Would solar sail get warm?

Question

Would solar sail get warm?

If yes, then approximately how fast could they be warming up?

That may be a weird question, but that's for a short story I'm considering to write, and it's actually very important for the plot :) .

Answer 1

It depends on the distance from the sun and the percentage of the radiation that is reflected (for a celestial body you would call it the albedo).

According to the Stefan-Boltzmann law, the flux $j$ of a black body with temperature $T$ is given by:

$$j = \sigma T^4$$

where:

$$\sigma=\frac{2\pi^5 k^4}{15c^2h^3} = 5.670374\ldots\times 10^{-8}\, \mathrm{W\, m^{-2}\,K^{-4}}$$

The solar constant, that is the value of $j$ on the Earth is given as $j_S=1{,}36\,kW/m^2$.

If the sail reflects, say, $70\%$ of the sun's radiation and absorbs the remaining $30\%$, then in the equilibrium situation, it will have such a temperature $T$ that $j=30\% j_S$

Plugging it all in:

$$30\% j_S=\sigma T^4$$

Solving for $T$:

$$T=\sqrt[4]{\frac{30\%j_S}{\sigma}}$$

Plugging in the values gives around $292K$ or $19°C$.

Obviously, taking different distances from the sun and different albedo values, we will get different temperatures. $j_S$ falls quadratically with the distance to the sun.