Hi Andy, Tony,
This problem of giving uncertainties for all orbits is something
I've been working on recently. It's one of the "things I should
clean up before I post an update" that I mentioned in my previous
e-mail.
I'd be wary of setting a constrained orbit and trusting the resulting
uncertainties. The problem is that we don't know that the eccentricity
is actually 0.35. If it _is_ 0.35, then the range of possible orbits
is greatly reduced, and the sigmas will be correspondingly decreased.
But we've no real reason to think it is. (I've had at least one
situation where sigmas for a constrained orbit were probably valid:
that of sungrazing comets. For these, I sometimes constrain the
orbit to be parabolic, e=1. Given what we know of sungrazing comets,
this is a reasonable thing to do... for them.)
There are a lot of different ways to determine orbital uncertainties.
The sigmas Find_Orb shows are "formal covariance" uncertainties, the
sort you get when you solve a least-squares problem. These are
absolutely wonderful in many ways, but they do require (at minimum)
that you be able to get the least-squares fit to converge to a
reasonable answer.
Andy's test case, for the 2014 October 22 observations of 2004 YW5,
is a good one. You actually _can_ get a convergent least-squares fit
here. Just hit 'full step' until the values settle down. You'll get
something along the lines of
Orbital elements: 2004 YW5
Perihelion 2014 Sep 3.665189 +/- 129 TT = 15:57:52 (JD 2456904.165189)
Epoch 2014 Oct 22.0 TT = JDT 2456952.5 Earth MOID: 0.0084 Ju: 0.2563
q 0.93781127 +/- 1.08 Sa: 0.9977 Find_Orb
H 18.9 G 0.15 Peri. 330.86147 +/- 110
Node 357.93764 +/- 240
e 1.9192883 +/- 7.67 Incl. 9.31418 +/- 80
From 6 observations 2014 Oct. 22 (3.4 hr); mean residual 0.097
So you get an answer, but it's not a very helpful one. It tells you
that the eccentricity, for example, could be between about -5 and +10.
Since we know that "real-world" natural objects have eccentricities
between zero (circular orbit) and one (parabolic), that's not of much
use. The nominal orbit itself is totally wacky, which is why it's shown
in red. The inclination tells us that the object is probably (though not
definitely) prograde.
So the covariance matrix method is not really giving us good
uncertainties. About the only method that works well here (and for
similar very short arcs) is that of statistical ranging, or SR.
SR is a somewhat brute-force approach; it looks for a representative
sampling of orbits that fit your data. It's the method MPC uses for
NEOCP; when you see NEOCP scatterplots or look at the variant
"virtual asteroid" orbital elements, you're seeing SR results.)
Here are a thousand orbits SR came up with for your six observations :
http://www.projectpluto.com/temp/sr_elems.txt
My plan is that, in situations where covariance uncertainties
fail, Find_Orb will try SR to generate a list such as the above,
then use the variants to determine orbital element and ephemeris
uncertainties, with a warning that the orbital element uncertainties
can be quite wacky and mostly just tell you: "We don't know the
elements for this object very well."
-- Bill