Geoscience Reference
In-Depth Information
know the exact mass of the earth, how can we make
M
rigorously equal to
M
?
Subsequently, we will see that the first-degree harmonic can always be
assumed to be zero. Under this assumption, we can substitute (2-331) into
(2-328) and express
T
by the conventional Stokes formula (2-304). Thus we
obtain
∆
gS
(
ψ
)
dσ .
T
=
GδM
R
R
4
π
+
(2-333)
σ
This is the generalization of Stokes' formula for
T
. It holds for an arbitrary
reference ellipsoid whose center coincides with the center of the earth.
First-degree terms
The coecients of the first-degree harmonic in the potential
W
are, according
to (2-85) and (2-87), given by
GM x
c
,GMy
c
,GM
c
,
(2-334)
where
x
c
,y
c
,z
c
are the rectangular coordinates of the earth's center of grav-
ity. For the normal potential
U
, we have the analogous quantities
GM
x
c
,GM
y
c
,GM
z
c
.
(2-335)
As
x
c
,y
c
,z
c
are very small in any case, these are practically equal to
GM x
c
,GMy
c
,GMz
c
.
(2-336)
The coecients of the first-degree harmonic in the anomalous potential
T
=
W
−
U
are, therefore, equal to
x
c
)
,GM
(
y
c
−
y
c
)
,GM
(
z
c
−
z
c
)
.
GM
(
x
c
−
(2-337)
They are zero, and
there is no first-degree harmonic T
1
(
ϑ, λ
)
if and only if
the center of the reference ellipsoid coincides with the center of gravity of
the earth
. This is usually assumed.
In the general case, we find from the first-degree term of (2-76), on
putting
r
=
R
and using the coecients (2-85) together with (2-87),
R
2
(
z
c
−
T
1
(
ϑ, λ
)=
GM
z
c
)
P
10
(cos
ϑ
)+(
x
c
−
x
c
)
P
11
(cos
ϑ
)cos
λ
(2-338)
y
c
)
P
11
(cos
ϑ
)sin
λ
.
+(
y
c
−
If the origin of the coordinate system is taken to be the center of the reference
ellipsoid, then
x
c
=
y
c
=
z
c
=0.With
P
10
(cos
ϑ
)=cos
ϑ, P
11
(cos
ϑ
)=sin
ϑ
,
