Super-Earth Atmospheric Calculations

[The Hawk]

Help me, o wise forum sages!

I have spent much of my long weekend reading up on super-earths (which definitely doesn't have anything to do with noodling an idea for a Kopernicus mod planet. ) Starting with Wikipedia's super-earth page, my concept was basically a world on the scale of COROT-7b or Kepler-10b -- in other words, at the upper end of terrestrial. Unlike those worlds, however, I was envisioning something far enough away from its parent star to not have lost its primordial hydrogen/helium envelope from ultraviolet exposure.

I found my way to this paper, which contains this quote:

"...if we assume a more typical low-mass planet with a 5 [Earth-mass] core, then to be 2.0 [Earth-radii] it would need 0.5% of its mass in a H/He envelope. This may not sound like much, but it corresponds to ~20 kbars of hydrogen and helium, ~20x higher than the pressure at the bottom of the Marian[]as Trench. Moreover, the temperature at the bottom of such an envelope would be >~3000K... We believe that such a planet is more properly classified as a sub-Neptune."

Well, that's obviously more than I'm going for. So after ruminating a bit and consulting the tables in that paper, I decided on the following planetary characteristics:

Mass: 4.2 MKerbin (2.222463306 x10^23 kg)

Equatorial Radius: 1.5 RKerbin (900,000 m)

From that I derived:

Gravity: 18.308158834093 m/s^2 (~1.867 G)

So far, so good. I also decided on .01% of the planetary mass tied up in the atmosphere. From that I derived:

Surface-level pressure: ~39974624 Pa = ~394.5 atm

Heady, but survivable by human-made craft, although compared to Eve's 5 atmospheres at sea level, getting back is going to be... challenging. I also arbitrarily decided on 3.5 g/mol for the mean molecular weight of the planetary atmosphere, on the theory that volcanism, and probably atmospheric processes, would dump some heavier elements into the atmosphere, but it would remain largely H/He. (I have no idea how realistic a figure this is, but compare to 2.07 for Saturn, 2.64 for Uranus.)

Unfortunately, that's where my very limited command of physics hits the wall. I have no idea how to derive the density, (probably enormous) surface-level temperature, or volume of this atmosphere, but I am given to believe that without one of those three things I can't calculate the others. If I have to make up one of those values, I don't know what a good starting point would be. If I can get temperature, I understand I can figure out the scale height and, from there, a reasonable extrapolation of the height of the atmosphere.

The other question I have is, what would conditions on this theoretical world be like? Given the described pressure and atmosphere (and whatever the derived temperature is), would the atmosphere still be gaseous at surface level? Would landing using stock parts even be feasible, or would the temperature be too intense (or would moving through such a thick atmosphere at any appreciable speed -- like that necessary to achieve orbit -- cause such significant overheating as to destroy the craft)? If landing is feasible, what about liftoff?

Assuming the goal is to have a rocky surface (rather than one encased in a sheath of liquid hydrogen), what would the byplay between the described atmosphere and the surface be? Would the hydrogen be pulled out of the atmosphere into compounds? What impact would typical (terrestrial) levels of volcanism have on the atmospheric composition? Does that make my mean molecular weight way too low?

At the upper end, would the atmosphere be layered (and if so, how)? How would the planet go about retaining its atmosphere? Assuming a distance from Kerbol comparable to real-world TNOs' distance from the sun, and therefore very cold corresponding temperatures of perhaps 12K-40K, would hydrogen liquefy and precipitate at some altitude? Or would the retained heat of the planet, and any geological activity, keep overall atmospheric temperatures up?

Thanks in advance for your thoughts!

[K^2]

The atmosphere will be a supercritical fluid. Which means, there is no longer a difference between it being a gas or a liquid. It also means that, unfortunately, most formulae you could have used for scale height are going to be useless. This sort of fluid is very far from ideal gas. The relationship between pressure, volume, and temperature is dominated by molecular interactions rather than ballistic movement of particles in an ideal gas.

