The Effect of the Mutual Gravitational Interactions on the Perihelia Displacement of the Orbits of the Solar System’s Planets

The classical $N$-body problem in the case when one of the bodies (the Sun) has a much larger mass than the rest of the mutually gravitating bodies is considered. The system of equations in canonical Delaunay variables describing the motion of the system relative to the barycentric coordinate system is derived via the methods of analitical dynamics. The procedure of averaging over the fast angular variables (mean anomalies) leads to the equation describing the evolution of a single Solar system planet’s perihelion as the sum of two terms. The first term corresponds to the gravitational disturbances caused by the rest of the planets, as in the case of a motionless Sun. The second appears because the problem is considered in the barycentric coordinate system and the orbits’ inclinations are taken into account. This term vanishes if all planets are assumed to be moving in one static plane. This term contributes substantially to the Mercury’s and Venus’s perihelion evolutions. For the rest of the planet this term is small compared to the first one. For example, for Mercury the values of the two terms in question were calculated to be 528.67 and 39.64 angular seconds per century, respectively.