Given perfect conduction (uniform surface temperature, bright side and dark side have the same temperature at all times), https://en.wikipedia.org/wiki/Black-body_radiation#Temperature_relation_between_a_planet_and_its_star applies : temperature does not depend on rotation speed. Then T = T_sun sqrt(R_sun/(2D)) ; it is the temperature T that balance incoming radiation P_inc = pi (R_planet^2) (R_sun^2) (T_sun^4)/(D^2) and emitted radiation P_em = 4 pi (R_planet^2) * T^4
Let’s suppose no conduction at all. The bright side and the dark side does not exchange heat at all. Let’s take two limiting cases : tide-locked planet, and an “infinitely fast” fliping planet.
In the first case, the dark side of the planet is at absolute 0. The bright side of the planet receives the same incoming radiation but emit half its radiation (halved surface) -- change 4 pi to 2 pi in P_em. Its temperature is T_bright = 2^(1/4) T. Average temperature of the planet is (0+T_bright)/2=2^(-3/4) T
In the second case, each side gets half the incoming power from the sun and radiates half the energy (surface halved). Average surface temperature of the planet is the same that the average surface temperature of any side, which is the same temperature that the perfectly conducting planet, T (the .5 from halved incoming power and .5 from halved outgoing radiation cancel each other).
So : rotation raises temperature of a non-perfectly-heat-conducing planet, bringing its surface temperature closer to the perfectly-heat-conducting planet surface temperature.
You’re right, the 2^(-3/4) (and the 2^1/4) is probably quantitatively wrong (unless each side is perfectly heat-conducing but both are isolated from each other. Or if the planet is a coin facing the sun. You know, spherical cows in a vacuum…). But I don’t think that changes the qualitative conclusion, which hold as long as the bright side is hotter but not twice as hotter than the perfectly-heat-conducing planet.
So consider an extreme: a tide-locked planet in that same orbit. if it begins to rotate, does the average temperature increase or decrease? I don’t actually know, and can’t think of a reason it would change. The sunrise will now have access to surface area not yet at the inferno equilibrium, so it’ll be absorbing heat. But the sunset is doing the opposite: radiating all that heat that’s no longer being replenished.
Does it matter if the planetary material absorbs and radiates heat at different rates? Is that even possible?
Edit: extend this idea. Take a tide-locked planet at equilibrium. Magically flip it so the cold side is toward the sun and the hot side toward space. Give it time to come back to equilibrium. Average surface temperature at t and t is the same, right? During this period, does the average rise and then fall back, or does it drop and then rise back?
I think I recall that hot things radiate faster than cold thing absorb heat, which implies it’ll overall cool and then come back up after the now-cold side reaches minimum temperature before the now-hot side reaches maximum. which implies that spinning faster makes the average cooler.
I guess there may be some energy effects of the rotation itself, generating a bit of heat from internal differentials.
I don’t know the answer but let me write down some thoughts.
If I’m a point on the surface of a planet then I spend the day slowly getting warmer as I absorb sunlight and the night slowly getting cooler as I radiate my heat into space. If the days are longer then the temperature I reach during the day is higher, which means that when the night begins I radiate heat more rapidly into space. But conversely by the end of the night I’ve cooled down to a lower temperature so I absorb more heat from the sun at the star of the day.
In fact none of this should matter. Can’t we say that the space at that distance from the sun has a particular temperature, and both planets are in thermal equilibrium with that space, so they have the same temperature? That’s not such a convincing argument, since the space near a star is not a typical thermodynamic system.
What about atmospheres? They should help warm up the (solid) surface of the planet via the greenhouse effect. I guess the faster spinning planet has a thinner atmosphere, because of centrifugal force, so maybe it’s colder.
I think the rate of cooling depends on temperature much more than the rate of warming up, because T_sun—T_planet >> T_planet—T_space. So a faster rotating planet should be warmer.
And this explains a lot. The so called Faint Sun Paradox is then not a problem at all.
Early Earth was much warmer despite of a fainter Sun mostly thanks to its faster rotation. Partly also because of a smaller distance to the Sun back then, but mostly because of its faster rotation.
A new problem:
https://protokol2020.wordpress.com/2017/05/22/and-another-physics-problem/
Given perfect conduction (uniform surface temperature, bright side and dark side have the same temperature at all times), https://en.wikipedia.org/wiki/Black-body_radiation#Temperature_relation_between_a_planet_and_its_star applies : temperature does not depend on rotation speed. Then T = T_sun sqrt(R_sun/(2D)) ; it is the temperature T that balance incoming radiation P_inc = pi (R_planet^2) (R_sun^2) (T_sun^4)/(D^2) and emitted radiation P_em = 4 pi (R_planet^2) * T^4
Let’s suppose no conduction at all. The bright side and the dark side does not exchange heat at all. Let’s take two limiting cases : tide-locked planet, and an “infinitely fast” fliping planet.
