Geology Reference
In-Depth Information
so with binomial expansion of
R
3
,wehave
π
1
3
m
2
R
2
R
1
sinθ
d
θ
2
3
V=
+
3
mR
1
+
+
+···
0
2π
m
π
0
4
3
+
=
R
1
sinθ
d
θ
2π
m
2
π
0
R
2
R
1
sinθ
d
θ
+···
,
+
+
(5.12)
showing that the volume is given correctly by the term of zeroth order in
m
.The
corresponding equipotential expressed in dimensionless variables is
−
1, and
dimensionless gravity, everywhere uniform on the spherical surface (5.9), is
1.
A theory of the figure, correct to terms of first order in
m
, must include Legendre
polynomials of at least second degree in cosθ, since in dimensionless variables
expression (5.2) for the centrifugal potential becomes
+
1
1
3
mr
2
3
mr
2
P
2
(cosθ).
W
=−
+
(5.13)
Thus,
R
1
(θ) in (5.8) has terms up to and including
P
2
(cosθ). However, since it
must be even in cosθ, its
P
1
(cosθ) term is missing, and its
P
0
(cosθ) term must
be zero, in order to correctly give
in the expansion (5.12). If
a
is the equatorial
radius of the true reference surface, and
c
its polar radius,
V
m
2
R
2
(π/2)
a
=
1
+
mR
1
(π/2)
+
+···
,
(5.14)
m
2
R
2
(0)
c
=
1
+
mR
1
(0)
+
+···
.
(5.15)
By binomial expansion,
1
a
=
m
2
R
1
(π/2)
R
2
(π/2)
1
−
mR
1
(π/2)
+
−
+···
.
(5.16)
The remaining coe
cient of
P
2
(cosθ)in
R
1
(θ) is then determined by the flattening
−
a
c
f
=
a
m
R
1
(π/2)
R
1
(0)
=
−
m
2
R
2
(π/2)
R
1
(π/2)
R
1
(0)
+···
.
R
1
(π/2)
+
−
R
2
(0)
−
+
(5.17)
For the flattening to be given correctly to first order in
m
,wemusthave
2
3
f
m
P
2
(cosθ),
R
1
(θ)
=−
(5.18)
showing that
f
is of order
m
.
To complete the first-order theory, the equipotential is now augmented by the
second-degree term of order
m
in the gravitational potential and by the centri-
fugal potential expressed in (5.13). Outside the Earth, the gravitational potential
Search WWH ::
Custom Search