The Earth is assumed to be a sphere.
Let's define a right handed cartesian co-ordinate system with it's origin at Earth's centre, positive x-axis passing through equator at 0 deg longitude, positive y-axis passing through equator at 90 deg E longitude and postive z-axis through North Pole coincident to rotational axis of earth. And i, j and k be the unit vectors along positive x, y, and z axis respectively.
Let r be the radius of Earth and r'be the radius of Satellite (i.e. Distance of Satellite from Earth's centre).
Let r1 be the position vector of a point on the surface of Earth and r2 be the position vector of Satellite.
r1 = r Cos L cos N i + r cos L Sin N j + r Sin L k
r2 = r' cos s i + r' sin s j
The angle Theta is the elevation(E). Phi is the angle between vector r1 and vector (r2 - r1).
(r2-r1) . r1 = |r2-r1| |r1| Cos (phi)
( r2 -
r1) . r1
Cos (phi) = ----------------
|r2-r1|
|r1|
Since (phi) = (PI / 2) - Theta (E) and Cos {(PI/2) - E} = Sin E
(r2 -
r1) . r1
Sin E = -----------------
|r2 - r1|
|r1|
r2-r1 = (r'cos s - r cos l cos n) i + (r' sin s - r cos l sin n) j - r sin l k
====================================
(r2 - r1). r1 = (r' Cos S - r Cos L Cos N) r Cos L Cos N + (r' Sin S - r Cos L Sin N) r Cos L Sin N - r2 Sin2L
= r' r Cos S Cos L Cos N - r2 Cos2L Cos2N + r' r Sin S Cos L Sin N - r2 Cos2L Sin2N - r2 Sin2L
= r' r Cos L (Cos S Cos N + Sin S Sin N) - r2 Cos2L (Sin2N + Cos2N) - r2 Sin2L
= r' r Cos L Cos(S - N) - r2 (Sin2L + Cos2L)
= r' r Cos L Cos G - r2
================================================
|r2 - r1| = {(r' Cos S - r Cos L Cos N)2 + (r' Sin S - r Cos L Sin N)2 + r2 Sin2L}1/2
= {r'2 Cos2S + r2 Cos2L Cos2N - 2 r r' Cos S Cos L Cos N + r'2 Sin2S + r2 Cos2L Sin2N - 2 r r' Sin S Cos L Sin N + r2 Sin2L}1/2
= {r'2 (Cos2S + Sin2S) + r2 Cos2L (Cos2N + Sin2N) - 2 r r' Cos L (Cos S Cos N + Sin S Sin N) + r2 Sin2L}1/2
= {r'2 + r2 Cos2L + r2 Sin2L - 2 r r' Cos L Cos G}1/2
= {r'2 + r2 - 2 r r' Cos L Cos G}1/2
|r1| = r
===========================================
r' r Cos G Cos L - r2
Sin E = -----------------------------------------
r {r'2 + r2 - [2 r r' Cos G Cos L]
}1/2
r' Cos G Cos L - r
Sin E = ---------------------------------------
{r'2 + r2 - [2 r' r Cos G Cos L]
}1/2
r' Cos G Cos L - r
Sin E = -------------------------------------------------
{r'2 [1 + (r / r')2 - 2 (r / r') Cos G Cos
L] }1/2
r' Cos G Cos L - r
Sin E = ---------------------------------------------
r'{1 + (r / r')2 - 2 (r / r') Cos G Cos
L}1/2
Cos G Cos L - (r / r')
Sin E = ---------------------------------------------
{1 + (r / r')2 - 2 (r / r') Cos G Cos L
}1/2
Cos2G Cos2L + (r / r')2
- 2 (r / r') Cos G Cos L
Sin2E = -------------------------------------------------------
1 + (r / r')2 - 2 (r / r') Cos G Cos L
Since
Sin E
Tan E = ------------------
{1 - Sin2E}1/2
Therefore,
=========================================
Cos G Cos L - (r/r')
Tan E =
-------------------------------------------------------------------------
(1 + (r/r')2 - 2 (r/r) Cos G Cos L)1/2 {1-
(Cos2G Cos2L + (r/r')2 - 2 (r/r') Cos G Cos
L) / (1 + (r/r')2 - 2 (r/r') Cos G Cos L)}1/2
Cos G Cos L - (r/r')
= -------------------------------------------------------------------------------
{ (1 + (r/r')2 - 2 (r/r') Cos G Cos L) [1- {(Cos2G
Cos2L + (r/r')2 - 2 (r/r') Cos G Cos L) / (1 +
(r/r')2 - 2 (r/r') Cos G Cos L)} ] }1/2
Cos G Cos L - (r/r')
= -------------------------------------------------------------------------------
{ (1 + (r/r')2 - 2 (r/r') Cos G Cos L) - (Cos2G
Cos2L + (r/r')2 - 2 (r/r') Cos G Cos L)
}1/2
Cos G Cos L - (r/r')
= ---------------------------
{1 -Cos2G Cos2L }
Deriving...
Last revision: June 25, 2009
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