# Black body radiation -- Spherical shell surrounding a star

• Eitan Levy
In summary, the problem is not correctly solved because at equilibrium the temperature of the shell is not the temperature of the star.

#### Eitan Levy

Homework Statement
A spherical shell with a radius of R surrounds a star with temperature T.

Find the amount of energy that the shell from the star in an hour.
Relevant Equations
$$P=\sigma*A*T^4$$
I don't understand how this can be solved.

The official solution was:

$$F=\sigma*T^4$$

$$E=F*4\pi R^2*60*60$$

This doesn't make sense to me, as it seems to imply that the energy that the black body radiates depends on the radius of the shell. For a very large shell the body will reflect "infinity" energy.

Thank you.

If a body emits a total of 100 Joules per second and you enclose it completely with a shell, 100 Joules per second will pass through the entire shell no matter how large the shell is.

• Eitan Levy
kuruman said:
If a body emits a total of 100 Joules per second and you enclose it completely with a shell, 100 Joules per second will pass through the entire shell no matter how large the shell is.

That's what I figured! Is the solution wrong?

You should previously know the star's radius.

• hmmm27
Gordianus said:
You should previously know the star's radius.
The radius R of the surrounding shell is given. That ought to be enough.

• hmmm27
But at equilibrium the temperature of the shell is not the temperature of the star. That is the point and the OP is correct.

hutchphd said:
But at equilibrium the temperature of the shell is not the temperature of the star. That is the point and the OP is correct.
True, but there is no mention of equilibrium in the statement of the problem. My interpretation is that the shell is something like a Gaussian surface and not a material object.

Gaussian surfaces do not radiate. The ##\sigma T^4## is for the surface that radiates and is emitted power per area. You are making the same mistake as the Prof I fear.

• kuruman
Perhaps the homework has a wording problem. I think the sphere of radius R is a sort of Gaussian surface that encloses the star (that radiates to the 2.7 K background). Thus, we should know the star's radius (call it a)

• hutchphd
You need to know the the temperature of some physical object of known radius. The problem as stated is not correctly solved.
No hand waving required. As pointed out astutely by the OP this leads to infinite power Dyson spheres.

hutchphd said:
Gaussian surfaces do not radiate. The ##\sigma T^4## is for the surface that radiates and is emitted power per area. You are making the same mistake as the Prof I fear.
You fear correctly. Goes to show that I shouldn't be trying to do problems in my head.

• hutchphd