Don't think of the Earth as a solid sphere. It's only solid on the surface. For the most part, the Earth is a molten ball of rock. Tall structures weigh a lot. Think about ice flows. There's some terrain to them, but structures in the thousands of feet do not exist. They just sink lower as their weight increases.
Solid rock on top of liquid rock acts similarly, but structures can grow into the 10's of thousands of feet.
Mathamatically.... There's some formula for what percentage of an iceberg is above water, and what percentage is below water. The formula is derived from the density of frozen water compared to the density of liquid water. With knowledge of the densities of liquid rock vs. solid rock, a similar formula should exist.
It's possible to have higher peaks than Mt.Everest.Incidentally,there used to be said that,due to the continuous advancement of the Indian tectonic plate into the Asian continent (subduction (?)),Mt.Everest would be not only growing taller,but also moving its peak towards NE (i.e.into China).
I dunno whether the latest accurate geological info still claim this thing.
Several levels of explanation (science, BTW, not maths; the math is just a tool):
- real mountains are made of real rocks; rocks can only be so strong, you put more rocks on top of a (max height) mountain and it collapses
- mountains on Earth can only be made by a small number of geophysical processes (e.g. volcanism, plate tectonics); there are limits on how fast each process can make mountains, bearing in mind that there are processes which wear mountains down (e.g. erosion), and this leads to a limit on how high a mountain can be
- Earthly mountain building processes in turn are driven by the bulk composition of the Earth (esp the lack of H and He, and abundance of radionuclides) and its history (esp accretion and the collision that created the Moon); these limit the strength of volcanism and plate tectonics today
Note that if the main 'mountain building' process found on most solid bodies in the solar system were primary on Earth (i.e. that we had essentially no erosion or plate tectonics), then we'd be talking about craters and basins, like Mare Orientale, Caloris, or Valhalla.
I think there may be another limitation also, I'm not sure how much of an influence it is. As Tony mentioned, the crust floats on the asthenosphere, a partially molten layer of the mantle. Beneath mountain ranges the crust protrudes into the asthenosphere in order to maintain bouyancy. I would think that at a certain depth this crust begins to melt, and the mountains will sink.
Just a thought, not sure if I'm right.
For starters the Astenosphere is not nearly molten let alone partially molten. It's just a wee bit more ductile than the environment like the crust and the upper mantle. The lower mantle is even more plastic again.
As far as I know, there is no theoretical limit on the hight of a mountain. Those things get a bit too complicated to model mathematically. But most factors are mentioned. Buoyancy is probably prevailing. Try to think of Earth as being fluid like water. Due to rotation and gravity forces it will attain a geodic shape, now, throw an ice cube on it and there is your mountain, the emersed part of the ice. On million years time scale Earth behaves like fluid. As the crust is much less dense than the mantle it is assumed that, in equilibrium, the dept of the crust is reflected on it's elevation, like the floating ice cube. Most of the ice is submerged. In analogy, oceanic crust is assumed to be very thin. Continental crust is much thicker to have the land float higher like the ice cube.
Now, if two tectonic plates collide, one will subduct, the other will pass overhead but the local thickness of the (light) double crusts increases significantly, playing bouyant ice cube on the (heavy) ductile mantle, consequently the mountain range is build. And mountains literally have deep roots of crust material (presumably).
What would limit that mountain buiklding process? Outflowing of the mountain like ice sheets do? Erosion? It's all speculation.
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