# Yearly changes of solar irradiation and atmospheric feedback

I think I understand the mechanism of water vapor feedback in climate change pretty well: a (theoretical) temperature increase due to some isolated factor (e.g. increase in $$CO_2$$ concentration) causes a shift in water vapor saturation density and an additional evaporation rate. The consequential shift in actual water vapor concentration causes additional absorption together with isotropic re-emission, slowing down the radiation transport. So the theoretical temperature rise $$\Delta T_{nofb}$$ without consideration of feedback is "amplified" by some factor $$\beta$$, giving the real temperature rise, with feedback

$$\Delta T_{wfb} = \beta \Delta T_{nofb}$$

Of course I know that this argument has to be replaced by solving a differential equation in a more rigorous treatment. There is no "before" and "after", there is only the combined effect.

But what if we consider the "theoretical" temperature change due to the change in solar irradiation from orbital eccentricity? Shouldn't this be amplified by the same water vapor feedback mechanism as well?

Solar irradiation varies over the year between $$1310$$ $$W/m^2$$ and $$1420$$ $$W/m^2$$. According to Stefan-Boltzmann's Law in differential form we would have a heating of the ground by direct absorption of sunlight

$$\frac{dP}{P}=4 \frac{dT}{T}$$

or

$$\frac{\Delta P}{P}\approx 4 \frac{\Delta T}{T}$$

So I would expect the temperature change due to yearly changes in solar irradiation (but without the effect of water vapor feedback) to be about

$$\Delta T \approx \frac{T}{4} \frac{\Delta P}{P} = \frac{290 K}{4} \frac{90 W/m^2 }{1366 W/m^2}\approx 4.8 K$$

The favorite answer to this question shows a plot that seems to indicate that the order of magnitude is indeed like that, the global line shows a variation of about $$3.8$$ $$K$$. I don't know if this was derived from measurements, but I assume that it is somehow substantiated.

But if we now adopt the assumption that these ~5 Kelvin get amplified by the water vapor feedback as well, and take into account that this amplification is said to be in the order of magnitude ~2 in the context of $$CO_2$$ entry by mankind, I would expect the amplified yearly temperature change due to the orbital eccentricity of earth to be $$~10$$ $$K$$.

Is the linked answer wrong, or is there some flaw in the argument?