<p>Out of the four <a href="https://en.wikipedia.org/wiki/Stellar_structure#Equations_of_stellar_structure" rel="nofollow noreferrer">stellar structure ODEs</a>, I would like to understand why the mass continuity equation was named this way. It reads
<span class="math-container">$$ \frac{dm}{dr}=4\pi r^2\rho \tag{1} $$</span>
and I understand what it means, but in the Wikipedia entry they also reference the (mass) <a href="https://en.wikipedia.org/wiki/Conservation_of_mass#Formulation_and_examples" rel="nofollow noreferrer">continuity equation</a> given as
<span class="math-container">$$ \frac{\partial\rho}{\partial t}+\nabla\cdot(\rho\vec{v})=0 \tag{2} $$</span></p>
<p>I do not understand how these two are related - which assumptions do I have to make to transform eq. <span class="math-container">$(2)$</span> into eq. <span class="math-container">$(1)$</span>?</p>

CC BY-SA 4.0
2023-03-21T17:27:18.607Z
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).
AtmosphericPrisonEscape
CC BY-SA 4.0
2023-03-21T17:52:28.273Z
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?
user358572
CC BY-SA 4.0
2023-03-21T18:13:26.610Z
I would say that is essentially correct. It's a misnomer. Maybe @ProfRob has a different view.
AtmosphericPrisonEscape
CC BY-SA 4.0
2023-03-21T20:18:09.303Z
I don't believe the first equation is commonly known as the mass continuity equation. It more normally called a mass conservation equation.
ProfRob
CC BY-SA 4.0
2023-03-21T20:50:48.303Z
@ProfRob I quoted its name straight from Wikipedia. This page (http://www.vikdhillon.staff.shef.ac.uk/teaching/phy213/phy213_masscons.html) calls it the equation of mass conservation. But I still cannot see what it has to do with mass conservation. IIUC It tells us how much mass is enclosed in a very thin shell with a given mass density, but calling it mass conservation seems a bit off unless I am mistaken. Please feel free to correct me!
user358572
CC BY-SA 4.0
2023-03-21T21:55:48.263Z
Wikipedia gets stuff wrong.
ProfRob
CC BY-SA 4.0
2023-03-21T22:00:03.250Z
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?
user358572
<astrophysics><stellar-astrophysics>
Stellar structure equations - mass continuity
user358572
# Stellar structure equations - mass continuity

(Asked by: username_0 on 2023-03-21)

Out of the four [stellar structure ODEs](https://en.wikipedia.org/wiki/Stellar_structure#Equations_of_stellar_structure), 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](https://en.wikipedia.org/wiki/Conservation_of_mass#Formulation_and_examples) 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. $(1)$?

**Tags:** astrophysics, stellar-astrophysics
> **Comments:**
> **username_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).  
> **username_0**: 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?  
> **username_1**: I would say that is essentially correct. It's a misnomer. Maybe @ProfRob has a different view.  
> **username_2**: I don't believe the first equation is commonly known as the mass continuity equation. It more normally called a mass conservation equation.  
> **username_0**: @ProfRob I quoted its name straight from Wikipedia. This page (http://www.vikdhillon.staff.shef.ac.uk/teaching/phy213/phy213_masscons.html) calls it the equation of mass conservation. But I still cannot see what it has to do with mass conservation. IIUC It tells us how much mass is enclosed in a very thin shell with a given mass density, but calling it mass conservation seems a bit off unless I am mistaken. Please feel free to correct me!  
> **username_2**: Wikipedia gets stuff wrong.  
> **username_0**: 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?  

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## 0 answers

