How long can this planet hold its atmosphere?

Question

Humans are looking for a permanent home. They colonized Mars and later terraformed it, temporarily, before being forced out of the solar system entirely when the Sun went all Red-Giant on them a few hundred years later.

Since then they've been planet hopping and hibernating between stops. Their terraforming ability, while impressive, is limited. Without a planet that more closely resembles Earth than, say, Mars, they keep running into issues for long term human viability. They knew Mars was a temporary solution, a pit stop to gather materials and experience to gear up for the inevitable death of Earth. Even having terraformed it, and assuming the Sun had taken longer than expected to expand, they knew that Mars would not hold an atmosphere for long. A replenishment of the atmosphere might have been possible, but repeating the terraforming process only delays the inevitable. They wanted a more permanent solution.

Other planets they've stopped at along the way have had other issues. Too much gravity, too close or too far from the star, atmosphere instabilities, radiation, etc.

Each time they would terraform it the planet, both to make the stop more comfortable, as well as to gain knowledge and experience of the terraforming process. While at each stop they would also replenish their resources (a long process) needed for the next terraform.

The latest stop is this planet:

1. Earthlike crust mineral content (sans water, which will be provided by the terraforming process).
2. Earthlike magnetosphere.
3. Radius of 2142 km.
4. Mass of 6.594e^23 kg.
5. Orbiting a star with properties close enough to Sun-like for the differences to be negligible for the purposes of this question.
6. If I've done my calculations correctly, and taken into consideration all relevant factors, this gives the planet an escape velocity of 3.983 miles per second (compared to Earth's 6.95 miles per second).

Unfortunately, I've been unable to find a calculator or other math reference information for determining the actual rate of atmospheric losses as a function of escape velocity, so I don't know where to start to attempt that calculation.

Note: If I've left off any important data, let me know in comments. I almost certainly have it available and just forgot to include it or didn't realize it was relevant.

Once the terraforming process is complete, and an Earth-analogue atmosphere is established, and assuming humans care for it, in the sense of intentionally avoiding significant damage to it, but not actively replenishing it, how long will the atmosphere remain hospitable to human inhabitants, before too much of the atmosphere bleeds off in to space due to the low escape velocity of this planet?

Answer 1

Here is a start on an answer for you. I quote from my own answer dated Jan. 20, 2022 to my question:

How to make an Old Solar System scientifically possible

Habitable Planets for Man, Stephen H. Dole, 1964, discusses the requirements for human habitability.

https://www.rand.org/content/dam/rand/pubs/commercial_books/2007/RAND_CB179-1.pdf

On page 35 Dole describes the escape velocity requirements for a world to retain a gas in its atmosphere. The ratio between the escape velocity of the world and the root-mean-square of the velocity of the gas particles in the exosphere, the outer atmosphere where particles escape from the planet, is vital.

According to table 5 on page 35, the e -1 life of an atmosphere will be zero if the ratio is 1 or 2, a few weeks if the ratio is 3, several thousand years if the ratio is 4, approximately a hundred million years if the ratio is 5, and approximately infinite if the ratio is 6 or higher.

In your question you describe the latest terraforming candidate as:

Radius of 2142 km. Mass of 6.594e^23 kg

And as:

If I've done my calculations correctly, and taken into consideration all relevant factors, this gives the planet an escape velocity of 3.983 miles per second (compared to Earth's 6.95 miles per second).

A radius of 2,142 kilometers gives a diameter of 4,284 kilometers. Earth has a mean radius of 6,371 kilometers, so your planet has 0.3362109 of the radius of Earth, 0.1130377 of the surface area, and 0.0380045 of the volume of Earth. So if you planet had the same overall density as Earth 5.514 grams per cubic centimeter, it would have 0.0380045 the mass of Earth.

Since the mass of Earth 1s 5.97237 times 10 to the 24th power kilograms, 0.0380045 times that would be 2.269769 times 10 to the 23rd power kilograms. You give the mass of the planet in question as 6.594 times 10 to the 23rd power kilograms. That is 2.9051414 times the mass if your planet has Earth's overall density of 5.514 grams per cubic centimeter.

Therefore your planet should have 0.1104084 the mass of Earth and an overall density of 16.018949 grams per cubic centimeter. The natural elements with densities higher than 16.018949 include tantalum, neptunium, gold, tungsten, americium, uranium, rhenium, platinum, iridium, and Osmium, plus a number of elements created in laboratories which quickly decay into other elements.

You might want to look at my answer to my question, linked to above, where I go on to design small worlds with high densities composed of heavy elements and calculate their surface gravities and escape velocities.

Your planet should have a surface gravity of 0.98 g and an escape velocity of 6.41 kilometers per second.

https://philip-p-ide.uk/doku.php/blog/articles/software/surface_gravity_calc

https://www.omnicalculator.com/physics/escape-velocity

6.41 kilometers per second is about 3.9829893422 miles per second, which is close to your figure of 3.983 miles per second.

