cooling

Cooling Thinsats

Thinsats 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 \equiv Power flow (W)

\epsilon \equiv emissivity ( 0.0 = white/shiny, 1.0 = black )

\sigma \equiv 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 thinsat 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.

Solar cells

Early thinsats 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 = 0.5 * 1156W/m2 / \sigma +0.25*250K^4 , $ T_{cell} = 325K .

Later thinsats may 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 with 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. The waste heat is now 1156-482W = 674W, radiated as before, resulting in $ T_{cell} = 288K .

The above cases are ideal - in real life, emissivities will be less than perfect, filters will be less than perfect, etc. So the above numbers are best case. Note that for very thin solar cells in direct bandgap materials, a high temperature does not degrade performance as much as with silicon, while the temperature may help anneal radiation damage.

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. In this case, a heat spreader and low power chips can result in arbitrarily low temperatures, depending on the power level. In front of the earth, the weighted sky temperature at m288 is 210K, and if the power density is low enough, the chips can approach that. If the thinsat is 90 degrees from the sun, then the weighted sky temperature drops to 177K, and at 150 degrees ( just before eclipse ) the sky temperature drops to 2.7K.

Heat spreaders for thinsats will be very thin, so they will not spread the heat very far. In the near term, they will be graphite, and in the longer term graphene. So, a small die might spread heat to a centimeter squared. With a 210K average sky temperature,

Again, if we can cover the front surface with a filter that is reflective on-axis to the sun, and highly emissive off-axis, we can emit heat on both sides, and the average sky temperature will always be 177K (because the earth is always in the sky somewhere).

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