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Suppose that I have a planet very similar to Earth. It has the same level of gravity, water-land ratio, temperature, air composition, rotation and revolution speed, tectonic activities, temperature. It has ice caps on the poles just like Earth. It also have life on it with plants, animals and humans, although they can be different than what we have here.

How small this planet can be in diameter while still retaining those properties? I suspect that such planet would have to be denser so that it retains the same amount of mass which will then affect the gravitational pull. But just how small is the limit so that it still have enough time (that is the core is still active long enough) for life to bloom on it and evolve into our level (bipedal humanoid with intelligence, if possible).

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    $\begingroup$ Lighter materials but denser? Care to run that by us again? $\endgroup$ – JDługosz Nov 28 '16 at 7:29
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    $\begingroup$ Ah yes, it is contradictory. I'll edit it out. $\endgroup$ – 絢瀬絵里 Nov 28 '16 at 7:39
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    $\begingroup$ As far as I know, we live on the densest planet in our solar system. And smaller planets are less dense mostly because of the lack of gravitational compression. Do you need your planet to form naturally, or just to be and to heel with how? $\endgroup$ – Mołot Nov 28 '16 at 8:01
  • $\begingroup$ I'd like it to be natural if possible. Smaller planet has less mass, thus less gravity, no? I mean they have less gravity because they have less mass and not the other way around, right? $\endgroup$ – 絢瀬絵里 Nov 28 '16 at 8:14
  • $\begingroup$ It works both ways. At some point adding mass to a planet causes it grow smaller. Even before that point, for the same mass, denser material means higher surface gravity, higher mean gravity planet matter is under, and material going even denser due to compression, making raidus smaller and surface gravity still higher. $\endgroup$ – Mołot Nov 28 '16 at 8:27
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Surface gravity

Surface gravity is really the most important quantity when it comes to determining many of your planet's properties. It can be used to constrain atmospheric composition, planetary mass and radius, composition, and more.

A planet with mass $M$ and radius $R$ has a surface gravity of $g\equiv GM/R^2$. Therefore, all planets with Earth's surface gravity obey the following mass-radius relation: $$\frac{M}{M_{\oplus}}=\left(\frac{R}{R_{\oplus}}\right)^2\tag{1}$$ where $_{\oplus}$ denotes a parameter of Earth.

Theorists have come up with additional mass-radius relations that depend on a planet's composition. Seager et al. 2008 came up with mass-radius relations for a number of rocky planet compositions. I've plotted their results for several different planet types (iron, water, silicate, and graphite), as well as the criterion from $(1)$:

Plot of mass-radius relations

Assuming that iron is the densest likely composition, we see a lower radius limit of $R\approx0.5R_{\oplus}$, corresponding to $M\approx0.2M_{\oplus}$. We appear to have found exoplanets with higher densities (e.g. Kepler-36b), but those may be due to measurement errors; they seem unphysical.

Now, iron planets are often thought to form through giant impacts - collisions between protoplanets (Marcus et al. 2010); Mercury is thought to have been affected by such a collisions. We can calculate the mass of the largest remnant of such a collision by the formula $$M_{\text{rem}}=\left[-1.2(f_{\text{Fe}}-0.33)^{1/1.65}+1\right]M_0(1+\gamma)$$ where $f_{\text{Fe}}$ is the final iron mass fraction, $M_0$ is the initial mass of the protoplanet, and $\gamma$ is the mass ratio of the impactor to the protoplanet. Typically, an upper limit of $f_{\text{Fe}}\sim0.8$ is assumeto be the theoretical limit. If we ignore this and set $f_{\text{Fe}}=1$ and $M_{\text{rem}}=0.2M_{\oplus}$, we find that $M_0(1+\gamma)=3.41$, which can be satisfied by, say, a $2M_{\oplus}$ protoplanet and a $1.4M_{\oplus}$ projectile - not unreasonable. For $f_{\text{Fe}}=0.8$, we can afford to have smaller bodies. Either way, producing an iron planet of the desired mass is quite easy.

Plate tectonics

Plate tectonics depend on a number of factors, including a planet's size and composition. The smaller the planet, the quicker the cooling rate, meaning that this body is likely to cool quickly, making plate tectonics unfeasible. Arguably, though, our stripped-mantle iron planet lacks a mantle or crust, and so plate tectonics as we know it cannot exist at any point in time.

Temperature

Assuming a solid iron surface, with water constituting the liquid part, I see no reason why the temperature couldn't be similar to Earth, assuming a similar atmosphere. The albedo should be the same, and if the planet is as far from a Sun-like star as Earth is, its effective temperature should be the same. Depending on the atmosphere you end up with, you can vary the orbital and stellar parameters as you choose.

