# How large can differences in gravity be without being overly dangerous to humans?

Let's imagine a planet that is 5000 km in diameter but has the mass of the Earth. I'd imagine standing on that planet might feel similar to standing on the Earth, because the gravity would be comparable.

However, at smaller and smaller volumes, there would be a greater difference in gravity with a smaller change in distance from the planet's center. Standing on a spherical object only a few meters in diameter with the mass of the Earth, the bottoms of your feet might experience Earth-like gravity, but your head would experience hardly any gravity at all. I'd imagine this would cause some serious health problems.

I imagine that no matter what, there might be some health problems when changing the volume of the Earth-like mass; however, I want to know what the smallest volume could be that has no immediate (no problems for at least a few years) health problems.

# How small can a planet or object of Earth-like mass be without being overly dangerous?

Worldbuilding context: I am wondering what the smallest object can be that has Earth-like gravity. It would be interesting to be able to run around the world very quickly.

NOTE: I am only interested in the gravitational affects on the human body, so please ignore any other affects. I don't care about retaining Earth's qualities. The question How small in diameter a planet can be while retaining most of Earth's properties? is asking for much more than my question.

• Should answers assume the mini-earth is spinning fast enough that an average person's head experiences somewhere in the neighborhood of 1G? For example, the 2.5TG differential between feet and head of a 3m sphere won't really matter when the head is still experiencing 1.5TG -- a person is too busy being flattened to worry about being torn apart. – manveti Nov 18 '19 at 23:13
• Note that a smaller object whose total mass is like Earth's will have stronger gravity on the surface. Surface gravity diminishes with the square of the body's radius. – Cadence Nov 18 '19 at 23:18
• Interesting side note: if the 3m-radius mini-earth were spinning fast enough for a person's head to experience 1G, that person's head would be traveling roughly .01c. Were they to somehow live out a 75-year life there, their feet would be a little more than 10 hours older than their head. – manveti Nov 18 '19 at 23:36
• super-relevant xkcd – Punintended Nov 19 '19 at 0:33
• With a diameter of 5000km, your planet would need a greater density to have Earth-like surface gravity. The easiest way to have this occur is for your planet to have a greater ratio of core mass to mantle mass. Perhaps something more akin to Mercury? This will of course have major implications for the internal geology of your planet... But they may actually be positive ones. A larger core means more radiogenic heating, which counteracts the accelerated cooling of the planet due to having a smaller surface-volume ratio. – Arkenstein XII Nov 19 '19 at 3:21

The assertion that an earth mass only a few meters in diameter having a surface gravity equal to that of earth is wrong. Surface gravity can be expressed by $$g = \frac{GM}{r^{2}}$$ so if you reduce $$r$$ from ~ six million metres to, say, six, the surface gravity is multiplied by the inverse square of that reduction. So the surface gravity of a six-metre earthlike mass would be $${earth gravity} * {10}^{12}$$.
What you're looking for is a much less massive object, but one where the escape velocity is still sufficient to keep your human from launching themselves off the surface. If we let the object be a kilometre in diameter, then a mass of ~200 quadrillion kg gives us (Wolfram gravitational calculator link) a surface gravity approximately that of Earth, and a difference of $$2m$$ in $$r$$ (a tallish human) would make no effective difference. An escape velocity of $$162 m/s$$ isn't going to be something your runner can achieve unaided.
It's worth noting that to have a 1km radius sphere with a mass of $$2 * 10^{17} kg$$, you'd need an average density of $$\frac{2 * 10^{17} kg}{\frac{4}{3}\pi m^{3}} = 4.77 * 10^{7} kg/m^3$$ or about fifty million times denser than water. You're definitely looking at some peculiar matter to build your Petit Prince planetoid out of.