<p>Answers will be different because they must be tied to a model of solar evolution, and all models are a bit different.</p>

<p>So to answer your question we have to select a model. A pretty standard and trusted one, is the one used in the paper <a href="https://academic.oup.com/mnras/article/298/2/525/1056616" rel="nofollow noreferrer">Stellar evolution models for Z = 0.0001 to 0.03</a>. Where Z stands for the <a href="https://en.wikipedia.org/wiki/Metallicity" rel="nofollow noreferrer">metalicity</a> of the star, that for the Sun they indicate would be 0.0188.</p>

<p>In figure 2, they show the trajectory of a star like the sun in the <a href="https://en.wikipedia.org/wiki/Hertzsprung%E2%80%93Russell_diagram" rel="nofollow noreferrer">H-R diagram</a> (look for the line labeled "1.0").</p>

<p><a href="https://i.sstatic.net/1i9Sk.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/1i9Sk.png" alt="enter image description here"></a></p>

<p>The data output of this model is available at the <a href="http://vizier.u-strasbg.fr/viz-bin/VizieR-3?-source=J/MNRAS/298/525/summary" rel="nofollow noreferrer">VizieR catalog</a>. I just downloaded the data for starts of one solar mass and Z=0.02 (like the Sun), and computed the <a href="https://en.wikipedia.org/wiki/Effective_temperature" rel="nofollow noreferrer">effective temperature</a> of Earth using the formula in the linked question</p>

<p><span class="math-container">$\Large \frac{T}{T_0}=\left(\frac{L}{L_0}\right)^{\frac{1}{4}}$</span></p>

<p>Using a current effective temperature <span class="math-container">$T_0$</span> of -19°C (you will find values between -18 and -21°C), we get that the plot of Luminosity and temperature versus time looks like this</p>

<p><a href="https://i.sstatic.net/2pS6W.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/2pS6W.png" alt="enter image description here"></a></p>

<p>And given that you are interested in specific values here is some tabulated data including also the solar radius. Note that most of these points are linear interpolations of the original data, that had only six points in this age range.</p>

<pre><code>Age         Radius      Lumin.  Temperature
[Billion    [solar      [Solar  [°C]
 years]     radius]     Lumin.]
0.00        0.89        0.7     -41.4
0.25        0.90        0.7     -40.2
0.50        0.91        0.7     -39.0
0.75        0.91        0.7     -37.8
1.00        0.92        0.8     -36.6
1.25        0.93        0.8     -35.4
1.50        0.94        0.8     -34.2
1.75        0.95        0.8     -32.9
2.00        0.95        0.8     -31.7
2.25        0.96        0.8     -30.4
2.50        0.97        0.8     -29.2
2.75        0.98        0.9     -27.9
3.00        0.99        0.9     -26.7
3.25        1.00        0.9     -25.4
3.50        1.00        0.9     -24.1
3.75        1.01        0.9     -22.8
4.00        1.02        1.0     -21.5
4.25        1.03        1.0     -20.2
4.50        1.04        1.0     -18.9
4.75        1.05        1.0     -17.6
5.00        1.06        1.0     -16.3
5.25        1.07        1.1     -15.0
5.50        1.08        1.1     -13.7
5.75        1.08        1.1     -12.3
6.00        1.09        1.1     -11.0
6.25        1.10        1.2     -9.6
6.50        1.11        1.2     -8.3
6.75        1.12        1.2     -6.9
7.00        1.13        1.2     -5.5
7.25        1.14        1.3     -4.1
7.50        1.15        1.3     -2.8
7.75        1.16        1.3     -1.4
8.00        1.17        1.3     0.0
8.25        1.18        1.4     1.4
8.50        1.19        1.4     2.9
8.75        1.20        1.4     4.3
9.00        1.21        1.4     5.7
9.25        1.22        1.5     7.2
9.50        1.26        1.5     9.2
9.75        1.32        1.6     11.9
10.00       1.38        1.6     14.7
10.25       1.44        1.7     17.5
10.50       1.51        1.8     20.3
10.75       1.58        1.8     23.1
11.00       1.66        1.9     25.9
11.25       1.74        2.0     28.8
11.50       1.82        2.1     31.7
11.75       1.90        2.2     34.7
12.00       2.40        3.1     65.0
12.25       4.08        7.8     151.9
12.50       6.94        19.6    261.2
12.75       11.68       57.8    427.2
</code></pre>

