Basic GPS Concept - Fundamentals of Global Positioning System Receivers: A Software Approach - page 21

From the triangle OPC and the law of sine, one can write

sin D
OC

sin (π − L)

= ( 2 . 36 )

r

From Equation (2.35),

e e OE e e r 0 cos L co

OC = = ( 2 . 37 )

but

L co = −

L D 0 ( 2 . 38 )

Therefore,

e e r 0 cos (L e e r 0 ( cos L cos D 0 + sin L sin D 0 ) OC = − D 0 ) = ( 2 . 39 )

From Equation (2.23), the ellipticity e p of the earth is

a e − b e
a e

e p = ( 2 . 40 )

The eccentricity and the ellipticity can be related as

a e − b e
a e

(a e − b e )
a e

(a e + b e )
a e

e p ( 2 a e − a e + b e )
a e

e e =

e p )
( 2 . 41 )
Substituting Equations (2.39) and (2.41) into Equation (2.36), the result is

= = = e p ( 2 −

1
2 sin 2 L cos D 0 +

sin 2 L sin D 0 2 e p 1

e p
2

r 0
r 0 +

sin D = − ( 2 . 42 )

h

In the above equation the relation r h is used. Since D and D 0 are both

very small angles, the above equation can be written as

= r 0 +

1
2 sin 2 L

D 0 sin 2 L 2 e p 1

e p
2

r 0
r 0 +

D = − + ( 2 . 43 )

h

The relations

sin D ≈ D ; s p sin D 0 ≈ D 0 ; s p cos D 0 ≈ 1 ( 2 . 44 )

are used in obtaining the results of Equation (2.43). If the height h = 0, then

from Figure 2.7 D D 0 . Using this relation Equation (2.43) can be written as

D 0 1 − 2 e p 1 −

=

2 sin 2 L

e p 1 −

2 sin 2 L or

e p e p

=

D 0 = e p sin 2 L + 1 (2.45)