Perihelion precession of Mercury

As described in Appendix B, if the calculation described in the previous section is carried out more accurately, taking into account the slight eccentricities of the planetary orbits, as well as their small mutual inclinations, then the perihelion precession rate of the planet Mercury is found to be $5.32$ arc seconds per year. However, the observed precession rate is $5.75$ arc seconds per year. It turns out that the cause of this discrepancy is a general relativistic correction to Newtonian gravity.

General relativity gives rise to a small correction to the force per unit mass exerted by the Sun (mass $M$) on a planet in a circular orbit of radius $r$, and angular momentum per unit mass $h$. This correction is due to the curvature of space in the immediate vicinity of the Sun. In fact, the modified formula for $f$ is (Rindler 1977)

$$f \simeq -\frac{G\,M}{r^{\,2}}-\frac{3\,G\,M\,h^{\,2}}{c^{\,2}\,r^{\,4}},$$ (5.26)

where $c$ is the velocity of light in vacuum. It follows that

$$\frac{r\,f'}{f} \simeq - 2\left(1+ \frac{3\,h^{\,2}}{c^{\,2}\,r^{\,2}}+ \cdots\right),$$ (5.27)

to first order in $h^{\,2}/(c^{\,2}\,r^{\,2})\ll 1$. Hence, from Equation (5.19), the apsidal angle is

$$\psi \simeq \pi\left(1+ \frac{3\,h^{\,2}}{c^{\,2}\,r^{\,2}}\right).$$ (5.28)

Thus, the perihelion advances by

$$\delta\varpi \simeq \frac{6\pi\,G\,M}{c^{\,2} \,r}$$ (5.29)

radians per revolution due to the general relativistic correction to Newtonian gravity. Here, use has been made of $h^{\,2}=G\,M\,r$. It follows that the rate of perihelion precession due to the general relativistic correction is

$$\dot{\varpi} \simeq \frac{0.0383}{a\,T}$$ (5.30)

arc seconds per year, where $a$ is the mean orbital radius in astronomical units, and $T$ the orbital period in years. Hence, from Table 4.1, the general relativistic contribution to $\dot{\varpi}$ for Mercury is $0.41$ arc seconds per year. It is easily demonstrated that the corresponding contributions are fairly negligible for the other planets in the solar system. If the preceding calculation is carried out sightly more accurately, taking the eccentricity of Mercury's orbit into account, then the general relativistic contribution to $\dot{\varpi}$ becomes $0.43$ arc seconds per year. (See Exercise 2.) It follows that the total perihelion precession rate for Mercury is $5.32+0.43= 5.75$ arc seconds per year. This is in exact agreement with the observed precession rate. Indeed, the ability of general relativity to explain the discrepancy between the observed perihelion precession rate of Mercury, and that calculated from Newtonian mechanics, was one of the first major successful tests of this theory.