il27
Homework Statement
In the grey atmosphere radiative energy balance model, we replace the multi-layer approximation used above with still simplified but significantly more realistic model involving a continuous atmosphere with a continuously varying temperature. The variation with temperature is a function of optical depth.
a) Find the total optical thickness of the atmosphere of Venus using the gray atmosphere model. Assume the same emission and ground temperatures as in the previous question.
b) Find the temperature difference between the ground temperature Tg and the atmospheric temperature at ground level.
Homework Equations
$$ T_g ^4 = T_e ^4(1 + \frac{3}{4} \tau) $$
There is another equation for optical depth, but I am not sure how to derive this:
$$ \tau = \frac{T_g ^4}{T_e ^4} - 1 $$
$$ T_e = (\frac{(1-\alpha)}{4 \sigma} F_0)^.25 $$
The Attempt at a Solution
I am thinking I can calculate the emission temperature with the 3rd equation, but I am stuck on finding the ground temperature.
Also which optical depth equation is correct?