Further physics - How high could a mountain be?
Tong Shiu-sing & Hui Pak-ming
(Translation by Tong Shiu-sing & Janny Leung)

Introduction

Victor Weisskopf (MIT) is a famous contemporary physicist. He has made significant contributions in nuclear physics and quantum electrodynamics. In 1960-1965, Weisskopf was the director of the CERN (The European Organization for Nuclear Research). Every summer, CERN organizes visits for local high school students. Weisskopf has delivered a series of popular lectures, in which some profound problems in modern physics were elaborated in a simple manner for high school students. Weisskopf emphasized the importance of using basic knowledge in physics as a means to make reasonable estimates. Here we shall consider a problem he proposed.

Question

The highest mountain in the world, Mount Everest, has a height of about 9 km. Have you thought about this question: Why does the highest mountain on earth has a height of the order of 10 km, but not 100 km or higher, or below 1 km? The problem we have is: Consider the problem in terms of energy to give a rough estimate on the maximum height that the mountains on earth can attain, and also discuss the height of mountains on other celestial bodies (e.g. Mars).

Hint

A mountain of simplified shape
Fig. 1   A mountain of simplified shape.

For the sake of convenience, let us simplify the shape of a mountain as shown in the figure. Why can't a mountain be too high? A mountain will become very heavy if it is really high, and the weight may cause the mountain to sink. When the mountain is lowered by a height of x, it loses some potential energy. According to Weisskopf, if the released potential energy can melt part of the rock (assuming that it is solely made up of ), then the mountain will continue to sink. Therefore the height of a mountain can be estimated by using energy as a consideration. The answer can be calculated and expressed in terms of the following physical quantities:

A = The total no. of protons and neutrons in a molecule () of the rocks
(mass of a proton mass of a neutron)
(acceleration due to gravity on the earth's surface)

= the energy required to melt the rock per molecule 0.3 eV per molecules

Answer

Let M be the mass of the mountain. Thus when the mountain sinks by a height of x, the gravitational potential energy released = Mgx. Let a be the cross sectional area of the mountain, and be the latent heat absorbed per molecule of when the rocks melt, and n be the number of molecules per unit volume. So the energy required to melt a layer of rocks of height x would be . The energy released when the mountain sinks by a height of x must be smaller than the energy needed to melt the layer, otherwise the mountain will continue to sink. Therefore we require , or

    (*)


Then, we have to find out how the mass M and the height h of the mountain are related. Suppose the mass of a molecule is m, the total number of protons and neutrons in a single molecule is A, the mass of a proton mass of a neutron, then we have . It follows that

M = (Volume of the mountain) (No. of molecules per unit volume) (molecular mass of )
    =     (**)

Combining (*) and (**), we have




We know that , for , , the mass number of silicon atom is 28, that of oxygen is 16, so A = 28 + 216 = 60. For the earth, ,




which corresponds to the order of the height of the highest mountain on earth. On Mars, the acceleration due gravity is about 38% that of the earth, so

h < 13 km

The highest mountain on Mars is Olympus Mons, which has a height of about 25 km.

Reference: The Many Phases of Matter, a popular science book by A G. Venkataraman.