# Thread: Atmospheric composition of planets

1. ## Atmospheric composition of planets

I understand that the surface gravity of a planetary can dictate what gases can be retained for its atmosphere. Hydrogen, for example, is the lightest element and requires the greatest gravity to retain.

Is there a resource that lists, or a way to calculate, what level of surface gravity is required to retain the most common planetary gases?

2. EDIT: Ignore this post, look at the formulae I posted further down the thread:
http://bautforum.com/showpost.php?p=1011298&postcount=8
Last edited by EDG; 2007-Jun-30 at 10:07 PM. Reason: got the formulae wrong, use my posts below instead!

3. And then you need to know what the molecular weights of gases are:

Code:
```Molecular Hydrogen (H2)   2.00
Helium (He)               4.00
Methane (CH4)            14.00
Ammonia (NH3)            17.00
Water Vapour (H2O)       18.00
Molecular Nitrogen (N2)  28.00
Carbon Monoxide (CO)     28.00
Molecular Oxygen (O2)    32.00
Hydrogen Sulphide (H2S)  34.10
Hydrogen Chloride (HCl)  36.50
Argon (Ar)               39.00
Carbon Dioxide (CO2)     44.00
Ozone (O3)               48.00
Sulphur Dioxide (SO2)    64.10
Chlorine (Cl2)           71.00
Sulphur Trioxide (SO3)   80.10
Xenon (Xe)              131.30```

4. EDG, what an outstanding response. thank you very much for the detailed explanation...

I'd like to make sure I fully understand, so I'm going to try an example...

Given
m = 36kTR/GM(mh)

where:
(reducing Joules & Newtons to more elementary units)
m = minimum molecular weight retained by planet
k = Boltzmann constant (1.38 x 10e-23 kg m2 s-2 K-1).
T = Base Temperature in K.
R = Radius of planet in m.
G = Gravitational Constant (6.67 x 10e-11 m3 kg-1 s-2).
M = Mass of planet in kg.
mh = Mass of hydrogen atom (1.67 x 10e-27 kg)

and using Earth as my example:
R = 6372797 m
M = 5.9736 x 10^24 kg
T = 280K (derived from sun's luminosity and distance)

(for readability, I've placed all units within brackets)

Substituting:
Code:
```    36*(1.38*10^-23)*[kg]*[m2]*[s-2]*[K-1]*280*[K]*(6.372797*10^6)*[m]
m = ------------------------------------------------------------------
6.67*10^-11*[m3]*[kg-1]*[s-2]*(5.9736*10^24)*[kg]*(1.67*10^-27)*[kg]```
Simplifying to:
Code:
```     88.648*10^-14*[kg]*[m3]*[s-2]
m =  ------------------------------
66.54*10^-14*[kg]*[m3]*[s-2]```
m = 1.33224

So, give the Sun's luminosity, as well as Earth's mass, radius & distance from the Sun, we can calculate that the Earth is able to gravitationally retain, long-term, any gas with a molecular weight of at least 1.33... Is that correct?

5. Originally Posted by baric
EDG, what an outstanding response. thank you very much for the detailed explanation...
Yer welcome

So, give the Sun's luminosity, as well as Earth's mass, radius & distance from the Sun, we can calculate that the Earth is able to gravitationally retain, long-term, any gas with a molecular weight of at least 1.33... Is that correct?
Yep. Theoretically.

If I use 150 instead of 36 then I get 5.52, which makes more sense since Earth clearly can't hold onto hydrogen for very long (or we'd still have a lot in our atmosphere). So I'm not sure about those constants. But the rest of it's correct...

6. Originally Posted by EDG_
Yer welcome

Yep. Theoretically.

If I use 150 instead of 36 then I get 5.52, which makes more sense since Earth clearly can't hold onto hydrogen for very long (or we'd still have a lot in our atmosphere). So I'm not sure about those constants. But the rest of it's correct...
I'm pretty sure the constants are correct, although I did not calculate the 270K myself. But even if you assume 300K for Earth, a temperature that certainly requires atmosphere warming to achieve, m only rises 1.425, still clearly below the molecular weight of H2.

So several possibilities might explain the dearth of H2 in our atmosphere:

1) the 36x factor is insufficient, and a larger factor such as 150 is more appropriate
2) H2 in the inner solar system was cleared out before the rocky planets acquired enough mass to capture it
3) an early event unique to Earth raised temperatures sufficiently to radiate away any H2 in the atmosphere

Regardless, it's interesting food for thought.

