# orbital elements

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## orbital elements

(**or**-bă-tăl) The parameters that specify the position and motion of a celestial body in its orbit and that can be established by observation (see illustration).

*Osculating elements*specify the instantaneous position and velocity of a body in a perturbed orbit (see osculating orbit).

*Mean elements*are those of some reference orbit that approximates the actual perturbed orbit. The shape and size of the orbit are specified by the eccentricity,

*e*, and for an elliptical orbit by the semimajor axis,

*a*. The orientation of the orbit in space is specified firstly by the inclination,

*i*, of the orbital plane to the reference plane, usually that of the ecliptic, and secondly by the

*longitude of the ascending node*, ω; the latter is the angular distance from the dynamical or vernal equinox, γ, to the ascending node, N. The orientation of the orbit in the orbital plane is usually specified by the angular distance, ω, between the periapsis, P (see apsides), and the ascending node. For an orbit round the Sun the periapsis is the perihelion and the angular distance ω is the

*argument of perihelion*. It is measured along the direction of motion. The angular distance equal to the argument of perihelion plus the longitude of the ascending node is called the

*longitude of perhelion*, which is also used as an orbital element.

The eccentric anomaly is used to determine the position of the body in its orbit. To calculate the position as a function of time an additional orbital element is used. This is the *time of periapsis* (or for a solar orbit *perihelion*) *passage*, *t *_{0}, by means of which Kepler's equation (see anomaly) can be solved. Analogous elements are used to describe the orbits of binary stars. To determine the orbit of a binary star system of unknown mass, the period must be established.

The Earth's orbital elements vary with time due to gravitational effects of the Moon and planets. The changes, approximately periodic, during the past several hundred thousand years have been calculated very accurately.

## Orbital Elements

in astronomy, a system of quantities (parameters) that define the orientation of the orbit of a celestial object in space, the dimensions and shape of the orbit, and the position of the body in orbit at some fixed moment of time. The unperturbed orbit, in which a body moves in accordance with Kepler’s laws, is defined by six orbital elements.

(1) The inclination of the orbit *i* to the plane of the ecliptic or to the plane of the earth’s equator (in the case of artificial earth satellites). The inclination may have values of 0° to 180°. It is less than 90° if the body appears to be moving counterclockwise to an observer located at the north ecliptic pole or the north celestial pole and greater than 90° if the body appears to be moving in the opposite direction.

(2) The longitude of the (ascending) node ☊ or the right ascension of the (ascending) node α_{☊} (in the case of artificial earth satellites); it may have values of 0° to 360°.

(3) The semimajor axis *a* of the orbit. The average motion of a body in an orbit *n* is sometimes used instead of the semi-major axis; in the case of unperturbed motion, the average motion is uniquely dependent on the semimajor axis.

(4) The orbital eccentricity *e*.

(5) The argument ω of perihelion or perigee (in the case of the moon or an artificial earth satellite); it may have values of 0° to 360°.

(6) The epoch (time) *T* at which the body is located at a certain point in the orbit, for example, at the ascending node or at perihelion or perigee. The start of the day is sometimes selected as the epoch; in this case, the orbital position is given by the mean anomaly *M*_{0} at this epoch.

In the case of a perturbed orbit, the orbital elements are considered as functions of time and are usually represented as the power series

*A* = *A*_{0} + *A*_{1} (*t* – *T*_{0}) + *A*_{2} (*t* – *T*_{0})^{2} + . . .,

where *A*_{0} is the value of an orbital element *A* at time *T*_{0}.

N. P. ERPYLEV