Mean orbital elements

Does anyone have a clear definition of 'mean' used in this context?
Obviously, it's some kind of average over time but what I would like to
know is how to calculate this averaging from, say, long-term position
and velocity data. I understand and can calculate osculating (two-body)
elements and maybe you can do the same sort of averaging directly with
these. Any ideas?

John.

In article ,
John Irwin wrote:

Does anyone have a clear definition of 'mean' used in this context?
Obviously, it's some kind of average over time but what I would like to
know is how to calculate this averaging from, say, long-term position
and velocity data. I understand and can calculate osculating (two-body)
elements and maybe you can do the same sort of averaging directly with
these. Any ideas?

The classical mean orbital elements had to be obtained by a quite
complex process. First, an analytical theory of the motion of the
planet had to be constructed. Once that was accomplished, the "mean
orbital elements" was the orbital elements of the motion of the
planets when all perturbation terms with a sufficiently short period
had been removed. Now, don't confuse this with osculating orbital
elements - such elements are a "snapshot view" of a planet's precise
motion at some specific moment, while the mean orbital elements are
an approximation of the planet's motion over a long time interval
without sharp limits. Variations in time of the mean orbital
elements was allowed if these variations could be expressed as simple
polynomials while still being accurate over a long time period.

Analytical theories are available only for a few celestial bodies:
the major planets except Pluto, our Moon, and Jupiter's Galilean
satellites. Attempts were made at a few other celestial bodies
(Pluto, Ceres, and some Saturnian satellites) but they weren't
completed. Constructing an analytical theory of some celestial body
is a lot of work: E.W. Brown spent a lifetime refining his theory of
the Moon's motion, but then his theory remained our most accurate
description of the Moon's motion for over 70 years.

Anyway, this means that mean orbital elements are unavailable for
most celestial bodies -- they're available only for those few bodies
for which an analytical theory has been constructed. Today, when
high-speed computers has made numerical integration quite effortless,
people try to make shortcuts. "Semi-analytical" theories have been
constructed, often based on e.g. a Fourier analysis of the results of
a numerical integration. Or people try to obtain "mean elements" by
averaging some set of osculating elements in some way. But the
resulting "mean elements" will then depend both on how the averaging
was done (the arithmetic mean isn't the only way to do it), and what
time interval was chosen for the averaging.

"But can't digital computes be used for constructing these analytical
theories much faster, just like they do numerical integration much
faster?" perhaps you ask. But then two problems will pop up: first,
it's much harder to do symbolic math than number crunching on a
computer. And second, each new celestial body will present a new
situation. Remember that the analytical theories aren't exact --
they depend on series expansion and then truncating the infinite
series at some point. And the decision where to truncate them, and
which series to expand at all, and how to expand them, requires human
judgement. Therefore it's not possible to automate this process with
a computer program. Analytical theories also work well only for
low-eccentricity low-inclination orbits, and the most interesting
celestial bodies moving in such orbits are those who already have
analytical theories.

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