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

Appendix E / Earth Geometry
_ _
18.5 Earth Curvature in the Plane of the Meridian
Curvature in the meridian at point (d) from the standard curvature equation,
2
−1
2
⎛ d z ⎞ ⎛ dz ⎞
R md : = ⎜ ⎟
⎜
⋅ 1 +
2
⎜ ⎟
dx ⎟
⎝ dx ⎠
⎝
18-10
⎠
Equation 18.5-1
Substituting for the derivatives and expanding the cotangent function,
R
md
: =
z ⋅ sin
2
λ
d
⋅
1 − e
2
2 ( 1 + cot λ )
⋅ sin
2
λ
d
d
3
: =
z ⋅ cos ecλ
2
1 − e ⋅ sin
Combining like terms after substituting for (z) and the Earth radius,
R
md
: =
P
r,
d
⋅
d
2
λd
Equation 18.5-2
2 2 2
2 2 2
( 1 − e + e ⋅ sin λd
) R ⋅ ( 1 − e + e ⋅ sin λ )
1 − e
2
⋅ sin
Expanding ignoring powers of (e) > 2,
R
md
: =
R
md
: =
2
λ
R
2
R A ⋅ ( 1 − e )
2 2
( 1 − e ⋅ sin λ ) ⋅ 1 −
d
d
A
⋅
: =
A
1 − e
2 2 2
( 1 − e + e ⋅ sin λ )
1 − e
e
2
2
⋅ sin
⋅ sin
2
λ
d
2
λ
d
: =
d
2
⋅ sin
Curvature in the meridian at point (p) close to the Earth's surface,
R : = R + P
λ d
md
ZG
d,
m
The variation of the meridian radius with latitude in (m/rad),
∂ R
∂ λ
md
d
: =
2
3 ⋅ R md ⋅ e ⋅ sin λd
⋅ cos λ
2 2
1 − e ⋅ sin λ
d
2
λ
Equation 18.5-3
Equation 18.5-4
( )
( ) 3
2
R A ⋅ 1 − e
2 2
1 − e ⋅ sin λ
d
Equation 18.5-5
Equation 18.5-6
Equation 18.5-7
d
d
d