Just spent a couple of hours trying to derive 'natural' temperature of the planet in the absence of greenhouse gas effects from basic Stefan-Boltzmann equation and inverse square law.

My first engineering approximation : a small rotating sphere of something approximating to graphite gave me a temperature of 278.5 K which is not bad perhaps, but considerably higher than the figure 255 K I've seen commonly quoted.

Correcting the solar influx for albedo figures of no particularly strong provenance hit the nail pretty well on the head for earth, but not so well for the moon - 270.3 K is still 20 too high.

Can anyone point me towards a bona fide source reference or maybe just yea or nay the attached back of envelope calcs.

Best regards, and thanks in advance for your kind attention. 

 

 

'Planetary Natural Black Body Temperature.pdf

11 hours ago, sethoflagos said:

Can anyone point me towards a bona fide source reference

https://energyeducation.ca/encyclopedia/Earth_Temperature_without_GHGs#

5 hours ago, sethoflagos said:

Can anyone point me towards a bona fide source reference or maybe just yea or nay the attached back of envelope calcs.

Hi Seth,

It is difficult to estimate the as it depends heavily on your model.

Most folks just quote a figure worked out by somebody else (perhaps with a reference).

Anyway the most baseline figure is what you get with the simplest heat balance of what would happen if the Earht's atmousphere suddenly disappeared, but everything else remained the same and is done in the Cambridge book for the whole solar system.

It is called the equilibrium temperature.

Thermodynamics of the Earth and Planets  A P Douce  Cambridge 2011

Pages 630 ff

 

However there are wrinkles in this.

The Earth's axis is tilted at about 23.5 degrees and also the distance from the Sun varies, both during one year and from year to year.

So the surface (even without the effect of the atmousphere) receives uneven insolation.
And of course half of it is is shaded by night at any one time.

You might think that the Moon was therefore a better candidate as it has no atmousphere to speak of.

But, whilst it is true that the Moon rotates so that all its surface gets insolation,  the shading effect of the Earth on the Moon is far greater than the shading effect of the Moon on the Earth owing to their size difference.

So the Moon receives less insolation than your calculations would suggest.

 

Coming back to the Earth's atmousphere, it affects the fugure's in several ways and more sophisticated models need to take these into acount.

Both the atmousphere and ocean distribute energy from wrmer zones to colder ones, thus modifying the average temperature difference for Stefan energy flows.

The atmousphere absorbs/ reflects some of insolation (Beer Lambert) so reducing the actual values.

The atmousphere also reflects some surface emission back (greenhouse effect) again modifying the surface temperatures.

 

A more sophisticated model , therefore, splits the surface into zones and applies appropriate insolation and emission values, spread over time then integrated over time and averaged, and then finally all collected together and again averaged.

 

2 hours ago, swansont said:

https://energyeducation.ca/encyclopedia/Earth_Temperature_without_GHGs#

Thanks, yes. Very slight differences in the numbers, but same formulae

46 minutes ago, sethoflagos said:

Thanks, yes. Very slight differences in the numbers, but same formulae

Right. They just used the measured solar irradiance at the top of the atmosphere rather than calculating it. And a slightly different albedo.

3 hours ago, studiot said:

whilst it is true that the Moon rotates so that all its surface gets insolation

Roughly once a month, which you could probably ignore for this calculation, but it also means you can’t just average over the whole surface, so it needs the more sophisticated model you suggest.

Just now, swansont said:

Right. They just used the measured solar irradiance at the top of the atmosphere rather than calculating it. And a slightly different albedo.

Roughly once a month, which you could probably ignore for this calculation, but it also means you can’t just average over the whole surface, so it needs the more sophisticated model you suggest.

“All models are wrong, but some are useful”. George E. P. Box

On 3/27/2025 at 7:53 PM, swansont said:

Right. They just used the measured solar irradiance at the top of the atmosphere rather than calculating it. 

Sometimes one forgets that for large, established swathes of physics, the empirical matches the theoretical with extraordinary precision.

A message easily lost amongst the blizzard of disingenuous clickbait media.

The lunar diurnal variation has been gnawing away at me. I approximated the lunar surface temperature to be 251 +/- D Kelvin for daytime/nighttime average temperature values and calculated for D.

Turns out that a D value as low as 60 K was enough to account for the temperature discrepancy of 20 K I'd run into in the opening post.