Problem $3$: $(20$ pts$)$

In this problem we want to better understand how the sun heats our planet.

$-$ The energy flux associated with solar radiation incident on the outer surface of the earth's

atmosphere has been accurately measured and is known to be $1368$ W$/$m$^2.$

$-$The diameters of the sun and earth are $1\mathrm{.}39\backslash$times $109$ and $1\mathrm{.}27\backslash$times $107$ m$,$ respectively, and the distance between the sun and the earth is $1\mathrm{.}5\backslash$times $10.$

Find:

a$)$ What is the emissive power of the sun?

b$)$ Approximating the sun's surface as black, what is its temperature?

c$)$ At what wavelength is the spectral emissive power of the sun a maximum?

d$)$ Assuming the earth's surface to be black and the sun to be the only source of energy for the earth, estimate the earth's surface temperature.