Tangential follow up to my earlier question about Chthonian planets.
What I am wondering is if, given a planets mass and radius, you could calculate its likely atmospheric mass and density, and therefore also things like surface pressure, surface atmospheric density, etc.
I figured that a more massive body is more likely to capture and hold an atmosphere, so there has to be some relation between it.
Radius: 21.948km
Mass: 2,334533e26kg (39,09 Earth masses)
No, with just radius and mass anything done about the atmosphere is pure speculation.
If you know the distance from the star and can estimate a temperature range you might determine which gases are more likely to be retained in the planet gravitational well, following the notorius Wikipedia chart,
but that won't tell you their abundances, so you won't be able to get a composition out of that.
Just look at our Earth: mass and radius haven't changed, but the atmospheric composition has, from the moment of its formation to the present day.
You need to know the history of the system where the planet is located to be able to make some educated estimate on the atmospheric composition.
There isn't a direct correlation between planetary mass and atmospheric composition/mass/density. While some general trends have been observed which could give you a fair answer in this case, I can state with fair confidence that this planet quite likely cannot exist.
If modeled as a sphere — which it isn't, but the approximation is close enough — this planet has a density of about 5,271 kilograms per cubic meter, comparable to Earth. The problem is that it has Earth-like density but the mass of a fairly substantial ice giant — not even a mini-Neptune, because it's more massive than Neptune and Uranus combined. The mass of planets of this size means they invariably acquire large gas envelopes that reduce their density to something below 2,000 kilograms per cubic meter. Here is a complicated answer by HDE 226868 which explains this relationship better than I can; the long and short of it is that the bigger the planet is the better it is at holding onto fluffy, lightweight things like hydrogen, helium, nitrogen, and the like.
Note that I say "gas envelope", not atmosphere. See, the thing about really big planets is that the difference between the surface and the atmosphere stops existing. On Jupiter, for instance, there's no boundary between "air" and "ground", nor is there ground in the first place; the atmosphere simply gets thicker and thicker the further down you go, and at some point it likely starts being a supercritical fluid instead of a gas.
So this planet might have an enormous rocky core that indeed masses 39.09 Earth masses and has a radius of 3.5 Earth radii, but that enormous rocky core is going to become wrapped in a gigantic blanket of what will most likely be water vapor, methane, and ammonia, and to some lesser degree hydrogen and helium.
Now, it is possible for you to have a rocky planet that's larger than the Earth. Super-Earths are generally accepted to exist and mega-Earths or supermassive terrestrial planets are hypothetically possible. However, even the largest proposed mega-Earths, barring freakish exceptions which may be due to data processing errors and not actually exist, are somewhere around Neptune-sized. Below are some tables detailing the expected compositions of planets with certain radii and masses, and here's a video with a more detailed table.
As these tables can show you, it is, to some extent, possible to make an educated guess about some atmospheric characteristics based solely on planet size. But I would have a hard time believing something with the mass of your planet is going to have that density, and I can also state with a fair degree of confidence confidence that it'll be a Neptune- or Uranus-like planet.
Now, there is one exception to this — as in your previous question, this planet could be a Chthonian planet. However, it would have to have formed out of quite a large gas giant; Jupiter's core, if it is rocky, is perhaps 14 to 18 times the mass of the Earth, and this planet is larger. The range of bodies which would have a 39-ish Earth mass rocky core are somewhere between "Jupiter but larger" and "incredibly small brown dwarf". However, the solar flux needed to blast the gas envelope off a body that size quickly enough for life to evolve before the star in question dies would also likely be enough to bake the surface of the resulting planet (or even disintegrate it), so take this with a large grain of salt.
Venus has about the same mass and Radius as the Earth. But the atmosphere of Venus is about 100 times greater than that of the Earth. And due to higher temperature, Venus should have a sparser atmosphere, due to the percentage of escaping moleculas! So, size and mass, even with a temperature(or the relative brightness of the sun), are definitely not enough to determine or even to suppose the atmosphere's size.
Of course, if the planet is very small and close to the sun, there will be no atmosphere. But it is not your case. Such a heavy planet can have a significant escape percentage only being very close to the sun. Being far from the sun, your planet can easily consist of gas only, as Saturn.
