# What is the highest a mountain can be? Is there a limit to it?

As a mountain gets taller, it gets more massive. As it gets more massive, the pressure on the rock at its base increases. Eventually, this pressure would exceed the breaking strength of the rock.

That pressure could be written

```
P = rho g h
```

where *P* is the pressure on the base, *rho* is the density of the rock, *g* is the surface gravity of the planet, and *h* is the height of the mountain. If *P* is the breaking strength of the rock, you'll find a cool relation:

```
h g = P/rho
```

Since *P/rho* is just a constant, this relation tells us that **as the surface gravity of the planet in question increases, the maximum size of a mountain it can support decreases.**

This also tells us *h g* must also be a equal to a constant, which lets us relate the maximum height of mountains on planets of similar compositions but with different masses:

```
h_1 g_1 = h_2 g_2
```

You can do something *really cool* with this. If you take Mt Everest to be the tallest mountain that can be supported on earth, and if you know that Mars surface gravity is 2/5th of earth surface gravity, you can actually calculate the height of Olympus Mons, which is the tallest mountain on Mars, if you write

```
h_everest g_earth / g_mars = h_olympusmons
5/2 h_everest = h_olympusmons
```

Which is actually really close to the true value! This is even cooler because it argues that *both* Earth and Mars have mountains near the maximum possible height for the planet. Of course, a geologist may not like any of what I just said above. Mountains and tectonic plates and mantles are complicated beasts - this was just a first order approximation.

But, as one last fun fact, you can do something else with this approximation. We can predict the 'potato radius' - the maximum size a 'potato shaped' asteroid can be before its gravity becomes strong enough to pull it into a sphere. This is done by modeling the potato asteroid as a sphere with a *huge* mountain on it that must shrink as the asteroid grows in mass, until the mountain is smaller than the radius of the asteroid.

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