Lambda Perturbations of Keplerian Orbits in the Expanding Universe
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Lambda Perturbations of Keplerian Orbits in the Expanding Universe Yu. V. Dumin1, 2* 1
Space Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya str. 84/32, Moscow, 117997, Russia 2 Sternberg Astronomical Institute (GAISh), Lomonosov Moscow State University, Universitetskii prosp. 13, Moscow, 119234, Russia Received June 15, 2020; revised June 15, 2020; accepted June 29, 2020
Abstract—To estimate the influence of “dark energy” on Keplerian orbits, we solve the general-relativistic equations of motion of a test particle in the field of a pointlike mass embedded in the cosmological background formed by the cosmological constant with realistic cosmological Robertson–Walker asymptotics at infinity. It is found that under certain relations between three crucial parameters of the problem—the initial radius of the orbit, the Schwarzschild and de Sitter radii—a specific secular perturbation caused by Λ-term becomes significant, i.e., can reach the rate of the standard Hubble flow. This fact is interesting both by itself and may have important consequences for the long-term dynamics of planets and stellar binary systems. DOI: 10.1134/S0202289320040040
1. INTRODUCTION The question if the planetary orbits and other small-scale celestial systems are subject to cosmological influences (in particular, if they feel the universal Hubble expansion) was put forward by McVittie as early as in 1933 [1]; and this problem attracted attention of a number of other researchers during the subsequent decades [2–18]; quite a comprehensive review of these works was given by Bonnor [19]. The most frequent conclusion of such studies was that the effect of cosmological expansion at the planetary scales should be very small or absent at all. However, the particular estimates given by different authors substantially disagree with each other. Moreover, most of these estimates (excluding the most recent ones) are not applicable to the case where the cosmological background is formed by “dark energy,” (DE), i.e., in a reasonable approximation, if not precisely, the Λ-term in the Einstein equations, because it is distributed perfectly uniformly and insensitive to the local gravitational perturbations. For example, the most well-known argument against the local Hubble expansion is the so-called Einstein–Straus theorem [2]: Let us consider a uniform background cosmological matter distribution and, next, cut out a spherical cavity and concentrate all its mass at the central point. Then, the solution of the General Relativity equations will be given by the purely static Schwarzschild metric inside the cavity, and it will transform to the time-dependent *
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Friedmann–Robertson–Walker metric outside the cavity. In other words, there is no Hubble expansion in the local empty neighborhood of a pointlike massive body, but such an expansion appears in the regions of space filled with the cosmological background matter. Unfortunately, despite an apparent generality of this result, it is evidently inapplicable to th
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