The word is Latin for “kiss”. In mathematics, two curves osculate when they just touch, without crossing, at a point, where both have the same position and slope, i.e. the two curves “kiss”.
Kepler elements
An osculating orbit and the object's position upon it can be fully described by the six standard Kepler orbital elements, which are easy to calculate as long as one knows the object's position and velocity relative to the central body. The osculating elements would remain constant in the absence of perturbations. Real astronomical orbits experience perturbations that cause the osculating elements to evolve, sometimes very quickly. In cases where general celestial mechanical analyses of the motion have been carried out, the orbit can be described by a set of mean elements with secular and periodic terms. In the case of minor planets, a system of proper orbital elements has been devised to enable representation of the most important aspects of their orbits.
Perturbations
that cause an object's osculating orbit to change can arise from:
A non-spherical component to the central body.
A third body or multiple other bodies whose gravity perturbs the object's orbit, for example the effect of the Moon's gravity on objects orbiting Earth.
A relativistic correction.
A non-gravitational force acting on the body, for example force arising from:
An object's orbital parameters will be different if they are expressed with respect to a non-inertial reference frame, than if it is expressed with respect to a inertial reference frame. Put in more general terms, a perturbed trajectory can be analysed as if assembled of points, each of which is contributed by a curve out of a sequence of curves. Variables parameterising the curves within this family can be called orbital elements. Typically, these curves are chosen as Keplerian conics, all of which share one focus. In most situations, it is convenient to set each of these curves tangent to the trajectory at the point of intersection. Curves that obey this condition are called osculating, while the variables parameterising these curves are called osculating elements. In some situations, description of orbital motion can be simplified and approximated by choosing orbital elements that are not osculating. Also, in some situations, the standard equations furnish orbital elements that turn out to be non-osculating.
