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PostPosted: Sat Dec 21, 2013 9:31 pm 
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(this is a repost from another thread, just figured it's useful and comes up enough to have its own thread).

Atmosphere retention depends on the Luminosity of the star, the distance of the planet from the star, the mass of the planet, and the radius of the planet (the last two essentially can be combined into "the surface gravity of the planet"). The exact relationship is somewhat fuzzy and recursive though, since it technically really depends on the top of the planet's exosphere (from which gaseous molecules can escape) - which is in part influenced by the atmospheric gases that it holds - and also on things like whether the planet has a magnetic field and how effectively the solar wind coming in from the star can break up molecules and strip atoms away from the atmosphere.

But the "simple" way to do it is to use the following equation:

Code:
mmw = 150*0.0001322*((T*r)/M))

mmw = minimum molecular weight retained
M = mass of the planet ([b]in EARTH MASSES[/b]) (1 earth mass = 5.9742e24 kg)
T = blackbody temperature of planet (Kelvin)
r = radius of the planet ([b]in EARTH RADII[/b]) (1 earth radius = 6,378 km)


Addendum (from a later post):
I guess using units of km might be more intuitive, since not everyone's going to figure out the radius of their planet in earth radii... So here's an equation with more sensible units to use:

Code:
mmw = 150 * (2.074e-8) * [(T*r)/M]

150 = atmospheric retention constant (explained below)
T = blackbody temperature in Kelvin
r = radius of planet in km
M = mass of the planet in Earth masses.
(If you're not familiar with scientific notation, 2.074e-8 is 0.00000002074).


Or if you just want everything in metric units (which is the safest way to do it, IMO):

Code:
mmw = (150*k*T*r)/(G*M*h)

150 = atmosphere retention constant
k (the Boltzman constant) = 1.38E-23 J/K
T = blackbody temperature in Kelvin
r = radius of planet in [b]metres[/b] (1 earth radius = 6378000 m)
M = mass of planet in kg (1 earth mass = 5.9742e24 kg)
G (Gravitational Constant) = 0.00000000006672559 N (m/kg)²
h (mass of a hydrogen atom in kg) = 1.67e-27 kg



The minimum molecular weight (m) retained is determined by the ratio of the thermal escape velocity (Vt) of a gas to the gravitational escape velocity of the planet (Vg). If Vg < Vt, the gas is energetic enough to escape from the planet and be lost into space over some timescale. If Vg > Vt then it becomes harder for the gas as the ratio between the two velocities increases, until the gas cannot escape at all. This is because distribution of molecular velocities of the gas follows a Maxwellian distribution - it’s not that all the gas particles may have velocities less than the gravitational escape velocity, it’s just that the average velocity is less. This means that the fastest particles will have velocities greater than Vg, and so will be lost into space - so in practice, Vg has to be considerably greater than Vt in order to retain the gas over billions of years.

Most of the texts suggest that Vg > 6Vt would be fine to use for a gas to be retained for billions of years:

Vg > 6Vt: gas retained for billions of years.
Vg > 5Vt: gas retained for tens of millenia/a million years.
Vg > 4Vt: gas retained for decades/millennia.
Vg > 3Vt: gas retained for weeks/months.
Vg > 2Vt: gas is retained for hours/days.

The 150 in the equation I gave is related to this ratio, and assumes that Vg > 10Vt, but that's just my own preference, to be really conservative about it. If you want to calculate alternate constants then you just square the number before Vt and multiply that by 1.5 (10²*1.5=100*1.5=15, so if you use the 6Vt then you should get use 54 as the constant (6²*1.5=36*1.5=54).

The 0.0001322 in the equation is what the other constants (the Boltzmann constant k, the gravitational constant G, and the mass of the hydrogen atom) simplify to.

So that gives you an m value, which you then compare with the values on the table below for the molecular weights of the gas molecules:

Code:
Gas                     Molecular Weight
     
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


So let's say you get a result of m=30 for your planet, assuming you're looking at a timescale of billions of years (which is what you want for a long term atmosphere). That means that the lightest gas that it can retain has a molecular weight of 30 - anything with less than that will not be retained over that timescale (if you change the 150 constant in the equation to a smaller value and compare the results, you can use that to find out how quickly it will take gases to be lost to space). So with m= 30, the planet can retain molecular oxygen, carbon dioxide, sulphur dioxoide, etc.

Also, just because an atmosphere can hold onto a gas, doesn't mean it necessary HAS that gas in it. The gases with * next to them are common atmospheric gases, generally produced by volcanism. Things like oxygen and methane are generally reactive and tend to be actively produced (e.g. by life, or photolysis), so you have to be sure that the gases will not all react with eachother and get used up or turned into other gases.

It all gets rather complicated when you get in depth on the subject, but what I've describe here is a reasonable approximation to reality.

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PostPosted: Fri Jan 03, 2014 12:34 pm 
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Can I piggyback on this post with a related question?

GURPS Traveller: First In categorizes worlds based on size, MMW retained and location in the system, and one category of world is "subgiant". Is this a thing? Large enough to retain hydrogen but did not accumulate enough to become a full-fledged gas giant.

I wondered the same thing about "failed core" worlds in 2300.


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PostPosted: Fri Jan 03, 2014 2:13 pm 
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Not sure about the specific GURPS meaning, but a Sub-Giant could be a Neptune/Uranus sized gas giant OR it could mean a world that can retain Helium (He), but not molecular Hydrogen (H2).

In GURPS Space, they call these worlds LARGE as a size category.

It might also have been a term used to describe what are now being called "Super-Earths".

