Question

Mars has an albedo of 15\%

15%. Given that it doesn't have much of an atmosphere, estimate the mean surface temperature on Mars. Note: the semimajor axis of the orbit of Mars is 1.52 AU.

Answer 1

The illuminance generated by the Sun on the orbit of the Mars is

$E = \dfrac{L_{\odot}}{4\pi a^2}$ .

The energy gathered by the martian surface every second is $(1-A)E\cdot \pi r^2$ , where r is the radius of the Mars.

Due to the energy balance, the equal amount of energy is released every second. If we assume Mars to behave like a perfect black body, this energy is $4\pi r^2\sigma T^4$ (Stephen-Boltzmann law).

$\dfrac{L_{\odot}}{4\pi a^2} \cdot(1- A)\pi r^2 = 4\pi r^2\sigma T^4,\\ \, \\ \dfrac{L_{\odot}}{4\pi a^2} \cdot(1- A)= 4\sigma T^4,\\ \,\\ T = \sqrt[4]{\dfrac{(1-A)L_{\odot}}{16\pi \sigma a^2}} ,\\ T = \sqrt[4]{\dfrac{(1-0.15)\cdot4\cdot10^{26}}{16\pi \cdot5.67\cdot10^{-8}\cdot(1.52\cdot1.5\cdot10^{11})^2}} = 219\,\mathrm{K}.$