[ygnpgfnt] potential energy of a compressible ball

model a spherical solid as a lattice of atoms connected by springs.  let the solid's own gravity compress itself.  how much energy gets stored in the compressed springs?

the earth's nickel-iron core is denser than nickel-iron at room temperature, denser than nickel-iron meteorites.

aside: does this mean that destroying a planet requires less than its gravitational binding energy?  harness this internal potential energy.  but this requires more thought: gravitational potential energy balances spring energy.

to model the lattice of springs, we need spring potential energy to approach infinity as distance between atoms approaches zero.  Hooke's Law is not the right model for this.

aside: does it matter what lattice (e.g., cubic, rhombic dodecahedral) of springs?

we also need potential energy to head to infinity as distance increases to infinity so that atoms do not convert to gas and evaporate in our model.  Hooke's Law could work for this direction.

U = x^2/2 + 1/x.  first term is Hooke's Law; second is electrostatic repulsion.  the leading coefficient of 1/2 places the local minimum at x=1.

alternatively, a U-shaped function that has local minimum at x=1 and is infinite at x=0 and x=2: U = 1/x - 1/(x-2) = 2/(x*(2-x)).  the two expressions after U= are the two most straightforward ways of constructing a function with the two vertical asymptotes.  it is neat that they yield the same result.

consider general relativity: the potential energy of compressed strings exerts gravity, bends spacetime.