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I have read that Venus is roughly at radiation equilibrium - that is, heat out is approximately equal to heat in. I have also read that Venus began its heating-up around 700-750 million years ago.

My question is, around when did Venus reach approximate radiation equilibrium, where it stopped getting appreciably hotter in the Venusian atmosphere?

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    $\begingroup$ The approximate radiative timescale to reach equilibrium is a few hours on Venus - on Earth it is minutes due to the lower infrared optical depth. Hence the equilibrium state always follows other perturbations which move that state around on millions on years timescales. $\endgroup$ Sep 19, 2023 at 18:41
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    $\begingroup$ This definitely answers the question as asked. I guess I should have asked a question more in line with "at what point did the Venusian atmosphere get roughly as hot as it is now". $\endgroup$ Sep 19, 2023 at 19:04
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    $\begingroup$ Both is an interesting question. Maybe @AtmosphericPrisonEscape should write his comment as answer - and you ask a new question where you ask what you really meant to ask :) $\endgroup$ Sep 19, 2023 at 21:14

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If we assume that energy transport in atmospheres happens purely via radiation, then we can estimate a simple adjustment time to an exchange of exterior forcing factors (change in mean solar angle, orbital distance, aliens building giant shades, etc ...) of the atmosphere by using the internal energy equation of thermodynamics, coupled with the radiation transport equation.
I will omit the details here and merely state the result of the timescale as
$$\rm t_{adjustment} \approx \frac{c_v M_{atmo} \tau_{IR}}{R_p^2 \sigma_{SB} T^3},$$ where we have the mean specific heat capacity of the atmospheric gas $\rm c_v$, the total atmospheric mass $\rm M_{atmo}$, the infrared optical depth of the atmosphere $\rm \tau_{IR}$, the planetary radius $\rm R_p$, the Stefan-Boltzmann constant $\rm \sigma_{SB}$ and the temperature $T$ at which the atmosphere is cooling (NOT the surface temperature). For Earth, I get about 23 days (the number quoted in my comment above was apparently the same number but for the stratosphere, i.e. not the entire atmospheric mass), and for Venus about 60 Earth years.

Those numbers are however only the 'slowest' estimates, as in effect, convection acts to equilibrate the lower atmosphere much faster with the stratosphere on both planets. Still, with this caveat we can state that both planets follow any perturbations, be it solar or man-made much faster than millions-of-years timescales. Hence, the temperature state we currently see Venus at is a result of current atmospheric optical depth and forcing. Venus will therefore always be as hot as current geological and astrophysical conditions dictate.

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