I would say that a Failed Core in 2300 would be a Super-Earth world now. Back then, there were a lot of people that recogized that there was a gap in world sizes within our solar system. Something larger than the Earth, but not a gas giant. A variety of terms were used by a lot of different games to try to come up with a way to categorize something that no one had discovered yet. Panthelassic is another term for these larger worlds.

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PostPosted: Fri Jan 03, 2014 3:44 pm 
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Thanks, uh, "Sir". That fits with my semi-scientific understanding of it (I have a degree in pretend physics).

So, if I'm following along, inner system you have the potential for two types of very large worlds: 1) a gas giant that formed out-system and wandered in, or 2) a super-earth/panthelassic.

But a world large enough to retain hydrogen/helium in the outer system (past the snow line?) would eventually become a gas giant. It looks like, based on that equation above, a world has to be relatively small in order to not become a GG once you get out in the orbital boondocks.


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PostPosted: Fri Jan 03, 2014 4:39 pm 
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A 'subgiant' would be a terrestrial world that can retain hydrogen and helium - for whatever reasons (stellar wind? lack of materials to scoop up?) it didn't snowball enough to become a full-fledged gas giant. It'd be what we'd call a 'mini-Neptune' nowadays (as opposed to a Superearth, which would be more earthlike but bigger than Earth). It would most likely have a hundreds of km deep supercritical ocean on its surface under hundreds or thousands of atmospheres pressure, beneath a mostly hydrogen/helium atmosphere at least 1000 km thick.

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PostPosted: Sun Dec 18, 2016 3:29 am 
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EDG wrote:
A 'subgiant' would be a terrestrial world that can retain hydrogen and helium - for whatever reasons (stellar wind? lack of materials to scoop up?) it didn't snowball enough to become a full-fledged gas giant. It'd be what we'd call a 'mini-Neptune' nowadays (as opposed to a Superearth, which would be more earthlike but bigger than Earth). It would most likely have a hundreds of km deep supercritical ocean on its surface under hundreds or thousands of atmospheres pressure, beneath a mostly hydrogen/helium atmosphere at least 1000 km thick.


Digging this thread up as I trudge on with my system generator...

Sub giants/failed core planets located in the habitable or middle zones, would their atmospheres have enough greenhouse gases to capture warmth as venus does?

On a related note, a Titanian world in the middle zone, could it potentially be quite warm? Would the methane gas act as a greenhouse gas and warm the planet noticeably above it's BBT?


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PostPosted: Sun Dec 18, 2016 7:18 am 
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hiro wrote:
Sub giants/failed core planets located in the habitable or middle zones, would their atmospheres have enough greenhouse gases to capture warmth as venus does?

Some would, some wouldn't. The examples of Venus and Earth show that you can get planets that are structurally similar and about the same size but that have very different atmospheres.
Quote:
On a related note, a Titanian world in the middle zone, could it potentially be quite warm? Would the methane gas act as a greenhouse gas and warm the planet noticeably above it's BBT?

Umm. The average surface temperature on Titan is about 94K, which is between the melting point of methane ice (90.7 K) and the boiling point of liquid methane (111.7 K). So methane on Titan is like water on Earth, and what you have in Titan's atmosphere is not methane gas but methane vapour. Titan's atmosphere consists of 98.4% nitrogen, and I suspect that its methane is retained against Jeans escape partly because it condenses and rains out before it reaches the effective top of the atmosphere. So a Titanian world is one where the atmosphere consists substantially of nitrogen, water is a rock-forming mineral, and methane goes through a cycle like the hydrographic cycle.

Any body that is massive/cold enough to retain methane against Jeans escape must also retain water vapour, because water vapour's molecular mass (18) is higher than methane's (16). If you bring a Titanian world in to the middle or habitable zones (increasing its gravity to prevent methane from evaporating off to space, but not so much that you start retaining helium), the first thing that happens is that it warms up to the point where methane is not longer a liquid at the surface: the methane seas boil away, leaving you with a standard ice world having a nitrogen atmosphere with traces of methane. Coming in further you get to the point where the water-ice melts and water vapour becomes a significant constituent of the atmosphere. You get an ocean world or a greenhouse world.

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PostPosted: Sun Dec 18, 2016 7:35 am 
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EDG wrote:
So that gives you an m value, which you then compare with the values on the table below for the molecular weights of the gas molecules:

Code:
Gas                     Molecular Weight
     
Molecular Hydrogen (H2)     2.00
Helium (He)                 4.00
Methane (CH4)              14.00
Ammonia (NH3)              17.00
Water Vapour (H2O)         18.00*

The molecular weight of methane (CH4) is closer to 16 than to 14.

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PostPosted: Sun Dec 18, 2016 8:17 am 
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EDG wrote:
A 'subgiant' would be a terrestrial world that can retain hydrogen and helium - for whatever reasons (stellar wind? lack of materials to scoop up?) it didn't snowball enough to become a full-fledged gas giant.

What if it retained helium but not hydrogen?

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PostPosted: Sun Dec 18, 2016 4:15 pm 
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Agemegos wrote:
Umm. The average surface temperature on Titan is about 94K, which is between the melting point of methane ice (90.7 K) and the boiling point of liquid methane (111.7 K). So methane on Titan is like water on Earth, and what you have in Titan's atmosphere is not methane gas but methane vapour. Titan's atmosphere consists of 98.4% nitrogen, and I suspect that its methane is retained against Jeans escape partly because it condenses and rains out before it reaches the effective top of the atmosphere. So a Titanian world is one where the atmosphere consists substantially of nitrogen, water is a rock-forming mineral, and methane goes through a cycle like the hydrographic cycle.

On the other hand, Titan's equilibrium (black body) temperature is 85K, so there is some greenhouse warming going on. A world with a nitrogen-methane atmosphere, necessarily beyond the snow line, would still be warmer than an airless rockball in the same position.


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