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← Revision 12 as of 20210618 19:10:16 ⇥
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Deletions are marked like this.  Additions are marked like this. 
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For FF=0.2, A=0.024, the array spacing L = sqrt( N * A / FF ). The array contains N^3^ thinsats, the array edge is N^L^, the array mass M is 0.003kg*N^3^, the array power is approximately P = 3W*N^3^, and the main lobe ground spot diameter G at 10,000 km and 4mm wavelength is G = 40000 m^2^/N*L :  The array spacing is L = sqrt( N * A / FF ). The array contains N^3^ thinsats, the array Edge is (N1)*L, the array mass M is 0.003kg*N^3^, the array power is approximately P = 3W*N^3^, and the main lobe ground spot diameter Gspot at 10,000 km and 8mm (38 GHz) wavelength is approximately Gspot ≈ 1.22*distance*λ/D ≈ 1e5 m^2^/(N1)*L : 
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 N  N^3^  M(kg)  P  G (m)  L (m)  N*L (m)   10  1000  3  3kW  3636  1.10  11   20  8000  24  24kW  1291  1.55  31   32  32768  98  98kW  638  1.96  63   50  125000  375  375kW  326  2.44  123   64  262144  786  786KW  225  2.77  177   100  1000000  3000  3.0MW  115  3.46  346   128  2.1M  6300  6.3MW  80  3.92  502   700  343M  1030t  1.0GW  6  9.17  6420  
<7> Fill factor FF=0.2, Thinsat area A=0.024   N  N^3^  M(kg)  P  L (m)  Edge (m)Gspot (m)  10  1000  3  3kW  1.10  10  10000   20  8000  24  24kW  1.55  29  3440   32  32768  98  98kW  1.96  61  1640   50  125000  375  375kW  2.44  120  830   64  262144  786  786KW  2.77  180  560   100  1000000  3000  3.0MW  3.46  340  290   128  2.1M  6300  6.3MW  3.92  500  200  
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Note: we can always ''defocus'' the beam. The last entry, 700 thinsats cubed, is a solar power satellite equivalent of a full sized power plant. If it put 1GW into a 6 meter diameter area, it would heat it to 5000K  a weapon, if the heat did not spread out in the atmosphere, and if most of the energy did not end up in sidelobe power.  <7> Fill factor FF=0.01, Thinsat area A=0.024   N  N^3^  M(kg)  P  L (m)  Edge (m)Gspot (m)  10  1000  3  3kW  4.9  44  2300   20  8000  24  24kW  6.9  130  770   32  32768  98  98kW  8.8  270  370   50  125000  375  375kW  11.0  540  190   64  262144  786  786KW  12.4  780  130   100  1000000  3000  3.0MW  15.5  1530  65   128  2.1M  6300  6.3MW  17.5  2200  45  
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MORE LATER  
Because of the depth of the array, and the distortions caused by apogee skew, the ground spot will be compressed northeast and southwest and stretched northwest to southeast. If a launch can put 3146 kg of thinsats (1.049 million) into an M288 orbit, and these are deployed into 32 arrays spaced 11.25 degrees apart, then each array can contain 32768 thinsats, a distorted cube 32 thinsats on a side. . Note: Real arrays will probably be a more complicated, spherical array with tetrahedral arrangement and spacing and Golombruler offset dither. Too complicated to analyze and explain here. If thinsats are spaced 8 meters apart, the array is 250 meters on a side, the ground spot diameter is 400 meters, and the fill factor is 0.012 . The array rotates 360 degrees in about 4 hours, or 40 seconds per degree. A 20 cm wide thinsat will partially eclipse another thinsat directly behind it for about 1 minute out of every 4 minute orbit; we can probably arrange spacing and dithering so the thinsats are never precisely in line.  {{ attachment:shadow2.png   height=288 }}  {{ attachment:shadow3.png   height=288 }}  MoreLater 
Array Fill
Thinsats should be in continuous sunlight for maximum radio and computational power. If one thinsat shades or partially occludes the other, the result will be a loss of power, and nonuniform temperature changes. Thinsats will need to actively and cooperatively move to keep from shading each other.
