# How small can a planet be and still have Earth-like gravity?

A planet made from denser material than Earth might have equal gravity to Earth but a smaller radius. How small can a planet be and still have Earth gravity?

Ideally it should be habitable by humans, so not made of anything dangerous, radioactive or unstable.

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## 1 Answer

The surface gravity of a planet is very close to $$g=\frac{4\pi G}{3}\rho r.$$ With $g$ to be kept constant, and $\frac{4\pi G}{3}$ a constant, we need $\rho_Pr_P=\rho_Er_E$, or $$r_P=\frac{\rho_E}{\rho_P}r_E,$$ with $\rho_E=5.515 \mbox{ g}/\mbox{cm}^3$ the mean density of Earth, $r_E=6371.0 \mbox{ km}$ the mean radius of Earth, $\rho_P=22.59\mbox{ g}/\mbox{cm}^3$ the density of densest natural element osmium, and $r_P$ the radius of the fictive osmium planet.

Hence $$r_P=\frac{5.515}{22.59}r_E=0.2441~r_E=1555\mbox{ km}.$$

Some compression of the core of an osmium planet due to pressure is neglected.

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Wow that's really small on the planet scale. –  TheBluegrassMathematician Apr 30 at 1:56
Yeah, that is small; that's half the size of Mercury. –  LDC3 Apr 30 at 2:10
Notably, despite having the same surface gravity of Earth, this "osmium-earth" has less than 0.1 times the mass. –  zaratustra Apr 30 at 15:23