In the first case, the dark side of the planet is at absolute 0. The bright side of the planet receives the same incoming radiation but emit half its radiation (halved surface) -- change 4 pi to 2 pi in P_em. Its temperature is T_bright = 2^(1/4) T. Average temperature of the planet is (0+T_bright)/2=2^(-3/4) T
In the second case, each side gets half the incoming power from the sun and radiates half the energy (surface halved). Average surface temperature of the planet is the same that the average surface temperature of any side, which is the same temperature that the perfectly conducting planet, T (the .5 from halved incoming power and .5 from halved outgoing radiation cancel each other).
So : rotation raises temperature of a non-perfectly-heat-conducing planet, bringing its surface temperature closer to the perfectly-heat-conducting planet surface temperature.
This is clearly wrong. The bright side hasn’t an uniform temperature T.
You’re right, the 2^(-3/4) (and the 2^1/4) is probably quantitatively wrong (unless each side is perfectly heat-conducing but both are isolated from each other. Or if the planet is a coin facing the sun. You know, spherical cows in a vacuum…). But I don’t think that changes the qualitative conclusion, which hold as long as the bright side is hotter but not twice as hotter than the perfectly-heat-conducing planet.
Oh, yes, it does change, of course.
The result, that a faster rotating planet is warmer, is against the Al Gore’s theology about Climate Change, formerly known as the Global Warming.
As Scott Alexander said—the scientific community consensus was wrong, I was right.
I am not sure about him, I know I am right here and the “science community” as it is self-proclaimed—is wrong.
A faster rotating planet is warmer.
So consider an extreme: a tide-locked planet in that same orbit. if it begins to rotate, does the average temperature increase or decrease? I don’t actually know, and can’t think of a reason it would change. The sunrise will now have access to surface area not yet at the inferno equilibrium, so it’ll be absorbing heat. But the sunset is doing the opposite: radiating all that heat that’s no longer being replenished.
Does it matter if the planetary material absorbs and radiates heat at different rates? Is that even possible?
Edit: extend this idea. Take a tide-locked planet at equilibrium. Magically flip it so the cold side is toward the sun and the hot side toward space. Give it time to come back to equilibrium. Average surface temperature at t and t is the same, right? During this period, does the average rise and then fall back, or does it drop and then rise back?
I think I recall that hot things radiate faster than cold thing absorb heat, which implies it’ll overall cool and then come back up after the now-cold side reaches minimum temperature before the now-hot side reaches maximum. which implies that spinning faster makes the average cooler.
I guess there may be some energy effects of the rotation itself, generating a bit of heat from internal differentials.
Heat is actually being generated by the tidal effects here on Earth. The mechanical energy gradually becomes heat.
But those are miniscule amounts, irrelevant for our problem as stated.
I don’t know the answer but let me write down some thoughts.
If I’m a point on the surface of a planet then I spend the day slowly getting warmer as I absorb sunlight and the night slowly getting cooler as I radiate my heat into space. If the days are longer then the temperature I reach during the day is higher, which means that when the night begins I radiate heat more rapidly into space. But conversely by the end of the night I’ve cooled down to a lower temperature so I absorb more heat from the sun at the star of the day.
In fact none of this should matter. Can’t we say that the space at that distance from the sun has a particular temperature, and both planets are in thermal equilibrium with that space, so they have the same temperature? That’s not such a convincing argument, since the space near a star is not a typical thermodynamic system.
What about atmospheres? They should help warm up the (solid) surface of the planet via the greenhouse effect. I guess the faster spinning planet has a thinner atmosphere, because of centrifugal force, so maybe it’s colder.
I think the rate of cooling depends on temperature much more than the rate of warming up, because T_sun—T_planet >> T_planet—T_space. So a faster rotating planet should be warmer.
Cool (heh). Good thinking!
Of course.
And this explains a lot. The so called Faint Sun Paradox is then not a problem at all.
Early Earth was much warmer despite of a fainter Sun mostly thanks to its faster rotation. Partly also because of a smaller distance to the Sun back then, but mostly because of its faster rotation.
It’s quite elementary if you think about it.
For now, put the atmosphere aside.
The thing is, that the radiation power is proportional to the T^4. And that the peak temperature matters a lot. Don’t you agree?