Worksheet 11, Problem 2: Composition Change

11. 2 Problem: 3

Derive the third equation for mass conservation (dM/dr) starting with the integral equation that relates M(r) to r and \( \rho(r) \).

In this problem, we study how the mass of a star changes as the radius changes. The composition of star change drastically from their dense core to the more perfuse outer layers.

To begin, we can consider the mass of a star in terms of radius: \[ M(r) = \frac{4}{3} \pi r^3 \rho (r) \]
From this, we can establish an integral relationship by imagining infinitesimally small concentric shells for the sun with radii dr : \[ M(r) = \int_0^r { 4 \pi r^2 \rho (r) dr } \] Taking the derivative \( \frac{dM(r) }{dr}\) , we can resolve the way that mass changes over radius of a star. \[ \frac{dM(r) }{dr} = 4 \pi r^2 \rho (r) \]
Acknowledgements: Team EE