Where am I going wrong (energy transfer between black bodies)?

[Steve4Physics]

TL;DR Summary

Where am I going wrong (energy transfer between black bodies)?

I have a problem with a very basic ‘thought experiment’. I can’t see my mistake(s) - I’m pretty sure there must be at least one! So I’m accepting likely humiliation/embarrassment and asking if anyone can explain where I’m going wrong...

The surface of a black body (BB1) is at temperature T and radiates at R W/m².

An ‘optical’ system collects/redirects/focuses the radiated energy from 1m² of BB1 onto a smaller black body (BB2) of area 10⁻⁴m². We now have R watts directed onto 10⁻⁴m². That’s an incident intensity onto BB2 of 10⁴R W/m².

When BB2 reaches equilibrium, the power it receives (R watts) will be the same as the power it emits. So BB2 will emit R watts from an area of 10⁻⁴m². BB2’s surface is radiating at 10⁴R W/m².

Since a black body's radiated power/unit area is proportional to $T_{abs}^4$ this means BB2’s temperature must be 10T.

So energy is spontaneously flowing from an object at temperature T to one at temperature 10T. Err...

 

[Lnewqban]

You still only have 1 watt of radiated power to be transferred among bodies.
BB2 can't radiate more than it receives from the 1 m x 1 m of BB1 surface.

 

[Dale]

You appear to be assuming that all of the energy leaving a 1 square meter surface can be focused down to 1 square cm. It cannot. This is related to the conservation of etendue and view factors. I believe, without calculating it, that at most 1/10000 of the light can be focused that small.

 

[Steve4Physics]

You still only have 1 watt of radiated power to be transferred among bodies.
BB2 can't radiate more than it receives from the 1 m x 1 m of BB1 surface.

Not quite with you. The radiated power from 1m² BB1 is equal to the power received by (the much smaller) BB2. This, in turn is the same as the radiated power from BB2. Agreed,

But this means the power/m² radiated by BB2 is 10⁴ times the power/m² originally radiated by BB1. (Because BB2 is so much smaller than BB1.)

From the Stefan-Boltzmann law, this means BB2's temperature must be (⁴√(10⁴) =) 10 times that of BB1.

 

[Steve4Physics]

You appear to be assuming that all of the energy leaving a 1 square meter surface can be focused down to 1 square cm. It cannot. This is related to the conservation of etendue and view factors. I believe, without calculating it, that at most 1/10000 of the light can be focused that small.

Aha! I have indeed made that assumption. I'm not familiar with the conservation of etendue. I'll go and do some reading. Many thanks!

(Minor edit.)