Satellite Constellation - Fundamentals of Global Positioning System Receivers: A Software Approach - page 45

Once the E is found, the next step is to find the true anomaly ν . This relation
can be found by referring to Figure 3.7.

OP
a s

c s − PF

a s

c s + r cos ν

a s

cos E = = = ( 3 . 32 )

Now let us find the distance r in terms of angle ν . From Figure 3.6, applying
the law of cosine to the triangle GSF, the following result is obtained

r 2 r 2 + 4 c s = + 4 rc s cos ν ( 3 . 33 )

where r and r are the distance from the foci G and F to the point S .For
an ellipse,

r + r = 2 a s ( 3 . 34 )

Substituting this relation into Equation (3.33), the result is

c s
a s + c s cos ν =

a s − e s )

1 + e s cos ν

a s ( 1 −

r = ( 3 . 35 )

Substituting this value of r into Equation (3.32) the result is

e s + cos ν
1 + e s cos ν

cos E = ( 3 . 36 )

Solve for ν and the result is

cos E − e s

=

cos ν ( 3 . 37 )

1 − e s cos E

This solution generates multiple solutions for ν because cos ν is a multivalued
function. One way to find the correct value of ν is to keep these angles E and ν
in the same half plane. From Figure 3.7 one can see that the angles E and ν are
always in the same half plane.
Another approach to determine ν is to find the sin ν . ( 13 ) If one takes the square

on both sides of the above equation, the result is

−

e s ) 2

( cos E

cos 2 ν = 1 − sin 2 ν = ( 3 . 38 )

( 1 − e s cos E) 2

Solve for sin ν and the result is

1

− e s sin E

=

sin ν ( 3 . 39 )

1 − e s cos E