# Estimate of the greatest possible radius of a rocky planet

• kakaroth
In summary, your math suggests that the maximum pressure a planet of iron can withstand is when the pressure equals the bulk modulus. This limit increases as you add more iron, suggesting that the planet would eventually become too dense to be considered rocky material.f

## Homework Statement

"Given a certain rocky material, estimate the greatest possible radius of a planet made up of said material"

## Homework Equations

##P = \frac{2}{3}\pi G\rho^2R^2##

##R = \frac{1}{\rho}\sqrt{\frac{3P}{2\pi G}}##

## The Attempt at a Solution

I'm not quite sure of the validity of my attempt at a solution, but here it is:
First I calculated the pressure at the center of a sphere of uniform density ρ and radius R under its own gravity, getting
##P = \frac{2}{3}\pi G\rho^2R^2##
which gives ##R = \frac{1}{\rho}\sqrt{\frac{3P}{2\pi G}}##.

Assuming the material was Iron, I plugged in its density ρ = 7874 kg/m3 and for P i used its bulk modulus of 170 GPa. The result was 4.43 * 106 m.
Then I cheated and used the average density of the Earth ρ = 5513 kg/m3 and the same bulk modulus, which gave 6.32 * 106 m.

But both result are less than the radius of the Earth, so my solution is probably completely worthless. Any ideas on how to approach this problem differently?

Before you dive into the math, state clearly what limits the radius from being larger than a certain value and create some math to express that.

You first math, if I am following correctly, implies that the maximum allowed pressure at the center of an iron planet is when the pressure is equal to the bulk modulus of iron. Why would this limit the maximum size of an iron planet? What happens if you add a bit more iron?

Your second math using the density of the Earth I can't divine any reasoning from, not sure what you are picturing here.

I don't know what might limit the size of a rocky planet - strikes me as being somewhat unconstrained but it makes sense there must be some limit before the rocky material stops being rocky material.

You seem to assume that the density is constant, but this is not the case unless the bulk modulus is infinite.

@kakaroth is that really the entire problem statement? From what I can find on the internet, the problem as you state it does not seem constrained enough to have an objectively correct answer. Are there no relevant lecture notes or other context to bound it somehow?