Proper orbital elements

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.

[citation needed] Historically, various approximate analytic calculations were made, starting with those of Kiyotsugu Hirayama in the early 20th century.

At present the most prominent use of proper orbital elements is in the study of asteroid families, following in the footsteps of the pioneering work of Hirayama.

Distribution of the difference between proper and osculating orbital elements for asteroids with semi-major axes lying between 2 and 4 AU .
Osculating (left) and proper (right) orbital elements for asteroids in the asteroid belt . Note how asteroid family clumps are not discernible on the left.