We’re far away from the Sun. Heat radiating outwards is subject to the inverse-square law. Because the surface area of a sphere is a function of r², the intensity of radiation from the Sun at a distance r is proportional to 1/r². This means that at a distance of 2 solar radii from the centre of the Sun, the amount of radiation you get is 1/4 of what’s leaving the surface. We are so far away that the proportional amount we receive on Earth is tiny compared to the output of the Sun.
Now, the Sun is at millions of degrees. If the temperature went up by 5°C, that 5°C worth of extra heat gets spread out as it moves away from the Sun. By the time it reaches us, the difference is tiny. Another way to think about it is that that’s 5°C out of millions of degrees. That tiny fractional increase is like the fractional increase we’d get. Except we’re sitting at a cool 300K-ish (only a physicist deals in absolutes). So that tiny percentage increase amounts to much less than a degree in Earth.
5°C on the suns surface would be a change of roughly 0.1%. Since radiation energy rises with the 4th power of the temperature, it would rise by about 0.4%, or by about 5 W/m².
For comparison, a doubling of the CO2 consentration in the atmosphere is the equivalent of an extra 3.7 W/m².
So yes, a long-lasting rise of 5°C on the suns surface would have significant consequences.
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