Geoscience Reference
In-Depth Information
The components of the force of attraction are given by (1-7). For in-
stance,
(
ξ, η, ζ
)
l
X
=
∂V
∂x
∂
∂x
=
G
dξ dη dζ
v
(1-15)
1
l
dξ dη dζ .
=
G
v
(
ξ, η, ζ
)
∂
∂x
Note that we have interchanged the order of differentiation and integration.
Substituting (1-8) into the above expression, we finally obtain
G
v
x − ξ
l
3
X
=
−
dv.
(1-16)
Analogous expressions result for
Y
and
Z
.
The potential
V
is continuous throughout the whole space and vanishes at
infinity like 1
/l
for
l
. This can be seen from the fact that for very large
distances
l
the body acts approximately like a point mass, with the result
that its attraction is then approximately given by (1-6). Consequently, in
celestial mechanics the planets are usually considered as point masses.
The first derivatives of
V
, that is, the force components, are also contin-
uous throughout space, but not so the second derivatives. At points where
the density changes discontinuously, some second derivatives have a discon-
tinuity. This is evident because the potential
V
may be shown to satisfy
Poisson's equation
→∞
∆
V
=
−
4
πG,
(1-17)
where
∆
V
=
∂
2
V
∂x
2
+
∂
2
V
∂y
2
+
∂
2
V
∂z
2
.
(1-18)
The symbol ∆, called the
Laplacian operator
, has the form
∂
2
∂x
2
+
∂
2
∂y
2
+
∂
2
∂z
2
.
(1-19)
From (1-17) and (1-18) we see that at least one of the second derivatives of
V
must be discontinuous together with
.
Outside the attracting bodies, in empty space, the density
is zero and
(1-17) becomes
∆
V
=0
.
(1-20)
This is
Laplace's equation
. Its solutions are called
harmonic functions
. Hence,
the potential of gravitation is a harmonic function outside the attracting
masses but not inside the masses: there it satisfies Poisson's equation.
