Gravity field of the earth - Physical Geodesy - page 44

P

H
g

geoid
W= 0

level surface
W= constant

Fig. 2.2. Level surfaces and plumb lines

If we look at equation (2-7) for the gravity potential W ,wecanseethat
the equipotential surfaces, expressed by W ( x, y, z ) = constant, are rather
complicated mathematically. The level surfaces that lie completely outside
the earth are at least analytical surfaces, although they have no simple ana-
lytical expression, because the gravity potential W is analytical outside the
earth. This is not true of level surfaces that are partly or wholly inside the
earth, such as the geoid. They are continuous and “smooth” (i.e., without
edges), but they are no longer analytical surfaces; we will see in the next sec-
tion that the curvature of the interior level surfaces changes discontinuously
with the density.
The lines that intersect all equipotential surfaces orthogonally are not
exactly straight but slightly curved (Fig. 2.2). They are called lines of force ,
or plumb lines . The gravity vector at any point is tangent to the plumb line at
that point, hence “direction of the gravity vector”, “vertical”, and “direction
of the plumb line” are synonymous. Sometimes this direction itself is briefly
denoted as “plumb line”.
As the level surfaces are, so to speak, horizontal everywhere, they share
the strong intuitive and physical significance of the horizontal; and they share
the geodetic importance of the plumb line because they are orthogonal to
it. Thus, we understand why so much attention is paid to the equipotential
surfaces.
The height H of a point above sea level (also called the orthometric
height ) is measured along the curved plumb line, starting from the geoid