Fortunately, the goal isn't to get exact numbers, but rather something plausible. I would assume, for sake of calculations, that density isn't going to change much once your atmo hits critical pressure. You can't really model that in KSP, but we'll pretend for sake of temperature calculations. I also am not sure what the critical pressure of this mix would be, but critical pressure of hydrogen is 12.96 bar, so lets use that. Next, you need pressure at the upper cloud layer, because that's where radiation equilibrium with the star will establish. Do you know how to compute radiation equilibrium, by the way? Wikipedia's article on Black-Body Radiation has a section called "Temperature of Earth" that runs through an example. Use that to compute your T₀. You also need pressure at cloud layer, P₀. I have no idea what that should be, but I'd use pressure of upper cloud layer on smaller gas giants as a guess. From here, I would assume Adiabatic compression to your P₁, which is your critical pressure of 12.96 bar. Use that to get T₁, which you'll use as surface temperature. Now, this is technically wrong. Atmospheric compression isn't Adiabatic, but correct model depends on so many factors... And you need an estimate, which this will give you.

Finally, we get back to the question of scale height. Now, KSP actually has support for non-exponential atmosphere in its source code, but I don't know how to enable it for modded planets, or if it's even possible, so even though it's technically isn't right for supercritical atmo, you'll still need an H value. As I've explained above, anything you'd get from scale height formula is technically wrong. But what I'd do is get scale heights at T₀ and T₁, and use something in between. Real planet probably wouldn't be anything like that in terms of your ability to escape it, but it should, at least, give you something semi-plausible for landing probes. And that's probably all you'll be able to do with this world, so maybe that's good enough?

For atmospheric height, just see at what pressure they cut it off in KSP, and use your scale height formula to find the correct cutoff altitude. Again, it's not really correct, seeing how so many standard assumptions are incompatible with this world, but at least this value will be self-consistent with all the assumptions KSP makes.

[The Hawk]

Well, the goal is to have solid terrain on which a landing and return is at least theoretically possible even if exceedingly difficult (otherwise it may as well be a gas giant). I'm not married to the above figures if they conflict with that goal. It seems to me the way to accomplish that is either reduce the overall mass of the atmosphere so the pressure is reduced, or increase heavy elements in the composition of the atmosphere such that supercriticality is not an issue.

The Venusian atmosphere is said to be a supercritical fluid at surface level, and indeed the temperature and pressure there are well above the critical point for CO2, but it still seems to function substantially as a gas and did not prevent probe landings on the surface (though obviously the temperature was a pretty significant problem. ) I confess, though, that the thought of an atmosphere on the tender edge in terms of pressure, such that below a certain altitude there are "seas" (as KSP conceives of them) of supercritical fluid, with "islands" of higher terrain in the gaseous portion of the atmosphere, is interesting to me.

[K^2]

Why would liquid "atmosphere" prevent a probe from landing? What you don't want is a sharp boundary between gas and liquid. That can cause all sorts of problems to the probe during landing. Even that isn't an automatic disqualifier. But all you are really dealing with here are extreme pressure and density that's approaching that of a liquid.

[windows_x_seven]

With that much atmospheric pressure, I doubt that there would be life there...

[Xannari Ferrows]

You could study Jupiter to learn about fluidic atmospheres. The two are slightly different concepts, but it may be of some use.

[The Hawk]

Why would liquid "atmosphere" prevent a probe from landing? What you don't want is a sharp boundary between gas and liquid. That can cause all sorts of problems to the probe during landing. Even that isn't an automatic disqualifier. But all you are really dealing with here are extreme pressure and density that's approaching that of a liquid.

Unfortunately, if this is used as the basis for a mod planet, KSP only knows of a sharp boundary between gas and liquid; there has to be a sea level somewhere, and the game needs to know when you've transitioned from "flying" to "splashed down". But the goal is for there to be land and seas, not merely seas of liquid hydrogen at some suitable density of the fluid; as I say, if that's the case, it might as well be a gas giant.

With that much atmospheric pressure, I doubt that there would be life there...

Well, 394.5 atmospheres is about 40% of the pressure at the bottom of the Marianas Trench, and there's life at the bottom of the Marianas Trench. The bigger issue is probably temperature, and whether life is capable of developing in such an atmosphere at all. But the presence of life is not terribly important to the concept.

You could study Jupiter to learn about fluidic atmospheres. The two are slightly different concepts, but it may be of some use.

I have, but a fluidic atmosphere is really not the goal, merely a consequence of the parameters outlined above. Those can be changed.