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A recent question If the Sun were bigger but colder, Earth would be hotter or colder? asked - if the Sun got bigger and cooler, would the Earth heat up or cool down. I think the answer to that is mainly that it depends on the final luminosity.

However, what I want to know here (hypothetically), is if the Sun got larger and it's effective temperature decreased such that it's luminosity was unchanged; how would that affect the equilibrium temperature of the Earth? I suspect the answer may involve the wavelength dependence of the albedo, emissivity and atmospheric absorption of the Earth.

Another, less hypothetical, way of asking this is, if you put an Earth-like planet at different distances from stars with a variety of temperatures, such that the total flux incident at the top of the atmosphere was identical, how would the temperatures of those planets compare?

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    $\begingroup$ If you increase $R_\odot$ by a factor of 215, Earth will be on the Sun's surface. Its area then increases by a factor of 46,200. To maintain its luminosity, $T^4$ must decrease by the same factor, so it will be 120 ºC, in which case Earth should get hotter. $\endgroup$ – pela Dec 20 '15 at 10:19
  • $\begingroup$ @Pela Nice one. That's not really what I was thinking of. More the difference between the Earth-Sun, and the Earth being much closer to an M-dwarf for instance. So factors of a few. $\endgroup$ – Rob Jeffries Dec 20 '15 at 12:03
  • $\begingroup$ Yes okay, I see. $\endgroup$ – pela Dec 20 '15 at 15:03
  • $\begingroup$ I assume you are asking about something like $$ T_{p}^{4} = \frac{ r_{s}^{2} T_{s}^{4} }{ 4 \left( 1 - \alpha \right) a_{p}^{2} } = \frac{ \left( 1 - \alpha \right) \ E_{abs \ at \ p} }{ \sigma \ A_{emit, p} } $$ (which I know you already know)? I would tend to agree that the primary difference would arise due to the albedo term, $\alpha$, and possibly an emissivity term (not included above), both dependent upon wavelength/frequency. $\endgroup$ – honeste_vivere Aug 18 '16 at 12:47
  • $\begingroup$ You defiantly should ask more questions? $\endgroup$ – Muze the good Troll. Jul 7 '18 at 16:16
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To a first approximation if the Sun got bigger (by a relatively small factor, say a few times) but maintained its present luminosity, as the Earth would still be intercepting the same total energy per unit time the temperature would stay the same. Constant luminosity gives a constant intensity at the Earth's distance from the Sun (intensity is the energy crossing unit area in unit time) which would mean that the total solar energy intercepted by the Earth would remain the same.

However in this scenario the colour of the sun would change and we would get secondary effects due to the reflectivity of the Earth being frequency dependent so the fraction of the incident energy reflected rather than absorbed would change, which would result in a change in temperature. Which direction this would go in is difficult to say as an increase in temperature should result in more cloud which would increase reflectivity ...

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    $\begingroup$ Conrad, I knew all this. I want to know which way the temperature changes and why. $\endgroup$ – Rob Jeffries Dec 20 '15 at 12:02
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    $\begingroup$ Then its not a question about astronomy but about climatology. $\endgroup$ – Conrad Turner Dec 20 '15 at 15:39
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    $\begingroup$ Climatology may be involved. I suspect cloud formation is not important. The question of the location of a habitable zone around stars of different spectral type is most definitely on topic. $\endgroup$ – Rob Jeffries Dec 20 '15 at 21:21
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The key issue is the opacity of the atmosphere, because I presume the question is about the temperature at the solid surface of the Earth. The atmospheric opacity can be seen from https://physics.stackexchange.com/questions/135260/can-someone-explain-to-me-the-concept-of-atmosphere-opacity, where you can see that the "rainbow" of maximum heat flux from the Sun happens to hit a kind of hole in atmospheric opacity. That has a significant warming effect on the Earth, and is exacerbated by the Greenhouse effect. If sunlight was further into the infrared, the graph shows that much more of it would be intercepted in the atmosphere. That would make the surface significantly colder, though certainly not a factor of 2 colder.

No doubt the question is of more than passing interest, because M dwarfs are the most numerous main-sequence stars and are therefore interesting for life. To have life near an M dwarf, the planet would need to be closer than Earth is to the Sun, but the effect of moving the planet closer and shrinking and cooling the star would be similar to leaving Earth where it is and making the star cooler and larger. So the nature of atmospheric opacity for wet atmospheres must be of great significance for understanding the prospects for life around M dwarfs.

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