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Sorry been busy & didn't get back to your question on the previous Brian post .
I am surprised that , while several people commented that your question was treated in any elementary heat transfer physics course I believe only Ebel , other than myself , responded with an equation . Mine was the only equation I saw for a sphere as you specified . Your question is exactly the sort that must be quantitatively understood -- as judged by how well they correspond to quantitative experiment , if planetary temperature is to be understood at the 3rd and 4th decimal place range of the data . I've pointed out before the limitations of the internally heated ball analogy . At the Heartland conference a couple of speakers gave a value of about 0.02% ( Wikipedia , 0.03% ) of our external heating from the Sun . And external heating imposes the constraint that divergence = 0 over shells within .
I wish someone would do a "SteveGodard" on the experimental support for the mathematical abstractions , the crucial portions of which are 19th century . I'd really like to work with some others putting together an experimentally based , but with the theoretical understanding expressed in executable algorithms implementable , and therefore experimentally testable by the students , at the level of the PSSC course I had a half century ago .
This return to the discipline of the classic experiment><algorithmatise physics with the ability of succinct array programming languages as an aid to the process is the thrust of my Heartland presentation : http://cosy.com/Science/HeartlandBasicBasics.html . These Gedankenexperiment undoubtedly were approximated by the realeExperimente upon which the equations were abstracted . My standard reply in any assertion of qualitative effects is : " show me the equations " . In discussions like this , I want to ask " show me the experiments " .
You thought experiment raises several interesting points .
- Computations in energy are linear . ( Otherwise talking in terms of { % m ^ 2 3 } would make no sense . )
( Sorry , I'm going to indulge in expressing computations in notation very close to executable APL . )
Once you convert temperatures to energy via { t ^ 4 } , you can do simple arithmetic with the flows . Thus you have the basic { -/ ( t0 ; t1 ) ^ 4 } for net energy flow . And they are conserved . ( Not as Willis Eschenbach misspoke in his Steel Greenhouse post , w % m ^ 2 , rather , over closed surfaces . Thus , the divergence theorem says that the sum over spherical shells around a source will be equal . Therefore the energy density must decrease by the inverse square of the radius . )
- A energy flow isn't a temperature without a volume , and
therefore a surface . So the equations for spheres can always be
expressed in terms of shells . ( In the limit , a delta function ) . I
found this useful link I really need to study further with the
equations for steady state conduction : http://www.cdeep.iitb.ac.in/nptel/Mechanical/Heat%20and%20Mass%20Transfer/Conduction/Module%202/main/2.6.5.html.
I was looking for the equations for conduction because in these
"thermostatics" , for a sphere , temperature follows either the
inverse square-root law of radiation or the inverse law given in the
table .
- If a body is truly black , it by definition absorbs all the energy impinging on it . Thus there is no reflected energy to
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