A commenter on another site with the handle ‘Agent009’ has come up with an interesting formula for calculating the environmental lapse rate on three solar system bodies with atmospheres. Talkshoppers might offer some ideas as to why it works. H/T to Stuart ‘Oldbrew’ for flagging this one up.
I’ve been trying to solve a puzzle… dry adiabatic lapse rate is normally calculated as following:
Γ = g·M/cp
where Γ is lapse rate, g is surface gravity acceleration, M is mole mass and cp is molar heat capacity.
However, if you calculate this for Earth, you arrive at 9.77 K/km, but actual environmental lapse rate, as defined in the ISA, is 6.49 K/km, which is about 9.77 * 0.665. So, I decided to take a look at how this works on Venus and Titan – the only two other worlds in the Sol System that actually have tropospheres.
On Venus (assuming tropopause at 55 km), the average lapse rate is about 7.9 K/km, but the above formula gives you 10.46 K/km, which means that you must multiply the result by 0.756 to get the actual value. On Titan (assuming tropopause at 42 km), actual average lapse rate appears to be around 0.5 K/km, but predicted lapse rate is 1.26 K/km – which gives you the coefficient 0.427. So I’ve been trying to figure what this mysterious coefficient depends upon – and, I think, I’ve found it. The following expression gives you almost exactly those numbers (using SI units, that is):
³√(12·g·M·(1/R – 1/cp))
where R is the ideal gas constant.
Note, that M·(1/R – 1/cp) is what you must divide the temperature value by to get the square of the speed of sound (I call inverse of this number the “speed of sound factor” because it is constant for any arbitrary gas mixture and gives you the best idea how any changes in atmospheric composition would affect the speed of sound) .
I’m not sure why the heck this works, but the following formula predicts the actual mean tropospheric lapse rates for all 3 worlds with astonishing degree of accuracy:
Γ = (g·M/cp)·³√(12·g·M·(1/R – 1/cp))
For Earth this gives you exactly the standard ISA lapse rate – down to 0.01 K/km precision! For Venus, it predicts the average measured lapse rate with nearly the same degree of accuracy! For Titan, there is somewhat less actual data available and the situation seems to be complicated by the fact that mole fraction of methane apparently varies significantly with altitude, but all measurements reported so far seem to correspond with the result predicted by the above formula at least by order of magnitude.
A coincidence? Well, the lapse rate predicted for Titan (based on 95% N₂ / 5% CH₄ atmospheric composition) would be 0.42 K/km. We’ll see if this actually turns out to hold true once more data is available.
Basically, I was just looking at ISA and Venus mean values and it suddenly struck me that this relation was very close to the relation between the “speed of sound factor”. I’ve played a bit with numbers and it turned that the relation between cube root of this number and lapse rate was nearly the same. Then I took a look at Titan data, but the relation did not seem to hold. So, I though to myself: “what might Earth and Venus have in common that could matter for lapse rate relation that is very different on Titan?”. The obvious answer was g, because otherwise, atmospheres of Earth and Venus are radically different. So, I tried to introduce g into the equation and, I was astounded, when it turned out that factoring the relation by ³√g not only made sense in terms of Titan data, but also got me closer to the relation between Earth and Venus values – the relation was proportional by nearly the same number. I was even more amazed when I realized that this factor was almost exactly the cube root of 12! And I was completely shocked when I approximated this factor to ³√12 and result for Earth turned out to be exactly the ISA lapse rate – down to 2 decimal points precision! The results for Venus still appear to be a nearly perfect match of the actually measured values and the results for Titan seam to work reasonably well considering the scarcity and imprecision of the data we have so far. Probably this “factor 12” has some significance I do not understand yet… but since Earth, Venus and Titan are the only worlds we are currently able to study that actually have tropospheres, I cannot do much better than this for now.
Actually, it’s more like I have 2.5 data points (since data available for Titan is still quite controversial and imprecise right now). Unfortunately, it does not seem that we gonna have any more data points in any foreseeable future, so I have to do the best with what what we’ve got.
Regarding the units… g is m/s², M is kg/mol and (1/R – 1/cp) is mol·K/J as both R and cp are J/mol·K. Since J = kg·m²/s² the g·M·(1/R – 1/cp) would be K/m. Thus, as the result of ³√(12·g·M·(1/R – 1/cp)) should to be dimensionless, this mysterious “factor 12” would need to be 12 m/K. This might be some relation I do not yet understand, but it could also be just another universal constant, like R or G, for all I know.
More measurements from Titan and some data from outside the Sol System would certainly help, but I cannot dare to hope for that anytime very soon, I fear.