Is the Planet's Surface Hotter Than Its Shell?

[Tandem78]

well, not quite... imagine an earth-like planet totally enclosed in a shell of some material which conducts heat and has near black-body properties both inside and out.
The shell is at a distance of some 100s of kilometres from the planet's surface. The shell is light enough so that it has no significant gravitational mass, and totally encloses the atmosphere.
The planet orbits a sun-like star at a distance of one AU. It has no geothermal source of heat.

The question is - is the planet's surface hotter than the shell? Or put another way, is there a temperature gradient in the atmosphere?

 

[berkeman]

well, not quite... imagine an earth-like planet totally enclosed in a shell of some material which conducts heat and has near black-body properties both inside and out.
The shell is at a distance of some 100s of kilometres from the planet's surface. The shell is light enough so that it has no significant gravitational mass, and totally encloses the atmosphere.
The planet orbits a sun-like star at a distance of one AU. It has no geothermal source of heat.

The question is - is the planet's surface hotter than the shell? Or put another way, is there a temperature gradient in the atmosphere?

What is the context of the question? What do you know about the transmissivity and reflectance of "transparent" layers?

http://en.wikipedia.org/wiki/Reflectivity

.

 

[Tandem78]

What is the context of the question? What do you know about the transmissivity and reflectance of "transparent" layers?

http://en.wikipedia.org/wiki/Reflectivity

.

The context is a dispute as to whether the temperature gradient of an atmosphere is the inevitable consequence of the increasing pressure with depth, or whether a source of energy is required.

 

[cesiumfrog]

well, not quite... imagine an earth-like planet totally enclosed in a shell of some material which conducts heat and has near black-body properties [..] is there a temperature gradient in the atmosphere?

No.
And worse, life ends.

Life is much about entropy. The sun's energy is only of use to us because we receive it from such a hot reservoir and the rest of the sky presents to us such a cold reservoir. If the whole sky were a constant minus 20 Celsius (the default temperature at 1AU), then the whole Earth (and atmospheric column) would also eventually thermalise to the same temperature. Photosynthesis (the food chain) would be thermodynamically prohibited.

The key to the thermal gradient is having an energy source on the inside. So if we had a petawatt of nuclear electricity stations to go on working our climate-controlled cities and grow-houses (or if you just pedantically focus on the residual heat coming from the mantle), then yes, there would still be a thermal gradient (less steep in that case, due to nonlinearity of the Stefan-Boltzmann law). And if you added more and more of those black shells, each would make the planet hotter still, and there would also be a gradient from shell to shell.

If you make the shells more complicated by introducing a mechanism for some heat to escape directly from inner shells (without getting intercepted by the outer shells) then you can even make the outermost shells cooler than they would otherwise be, replicating another effect from Earth's atmosphere.

PS: Pressure has nothing to do with it.

 

[Tandem78]

Pressure has nothing to do with it.

That is what I thought too. Thanks.