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This is a drawing by my daughter which inspired one of the elements in my story/world:

Hand-drawn image of small planet

The perspective is a little bit mixed here (certainly I don't have a 15 meter-high volcano in this world), but for the sake of this question let's assume that such planet is between 1 and 2 kilometres in diameter or 500-1000 meters in radius. Planets like this are sold by various agencies in the world of my story.

Given the fact that the whole story is set in exactly the same universe, as ours (only a little bit into the future), with exactly the same physics and other laws, the question is, if such planet:

  • could exists naturally (what are the limits here) or
  • these companies are selling artificial creations.

What is the smallest possible planet that can exists naturally in our universe? Must I assume that such small planets cannot exist naturally?


Edit: Here are some answers to the questions given in comments. In short, think about compressing Earth to as smallest size as our current physics allows.

  1. The planet must be habitable, with gravity as close to Earth's one as possible.
  2. Any round rock orbiting the sun or other stellar body does the trick. Meeting IAU's or other definitions is not needed.
  3. Buildings required. Other stuff as well. Full recreation of Earth's look & feel very welcome. This should be private planet as we now understand a private island.
  4. Escape velocity and other physical, chemical or geological parameters as close to the Earth as possible. So, again, a rock in space, not a small gas body.
  5. Again, circling the Sun or other stellar body. No comets lost in space, please. Light and day-night conditions and yearly seasons as close to Earth as possible, please.
  6. No rain, snow or clouds needed, though nice to have. Breathable atmosphere is a must.
  7. This should be a habitable planet, but for a single person or a small group of people. As you can see in the image, we don't need more than 3-5 story buildings. So the fact that breathable atmosphere would be as thick as 500-1000 m above the ground isn't necessary a problem.
  8. Extremely dense core sounds like a good idea, but that might ruin the "as close to Earth physics as possible", if I am not mistaken. And it would also most likely fail under "habitable", as we would need some underground water sources etc.
  9. A black hole in a center of a planet certainly sounds good as long as above conditions are met.
  10. Asteroid might work as well as long as above conditions are met.

The planet must be habitable and easily accessible. So, if by any mean, planet's atmosphere would be filled with some orbiting rocks or other space trash, disallowing any easy navigation and landings, then this is out of question.

The company wants to sell a fully-featured product, where you can spend the rest of your life. Not just a rock in space, that you can show off on your pictures, but that you cannot land on and live on.

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    $\begingroup$ You ask, What is the smallest possible planet - This depends entirely on what definition you use for the word "planet". Clearly there are objects out there from the size of Jupiter to small specks of dust. What is your definition of "planet"? $\endgroup$ – chasly - supports Monica Jun 18 '20 at 20:29
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    $\begingroup$ What definition of planet do you want to use? Are you limited to the IAU's definition? Or, just any round rock orbiting the sun with the features you want? $\endgroup$ – Mathaddict Jun 18 '20 at 20:29
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    $\begingroup$ Do you want to be able to have buildings on this planet, the inhabitants to not feel drastic changes in gravity by walking upstairs? $\endgroup$ – illustro Jun 19 '20 at 11:47
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    $\begingroup$ As a point of comparison, 67P/Churyumov–Gerasimenko the comet that Rosetta's lander Philae attempted to touchdown on in 2014 was ~4km. The escape velocity of that comet is ~1m/s which means you could fall off it by walking. en.wikipedia.org/wiki/67P/Churyumov%E2%80%93Gerasimenko $\endgroup$ – David258 Jun 19 '20 at 13:44
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    $\begingroup$ Mandatory xkcd: what-if.xkcd.com/68 $\endgroup$ – Suppen Jun 19 '20 at 21:25
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Assuming by "planet" you mean a roughly spherical body - encompassing both dwarf planets and "true" planets - then the smallest naturally occurring body is somewhere between 1 Ceres (dwarf planet) and 4 Vesta (not a dwarf planet because it's insufficiently round). Ceres is about 900-950 km across; Vesta more like 450-600 km. An object of only a few kilometers' diameter would be much too small to round itself under its own gravity.

