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Decomposition of Tide Generating Potential
Let
,
,
be righthanded spherical coordinates in a nonrotating reference frame whose origin lies at the center of the planet,
and whose symmetry axis coincides with the planetary rotation axis. Thus, the vector position of a
general point is

(12.15) 
where
, et cetera.
Let the coordinates of the moon's center be
,
,
. It follows that

(12.16) 
Hence, from Equation (12.5),

(12.17) 
Now, according to the spherical harmonic addition theorem (Arfken 1985),
which implies that
Here,

(12.21) 
for
and
, denotes an associated Legendre function (Abramowitz and Stegun 1965). In particular,
Note that
.
Let
where
. According to Equations (12.14), (12.20), and (12.25)(12.27),
Here, we have neglected the unimportant constant term in Equation (12.14).
Next: Expansion of Tide Generating
Up: Terrestrial Ocean Tides
Previous: Tide Generating Potential
Richard Fitzpatrick
20160331