Venus if you will

Eli has been applying the rabbety art of kickboxing to some dolt named Goddard. This miscreant, writing at one of the many branch campuses of Climate Stupid, had suggested that Venus was hot not because its atmosphere was opaque to IR but because it weighed a lot. Naturally, the kind of dunce who comes up with these ideas never stops to consider the application of his notion to planets like Jupiter, Saturn, etc, much less doing any of the radiative transfer calculations.

Imagine my disappointment, then, when I saw that Lubos had bought into this nonsense. Oh dear! This is algebra and a little calculus, subjects Lumo probably mastered before he was toilet trained. This is our old friend Stefan’s law.

The temperature at the surface of the planet is 735 K and that surface is probably a fair approximation to a black body (but in any case, a body of known emissivity). The planet as a whole radiates with effective radiating temperature 220 K, meaning that its net emissivity is about epsilon = Teff^4/Tsurf^4 = (220/735)^4 =0.008. Why the difference? This is a straightforward (numerical) computation in radiative transfer, but it’s only dependence on atmospheric mass and density is through their effect on opacity.

Suppose you were to replace the 93 or so atmosphere’s worth of CO2 and other stuff on Venus with ¼ of an atmosphere of a (hypothetical but plausible) gas with equal net opacity. The surface temperature and the effective radiating temperature would be unchanged. For the pretty good approximation of local thermodynamic equilibrium (LTE) that calculation simplifies to:

It = Is*exp[-tau(s,t)] + int[alpha(z)*B(z)*exp[-int[alpha(z’),{z,t,dz’}]],{s,t,dz}]

Where “It” is radiated intensity at top of atmosphere, “Is” is radiated intensity at surface, alpha is absorption coefficient, tau = total opacity of atmosphere from s to t, and B is the Planck function. By int[f(z),{a,b,dz}] I mean the definite integral of f(z) from a to b.

Similar calculations (with possible corrections for non-LTE effects like scattering) are the basis for every satellite temperature measurement as well as nearly everything we know about stellar astrophysics. This, as we in the rocket science community like to say, is not string theory - or even brain surgery.

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