# The Nested Black Body Shells Model and Extreme Greenhouse Warming

**And Lessons from this model that show us how limited the Greenhouse Effect actually is**

_{S}and the power per unit area of surface it emits is P

_{S}. The first closely surrounding concentric shell has no power input except that radiated by the sphere. Its own temperature is T

_{O1}, which when the power to the sphere is turned on is 0 K and the shell is only then warmed by radiation from the sphere which travels at the speed of light to it. It radiates no photons toward the sphere, but does radiate photons as its temperature rises toward the second shell with power P

_{O1}. The second concentric shell has the same parameters but with a 2 in the subscript, rather than a 1. It also starts from T=0K. Only vacuum exists between the sphere and the planes so that there are no heat losses except by means of thermal radiation.

_{SI}, given by the Stefan-Boltzmann Law, since the sphere is at that instant surrounded by T = 0K.

_{SI}= σT

_{SI}

^{4}

_{SE}= P

_{O1E}= P

_{O2E }

_{SE}

^{4}– σ T

_{O1E}

^{4 }= σ T

_{O1E}

^{4}– σ T

_{O2E}

^{4}= σ T

_{O2E}

^{4}

_{SI}= T

_{O2E}

_{O1E}

^{4}= 2 T

_{O2E}

^{4}

_{SE}, we have

_{SE}

^{4}– T

_{O1E}

^{4}= T

_{O2E}

^{4}

_{SE}

^{4}– ( 2 T

_{O2E}

^{4}) = T

_{O2E}

^{4}

_{SE}

^{4}= 3 T

_{O2E}

^{4}or T

_{SE}= 3

^{0.25}T

_{O2E}= 3

^{0.25}T

_{SI}= 1.3161 T

_{SI}, since T

_{O2E}= T

_{SI}

_{SE}= 2

^{0.25}T

_{SI}= 1.1892 T

_{SI}

_{SE}= (N+1)

^{0.25}T

_{SI}

_{SE}= 1.8212 T

_{SI}

_{SE}= 3.1702 T

_{SI}

_{SE}= 5.6282 T

_{SI}

_{S}, and an emissivity to shells representing absorptions by a greenhouse gas, ɛ

_{G}. The equations for a two-shell model then become:

_{S}σ T

_{SI}

^{4}

_{S}σ T

_{SE}

^{4}– ɛ

_{G}σ T

_{O1E}

^{4}= ɛ

_{G}σ T

_{O1E}

^{4}– ɛ

_{G}σ T

_{O2E}

^{4}= ɛ

_{G}σ T

_{O2E}

^{4}

_{S}σ T

_{SI}

^{4}= ɛ

_{G}σ T

_{O2E}

^{4}, so T

_{O2E}= (ɛ

_{S}/ɛ

_{G})

^{0.25}T

_{SI}

_{O1E}

^{4}= 2 T

_{O2E}

^{4}

_{S}T

_{SE}

^{4}= 3 (ɛ

_{G}T

_{O2E}

^{4}) = 3 (ɛ

_{S}T

_{SI}

^{4})

_{SE}= 3

^{0.25}T

_{SI}, the same solution for the equilibrium sphere temperature we had for the black body emitters and absorbers.

_{SE}= (N + 1)

^{0.25}T

_{SI}just as with N black body shells.

_{SE}= 2

^{0.25}T

_{SI}= 1.189 T

_{SI }

_{SI}. If one takes T

_{SI}= 255 K, the radiative temperature of the Earth system as a whole with respect to space, then the change of temperature attributable to the greenhouse gas effect for our present atmosphere is

Source: https://objectivistindividualist.blogspot.com/2018/08/the-nested-black-body-shells-model-and.html