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

Appendix E / Earth Geometry
_ _
cos λ
rd
: =
18-7
1 +
1
2 2
( 1 − 2 ⋅ e ) ⋅ tan λd
Replacing (secλd) assuming that ( tan λ rd ≈ tan λd
),
d
d
Equation 18.2-15
2 2
( 1 + e ⋅ sin )
1
cosλ rd : =
: = cos λd
⋅
λ
2
2 2
sec λ − 2 ⋅ e ⋅ tan λ
Equation 18.2-16
Substituting from this equation into the following trigonometric identity,
2
rd
2
rd
2 2 2 2
( 1 + e ⋅ sin λ ) ⋅ cos d
sin λ : = 1 − cos λ : = 1 −
λ
Expanding ignoring powers of (e) greater than 2,
18.3 Geocentric Earth Radius
2
rd
2 2
2
( 1 + 2 ⋅ e ⋅ sin λd
) ⋅ cos d
sin λ : = 1 −
λ
2
rd
2 2
2
( 1 − 2 ⋅ e ⋅ cos λd
) ⋅ sin d
sin λ : =
λ
2
rd
2 2
2
( 1 − e ⋅ cos λd
) ⋅ sin d
sin λ : =
λ
d
Equation 18.2-17
Equation 18.2-18
Equation 18.2-19
Equation 18.2-20
The geocentric radius is the distance from point (r) to point (p). Substituting
for (x) and (z) in the general equation for an ellipsoid,
1
⎛ 2
2
cos λ λ
⋅ ⎜ rd sin
⎜
+ 2
⎝ R a R b
2
rd
: = Pr,
d
2
Re-arranging and taking the positive root of this equation,
P
r,
d
: =
cos
2
λ
rd
+
R
A
⎞
⎟
⎠
2 2
( R A R B ) ⋅ sin λrd
Equation 18.3-1
Equation 18.3-2
d