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Astrobiology Magazine's Amanda Doyle describes one process by which mini-Neptunes orbiting red dwarfs might evolve into broadly Earth-like worlds.

[W]hat if a gas giant migrated into the habitable zone? Astronomer Rodrigo Luger of the University of Washington, along with colleagues, have found that a certain kind of planet called a mini-Neptune with its atmosphere removed could, in fact, become a viable planet to life.

A mini-Neptune is a gaseous planet that is up to ten times the mass of the Earth. Such a planet would be engulfed in a thick atmosphere of gas and then would need to lose its envelope before becoming a water-rich world.

[. . .]

There are two ways in which a mini-Neptune could evict its atmosphere. The first is via a process known as hydrodynamic escape. Extreme radiation from the host star in the form of x-ray and ultraviolet rays bombard the planet, causing the atmosphere to heat up. The upper atmosphere then expands, forcing the gas to accelerate to supersonic speeds. This hydrodynamic wind is fast enough for the atmosphere to escape into space.

The second way for a mini-Neptune to shed its cloak of gas is for the atmosphere to become so extended that it is no longer gravitationally bound to the planet. The area around a star or planet where material is gravitationally bound is known as the Roche lobe. Once gas reaches the edge the of this teardrop-shaped lobe it can escape, and this is known as Roche lobe overflow. Roche lobe overflow couldn’t occur during planet formation, as it simply wouldn’t accrete the material in the first place. However, a planet migrating inwards will start to feel the effects of the star’s gravity more and more, and this can trigger the overflow.

Once the initial atmosphere is gone, the solid core left behind becomes a terrestrial planet. Assuming that a secondary atmosphere could form through a process such as volcanic outgassing, this core could become habitable, earning it the name “habitable evaporated core” (HEC).
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