What size/distance a planet should be from its star to have a dense atmosphere and not become a gas giant?

Question

I have recently learned that should Earth form a tad farther from the Sun (at about the distance of Mars) it would have enough gravity pull to accrete hydrogen and not enough heat radiation received from the Sun to wind it away, thus would eventually form a gas giant. Also there was a discussion elsewhere about zeppelins, that ended with a phrase "should Earth's atmosphere be triple density, the life on the planet would grow floating bladders". I tried to wrap my mind around this, but failed to actually come up with a range of conditions that would a) support any form of life, including hypothetical (say if liquid CO2 would be found out to support life in place of water, or at least create enough greenhouse effect to maintain surface-close temperatures to keep water liquid at least somewhere), b) would have its atmosphere dense enough to allow floating lifeforms, with whatever lifecycles, or dense enough to allow modified (localized) humans to fly on zeppelins with relative ease, and c) not end up a gas giant due to its own gravity. Googling about super-Earths did not reveal enough data to start the concept, as they are either too close to a very dark star (brown or red dwarf like TRAPPIST-1), so that the star's wind is not to scale with our system, or just are way too far away from their stars to have N and O solidified on the surface, so whatever atmosphere that planet could have is either a H2 layer too thin, or absent. And, I couldn't find how the density of a planet's atmosphere is dependent on its radius, surface temperature and surface gravity, while I believe there is some, however my calculations returned that a planet at Earth's position (with its temperature) but with mass of 150% Earth and radius of either 100% or 150% would turn into a gas giant. So I am kind of at a loss about possible parameters for such a world.

Can you please help determine the range(s) of (size, surface temperature) for a planet to have an atmosphere dense enough, be able to support life and still not be able to spontaneously become a gas giant? I would tolerate settings for any temperature+pressure settings that would allow a liquid on the planet's surface to act as local "water", be it water or carbon dioxide or whatever else, so that somehow a life could form in such a matter. "Enough density" I count as 3 to several hundreds of "technical atmospheres" or 300k to 10M Pa, to actually have two layers of atmosphere and hydrosphere.

Answer 1

Be a moon

This was initially an afterthought but as I developed the idea I thought it was so good that it deserved to be the first item.

You could design an earth-distance gas giant and a habitable moon with a mass higher than the local hydrogen-accretion limit, then your higher-mass gas giant would consume all of the locally available hydrogen in the early solar system as well as any of the planet's atmosphere which made it's way outside of the planet's roche lobe. In order for this to work without destroying the planet it would need to remain outside the roche limit but within the significant gravitational influence of the companion gas giant. Basicslly you could have a habitable planet of practically unlimited mass & therefore practically unlimited atmosphere density which far exceeds the mass required for runaway hydrogen capture in a normal planet.

Roche limit

The more massive the habitable planet compared to the host, the closer the roche limit would be, the steeper the gradient would be between the beginning of roche lobe atmosphere transfer and exceeding the roche limit altogether. The ideal conditions for non-destructive roche envelope transfer would require the habitable planet to be of much lower mass than the host and to orbit at a distance where the gravitational gradient is quite shallow. This could be addressed by making the host a substellar object of up to 80 Jupiter masses which is about as big as it can get without becoming a star.

If this works then it would put a hard limit on atmospheric density and allow you to exceed the hydrogen accretion limit by any amount you wanted just by varying the mass of the life-bearing planet and it's host / or the orbital distance between them.

Deuterium star

The host planet has the potential to become a deuterium star between 13-80 Jupiters, meaning you might have a binary star system for a few million years. The solar wind from the deuterium star would also have an atmosphere-stripping effect on the orbiting planet. It might also sterilise the planet if it were close enough to be within the roche lobe transfer limit so this is likely to be an either/or scenario (either roche transfer or solar wind). Maybe the deuterium star burned out early in the planet's life cycle and it then settled down to be a roche-transfer system / or captured the habitable planet after the deuterium burning phase was over.

Tidal forces

The planet will almost certainly be tidally locked to the host and you would get one day per each orbit of the host. It's probable that such a planet would have increased volcanic activity as a result of tidal heating & if the planet were too close to the host it could still be made uninhabitable by tidal heating.

How to actually calculate it

An easily pluggable equation for calculations of the roche lobe size of planets does not seem to exist. There are some calculators which rely on lookup tables for stars (https://github.com/denisleahy/roche-radius-calculator) but the computations of individual roche lobe size appear to rely heavily on real-world observations and as there is no such object available for us to observe we would need to use creative license or do some speculative math based on the roche limit and atmospheric density.

