# Stellar structure equations - mass continuity

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. $$(1)$$?

• 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). Mar 21 at 17:27
• 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? Mar 21 at 17:52
• I would say that is essentially correct. It's a misnomer. Maybe @ProfRob has a different view. Mar 21 at 18:13
• I don't believe the first equation is commonly known as the mass continuity equation. It more normally called a mass conservation equation. Mar 21 at 20:18
• 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? Mar 21 at 22:00