Abstract
High-precision observations require accurate modelling of secular changes in the orbital elements in order to extrapolate measurements over long time intervals, and to detect deviation from pure Keplerian motion caused, for example, by other bodies or relativistic effects. We consider the evolution of the Keplerian elements resulting from the gradual change of the apparent orbit orientation due to proper motion. We present rigorous formulae for the transformation of the orbit inclination, longitude of the ascending node and argument of the pericenter from one epoch to another, assuming uniform stellar motion and taking radial velocity into account. An approximate treatment, accurate to the second-order terms in time, is also given. The proper motion effects may be significant for long-period transiting planets. These theoretical results are applicable to the modelling of planetary transits and precise Doppler measurements as well as analysis of pulsar and eclipsing binary timing observations.
Highlights
Keplerian orbits are among the most important concepts in astronomy
The equatorial coordinate system is used throughout this paper, with all relevant quantities referring to the solar system barycenter (SSB)
We discuss some practical implications of the analytical results obtained in the preceding sections
Summary
Keplerian orbits are among the most important concepts in astronomy. Variations of the parameters of a Keplerian orbit lead to various phenomena in observational data. They are employed in studying geodetic intersatellite tracking (Cheng, 2002), in developing relationship between the perturbation of the conventional orbital elements and the perturbations of position and velocity (Casotto, 1993) They provide a theoretical basis for analytical calculation of the post-Keplerian effect in the radial velocity of binary systems (Iorio, 2017a) and the orbital time shift in binary pulsars (Iorio, 2017b) as well as the effect of the Lense-Thirring precession on BepiColombo mission (Iorio, 2018). Results of this study indicate that for stars with proper motion at the level 10–100 mas yr−1 the rate of variation of transit duration is comparable or exceeds effects from general relativity or stellar oblateness. These works consider only linear, first-order in time, terms of the orbital parameters variation.
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