# [amsat-bb] ANS-199 AMSAT News Service Special Bulletin - AMSAT Fox-1C Launch Opportunity Announced

Phil Karn karn at ka9q.net
Mon Jul 21 05:53:19 UTC 2014

```On 07/20/2014 04:58 PM, Robert Bruninga wrote:
> Hummh,
>
> We get an equilibrium of a cube to be about 55F (13C) when exposed to the
> sun on one side and all the other sides radiating to cold space. (assuming
> they are thermally connected).
>
> I wonder why the big difference between our calculations?

Well, here's mine again:

Stefan-Boltzmann law:

P = A*e*s*T^4, where

P = radiated thermal power in watts
A = radiating area in m^2
e = emissivity (dimensionless)
s = Stefan-Boltzmann constant: 5.6703e-8 W/(m^2 K^4)
T = absolute temperature, kelvins

Solving for T:

T = (P / (A*e*s)) ** (1/4)

The solar constant at 1 AU is 1367.5 W/m^2.

For a 10 cm cube with a = e = 1.0 (blackbody) at 1 AU, with the sun
normal on one surface, the absorbed solar thermal power is 1367.5 * .01
= 13.675 W. The radiating surface is six times the absorbing area, or
.06 m^2, because it radiates from all six sides (including the side
facing the sun).

Plugging into the previous formula:

T = (13.675 / (.06 * 1.0 * 5.6703e-8)) ** (1/4)
= 251.8 K = -21.4 C.

Shape matters a very great deal. If you have a blackbody sphere instead
of a cube, then the ratio of radiating to absorbing area is only 4:1
instead of 6:1 and the equilibrium temperature rises to 278.7K = +5.5C.
A thin flat plate (normal to the sun) is only 2:1 so it's even warmer:
331.38K = +58.2 C (quite toasty).

Here's why we may get different answers. If the sun shines on an edge of
the cube instead of normal to one surface, the effective absorbing area
increases to sqrt(2) times that of a single face. In that case, the
equilibrium temperature increases to 274.5K = +1.4C.

And if the sun shines directly on a corner, the absorbing area becomes
(3/4) * sqrt(3) times the area of a single face and the equilibrium
temperature is 268.8 K = -4.3C.

But without active attitude control you can't count on *any* specific
orientation with respect to the sun, and that's precisely the problem.
Even a random average is not necessarily useful if the cube is small
(with a low thermal mass) and turning very slowly. Add the ~90 min
periodic changes due to eclipses and the much slower periodic changes
with the seasons and you can see why the thermal design of a 1U
unstabilized cubesat is such a challenging problem.

It's much less of a problem on a larger spacecraft because you don't
have to cover every surface with solar cells to maximize power
generation, you have a much greater thermal mass to smooth out eclipse
(but not seasonal) variations, and you have room for insulation to
thermally isolate the external surfaces from the interior components.

--Phil
```