Other
Scientific paper
May 2010
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2010dda....41.0604m&link_type=abstract
American Astronomical Society, DDA meeting #41, #6.04; Bulletin of the American Astronomical Society, Vol. 41, p.931
Other
Scientific paper
Recent years have seen a particular interest in estimating orbital elements and ephemeris uncertainties from just two astrometric observations. Since 1996 the Minor Planet Center has used a two-observation method in which the coordinate system is rotated so that the reference plane passes through the two observations and the origin is at one of them. The selection of appropriate values for the topocentric distance ρ at the other observation then immediately provides, not only the components of the heliocentric position vector, but also two components of the heliocentric velocity vector. It therefore remains to select appropriate values for the third velocity component, the value zero yielding α, the largest reciprocal semimajor axis for the specified ρ. The Vaisala (or apsidal) value of the velocity component follows, together with values yielding the two lateral orbits (at most one of which can be elliptical), with the object on the orbital latus rectum. If α is positive, elliptical solutions exist for values of the velocity component out to ±√α.
It is also the case that α generally decreases with increasing ρ, though not necessarily monotonically. Indeed, for an object at opposition, distances corresponding to a lone parabolic solution readily follow from a cubic equation, there being one or three real roots according as to whether the apparent retrograde motion is greater than or less than some critical value. A very similar quadratic equation can be used to derive the distances corresponding to precisely circular solutions, when the apsidal velocity component is equal to the smaller of the lateral values. Corresponding equations are also derived to describe possible orbits near quadrature.
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