<p>How can you calculate planetary surface temperatures from first principles? The obvious application is for Earth, with a surface temperature of <span class="math-container">$288$</span>K. I want a calculation with <em>no fudge factors</em> (i.e., factors calculated knowing the final result) and that also works for Venus and Mars (with the adequate parameters). I've come across some derivations (listed below, and in other posts here) but they all employ a fudge factor at some point. Overall, what I'd like is a full set of equations (algebraic or differential, with well-justified boundary conditions) that give the correct answer.</p>
<p><strong>First attempt</strong>: Based on a one layer model of the atmosphere, with an emissivity of <span class="math-container">$\epsilon=0.8$</span>, we get</p>
<p><span class="math-container">$T_{p} = \left(\frac{2}{2-\epsilon}\right)^{1/4}T_{a} = 288$</span>K</p>
<p>where <span class="math-container">$T_{a}$</span> is the temperature at the top of the atmosphere given by the blackbody calculation.</p>
<p><em>Problems:</em></p>
<p>1a. The emissivity is a fudge factor we calculated knowing the final answer. Is there an independent estimate or measurement of the emissivity?</p>
<p>1b. If we take <span class="math-container">$\epsilon = 1$</span>, this gives as an upper bound on how much heating you can expect on any planet, namely <span class="math-container">$2^{1/4}$</span> times the blackbody estimate. This upper bound is clearly surpassed on Venus.</p>
<p><strong>Second attempt</strong>: The lapse rate of the lower atmosphere is <span class="math-container">$\Gamma = 6.5$</span> K/km, so</p>
<p><span class="math-container">$T_{p} = T_{a} + h\Gamma$</span></p>
<p>and we get <span class="math-container">$288$</span>K for <span class="math-container">$h=5$</span> km.</p>
<p><em>Problems:</em></p>
<p>2a. The height <span class="math-container">$h$</span> is a fudge factor. How do we know from first principles at what altitude the <span class="math-container">$255$</span>K is reached?</p>
<p>2b. The <span class="math-container">$255$</span>K temperature is actually reached at four different altitudes in the atmosphere (see figure). Near the top of the troposphere (at <span class="math-container">$11$</span>km) the temperature is <span class="math-container">$216$</span>K, so the <span class="math-container">$255$</span>K at <span class="math-container">$5$</span> kilometres is very much due to atmospheric effects.</p>
<p><img src="https://qph.cf2.quoracdn.net/main-qimg-9a4a2efeb74ce9178bfe65d9b2d458a9" alt="" /></p>
<p>2c. Lets say we take <span class="math-container">$216$</span>K at the top of the troposphere as a more natural reference point. This still requires justification because it's lower than the <span class="math-container">$255$</span>K, but I guess ozone reduces the UV radiation and hence the effective irradiance that the troposphere receives?</p>
<p>2d. Even with a more reasonable reference point, the calculation of the temperature still depends on the lapse rate. The lapse rate is determined by hydrostatic pressure, adiabatic expansion and water vapour (see last reference). I can see how atmospheric composition plays a role via molar properties (of air and water), but it's not obvious where spectral properties come in (absorption etc.). Where does the greenhouse effect feature?</p>
<p>2e. The usual logic I've seen is that an increase in greenhouse gases would raise the altitude <span class="math-container">$h$</span> at which <span class="math-container">$255$</span>K is reached, and with a constant lapse rate you get an increase in surface temperature. Again, the whole <span class="math-container">$255$</span>K reference point for the &quot;top of the atmosphere&quot; (actually mid troposphere) is unjustified. Also, I find it odd that a small addition of gases would expand the entire atmosphere (a <span class="math-container">$1.1$</span>K increase means a expansion of the atmosphere by <span class="math-container">$170$</span>m). I find it more likely that the lapse rate would change (from 6.5 to 6.6), which would result in higher surface temperatures.</p>
<p>Preferably, I'd like answers to all the questions. But, I know this is a long post so you can just give the right derivation, or point to a definitive reference. Given the topic, I just want to say this is a scientific matter, lets keep it as such. Some sources:</p>
<p><a href="https://en.wikipedia.org/wiki/Stefan%E2%80%93Boltzmann_law#Effective_temperature_of_the_Earth" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Stefan%E2%80%93Boltzmann_law#Effective_temperature_of_the_Earth</a>
<a href="https://www.acs.org/climatescience/atmosphericwarming/singlelayermodel.html" rel="nofollow noreferrer">https://www.acs.org/climatescience/atmosphericwarming/singlelayermodel.html</a></p>
<p><a href="https://www.youtube.com/watch?v=hUFOuoD3aHw" rel="nofollow noreferrer">https://www.youtube.com/watch?v=hUFOuoD3aHw</a></p>
<p><a href="https://phys.libretexts.org/Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book%3A_Heat_and_Thermodynamics_(Tatum)/08%3A_Heat_Capacity_and_the_Expansion_of_Gases/8.08%3A_Adiabatic_Lapse_Rate" rel="nofollow noreferrer">https://phys.libretexts.org/Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book%3A_Heat_and_Thermodynamics_(Tatum)/08%3A_Heat_Capacity_and_the_Expansion_of_Gases/8.08%3A_Adiabatic_Lapse_Rate</a></p>

