wba <wbackerman@gmail.com> A long time ago (a long, long, long time ago) I attended a physics lecture/colloquium at MIT titled "How High is a Mountain?" The lecture was a virtuoso performance of the kind of approximation and hand-waving that some professors are spectacularly good at. He used fundamental constants, like Planck's constant, the gravitational constant, particle masses, strength of the electromagnetic force, etc., to work out that, for "things" made of "matter" (that is, the kind of stuff that familiar matter seems to be made of--you know, atoms and molecules and all that) the product of the highest feature and the object's diameter would be 40,000 miles^2. E.g. for earth, 8000 miles * 5 miles.
--OK, I attempted to redo mystery-professor's work to find a formula for (max mountain height) * (planet radius). First of all, for Venus which as we saw has the greatest value, for Maxwell Montes, this product is (11 km) * (6052 km) = 66572 km^2 = 25703 mile^2 somewhat exceeding mystery-professor's claim 20000 mile^2. In terms of fundamental constants I find F*A^(-5/2)*epsilon0*(proton mass)^(-5/2)*(planck constant)^4*(electron mass)^(-3/2)*(elementary charge)^(-2)*G^(-1) = F * A^(-5/2) * 10^16 * meter^2 where A is the dimensionless mean atomic weight of the stuff the planet is made of, and F is a dimensionless fudge factor. If we take A=20 (for SiO_2) then the value of F needed to make the formula agree with Venus is F = 0.012. So in summary (max mountain height) * (planet radius) = 0.012 * A^(-5/2) * epsilon0*mp^(-5/2)*h^4*me^(-3/2)*e^(-2)*G^(-1) To derive this, Let the planet radius be R and the mountain height be H. Surface gravitational accel = AtomicDensity * R * G (up to some constant factor). Hence Pressure inside base of mountain = H * AtomicDensity^2 * R * G (ditto). Set that equal to strength of fairly strong material (with average atomic weight=A). Solve for H*R = StrengthMat / (AtomicDensity^2 * G). Finally consult my table https://dl.dropboxusercontent.com/u/3507527/WDSphysicsFormulas.html This verifies my table makes calculations like this easy :) If anybody can contribute more such formulas, be happy to add them to the table. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)