⇤ ← Revision 1 as of 20180716 17:24:33
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← Revision 2 as of 20180716 17:57:09 ⇥
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The polar radius is 6353 km; assume this radius out to the edge of Antarctica. A right triangle with a hypoteneuse of 42164 km and one side 6353 + 25 = 6378 km (in the lower polar stratosphere) will occur at arccos( 6378 / 42164 ) = 81.3 °S latitude, over unpopulated continental [[ http://www.nationsonline.org/maps/antarctica_map.jpg  Antarctica]].  The polar radius is 6353 km; assume this radius out to the edge of Antarctica. A right triangle with a hypoteneuse of 42164 km and one side 6353 + 25 = 6378 km (in the lower polar stratosphere) will occur at arccos( 6378 / 42164 ) = 81.3°S latitude, over unpopulated continental [[ http://www.nationsonline.org/maps/antarctica_map.jpg  Antarctica ]]. The distance from this point to GEO is 42164 sin( 81.3° ) = 41680 km. A triangle with a hypoteneuse of 6353 + 50 + 5 = 6408 km and a side of 6378 km defines where the edge of the beam hits the top of the stratosphere; arccos( 6378 / 6408 ) = 5.5° is added or subtracted from the latitude to find the latitude range ( 75.8°S to 86.8°S ) where we might expect some small amount of scattering to the surface below. The map shows that 75.8 °S crosses the Weddell and Ross Seas, and encompasses six research stations; however, most longitudes have nothing living between 75.8 °S and the pole, so the geostationary satellite should be located there. Some longitudes are more stable than others (MoreLater) and require less stationkeeping thrust to "anchor" a satellite. The beam center crosses 50 km altitude (inbound and outbound) at arccos( 6378 / 6403 ) = 5° from the 81.3° latitute center; latitude range ( 76.3°S to 86.3°S ). The beam traverses 2*6378*tan(5°) = 1100 km of stratosphere  much better than the 30 km path of a vertical beam. The beam width from the geosat is atan( 50 km / 41680 km ) = 0.07° = 1.2 milliradians. The beam height is 0.2 times that, or 0.014° = 0.24 milliradians. The transmitter aperture at GEO is 1.2 times the wavelength times the reciprocal of the radians; for a 10 cm wavelength (WAG), the transmitter aperture would be 500 meters high by 100 meters wide. MoreLater (refer to russian papers) 
Ozone Beam
A smaller scale version of space solar power might be able to pulse microwaves to create ozone between 20 km and 50 km altitude. If the beam was 10 km high and 100 km wide, the transmitter could be aimed from GEO towards the limb of the Earth over Antarctica, where scattering will have few biological consequences.
The polar radius is 6353 km; assume this radius out to the edge of Antarctica. A right triangle with a hypoteneuse of 42164 km and one side 6353 + 25 = 6378 km (in the lower polar stratosphere) will occur at arccos( 6378 / 42164 ) = 81.3°S latitude, over unpopulated continental Antarctica. The distance from this point to GEO is 42164 sin( 81.3° ) = 41680 km. A triangle with a hypoteneuse of 6353 + 50 + 5 = 6408 km and a side of 6378 km defines where the edge of the beam hits the top of the stratosphere; arccos( 6378 / 6408 ) = 5.5° is added or subtracted from the latitude to find the latitude range ( 75.8°S to 86.8°S ) where we might expect some small amount of scattering to the surface below.
The map shows that 75.8 °S crosses the Weddell and Ross Seas, and encompasses six research stations; however, most longitudes have nothing living between 75.8 °S and the pole, so the geostationary satellite should be located there. Some longitudes are more stable than others (MoreLater) and require less stationkeeping thrust to "anchor" a satellite.
The beam center crosses 50 km altitude (inbound and outbound) at arccos( 6378 / 6403 ) = 5° from the 81.3° latitute center; latitude range ( 76.3°S to 86.3°S ). The beam traverses 2*6378*tan(5°) = 1100 km of stratosphere  much better than the 30 km path of a vertical beam.
The beam width from the geosat is atan( 50 km / 41680 km ) = 0.07° = 1.2 milliradians. The beam height is 0.2 times that, or 0.014° = 0.24 milliradians. The transmitter aperture at GEO is 1.2 times the wavelength times the reciprocal of the radians; for a 10 cm wavelength (WAG), the transmitter aperture would be 500 meters high by 100 meters wide.
MoreLater (refer to russian papers)