Atmosphere

Atmospheric escape is going to be a problem; it's how Earth lost its early hydrogen/helium envelope. I covered that more thoroughly in an answer on Physics Stack Exchange, but the important equation here is for the Jeans flux for a particle of mass $m$, $\phi_J(m)$, which describes how many particles of mass $m$ will escape the atmosphere through thermal methods: $$\phi_J(m)\propto n_c\sqrt{\frac{2kT}{m}}\left(1+\frac{GMm}{kTr}\right)\exp\left(-\frac{GMm}{kTr}\right)$$ The important term here is $$\frac{GMm}{kTr}\approx\frac{gR}{kT}$$ where $r$ is the distance to the lower edge of the exosphere. Given that $g$ is the same as $g$ on Earth, and $R=0.2R_{\oplus}$, then at Earth-like surface temperatures, we should see substantially higher Jeans fluxes.

Jeans flux primarily impacts hydrogen, helium, and other light gases, so these gases may be lost entirely. It's also possible that oxygen, nitrogen, and related gases will be lost, although the main mechanisms for their loss are non-thermal. Still, you likely will have an atmosphere different from Earth's.

Rotation and revolution

These are essentially arbitrary. You can put the planet as close or as far from the star as you want (although I'd recommend keeping it in the habitable zone if you want life), so you can pick whatever values suit your purposes. Size, mass and surface gravity aren't important here for low rotation rates; as ben pointed out, for large angular speeds, the centrifugal force does indeed become important.

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    $\begingroup$ As you add more heavy metals like iron, you are likely also adding more radioactive metals like uranium, iridium, etc. which contribute to a planet's molten core. For Tectonics, I'd expect a denser planet to be much more volcanically active than Earth. But that activity would die down as the planet ages; so, an older smaller world might be similar to Earth. The more active core may also give it a stronger magnetic field for to compensate for the atmospheric retention problem; so, maybe there is a happy medium there. $\endgroup$ – Nosajimiki Apr 29 at 17:37
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    $\begingroup$ Nitpitck, but a higher rotation rate will counter surface gravity (most strongly at the equator) right? This planet is smaller in radius than Earth so it would have to be spinning rapidly for this to matter but rotation rate isn't entirely arbitrary. $\endgroup$ – ben Apr 29 at 18:24
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    $\begingroup$ @ben You're quite right; it might be interesting to check out how giant impacts can affect rotation rates. Perhaps iron planets formed like this could be at risk of becoming rapid rotators, depending on impact angle and collision speeds. $\endgroup$ – HDE 226868 Apr 29 at 19:03
  • $\begingroup$ @Nosajimiki I'd expect a small iron planet to cool over short timescales, meaning tectonic activity would stop rather quickly. I should do some more reading into that, though. Also, if the iron planet does originate from a giant impact, I don't think that there would be more radioactive metals - a higher proportion, yes, with the silicates mostly gone, but the same total amount that a non-stripped planet would have begun with. $\endgroup$ – HDE 226868 Apr 29 at 19:08
  • $\begingroup$ @HDE226868 Yes, that would be true of a planet made from a stripped core. I was picturing the result of a planet formed in nursery where the heavy metal ratios are higher from a larger concentration of nova debris. I think a stripped core planet would be much harder to match to earth-like properties though. $\endgroup$ – Nosajimiki Apr 29 at 21:24
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We can probably make a really small planet if we assume it is made of a very dense material like iridium or platinum or such - at least in the core.

However, there is another factor which we need - size of the biome. The smaller the biome, the slower it will develop. A sufficiently small one might even end up in an evolutionary stand-still - any advances which require a few less advantageous mutations to happen are impossible because the biome is too small for them to survive the unmutated competition long enough.

If you want a really small planet and high speed evolution, I suggest a water planet with frozen polar caps. Kind of like Europa, but in the habitable zone. Let it have lots of energy, say from being part of a binary planet system and close to a not-too-bright sun, so tidal forces heat up the core and create loads of underwater volcanoes.

This planet has a much lower gravity, but could otherwise be quite similar to Earth.

Plants could start to form as algae, growing into water lilies in small water lakes surrounded by ice, then spread over the rest of the ocean and eventually compete for height, forming grass or even bamboo like floating plants, connected with roots, so smaller animals can walk on them - like in a swamp.

As there is no real land, we have to evolve our humanoids from animals closer to the ocean - maybe similar to penguins, but eventually conquering the ice in a fashion close to the Innuit and taking plants with them using green houses or floating farms once they have technology.

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