There are certainly rocky bodies of that size in the solar system, and with care and attention you could shape them into spheres, but it would be very rare for them to be found that way in nature.

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    $\begingroup$ +1.. IOW, to more directly answer the OP's question, such "planets" are probably artificial. $\endgroup$ – Matthew Jun 18 '20 at 20:34
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    $\begingroup$ Could a planet somehow have an extremely dense core? Maybe some kind of super dense material, with its strong gravity, collected bits of dust and made the small, round planet around itself? Is that possible? $\endgroup$ – colmde Jun 19 '20 at 9:04
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    $\begingroup$ @colmde Not really possible with conventional matter. The Little Prince's planet has about 1/500th the radius of Ceres, so it has 1/125,000,000 the volume. There isn't any "normal" matter with 125 million times the density of rock - we'd be talking about a neutron star or a black hole at that point. $\endgroup$ – Nuclear Hoagie Jun 19 '20 at 14:18
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    $\begingroup$ @NuclearWang: It would not need quite that much mass, due to how gravity at the surface scales (the inverse square rule for gavity field strength opposes the scale factor for volume for requried mass meaning only factor of r, not r cubed). In fact the gravity would be far too high at the surface with the density your estimate, you only seem to need a few thousand times the density. See HDE's answer. $\endgroup$ – Neil Slater Jun 21 '20 at 10:27
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    $\begingroup$ @Taladris No, because gravity isn't the only force acting on it. In particular, once it cools to the point where its constituent material is solid, that material has its own compressive strength that resists further changes. $\endgroup$ – Cadence Jun 21 '20 at 21:32
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I'll start looking at this from a slightly different perspective than Cadence's: surface gravity. Let's say that we want the Little Prince's planet to have Earth-like surface gravity. This means that its mass and radius obey $$\frac{GM}{R^2}=g=9.8\;\text{m/s}^{2}$$ If we want $R=1000\;\text{m}$, we see that the planet needs to have a mass of about $M=10^{17}\;\text{kg}$, giving it a density of $\rho\approx35000\;\text{g/cm}^{3}$. For comparison, the density of Earth is approximately $5.5\;\text{g/cm}^{3}$; an iron planet would have a density of $\sim10\;\text{g/cm}^{3}$. The Little Prince's planet will be comparable in density to a white dwarf!

Let's go back to thinking about size. How low can we truly go and have our planet still be round? This is an ongoing topic of research; 400 km in diameter is a number that gets tossed around a lot - which, interestingly enough, is almost exactly the size of the moon Mimas. But this number really depends on the composition of the body, and I've heard even lower limits proposed: $\sim$200 km in diameter is the lowest I'm aware of. At Earth-like densities (likely an overestimate), this gives us a surface gravity of $0.15\;\text{m/s}^{2}$ - much lower than we're used to on Earth!

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    $\begingroup$ Ey, that's not bad. So if we carve a small chunk of a brown dwarf and somehow convince it to not expand rapidly immediately, we'll have Little Prince's homeworld in a jiffy. $\endgroup$ – John Dvorak Jun 19 '20 at 6:04
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    $\begingroup$ Considering planets can be artificially made, I think perfect roundness isn't a hard requirement (same as clearing its orbit). So the main thing we'd need to care about is gravity. Maybe a handwavium-based gravity generator could do the trick? $\endgroup$ – Teleporting Goat Jun 19 '20 at 8:20
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    $\begingroup$ One problem would be how the planet keeps its atmosphere. Our atmosphere is roughly 100 km thick, where the gravity is still very nearly 1 g. At 100 km, this microplanet's gravity is only 1/100.000 g. Even 5 km up - where our atmosphere has dropped off to half pressure - the gravity would be 1/25 g. A lot more air would be needed to provide surface pressure as on Earth, since the weight of a column of air would be far smaller (even though the 'column' would be more of a cone), and the upper parts would be much more likely to be blown off by the solar wind or escape due to molecular motion. $\endgroup$ – Klaus Æ. Mogensen Jun 19 '20 at 8:42
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    $\begingroup$ If your handwavium is producing 1g on the surface of the tiny planet, wouldn't that gravity also retain an atmosphere at roughly 760 mmHg (assuming, of course, that the gas to form that atmosphere is available in the first place)? $\endgroup$ – papidave Jun 19 '20 at 21:40
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    $\begingroup$ @papidave no, because the attractive force of gravity falls off with an R^2 relation. It's heavily dependent on the percentage of the planet's radius you are away from it. $\endgroup$ – illustro Jun 20 '20 at 12:54
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There are a few issues with having "planets" this small

