# If the Sun got larger, but maintained its luminosity, would the Earth get hotter or colder?

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?

• 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.
– pela
Dec 20, 2015 at 10:19
• @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. Dec 20, 2015 at 12:03
• Yes okay, I see.
– pela
Dec 20, 2015 at 15:03
• 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. Aug 18, 2016 at 12:47
• You defiantly should ask more questions?
– Muze
Jul 7, 2018 at 16:16