Three-body-interaction effects on the relativistic perihelion precession for the Sun-Jupiter-Saturn system

The relativistic perihelion precession due to the three-body interaction is derived. We consider a hierarchical coplanar three-body system, such as the Sun, Jupiter and Saturn, in which both the secondary object as the largest planet corresponding to Jupiter (mass $m_2$) and the third one corresponding to Saturn (mass $m_3$) orbit around the primary object corresponding to Sun (mass $m_1 \gg m_2 \gg m_3$), where the mean orbital radius of the third body is larger than that of the secondary one (denoted as $\ell$). We investigate the post-Newtonian effects on the motion of the third body (semimajor axis a, eccentricity e for the Keplerian orbital elements). Under some assumptions with a certain averaging, the relativistic perihelion precession of the third mass by the post-Newtonian three-body interaction is expressed as $6 G m_2 \ell^2 c^{-2} a^{-3} n (1+9e^2/16) (1-e^2)^{-3}$, where G and c denote the gravitational constant and the speed of light, respectively, and the mean motion for the third body is denoted as $n = 2\pi a^{3/2} G^{-1/2} (m_1+m_2)^{-1/2}$. For the Sun-Jupiter-Saturn system, it is $7.8 \times 10^{-6}$ arcsec/cy. This is larger than the Lense-Thirring effect by Sun but it cannot yet explain the recently reported value for the anomalous perihelion precession of Saturn as $-0.006 \pm 0.002$ arcsec/cy by Iorio (2009) based on the analyses by Pitjeva with the EPM2008 ephemerides.