1. When your planet has a radius of 1000 m, the height of an adult human, is a noticeable percentage of the radius (on the order of 0.15 - 0.2%) and small buildings are closer to 1% of the radius!

To work out the surface gravity at the surface, we need to use the following equation

$$g = \frac{GM}{R^2}$$

The units of $g$ are $\text{m/s}^{2}$.

The components of this equation are:

  • $G = 6.67259\text{ }*\text{ }10^{-11} \text{ ; units: }m^3\text{ }kg^{-1}\text{ }s^{-2}$
  • $R = 1000 \text{ ; units: }m$
  • $M = \rho\text{ } * \text{ Volume}\text{ ; units: }kg$
    • $\text{Volume} = \frac{4}{3} \pi R^3\text{ ; units: } m^3$

Substituting this all in we get:

$$g = \frac{6.67259\text{ }*\text{ }10^{-11}\text{ }*\text{ }\rho\text{ } * \frac{4}{3} \pi R^3}{R^2}\text{ ; units: } m/s^2 $$ $$ = 6.67259\text{ }*\text{ }10^{-11}\text{ }*\text{ }\rho\text{ }* \frac{4}{3} \pi R\text{ ; units: } m/s^2 $$ $$ = 6.67259\text{ }*\text{ }10^{-8}\text{ }*\text{ }\rho\text{ }* \frac{4}{3} \pi \text{ ; units: } m/s^2$$

So the key variable for targeting a particular gravity is $\rho$. If we want to target a $g$ close to that of Earth ($9.798\text{ }m/s^{2}$source: NASA factsheet), then we need a value of $\rho = 35,055 \text{ }g/cm^3$ (ie approximately the density of some black holes and white dwarf stars!). It also gives our planet a mass of $1.47 * 10^{17}$ kg (when we change the radius later for analysing surface gravity changes, we will need to keep the planetary mass constant).

If we go with that, then we run into a separate issue...that the force of gravity changes appreciably over scales as small as the human body (which would be an issue for small things like distribution of blood over the body).

For example, right at the surface $g = 9.798\text{ m/s}^{2}$, but, only 2m out from the planet's surface it changes to $g = 9.759\text{ m/s}^{2}$, and, were we to have a 2-3 story building, approximately 10m high, $g = 9.605\text{ m/s}^{2}$.

If we normalise $g$ so that it is instead $10\text{ } m/s^2$, to make these values easier to parse, then our required density becomes $35,778.07\text{ } g/cm^3$ and our comparison becomes:

For example, right at the surface $g = 10\text{ m/s}^{2}$, but, only 2m out from the planet's surface it changes to $g = 9.960\text{ m/s}^{2}$, and, were we to have a 2-3 story building, approximately 10m high, $g = 9.803\text{ m/s}^{2}$.

2. If we make the density of the planet low, to counteract this drastic change in surface gravity over different parts of the human body, we will make the escape velocity of the "planet" significantly lower

The equation to calculate escape velocity is:

$$v_\text{escape} = \sqrt{\frac{2GM}{R}}\text{ ; units: } m/s$$

If we were to change the density of our small planet, down to that of Earth ($5.51\text{ } g/cm^{3}$source: NASA factsheet), then our mass becomes $2.3 * 10^{13}$ kg and we get $g = 0.00154\text{ m/s}^{2}$.

