15
Hence
-r
R + (2h' -h)
and
r
2
- r'
2
"' (R
2
+ 2Rh) - (R
2
- 2Rh') = 2R (h - h') .
Obviously even the term 2h'-h in the first equation can be neglected and we end up with
= l.ig
do
after having denoted
/cos S' by q,.
Now the spatial distance of P and
24
14)
Discussion of the Molodenskij deflections
The Molodenskij deflections are indeed different from both kinds of
deflections used in classical geodesy.
I I,
6.3) we use:
1)
In classical geodesy (see Physical Geodesy
The gravimetric deflections (on the
21
gravity vector Yp' both at the surface.
few hundreds of second of arc.
These two definitions vary by at most a
They differ by the term due to the curvature of the
normal plumb-1 ine between the telluroid and the surface.
It can be shown that the compon
17
Here
L
Po
2R(h'-h)
3(h'-h) 2
>
p3
2p3
Hence in the
zero
approximation the two last
0
0
terms of the kernel may be neglected.
Neglecting cos
2
(3
, which can be done for a
flat terrain around the point of interest we can write the zero approximation as
19
for mountaineous regions due to the negligence of the term cos
2
s.
Treatment of the
Molodenskij integral equation for the surface density layer in mountaineous areas
remains still an open question.
12)
Evaluation of the height anomalies
We have learnt
25
and ya,yb are the values of normal gravity on the equator and at the poles.
Integrating 1/y-times the derivative above we get finally
81;:
f1
- R sin 2 H " -0. 17 11 sin 2H
where H is the height above the ellipsoid.
Evidently, this correction has to be
12
#'I.
.!._
21. dr.
P ah
t
0
I
p
r.
\
-21T<P coss
p 6 r.
outer side
+21T<P cosf3
=-
P a r.
inner side.
HereS is the maximum inclination of the tellurooid at P or, which is the same, the
angle between the normals tor. and to the ellipsoid.
Hence on the ou
3
+
+
+
+
div f = Vf = Vcfw_UVW) = VUVW + UL\W
div g = Vg = Vcfw_WVU) = VWVU + WL\U
+
+
Applying the Gauss formula successively to f and g we get
U
8n
dS =
=
W 8n dS
Jffvcfw_VWVU + WL\U) dV .
Substracting the second from the first formula yields:
\ cff s
1)
Introduction
The classical approach in geodesy is based on the use of two different
reference surfaces- the ellipsoid of rotation and the geoid.
To establish the
relation between the two surfaces is one of the main tasks of geodesy.
old problem of
11
T
9
Molodenskij has derived the following approximate expression for
the proof of which we are not going to give here:
dT
an-
dT
an
where llg refers to the telluroid cfw_or the terrain), y and
0
refer to the telluroid,
are the components of the deflection o
7
We may note that when defining
the telluroid or quasigeoid we do not
have to postulate any hypotheses.
Both
surfaces are purely conventional
(mathematical as opposed to physical in
case of the geoid) and are not even meant
to represent any physical prop
5
4) Molodenskij equation for disturbing potential
In order to get rid of the volume term, let us apply the 3rd Greens
identity to the normal potential U as we know it from classical geodesy.
We end
up with Molodenskij equation for U in following form:
-2