#format jsmath

= Satellite Distance =

|| Thinsat arrays will not always be directly above the ground antenna, but will be at an elevation angle $ e $ above the horizon.  If the array orbits at a radius $ A \bullet R_E $,  what is the distance $ D \bullet R_E $ between the array and the ground antenna?  Note that $ D = 1 $ when elevation angle $ e $ is $ \pi / 2 $ or 90&deg;.<<BR>><<BR>>Signals between satellite and ground are attenuated by distance, though at 70 GHz attenuation due a longer slant path through the atmosphere is more important.  For the M288 orbit, where A &approx; 2, $ D = \sqrt{ 3 } $ if $ e = 0 $, attenuating signals by -10 log 3 or -4.8 dB.  The atmospheric attenuation is far greater.  Unless there are multiple ground antennas pointed at different parts of the sky, a flat antenna pointed at zenith talking to a satellite at 20&deg; elevation will get -10 log( sin( 20&deg;)) or -4.7 dB less signal due to smaller projected signal capture area, while the distance attenuation is -3.1 dB .  See PathAttenuation for the sum of all effects. || {{ attachment:SatDistance.png | | width=200 }} ||

The sin law states that:

$ { { sin( d ) } \over D } = { { sin( a ) } \over A } = { { sin( c ) } \over 1 } $

$ sin( a ) = sin( e + { \pi \over 2 } ) = \cos( e ) $

$ sin( c ) = { { sin( a ) } \over A } = { { cos( e ) } \over A } $

$ cos( c ) = \sqrt{ 1 - \left( { cos( e )^2 } \over A \right)^2 } $

The sum of the corners of a triangle is $ \pi $, so 

$ \pi = a + c + d = e + { \pi \over 2 } + c + d \;\;\;\;\;\;\;\;\; { \pi \over 2 } = e + c + d \;\;\;\;\;\;\;\;\; d = { \pi \over 2 } - ( e + c ) $

$ sin( d ) = sin( { \pi \over 2 } - ( e + c ) ) \;\;\; = cos( e + c ) \;\;\; = cos( e ) cos( c ) - sin( e ) sin( c ) $

$ D = sin( d ) / sin( c ) \;\;\; = cos( e ) cos( c ) / ( cos( e ) / A ) - sin( e ) sin( c ) / sin( c ) \;\;\; =  A cos( c ) - sin( e ) $

$ D = A \sqrt{ 1 - \left( { cos( e )^2 } \over A \right)^2 } - sin( e ) \;\;\; = \sqrt{ A^2 - cos( e )^2 } - sin( e ) $

|| $ D = \sqrt{ ( A^2 - 1 ) + sin( e )^2 } - sin( e ) $  ||