From the perspective of our corporation, this is much more desirable, as they need to source a significantly smaller mass of material (by a factor of 10,000!).

However, if we have a surface gravity that low, then working through the numbers we end up with $v_\text{escape} = 1.755\text{ } m/s = 6.32\text{ } km/h$. This is low enough that a human would likely easily be able to reach that speed. Usain Bolt has achieved speeds of $10.44\text{ m/s}$ or $37.58\text{ km/h}$, so a speed of $6.32\text{ } km/h$ is certainly within the capability of a regular human.

Conclusion

The primary parameters that we would need to balance are the radius and the density of the planet. To mitigate the most severe gravitational and escape velocity problems, we would need our planet to be significantly larger and have a surface gravity pretty significantly lower than that of Earth.


Notes

The source of a number of the values I've used for Earth comparisons is the NASA Planetary Factsheet for Earth. For $g$ in particular is has this definition:

Equatorial gravitational acceleration at the surface of the body or the 1 bar level, not including the effects of rotation, in meters/(second^2)

Defined here.

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    $\begingroup$ Your 9.798 m/s^2 is overly precise. Earth surface-gravity can vary by several percent. I was taught 9.81 m/s^2 in school, but values from 9.76 to 9.83 can be measured, so unless you're referring to a specific place, more precision than 9.8 m/s^2 is unnecessary. "Standard gravity" is defined as 9.80665 m/s^2. $\endgroup$ – AI0867 Jun 19 '20 at 14:27
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    $\begingroup$ @AI0867 yes it is. The point of the precise number wasn't to quibble about the value, but to show the effect over scales on the order of the human body. If you do a similar analysis for length scales on the order of 10s of metres on Earth, the value of g changes beyond the level of precision I've stated in my answer (ie in the 5th or 6th decimal place). $\endgroup$ – illustro Jun 19 '20 at 14:29
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    $\begingroup$ You may want to normalize to g then. That also makes the size of the effect easier to see. The current phrasing seems to say that earth's gravity is 9.798 and that you'll pick a value close to it, rather than that the value you pick is 9.798, which is close to earth's gravity. $\endgroup$ – AI0867 Jun 19 '20 at 14:30
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    $\begingroup$ @AI0867 I'll have a think about doing that. Btw, the source for my figure for g (and the density of the earth) was this page from NASA: nssdc.gsfc.nasa.gov/planetary/factsheet/earthfact.html $\endgroup$ – illustro Jun 19 '20 at 14:32
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    $\begingroup$ (Ta for the edit btw...hadn't noticed the copying and pasting of incorrect units at the end!) $\endgroup$ – illustro Jun 19 '20 at 14:34
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This can be broken down into several sub questions

What constitutes a “planet”? By current standards such a small body could not be classified as a planet because it would not have a strong enough gravitational field to sweep its orbit clear of debris.

Can a spherical body of 200 km diameter form naturally? No at 200 km it is unlikely, as gravitational forces are barely sufficient and the object is likely to be at least slightly oblate or potato shaped like a large asteroid. But under some special circumstances I’m sure it could happen. A small asteroid might be diverted into an elliptical orbit close to the sun making it partially molten or more plastic and capable of pulling itself into a spherical shape over time. Such a body might then be ejected further out into the solar system by another planet.

Can such a body have an atmosphere? No it cannot have any meaningful atmosphere as the gravitational forces would be so low. Even by artificial means of using very dense metals like platinum and tungsten to build such an object the gravitational pull would still be very weak and insufficient to hold on to an atmosphere.

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  • $\begingroup$ Please, see the edited question. I hope I have clarified most of your doubts. Thank you for your contribution. $\endgroup$ – trejder Jun 20 '20 at 4:17
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Naturally occurring, the answer is no on SO many levels.

You will not get a round shape that small, you can't get a molten core on something that small, your Oceans and atmosphere will float away... I mean, the problems are so numerous this seems like it should be a hard no, but when it comes to building worlds, I'm no quitter; so, I will try to propose something that would at-least in theory work.