<p>Note that you can only know the <a href="https://en.wikipedia.org/wiki/Effective_temperature" rel="nofollow noreferrer">effective temperature</a>, the actual temperature will depend on the strength of the greenhouse effect, and modeling that is a whole new problem with huge uncertainties.</p>

<p>According to this model, <strong>the terminal age main sequence of the Sun would be 9.38 Billion years, and according to the data above, the effective temperature then would be 8.2 °C (281 K), that's 27.2 °C hotter than today.</strong></p>

CC BY-SA 4.0
2019-03-26T01:56:42.437Z
Thank you for the detailed answer. Can you suggest any recent papers which have modeled the actual temperature? or any books which discuss compare various models for actual temperatures.
sidharth chhabra
CC BY-SA 4.0
2019-03-26T02:01:55.490Z
I'm doubt anyone have dare to model greenhouse effect in multi-billion years timescale. Or at least I'm not aware of such research. I don't know why exactly do you need this number, but I think you are better off, figuring our at what temperature using normal green house effects, a runaway greenhouse effect would be inevitable.
Camilo Rada
CC BY-SA 4.0
2019-03-26T02:03:35.613Z
Or as a mental exercise, it would be possible to calculate what would be the temperature if we have the same greenhouse effect than today but the solar luminosity of four billion years in the future.
Camilo Rada
CC BY-SA 4.0
2019-03-26T02:10:23.767Z
I am trying to understand the future of Earth when the Sun becomes a red giant. Do you have any books or latest papers which discuss the future 5 billion years from now?
sidharth chhabra
CC BY-SA 4.0
2019-03-26T02:11:34.023Z
This article is nice https://academic.oup.com/astrogeo/article/42/6/6.26/293074
Camilo Rada
CC BY-SA 4.0
2019-03-26T08:18:37.967Z
I'm a bit confused about the term Billion. Do you mean 1 Billion=  1.000.000.000.000 or 1,000,000,000?
undefined
CC BY-SA 4.0
2019-03-26T14:08:47.777Z
@undefined 1 Billion = $10^9$=1,000,000,000
Camilo Rada
CC BY-SA 4.0
2019-03-30T18:33:05.370Z
@sidharthchhabra Modelling the temperature of Earth including greenhouse effect 9 billion years in the future is akin to predict the weather for March 23 2045 in Paris. You could run a model and get an answer but is such a chaotic system that your chances to get the right answer are nil.
If you look at the last billion years the temperature have changed wildly, and there is no reason to think it won't keep changing wildly in the future. It is a system as chaotic as the weather as such long timescales.
Camilo Rada
CC BY-SA 4.0
2019-03-26T01:44:35.283Z
2019-03-30T18:27:22.643Z
2019-03-30T18:27:22.643Z
Camilo Rada
<p>We know that presently Sun is 4.5 billion years into its main sequence. It has another 5 billion years before it enters the Red Giant phase. We also know that Sun's luminosity increases by 10% every billion years during the main sequence. I am interested in finding the temperature rise as we approach the end of main sequence. I got two different values for temperature on Earth. </p>

<p><a href="https://en.wikipedia.org/wiki/Future_of_Earth#Loss_of_oceans" rel="nofollow noreferrer">Wikipedia</a> entry says that temperature on Earth would be 422 k in 2.8 billion years. However, if we use the formula for effective temperature as discussed in this answer 
<a href="https://earthscience.stackexchange.com/a/4274/15299">https://earthscience.stackexchange.com/a/4274/15299</a> and L = 1.8 then, the temperature on Earth would be 330K. Also in <a href="https://www.springer.com/us/book/9783319649603" rel="nofollow noreferrer">this book</a>, the author does the same calculations on page 255. </p>

