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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. |
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Mean orbital elements
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. -- ---------------------------------------------------------------- Paul Schlyter, Grev Turegatan 40, SE-114 38 Stockholm, SWEDEN e-mail: pausch at stockholm dot bostream dot se WWW: http://www.stjarnhimlen.se/ http://home.tiscali.se/pausch/ |
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