Week 11: WS20 #1

1. The Earth resides in a "Goldilocks Zone" or habitable zone (HZ) around the Sun. At our semimajor axis we receive just enough sunlight to prevent the planet from freezing over and not too much to boil off our oceans. Not too cold, not too hot. Just right. In this problem we'll calculate how the temperature of a planet, T_p, depends on the properties of the central star and the orbital properties of the planet. 
a) Draw the Sun on the left, and a planet on the right, separated by a distance a. The planet has a radius R_p and temperature T_p. The star has a radius R_★ and a luminosity L_★ and a temperature T_eff. 

b) Due to energy conservation, the amount of energy received per unit time by the planet is equal to the energy emitted isotropically under the assumption that it is a blackbody. How much energy per time does the planet receive from the star? How much energy per time does the planet radiate as a blackbody? 

We know that the flux from the star at the distance a is given by \(F_\bigstar=\frac{L_\bigstar}{4\pi a^2}\). However, this is the total flux, and we want only the power received by the planet. If we multiply the flux by the cross-sectional area of the planet, we get the correct units for power (energy per time) and the proper dependence on the planet's radius.
\(P=\frac{L_\bigstar}{4\pi a^2}(\pi R_p^2)=\frac{L_\bigstar R_p^2}{4a^2}\)
In order to calculate how much energy the planet radiates as a blackbody, we simply use the Stefan-Boltzmann Law to find the flux, and then multiply by the planet's surface area in order to convert to power.
\(F_p=\sigma T_p^4\)
\(P=\sigma T_p^44\pi R_p^2\)

c) Set these two quantities equal to each other and solve for T_p. 

\(\frac{L_\bigstar R_p^2}{4a^2}=\sigma T_p^44\pi R_p^2\)
\(T_p=(\frac{L_\bigstar}{16\pi a^2\sigma})^{1/4}\)

d) How does the temperature change if the planet were much larger or much smaller? 

There's no dependence on the planet's radius in the above equation, so the temperature won't change if the size of the planet changes. This makes sense, because any extra radiation absorbed by the planet due to its increased cross section will be radiated away due to its increased surface area.

e) Not all of the energy incident on the planet will be absorbed. Some fraction, A, will be reflected back out into space. How does this affect the amount of energy received per unit time, and thus how does this affect T_p? 

Any reflected energy will correspond to an equivalent decrease in the energy absorbed by the planet, which will correspond to a decreased surface temperature.