Osculating orbit

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Osculating orbit (inner, black) and perturbed orbit (red) Enckes method-vector.svg
Osculating orbit (inner, black) and perturbed orbit (red)

In astronomy, and in particular in astrodynamics, the osculating orbit of an object in space at a given moment in time is the gravitational Kepler orbit (i.e. an elliptic or other conic one) that it would have around its central body if perturbations were absent. [1] That is, it is the orbit that coincides with the current orbital state vectors (position and velocity).



The word osculate is Latin for "kiss". In mathematics, two curves osculate when they just touch, without (necessarily) 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 (osculating 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 (as they have been for the major planets, the Moon, and other planetary satellites), 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:


An object's orbital parameters will be different if they are expressed with respect to a non-inertial reference frame (for example, a frame co-precessing with the primary's equator), than if it is expressed with respect to a (non-rotating) 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 (though not necessarily), 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 (and also the further condition that they have the same curvature at the point of tangency as would be produced by the object's gravity towards the central body in the absence of perturbing forces) 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 (Lagrange-type or Delaunay-type) equations furnish orbital elements that turn out to be non-osculating. [2]

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Proper orbital elements

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Kepler orbit

In celestial mechanics, a Kepler orbit is the motion of one body relative to another, as an ellipse, parabola, or hyperbola, which forms a two-dimensional orbital plane in three-dimensional space. A Kepler orbit can also form a straight line. It considers only the point-like gravitational attraction of two bodies, neglecting perturbations due to gravitational interactions with other objects, atmospheric drag, solar radiation pressure, a non-spherical central body, and so on. It is thus said to be a solution of a special case of the two-body problem, known as the Kepler problem. As a theory in classical mechanics, it also does not take into account the effects of general relativity. Keplerian orbits can be parametrized into six orbital elements in various ways.

Orbit modeling is the process of creating mathematical models to simulate motion of a massive body as it moves in orbit around another massive body due to gravity. Other forces such as gravitational attraction from tertiary bodies, air resistance, solar pressure, or thrust from a propulsion system are typically modeled as secondary effects. Directly modeling an orbit can push the limits of machine precision due to the need to model small perturbations to very large orbits. Because of this, perturbation methods are often used to model the orbit in order to achieve better accuracy.


  1. Moulton, Forest R. (1970) [1902]. Introduction to Celestial Mechanics (2nd revised ed.). Mineola, New York: Dover. pp. 322–23. ISBN   0486646874.
  2. For details see: Efroimsky, M. (2005). "Gauge Freedom in Orbital Mechanics". Annals of the New York Academy of Sciences. 1065: 346–74. arXiv: astro-ph/0603092 . Bibcode:2005NYASA1065..346E. doi:10.1196/annals.1370.016. PMID   16510420.; Efroimsky, Michael; Goldreich, Peter (2003). "Gauge symmetry of the N-body problem in the Hamilton–Jacobi approach". Journal of Mathematical Physics. 44 (12): 5958–5977. arXiv: astro-ph/0305344 . Bibcode:2003JMP....44.5958E. doi:10.1063/1.1622447.