CC BY-SA 4.0
2023-05-01T10:45:45.423Z
Have a look here for purely radiative solutions, generalized for all planets https://ui.adsabs.harvard.edu/abs/2010A%26A...520A..27G/abstract  and https://ui.adsabs.harvard.edu/abs/2012ApJ...757..104R/abstract  for generalized radiative-convective models. They test their models on various solar system planets. Those single-layer toy models which you have mentioned have their purpose only in giving you some intuition about the physics.
AtmosphericPrisonEscape
CC BY-SA 4.0
2023-05-01T10:52:12.770Z
As to fudge factors - the full solutions i have referred to give temperature as function of optical depth. The optical depth of your gas at the surface then still has to be 'fudged' from knowing the surface pressure, and how the opacities react to pressure. This is as far as analytics goes - beyond that you'd have to iterate solutions. Furthermore, you will usually keep the albedo as a fudge factor, because otherwise you'd have to solve an extremely expensive radiative scattering problem https://ui.adsabs.harvard.edu/abs/2014ApJS..215....4H/abstract .
AtmosphericPrisonEscape
CC BY-SA 4.0
2023-05-01T10:57:23.590Z
The division between surface and atmosphere is totally arbitrary: there are continuous mass exchanges (in gas and even in the liquid phase) between the two, so you are already not discussing first principle.
EarlGrey
CC BY-SA 4.0
2023-05-01T11:29:04.243Z
If evaporation, geothermal energy and other phenomena are crucial in determining the temperature then that should feature in the derivation (not necessarily explicitly but e.g. percentage of water vapour). Again, not looking for an exhaustive treatment but mostly something that doesn't leave cracks open. I suspect that the entire thermal history of the planet since formation might be necessary, but that's just a convoluted way of saying we don't know.
Sebby
CC BY-SA 4.0
2023-05-01T11:30:47.100Z
I suspect you'll face a problem here that planets (at least those with atmospheres) are too complex to straightforwardly model from first principles with no fudge factors. Witness the whole field of earth system modelling...
Semidiurnal Simon
CC BY-SA 4.0
2023-05-01T22:38:59.060Z
"not looking for an exhaustive treatment but mostly something that doesn't leave cracks open." That seems very contradictory to me. As for internal heat, that is completely irrelevant for the terrestrial planets at our current age, only for Uranus and Neptune is $T_{\rm irradiation} \approx T_{\rm internal}$.
AtmosphericPrisonEscape
CC BY-SA 4.0
2023-05-02T00:23:08.373Z
The division between the Earth's surface and the atmosphere was not my point of contention, the other person brought it up. Yes, I know the internal heat is irrelevant. Well, I mentioned all the cracks in the derivations above. If someone can address them that's pretty much it. A one equation model was inadequate (1st), a three equation model (2nd) does a better job but with some loose ends. Maybe a five equation model does the trick? An exhaustive treatment would be like 1000 equations for every conceivable phenomenon.
Sebby
CC BY-SA 4.0
2023-05-02T06:55:36.783Z
This is a complete misunderstanding. The fudge factors have nothing to do with measurements, they come from inadequate models and are calculated on a theoretical basis from the final answer you should get (circular reasoning). The thickness is explicitly considered in the 2nd attempt, same with layering (only the 2 lower levels of the atmosphere are relevant), pressure comes in hydrostatic equilibrium conditions and the adiabatic expansion, composition of gases and vapors features in the moist lapse rate (mostly via water content and latent heat). It's atmospheric science 101 (last reference).
Sebby
CC BY-SA 4.0
2023-05-02T13:36:41.020Z
You have to "@" users with their tags, when responding to them, otherwise ppl don't see your comments pop-up. Not so when others post in your post, that will pop-up without an "@". Anyway, why don't you look at Eqn. 27 in my first reference. That's a single-equation solution for the temperature structure as function of optical depth. It seems to me that this should be a sensible starting point for what you are looking for.
AtmosphericPrisonEscape
CC BY-SA 4.0
2023-05-02T14:51:30.007Z
@AtmosphericPrisonEscape The temperature scales in the figures are in thousands of Kelvin and applies to exoplanets. This seems rather exotic and outside the scales relevant to the inner planets of our solar system. I am looking for something applicable in the range of 200-300K, not 2000-3000K.
Sebby
CC BY-SA 4.0
2023-05-02T15:07:20.013Z
@Sebby: Sigh. Earth science ignorance mumble mumble. Those models are generally valid. Go through the derivation to confirm this for yourself.
AtmosphericPrisonEscape
CC BY-SA 4.0
2023-05-02T17:16:37.597Z
@AtmosphericPrisonEscape The question I asked is pretty simple. Those references seem like the most complicated possible resources you could have sent and feel like a wild goose chase. If you know the matter so well just give an answer already. I'm just going to say you're not being helpful at this point.
Sebby
CC BY-SA 4.0
2023-05-02T18:02:11.997Z
@Sebby: Those are simple models, the absolute minimum to get a physically sensible temperature profile involving greenhouse effects. As you see, radiation transport, in its simplest form, is not so straight forward. Don't worry, if you refuse to be helped, i will stop.
AtmosphericPrisonEscape
CC BY-SA 4.0
2023-05-02T18:41:00.470Z
Let us [continue this discussion in chat](https://chat.stackexchange.com/rooms/145754/discussion-between-sebby-and-atmosphericprisonescape).
Sebby
<climate-change><climate><temperature><climate-models><planetary-science>
Planetary surface temperature from first principles
Sebby
# Planetary surface temperature from first principles

