<p>Recently, I started writing a program to generate star systems, and I need a formula to find the approximate surface temperature of a planet. I know of several formulas for this, for example this one based on the formula for equilibrium temperature I found from a site called <a href="http://burro.case.edu/Academics/Astr221/SolarSys/equiltemp.html" rel="nofollow noreferrer">burro.case.edu</a>:</p>
<p><span class="math-container">$$T_{P}=(T_{\odot})(({1-a})^{1/4})\left( \sqrt{\frac{R_{\odot}}{2D}}\right)$$</span>
Where <span class="math-container">$T_{\odot}$</span> denotes the parent star's temperature in Kelvin, <span class="math-container">$a$</span> denotes the dimensionless bond albedo, <span class="math-container">${R_{\odot}}$</span> denotes the parent star's radius in AU (not kilometers/solar radii), <span class="math-container">$D$</span> denotes the semi-major axis in AU, and <span class="math-container">$T_{P}$</span> denotes the resulting computed planet temperate in Kelvin.</p>
<p>For example, if we plug  in the relevant numbers to Mars (<span class="math-container">$T_{\odot}=5778,a=0.25,R_{\odot}=0.0046547454,D=1.524$</span>), we get an estimated temperature of Mars as 210.127 K, which is nearly spot-on with Mars's real temperature of 210 K. But if I plug in the formula for Venus, it gives a wildly off answer of about 260 K. In fact, this is briefly mentioned on the linked page:</p>
<blockquote>
<p>Let's try this for Venus. Putting in the numbers (a=0.6, Tsun=5770K, Rsun=7x105 km, D=0.72 AU) we get the equilibrium temperature of Venus = 260 K.
The surface temperature of Venus = 740 K. Whoa! Where'd we mess up??
For that matter, the equilibrium temperature of the Earth is 255 K (or ~ -1 F). Something's wrong!</p>
</blockquote>
<p>Obviously this happens becausing trying to estimate a true temperature from the equilibrium temperature results in ignorance of the atmospheric pressure. Venus's atmosphere is about 93 atm, resulting in a runaway greenhouse effect and heating the temperature. On a smaller scale, the same thing occurs for Earth as it has a pressure of 1 atm. The temperature calculation for Mars worked well because it has a pressure of only 0.01 atm making the difference minimal.</p>
<p>So I started searching on the web for other sources which have bias terms accounting for the atmosphere, but I still can't find anything. All I currently have  is a very rough approximation based on this <a href="https://youtu.be/y3Kb_ik5f-I?t=21" rel="nofollow noreferrer">Youtube video</a>:</p>
<p><span class="math-container">$T_{P} = 255(\frac{L^{1/4}}{\sqrt{D}})+50\sqrt{P}-50a$</span></p>
<p>Where <span class="math-container">$L$</span> denotes the star's luminosity with respect to the Sun, <span class="math-container">$P$</span> denotes the planet's atmospheric pressure with respect to Earth, and all the other variables are the same as they were in the last approximation. But this is still far from perfect, Venus's temperature has a ~15 K deviation from the real temperature. So, is there any more accurate formula for the atmospheric temperature of a planet?</p>

CC BY-SA 4.0
2020-08-25T03:39:16.550Z
I would search scientific journals if you can or do a google scholarly search on the subject. Unless the Youtube video you saw had a reference I wouldn't trust it.
Natsfan
CC BY-SA 4.0
2020-08-25T04:01:10.187Z
The former (and most) equilibrium temperature do not account for any atmosphere and its greenhouse effect. The temperatures sound right for bodies without atmosphere. For accurate surface temperatures you need to consider those - yet that means to complicate matter by orders of magnitude as you see in the difference of the atmospheric effect on earth and venus. You need elaborate models for radiative transfer and atmospheric chemistry to get more precision
planetmaker
CC BY-SA 4.0
2020-08-25T04:04:25.940Z
That would basically mean modelling planet by planet because of its atmosphere (composition especially). In my modest opinion one cannot attain a simple general formula. I also think that 15 K is not that bad...
Alchimista
CC BY-SA 4.0
2020-08-25T06:24:05.833Z
You are not going to find a simple formula that is anywhere close to correct. The equation from the YouTube video is nonsense.
David Hammen
CC BY-SA 4.0
2020-09-05T03:43:49.823Z
How accurate does it need to be? Can you just use the effective temperature from the formula you gave then guess a greenhouse multiplier based on atmospheric conditions. You can use planets and moons of our solar system to have sample greenhouse numbers.
Jack R. Woods
CC BY-SA 4.0
2021-03-01T04:11:51.697Z
Any further thoughts on this? There are several comments but so far no responses.
uhoh
<star><planet><planetary-atmosphere><temperature>
How to find the temperature of a planet accounting for the atmosphere?
Nirvana
# How to find the temperature of a planet accounting for the atmosphere?

