# [LEAPSECS] Mean ... Orbits

Finkleman, Dave dfinkleman at agi.com
Wed Feb 2 22:03:09 EST 2011

I know that I'm getting in over my head, but the thread is sounding a
lot like statistical filtering, orbit determination, and tracking.

Filtering is estimation into the future based on mathematically
redundant observations subject to minimizing a cost function. Think
least squares. (I just know that I am selling ice to the Eskimos.) It
is finding the best fit to the observations of a hypothesis with free
parameters. Smoothing is going backwards with the same process --
perhaps with an improved hypothesis from filtering. (Think, perhaps,
prolepsis.) I am sure that many have applied these principles to
estimating the "state" of the Earth, where the state vector includes
things such as nutation and the deceleration of the Earth's rotation.
So much for teaching astronomers algebra.

Most such estimations are the outcome of series expansions of governing
independent variables, dependent variables, or similarity parameters.
Think Fourier series -- actually, Jacobi Polynomials, but who is keeping
track. Think also of the moments of an assumed statistical
distribution. The lowest order (DC component, if you will) is called
the mean. In Fourier series, it would be the state averaged over the
period of the trigonometric functions. For an assumed Gaussian, it is
really the Mean. (If the statistics are Gaussian, the first two moments
specify the distribution completely.) Otherwise, that is not really a
time averaged mean -- partly because the independent variable isn't
always time. In optics expanded in Zernike Polynomials (a complete,
orthonormal series over the unit circle), it is called "PISTON,"
translating a lens back and forth.

The result in orbit determination is called the "mean orbit," but that
doesn't imply that it is averaged over time. It is usually the
Keplerian, two-body orbit, which is certainly time varying. The second
order (linear terms) are called "secular," or in optics "tilt." The
quadratic terms in optics are called focus (the cross product term is --
guess what -- astigmatism.)

Now that I've exposed my lack of understanding, the conclusion is that
the "mean" solar second is the outcome of a statistical estimation of
governing state variables fitted against observations. Which "means"
that it is really not necessarily an average over time.

Now, attack!!!

Dave Finkleman
Senior Scientist
Center for Space Standards and Innovation
Analytical Graphics, Inc.
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Colorado Springs, CO 80920

Phone: 719-510-8282 or 719-321-4780
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