The thing I miss most is just being independent.

Ian Burkhart John Campbell, a solid state physicist at the University of Canterbury, responds: With qualifications, yes. The maximum height of mountains is limited by the strength of the rocks at the bottom.

If there is too much pressure because of the weight of the mountain above, the rock fails. We can quickly estimate this height, and determine what is important, with this back-of-the-envelope calculation: Consider a column of rock, of density d and of height h above the surrounding rock. The pressure (P) at the base of the column is equal to dgh, where g is the acceleration because of gravity.

Hence the maximum height of a column of rock under gravity is the failing pressure (Pf) of the rock divided by the density of the rock and the acceleration due to gravity.

Now put in typical numbers. Rock has a typical density of about 2700 kilograms per cubic metre and a failing pressure of about 200 megapascals. At the Earth’s surface, the acceleration resulting from gravity is about 10 metres per second per second.

This rough calculation shows that mountains on Earth cannot be more than about 8km above their surrounding planes.

How does this compare with observation? The highest mountain on Earth is Mt Everest at 8.8km above sea level, or about 7.8km above the Indian planes south of it.

A more accurate calculation would require knowing what types of rocks the mountain was made of (but the density of typical rocks varies only from about 2500 to 2900kg per cubic metre), what is the temperature the interior of the base of the mountain (because the failing pressure depends on temperature), and other things including geologically different layers at an angle across the mountain (these could sheer).

Other planets have different accelerations because of the gravity at their surfaces.

Redoing the rough calculation shows that the limiting height for mountains on Venus is estimated to be 7.5km (compared with the actual measured height of 11km), Mars 18km (compared with 21km) and Io 37km (compared with 17km). This is good agreement for the approximate numbers we have used. We should use the actual density, failing pressure and temperature for the rock under each mountain, and also whether or not geological upthrust had ceased long ago plus what erosion processes are appropriate. For planets we haven’t yet landed on, we could turn this around to use the maximum height observed to estimate what type of material its surface is made of.

We should also note that most mountains aren’t the maximum allowed because they have eroded over time. Some, such as our Southern Alps, are eroding at about the same rate they are being forced upwards.

I often get students to work out the maximum pressure between atoms in a solid.

This is some 10,000 times higher than the failing pressure of rocks because rocks fail because of the grains and different minerals making up the rock sliding past each other, not by the bonds between atoms being wrenched asunder.

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