LW - 5 Physics Problems by DaemonicSigil

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Welcome to The Nonlinear Library, where we use Text-to-Speech software to convert the best writing from the Rationalist and EA communities into audio. This is: 5 Physics Problems, published by DaemonicSigil on March 18, 2024 on LessWrong.
Muireall and DaemonicSigil trade physics problems. Answers and discussion of the answers have been spoilered so you can try the problems yourself. Please also use spoiler formatting (type "
>!") in the comments.
Smeared Out Sun
Okay, so the first problem is from Thinking Physics. (I promise I'll have some original problems also. And there's a reason I've chosen this particular one.)
It's the problem of the Smeared out Sun (caution! link contains spoilers in the upside-down text).
The problem goes as follows: The sun is far enough away that we could replace it with a disc of equal radius at the same temperature (and with the same frequency-dependent emissivity), and so long as the plane of the disc was facing the Earth, there would be little difference in the way it was heating the Earth. While scientists would certainly be able to tell what had happened, there would be little effect on everyday life. (Assume no change to the gravitational field in the solar system.)
Now, suppose that after turning into a disc, the sun is spread into a sphere of radius 1AU surrounding Earth. We'd like to keep the spectrum exactly the same, so we'll imagine breaking the disc into many tiny pieces, each perhaps the size of a dime, and spreading these pieces out evenly across the 1AU sphere. Between these sun-dimes is empty space.
The goal of this exercise is to keep the incoming radiation as similar as possible to that which is given to us by the sun. The spectrum is the same, the total energy delivered is the same, the only difference is that it now comes in from all directions. The question is: What happens to the average temperature of the Earth after this has happened: Does it heat up, cool down, or stay the same?
I think this question is basically asking about the convexity of the relationship between total radiated power and temperature. It's T4 (some law with a name that I forget), which is strictly convex, so for the Earth to be in power balance again, the average temperature needs to be hotter than when there was a wider spread of temperatures.
(If the Earth had a cold side at absolute zero and a hot side at T, with an average temperature of T/2 and average radiated power like T4/2, then with the Earth at a single temperature you'd need it to be T/21/4, which is hotter.)
That should be the main effect. The Earth sees the same amount of hot vs cold sky, so if we ignore how the Earth equilibrates internally, I think there's no change from moving pieces of the Sun disc around.
Yes, exactly, the Earth gets hotter on average after the sun is spread out over the sky.
The name of the T4 radiation law is the Stefan-Boltzmann Law, in case a reader would like to look it up. As things get hotter, the amount they radiate increases more than you'd expect by just extrapolating linearly. So things that are hot in some places and cold in others radiate more than you'd expect from looking at the average temperature.
Interestingly, Epstein's answer in Thinking Physics is that the average temperature of the Earth stays the same, which I think is wrong. Also, in his version the sun becomes cooler and cooler as it spreads out, rather than breaking into pieces. We can still model it as a blackbody, so this shouldn't change the way it absorbs radiation, but then greenhouse-type effect might become important. I didn't want to have to think about that, so I just broke the sun into pieces instead.
Measuring Noise and Measurement Noise
I agree that your version is cleaner, and I'm not really sure what Epstein was getting at - I don't really have any conflicting intuitions if he's treating the Earth as at a single temperature to begin with. I do think there's an interesting line of questions here that leads to something like [r...