Note on the perihelion/periastron advance due to cosmological constant

We will comment on the perihelion/periastron advance of celestial bodies due to the cosmological constant $\mathrm{\Lambda }$. It is well known that the cosmological constant $\mathrm{\Lambda }$ 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 $\Delta \omega_{\Lambda} = (\pi c^2 \Lambda a^3/(GM))\sqrt{1 - e^2}$isthegeneralformulaforanyorbitaleccentricity$e$andtheexpression$\Delta \omega_{\Lambda} = (\pi c^2 \Lambda a^3/(GM))(1

• e^2)^3$comesfromthenearlycircular($e \ll 1$) approximation.