7. I suspect it's the constants that are the problem. Though even if it's set to 54 (36 x 1.5) the MMW retained is still 1.99. Most vexing.

8. AHA!

Thanks to Himanshu Raj, I've solved the errant constant problem.

It looks like one of the equations I used was incorrect. It should go as follows:

Basically, it's a battle between gravity and temperature. If the molecules of a gas are moving faster than the escape velocity of the planet, then the gas will escape - how fast they move is determined by their temperature.

Code:
`0.5mV^2 = (1.5)kT`
(the 1.5 on the RHS was missing in my original post)
where m is the mass of the particle and Vt is the average velocity of the particles, k is the Boltzmann constant and T is the temperature. So:

Code:
`Vt = SQRT(3kt/m)`
for the atmosphere to escape, Vt (the "thermal velocity") has to be bigger than the escape velocity (Ve).

If you set these as equal, you get

Code:
`SQRT(3kT/m) = SQRT(2GM/R)`
Make T the subject and you get:

Code:
`T = GMm/3kR`
Make m the subject and you get

Code:
`m = 3kTR/2GM(mh)`
which can be re-written as:

Code:
`m = (3/2)kTR/GM(mh)`
where m = (minimum molecular weight retained by planet).
k = Boltzmann constant (1.38 x 10e-23 J/K).
T = Base Temperature in K.
R = Radius of planet in m.
G = Gravitational Constant (6.67 x 10e-11 N m2 kg-2).
M = Mass of planet in kg.
mh = Mass of hydrogen atom (1.67 x 10e-27 kg)

so you can use that to figure out the MMW (minimum molecular weight) retained by the object.

But because the velocity of the particles follow what's called a maxwellian distribution (which looks like an offset bell curve), that means that while most particles will have the calculated velocity at a given temperature, a smaller number will have higher and lower velocities too (the higher velocities are in a long "tail"). So you have to put in a "fudge factor" to account for that.

The general consensus is that for a planetary body to be able to hold onto an atmosphere for billions of years, Ve > 6 Vt

Martyn Fogg (something of an expert in terraforming) says that:

"As a rough guide the exosphere leaks a particular gas to 1/e of its former concentration in ~days when Vg/Vt=3; ~decades when Vg/Vt=4; ~tens of millennia when Vg/Vt=5; and billions of years for Vg/Vt=6."

So roughly speaking:

Ve > 1 Vt: gas can be retained for seconds
Ve > 2 Vt: gas can be retained for hours
Ve > 3 Vt: gas can be retained for days
Ve > 4 Vt: gas can be retained for years/decades
Ve > 5 Vt: gas can be retained for millennia
Ve > 6 Vt: gas can be retained for billions of years

If you plug those numbers into the "m" equation above then:

Code:
`m = 36.(3/2).kTR/GM(mh) if Ve > 6 Vt`
Which means:

Code:
`m = 54.kTR/GM(mh) if Ve > 6 Vt`
So that's the minimum molecular weight that can be retained over billions of years.

So that explains where I got my 150 constant from. I was assuming that Ve > 10 Vt. 150 is 1.5 * (10*2). So I was right after all .

9. LOL.. I pointed this out on the other thread... Ignore that post!

10. One more thing to keep in mind: at the top of the atmosphere, there are several processes other than "collisions with other atmospheric molecules" which can give a molecule a substantial kick: high-energy gamma rays and cosmic-ray particles can slam into a molecule, too. These rare events could create a small tail in the distribution of molecular speeds, which, over the course of eons, might be important.

The temperature of the atmosphere is important, yes, but it may not be the only factor one must account for when looking at the lightest gases ...

11. Here is a temperature gradient chart.

12. Originally Posted by StupendousMan
One more thing to keep in mind: at the top of the atmosphere, there are several processes other than "collisions with other atmospheric molecules" which can give a molecule a substantial kick: high-energy gamma rays and cosmic-ray particles can slam into a molecule, too. These rare events could create a small tail in the distribution of molecular speeds, which, over the course of eons, might be important.

The temperature of the atmosphere is important, yes, but it may not be the only factor one must account for when looking at the lightest gases ...
