Errors and Exospheres

First off, sorry about the delay. Looking back over part IV of my Yaccatrice workflow, I realized I screwed up almost every calculation I made. Trouble is, every time I try to recalculate things seem to come up different, so I’m setting my computer up plug and chug the calcs automagically. All of my formula look right and proper, so it’s just the numbers that are, “suspect.” It still looks like Sky Moon and Yaccatrice survive the revised numbers. To save myself from my own ineptitude and possibly make future efforts in this vein a little easier, I decided to automate some of the process in Mathematica. This slowed things down further.

While tracking down formulae I became curious about the minimum molecular weight retained(MMWR) calculation. Sometimes I like to actually know what I am doing and understand the rationale behind it.  One of the things I was curious about was how exosphere temperature was determined in GURPS Space.

I knew the basis of the MMWR calculation(in real world terms) was

Vesc > f Vrms,

where:

Vesc is the escape velocity of the body and Vrms is the root mean square thermal velocity and f is the ratio between escape velocity and RMS velocity.

While I was researching this, I discovered a recent thread on the sjgames board on this very subject. Following is the text of my reply.

I’ve been working on the math here. Trying to figger out how the authors of GURPS Space derived MMWR.Here is what I came up with.

SQRT(2 G M / r) > f SQRT(3 Kb T / m mH)

Where:
G: Gravitational constant ~ 6.673E-11 m^3 s^-2 kg^-1
M: Planetary mass (kg)
r: Planetary radius (m)
kb: Boltzmann constant ~ 1.381E-23 J K^-1
T: Temperature(at the exosphere) (K)
m: molecular weight
mH: mass of a single particle of molecular weight 1 ~ 1.661E-27 kg
f: the ratio of escape velocity to RMS velocity to retain the gas component for a given period of time.

To make the format more GURPSy, I must make some substitutions:

r = D d0 / 2
q = K q0
M = V q
V = 4/3 pi r^3
T = C B

r: radius of the planet is equal to the diameter of the planet in Earth diameters times the diameter of the Earth divided by two.

d0:Diameter of the Earth ~ 12,762,098 m

q: density of the planet is equal to the product of the density of the planet in Earth densities and the density of the Earth.

q0:Density of the Earth ~ 5520 kg m^-3

V:Volume of the planet (m^3)

T: Exospheric temperature. GURPS clearly assumes a linear function of planetary blackbody temperature. As we can see from Table 1 on page 7 of Mueller-Wodarg(2004), this isn’t necessarily a good assumption(note how the exospheric temperature of Venus is not only less than that of Earth, but less than the temperature of Mars’ exosphere… 0_0 DUDE!). I have my own ideas about this: I think that planets of hotter stars will have hotter exospheres. It’s also clear that atmospheric composition has a very dominant effect on the temperature. Since I’m interested in deriving the GURPS formula and since I figure the assumption is good for Earthlike atmospheres and Sunlike stars, I’ll go with it for now.

Giving all the necessary substitutions and solving for m(that is, mu), I get:

m > f^2 [ (9 kb) / (2 G pi d0^2 q0 mH) ] C [ B / (K D^2) ]

I’m being egregious with the parentheses for clarity. According to Table 5 on page 35 of Habitable Planets for Man(available free, here), a ratio of escape velocity to RMS velocity(f) of 6 is sufficient for an essentially permanent atmosphere. So we’ll substitute 36 for f^2.

m > [ (162 kb) / (G pi d0^2 q0 mH) ] C [ B / (K D^2) ]

To make the format more like that in GURPS Space, I modify this further:

m > B / [ ([G pi d0^2 q0 mH]/[162 kb C]) K D^2 )

m >~ B /( [140/C] K D^2 )

Since GURPS assumes 140/C == 60, we can figure an exospheric temperature for Earth of about 648 K, much less than Earth’s actual value averaging 1200 K. A value for the constant of 30 to 35 may be more realistic for an Earthlike atmosphere.

The lower value that GURPS assumes might still be a better approximation to general planetary atmospheres and may give a better sense of the historical situation of an Earthlike atmosphere as its composition evolves.

For my own purposes, and moving away from the GURPS standards, I assume an exospheric temperature of about

1.75 B^0.7 Tstar^0.3,

for an MMWR of

m > (1.75 B^0.7 Tstar^0.3) / (140 K D^2),

where:

Tstar is the effective surface temperature of the star, about 5770 K for the Sun.

This is strictly for planets with an Earthlike atmosphere, but this takes into account my own assumption that planets of smaller, redder will find it easier to retain heavy atmospheres. This gives an exospheric temperature for Earth of about 1208 K, good enough for me. m(mu) in this case would be about 8.64.

For other sorts of planets I would probably use a slightly modified version of the GURPS formula ’cause I ain’t about to get into figuring out ionization potentials and all the other crap you’d have to work out to make a generalized formula for atmospheres as different as those of Venus and Earth and Jupiter and Saturn. I’ll leave that to more egg-like heads than my own. I do pray if anyone comes up with a reasonable solution that they post it, though…

Hopefully, this will be of use to my readers. That last formula, though with real world units is what I will use to determine the MMWR for Yaccatrice.

As a further note, due to the trouble I have had of late in keeping up with this blog, which is an important thing to me, I have decided to limit myself to weekly posts until I have a queue of at least six weeks of posts. I’d like for my blog to be a more reliable source of interesting information.

My next post will be my fixed information for Cintilla, Sky Moon and Yaccatrice. After that I will get back to my workflow.

Thank you for your understanding,

The Astrographer

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One Response to Errors and Exospheres

  1. Astrographer says:

    Seems like when I plug a temperature of 1200 K directly into the MMWR calculation, Earth just barely retains water. Very little water (and even less methane and ammonia) gets up as high as the escape layer, the temperature inversion above the tropopause makes a pretty good lid. In retrospect, I think I’ll assume an effective exosphere temperature roughly the same as what GURPS assumes. If I use ^0.7 * ^0.3 that gives an EFFECTIVE exosphere temperature of ~690 K and an MMWR of ~9.93 gmol. This remains a kludgy-ass way of handling some fairly complicated modeling issues of atmospheric stratification, but it will need to be sufficient. I begin to understand the reasoning behind the choices that the authors of GURPS Space made, the more I bang my head into this the more complicated it gets.

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