be the angular velocity of precession of Mercury's eccentricity vector, . It follows that [see Equation (A.96)]
which can be inverted to give
Here, we have assumed, without loss of generality, that
is normal to (because any parallel component of
have no effect on the time evolution of ).
Consider a Cartesian coordinate system , , that is instantaneously aligned with the orbit of Mercury, in such a manner
. Thus, the orbital plane coincides with the - plane, and
the perihelion lies (instantaneously) on the -axis.
Because the eccentricity vector is always directed toward Mercury's perihelion, the rate of perihelion precession is equivalent
to the component of
that is normal to the orbital plane. In other words,
Note that any component of
that lies in the orbital plane would cause the orientation of this plane to evolve in time.
In the following, we shall neglect this effect because it has a negligible influence on Mercury's perihelion precession rate
(given Mercury's small orbital eccentricity, as well as its small orbital inclination with respect to the orbits of the other planets
in the solar system.)
Let , , be a cylindrical coordinate system in the (, , ) frame. Mercury's orbital
plane thus corresponds to . Moreover, we can
write (see Section 4.4)
as well as
where the Cartesian components of the unit vectors ,
, and are
where is Mercury's orbital major radius. (See Section 4.8.)
Incidentally, the angle is
Mercury's true anomaly. (See Section 4.11.).
It follows from Equations (B.5) and (B.9)–(B.11) that
Moreover, Equations (B.8) and (B.12)–(B.13) yield
However, differentiation of Equation (B.16) with respect to time gives
Thus, we obtain
Finally, from Equation (B.11),