Cooling Server Satellites
Server satellites are heated by the sun and by power dissipation, and cooled by black body radiation.
For a uniform source object surrounded by a uniform radiation heat sink, black body heat radiation power is proportional to the emissivity multiplied by the difference between the temperatures of source and sink to the fourth power:
P = ~ \epsilon ~ \sigma ~ ( T_{source}^4 - T_{sink}^4 )
P == Power flow (W)
\epsilon == emissivity ( 0.0 = white/shiny, 1.0 = black )
\sigma == Stefan-Boltzmann black body constant, = 5.67 × 10−8 W m−2 K−4
The solid angle of the entire surroundings is \Omega = 4 \pi . One hemisphere of the sky is \Omega = 2 \pi . At m288, the earth occupies a 120 degree swath of the sky, and the solid angle is \Omega = 2 \pi ( 1 - cos( \theta ) ) = 2 \pi ( 1 - cos( 60^o ) ) = \pi .
As an example, the Earth behaves as a black body radiator, but a complicated one. The Earth intercepts 1360 watts per square meter of sunlight, and absorbs approximately 900 watts of that in the atmosphere and on the surface. The "absorbing surface" is the disk facing the sun, \pi R^2 , and the emissive surface is the whole surface, 4 \pi R^2 . In equilibrium, the average power is 225 W/m2 . This corresponds to a black body temperature of approximately 250K . The actual numbers are complicated by different emissivities at different wavelengths and altitudes.
The server-sat heat sink is not uniform - the sun is very hot (and visually small), the 250K earth fills a quarter of the sky (at m288), and deep space is 2.7K (effectively zero). The heat sink is now the sum of the solid angles of these different heat sinks. It is easier to treat the sun as a point that delivers heat (negative power flow) to the exposed area of the object.
Since the glass substrate is a very poor thermal conductor, electronics and solar cells have different temperatures (note: check differential thermal expansion issues, bulging, etc.). These regions will have different front and back sides, and the orientation in relation to the 250K earth will result in different temperatures for these regions at different times.
Solar cells
Early server satellites will have no filtering between the sun and the cell. Assuming 15% efficient InP cells, they produce 210W/m2 of power and 1156W/m2 of heat, emitted on both sides. The ambient environment of the cells is 2.7K for 3/4 of the sky, and 250K earth for 1/4 of the sky. The temperature of the solar cells in sunlight will be T_{cell}^4 = 1156W/m2 / \sigma +0.25*250K^4 , $ T_{cell} = 382K .
Later server satellites will have an anisotropic infrared filter on the sunward side of Indium Phosphide solar cells, passing light with energies higher than the 1.35 eV bandgap ( < 920nm ), and reflecting the longer infrared wavelengths on axis to the sun, while passing infrared (as heat) off axis. With ideal filters, 884 W/m2 passes through the filter, and 482 W/m2 is reflected.
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Electronics
Electronics and other heat sources have low emissivity reflectors on the sun side, and high emissivity black surfaces on the dark side of the cell.
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