Note on the perihelion/periastron advance due to cosmological constant

TL;DR

This paper clarifies the correct formula for the perihelion/periastron advance caused by the cosmological constant, resolving discrepancies among previous derivations and providing a general expression valid for any orbital eccentricity.

Contribution

It presents a unified formula for the perihelion shift due to the cosmological constant applicable to all eccentricities, correcting previous approximations.

Findings

The general formula for perihelion shift is Δω_Λ = (π c^2 Λ a^3 / GM)√(1 - e^2).

The nearly circular orbit approximation yields Δω_Λ = (π c^2 Λ a^3 / GM)(1 - e^2)^3.

Discrepancies in earlier formulas are due to different assumptions about orbital eccentricity.

Abstract

We will comment on the perihelion/periastron advance of celestial bodies due to the cosmological constant $\Lambda$. It is well known that the cosmological constant $\Lambda$ causes the perihelion/periastron shift; however, there seems to still exist a discrepancy among the various derived precession formulae. We will point out that the expression $\Delta \omega_{\Lambda} = (\pi c^2 \Lambda a^3/(GM))\sqrt{1 - e^2}$ is the general formula for any orbital eccentricity $e$ and the expression $\Delta \omega_{\Lambda} = (\pi c^2 \Lambda a^3/(GM))(1 - e^2)^3$ comes from the nearly circular ($e \ll 1$) approximation.