Out of the four stellar structure ODEs, I would like to understand why the mass continuity equation was named this way. It reads $$ \frac{dm}{dr}=4\pi r^2\rho \tag{1} $$ and I understand what it means, but in the Wikipedia entry they also reference the (mass) continuity equation given as $$ \frac{\partial\rho}{\partial t}+\nabla\cdot(\rho\vec{v})=0 \tag{2} $$

I do not understand how these two are related - which assumptions do I have to make to transform eq. $(2)$ into eq. $(1)$?

  • $\begingroup$ It's more a coordinate transformation than the continuity equation - if you treat the continuity equation properly, you end up with $\partial(4\pi r^2 \rho)/\partial r = 0$, i.e. not the same as (1). $\endgroup$ Mar 21 at 17:27
  • $\begingroup$ OK, but then why do they call $(1)$ the mass continuity equation, when it has little to do with the proper CE? To me, there is nothing in $(1)$ that says this and that is a conserved quantity, or is it? $\endgroup$
    – user358572
    Mar 21 at 17:52
  • $\begingroup$ I would say that is essentially correct. It's a misnomer. Maybe @ProfRob has a different view. $\endgroup$ Mar 21 at 18:13
  • $\begingroup$ I don't believe the first equation is commonly known as the mass continuity equation. It more normally called a mass conservation equation. $\endgroup$
    – ProfRob
    Mar 21 at 20:18
  • 1
    $\begingroup$ No, what I was referring to was the fact that this equation carries a rather misleading name. To be clear: $dm/dr=4\pi r^2\rho$ is called eq. of mass conservation, but why does it conserve mass, exactly? $\endgroup$
    – user358572
    Mar 21 at 22:00


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