Is a bigger planet than Earth with the same density possible?
Let's say there's a planet with 18 times the radius of Earth, but has the same density as Earth. If so, how would it be possible for a big planet to such a density of Earth? Would it have the same gravity as Earth?
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1 answer
That's no planet!
Let's say that the radius is $18R_{\oplus}$, where $R_{\oplus}$ is Earth's radius. The volume will then be $$\frac{4}{3}\pi(18R_{\oplus})^3\bar{\rho}=5832\left(\frac{4}{3}\pi R_{\oplus}^3\bar{\rho}\right)=5832M_{\oplus}$$ where $M_{\oplus}$ is the mass of Earth and we assume the same mean density $\bar{\rho}$ for both bodies. That's roughly $18.4$ times the mass of Jupiter. Something this massive isn't a rocky planet, and it isn't even a gas giant. It's like a decent-sized brown dwarf.
Brown dwarfs generally have higher densities than gas giants; their central densities can reach anywhere from $\sim10$ to $10^3\text{ g/cm}^3$, much greater than Earth's density of about $5.5\text{ g/cm}^3$. The brown dwarf is less dense towards its surface, and there's not really a clear boundary (as is the case with stars, as they're gaseous), so some parts will be more dense than Earth (on average), while others will be less dense.
I see that Mormacil's answer mentioned a question I answered two years ago and have, I think, cited a couple of times since. An important takeaway is that you can't simply add more and more mass to rocky planets and expect them to stay rocky. One group (Lammer et al. (2014)) found that at around $2M_{\oplus}$, rocky bodies will retain hydrogen/helium envelopes, entering a class of objects that are more like gas planets than rocky planets. Your $18M_{\oplus}$ "planet" certainly won't be terrestrial in nature. Based on Seager et al. 2008, the maximum achievable radius for a terrestrial planet is, optimistically, $4\text{-}5R_{\oplus}$, assuming a pure ocean world (which should be less dense than a silicate Earth-like planet of the same mass).
Surface gravity
The surface gravity $g$ is related to the radius $R$ by $$g\propto\bar{\rho}R$$ Given that $R=18R_{\oplus}$, the surface gravity will be 18 times that of Earth. Again, though, it's not clear where the surface of a brown dwarf actually is, so take this figure with a grain of salt.
Life
Life on brown dwarfs has been discussed in Can life arise on a brown dwarf?, among other places. Essentially, there are some big problems you'd need to overcome, including high temperatures and possible radiation.
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