SpacePioneer wrote:Source of the post Could someone show the formula for the maximum height of mountains on a planet with, I'm guessing, some variables related to density, possibly rotational period, etc..

The physics behind the maximum height of terrain on a planet gets quite complex and is a big part of a course in planetary surface processes, but it is possible to come up with a simple formula that gets approximate answers. We can say that t

he maximum height of terrain supportable on a planet is roughly where F

_{c} is the compressive strength of the material, ρ is the density of the material, and g is the surface gravity of the planet. Alternatively, expressing g in terms of the planet's mass M and radius R (g=GM/R^2), it becomes

^I like putting it in this form because it distinguishes the properties of the planet (R and M) and the properties of the material (F

_{c} and ρ). And you mentioned using Mass and Radius anyway.

How could one come up with this formula from scratch? In fact I do not have this formula in my memory, so if you're interested, this is how I figure it out.

The technique is "dimensional analysis", which is very powerful, and I think worth spending an in-depth post about. Much insight into difficult problems of physics was gained through this method, from figuring out how atmospheric scattering works to estimating the yield of the first nuclear tests.► Dimensional Analysis

First, think of what factors ought to be involved in the relation. Stronger rock (greater resistance to being crushed) should allow taller mountains. Surface gravity should play a role as well. And the density of the material, since a mountain built of denser material will have more weight and therefore more force at its base. Rotation has an effect as well, but it is usually quite small and we can safely ignore it -- especially since this is only an order of magnitude estimate anyway.

So we may try to model the relation using these three variables, F

_{c}, ρ, and g. But how do they show up in the formula? Maybe h

_{max} is proportional to the compressive strength

*squared*. Or maybe it is inversely proportional to the square root of the surface gravity. Let's write it out in the most general way, where each of these variables may have

*any* exponent:

where alpha, beta, gamma are the unknown exponents for F

_{c}, ρ, and g, respectively.

How do we determine those exponents? Use the units, or "dimensions", of each of the variables! Because at the end of the day, the formula must be dimensionally consistent.Compressive strength F

_{c} has units of pressure (Pascals), which is force per area, or kg*m

^{-1}*s

^{-2}Density ρ is mass per volume, or kg*m

^{-3 }Surface gravity is an acceleration, or m*s

^{-2}However these terms appear on the right hand side of the formula, their dimensions must equate to those on the left hand side, which is a height (meters). So let's equate these units:

The left hand side is meters to the 1st power. No kilograms, no seconds. On the right hand side we have all three dimensions kilograms, meters, and seconds, with the unknown powers alpha, beta, gamma. So this is actually a set of 3 linear equations in alpha, beta, and gamma. The first one (from equating the units of meters) says that 1 = -1*alpha - 3*beta +1*gamma. The second (equating units of kg) says that 0 must equal 1*alpha +1*beta + 0*gamma. And last, the lack of seconds on the left hand side says that 0 = -2*alpha -2*gamma. Three equations, three unknowns.

Solve this set of equations using any preferred method. I like to work in terms of matrices:

Row reducing this matrix yields

Therefore alpha = 1, beta = -1, and gamma = -1.

This tells us that the exponent for F

_{c} is 1, the exponent for ρ is -1, and exponent for g is -1. Hurrah! We have determined the formula:

**A cautionary note:**Remember this formula captures only a part of the physics of terrain support. It is not a precision tool, it is meant to give an intuition and a practical way to get an estimate.

A more careful calculation should consider that terrain is supported by the underlying rock or mantle, where changes in density, composition and temperature may be important. (A mountain of width w probes the strength of the mantle up to a depth of about w/3 beneath it.) Then we must consider mechanisms that both build and erode terrain. But accounting for all of this gets very complicated. (The physics of planets is hard!) **Applying the formula to Earth:**The densities and compressive strengths of many common substances can be found online. Granite has a strength of about 200Mpa and a density of about 3000kg/m

^{3}. The Earth's surface gravity is about 10m/s

^{2}. (Or use mass = 6x10

^{24}kg and radius 6371km). Plugging these numbers in (and be sure to convert the units correctly;

Wolfram Alpha can do this automatically) gives a max height of about 7km. Fairly good agreement with the highest mountains on Earth.

Also, consider that most mountain elevations do not correspond to how much they actually rise above their surroundings. A better measure, common in mountaineering, is their

*prominence*. Mount Evans for example has an elevation of 4350m, but a prominence of only 840m, because it rises up from the Colorado plateau. Whereas Mount Rainier, a large volcano, has an elevation of 4390m and a prominence of 4026m, since it rises up from nearly sea level.

Mount Everest has a prominence equal to its elevation by the definition (a weakness of how prominence is defined), but in terms of how much it rises from the immediate surroundings is more like 4000 meters.

**Applying the formula to other objects:**Mars has a much lower surface gravity than Earth, of 3.7m/s

^{2}, and all else being equal this formula implies the max supportable terrain there should be about 18km. This agrees fairly well with Olympus Mons, which rises around 20km above the surroundings.

For Venus it predicts a max height comparable to Earth. Yet we observe mountains on Venus that rise upwards of 13km! This is very surprising. In fact planetary scientists are still not certain exactly how the crust and mantle of Venus is able to support such topography. (A common assumption is that it involves the lack of water.)

Where might this formula break down? It is particularly bad for smaller bodies (moons and asteroids). Applying the formula to the Moon (surface gravity 1.6m/s

^{2}, density 3000kg/m

^{3} and strength about 150Mpa) yields a max height of around 31km. This is pretty crazy high, and of course we do not see terrain like that (the highest mountains on the Moon are similar to Earth's). Applying to asteroids gives even crazier results. At this scale we're basically hitting the limit where objects crush themselves into spheres (the

Potato Radius).

So that's the simple max mountain height formula in a nutshell, and how to find it. Hopefully you find this helpful and interesting!