At What Wavelength is the Peak of the Planet's Blackbody Intensity Function?

[TheCammen]

Homework Statement

Imagine you are trying to detect a Jupiter-like planet located 7.5E11 m from a Sun-like star. At what wavelength is the peak of the planet's Planck blackbody intensity function (assume the planet is a perfect blackbody)?

Radius of Planet r = 7E7 m
Radius of Star R = 7E8 m
Temp of Star = 5800 k
Luminosity of Star = 3.8E26 J/s

Homework Equations

Stefan-Boltzmann law j = σT^4
Total Incident Flux (σ T^4 r^2) /distance^2

The Attempt at a Solution

Now, my professor gave me the equation for Stefan's law. He then gave us the task of determining the incident flux upon the planet given by the star on the planet. If the planet is a perfect black body, then the planet will absorb and then emit all this energy. I can go through the steps to show you how I determined the Total Incident Flux equation. But I believe that it's correct.

The problem I am having now is how to change this emission to a wavelength. I feel like I need a nudge in the right direction. Do I need to rearrange Planck's law and solve for wavelength at the temperature of the planet?

 

[Dick]

What you've got there looks like a flux density at the distance of the planet. It's not a total incident flux. Remember the planet only absorbs the flux crossing through it's cross section. Then, sure, equate that to outgoing flux over the whole surface of the planet, assuming it's at a constant temperature. Then you should be able to find the temperature of the planet. Then, also sure, use Planck's law to find the wavelength at the peak.

 

[TheCammen]

Thanks! I see how to do it now. I "knew" how to solve it. The concepts just weren't clicking. Gracias!