On Nearly Circular Orbits in Stellar Dynamics

We analyze the motion of a test particle in a stationary and axially symmetric potential along a nearly circular orbit. The dependence of the distance on time is studied. The solution is obtained by applying energy conservation, which requires to find approximate relations valid for nearly circular motion between the pair energy and angular momentum and the one consisting of the mean distance and eccentricity. The results obtained in this way are then subjected to a more general approximation which includes a power law dependence of the circular speed on distance, but without assuming a priori a low eccentricity. The sinusoidal dependence of distance on time, though confirmed as the first approximation, is not sufficient. Therefore, in the present paper an amendment is proposed. The difference of the squares of angular momenta which concern a purely circular orbit and the corresponding (same mean distance) nearly circular orbit is approximately proportional to the eccentricity square. The difference of the speed square at the mean distance divided by the circular speed from unity is, in the case of increasing circular speed, in the vicinity of the mean distance, approximately proportional to the eccentricity square. Otherwise it is negligible. The influence of the local circular speed slope on the period with respect to distance is much more significant than that of eccentricity. This conclusion is rather confirmed also after studying realistic galactocentric orbits, even of not too low eccentricity. The basic drawback of the sinusoidal dependence of distance is that it predicts equal time intervals between the moments of pericentric distance and mean distance and between those of the mean and apocentric distances, which is incorrect even for very low eccentricities.

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Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023)