How Big Can a Habitable Planet Get Before Its Gravity is More Than 0.8 m/s² above Earth's 9.81 m/s²?

This is NOT a repeat of this question.

I am NOT looking to see the smallest planet possible that has less gravity than Earth. I am looking for how big a perfectly habitable planet (Habitable meaning capable of supporting Earth-like lifeforms, in terms of temperature, magnetic field, and atmospheric pressure) can get, WHILE having a gravity field no more than 0.8 m/s² more than Earth's 9.81 m/s², if not the same as Earth's.

This would mean I want the diameter (at the planet's equator) for the biggest possible planet with a magnetic field strong enough to support Earth animals, AND the atmosphere to do so, while having a gravity strength between 9.81 to 10.61 m/s². Nothing else about the planet - moons, topography, atmospheric contents - is important here. I just want to know how big a planet can get before its gravity exceeds 10.61 m/s², while having a magnetic field and the atmosphere needed for supporting life.

A good answer tells me the maximum size of this Mega Earth with habitability while keeping within the 10.61 m/s² upper limit.

A great answer also tells me the materials this planet would be made of, AND keeps it super close to 9.81 m/s² with the magnetic field.

• Thank you, sir. Mar 17 '21 at 17:50
• Are we allowed to assume that this planet was built (ie. wildly implausible materials), or should the assumption be that it occurred naturally? (Also, why the fixation on 0.8m/s/s?) Mar 17 '21 at 17:54
– L.Dutch
Mar 17 '21 at 18:01
• A helpful planetary calculator Mar 17 '21 at 18:02
• @Rottweileronmarket-day. - he said no artificial constructs, but got zapped for language. Mar 17 '21 at 18:57

You want a magnetic field...

This means, according to our latest theories on how planets produce their magnetic field, that in the core you need to have some metal. Considering the abundances of elements in the universe, I guess your choices are either metallic hydrogen or iron-nickel.

Metallic hydrogen is thought to be present in conditions present in the core of heavy-weights like Jupiter, therefore I guess it is disqualified. You are left then with iron-nickel, like our Earth.

Then the size of the planet starts to play a role: too small and it will quickly, in astronomical times, cool down, stopping the dynamo (ask Mars for info) and killing the magnetic field.

Of the 4 data points that we have in our statistical series (Mercury, Venus, Earth and Mars) only one has sufficient geological activity to signify a significant molten core, and that is Earth. Venus has signs of volcanism but no magnetic field, so it is sort of an odd ball. Let's go with a core size about the one of Earth. How can we make the planet bigger without getting too away with gravity?

We can use lighter elements, but there we are also constrained by the abundance of elements. The most plausible choice seems to be a carbon rich crust, way richer than Earth, which is at 3000 parts per million of Carbon and 653 parts per million of Silicon, with 9500 parts per million of Oxygen.

If you manage to pull down the average density to 5 g/cc, you can get to a radius of 7500 km with a gravity of 10.48 $$m/s^2$$. (Earth is at 5.5 g/cc with a radius of 6300 km). With 4.5 g/cc you can get to about 8000 km radius.

However mind that it can be tricky to reconcile a carbon rich planet with an oxygen rich atmosphere.

• Hmm, if you try to increase the radius by just swapping in lots of carbon, wouldn't you fairly quickly run into the problem of it getting compressed to some kind of high density diamonds? Mar 18 '21 at 7:45
• @Kaithar Diamonds have a standard density of ~3.5 g/cm^3, which is still a lot less than Earth's average density. Mar 18 '21 at 11:57
• Don't forget the effect of the Moon on the Earth. Tidal stresses produce heat too I believe. Mar 18 '21 at 12:02
• @Corey facepalm I totally forgot diamond is particularly incomprehensible, but that wasn't my point. What I meant is, if you try to reduce the density with something like coal or graphite, the pressure should start converting the lower layers to the higher density diamond. I was wondering something might be better for the lower level. Mar 19 '21 at 16:16

If the planet were made out of light material, sans large iron core, it could be much larger. The trick is the magnetic field...

Take a look at Ganymede: it has a magnetic field, but it's buried within the far larger field of Jupiter. I was thinking there was a moon with an "induced" magnetic field from its parent, but maybe that's out of date. But this works: it shows that a large body can be shielded by the magnetic field of its primary, so doesn't need to generate one itself!

Make the body, larger than Earth, orbit a super-Jovian planet or brown dwarf that has a magnetic field that works to deflect the solar wind and all that good stuff.

Thanks to M.A.Golding for finding this scientific work which investigates this exact idea: Magnetic Shielding of Exomoons Beyond the Circumplanetary Habitable Edge.

• Jupiter's magnetic field has...unpleasant...side effects such as a ferociously strong radiation belt.
– Mark
Mar 18 '21 at 22:03
• @Mark under some conditions, the magnetic field of a Jupiter sized planet can shield its moons from the solar wind without dangerous radiation. See researchgate.net/publication/… Mar 20 '21 at 19:32

If you want a habitable planet that has a gravity similar to that of Earth and a magnetic field and atmosphere similar to Earth then that planet will be of similar size and mass to Earth.

There are a number of interrelated properties which dictate why there won't be much deviation.

