Satellite Constellation - Fundamentals of Global Positioning System Receivers: A Software Approach - page 43

FIGURE 3.7 Fictitious and actual orbits.

Therefore, the area PSV can be obtained from area PQV as

b s
a s

b s
a s ( area OQV

area PSV = area PQV = − area OQP)

1

2 a s sin E cos E

b s
a s

1 a s b s

2 2 a s E

= − = (E − sin E cos E)

(3.23)

where the angle E is called eccentric anomaly. The area of triangle A 2 is

1
2 SP

1
2

b s
a s QP

1
2

b s
a s a s sin E(c s −

A 2 = × PF = × PF = a s cos E)

b s
2

a s b s
2

= sin E(e s a s − a s cos E) = (e s sin E − sin E cos E) (3.24)

In the above equation, the relation in Equation (3.20) is used. Substituting
Equations (3.23) and (3.24) into (3.21) the area A 1 is

a s b s
2

A 1 = (E − e s sin E) ( 3 . 25 )

Substituting this result into Equation (3.17), Kepler's second law, the result is

a s

A 1 T
πa s b s =

T
2 π (E

t − t p = − e s sin E) =

µ (E

− e s sin E) ( 3 . 26 )