The Geoid and Different Height Systems - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 140

s

ð n k Þ !

P nk cos ð ʸ Þ¼ 22n ð þ 1 Þ P nk cos ð ʸ Þ ð k > 0 Þ :

ð n þ k Þ !

Since the values of the associated Legendre polynomials differ very much from
each other when the degrees of the polynomials vary considerably, such as
P 21 (cos 58 ) 1.3482, and P 88 (cos 58 ) 542279, the results of the high-

degree polynomials computed using recursion fo rm ulae may lead to relatively
la rge

¼ ¼

P 21 cos 58 accumulative err o rs, whereas ð Þ¼ 1 : 7405, and

P 88 cos 58 6913. J n , J nk , S nk are the normalized coefficients. Comparing

( 4.19 ) with ( 4.17 ), we can obtain:

ð Þ¼ 0 :

a n0
GMa n

J n ¼

t

n a nk

GMa n

ð þ k Þ !

J nk ¼

22n ð þ 1 Þ ð n k Þ !

t

n b nk

GMa n

ð þ k Þ !

S nk ¼

22n ð þ 1 Þ ð n k Þ !

Adopting this set of coefficients, the differences between the values of the
polynomials changing along with n and k (degree and order) are slight, making it
convenient to use. The coefficient that is independent of longitude in ( 4.19 ) is called
the coefficient of zonal harmonics, while the coefficient that is longitude dependent
is called the coefficient of tesseral harmonics.
As the potential of the gravity field W is the sum of the potential of the
gravitational field V ( 4.19 ) and that of the centrifugal field Q ( 4.12 ), the mathemat-
ical expression of W is referred to as the Earth gravity field model.

4.1.3 Level Surface and the Geoid

We know that every point P on the Earth will be acted upon by the inertial
centrifugal force ! and the Earth's gravitational force ! (Fig. 4.6 ). The resultant
of these two forces ! for a unit mass is called gravity. The action line of gravity is
referred to as the plumb line. The direction of the force of gravity is the direction of
the plumb line. Given the inhomogeneous distribution of matter within the Earth
and the undulating form of the Earth's surface, the changes in the direction of the
plumb line at each point are irregular and the plumb line is far from a straight line.
When a fluid is in a state of equilibrium, everywhere on its surface is normal to
the direction of gravity, otherwise the liquid will flow. We call this the equilibrium
surface of the liquid level surface. Gravity exists everywhere on the Earth and in