CC BY-SA 4.0
2019-03-26T01:08:52.707Z
@arkaia I have shared another answer in the question which is similar and was answered by experts here. I request you to let my question stay here as there are many intelligent folks here who may resolve this question quickly.
sidharth chhabra
CC BY-SA 4.0
2019-03-26T02:14:58.597Z
@arkaia It is true that this question has a lot of astronomy, but heat balance and effective temperature is Earth Science-ish, and if somebody masters an answer including greenhouse effect, that would definitely be Earth Sciences.
Camilo Rada
CC BY-SA 4.0
2019-03-26T03:43:30.987Z
I understand your point. Close vote removed
arkaia
<geophysics><hypothetical><astronomy><energy-balance>
What will be the temperature on Earth when Sun finishes its main sequence?
sidharth chhabra
# What will be the temperature on Earth when Sun finishes its main sequence?

(Asked by: username_0 on 2019-03-25)

We know that presently Sun is 4.5 billion years into its main sequence. It has another 5 billion years before it enters the Red Giant phase. We also know that Sun's luminosity increases by 10% every billion years during the main sequence. I am interested in finding the temperature rise as we approach the end of main sequence. I got two different values for temperature on Earth.

[Wikipedia](https://en.wikipedia.org/wiki/Future_of_Earth#Loss_of_oceans) entry says that temperature on Earth would be 422 k in 2.8 billion years. However, if we use the formula for effective temperature as discussed in this answer
<https://earthscience.stackexchange.com/a/4274/15299> and L = 1.8 then, the temperature on Earth would be 330K. Also in [this book](https://www.springer.com/us/book/9783319649603), the author does the same calculations on page 255.

**Tags:** geophysics, hypothetical, astronomy, energy-balance

> **Comments:**
> **username_0**: @username_2 I have shared another answer in the question which is similar and was answered by experts here. I request you to let my question stay here as there are many intelligent folks here who may resolve this question quickly.  
> **username_1**: @username_2 It is true that this question has a lot of astronomy, but heat balance and effective temperature is Earth Science-ish, and if somebody masters an answer including greenhouse effect, that would definitely be Earth Sciences.  
> **username_2**: I understand your point. Close vote removed  

---

## 1 answers

### Answer by username_1 on 2019-03-26. Score: 4

Answers will be different because they must be tied to a model of solar evolution, and all models are a bit different.

So to answer your question we have to select a model. A pretty standard and trusted one, is the one used in the paper [Stellar evolution models for Z = 0.0001 to 0.03](https://academic.oup.com/mnras/article/298/2/525/1056616). Where Z stands for the [metalicity](https://en.wikipedia.org/wiki/Metallicity) of the star, that for the Sun they indicate would be 0.0188.

In figure 2, they show the trajectory of a star like the sun in the [H-R diagram](https://en.wikipedia.org/wiki/Hertzsprung%E2%80%93Russell_diagram) (look for the line labeled "1.0").

[![enter image description here](https://i.sstatic.net/1i9Sk.png)](https://i.sstatic.net/1i9Sk.png)

The data output of this model is available at the [VizieR catalog](http://vizier.u-strasbg.fr/viz-bin/VizieR-3?-source=J/MNRAS/298/525/summary). I just downloaded the data for starts of one solar mass and Z=0.02 (like the Sun), and computed the [effective temperature](https://en.wikipedia.org/wiki/Effective_temperature) of Earth using the formula in the linked question

$\Large \frac{T}{T\_0}=\left(\frac{L}{L\_0}\right)^{\frac{1}{4}}$

Using a current effective temperature $T\_0$ of -19°C (you will find values between -18 and -21°C), we get that the plot of Luminosity and temperature versus time looks like this

[![enter image description here](https://i.sstatic.net/2pS6W.png)](https://i.sstatic.net/2pS6W.png)

