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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

(B.39) 
where

(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,

(B.41) 
where
and
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,
and
, where
and
are the mean orbital angular velocities of
Mercury and the perturbing planet, respectively. (See Section 4.11.) Furthermore,
is a constant of the perturbing planet's motion. (See Section 4.8.)
Hence,
.
Thus, we obtain
where
,
,
,
, and
are specified in Equations (B.31), (B.32), (B.35), (B.36), and (B.41) respectively. Here,
arc seconds per year (where
is measured in astronomical units).
The previous expression can be simplified somewhat, because some terms are obviously annihilated by the integration in
. In fact,
Table B.1:
Contributions to the secular precession rate of Mercury's perihelion from the other planets in the solar system.


Planet 



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
arc seconds per year.
Next: Yielding of an elastic
Up: Perihelion precession of Mercury
Previous: Evaluation of disturbing function
Richard Fitzpatrick
20160331