Alien Worlds Might Be Covered in Enormous Mountains

A classic question in Order of Magnitude Physics classes is to calculate the maximum height of a mountain that a planet can support. There are a couple of solutions to this around the internet [1--2], but the main result is that the maximum height is inversely proportional to the surface gravity. At constant density, the surface gravity, in turn, is proportional to radius. So the maximum height of a mountain is inversely proportional to radius and you would expect that smaller planets have taller mountains.

You can take this to one extreme and ask how large an asteroid can be before it becomes spherical under its own gravity. This essentially means that the tallest mountain must be much less than the radius of the asteroid. If you work out the math it turns out that the largest non-spherical asteroid should be around a few hundred km in size, which agrees well with observations.

At another extreme you can perform this calculation for a neutron star. Although the radius is very small (which would seem to imply that it can support tall mountains), the density is much different from ordinary matter. As a result, the tallest mountains a neutron star can support is on the order of a few millimeters! Neutron stars are extremely spherical. (More correctly, due to spin, neutron stars are extremely spheroidal.)

[1]: http://www.phy.cuhk.edu.hk/phyworld/articles/mount_high/moun...

[2]: https://talkingphysics.wordpress.com/2011/09/08/how-high-can...

Would it be plausible for a mountain on an alien world to reach high enough to aid in space exploration? High enough to make launches cheaper from its peak?

It's hard to express how sad it makes me that I'll never see photos of the surface of another inhabited world.