Biggest and smallest possible "walkable" celestial body

Question

What are the biggest and the smallest possible sizes for a celestial body such that a human (in a not-too-advanced spacesuit if needed) could walk on like on Earth?

More precisely, the celestial body should have the following requirements:

• Roughly spherical shape - it shouldn't have anything non spherically symmetric to help its structure, such as significant rotation combined with a cylindrical or toroidal shape
• Stable over a typical human lifespan
• A surface which can safely support a standing human indefinitely
• Atmospheric pressure (if any) and temperature manageable by early 21st century spacesuits
• Earth's gravity at the surface
• No lethal radiation

Size is defined as the radius at the surface to be walked on.

Formation of the celestial body is irrelevant - it just has to be able to exist satisfying the above requirements. Also, it can be composed out of anything known and anything that is seriously being hypothesized, so even the exotic stuff is fine as long as it is described in peer-reviewed papers and considered plausible to exist.

First thing one could think of is making a ball made out of heavy elements for the small body (maybe surrounding it with a thin insulating layer if something radioactive would be used) and a ball made out of light but solid elements for the big one. However, this wouldn't cover a wide range of sizes. It would be interesting to see something roughly comparable to the asteroids from The Little Prince for the small body and something comparable to Jupiter or maybe even a Dyson sphere for the big body.

Celestial bodies of these sizes sizes certainly do exist, but they don't satisfy the "walkability" requirements (e.g. neutron stars are very small, but also very massive, resulting in an extreme surface gravity among other issues). Maybe a bit of something exotic could be combined with something more mundane in the composition to allow for a wider range of sizes of walkable celestial bodies.

Answer 1

Smallest Possible:

The highest density you can get with a naturally forming planet would be from one that forms in an environment that averages about 4600-5000°C. This will boil away everything else leaving just a molten mass of Tungsten, Osmium, Rhenium, and Tantalum. If something were to then happen that pulls or pushes the planet farther away from the heat source, you would be left with a round solid heavy metal world with a density of somewhere between 16.65-22.59 g/cm³ depending on the ratios of these 4 remaining elements. Since you won't get a purely Osmium world this way, your actual density cap is probably going to be somewhere around 20 g/cm³. (Technically a purely Rhenium planet could be 21 g/cm³ but its boiling point is so close to the less dense Tungsten that boiling off Tungsten without also losing your Rhenium is unfeasible). This would give you a radius of about 1750km

If your planet is artificially formed from natural elements, you could make it out of pure Osmium for a maximum density of 22.59 g/cm³ and a radius of 1550km. This would not happen in nature because of boiling points and co-genesis of these elements from the same sorts of astrological events.

For a an artificial structure that relies on purely theoretical science, you could build a shell around a primordial black hole (if they exist), but you have to make sure it is not so small that it would just melt from the black hole's hawking radiation. For this I would suggest you use a black hole that is ~6e13kg at a radius of about 20m. Now, something this small just containing a black hole of this size would probably just melt... unless you are doing something useful with all of that heat. With some clever engineering you can treat this "world" as a tiny power plant. At a power output of 100KW, this black hole is 10,000 times weaker than the average nuclear power plant which requires about 2.6million m² of land area meaning this power plant only needs about 260m². Since your tiny world has about 5027m² of surface, you have plenty of room for both the power plant and any additional stuff you may need to expend or transfer the energy off world and radiate off the unused heat. With cleaver engineering, you could probably even go a few meters smaller, but every meter you contract, the smaller you need to make the black hole and the hotter it gets, and the less room you have for your reactor... this means the issue of heat goes up exponentially VERY quickly at a smaller scale.

Biggest possible:

The biggest possible natural world is really hard to solve for because it is so hard to predict how loosely elements can pack under unknown circumstances. For example, Hyperion is a moon with what appears to have a rocky crust and maybe a highly porous icy core, but we don't really understand it that well. All we know for sure is that it has a density of only 0.5 g/cm³. Since we can't explain how it was formed or why its density is so low, we cannot extrapolate for sure if this phenomenon could apply to larger worlds. But if we work off of the assumption that it might, we could get a radius of about 70,000km.

