Fundamentals of Satellite Navigation - Understanding GPS: Principles and Applications - page 35

where ∇ is the gradient operator, defined as follows:

∂
∂
∂
∂
∂
∂

V
x
V
y
V
z

∇= V

def

Notice that for two-body motion, V = µ /r :

∂ ∂

∂

() ( )

1 2

r

− 1

x

2

++ yz

2 2

∂
∂
∂
∂
∂

x x

µ ∂

∂
∂
∂

() ( )

1 2

()

∇= µ r µ r

− 1

=− xyz

2

++

2 2

2

y r y

() ( )

1 2

− 1 2 2 2

r xyz ++

z z

2
2
2

x
y
z

x
y
z

µ µ

=− r 3

( )

1 2

−

2 2 2

=− xyz + + =− r

2 r

2

r

3

/r , (2.5) is equivalent to (2.4) for two-body motion. In the
case of true satellite motion, the Earth's gravitational potential is modeled by a
spherical harmonic series. In such a representation, the gravitational potential at a
point P is defined in terms of the point's spherical coordinates ( r ,

Therefore, with V = µ

φ ′ , ) as follows:

l

µ

∞ l

a
r

∑ ∑

( )( )

V = 1 + P sin φ ′ C cos m α + S sin m α (2.6)

lm lm lm

r

l = 2 m = 0

where:

r = distance of P from the origin

φ ′ = geocentric latitude of P (i.e., angle between r and the xy -plane)

= right ascension of P

a mean equatorial radius of the Earth (6,378.137 km in WGS 84)

P lm =

=

associated Legendre function

C lm = spherical harmonic cosine coefficient of degree l and order m

S lm = spherical harmonic sine coefficient of degree l and order m

Notice that the first term of (2.6) is the two-body potential function. Also notice
that geocentric latitude in (2.6) is different from geodetic latitude defined in Section
2.2. WGS 84 not only defines a reference coordinate frame and ellipsoid, but it also
has a companion geopotential model called WGS 84 EGM96. This model provides