Isostasy and Mountain Heights

Earth's mantle is mostly silicate rock and behaves as a viscous fluid in geological time (think large time scales). Oceanic crust and continental crust is primarily formed by partial melting of the mantle. There's a gravitational equilibrium between the crust and the mantle and different topographical heights exist on earth because of this static equilibrium. This equilibrium depends on the thickness and density of the crust.

Apparently, the Himalayas are not in isostatic equilibrium (which is a fancy way to say that they're still rising) and mountains primarily rise that high because they're low density. Think of this as them being huge, but having less mass, so they don't crumble. One important point we'll come back to is that mountains fail at their bases. When a mountain gets too high, they suffer plastic deformation at the base.

There are two major theories of isostasy. One that treats the lithosphere (the crust basically, but the more you read about it, the more you want to use the scientifically precise term) as a sheet floating on the mantle with different densities at different heights. This theory straight up applies Pascal's Law (hydrostatic pressure is the same at the same elevation for a fluid in static equilibrium) to all points in the crust. The second treats the lithosphere as an elastic plate that bends and thereby distributes loads.

If the maximum height a mountain can reach is $H$, this is the height at which the energy lost as the mountain "falls" equals the energy needed to cause plastic deformation at the base. Doing the computations for each molecule, we have $$mgh=E_p,$$ where $m=AM$, the atomic mass units in the molecule, $h=H$, the maximum height before deformation, and $E_p$ being the energy needed to cause plastic deformation. Note that Weisskopf uses $E_1$ for this. To cause plastic deformation, we need to liquefy the molecule without breaking the atomic structure. This will require a small fraction ($\epsilon$) of the binding energy $B=\gamma Ry$ where $Ry$ is the Rydberg unit. If we assume that the mountain is a block of SiO$_2$, the atomic mass will be 60 amu. And so the maximum height will be $$H=\epsilon\frac{\gamma Ry}{AMg}\sim \boxed{10\text{ km}}.$$ Later in the paper referenced below, Weisskopf shows a wonderful argument about the maximum height ($H$) of a topological structure on a planet being inversely proportional to the third root of the number of nucleons ($N_p$) in the planet. $$H\propto\frac{1}{N_p^{1/3}}.$$

Update: it turned out that this nucleon argument is neither as fancy or convuluted as I first thought it would be. Rahul Goel pointed out that for any object, $V\times\rho=m$ and mass is just the number of nucleons in an object, and volume is length cubed. And therein lies the intuition for the above equation. Other questions that were raised in the discussion were 1) How do you know that plastic deformation is what occurs when mountains fail at their bases and 2) Is it true that huge mountains usually grow isometrically?

References: Mantle (geology) | wikipedia.org, Isostasy | wikipedia.org, Of Atoms, Mountains, and Stars: A Study in Qualitative Physics | science.org by Weisskopf.