The sun is 1,391,980 km in diameter, and 149,600,000 km away (on average). Its angular size is 9.3E3 radians, or 0.53 degrees. A thinsat sized disk 20cm in diameter casts a black shadow up to 22 meters behind it, with a shaded penumbra in a narrow 0.53 degree cone widening to infinity. The penumbra cone is a meter wide 107 meters behind a thinsat, and 10 meters wide 1070 meters behind. However, the amount of shading diminishes with distance, too, though more thinsats in the front will contribute their shadows, too. Suffice it to say that thinsats in the back of the array will get somewhat less light, on average, than those in the front, and the amount of light will have ripples in it. For that reason, we cannot pack an array too densely.
The three dimensional spacing functions used in deployment will require a lot of research to optimize; they must also be compatible with a dither function that smears out grating lobes. There are some great opportunities for research here, and I hope the best functions will be developed for the public domain. However, mathematical functions are difficult to patent, and hopefully there will be an infinite number of suitable patterns, so the patent trolls will need to spend an infinite amount of filing fees to cover all of them.
We can characterize these functions by a "fill factor". Assuming a thinsat area of A = 0.024 m^{2}, and a three dimensional array of N x N x N or N^{3} thinsats, and a spacing of L, then the fill factor is defined as FF = N^{3}*A/(N*L)^{2} = N*A/L^{2}. For larger arrays, with many overlapping penumbras in the back, the average illumination in the back is (1FF), and the temperature of the thinsats will be proportional to the 4th root of that. So if the average thinsat temperature is 330K under normal conditions, it might be 310K in the back of the array with a fill factor of 0.2, and 322K with a fill factor of 0.05 . We can probably survive a fill factor of 0.2, though we must manage thermals carefully and avoid full shading.
The array spacing is L = sqrt( N * A / FF ). The array contains N^{3} thinsats, the array Edge is (N1)*L, the array mass M is 0.003kg*N^{3}, the array power is approximately P = 3W*N^{3}, and the main lobe ground spot diameter Gspot at 10,000 km and 8mm (38 GHz) wavelength is approximately Gspot ≈ 1.22*distance*λ/D ≈ 1e5 m^{2}/(N1)*L :
Fill factor FF=0.2, Thinsat area A=0.024 

N 
N^{3} 
M(kg) 
P 
L (m) 
Edge (m) 
Gspot (m) 
10 
1000 
3 
3kW 
1.10 
10 
10000 
20 
8000 
24 
24kW 
1.55 
29 
3440 
32 
32768 
98 
98kW 
1.96 
61 
1640 
50 
125000 
375 
375kW 
2.44 
120 
830 
64 
262144 
786 
786KW 
2.77 
180 
560 
100 
1000000 
3000 
3.0MW 
3.46 
340 
290 
128 
2.1M 
6300 
6.3MW 
3.92 
500 
200 
Fill factor FF=0.01, Thinsat area A=0.024 

N 
N^{3} 
M(kg) 
P 
L (m) 
Edge (m) 
Gspot (m) 
10 
1000 
3 
3kW 
4.9 
44 
2300 
20 
8000 
24 
24kW 
6.9 
130 
770 
32 
32768 
98 
98kW 
8.8 
270 
370 
50 
125000 
375 
375kW 
11.0 
540 
190 
64 
262144 
786 
786KW 
12.4 
780 
130 
100 
1000000 
3000 
3.0MW 
15.5 
1530 
65 
128 
2.1M 
6300 
6.3MW 
17.5 
2200 
45 
Because of the depth of the array, and the distortions caused by apogee skew, the ground spot will be compressed northeast and southwest and stretched northwest to southeast.
If a launch can put 3146 kg of thinsats (1.049 million) into an M288 orbit, and these are deployed into 32 arrays spaced 11.25 degrees apart, then each array can contain 32768 thinsats, a distorted cube 32 thinsats on a side.
 Note: Real arrays will probably be a more complicated, spherical array with tetrahedral arrangement and spacing and Golombruler offset dither. Too complicated to analyze and explain here.
If thinsats are spaced 8 meters apart, the array is 250 meters on a side, the ground spot diameter is 400 meters, and the fill factor is 0.012 . The array rotates 360 degrees in about 4 hours, or 40 seconds per degree. A 20 cm wide thinsat will partially eclipse another thinsat directly behind it for about 1 minute out of every 4 minute orbit; we can probably arrange spacing and dithering so the thinsats are never precisely in line.

