osculating orbit

the orbit that a celestial body would assume if at some instant in time the body were freed from all perturbing forces. An osculating orbit can be determined for any instant, and in general, different osculating orbits correspond to different instants. The elements that determine an osculating orbit are called osculating elements, and the instant for which these elements are computed is called the instant of osculation. An osculating orbit may be an ellipse, parabola, or hyperbola, and one that is calculated for an instant in the middle of a period of observation very closely corresponds to the actual motion of a celestial body over a certain length of time; in the case of a small planet or comet, this length of time can be several weeks or even months.

The actual motion of a celestial body may be considered as motion along an osculating orbit whose elements change continuously with time. Thus, at the instant of osculation the osculating orbit touches on the actual trajectory of the body, and for this reason it is also called the instantaneous orbit. Although osculating orbits may be used to study any type of orbital motion, their application in the method of variation of parameters (which in this case are the osculating elements) is most effective when the perturbing forces are small compared with the attraction of the central body. The fundamentals of the method of variation of parameters were first presented by I. Newton, and this method was later worked out in detail by J. Lagrange.