(Asked by: username_0 on 2020-08-25)

Recently, I started writing a program to generate star systems, and I need a formula to find the approximate surface temperature of a planet. I know of several formulas for this, for example this one based on the formula for equilibrium temperature I found from a site called [burro.case.edu](http://burro.case.edu/Academics/Astr221/SolarSys/equiltemp.html):

$$T\_{P}=(T\_{\odot})(({1-a})^{1/4})\left( \sqrt{\frac{R\_{\odot}}{2D}}\right)$$
Where $T\_{\odot}$ denotes the parent star's temperature in Kelvin, $a$ denotes the dimensionless bond albedo, ${R\_{\odot}}$ denotes the parent star's radius in AU (not kilometers/solar radii), $D$ denotes the semi-major axis in AU, and $T\_{P}$ denotes the resulting computed planet temperate in Kelvin.

For example, if we plug in the relevant numbers to Mars ($T\_{\odot}=5778,a=0.25,R\_{\odot}=0.0046547454,D=1.524$), we get an estimated temperature of Mars as 210.127 K, which is nearly spot-on with Mars's real temperature of 210 K. But if I plug in the formula for Venus, it gives a wildly off answer of about 260 K. In fact, this is briefly mentioned on the linked page:

> Let's try this for Venus. Putting in the numbers (a=0.6, Tsun=5770K, Rsun=7x105 km, D=0.72 AU) we get the equilibrium temperature of Venus = 260 K.
> The surface temperature of Venus = 740 K. Whoa! Where'd we mess up??
> For that matter, the equilibrium temperature of the Earth is 255 K (or ~ -1 F). Something's wrong!

Obviously this happens becausing trying to estimate a true temperature from the equilibrium temperature results in ignorance of the atmospheric pressure. Venus's atmosphere is about 93 atm, resulting in a runaway greenhouse effect and heating the temperature. On a smaller scale, the same thing occurs for Earth as it has a pressure of 1 atm. The temperature calculation for Mars worked well because it has a pressure of only 0.01 atm making the difference minimal.

So I started searching on the web for other sources which have bias terms accounting for the atmosphere, but I still can't find anything. All I currently have is a very rough approximation based on this [Youtube video](https://youtu.be/y3Kb_ik5f-I?t=21):

$T\_{P} = 255(\frac{L^{1/4}}{\sqrt{D}})+50\sqrt{P}-50a$

Where $L$ denotes the star's luminosity with respect to the Sun, $P$ denotes the planet's atmospheric pressure with respect to Earth, and all the other variables are the same as they were in the last approximation. But this is still far from perfect, Venus's temperature has a ~15 K deviation from the real temperature. So, is there any more accurate formula for the atmospheric temperature of a planet?

**Tags:** star, planet, planetary-atmosphere, temperature
> **Comments:**
> **username_1**: I would search scientific journals if you can or do a google scholarly search on the subject. Unless the Youtube video you saw had a reference I wouldn't trust it.  
> **username_2**: The former (and most) equilibrium temperature do not account for any atmosphere and its greenhouse effect. The temperatures sound right for bodies without atmosphere. For accurate surface temperatures you need to consider those - yet that means to complicate matter by orders of magnitude as you see in the difference of the atmospheric effect on earth and venus. You need elaborate models for radiative transfer and atmospheric chemistry to get more precision  
> **username_3**: That would basically mean modelling planet by planet because of its atmosphere (composition especially). In my modest opinion one cannot attain a simple general formula. I also think that 15 K is not that bad...  
> **username_4**: You are not going to find a simple formula that is anywhere close to correct. The equation from the YouTube video is nonsense.  
> **username_5**: How accurate does it need to be? Can you just use the effective temperature from the formula you gave then guess a greenhouse multiplier based on atmospheric conditions. You can use planets and moons of our solar system to have sample greenhouse numbers.  
> **username_6**: Any further thoughts on this? There are several comments but so far no responses.  

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## 0 answers