(Asked by: username_0 on 2023-05-01)

How can you calculate planetary surface temperatures from first principles? The obvious application is for Earth, with a surface temperature of $288$K. I want a calculation with *no fudge factors* (i.e., factors calculated knowing the final result) and that also works for Venus and Mars (with the adequate parameters). I've come across some derivations (listed below, and in other posts here) but they all employ a fudge factor at some point. Overall, what I'd like is a full set of equations (algebraic or differential, with well-justified boundary conditions) that give the correct answer.

**First attempt**: Based on a one layer model of the atmosphere, with an emissivity of $\epsilon=0.8$, we get

$T\_{p} = \left(\frac{2}{2-\epsilon}\right)^{1/4}T\_{a} = 288$K

where $T\_{a}$ is the temperature at the top of the atmosphere given by the blackbody calculation.

*Problems:*

1a. The emissivity is a fudge factor we calculated knowing the final answer. Is there an independent estimate or measurement of the emissivity?

1b. If we take $\epsilon = 1$, this gives as an upper bound on how much heating you can expect on any planet, namely $2^{1/4}$ times the blackbody estimate. This upper bound is clearly surpassed on Venus.

**Second attempt**: The lapse rate of the lower atmosphere is $\Gamma = 6.5$ K/km, so

$T\_{p} = T\_{a} + h\Gamma$

and we get $288$K for $h=5$ km.

*Problems:*

2a. The height $h$ is a fudge factor. How do we know from first principles at what altitude the $255$K is reached?

2b. The $255$K temperature is actually reached at four different altitudes in the atmosphere (see figure). Near the top of the troposphere (at $11$km) the temperature is $216$K, so the $255$K at $5$ kilometres is very much due to atmospheric effects.

![](https://qph.cf2.quoracdn.net/main-qimg-9a4a2efeb74ce9178bfe65d9b2d458a9)

2c. Lets say we take $216$K at the top of the troposphere as a more natural reference point. This still requires justification because it's lower than the $255$K, but I guess ozone reduces the UV radiation and hence the effective irradiance that the troposphere receives?

2d. Even with a more reasonable reference point, the calculation of the temperature still depends on the lapse rate. The lapse rate is determined by hydrostatic pressure, adiabatic expansion and water vapour (see last reference). I can see how atmospheric composition plays a role via molar properties (of air and water), but it's not obvious where spectral properties come in (absorption etc.). Where does the greenhouse effect feature?

2e. The usual logic I've seen is that an increase in greenhouse gases would raise the altitude $h$ at which $255$K is reached, and with a constant lapse rate you get an increase in surface temperature. Again, the whole $255$K reference point for the "top of the atmosphere" (actually mid troposphere) is unjustified. Also, I find it odd that a small addition of gases would expand the entire atmosphere (a $1.1$K increase means a expansion of the atmosphere by $170$m). I find it more likely that the lapse rate would change (from 6.5 to 6.6), which would result in higher surface temperatures.

