The Geoid and Different Height Systems - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 146

The normal gravity field is a close approximation to the actual Earth's gravity
field. In order to narrow down the difference between the two, we select the normal
ellipsoid in accordance with the requirements below:

1. The spin axis of the normal ellipsoid coincides with the Earth's axis of rotation,
and with equivalent angular velocity.
2. The center of the normal ellipsoid is at the Earth's center of mass. The coordi-
nate axis coincides with the Earth's principal axis of inertia.
3. The total mass of the normal ellipsoid is equal to that of the actual Earth.
4. The sum of squares of the deviations of the geoid from the normal ellipsoid is the
least.

The normal ellipsoid is defined by these four basic parameters: semimajor axis
of the ellipsoid a, flattening f, total mass M of the ellipsoid, and the angular velocity
ω of the ellipsoid rotating about the minor axis. The former two parameters specify

the geometric shape of the ellipsoid and the latter two identify the physical
properties of the ellipsoid.
The normal ellipsoid is regular, so obviously its gravitational potential is inde-
pendent of

in ( 4.17 ). The gravitational potential
of the normal ellipsoid is symmetric with respect to the equator. Find the cosine of

ʻ and is only a function of ˁ and ʸ

ʸ

and 180 for the two points that are symmetric to the equator with opposite

signs. Therefore, in the spherical harmonics series expansion of the Earth's grav-
itational potential, there are only even zonal harmonics. Hence, the gravitational
potential V of the normal ellipsoid at an exterior point can be obtained from ( 4.19 ),
given by:

ʸ

" #

:

2n

1

GM
ˁ

a
ˁ

V ˁ;ðÞ ¼ 1 J 2n P 2n cos ð ʸ Þ ð 4 : 23 Þ

n ¼ 1

Equation ( 4.23 ) can be determined because J 2n is the constant coefficient related
to the normal ellipsoid parameters.
Normal gravity can be obtained from the derivative of the normal gravity
potential due to their relations. With deviations omitted, the formula for normal
gravity value ʳ 0 on the normal ellipsoid is simplified as:

,

sin 2 B ʲ 1 sin 2 2B ʳ 0 ¼ ʳ a 1 þ ʲ ð 4 : 24 Þ

where ʳ a is the value of gravity at the equator, B is the geodetic latitude of the

computation point, and coefficients ʲ , ʲ 1 , and equatorial gravity ʳ a are given by: