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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
  • $\begingroup$ 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

A planet with mass $M$ and radius $R$ has a surface gravity of $$g=\frac{GM}{R^2}$$ If we assume it is roughly spherical, then it has a mean density $\bar{\rho}$ of $$\bar{\rho}=\frac{3M}{4\pi R^3}$$ and therefore we can make the substitution $$\frac{M}{R^2}=\frac{4\pi}{3}\bar{\rho}R\to g=\frac{4\pi G}{3}\bar{\rho} R$$ Therefore, for constant surface gravity $g'$, we have a relationship between $\bar{\rho}$ and $R$: $$\bar{\rho}(R)=\frac{3g'}{4\pi G}\frac{1}{R}$$ A greater density means a smaller radius. Therefore, we need a very dense planet. Now, the densest planets - such as PSR J1719-1438 b - are actually quite massive, often more massive than Jupiter, but still larger than Earth. I went to exoplanets.org to get a plot of density vs. radius for planets we know about. This is what I got:

enter image description here

The densest exoplanet there is Kepler-131c, with a projected density of $77.7\pm55.5\text{ g/cm}^3$ and a radius of $0.84R_{\oplus}$. I would say that we could use its composition for our planet, but the discovery paper (Marcy et al. (2014)) states

The density is unphysically large, and hence the mass is too large, indeed a detection at less than 2$\sigma$.

The second-most densest planet, Kepler-37c is also promising (with a radius of $0.742R_{\oplus}$; however, data on its mass is not easy to find. Large uncertainties are present in most of the cases of dense planets. A density of about $20\text{ g/cm}^3$ might be reasonable, though - within the range of Kepler-68c. This gives me a radius of about $1753\text{ km}$, or $0.275R_{\oplus}$:

enter image description here

Kepler-37b, another planet in the system which is even smaller, has a much larger density (see Marcy et al.), but that must be due to some sort of measurement error. The paper reports a density of $548.8\pm700.0\text{ g/cm}^3$, which is absolutely ridiculous. When the error is larger than the measurement, you know there's a problem.

Water-to-land ratio

The composition of Kepler-37c is unknown, but it is almost certainly rocky and could even contain a significant amount of iron. Both of these planets are smaller than Earth, and they happen to orbit far too close to their parent star to allow liquid water to stay there. However, if we just imagine a planet with the same density as Kepler-37c but in an orbit more favorable to life, this problem goes away.

Having water on the planet would lower the mean density - its density is roughly $1\text{ g/cm}^3$ - but I don't think that would be too much of an issue. Water accounts for something like 0.02% of Earth's total mass, yet there's more water than land on the surface. You could cover the entire surface with water and likely still have roughly the same surface gravity - not that there would be much of a surface to stand on.

Whether or not water could develop on a world with this high density is another thing entirely. Pure iron planets are unlikely to have liquid water on the surface. However, if the planet isn't fully iron, and has a non-negligible amount of silicates, it could harbor liquid water. You'd still have to explain the extremely large iron core, but again, we don't know that Kepler-37c is necessarily made of iron in any large quantity.

Temperature

Assuming a silicate solid surface, with water constituting the liquid part, I see no reason why the temperature shouldn'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.

Atmosphere

The above point assumes that the planet can hold onto an atmosphere similar to Earth's. This is not necessarily the case. 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, $\phi_J(m)$, 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 $$\frac{GMm}{kTr}$$ where $r$ is the distance to the lower edge of the exosphere. Given that $g$ is the same as $g$ on Earth, and the distance to the exosphere is smaller on this planet than on Earth, this term will be smaller, and so there will be a greater flux of gases.

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, meaning that the greenhouse effect will be different, and temperatures could be much lower. Liquid water might be less common.

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 don't matter here.

Tectonic activity

This is the one section where I'll say that I have no clue as to the answer. Some exoplanets are known to have plate tectonics, but I don't know how plate tectonics would behave on this particular type of planet.

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