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satisfied approximately, but not exactly. A similar demonstration that no exact static
equilibrium is possible in a rotating system, for the case where radiative transfer is
important, was given by Von Zeipel (1924), and the theorem is sometimes associ-
ated with his name.
When the density is supposed to depend only on the pressure, the equation of
state is said to be
barotropic
and the di
culties raised by the theorem are avoided,
since the heat energy equation (5.106) is then excluded from explicit consideration.
For example, if only adiabatic or isothermal processes are contemplated, the equa-
tion of state is barotropic and static equilibrium is possible. Much of the classical
or modern literature on the equilibrium theory of rotating, gravitating fluid bodies
is based on the barotropic assumption.
While the centrifugal potential retains its representation (5.2) in Legendre poly-
nomials in the interior of the Earth, it becomes more di
cult to give such a repres-
entation for the gravitational potential there. In integral form the latter is
G
ρ
(
r
)
|
V
(
r
)
=−
d
V
,
(5.114)
r
|
r
−
the integration with variable
r
extending over the whole of the Earth's interior.
The
generating function
for Legendre polynomials derives from the binomial
expansion
1
1/2
1
2
hz
−
1/2
∞
z
2
h
2
h
n
P
n
(
z
)f r
+
−
=
|
h
|
<
z
±
−
(5.115)
n
=
0
(Copson (1955)[pp. 277-278]), where the Legendre polynomials are generated as
the coe
1, the radius of convergence of
the expansion is unity. By the law of cosines, the denominator of the integrand in
(5.114) can be expressed as
cients of the expansion. For
−
1
≤
z
≤
1
1
D
=
1
r
|
=
Θ
1/2
.
(5.116)
r
2
|
r
−
r
2
2
rr
cos
+
−
Applying the generating function, we have the uniformly convergent expansions
r
r
n
∞
1
D
=
1
1
r
), for
r
<
r
,
1/2
=
P
n
(cos
Θ
(5.117)
r
1
r
r
2
2
r
r
cos
+
−
Θ
n
=
0
r
r
n
∞
1
D
=
1
1
r
), for
r
<
r
, (5.118)
1/2
=
P
n
(cos
Θ
r
1
r
2
2
r
cos
+
−
Θ
n
=
0
is the angle between
r
and
r
.
where
Θ
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