Osculating orbit

In astronomy, and in particular in astrodynamics, the osculating orbit of an object in space (at a given moment of time) is the gravitational Kepler orbit (i.e. ellipse or other conic) that it would have about its central body (corresponding to its actual position and velocity for that given moment of time) if perturbations were not present.^[1]

An osculating orbit and the object's position upon it can be fully described by the six standard Keplerian 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. However, 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.

The word "osculate" derives from a Latin word meaning "to kiss". Its use in this context derives from the fact that, at any point in time, an object's osculating orbit is precisely tangent to its actual orbit, with the tangent point being the object's location – and has the same curvature as the orbit would have in the absence of perturbing forces.

Perturbations that cause an object's osculating orbit to change can arise from:

• A non-spherical component to the central body (when the central body can be modeled neither with a point mass nor with a spherically symmetrical mass distribution, e.g. when it is an oblate spheroid).
• 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 non-gravitational force acting on the body, for example force arising from:
  • Thrust from a rocket or ion engine
  • Releasing, leaking, venting or ablation of a material
  • Collisions with other objects
  • Atmospheric drag
  • Radiation pressure
  • Solar wind pressure
  • Switch to a non-inertial reference frame (e.g. when a satellite's orbit is described in a reference frame associated with the precessing equator of the planet).

The form of expression of an object's orbital parameters is different in general if it is given with respect to a non-inertial frame of reference (for example, to 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 sharing one of their foci. 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]

See also

• Kepler orbit
• Eccentricity vector
• Orbital elements
• Contact (mathematics)
• Osculating circle
• List of orbits

References

1. ^ F R Moulton, 'Introduction to Celestial Mechanics', (1902, Dover reprint 1970), at pp.322-3.
2. ^ For details see: Efroimsky, M. 2006. ``Gauge Freedom in Orbital Mechanics." Annals of the New York Academy of Sciences, Vol. 1065, pp. 346 - 374 (astro-ph/0603092); Efroimsky, M., and Goldreich, P. 2003. ``Gauge Symmetry of the N-body Problem in the Hamilton-Jacobi Approach." Journal of Mathematical Physics, Vol. 44, pp. 5958 - 5977 (astro-ph/0305344).

External links

• Diagram of a sequence of osculating orbits for the escape from Earth orbit by the ion-driven SMART-1 spacecraft: http://sci.esa.int/science-e/www/object/index.cfm?fobjectid=35722
• A sequence of osculating orbits for the approach to the Moon by the SMART-1 spacecraft: http://sci.esa.int/science-e/www/object/index.cfm?fobjectid=36359
• Osculating orbits in a restricted 3-Body problem (video, youtube)
• Osculating orbits in a 3-Body Lagrange problem (video, youtube)
• Osculating orbits in a 4-Body Lagrange problem (video, youtube)
• Osculating orbits in the Pythagorean 3-Body problem (video, youtube)