Secular perihelion precession rate

According to Equations (B.23), (B.38), and (B.39), Mercury's instantaneous perihelion precession rate, due to the perturbing influence of some other planet in the solar system, takes the form

$$\dot{\varpi} =\frac{m'}{M+m}\,r^{\,2}\,r'\,\frac{\skew{5}\dot{\th... ...ta\,\beta\right]\left(\frac{1}{\delta^{\,3}}-\frac{1}{r'^{\,3}}\right)\right\},$$ (B.39)

where

$$\delta(\theta,\theta') = (r^{\,2} + r'^{\,2}-2\,r\,r'\,\beta)^{1/2}.$$ (B.40)

However, we are only interested in the secular precession rate; that is, the mean rate over some timescale that is much longer than either the orbital period of Mercury or that of the perturbing planet. We can obtain a formula for the secular precession rate by averaging the previous expression over the orbits of the two planets. In other words,

$$\langle \dot{\varpi}\rangle = \oint\oint \dot{\varpi}\,\frac{d{\cal M}}{2\pi}\,\frac{d{\cal M}'}{2\pi},$$ (B.41)

where ${\cal M}$ and ${\cal M}'$ are the mean anomalies of Mercury and the perturbing planet, respectively. (See Section 4.11.) Note that we must average in terms of the mean anomalies, rather than the true anomalies, because the former increase uniformly in time (unlike the latter). In fact, $d{\cal M} = n\,dt$ and $d{\cal M}'=n'\,dt$, where $n$ and $n'$ are the mean orbital angular velocities of Mercury and the perturbing planet, respectively. (See Section 4.11.) Furthermore, $h'\equiv r'^{\,2}\,\skew{5}\dot{\theta}'=(1-e'^{\,2})^{1/2}\,n'\,a'^{\,2}$ is a constant of the perturbing planet's motion. (See Section 4.8.) Hence, $d{\cal M}'= (1-e'^{\,2})^{-1/2}\,(r'/a')^2\,d\theta'$. Thus, we obtain

$$\begin{multline} \langle \dot{\varpi}\rangle = n\,\frac{m'}{M+m}\left(\frac{a}{a... ...,3}}-1\right)\right\}\frac{d\theta}{2\pi}\,\frac{d\theta'}{2\pi}, \end{multline}$$

where $r(\theta)$, $r'(\theta)$, $\beta(\theta,\theta')$, $\gamma(\theta,\theta')$, and $\delta(\theta,\theta')$ are specified in Equations (B.31), (B.32), (B.35), (B.36), and (B.41) respectively. Here, $n=1296000/a^{\,3/2}$ arc seconds per year (where $a$ is measured in astronomical units). The previous expression can be simplified somewhat, because some terms are obviously annihilated by the integration in $\theta'$. In fact,

$$\begin{multline} \langle \dot{\varpi}\rangle = n\,\frac{m'}{M+m}\left(\frac{a}{a... ...}\right)\gamma\right]\frac{d\theta}{2\pi}\,\frac{d\theta'}{2\pi}, \end{multline}$$

Table B.1: Contributions to the secular precession rate of Mercury's perihelion from the other planets in the solar system.

Planet	$\langle\dot{\varpi}\rangle(''\,{\rm yr}^{-1})$
Venus	2.7745
Earth	0.9084
Mars	0.0248
Jupiter	1.5400
Saturn	0.0731
Uranus	0.0014
Neptune	0.0004
Total	5.3226

Table B.1 shows the contributions to the secular precession rate of Mercury's perihelion, due to the perturbing influence of the other planets in the solar system, calculated from Equation (B.44), using the planetary mass and orbital element data given in Table 4.1. It can be seen that the dominant contributions to the precession rate come from Venus, Earth, and Jupiter. Furthermore, the total precession rate is $5.32$ arc seconds per year.