Calculating the Atmospheric Properties of an Exoplanet

Question

I am developing a double-star system filled with several planets, a few of which will be habitable. I also have a spreadsheet file full of all the planets information, from Semi-Major Axis, orbital inclination, and Argument of Periapsis, to planet's mass, blackbody temperature, surface gravity and so on. What I am trying to figure out is how to calculate the atmospheric pressure on each of the main planets. I have looked into several other posts here as well as multiple websites and have not been able to find anything that can allow me to determine the overall mass and pressure of the atmosphere.

As an example, one of the planets is as follows:

• Mass: 1.3507 Earths
• Radius: 1.1024 Earths
• Blackbody Temperature: 273.85K
• Scale Height: 7.247km
• Atmospheric Composition: Nitrogen:21.208%, Oxygen: 21.208%, Argon: 0.896%, etc.
• Atmospheric Mean Molar Mass: 0.02884 kg/mol

Using the scale height, I calculated that at 108.7km there will be 99.99998% of the atmosphere below. What I am trying to figure out is how to take the 0.02884kg/mol and the estimated atmospheric volume of 8.2086x10^19 cubic meters, and get a Surface Pressure it kPa, as well as a mass of the entire atmosphere. I am assuming roughly equal stellar wind pressures and an equal magnetosphere to that of Earth.

I am hoping to find some formula that can use the numbers I have available, but I am open to anything. Thanks everyone.

Answer 1

You can make a decent approximation of the surface pressure $p_0$ of a planet just knowing its mass $M$, radius $R$ and atmospheric mass $M_{\text{atm}}$: $$M_{\text{atm}}=\frac{p_0}{g}4\pi R^2=\frac{4\pi R^4p_0}{GM}\implies p_0=\frac{GMM_{\text{atm}}}{4\pi R^4}$$ Therefore, we can split your work in half and just ask what the mass of the atmosphere will be for a given planet, and we should get $p_0$ for free.

On the other hand, you definitely don't get the mass of the atmosphere for free - far from it. Determining the evolution of a rocky planet's atmosphere entails considering a lot of things (see e.g. Lammer et al. 2018, Elkins-Tanton & Seager 2008, Seager & Deming 2010), including:

• The initial mass of the envelope
• Atmosphere loss due to hydrodynamic escape
• The role of stellar winds and magnetic fields
• Outgassing
• Contributions from impact events
• Activity of the parent star

. . . and more.

Add to this the fact that our observational understanding of exoplanet atmospheres is not stellar - particular when it comes to low-mass terrestrial planets, although hopefully it will improve with the launch of JWST and better spectroscopic possibilities. This means it's also difficult to try an alternative approach: looking at existing data points and trying to extrapolate from that, as opposed to a calculation from first principles.

There are certainly some starting points. Lammer et al. discuss how much of a molecular hydrogen atmosphere a rocky core will accrete early on (see Fig. 2; the true mass of the planet will of course be larger than just the initial core mass). Most of that will be lost through hydrodynamic escape, in the case of Earth-like planets. Assuming you care about the long-term composition and evolution, Elkins-Tanton & Seager present some models which might help you estimate the atmosphere the planet will develop later on through degassing; there are too many details for me to present here. As an example, though, a planet with a composition of the CV chondritic class can develop an atmosphere which is composed of $0.0138M_p$ of water, $0.0027M_p$ of hydrogen, $0.0001M_p$ of nitrogen, and $0.0062M_p$ of carbon, adding up to a rather high atmospheric mass of $0.0228M_p$, with $M_p$ the mass of the planet.