<p>The <a href="http://en.wikipedia.org/wiki/Surface_gravity">surface gravity</a> 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 <a href="http://en.wikipedia.org/wiki/Earth">Earth</a>, $r_E=6371.0 \mbox{ km}$ the mean radius of Earth, $\rho_P=22.59\mbox{ g}/\mbox{cm}^3$ the density of densest natural element <a href="http://en.wikipedia.org/wiki/Osmium">osmium</a>, and $r_P$ the radius of the fictive osmium planet.</p>

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

<p>Some compression of the core of an osmium planet due to pressure is neglected.</p>

CC BY-SA 3.0
2015-04-21T09:03:02.127Z
There's probobly not anything denser than Osmium in this context.  Certainly there's denser material, even in our solar-system.   The density at the core of our sun is about 150 g/CM3, but that's only possible under enormous pressure.   The density of a white dwarf is much greater than that and the density of a neutron star, much greater still, but if you're talking about 1 earth gravity, neutron star material or white dwarf material would probobly be unstable and fly apart.   There are some theories that say it could maintain integrity, but that's uncertain and I think unlikely
userLTK
CC BY-SA 4.0
2020-02-06T12:56:22.853Z
On that osmium planet with the Earth's gravity and about a quarter its radius there would be quite a difference between the weight of your feet and the weight of your head, wouldn't it?
user30007
CC BY-SA 4.0
2020-02-07T13:26:39.750Z
@user30007 not really, the difference between your head and feet is still only ~0.00001%.
Tim
CC BY-SA 3.0
2014-04-30T01:56:35.827Z
Wow that's really small on the planet scale.
TheBluegrassMathematician
CC BY-SA 3.0
2014-04-30T02:10:24.223Z
Yeah, that is small; that's half the size of Mercury.
LDC3
CC BY-SA 3.0
2014-04-30T15:23:27.847Z
Notably, despite having the same surface gravity of Earth, this "osmium-earth" has less than 0.1 times the mass.
zaratustra
CC BY-SA 3.0
2014-04-30T01:51:19.680Z
2014-04-30T01:51:19.680Z
Gerald
<p>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?</p>

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

<planet><earth-like-planet>
How small can a planet be and still have Earth-like gravity?
Spinmeister P
# How small can a planet be and still have Earth-like gravity?

(Asked by: username_0 on 2014-04-30)

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.

**Tags:** planet, earth-like-planet

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## 1 answers

### Answer by username_1 on 2014-04-30. Score: 9

The [surface gravity](http://en.wikipedia.org/wiki/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](http://en.wikipedia.org/wiki/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](http://en.wikipedia.org/wiki/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.

> **Comments:**
> **username_5**: Wow that's really small on the planet scale.  
> **username_6**: Yeah, that is small; that's half the size of Mercury.  
> **username_7**: Notably, despite having the same surface gravity of Earth, this "osmium-earth" has less than 0.1 times the mass.  
> **username_2**: There's probobly not anything denser than Osmium in this context.  Certainly there's denser material, even in our solar-system.   The density at the core of our sun is about 150 g/CM3, but that's only possible under enormous pressure.   The density of a white dwarf is much greater than that and the density of a neutron star, much greater still, but if you're talking about 1 earth gravity, neutron star material or white dwarf material would probobly be unstable and fly apart.   There are some theories that say it could maintain integrity, but that's uncertain and I think unlikely  
> **username_3**: On that osmium planet with the Earth's gravity and about a quarter its radius there would be quite a difference between the weight of your feet and the weight of your head, wouldn't it?  
> **username_4**: @user30007 not really, the difference between your head and feet is still only ~0.00001%.  

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