system
Given the following post, respond as if you're commenting on Reddit:
human
I just found out that while mars has about 1/10th the mass of earth it has 38% of the gravity. This suggests a non-linear relationship between mass and gravity at the surface. What causes this effect? [Physics]

Follow up question: Does this effect mean that the gravity on the surface of planets that are significantly larger than earth would be survivable for humans?
gpt
Gravity is a function of both the mass of the object as well as the distance to it. The force of gravity is proportional to the mass, but falls of with the square of the distance.

In the case of a planet (or spherical body in general), the "distance" refers to the distance to the center of the planet, so for something on the surface this would be the radius of the planet.

If 2 bodies are made of the same materials and therefore have the same density, then the mass scales with the cube of the radius: M ~ r^(3). So gravity scales with the cube of the radius divided by the square of the radius: F ~ r^(3) / r^(2) or F ~ r.

If you'd take the average density of the Earth and made a new planet with half the radius, but the same density, then this mini-Earth would have 1/8th the mass but still half the gravity of Earth. Note that Mars has a radius that's about half of the Earth's radius. The relative figures for Mars' mass and gravity are fairly close to those of our hypothetical mini-Earth. The discrepancies are caused by the fact that Earth has a higher density than Mars.