I note that Mars has an escape velocity 5.03 kilometers per second or 3.125497 miles per second. So your humans would have terraformed and colonized a planet with a lower escape velocity than the one in your question, and would have thought that it would retain an atmosphere long enough to be worthwhile. Though of course maybe they only planned to live there for the few hundred years you mention.

Going back to Habitable Planets for Man, Stephen H. Dole, 1964, his discussion on page 54 suggests that habitable planets, which would have to have surface temperatures similar to Earth's, would probably usually have atmospheric temperatures in their exospheres similar to those on Earth.

We do not yet fully understand all the factors which are involved in producing he extremely high temperatures in Earth's exosphere (approximately 1000 degrees K to 2000 degrees K). However, if we take as a rough approximation that maximum exosphere temperatures as low as 1000 degree K are not incompatable with the required surface conditions of a habitable planet, then the escape velocity of the smallest planet capable of retaining atomic oxygen may be as low as 6.25 kilometers per second (5 x 1.25).

This means that Dole caculated (or looked up) the root-mean-square velocity of atomic oxygen at a temperature of 1000 degrees K and found it was about 1.25 kilometers persecond, and then looked at table 5 on page 35 to see that an escape velocity five times that, or 6.25 kilometers per second, would enable the world to retain atomic oxygen for about a hundred million years.

So the escape velocity of your planet, 6.41 kilometers per second, is somewhat above 6.25 kilometers per second, and so your planet should be able to retain an oxygen atmosphere for about one hundred million years. If the maximum temperature it its exosphere is less than 1000 degress K.

But apparently the temperatures in the exosphere of Earth vary between 1000 K and 2000 K, and the same would probably be true for any Earth like planets orbiting stars similar to the Sun at distances similar to one AU. Thus with a higher range of temperatures in the exospheres of those planets they might be able to retain oxygen rich atmospheres for "only" 98,251,302 years, or 55,457,816 years, or 10,493,227 years, or something.

Unfortunately, I do not know how to calculate how long the planet will retain ample oxygen with various combinations of exosphere temperature (and thus root-mean-square velocity of oxygen) and escape velocity.

And if that is a problem for your characters, then maybe the small planet they choose to terraform will be farther from its star and receive less heat and light from it. So the radiation it gets from its star will give it a lot colder surface and exosphere temperatures. Then the humans will have to find a way to heat up the surface of the planet and its lower atmosphere without heating up the exopshere. Possibly the artificially heated surface of the planet will radiate the extra heat in infrared radiation, which will not be absorbed in the exosphere or heat it up.

You will also have to consider the effect of the stellar wind of your star knocking air particles out of the upper atmosphere. Your planet should have a magnetosphere to deflect most solar wind particles and protect the atmosphere. That magnetosphere could be naturally generated by the planet, and/or artifically geernated by the humans.

I don't know how to calculate how long the planet will retain its atmosphere without a magnetosphere.

I hope this helps you.

Added January 25, 2022

I just wrote a new answer with a different solution for ensuring a small planet will keep its atmosphere for a long time.

Answer 2

Part One of Six.

Here is another answer to the question about how long a planet - a planet with a radius of 2,142 kilometers, mass of 6.594 X 10 the 23rd power kilograms or 0.1104084 the mass of Earth, and thus a surface gravity of 0.98 g and an escape velocity of 6.41 kilometers or 3.983 miles per second - could retain its atmosphere.

In my previous answer I wrote that if the oxygen in the exosphere of the planet's atmosphere, where gases escape into space, had a temperature of 1000 degees K or less, and a root-mean-square speed of 1.25 kilometers per second or slower, the planet could hold an oxygen atmosphere for about a hundred million years if it had an escape velocity of at least 6.25 kilometers per second.

Since your planet has an escape velocity of 6.41 kilometers per second, it may be able to hold on to an oxygen atmosphere for a hundred million years.

My previous answer also said that with the radius and mass specified in the question, the planet has an overall density of 16.018949 grams per cubic centimeter, which is a lot denser than most naturally occuring elements. Your planet would have to be composed almost entirely out of one of the densest known elements, which would be rather unlikely to happen naturally.

So perhaps I should suggest an alternate method of enabling a small planet with a radius of only 2,142 kilometers to retain its artifically produced atmosphere for long periods of time, without giving that planet extra mass requiring an improbable density.

Part Two: Percival Lowell's Main Error.

What was Percival Lowell's big error in his theory of the Martian canals?

Lowell believed that the planet Mars was slowly losing its water. Molecules of water vapor in the upper atmosphere would have been broken up by ultra violent ultraviolet rays into atoms of hydrogen and oxygen. The ultra light hygrogen would move too fast compared to the escape velocity of Mars and would escape from the planet, never again to combine with oxygen to make more water.

This process happens in Earth's atmosphere, and probably was a main cause of the actual lack of water on Venus and Mars.

So Lowell believed the hypothetical Martians would make the best of it by using their dwindling water supplies efficiently, distributing snow melt water from the polar caps all over the planet with a system of canals, delaying their inevitable doom.

