Is it realistic to have a livable planet with a quarter the size of Earth, but with the same gravity?

So in theory, I would want this planet to have the same everything as earth besides the radius (and anything that affects how radius plays into gravity). I know the density of this Planet X would have to be quite high based on some quick number crunches. But I'm not sure how realistic this would be. Would the planet be livable? How different would the temperatures of the planet be? Could it be mostly water (how does water affect the density)? Is there a way to have that high a density core?

Realistically how close could an average moon be to this planet? Could this planet have rings?

I have a fair idea of how planets work but not enough knowledge to actually know how realistic this might be.

• Not really, it'd need 4 times the mean density of Earth. A planet of iridium isn't realistic. – PM 2Ring Aug 29 at 6:02
• If you wanted to get creative in a world-building kind of way (bending the science), imagine that a chunk of white dwarf got blown off but remained stable and you have a chunk of dense white dwarf matter in the core. (There's a thousand reasons why that's unlikely as it would require an undiscovered type of matter, but it's just barely possible, like scratching the winning ticket in lotto while being bitten by a lion and being struck by lightning at the same time unlikely) . . . but you could pretend. – userLTK Aug 29 at 18:54
• Also, a planet with Earth's surface gravity but a quarter the radius has a surface escape velocity of about 5.6 km/s (vs 11.2 km/s for Earth). At about Earth temperature, it'll lose its atmospheric water vapor over the billions of years through Jeans escape, which will probably lead to the loss of any oceans. – notovny Aug 29 at 20:21
• Hi Dom, welcome to Worldbuilding! If you're still around, I think we'll need a little more clarification to give you the best answers. You're looking for a planet with a size (radius) one quarter that of Earth's, i.e. 1600km? You may be interested in our question here which approaches a similar topic. – Dubukay Sep 9 at 16:57
• By "a quarter the size" do you mean the diameter is 4x less? That means the volume is 64x less. Even if it was made of osmium, the densest regular material known to science (not neutronium or black hole), it would still not be massive enough. – Florin Andrei Sep 9 at 17:52

I'm going to use a few approximations here but:

To get the same gravity you need the same mass, however gravity also diminishes as a factor of r.

Earth has a mass of 5.97*1024 kg

Earth density is 5.51 g/cm3 with a radius of 6371km.

Assuming by 1/4 of the size you mean 1/4 of the radius then you need to look at the equation for volume of a sphere. The good news is that all the factors except for the radius cancels out. The bad news is that the radius is cubed.

So a sphere with 1/4 the radius has 1/64th of the volume.

However the good news is that because gravity reduces by radius r this does cancel out one factor of the volume.

So to get the same surface gravity you need 16 times the density to give you 1/4 of the mass at 1/4 of the radius.

5.51 * 16 = 88 g/cm3

One of the heaviest materials we have is gold with a density of 19.3g/cm3. Most other materials don't even come close and none are significantly heavier.

So even if your planet was made entirely of solid gold you would have a density only 4 times that of earth. You are 4 times too light. A solid gold planet 1/2 the radius of earth is about as small as you can go.

The only way would be as others have suggested putting something massive in the center such as a neutron star or black hole. It is hard to see how that could be done in a stable way though as you would expect it to eat the planet.

You can't take "neutron star material" and put it in the center of the planet as without the neutron stars gravity to hold it together we can expect it to expand, probably extremely rapidly.

• Gravity is also a function of distance to the center of mass. With 1/4 radius and same mass you'd experience 16x Earth's gravity. I think you'd only need to 4x Earth's density. – Renan Sep 12 at 12:35
• @Renan You're on the right track, but you need 16x the density, not 4x. Mass decreases with $r^3$, while gravity increases with $r$, so the net effect is that gravitational force decreases with $r^2$. A planet 1/4 the radius needs to be 16 times as dense to have the same surface gravity, which puts this well outside the density range of terrestrial materials. – Nuclear Wang Sep 12 at 13:00
• I am glad we can always count on WB to nerd things into correctness :) Thank you! – Renan Sep 12 at 13:09
• @NuclearWang Thanks. You are right of course, that's what I get for not paying attention. I've updated the answer. – Tim B Sep 12 at 13:35
• @Renan Thanks, you are right that I'd missed that factor - which I should have known better but it's been a while :) – Tim B Sep 12 at 13:36

My first instinct is saying yes; to keep the gravity ratio all you need is up the density as you lower the size of the planet; but a quarter is simply too much. The only way I see it happening is if something is artificially augmenting the gravity. Maybe there is an alien gravity generator in the core of the planet or it was artificially made out of a very dense material. Certainly current off-the-shelf technology wouldn't cut it, so it's gonna be a "NO" on my part. Sorry.

• There could be planetary-mass black hole somehow kept in the middle of planet in some enclosure that keeps the rock above from falling into it. Still, not realistic. – Juraj Sep 11 at 16:18

If you want earth like conditions, on a planet quarter the size, at least the crust has to be same as earth's. Which means the core of the planet is absurdly dense. This points to more metal in the core. Heavier metals. The more you shrink the planet, the bigger this metal core has to be. So as you reduce the size of the planet, assuming crust thickness remains the same, you need to calculate at what point there is only the core and the crust left, nothing in between. That is the smallest your planet can be.

One side effect you need to consider is that a smaller, denser planet will probably spin faster causing the planet to become more oblong along equator, basically a flatter earth. If the rate of rotation remains the same, the smaller radius means linear velocity of the surface would slow down instead, causing change in atmospheric winds etc. Finally bigger denser metal core would possibly up the magnetic field around the planet.

• Why would a smaller planet have to spin faster? I don't follow your point about the rate of rotation and atmospheric winds - the atmosphere will spin at roughly whatever speed the planet does. I could see both of those being the case if the earth was somehow shrunk in place to 1/4 the size, but we're talking about an entirely new planet. The lower limit on planet size isn't really meaningful, as you'll require exotic, super dense matter (the kind of stuff that planets aren't made of) to make up the missing mass. It's unclear why the crust has to stay the same 40 miles down to be earthlike. – Nuclear Wang Sep 12 at 13:06
• The change in rotation speed is to account for and maintain the proper centrifugal force. Thus I said probably. I compared it to spinning a thread with a weight attached in my head. As the threads becomes smaller, it must spin faster. – Deepon GhoseRoy Sep 12 at 13:34
• @DeeponGhoseRoy planets don't seem to have that correspondence. Jupitor and Saturn both have 10 hour days, Mars has similar length to ours but Venus 116 days! – Tim B Sep 12 at 13:39
• Also I tried to maintain similar crust size due to plate tectonics, otherwise the geographical feature would probably wbe two different. I am no expert so probably. – Deepon GhoseRoy Sep 12 at 13:45
• @TimB, you are right about that, but then at similar rotation rates, linear velocity of surface of smaller planet will be less due to smaller circumference. Don't we need to maintain that for earth like conditions ? – Deepon GhoseRoy Sep 12 at 13:51

You would need to have some kind of super dense material at the core of the planet, then the rest of the planet could just be normal earth and water.

For example you could have an Osmium core or Quark matter core. I am not sure how realistic these are, and whether or not these would need enormous pressure to keep them at the required densities. However something should meet the requirements.