Oxygen apparently gets ionised easily too, and the planet's own magnetic fields can cause sputtering to strip off molecules and heat up the exosphere too. So the formula is very much an approximation, but it's generally a good one.

13. Originally Posted by StupendousMan
One more thing to keep in mind: at the top of the atmosphere, there are several processes other than "collisions with other atmospheric molecules" which can give a molecule a substantial kick: high-energy gamma rays and cosmic-ray particles can slam into a molecule, too. These rare events could create a small tail in the distribution of molecular speeds, which, over the course of eons, might be important.

The temperature of the atmosphere is important, yes, but it may not be the only factor one must account for when looking at the lightest gases ...
On a related note, I found this link:
http://www.dangermouse.net/gurps/science/temps.html

It's a great general overview of related topics, but also includes these very interesting paragraphs:

Now let's consider the problem of outgassing or leakage of a hot atmosphere to space. This depends on the escape velocity of the planet, and the thickness and composition of the atmosphere.

The root-mean-square velocity of a gas particle is roughly √(3kT/m) where k is Boltzmann's constant, and T and m are the gas temperature and particle mass. For oxygen this is about 28 √(T) in SI units. For Earth the escape velocity is about 11,000 metres per second, which corresponds to 162,000 Kelvin for oxygen. If the air was this temperature, we would lose most of our oxygen within a matter of minutes. Luckily for us, the air never gets this hot!

This RMS velocity is a sort of average gas particle velocity, so there will be some gas particles with substantialy higher and lower velocities. What this means is that even though the gas temperature may be well below that required to make the RMS velocity near escape velocity, there will still be a few particles with velocities high enough to escape the atmosphere. If the difference between RMS and escape velocities is large, the gas will trickle away very slowly, taking billions of years to make any substantial difference. This is the case for oxygen on Earth.

For hydrogen the temperature comes to about 10,000 Kelvin. The air never gets anywhere near this hot either... at ground level. But up in the ionosphere, where it is heated by the full blast of the sun's radiation and particle streams, it gets close. So all of our hydrogen has leaked away on a timescale that is short compared to the age of the Earth.
The last part about the temperature of the ionosphere is what intrigues me. Since it will have a significant effect on the long-term losses of the lighter gases, what is a good approximation for this temperature and why is it different from the temperature calculated by the black-body radiation from the Sun?

14. Originally Posted by baric
On a related note, I found this link:
http://www.dangermouse.net/gurps/science/temps.html

It's a great general overview of related topics, but also includes these very interesting paragraphs
Heh, the author of that is an acquaintance of mine, we wrote a GURPS book about oceans on Earth and other planets for the Transhuman Space RPG .

I can't help but wonder if the people that know most about planetary science as a whole are people like me and him, who build worlds for fun for scifi settings and have to research everything to do it .

The last part about the temperature of the ionosphere is what intrigues me. Since it will have a significant effect on the long-term losses of the lighter gases, what is a good approximation for this temperature and why is it different from the temperature calculated by the black-body radiation from the Sun?
I think it's because the gases are getting heated up there by collisions from cosmic rays and solar wind and so on. Though the gas is so rarefied up there that I don't think you'd actually be incinerated by it if you were there.

15. Originally Posted by baric
The last part about the temperature of the ionosphere is what intrigues me. Since it will have a significant effect on the long-term losses of the lighter gases, what is a good approximation for this temperature and why is it different from the temperature calculated by the black-body radiation from the Sun?
The temperature of the Earth's exosphere is 1500-2000K; it varies according to solar activity. That drives hydrogen escape with a very short time constant, but lets us retain oxygen and nitrogen for billions of years.
The reason the temperature is so high is because rarified gas doesn't behave like a black body: it has only certain discrete modes of obtaining or losing energy. In Earth's case, one of the modes of gaining energy is by oxygen absorbing ultraviolet photons; it's a mechanism that's lacking in the exospheres of other planets that lack free oxygen, and IIRC their exosphere temperatures are correspondingly lower.

Grant Hutchison

16. Originally Posted by grant hutchison
The temperature of the Earth's exosphere is 1500-2000K; it varies according to solar activity. That drives hydrogen escape with a very short time constant, but lets us retain oxygen and nitrogen for billions of years.
