# How much variation is possible among planets with “Earth-like” surface gravity? [closed]

What is the possible range of variation among planets with Earth-like surface gravity, and what limitations could we expect on this variation assuming the planets were formed by natural processes? For example, could a planet of Moon-like radius have Earth-like surface gravity, and what would that be like (are there any special reasons worlds at the extremes of this range of variation would be enimical to life, would their escape velocities vary significantly?)

## closed as off-topic by sphennings, Aify, Thucydides, Bellerophon, VincentAug 20 '17 at 23:30

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question does not appear to be about worldbuilding, within the scope defined in the help center." – sphennings, Aify, Thucydides, Bellerophon
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• Earth is already made out of iron mostly, I doubt you can get much smaller with the same gravity naturally. Your question is super broad currently and I suspect it won't stay open very long in its current form. Could you maybe define variation? Do you want to know about sizes only? "What would that be like" I think is a question for another topic. Once you have decided on a size for your world, you should consider asking that question in a new thread – Raditz_35 Aug 20 '17 at 20:21
• The question is intended to be very general. IMO, an ideal answer would involve a plot of 1G on a density vs. radius graph and a discussion of problems and weirdnesses at the extremes of the curve. I think it's a very well-defined question, you just want it to be about a single planet rather than a range of possibilities. It's definitely about the latter. – SudoSedWinifred Aug 20 '17 at 20:28
• Now you are asking a well defined question, but I wouldn't have guessed that from your original post in a million years. I think you should edit that in. But now I wonder if you care about how that density is obtained (chemical composition of the planet) or just want to see that plot? Btw what do you mean by extremes of the curve? What kind of relationship do you suspect? – Raditz_35 Aug 20 '17 at 20:31
• The possible compositions would bear on the characteristics of the planet at the extremes (i.e. "at this end of the extreme it's all heavy metals so life will have a rough time" or "at this end of the extreme there's no solid surface because...") By the extremes of the curve I literally just mean the lowest densities and the highest densities (though this could also be in terms of radius, since they're related for the plot of 1G surface gravity, as defined in the question by the specification of Earth-like surface gravity.) – SudoSedWinifred Aug 20 '17 at 20:41
• All can be found here btw en.wikipedia.org/wiki/Surface_gravity . You can see that we have a r ~ 1/rho relationship here. Such a plot is very easy to do. I'm a big fan of people doing what they are able to do first before asking for help. I think you can solve the first part of your problem quite easily and then ask about problems with that curve, for example when the planet becomes a gas giant or needs a density too high to form naturally – Raditz_35 Aug 20 '17 at 20:53

That depends on how much the planet needs to have "Earth-like" surface gravity, and what range of variation in the strength of gravity still counts as "Earth-like".

The strength of gravity scales linearly with mass, and with the inverse square of radius. So, a planet with half the radius of the Earth would need to have 1/4 the mass to maintain the same surface gravity. It's volume would only be 1/8th as much as the Earth, however, so the density would have to double. Similarly, a planet with twice the radius of the Earth would need 4 times as much mass, but half the density.

Escape velocity, on the other hand, scales with the square root of mass and the inverse square root of radius, so the escape velocity will indeed be quite different for each of these hypothetical worlds. A planet with half the Earth's radius and equal gravity would have an escape velocity about 70% ($\frac{1}{\sqrt{2}}$) as large as Earth's, while a planet with twice the Earth's radius but equal gravity would have an escape velocity about 40% larger.

The average density of the Earth is $5.51g/cm^3$. The upper mantle has an average density of $3.4g/cm^3$, while the inner core has a density of $12.8g/cm^3$. If we take those as reasonable minimum and maximum values for the average density of a naturally formed rocky planet, made entirely out of relatively light-weight rocks at one extreme, and a solid ball of iron at the other extreme, that means you could have a planet up to about 1.62 times larger than Earth in radius, with a higher escape velocity, or as small as 0.43 times Earth's radius, with a lower escape velocity.

Both of these extremes may present problems for habitability. The large world, with no metallic core, would have a harder time supporting a planetary magnetic field, leaving the atmosphere vulnerable to hydrodynamic stripping by the solar wind. Given the higher escape velocity, and depending on how much atmosphere it starts with, that may or may not be an insurmountable problem. The smaller world, however, with a lower escape velocity, will have a harder time holding on to light gasses despite having a magnetic field. This may be survivable with the right atmospheric composition to keep exospheric temperatures down.

Note, however, that there are a lot more factors that will influence habitability. Venus, for example, is almost exactly the same size as Earth, has almost exactly the same surface gravity and almost exactly the same escape velocity... and it's an acidic, metal-melting hell.

