<p>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.</p>

<p>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 $T_E \propto L^{1/4}$.</p>

<p>So the answer to your question depends on by how much you increase the radius compared with the decrease in temperature.</p>

<p>There will be second order effects that do depend on the spectrum of radiation from the Sun (and therefore its temperature) compared with the wavelength dependence of the albedo and emissivity of the Earth. So I will post a better question...</p>

CC BY-SA 3.0
2015-12-20T22:55:28.510Z
@CarlosVazquezMonzon I will add something
ProfRob
CC BY-SA 3.0
2015-12-20T22:56:54.580Z
"The equilibrium temperature of the Earth scales roughly as L^1/4..." How do you know that?? Are you saying that the equilibrium temperature is the effective temperature of the Earth?
Carlos Vázquez Monzón
CC BY-SA 3.0
2015-12-20T08:17:06.087Z
2015-12-20T23:08:51.217Z
2015-12-20T23:08:51.217Z
ProfRob
<p>That is the question. I know the concepts of luminosity <span class="math-container">$(L=4\pi R^2F)$</span> and the flow <span class="math-container">$F=\sigma· T^4$</span>, with <span class="math-container">$T$</span> the temperature in its surface. But how I use that to know if the Earth would get warmer or not if <span class="math-container">$R$</span> increases and <span class="math-container">$T$</span> decreases?</p>

CC BY-SA 3.0
2015-12-19T15:09:33.560Z
This depends in part on the change in $R$ and $T$.
HDE 226868
CC BY-SA 3.0
2015-12-19T15:41:27.470Z
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
CC BY-SA 3.0
2015-12-19T16:02:58.637Z
@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
CC BY-SA 3.0
2015-12-19T20:22:25.430Z
Yes, I know that, but more luminosity implies more heat? That's the question
Carlos Vázquez Monzón
CC BY-SA 3.0
2015-12-19T23:10:29.977Z
@CarlosVázquezMonzón Yep, via the formula for [effective temperature](https://en.wikipedia.org/wiki/Effective_temperature).
HDE 226868
<astrophysics><luminosity><spectrometry>
If the Sun were bigger but colder, Earth would be hotter or colder?
Carlos Vázquez Monzón
# If the Sun were bigger but colder, Earth would be hotter or colder?

(Asked by: username_0 on 2015-12-19)

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?

**Tags:** astrophysics, luminosity, spectrometry
> **Comments:**
> **username_1**: This depends in part on the change in $R$ and $T$.  
> **username_0**: Yeah but  the question is: the bigger the luminosity, the most the Sun warms the Earth? Or it depends only on the flow?  
> **username_2**: @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.  
> **username_0**: Yes, I know that, but more luminosity implies more heat? That's the question  
> **username_1**: @CarlosVázquezMonzón Yep, via the formula for [effective temperature](https://en.wikipedia.org/wiki/Effective_temperature).  

---

## 1 answers

### Answer by username_3 on 2015-12-20. Score: 3 | Accepted answer

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 $T\_E \propto L^{1/4}$.

So the answer to your question depends on by how much you increase the radius compared with the decrease in temperature.

There will be second order effects that do depend on the spectrum of radiation from the Sun (and therefore its temperature) compared with the wavelength dependence of the albedo and emissivity of the Earth. So I will post a better question...

> **Comments:**
> **username_3**: @CarlosVazquezMonzon I will add something  
> **username_0**: "The equilibrium temperature of the Earth scales roughly as L^1/4..." How do you know that?? Are you saying that the equilibrium temperature is the effective temperature of the Earth?  

---

