Star absorbed by spherical dust cloud - find effective temp

[AStaunton]

problem:

The radiation from an old evolved star of radius 10^6km is obsorbed by a spherical dust and gas cloud, ejected previously by the star, of radius 4*10^8km. If all the radiation from the star is re-emitted by the cloud, which is observed to have a temperature of 500K, what is the effective temperature of the star?

My solution:

basically, this is Stefan-Boltzman law:

$$L=4\pi R^{2}\sigma T^{4}$$

first sub in values for cloud, so R=4*10^11m and T=500K, solving for luminosity gives:

L=7.12*10^27W

Now re-use stef-boltz law for the star, now that L is known:
plug R=10^9m L=7.12*10^27W into the equation and solve for T.

I found T to be approximately equal to 10,000K.

My query:
Can anyone comment on the method I used to find this result, and perhaps are there areas where it could be improved? Also, please suggest any real life examples of such a phenomenon.

 

[anathema]

Your method seems sound and correct. The only area where it could be improved is perhaps by taking into account the absorption and re-emission of radiation by the dust and gas cloud. This could affect the effective temperature of the star, as some of the radiation may be absorbed and not re-emitted. However, in this scenario, the difference would likely be minimal.

An example of a similar phenomenon in real life is the reflection and absorption of sunlight by Earth's atmosphere and clouds. This results in the effective temperature of the Earth being higher than it would be without the atmosphere, as the atmosphere traps some of the heat and re-emits it back to the surface.