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Let
be the angular velocity of precession of Mercury's eccentricity vector,
. It follows that [see Equation (A.96)]
which can be inverted to give

(B.7) 
Here, we have assumed, without loss of generality, that
is normal to
(because any parallel component of
would
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
that
and
. 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

(B.11) 
where the Cartesian components of the unit vectors
,
, and
are
Furthermore,

(B.15) 
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

(B.16) 
Moreover, Equations (B.8) and (B.12)(B.13) yield

(B.17) 
However, differentiation of Equation (B.16) with respect to time gives

(B.18) 
Thus, we obtain

(B.19) 
Finally, from Equation (B.11),
which yields

(B.22) 
Next: Evaluation of disturbing function
Up: Perihelion precession of Mercury
Previous: Time evolution of eccentricity
Richard Fitzpatrick
20160331