Once again, Daistallia’s World Building/Non-linguistic Resources thread over at the zompist board has come to the rescue. Apparently, the nice folks at Rand have decided to freely distribute Stephen H. Dole’s seminal Habitable Planets for Man as a pdf. This is free, grab it. I think I have an Amazon link for the hardcopy version if you really wanna help my finances a tiny bit… 😉
Previously, I wrote about a formula I once had access to for determining the residence times for various atmospheric constituents, based on exospheric temperature, molecular weight and escape velocity. Turns out a couple variations of the formula live on page 34 of Habitable Planets for Man:
t = (vBar^3/(2 * g^2 * R)) * e^(3*g*R/vBar^2)
t = time in seconds required for the abundance of the given constituent to decrease by a factor of e, or to 0.368 of its original abundance.
vBar = root-mean-square velocity of the molecules
R = planetary radius
g = surface gravity
If vBar is in meters per second, R is in meters, and g is in meters per second per second then t will be in seconds, if they are in cm/sec, cm and cm/sec^2, then… t will be in seconds. You just have to assure that basic length and time units are equivalent.
After a bit of simple algebra, that I don’t think I messed up :), this reduces to:
t = ((2 * vBar^3 * R) / Vesc^4) * e^(3/2 * Vesc^2 / vBar^2)
Vesc = the planet’s escape velocity = sqrt(2GM/R)
with G being the Gravitational constant: G = 6.6720 x 10^-11 N m^2 kg^-2, which mires us hopelessly in meters, kilograms and seconds as units. M, by the way is the planet’s mass. With this value for G, M is clearly in kilograms.
Now all of these measurements(radius, gravity, vBar, Vesc, etc) should really be measured at the exosphere, but for simplicity, and especially for work on a science fiction story- or game-world, surface values should be sufficient.
vBar, unless there’s a flaw somewhere in my math or my memory of physics class, should be thus:
vBar = sqrt(R*T / mu), where
R = molar gas constant = 8.31441 J mol^-1 K^-1
T = exospheric temperature in kelvins. I’m assuming, 2000 K * [Luminosity of star] / [Luminosity of Sun] / [Distance of planet from star in au’s]^2, but it may be closer to 2000 K * [Surface temperature of star] / [Surface temperature of Sun]. I’m not sure how exosphere temperature scales.
mu = the molecular weight of the given atmospheric constituent.
Now if you have read this far, you are probably asking yourself, “How is this useful to me?” Well, I’ve been bouncing this around in my head for a few years now and I’m starting to think it isn’t, but… If you knew or could as the omniscient narrator/god know the composition and proportions of some primordial atmosphere of your planet, you could calculate the rate of loss after a time, T, of each component as:
qa = qa0 e^-(T/t), where qa is the final quantity of component a and q0a is the initial content of component a in the primordial atmosphere.
I don’t know how useful this really is to most conworlders including myself, but… hey… if the conlang people(I’m looking at you Mr. Tolkein) can spend all kinds of sweat and blood working out the protolanguages from which their imaginary languages are derived, why can’t us rockheads do something equivalent?
As always, any comments are welcome. Especially if you find a math error. It’s been awhile since I’ve used algebra in anger.