# If the Sun were bigger but colder, Earth would be hotter or colder?

That is the question. I know the concepts of luminosity $$(L=4\pi R^2F)$$ and the flow $$F=\sigma· T^4$$, with $$T$$ the temperature in its surface. But how I use that to know if the Earth would get warmer or not if $$R$$ increases and $$T$$ decreases?

• This depends in part on the change in $R$ and $T$. – HDE 226868 Dec 19 '15 at 15:09
• Yeah but the question is: the bigger the luminosity, the most the Sun warms the Earth? Or it depends only on the flow? – Carlos Vázquez Monzón Dec 19 '15 at 15:41
• @Carlos: As HDE said it depends on the precise values. Without pluggin in numbers you cannot know, as expansion and cooling have two competing effects on the luminosity. – AtmosphericPrisonEscape Dec 19 '15 at 16:02
• Yes, I know that, but more luminosity implies more heat? That's the question – Carlos Vázquez Monzón Dec 19 '15 at 20:22
• @CarlosVázquezMonzón Yep, via the formula for effective temperature. – HDE 226868 Dec 19 '15 at 23:10

The equilibrium temperature of the Earth, $T_E$, scales roughly as $L^{1/4}$, which is proportional to $R^{1/2} T$, where $L$, $R$ and $T$ are the solar luminosity, radius and temperature.
The actual approximate relationship is derived by equating the power received by the Earth, which is proportional to the solar luminosity $L$, with the power radiated by the Earth, which is proportional to $T_E^4$ for a blackbody. Hence $T_E \propto L^{1/4}$.