**Great circle angle between an earth station and a satellite**
N_{gc} = cos^{-1}(sin(P)*sin(I)+cos(P)*cos(I)*cos(U_{s}-U_{e}))
Notes:
P is the earth station latitude
U_{e} is the earth station longitude
I(t) is the satellite (instantaneous) inclination
U_{s}(t) is the (instantaneous) longitude of the sub-satellite point
**Earth station elevation angle to the satellite**
E_{e} = tan^{-1}((cos(N_{gc})-R/(H+R))/sin(N_{gc}))
Notes:
R is the radius of the earth (6378*10^{3}m)
H(t) is the (instantaneous) altitude of the satellite (geostationary satellite: 35786*10^{3}m)
N_{gc} is the great circle angle
**Earth station azimuth angle to the satellite**
A_{e} = tan^{-1}(sin(U_{sat}-U_{es})/(sin(P)*cos(U_{s}-U_{e})-tan(I)*cos(P))
Notes:
It is necessary to investigate the signs of the numerator and denominator to find the
quadrant in which the azimuth angle is.
**Earth station off-axis angle between the principal axis of the antenna and the direction
of a second satellite**
N_{e} = cos^{-1}(sin(E_{e1})*sin(E_{e2})+cos(E_{e1})*cos(E_{e2})*cos(A_{e1}-A_{e2}))
Notes:
Suffixes 1 and 2 denote angles to the first and second satellites.
**Slant range between an earth station and satellite**
S = (-)R*sin(E_{e})+((R*sin(E_{e}))^{2}+H*(H+2*R))^{0.5}
metres
S = R*sin(N_{gc})/cos(E_{e}+N_{gc}) metres
**Geostationary satellite azimuth angle where the earth's axis of rotation defines zero
degrees azimuth**
A_{s} = sin^{-1}(R*cos(P)/Z*sin(U_{s}-U_{e}))
A_{s} = tan^{-1}(R*cos(P)*sin(U_{s}-U_{e})/(R+H-R*cos(U_{s}-U_{e})*cos(P)))
Notes:
Z=(R^{2}*cos^{2}(P)+(R+H)^{2}-2*R*cos(P)*(R+H)*cos(U_{s}-U_{e})^{0.5})^{
}Z=(S^{2}-R^{2}*sin^{2}(P))^{0.5
}I (satellite inclination angle) is zero.
**Geostationary satellite elevation angle where the earth's equatorial plane defines zero
degrees elevation**
E_{s} = sin^{-1}(R*sin(P)/S)
E_{s} = tan^{-1}(sin(A_{s})*tan(P)/sin(U_{s}-U_{e}))
Notes:
I (satellite inclination angle) is zero.
**Satellite off-axis angle between the principal axis of the satellite antenna and the
direction of the earth station**
N_{s} = cos^{-1}(sin(E_{se})*sin(E_{sb})+cos(E_{se})*cos(E_{sb})*cos(A_{se}-A_{sb}))
Notes:
E_{sb} is the satellite elevation angle in the direction of the boresight.
E_{se} is the satellite elevation angle in the direction of the earth station.
A_{sb} is the satellite azimuth angle in the direction of the boresight.
A_{se} is the satellite azimuth angle in the direction of the earth station.
**Beamwidth of a satellite antenna with an elliptical beam in the direction of an earth
station**
P_{hio} = (A^{2}*B^{2}/A^{2}(sin(Q))^{2}+B^{2}(cos(Q))^{2}))^{0.5}
Notes:
A is the major axis of the ellipse
B is the minor axis of the ellipse
Q=tan^{-1}((E_{se}-E_{sb})/(A_{se}-A_{sb})) - O
O is the orientation of the ellipse defined as the counter clockwise angle from a line
parallel to the equatorial plane to the major axis of the ellipse.
**Earth station latitude expressed in terms of satellite azimuth and elevation angles**
P = sin^{-1}(tan(E_{s})/R/cos(A_{s})*(R+H-R*cos(N_{gco})))
Notes:
cos(N_{gc})=cos(U_{s}-U_{e})*cos(P)
cos(N_{gc})=cos(sin^{-1}((R+H)/R*sin(N_{so}))-N_{so})
N_{gco} is the great circle angle between an earth station and a geostationary
satellite (can be found by letting I=0 in our first equation in this section).
N_{so}=cos^{-1}(cos(E_{s})*cos(A_{s}))
N_{so} is the angle between the sub-satellite point and the earth station as
determined from the geostationary satellite (can be found by letting E_{sb}=0 in our
equation for N_{s} above).
**Earth station longitude expressed in terms of satellite azimuth and elevation angles**
U_{s}-U_{e} = sin^{-1}(sin(A_{s})*tan(P)/tan(E_{s})) |