# Mountain heights - theoretical limits

I have a country bounded on three sides by fold mountains. Due to a magical plot device, the force acting to drive the upward motion of the mountains persists indefinitely, acting on the plates outside of the one my country sits on. The same magic prevents any damage (weathering, melting, etc) to the central plate. I believe that as the mountains increase in height they will weigh more and therefore sink deeper into the mantle where the base would melt. This should lead to a balance eventually being reached dependent on the type of rock, the rate of mountain top weathering, the speed at which the tectonic plates are forced up, the temperature of the mantle etc.

Playing through the various factors leads me to believe at doubling the speed would not double the height attained and there would be an absolute limit that could be attained.

Am I right in coming to that conclusion that the growth rate influences the maximum height of a mountain and if so are you able to suggest what height that would be? (limitations: Assume this takes place on earth with the same crust composition and gravity)

• There are already several questions here asking about the maximum height of mountains. To avoid being closed as duplicate, can you explain how is this different from those? – L.Dutch - Reinstate Monica Mar 4 at 10:23
• Natural systems are driven by continental drift and weathering. Fold mountains formed when two continents are pushing against each other create a very wide base and so the melting base is very large and they form broad mountains with a lot of weathering that will constantly work to reduce the height of the mountains. I believe the other questions come to a momentary maximum, not a steady-state maximum. – Hukk2010 Mar 4 at 10:29
• None of the answers there mention the erosion rate being higher than the growth rate as limiting factor for the maximum theoretical height – L.Dutch - Reinstate Monica Mar 4 at 11:17
• "magic prevents any damage (weathering, melting, etc) to the central plate" Then your mountain can be as many lightyears tall as you want. In a real world, the limit is when the compressive strength of the material of the mountain is exceeded by the weight on top of it, which causes the rock to break. Unbreakable rock voids this limitation. – PcMan Mar 4 at 12:54
• @PcMan The magic prevents damage to the central plate, the incoming plate is forming the mountain so that is susceptible. For the usual fold mountains, both plates crumple upwards but in this one, only one of them does and so the mountain base would be much smaller... I will grant you, the more I am thinking about this the weirder the effects of this plot device! – Hukk2010 Mar 4 at 13:08

As pointed out in this answer

A mountain is a lot of rock placed atop other rock. So, you need for the lowest layer of the rock to not crumble and flow outward (beyond a certain point, the rock will behave like a slow-flowing liquid); you want a very high compressive strength.

Since you seek to maximize the (roughly speaking) mass of the mountain and the F=ma equation tells us that m = F/a, you not only want to maximize the compressive strength (which equates F) but also minimize a, which in this case is the gravitational acceleration "g".

Then again you do not want to maximize the mass, you want height, so, a huge volume for any given mass. You want a mountain that is not too dense.

The weight of the mountain is proportional to density multiplied by the volume, which is $$1/3 \cdot S \cdot h$$ for a conical mountain with base S. The downward pressure is then $$\rho \cdot g \cdot h/3$$ and we want it to equate the material's compressive strength:

$$\rho gh/3 = c$$

so $$h = 3c/(\rho g)$$

with c = compressive strength, $$\rho$$ = density, g = surface gravity.

Simply plug in the parameters for the material (c and $$\rho$$) and the planet's surface gravity and you ought to be done. With c measured in Newton over square meters, $$\rho$$ in kilograms over cubic meters and g in meters over seconds squared, you will get the maximum height expressed in meters.

Note the maximum height doesn't depend on the growth or erosion rate, but solely on the resistance of the rock against the rock weight.

If that wasn't the case, we would see some pretty majestic mountain somewhere, as a result of asteroid impacts, either on Earth or on some rocky planet in our solar system.

Asteroid impact is probably the fastest way to displace up a lot of rock: the transient peak quickly collapses under its own weight when it goes past the limit calculated above.

• Really sorry. I completely missed this question when reading around. – Hukk2010 Mar 4 at 11:43
• @Hukk2010: With the observation that here on Earth your choice of materials for mountain building is basically either basalt or granite. We don't really have any other high-strength material available in the required quantity. – AlexP Mar 4 at 12:48