How to do it artificially

So this is a bit of a frame challenge since this is not a near future tech solution, but if you are a really advanced civilization there might be a way. First you will need something with the gravity of Earth, but smaller than 1km in radius. Using https://planetcalc.com/1758/ I have estimated that for a 1000m radius world to have 1 Earth Gravity at its surface, it would need a mass of about 1.5e17 kg, but a planet made out of the same stuff as Earth would only have a mass of 2.3e13 kg which would result in only 0.00015G... not nearly enough to have a nice usable world. A neutron star however has a density of at least 3.7e17 kg/m3 meaning that if you were to extract about 1/2 of a cubic meter of pure neutrons from neutron star, and if you could stabilize this mass without it suddenly exploding (BIG IF: see comments), you could use it to make a planetary core able to produce Earth like gravity at a 1000m radius. Then you just start piling on good old fashioned rocks and stuff.

If you want your planet to have tides, axial wobble, etc, you can repeat the same process to give it a small moon.

As for volcanic activity, that will just be a matter of introducing the right amount of radioactive elements to maintain a molten mantel, but without melting the crust.

I also noticed your world only has one ice cap which suggests that your planet is at least partially tidal locked toward the sun. This means that your "north pole" will be stuck in perpetual day light. Perhaps the extra tidal forces explain why there are more volcanoes here. Then your tropical zone will be more of a perpetual twilight; though, with enough wobble, you could still have a sort of day/night cycle here. Then the South pole would be always in darkness.

Lastly, there is the issue of an atmosphere. Escape velocities are not your friend here meaning that even if you have Earth like gravity at your surface, that gravity will fall off way too quickly to hold an atmosphere. To solve for this you will need to basically install a giant fish bowl surrounding the planet to hold the air in.

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    $\begingroup$ Getting that small chunk of neutron star to remain than dense will be...difficult, as the gravitational binding energy for something that small is now, no longer, greater than the repulsive forces at the atomic and subatomic levels. What you would end up with is a very large bomb if you didn't have some incredibly advanced solution to contain it. $\endgroup$ – illustro Jun 19 '20 at 15:02
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    $\begingroup$ Yes it would expand a little bit, but even without intense gravitational pressures, pure neutrons should remain incredibly dense because there are not all of those annoying protons and electrons pushing things apart. I believe you should be left with something of the approximate density of a normal atomic nucleolus which is about 2.3×e17 kg/m3 $\endgroup$ – Nosajimiki Jun 19 '20 at 15:10
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    $\begingroup$ That is unless the weak force will cause it to instantly turn into a nuclear explosion... can't really say I'm enough of an expert in nuclear physics to say which way that would go. But if we are going to hand-wave in the tech to scope matter out of a neutron star, having the tech to keep it from exploding does not seem that far fetched IMO. $\endgroup$ – Nosajimiki Jun 19 '20 at 15:14
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    $\begingroup$ Someone has looked at that, see here: astronomy.com/magazine/ask-astro/2018/08/… $\endgroup$ – illustro Jun 19 '20 at 15:16
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    $\begingroup$ The salient quote from the article: "A spoonful of neutron star suddenly appearing on Earth’s surface would cause a giant explosion, and it would probably vaporize a good chunk of our planet with it." $\endgroup$ – illustro Jun 19 '20 at 15:17
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Naturally occurring solid spherical objects of the requested size are very rare. There are likely some somewhere in our vast universe, but none have been discovered. Solid objects of that size do not have enough gravity to make themselves spherical. They also cannot hold an atmosphere. It would be easier to build a solid sphere of that size than it would be to find one.

If very low gravity and lack of atmosphere are acceptable, the companies selling these objects would likely make them by crunching some small asteroids into spherical shapes. If they must have Earth-like surface gravity, these solid spheres would have to have an extremely high density, higher than that of a white dwarf. The gravity would be far too weak for stabilizing normal matter at this density, so the object would explosively decompress itself. Matter composed of different quarks might be stable at such high densities. It would not be very realistic to have a solid sphere of the requested size with Earth-like surface gravity.