In order to roughly estimate the roche lobe for an exoplanet, perhaps you can calculate the atmospheric density (https://www.mide.com/interplanetary-air-pressure-at-altitude-calculator) at varying altitudes and plug these densities into a roche limit calculator (https://calculator.academy/roche-limit-calculator/)

The density at which the roche limit is exceeded would be the atmospheric altitude at which atmosphere will be stripped away from the planet. In the real world this atmospheric stripping will effect the density of the atmosphere at lower levels giving rise to complicated interactions between roche envelope & atmospheric density.

Magnetic field interactions

This is further complicated by magnetic field interactions. For a start the host planet's magnetic field is strong enough that it completely diverts the solar wind before it reaches the planet so solar wind interactions are not relevant anymore, instead there are host-moon interactions.

If the host magnetic field is strong enough to form a combined field with the moon then charged particles from the moon will be directed into the host magnetic field even from within the roche lobe, but if the habitable moon has a strong enough magnetic field of it's own then this would divert escaped ions along the moon's magnetic field lines. The moon probably pumps a massive number of charged particles into the host's magnetosphere & there would be a polar wind driving atmosphere escape from within the planet's magnetic field, the moon would probably eject plumes of charged particles from it's poles. In this case particle escape would also be determined by magnetic interactions / the shape of the combined magnetic field.

At the moment this is just a nascent idea and it may turn out to be impractical, I might be back later if I can actually work it out.

Just change the starting conditions

Less water

Atmospheric pressure on an earth-like planet at an atmospheric depth of 150 miles (1.5x current surface depth) would be > 3 atmospheres. All you would need is less starting water to fill some deep parts of the surface.

More air

Air will escape, but the speed at which it can escape is limited not just by the mass of the planet but by the strength of the solar wind which for an earth-like planet is not significant enough to result in the substantial escape of heavier elements.

If you start with more air, particles such as oxygen and carbon will not be easily dislodged by the solar wind and will mostly accumulate in the atmosphere or via sequestration into dirt and rocks.

Bigger magnet

According to some models the core dynamo creates an electromagnetic shield which prevents the solar wind from energising the upper atmosphere & reduces the number of particles which can escape.

It has been shown that atmospheric escape may not be heavily incluenced by magnetic field: https://www.aanda.org/articles/aa/pdf/2018/06/aa32934-18.pdf

But this does not seem to have been modelled with alternate (ie entirely fictional) planetary configurations.

More mass

This one is tricky. Earth is near the upper range for a water-retaining planet, much cooler or much larger and it will begin to accrete hydrogen = gas giant. In order to defeat this the planet will need to have a higher surface temperature, meaning it will need to be closer to the Sun. The closer to the Sun you get the more massive the planet can be without accreting hydrogen, but you again have the problem of water retention.

The reason why much colder worlds such as Io and Titan have not become gas giants is most likely their close proximity to gas giants, so the conditions on these worlds is not necessarily a useful model when considering a free orbiting world unless you want to be in orbit around a gas giant (goto 10)

Answer 2

To determine whether a significant quantities of a gas could be retained in a planet's atmosphere over geological timescales, you can set the escape velocity of the planet equal to be much larger greater than the thermal speed of the gas. The critical point is when the ratio between thermal speed and escape velocity is approximately 1/6: $$v_{\text{esc}}=\sqrt{\frac{2GM}{R}}\approx 6v_{\text{therm}}=6\sqrt{\frac{3k_BT}{m}}$$ where $M$ is the mass of the planet, $R$ is the radius, $T$ is the temperature and $m$ is the mass of a particle of gas. Substituting in the parameters of the Earth and setting $m$ to be the mass of hydrogen, we see that Earth would need to have a surface temperature of somewhere around 140 Kelvin to retain a hydrogen envelope.

The surface temperature of the planet in turn scales roughly as $T\propto r^{-1/2}$ with $r$ the distance to the planet. The current mean surface temperature of Earth is about 287 Kelvin, so running the numbers means we'd need to be a bit further than 4 AU from the Sun to become a gas giant.

You can estimate the atmospheric density at Earth's surface from the ideal gas law and some estimations. Assuming the acceleration due to gravity changes by very little with altitude, we can relate the mass of the atmosphere $M_{\text{atm}}$ to the surface pressure $p_0$ by $$M_{\text{atm}}=\frac{p_0}{g}4\pi R^2=\frac{4\pi R^4p_0}{GM}$$ Inserting this into the ideal gas law gives us a surface density of $$\rho_0=\frac{mp_0}{k_BT}=\frac{GMM_{\text{atm}}m}{4\pi k_B R^4T}\propto\frac{M_{\text{atm}}}{T}$$

If we place Earth closer to the region of the Solar System where it could retain hydrogen, we see that its surface density would be higher by a factor of 2 due to the decreased temperature and a factor of several due to the fact that it would retain a substantially larger hydrogen/helium envelope. So I think a surface density about 10 times-ish that of Earth is quite achievable while staying away from that critical distance of 4 AU .