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

Question

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?)

Answer 1

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.

Answer 2

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.

Added:

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.

Answer 3

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).