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Theproper orbital elements orproper elements of anorbit are constants of motion of an object in space that remain practically unchanged over an astronomically long timescale. The term is usually used to describe the three quantities:
The proper elements can be contrasted with theosculating Keplerianorbital elements observed at a particular time orepoch, such as thesemi-major axis,eccentricity, andinclination. Those osculating elements change in aquasi-periodic and (in principle) predictable manner due to such effects as perturbations from planets or other bodies, and precession (e.g.perihelion precession). In theSolar System, such changes usually occur on timescales of thousands of years, while proper elements are meant to be practically constant over at least tens of millions of years.
For most bodies, the osculating elements are relatively close to the proper elements because precession and perturbation effects are relatively small (see diagram). For over 99% ofasteroids in theasteroid belt, the differences are less than 0.02 AU (forsemi-major axis a), 0.1 (foreccentricity e), and 2° (forinclination i).
Nevertheless, this difference is non-negligible for any purposes where precision is of importance. As an example, the asteroidCeres has osculating orbital elements (atepoch November 26, 2005)
| a | e | i |
|---|---|---|
| 2.765515 AU | 0.080015 | 10.5868° |
while its proper orbital elements (independent of epoch) are[1]
| ap | ep | ip |
|---|---|---|
| 2.767096 AU | 0.116198 | 9.6474° |
A notable exception to this small-difference rule areasteroids lying in theKirkwood gaps, which are in strong orbital resonance with Jupiter.
To calculate proper elements for an object, one usually conducts a detailed simulation of its motion over timespans of several millions of years. Such a simulation must take into account many details of celestial mechanics including perturbations by the planets. Subsequently, one extracts quantities from the simulation which remain unchanged over this long timespan; for example, the mean inclination, mean eccentricity, and mean semi-major axis. These are the proper orbital elements.[citation needed]
Historically, various approximate analytic calculations were made, starting with those ofKiyotsugu Hirayama in the early 20th century. Later analytic methods often included thousands of perturbing corrections for each particular object. Presently, the method of choice is to use a computer to numerically integrate the equations ofcelestial dynamics, and extract constants of motion directly from a numerical analysis of the predicted positions.
At present the most prominent use of proper orbital elements is in the study ofasteroid families, following in the footsteps of the pioneering work of Hirayama.
AMars-crosser asteroid132 Aethra is the lowest numbered asteroid to not have any proper orbital elements.