The reason the temperature is so high is because rarified gas doesn't behave like a black body: it has only certain discrete modes of obtaining or losing energy. In Earth's case, one of the modes of gaining energy is by oxygen absorbing ultraviolet photons; it's a mechanism that's lacking in the exospheres of other planets that lack free oxygen, and IIRC their exosphere temperatures are correspondingly lower.

Grant Hutchison
ahh... so this is an Earth-specific mechanism due to our O2 & O3, which is the result of biological activity?

17. Originally Posted by baric
ahh... so this is an Earth-specific mechanism due to our O2 & O3, which is the result of biological activity?
Yes. But I don't know if it entirely accounts for the difference between Earth's exosphere and that of other planets, or if some other mechanisms are also at work.
There is, however, a significant difference between Earth and elsewhere, which needs some explanation: estimates for the exosphere temperatures for Venus and Mars seem to be around 300K and 200K.

Grant Hutchison

18. Newbie
Join Date
Sep 2007
Posts
4
I was just wondering if it were ok if someone could just verify something for me please. I found it all a little hard to understand but does the formula work like this:

M = 150*kTR/GM(mh)

...1.67e-27 is the mass of Hydrogen molecule..

I get 5.55... when I calculate it. Does that mean that
according to:

Ve > 1 Vt: gas can be retained for seconds
Ve > 2 Vt: gas can be retained for hours
Ve > 3 Vt: gas can be retained for days
Ve > 4 Vt: gas can be retained for years/decades
Ve > 5 Vt: gas can be retained for millennia
Ve > 6 Vt: gas can be retained for billions of years

Because 5.55 is greater than 5 it would stay in the atmosphere for a mellennia?

19. I answered via PM too, but here's what I said there:
No... it means that the minimum molecular mass retained by the planet is 5.555. So it can basically hold onto any gas except helium (mass = 4) and hydrogen molecules (mass = 2).

The "Ve > (number)" bit comes in when you're calculating the multiplier in the formula. In this case, you're using a multiplier of 150 in the equation, which corresponds to Ve > 10 (I think I explained how you get 150 from that in that thread). So it means that you can hold onto gases with molecular mass over 5.55 basically forever, assuming that the temperature and so on don't change.

Using a constant of 150 is a bit overkill, but you can get away with using Ve > 6 (which means a constant of 54, IIRC) and that would probably be more realistic.

Is that any clearer? If not, just ask more questions

20. Newbie
Join Date
Sep 2007
Posts
4
Ah awesome, thats so much, I fully understand now

21. you're welcome

22. Newbie
Join Date
Apr 2008
Posts
1

## My attempt

I apologize for this but I am trying to get an approximation for MMWR. So if anyone could help me, I would like to create a simple formula that I can plug into a speadsheet.

If I assume that Ve > 6 Vt (over billions of years) then could I approximate with

Calculation for MMWR:
= B / (50 x D^2 x K)

Variables
B Blackbody temp
D Diameter of Planet in Earth Diameters
K Planetary Density In Earths (which = 5.515 g/m^2)

The value of 50 I found in a gaming book for the calculation of MMWR (in the book it was 60 but I found that this produced a MMWR value for too high for present day Earth). In fact, I was thinking this constant is still too high but an not sure what this constant represents.

and Blackbody Temp (B)
= (278 * (L^-4)/(R^-2)

Variables
L Luminosity of Star (in Sols)
R Orbital radius of Planet (in AUs)

This gives me a present day MMWR value of 5.6 for Earth.

23. ## May be of interest

Originally Posted by baric
I understand that the surface gravity of a planetary can dictate what gases can be retained for its atmosphere. Hydrogen, for example, is the lightest element and requires the greatest gravity to retain.

Is there a resource that lists, or a way to calculate, what level of surface gravity is required to retain the most common planetary gases?
I use sometimes this site : "Planet Designer"

http://www.transhuman.talktalk.net/iw/Geosync.htm

24. Originally Posted by baric
But even if you assume 300K for Earth...
The temperature should be that at the altitude from which the molecules are able to escape. That's high up in the atmosphere (lower down even fast-moving molecules will collide with others and change direction before they can escape).

I believe a figure of 1000 K is typical for the Earth.

25. Newbie
Join Date
Oct 2011
Posts
1
New and improved atmospheric gas molecular weight table.