If you don't require the surface to be solid, you can of course have much larger planets with Earth-like gravity on a liquid or gaseous surface, with much lower densities. Uranus, for example, has lower "surface" gravity (at the cloud tops) than Earth, while Saturn and Neptune have cloud-top gravity only slightly higher.

If you don't need to have Earth-like gravity over the entire surface of the planet, you can go larger still. Spinning the planet sufficiently rapidly can noticeably reduce the effective gravity at the equator. It may not be particularly likely for a world with, say, 20 times Earth's mass to form naturally with a high enough spin to produce 1g at the equator, but at the extreme you can theoretically get planets like Hal Clement's Mesklin, with a mass 16 times greater than Jupiter, 3g at the equator (which could be even lower if you spun it a little faster) and somewhere between 200 and 700g at the pole.

• This is great and I may eventually accept it (already upvoted), I hadn't even personally considered the effect of rotation on surface gravity, but now that you've mentioned it I see that this is a much more complex question than I previously estimated. – SudoSedWinifred Aug 20 '17 at 21:06

Surface gravity, as a ratio to earth gravity, is determined by two factors: The planet's Mass (in Earth masses, call that $M$), and the planet's Radius (in Earth Radii, call that $R$). The formula is simple: The surface gravity $G=\frac{M}{R^2}$. Here is a helpful page for that from Stanford.

Here is a Nasa spreadsheet with everything specifically Relative to Earth, and another of Raw Metric values.

So, for example, The Moon's mass is only 0.0123 of Earth (1.23%), but its diameter is 0.2724 that of Earth. So it's surface gravity is $0.0123 / (0.2724^2) = 0.1658$ of Earth gravity; about one sixth. That computation is on the NASA sheets too.

So look at Saturn: It's mass is 95x that of Earth, but because of its huge radius, it's surface gravity is actually less than Earth.

What matters is basically the radius; a very high density material (this is measured in grams per cubic centimeter) can have a smaller radius: To answer your question, for the mass of the Moon, you would need that amount of Platinum (or iridium, or uranium) to have a small enough radius to match the gravity of the Earth.

But obviously very large planets, with low density, could also have the same surface gravity as the Earth. What you would need to do is pick a size, work backward from the formula to compute a density, and see if there are materials that could be used. If I have time later, I will edit and add some examples.

Here are the elements sorted by density.

I deleted my error on escape velocity and bow to greater expertise.

Recall we computed the Moon's surface gravity at $G=0.0123 / (0.2724^2) = 0.1658$. If we wanted the Moon to have Earth's gravity without changing its radius, we would have to multiply its mass of 0.0123 by $\frac{1}{0.1658}=6.03$. From the NASA fact sheet, the density of the Moon is 3340 kg/m$^3$. We need to convert that to normal elemental densities of g/cm$^3$, so we have $3,340,000~ g/(100 cm)^3 = 3.340~ g/cm^3$. Multiplied by 6.03, we need a material that is 20.1402 g/cm$^3$. Looking at the elements sorted by density, we see five that match that limit: Uranium (20.2), Rhenium (21.04), Platinum (21.45), Iridium (22.4) and Osmium (22.6). Plus many of the strange or ephemeral elements from super-colliders for which we have no density, but are probably candidates too. Platinum and Iridium are non-toxic in bulk solid form (as a dust they might be harmful). But a planet the size of the Moon made of some mix of these very dense elements, with just a small amount of less dense elements (like oxygen) could have exactly the same surface gravity as Earth.

It would be difficult to get much smaller than the moon, however. You could get bigger if your planet was made of very low density material; like silicon and aluminum.

• Escape velocity is given by the square root of 2GM/r, so it definitely diverges from surface gravity (m/r^2, as you note) at some parts of the density vs. radius curve. (It's worth noting that the dimensions are also wrong: escape velocity is velocity, surface gravity is acceleration.) – SudoSedWinifred Aug 20 '17 at 20:47
• You believe wrong. Escape velocity scales with the square root of mass over radius, so larger planets with equal gravity will have higher escape velocities. Hence why Saturn has a much higher escape velocity than Earth despite having similar "surface" gravity. – Logan R. Kearsley Aug 20 '17 at 20:48

To put it in another way: Earth is already almost the smallest planet that can have "Earth-like" gravity because it already has a fairly heavy nucleus (mostly Ni/Fe, but it also has a Si/Al part, so it's not really on boundary).

To have smaller planets with same surface gravity you would need a nucleus almost completely composed by Fe, which is unlikely.

To have Moon-size planets with Earth surface gravity you need a nucleus of very dense metal, very unlikely.

Moving in the other direction, OTOH, is quite easy; if Earth would be fully composed of crust materials surface gravity would be about a fourth (or, to have the same gravity it should have a radius about six time larger).