., tried out some more possible orbits). With some searching,
I found one with i=4, q=.57, Q=1.89, though the residuals are
a bit high (rms error = .705 arcseconds).

One _could_ just look for the orbit with the lowest residuals.
In this case, with only three observations, the lowest residuals
would be exactly zero. As has been pointed out in this thread,
that leads to a somewhat exotic orbit with i=55, q=.09.

There are a couple of problems with doing this. For one thing,
with a very short arc like this, it's common for the lowest-residual
orbit to be something _really_ exotic, like a Warp 9 orbit out past
Alpha Centauri. Sometimes, it diverges completely. Or, it
converges on something pretty weird.

The usual way to tackle this sort of very short arc problem
would be with statistical ranging. SR says: We compute a slew
of possible orbits at various ranges and radial velocities, and
keep those with non-hyperbolic orbits. We also add in some "noise"
that reflects our guess as to how exact our observations are. All
the resulting "virtual asteroids" are at least _possible_ places
the object could be. (Though if both main-belt solutions and
exotic solutions appear, it'll probably be a main-belter: again,
expect horses, not zebras.)

This is the approach used for the NEOCP uncertainty charts: each
of triangle in the chart represents a possible orbit that fits the
observations tolerably well. Dave Tholen's KNOBS software, I gather,
does the same thing, and I think OrbFit and CODES have this
capability. As does Find_Orb... except %$!*^ it, I tried it on this
object and got three orbits, and then it failed. I'll have to check
that out. (I only added SR recently; it's a "work in progress".)

-- Bill