Instead, a shell could be constructed around a black hole with sufficient mass. This would have to be artificial. It would still have difficulty holding an atmosphere, but since there are shells anyway, maybe another shell could be added to hold in the atmosphere. The shell(s) would require a system for adjusting their position to keep the black hole in the center. Despite black holes being thought of as scary, this setup would be quite safe if implemented correctly.

One difficulty with the black hole idea would be transporting the planet-like thing. You couldn't just attach an engine to it because the black hole cannot be attached to anything. Moving the outer shell would not move the black hole since they are not attached. It might be possible to magnetize the black hole, but something that would definitely work is a gravitational tug. A massive object orbiting the planet-like thing could accelerate slowly, and the planet-like thing, including the black hole, would be accelerated too because of gravity. Alternatively, the planet-like thing could be built at its destination, avoiding the problem of moving the black hole to the destination. Another difficulty would be creating the black hole, but I believe a civilization advanced enough to have a market for personal tiny planets would have technology capable of doing this.

In conclusion, they would be artificial and either take the form of small spheres with little gravity made of rock taken from asteroids, or spherical shells with black holes at the center. They would have a shell, or extra shell, to hold in the atmosphere if the atmosphere is desired.

Edit: I put the information in paragraphs. Also, hawking radiation would not be a concern for the black holes that I mentioned. To have Earth's surface gravity at the minimum requested radius, 500 meters, the black hole would require a mass of about 3.7x10^16 kilograms. A black hole of this mass would have a luminosity of about 0.26 watts, and take about 1.3x10^26 years to evaporate.

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  • $\begingroup$ Good answer which could be improved by paragraphs, as it's somewhat of a wall-of-text at the moment and difficult to read. (From review). $\endgroup$ – Tantalus' touch. Jun 20 '20 at 4:31
  • $\begingroup$ "Despite black holes being thought of as scary, this setup would be quite safe if implemented correctly." Says who? Such tiny black holes would be incredibly unstable due to Hawking radiation, for one. $\endgroup$ – eps Jun 20 '20 at 18:30
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Short Answer:

No, such a tiny world can not be habitable for humans naturally. And by far the easiest way to artificially make a world of that size that is habitable for humans is to build an inside out version, a hollow cylinder that rotates to provide simulated gravity and uses its walls to retain its atmosphere.

Long Answer:

If you ask about the minimum size and mass a world needs to naturally become roughly spherical, you will learn that it is about a million times the volume and mass of your little worlds. The vast majority of tiny worlds in the question are much too irregular in shape to look spherical. So a tiny world of that size would have to be artificially shaped by an advanced civilization to become spherical enough for your purposes.

After shaping such a tiny world into the proper shape, the next step would have to be provide it with an artificial breathable atmosphere.

How long could such a tiny world retain an artificial breathable atmosphere once it was created?

You should obtain a paper or electronic copy of Habitable planets for Man, 1964, by Stephen H. Dole if you plan to write a lot of plausible science fiction set on habitable exoplanets.

Section added June 28, 2020

In chapter Four The Astronomical Parameters the section on planetary properties on pages 53 to 67 discusses the property of the planet necessary for human habitability.

Dole says that planet needs to have a surface gravity of less than 1.5 g to be habitable, which according to figure 9 on page 31 corresponds to a planet with a mass of 2.35 Earth, a radius of 1.25 Earth, and an escape velocity of 15.3 kilometers per second. (page 53).

I note that you specify the surface gravity of your planet, but not its escape velocity. The ability of a planet to retain whatever atmosphere it acquires depends of the chemical composition of that atmosphere, the escape velocity at the outer edges of the atmosphere where gases escape, and on the average velocity of the air particles in the escape lawyers of the atmosphere.

Dole says that in order for a planet to retain atmospheric oxygen, its escape velocity should be:

"of the order of five times the root-mean-square velocity of the oxygen atoms in the exosphere".