Code:
```Gas or Vapor                    Formula Molecular Weight
Hydrogen (Atomic)               H       1.008
Deuterium                               2.014
Hydrogen (Molecular)            H2      2.016
Helium                          He      4.02
Nitrogen (Atomic)               N       14.0067
Oxygen (Atomic)                 O       15.9994
Methane                         CH4     16.044
Ammonia (R-717)                 NH3     17.031 / 17.02
Water Vapor - Steam             H2O     18.016
Neon                            Ne      20.179
Carbon Monoxide                 CO      28.011
Nitrogen (Molecular)            N2      28.0134
Nitric Oxide                    NO2     30.006
Oxygen (Molecular)              O2      31.9988
Hydrogen Sulfide                H2S     34.076
Argon                           Ar      39.948
Carbon Dioxide                  CO2     44.01
Nitrous Oxide                   N2O     44.012
Nitrogen Dioxide                NO2     46.006
Ozone                           O3      47.998
Sulfur Dioxide                  SO2     64.06
Sulfur Trioxide                 SO3     80.062
Krypton                         Kr      83.80
Xenon                           Xe      131.30```
Bigger table that includes hydrocarbon and man-made gasses.
Code:
```Gas or Vapor                    Formula Molecular Weight
Deuterium                               2.014
Hydrogen (Molecular)            H2      2.016
Helium                          He      4.02
Methane                         CH4     16.044
Hydroxyl                        OH      17.01
Ammonia (R-717)                 NH3     17.031 / 17.02
Water Vapor - Steam             H2O     18.016
Natural Gas                             19.00
Neon                            Ne      20.179
Acetylene                       C2H2    26.04
Carbon Monoxide                 CO      28.011
Nitrogen (Molecular)            N2      28.0134
Ethylene                        C2H4    28.054 / 28.03
Air (on Earth)                          28.966
Nitric Oxide                    NO2     30.006
Ethane                          C2H6    30.07
Oxygen (Molecular)              O2      31.9988
Sulfur                          S       32.02
Methyl Alcohol                          32.04
Hydrogen Sulfide                H2S     34.076
Hydrogen Chloride                       36.461
Hydrochloric Acid               HCl     36.47
Fluorine                                37.996
Argon                           Ar      39.948
Propene (propylene)             C3H6    42.08
Carbon Dioxide                  CO2     44.01
Nitrous Oxide                   N2O     44.012
Propane                         C3H8    44.097
Nitrogen Dioxide                NO2     46.006
Ethyl Alcohol                           46.07
Ozone                           O3      47.998
Sulfuric Oxide                  SO      48.063
Methyl Chloride                         50.488
1-Butene                                56.108
cis -2-Butene                           56.108
Iso-butene                              56.108
trans-2-Butene                          56.108
Butylene (Butene)               C4H8    56.11
Butane                          C4H10   58.1
Iso-Butane (2-Metyl propane)            58.12
N-Butane                        C4H10   58.12
R-611                                   60.05
Nitrous Trioxide                N03     62.005
Sulfur Dioxide                  SO2     64.06
Ethyl Chloride                          64.515
Chlorine                        Cl2     70.906
Iso-Pentane                             72.15
Methyl Butane                           72.15
N-Pentane                               72.15
Carbon Disulphide                       76.13
Benzene                         C6H6    78.11
Sulfur Trioxide                 SO3     80.062
Krypton                         Kr      83.80
Cyclohexane                             84.16
Hexane                                  86.17
R-22                                    86.48
Toluene                         C7H8    92.141 / 92.13
N-Heptane                               100.20
R-134a                                  102.03
N-Octane                                114.22
N-Octane                                114.22
R-12                                    120.92
Xenon                           Xe      131.30
R-11                                    137.37
R-123                                   152.93
R-114                                   170.93```

26. Established Member
Join Date
Feb 2009
Posts
2,206
What are the observed exosphere temperatures of Titan, Triton and Pluto?

27. Newbie
Join Date
May 2014
Posts
4
Originally Posted by EDG
I suspect it's the constants that are the problem. Though even if it's set to 54 (36 x 1.5) the MMW retained is still 1.99. Most vexing.
It's not the constants -- you have to use exosphere temperature, which is something like 1500K for earth. The temperatures you are using are down near the surface of the planet, you need to think in terms of where the boundary of space is.

#### Posting Permissions

• You may not post new threads
• You may not post replies
• You may not post attachments
• You may not edit your posts
•