And given that you are interested in specific values here is some tabulated data including also the solar radius. Note that most of these points are linear interpolations of the original data, that had only six points in this age range.

```
Age         Radius      Lumin.  Temperature
[Billion    [solar      [Solar  [°C]
 years]     radius]     Lumin.]
0.00        0.89        0.7     -41.4
0.25        0.90        0.7     -40.2
0.50        0.91        0.7     -39.0
0.75        0.91        0.7     -37.8
1.00        0.92        0.8     -36.6
1.25        0.93        0.8     -35.4
1.50        0.94        0.8     -34.2
1.75        0.95        0.8     -32.9
2.00        0.95        0.8     -31.7
2.25        0.96        0.8     -30.4
2.50        0.97        0.8     -29.2
2.75        0.98        0.9     -27.9
3.00        0.99        0.9     -26.7
3.25        1.00        0.9     -25.4
3.50        1.00        0.9     -24.1
3.75        1.01        0.9     -22.8
4.00        1.02        1.0     -21.5
4.25        1.03        1.0     -20.2
4.50        1.04        1.0     -18.9
4.75        1.05        1.0     -17.6
5.00        1.06        1.0     -16.3
5.25        1.07        1.1     -15.0
5.50        1.08        1.1     -13.7
5.75        1.08        1.1     -12.3
6.00        1.09        1.1     -11.0
6.25        1.10        1.2     -9.6
6.50        1.11        1.2     -8.3
6.75        1.12        1.2     -6.9
7.00        1.13        1.2     -5.5
7.25        1.14        1.3     -4.1
7.50        1.15        1.3     -2.8
7.75        1.16        1.3     -1.4
8.00        1.17        1.3     0.0
8.25        1.18        1.4     1.4
8.50        1.19        1.4     2.9
8.75        1.20        1.4     4.3
9.00        1.21        1.4     5.7
9.25        1.22        1.5     7.2
9.50        1.26        1.5     9.2
9.75        1.32        1.6     11.9
10.00       1.38        1.6     14.7
10.25       1.44        1.7     17.5
10.50       1.51        1.8     20.3
10.75       1.58        1.8     23.1
11.00       1.66        1.9     25.9
11.25       1.74        2.0     28.8
11.50       1.82        2.1     31.7
11.75       1.90        2.2     34.7
12.00       2.40        3.1     65.0
12.25       4.08        7.8     151.9
12.50       6.94        19.6    261.2
12.75       11.68       57.8    427.2

```

Note that you can only know the [effective temperature](https://en.wikipedia.org/wiki/Effective_temperature), the actual temperature will depend on the strength of the greenhouse effect, and modeling that is a whole new problem with huge uncertainties.

According to this model, **the terminal age main sequence of the Sun would be 9.38 Billion years, and according to the data above, the effective temperature then would be 8.2 °C (281 K), that's 27.2 °C hotter than today.**

> **Comments:**
> **username_0**: Thank you for the detailed answer. Can you suggest any recent papers which have modeled the actual temperature? or any books which discuss compare various models for actual temperatures.  
> **username_1**: I'm doubt anyone have dare to model greenhouse effect in multi-billion years timescale. Or at least I'm not aware of such research. I don't know why exactly do you need this number, but I think you are better off, figuring our at what temperature using normal green house effects, a runaway greenhouse effect would be inevitable.  
> **username_1**: Or as a mental exercise, it would be possible to calculate what would be the temperature if we have the same greenhouse effect than today but the solar luminosity of four billion years in the future.  
> **username_0**: I am trying to understand the future of Earth when the Sun becomes a red giant. Do you have any books or latest papers which discuss the future 5 billion years from now?  
> **username_1**: This article is nice https://academic.oup.com/astrogeo/article/42/6/6.26/293074  
> **username_3**: I'm a bit confused about the term Billion. Do you mean 1 Billion=  1.000.000.000.000 or 1,000,000,000?  
> **username_1**: @username_3 1 Billion = $10^9$=1,000,000,000  
> **username_1**: @username_0 Modelling the temperature of Earth including greenhouse effect 9 billion years in the future is akin to predict the weather for March 23 2045 in Paris. You could run a model and get an answer but is such a chaotic system that your chances to get the right answer are nil. If you look at the last billion years the temperature have changed wildly, and there is no reason to think it won't keep changing wildly in the future. It is a system as chaotic as the weather as such long timescales.  

---