(page 54)

Dole calculates that the escape velocity of the smallest planet capable of retaining atmospheric oxygen can be as low as 6.25 kilometers per second. According to figure 9 that corresponds to a planet:

"having a mass of 0.125 Earth mass, a radius of 0.63 Earth radius, and a surface gravity of 0.49 g. Under the above assumptions, such a planet could theoretically hold an oxygen-rich atmosphere, but would probably be much too small to produce one, as will be seen below."

(page 54)

I note that a surface gravity of 0.49 g is 4.9 times as much as the 0.1 g you specified.

Dole then makes two separate rough calculations of the minimum sized planet necessary to produce an oxygen-rich atmosphere.

Dole calculates 0.25 Earth mass in one calculation, which he considers too low, and in the other calculation 0.0.57 Earth mass, which he considers too high.

"With 0.25 being too low and 0.57 being too high, the appropriate value of mass for the smallest habitable planet must lie between these figures, somewhere in the vicinity of 0.4 Earth mass."

(page 56).

"Since it is not possible to obtain a more precise determination of the minimum mass of a habitable planet, for our purposes the value of 0.4 Earth mass will be adopted as the minimum mass. This corresponds to a planet having a radius of 0.78 Earth Radius and a surface gravity of 0.68 g."

(page 57).

I note that a surface gravity of 0.68 g is 6.8 times the 0.1 g you specify.

End of section added on June 28, 2020

Since 1964 there are two developments which may affect the minimum mass of a naturally habitable planet.

Titan, the large moon of Saturn, which is much smaller than Dole's minimum mass, has been discovered to have a dense atmosphere with a surface pressure higher than Earth's.

And there is a new theory that Earth might be as small as is possible for habitable planet. Earth has plate tectonics. Venus, which is slightly smaller than Earth, does not. So if, repeat if, plate tectonics are vital for a planet to be habitable, Earth is about as small as a habitable planet can get.

It may not matter whether the minimum size and mass of a naturally habitable planet is that of Titan or that of Earth, since both Titan and Earth are literally billions of times as massive as the tiny worlds asked about in the question.

So those tiny worlds could never be massive enough to be naturally habitable.

Forget about naturally habitable. Since those tiny worlds have to be artificially reshaped to become spherical, terraforming them by adding artificial breathable atmospheres would not be too much more trouble.

But how long could such tiny terraformed worlds keep their artificial breathable atmospheres? I once read that if the Moon was given a breathable atmosphere, it would lose it into space in a thousand years. And the Moon is billions of times as massive as the tiny worlds in the question.

I doubt that they would retain artificial atmospheres long enough that providing those artificial atmospheres would seem worthwhile.

Their ability to retain their atmospheres would have to be increased by millions or billions of times to make providing artificial atmospheres worthwhile.

One method of doing that would be to find tiny worlds made of super dense material, and then put thin lawyers of normal material on top of them while terraforming those worlds.

And in fact, there is a classic science fiction story where that is done. In Jack Vance's "I'll build your dream Castle, 1947, the protagonist finds tiny asteroids made of white dwarf degenerate matter and terraforms them into tiny habitable worlds.

http://www.isfdb.org/cgi-bin/title.cgi?57659[1]

Of course white dwarf star degenerate matter is highly compressed because of all the matter on top of it. Once that matter is removed, the white dwarf matter would expand into normal matter. I think there was a question a week or two ago where it was established that there was a minimum amount of degenerate matter necessary to avoid expansion. So you should look that up.

This question is about a story idea similar to "I'll build Your Dream Castle":

https://worldbuilding.stackexchange.com/questions/117195/5km-artificial-planet-with-same-gravity-as-on-earth/117208#117208[2]

And some of the answers should be informative.

A comparatively low mass black hole within the tiny world would gradually swallow all of its matter, but would also increase the surface gravity and escape velocity, perhaps making the world spherical and enabling it to retain an atmosphere. I have not calculated whether a world of your desired radius could have a black hole of the right mass inside it for a period long enough to be worthwhile before being swallowed and destroyed by the black hole.

