Thesis - Leigh Moody.pdf - Bad Request - Cranfield University

Appendix E / Earth Geometry
_ _
Introducing eccentricity and expanding ignoring powers of (e) > 2,
P
r,
d
: =
cos
2
λ
R
rd
A
2
sin λ
+
1 − e
rd
2
18-8
: =
cos
2
λ
rd
+
R
A
2 2
( 1 + e ) ⋅ sin λrd
Equation 18.3-3
Replacing geocentric latitude, and expanding ignoring powers of (e) > 2,
P
r,
d
: =
1 + e
R
2
A
⋅ sin
Expanding to 1 st order, replacing geocentric with geodetic latitude,
P : = R ⋅ 1 − e ⋅ sin λ
r,
d
Lambert [L.9] provides a more accurate value for the Earth radius,
∆P
r,
d
: =
r,
d
A
A
2
2
λ
rd
( 0.
998320047 + P )
P : = R ⋅
∆
1.
683494 x10
−3
⋅ cos
2
d
r,
d
Equation 18.3-4
Equation 18.3-5
Equation 18.3-6
−6
( 2 ⋅ λ ) + 3.
549 x10
⋅ cos ( 4 ⋅ λ ) + L
The variation of the Earth radius with geodetic latitude,
∂ P
∂ λ
r,
d
d
: =
P
−
r,
d
d
2
⋅ e ⋅ sin λd
⋅ cosλ
2 2
1 + e ⋅ sin λ
Relating this expression to the geodetic velocity at point (d),
∂ P
∂ P&
r,
d
G
m
: =
∂ P
∂ λ
r,
d
d
∂ λ
⋅
∂ P&
d
G
m
: =
⎛
⎜
⎝
0
,
1
R
d
λd
,
d
⎞
0
⎟
⎠
d
Equation 18.3-7
Equation 18.3-8
T
∂ P
⋅
∂ λ
r,
d
d
Equation 18.3-9