Surface gravity, is determined by the radius of the planet and its mass.

$$g = GM/r^2$$

Where:

• $$g$$ is the surface gravity ($$\sf{m/s^2}$$)
• $$M$$ is the mass of the planet ($$\sf{kg}$$)
• $$r$$ is the radius of the planet ($$\sf{m}$$)
• G is gravitational constant, with a value of $$\sf{6.647 x 10^{-11} m^3.kg^{-1}.s^{-2}}$$

Average density of the planet is simply the mass divided by the volume, $$\rho = M/V$$

The properties for Earth are:

• Mass, $$\sf{5973.6 \ x \ 10^{21} \ kg}$$
• Radius, $$\sf{6371 \ km}$$
• Volume, $$\sf{1083.207 \ x \ 10^9 \ km^3}$$
• Average density, $$\sf{5.514 \ g/cm^3}$$
• Gravity, $$\sf{9.7803 m/s^2}$$

If the radius of your planet was 1.1 Earth radii ($$\sf{7008.1 \ km}$$) and the mass was 1.3072 Earth masses ($$\sf{7808.69 \ x 10^{21} \ kg}$$), its density would be $$\sf{5.416 \ g/cm^3}$$, which is slightly less than that of Mercury ($$\sf{5.427 \ g/cm^3}$$). The surface gravity would be $$\sf{10.61 \ m/s^2}$$.

If the radius was 1.5 Earth radii and the mass was 2.431 Earth masses, the gravity $$\sf{10.61 \ m/s^2}$$, but the average density would only be $$\sf{3.972 \ g/cm^3}$$, which would be very similar to the density of Mars ($$\sf{3.934 \ g/cm^3}$$). This is comparatively low density which would suggest a lighter weight core of the planet and possibly a smaller magnetic field.

You would want a planet with a lower density than Earth. If one were to increase the radius of a planet while keeping density the same, mass would increase in a cubic relationship, and as gravity is mass times the gravitational constant divided by the square of radius, that means that surface gravity will increase in an linear function. A planet with a density of Earth's, and a gravity of 1.08 g, which is the uppermost limit, would have a radius of 1.08 Earth radii. We would need to use less-dense materials, and let's assume that said planet is not a gas giant. Pallas, a dwarf planet made mostly of ice, has a density of 2890 kg/m^3, which would be about 1.91 times less dense than Earth.

If we had an planet with Earth's radius, and Pallas' density, its surface gravity would be .524 Earth radii. That means that the uppermost radius that would give a surface gravity within your specified range would be 2.06 Earth radii. Anything larger, and you would probably have to make it a gas giant to keep surface gravity just low enough.

• This fails the requirements of having a magnetic field - you wont' get a magnetic field with a pure iceball, and the larger planet is going to mean more pressure at the core anyway - which will apply state-changes and compression that will increase its overall density. Mar 18 '21 at 17:34

An Earthlike magnetic field implies a conductive liquid core (doesn't have to be metal; any conductive liquid that circulates in the correct manner will produce magnetism). The density of this core, the mantle, and crust material will determine how large the radius can be and fall within your gravitational limits (note that the larger the radius, the larger the mass can be, so we don't need to limit the planet to no more massive than Earth).

If we eliminate the possibility of artificial construction (a surface that's a thin shell supported by the narrowest possible pillars on top of a much smaller, denser body to provide the magnetic field requirement), we'll need to least dense material possible to predominated in the mantle and crust.

That least dense material (that can stand the weight of the layers above) is probably some form of pressure ice. There are other questions (and even an XKCD, as I recall) concerning how big and dense a pure-water planet could be; this wouldn't be pure water (must be conductive, which implies a certain minimum level of dissolved ions -- sea water works pretty well here, at 2.5% denser than pure), but it won't change things much.

What I recall from those is that you'll wind up with a liquid or solid surface (depending on the atmosphere), a deep layer of liquid water, a layer of hot pressure ice, and a superhot liquid core. The overall density will be no higher than a third of what you'd get from the silicate and ultramafic rocks of Earths' crust and mantle overlaying an iron-nickel core, which allows the planet to be roughly three times the diameter of Earth to have the same gravity.

The problem that arises is that water won't be water at the core of such a mass; there's a point down there somewhere at which it becomes a loose "soup" of unbonded oxygen and hydrogen, and as such may become non-conductive. As far as I know, no one has been able to investigate what water does under much more than a million atmospheres, and you'd pass that pressure a few hundred kilometers down.

How such a body could form is another interesting question, since even comets have some dust and rock in their nuclei, and would thus contribute to a small rocky core (which likely wouldn't be liquid, as there isn't as much core heat available as in a rocky planet). We may be back to an artificial construct after all...

Saturn.

Saturn has gravity of 10.44 m/s^2, just barely below your limit. It is made mostly of hydrogen with a little helium - and it's hard to see a way to refine helium out of a planet. You can suppose a development that rips out the small rocky core, and you can heat it a little, because right now the habitable zone (300 K, liquid water) is around 15 atm, and you want it closer to 1 atm. But the depths of the planet are tremendously hot, so you would be expanding at most a few hundred km of the atmosphere by two-fold or less, with a 2 pi effect on circumference. Unless the core of Saturn turns out to be substantial, to three significant figures, I think Saturn's 366000 km circumference is as far as you're going to get.

There is also a magnetic field. As for surface, well, that's another story... :)

• I did idly wonder about Saturn, but the requirement to have an atmosphere suitable for supporting Earth's animals rather scuppered the idea. It is difficult to get free oxygen in a hydrogen rich atmosphere, after all, and impractical to find a way to create a nice poofy gas giant with comfortable cloudtops that isn't hydrogen rich. Mar 17 '21 at 22:06
• ...but having said that, it turns out that Gliese 436b is a theoretical "helium planet", with a hydrogen-poor atmosphere. Now if only there was some way to boil off all that hydrogen without needing a torch orbit... Mar 17 '21 at 22:13
• Hmmm, I see what you mean. But then the question said "atmospheric contents" weren't important. I have my own crazy notions for how to make that atmosphere work (cleaving water using energy from organisms that extract mechanical energy from wind turbulence) but that's for another post or six. Mar 18 '21 at 2:08