And apparently Lowell never thought that a better method for the Martians would be to keep the water from being lost from Mars by stopping the breaking up of water molecules and the escape of hydrogen from Mars.

How could the Martians do that?

How would humans living in hypothetical Moon bases prevent their air and water from escaping into outer space? By building totally enclosed Moon bases where air and water could not get out and would be endlessly recycled.

So the hypothetical Martians could have solved the problem of Mars slowly losing water by building larger and larger totally self contained "Moon bases" on Mars, perhaps eventually totally covering all of Mars with an airtight roof to prevent air, escpecially the hydrogen necessary for water, from escaping.

Astronomers in Lowell's time didn't have very good views of Mars. There was no way they could have been able to tell if they were seeing the actual natural surface of Mars or a planet wide roof made of more or less transparent or opaque material covering all of Mars and holding the atmosphere in.

Part Three: The Roof of the World.

So possibly the humans in your story decide to terraform a small planet with a radius of 2,142 kilometers which doesn't have the extreme density you ask for. 2,142 kilometers is a little less than the radius of Callisto (2,410 kilometers), which has an escape velocity of 2.440 kilometers per second. But Callisto has a very low density compared to Earth. Mercury is a bit larger (2,439 kilometers) but has a density much closer to Earth's density and has an escape velocity of 4.25 kilometers per second.

So it might be possible for your humans to find a planet with a radius of 2,142 kilometers which is massive and dense enough to have an escape velocity of about 4.00 kilometers per second, without the planet having a core made out of gold or some other rare element. And that might not be a high enough escape velocity to retain the artificial oxygen rich atmosphere they plan to produce when they terraform that planet for long enough.

So they could build a roof over the entire planet, with giant airlocks to allow spaceships to land. And fill the space below the roof with the artifical atmosphere they produce. That would be a massive project, but terraforming a planet is a massive project.

Part Four: A Roof of Nano Machines.

I remember once looking at, but not reading, a science fiction novel by Arthur C. Clarke and a collaborator. That might be the one that mentioned life forms in space which evolve to attack and consume parts of spacecraft. Anyway, in that story the Moon had been terraformed and given a breathable atmosphere. To prevent the atmosphere from escaping into space a sort of a roof made of gazillions of nano macheines was built over the Moon. Each of the tiny nano machines was linked to its neighbors, and the spaces between were smaller than molecules. So air particles which hit the "roof" were bounced backed instead of escaping into space.

Part Five: Planetary Roofs Supported by Air Pressure.

According ot Wikipedia's list of theoretical megastructures:

Shellworlds or paraterraforming are inflated shells holding high pressure air around an otherwise airless world to create a breathable atmosphere.[8] The pressure of the contained air supports the weight of the shell.

https://en.wikipedia.org/wiki/Megastructure#Planetary_scale

A shellworld13 is any of several types of hypothetical megastructures:

A planet or a planetoid turned into series of concentric matryoshka doll-like layers supported by massive pillars. A shellworld of this type features prominently in Ian M. Banks' novel Matter.

A megastructure consisting of multiple layers of shells suspended above each other by orbital rings supported by hypothetical mass stream technology. This type of shellworld can be theoretically suspended above any type of stellar body, including planets, gas giants, stars and black holes. The most massive type of shellworld could be built around supermassive black holes at the center of galaxies.

An inflated canopy holding high pressure air around an otherwise airless world to create a breathable atmosphere.4 The pressure of the contained air supports the weight of the shell.

Completely hollow shell worlds can also be created on a planetary or larger scale by contained gas alone, also called bubbleworlds or gravitational balloons, as long as the outward pressure from the contained gas balances the gravitational contraction of the entire structure, resulting in no net force on the shell. The scale is limited only by the mass of gas enclosed; the shell can be made of any mundane material. The shell can have an additional atmosphere on the outside.[5][6]

https://en.wikipedia.org/wiki/Shellworld

Having the roof of a shell world supported by air pressure is not some wild fantasy.

An air-supported (or air-inflated) structure is any building that derives its structural integrity from the use of internal pressurized air to inflate a pliable material (i.e. structural fabric) envelope, so that air is the main support of the structure, and where access is via airlocks.

https://en.wikipedia.org/wiki/Air-supported_structure

Some quite large structures are air supported, although they are much smaller than air supoorted shells around an entire asteroid or around an entire planet. But any story involving terraforming a planet involves mega projects.

Part Six: Force Fields.

In some science fiction stories force fields of various types might have various practical uses. Perhaps in some stories force fields could keep the molecules of gas from escaping from tiny worlds which don't have enough escape velocity.

Those force shields might make it impossible for spaceships to land or take off. Or there might be giant air locks which stick up through the force shields that spaceships can take off and land in. Or maybe the force fields stop particles moving at the velocities of atmospheric gases, but let spaceships land and take off if they travel faster or slower than atmosphereic gases.

A famous example of a force field holding in an atmosphere is in Isaace Asimov's classic story "Not Final!" Astounding Science Fiction, October, 1941.

https://archive.org/details/Astounding_v28n02_1941-10/page/n47/mode/2up?view=theater