Another method to retain atmosphere would be to have have some hypothetical artificial gravity generators, to give the tiny worlds high enough surface gravity to be healthy for humans for long periods of time, and to increase their escape velocities enough to retain dense breathable atmospheres for long enough for the purposes of the story.

I believe that in the classic science fiction novel The Legion of Space, 1934, by Jack Williamson, many worlds in the solar system were terraformed, given artificial breathable atmospheres, and used generated gravity for human comfort and to retain those atmospheres.

Another way to retain the atmospheres would be to generate some sort of force field around a world that would prevent air molecules from passing though it somehow.

I note that another factor which causes worlds to lose atmosphere is sputtering, being hit by particles of solar wind that knock particles out of the atmosphere. A strong planetary magnetosphere helps block the solar wind and helps retain atmosphere. I note that a stronger magnetosphere tends to be associated with a higher mass almost as much as the escape velocity does.

So your tiny worlds would have to have artificially generated magnetospheres to repel solar wind. Possibly those magnetospheres would have different generators from the generators for the force fields holding in the air and the generators for the artificial gravity, but possibly the generators could be combined.

Another way to retain atmosphere might be to put a shell of linked nano machines around the world. I think I remember reading about the Moon have a shell of linked nano machines to hold in an artificial atmosphere in a story somewhere.

Of course a regular roof supported by columns could be build around such a tiny world as in this question.

And that idea leads back to the idea of building a cylindrical space habitat that spins to imitate Earth's surface gravity and relies on its walls to hold in and retain the atmosphere.

Added June 28, 2020: The answers, including mine, to this question may be of interest:

https://worldbuilding.stackexchange.com/questions/179529/what-is-the-most-energy-efficient-mode-of-travel-by-a-land-animal-on-a-terrestri/179565#179565[3]

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That probably doesn't directly answer your question, but as far as science-based goes, the Little Prince's planet is actually an asteroid: « l'astéroïde B 612 » (as the grownups call it).

One condition nowadays to be called a planet is that it needs to have cleared its orbit from everything else. That's important for your consideration because it means it would be dangerous to place (if it was artificially made) more than one such "planets" graviting around the same celestial body, as there are risks of collision!

Another criteria handwaved in the Little Prince is the atmosphere: such asteroïds cannot maintain one because the gravity is too low. And even if it had one, only your feet could breathe, atmospheres are usually thin compared to a planet's diameter..

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    $\begingroup$ There is nothing that forces a atmosphere's height to a certain relation compared to the planet's diameter. In the extreme case look at the gas giants, which consist mostly of atmosphere. $\endgroup$ – Paŭlo Ebermann Jun 20 '20 at 2:23
  • $\begingroup$ @PaŭloEbermann Thanks for the insight, thought there were some kind of relation, but if we exclude gas giants (that would not suit for the question), they are usually thin, right? $\endgroup$ – Kaddath Jun 22 '20 at 7:11
  • $\begingroup$ I think it's more related to the total mass and/or surface gravity which limits the maximal size of an atmosphere, than the diameter itself. (So a tiny planet with a (stable) micro-black hole in the middle might have quite some atmosphere.) On the other hand, even then it's not guaranteed. I.e. Venus has a lot denser (and I think also higher) atmosphere than Earth, which in turn has much more atmosphere than Mars. Mercury and our Moon, in turn, have almost no atmosphere. $\endgroup$ – Paŭlo Ebermann Jun 23 '20 at 17:46
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    $\begingroup$ The higher the surface gravity of a planet, the more it squashes the atmosphere down. The scale height of an atmosphere is a measure of the height necessary for specified difference in atmospheric pressure. According to this list - en.wikipedia.org/wiki/Scale_height#Planetary_examples - Earth's atmosphere has the lowest scale height of any in the solar system; so all other atmospheres decrease in pressure more slowly with height, including the atmospheres of smaller worlds. $\endgroup$ – M. A. Golding Jun 28 '20 at 21:55

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