For an artificial solution, the lightest known solid material that that can survive in 1G is a substance called graphene aerogel. With a density of .00016 g/cm³ you could have a radius of about 218,000,000km meaning you could make a solid planet just a bit larger than the orbit of the Earth around the sun with one surface G.

Now keep in mind that this math assumes a solid, homogeneous, aerogel structure; so, engineering limitations might force you to use a denser aerogel as you get to the core to compensate for pressure, but as an engineered structure, you also don't have to make it perfectly solid either; so, after all engineering variables are solved for, you might have an aerogel lattice structure or hallowed out structure over a billion kilometers across or you might be forced to make something much smaller. This is really hard to predict for certain without working out every possible engineering solution to the problem in detail, but either way; something on the dyson sphere scale certainly seems possible.

All calculations are approximations based off of these calculators:

• https://planetcalc.com/1758/
• https://www.ericjamesstone.com/blog/home/gravity-calculator-for-astronomical-bodies-based-on-radius-and-density/
• https://www.vttoth.com/CMS/physics-notes/311-hawking-radiation-calculator

Answer 2

Small Body: About 100km (Actually 1000km) Radius.

The problem with small bodies is (a) Limits on matter density and (b) tidal forces. For example more gravity at your feet than head is a recipe for a bad time. Let's do some rough calculations to see how small the body needs to be for double gravity at the feet.

Say the planet has radius is $r$ metres and mass $M$ kg, the person is $2m$ tall and has mass $m$ kg. The gravity $F_2$ at their head is half the gravity $F_1$ at their feet. Then for the gravitational constant $G \simeq 5 \times 10^{-11}$ we have

$$F_1 = \frac{GMm}{r^2} \qquad F_2 = \frac{GMm}{(r+2)^2} = \frac{F_1}{2}$$

Solve to see $r = \frac{2}{\sqrt 2-1} \simeq 5$ metres. We also want the force at the feet to be the same as Earth which is about $10$. So we solve $F_1 =10m$ to get

$$F_1 =10m \implies \frac{GM}{25} =10 \implies M = \frac{2}{5G} \simeq \frac{2 }{25} 10^{11}$$

So we need that much mass in a 5 metre ball. So what material is dense enough for that? Well the mass is $\rho (4/3) \pi r^3 \simeq 4\rho r^3 = \simeq 4\rho 125 = 500 \rho $ for $\rho$ the density. So we need $\rho \simeq 10^9$ kg per cubic metre. This is much denser than the densest element Osmium which is about 22 590 kg/m³.

The conclusion is you cannot build something small enough out of ordinary matter to make the tidal forces relevant. So let's just take a solid sphere of osmium. Eric James Stone says that means we need a radius of about 1500km. Of course there are errors for how the matter is more dense at the centre. So lets say we need a radius of 1000km or so. That's certainly larger than 5m.

Added later: The "largest possible" question is harder to answer. What you want to do is take a small dense object and build a hollow superstructure around it that is strong enough to support itself and the gravitational effect of the singularity. I'll go out on a limb and say the cube-square law prohibits such a large structure of any significant size.

There is a hard limit at about twice the mass of Jupiter where the structure turns into a star.

I'll guess the true upper limit is about nine times the diameter of Earth or 70,000km. Eric James Stone says a sphere of ice that size has about 1g gravity. We know there's no problem with that since Neptune is mostly ice.

You could in principle build something larger using a less dense material. For example solid hydrogen has a very low density of about 0.08 grams/cubic cm. So you could try to build a ball of solid hydrogen. You need 1 million km diameter to get the right surface gravity. But that's much bigger than Jupiter so I imagine gravitational effects come into play long (LONG) before that to make the planet more compact.

So let's say the size of Neptune.

Answer 3

Regarding the largest body. if you include artificially made bodies, the Birch planet ( the concept invented by Paul birch coined by youtuber Isaac Arthur) is a shellworld surrounding a super massive black hole with the option of making the gravity on the outer shell normal earth gravity it could in theory be a light year accross and still be considered one body

Edit: Just to add that this is basically an upscaled shellworld. A shellworld can be built around gas giants to simulate earth gravity on the outer shell