Preferably, I'd like answers to all the questions. But, I know this is a long post so you can just give the right derivation, or point to a definitive reference. Given the topic, I just want to say this is a scientific matter, lets keep it as such. Some sources:

<https://en.wikipedia.org/wiki/Stefan%E2%80%93Boltzmann_law#Effective_temperature_of_the_Earth>
<https://www.acs.org/climatescience/atmosphericwarming/singlelayermodel.html>

<https://www.youtube.com/watch?v=hUFOuoD3aHw>

<https://phys.libretexts.org/Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book%3A_Heat_and_Thermodynamics_(Tatum)/08%3A_Heat_Capacity_and_the_Expansion_of_Gases/8.08%3A_Adiabatic_Lapse_Rate>

**Tags:** climate-change, climate, temperature, climate-models, planetary-science

> **Comments:**
> **username_1**: Have a look here for purely radiative solutions, generalized for all planets https://ui.adsabs.harvard.edu/abs/2010A%26A...520A..27G/abstract  and https://ui.adsabs.harvard.edu/abs/2012ApJ...757..104R/abstract  for generalized radiative-convective models. They test their models on various solar system planets. Those single-layer toy models which you have mentioned have their purpose only in giving you some intuition about the physics.  
> **username_1**: As to fudge factors - the full solutions i have referred to give temperature as function of optical depth. The optical depth of your gas at the surface then still has to be 'fudged' from knowing the surface pressure, and how the opacities react to pressure. This is as far as analytics goes - beyond that you'd have to iterate solutions. Furthermore, you will usually keep the albedo as a fudge factor, because otherwise you'd have to solve an extremely expensive radiative scattering problem https://ui.adsabs.harvard.edu/abs/2014ApJS..215....4H/abstract .  
> **username_2**: The division between surface and atmosphere is totally arbitrary: there are continuous mass exchanges (in gas and even in the liquid phase) between the two, so you are already not discussing first principle.  
> **username_0**: If evaporation, geothermal energy and other phenomena are crucial in determining the temperature then that should feature in the derivation (not necessarily explicitly but e.g. percentage of water vapour). Again, not looking for an exhaustive treatment but mostly something that doesn't leave cracks open. I suspect that the entire thermal history of the planet since formation might be necessary, but that's just a convoluted way of saying we don't know.  
> **username_3**: I suspect you'll face a problem here that planets (at least those with atmospheres) are too complex to straightforwardly model from first principles with no fudge factors. Witness the whole field of earth system modelling...  
> **username_1**: "not looking for an exhaustive treatment but mostly something that doesn't leave cracks open." That seems very contradictory to me. As for internal heat, that is completely irrelevant for the terrestrial planets at our current age, only for Uranus and Neptune is $T_{\rm irradiation} \approx T_{\rm internal}$.  
> **username_0**: The division between the Earth's surface and the atmosphere was not my point of contention, the other person brought it up. Yes, I know the internal heat is irrelevant. Well, I mentioned all the cracks in the derivations above. If someone can address them that's pretty much it. A one equation model was inadequate (1st), a three equation model (2nd) does a better job but with some loose ends. Maybe a five equation model does the trick? An exhaustive treatment would be like 1000 equations for every conceivable phenomenon.  
> **username_0**: This is a complete misunderstanding. The fudge factors have nothing to do with measurements, they come from inadequate models and are calculated on a theoretical basis from the final answer you should get (circular reasoning). The thickness is explicitly considered in the 2nd attempt, same with layering (only the 2 lower levels of the atmosphere are relevant), pressure comes in hydrostatic equilibrium conditions and the adiabatic expansion, composition of gases and vapors features in the moist lapse rate (mostly via water content and latent heat). It's atmospheric science 101 (last reference).  
> **username_1**: You have to "@" users with their tags, when responding to them, otherwise ppl don't see your comments pop-up. Not so when others post in your post, that will pop-up without an "@". Anyway, why don't you look at Eqn. 27 in my first reference. That's a single-equation solution for the temperature structure as function of optical depth. It seems to me that this should be a sensible starting point for what you are looking for.  
> **username_0**: @username_1 The temperature scales in the figures are in thousands of Kelvin and applies to exoplanets. This seems rather exotic and outside the scales relevant to the inner planets of our solar system. I am looking for something applicable in the range of 200-300K, not 2000-3000K.  
> **username_1**: @username_0: Sigh. Earth science ignorance mumble mumble. Those models are generally valid. Go through the derivation to confirm this for yourself.  
> **username_0**: @username_1 The question I asked is pretty simple. Those references seem like the most complicated possible resources you could have sent and feel like a wild goose chase. If you know the matter so well just give an answer already. I'm just going to say you're not being helpful at this point.  
> **username_1**: @username_0: Those are simple models, the absolute minimum to get a physically sensible temperature profile involving greenhouse effects. As you see, radiation transport, in its simplest form, is not so straight forward. Don't worry, if you refuse to be helped, i will stop.  
> **username_0**: Let us [continue this discussion in chat](https://chat.stackexchange.com/rooms/145754/discussion-between-username_0-and-username_1).  

---

## 0 answers

