The Habitable Zone Atmosphere (HZA) is a measure of the potential of an exoplanet to hold a habitable atmosphere. Most life forms requires some basic atmospheric ingredients like carbon dioxide and oxygen. These gases thermally escape the gravity of a planets unless they have the right mass, size, and temperature. In general, small and hot planets will be airless and those big and cold will have dense atmospheres (assuming not cold enough for atmospheric collapse). Nitrogen, the next volatile after hydrogen and helium, is usually neglected in these considerations. Nitrogen is necessary for all life and mostly available to plants by microorganisms in a process knows as nitrogen fixation. Without atmospheric nitrogen (some available in rocks too) the food chain will break and a planet will probably be only habitable for microbial life.
Here we used the thermal escape limits between atomic nitrogen (N) and atomic hydrogen (H) to define the planetary mass-radius boundaries for the atmospheres of a habitable planets (Figure 1). Planets able to hold atomic N will also be able to hold all the other heavier volatiles necessary for life such as carbon dioxide (CO2), oxygen (O2), methane (CH4), ammonia (NH3), and water (H2O). The ability of a planet to hold atomic hydrogen, the most abundant element in the universe, will probably end up in a dense high-pressure atmosphere making carbon dioxide or water solid (or a supercritical fluid) at the surface level, and therefore non habitable.
A mass-radius relationship for the habitable atmospheric limits of a planet is simply derived by equating the planet's escape velocity with a conservative 10 times (Jeans parameter) the thermal escape speed of atomic N and H as:
where m and r are the mass and radius of the planet in Earth units, respectively, Teq is its equilibrium temperature in kelvins (calculated with an albedo of 0.3), Mw is the atomic weight of N (14 g/mol) or H (1 g/mol), and z is a constant equal to 2x10-2 mol g-1 K-1. The HZA is given by:
where MwN = 14 g/mol (atomic nitrogen) and MwH = 1 g/mol (atomic hydrogen).
Figure 1. Plot defining the limits of planets with a potential habitable interior and atmospheric composition (dark green area). Exoplanets in red dots and Solar System planets in black dots. The Habitable Zone Atmosphere (HZA) and the Habitable Zone Composition (HZC), complement the Habitable Zone Distance (HZD) as criteria for potential habitable exoplanets. The seven exoplanets that match the habitable interior and atmosphere composition does not mach the criteria for liquid water given by the HZD.
The HZA was only based on atmospheric thermal escape (Jeans escape) but there are many other process controlling planetary atmosphere such as the initial inventories, stellar wind erosion, impact weathering, sequestration, and the presence of magnetic fields. These processes could also be included in similar formulations.