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The proper orbital elements or proper elements of an

orbit In celestial mechanics, an orbit (also known as orbital revolution) is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an ...

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: *''proper semimajor axis'' (''a_p''), *''proper eccentricity'' (''e_p''), and *''proper inclination'' (''i_p''). The proper elements can be contrasted with the osculating Keplerian

orbital elements Orbital elements are the parameters required to uniquely identify a specific orbit. In celestial mechanics these elements are considered in two-body systems using a Kepler orbit. There are many different ways to mathematically describe the same o ...

observed at a particular time or

epoch In chronology and periodization, an epoch or reference epoch is an instant in time chosen as the origin of a particular calendar era. The "epoch" serves as a reference point from which time is measured. The moment of epoch is usually decided b ...

, such as the

semi-major axis In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the two most widely separated points of the perimeter. The semi-major axis (major semiaxis) is the longe ...

eccentricity Eccentricity or eccentric may refer to: * Eccentricity (behavior), odd behavior on the part of a person, as opposed to being "normal" Mathematics, science and technology Mathematics * Off-Centre (geometry), center, in geometry * Eccentricity (g ...

, and

inclination Orbital inclination measures the tilt of an object's orbit around a celestial body. It is expressed as the angle between a reference plane and the orbital plane or axis of direction of the orbiting object. For a satellite orbiting the Eart ...

. Those osculating elements change in a quasi-periodic and (in principle) predictable manner due to such effects as perturbations from planets or other bodies, and precession (e.g. perihelion precession). In the

Solar System The Solar SystemCapitalization of the name varies. The International Astronomical Union, the authoritative body regarding astronomical nomenclature, specifies capitalizing the names of all individual astronomical objects but uses mixed "Sola ...

, 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% of

asteroid An asteroid is a minor planet—an object larger than a meteoroid that is neither a planet nor an identified comet—that orbits within the Solar System#Inner Solar System, inner Solar System or is co-orbital with Jupiter (Trojan asteroids). As ...

s in the

asteroid belt The asteroid belt is a torus-shaped region in the Solar System, centered on the Sun and roughly spanning the space between the orbits of the planets Jupiter and Mars. It contains a great many solid, irregularly shaped bodies called asteroids ...

, the differences are less than 0.02 AU (for

''a''), 0.1 (for

''e''), and 2° (for

''i''). Nevertheless, this difference is non-negligible for any purposes where precision is of importance. As an example, the asteroid Ceres has osculating orbital elements (at

November 26, 2005) while its proper orbital elements (independent of epoch) are A notable exception to this small-difference rule are

s lying in the

Kirkwood gap A Kirkwood gap is a gap or dip in the distribution of the semi-major axes (or equivalently of the orbital periods) of the orbits of main-belt asteroids. They correspond to the locations of orbital resonances with Jupiter. The gaps were first n ...

s, 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. Historically, various approximate analytic calculations were made, starting with those of Kiyotsugu 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 of celestial dynamics, and extract constants of motion directly from a numerical analysis of the predicted positions. Asteroid osculating vs proper elements

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. A Mars-crosser asteroid 132 Aethra is the lowest numbered asteroid to not have any proper orbital elements.

References

External links

Latest calculations of proper elements for numbered minor planets at astDys

Asteroid proper orbital elements dataset at Asteroid Families Portal
{{Portal bar, Astronomy, Stars, Spaceflight, Outer space, Solar System Orbits

See also

References

Further reading

External links