Hello fellow students of physics,
I'm beginning my study on 'Stellar Structure and Evolution' and in the book I'm studying the author(s) begin(s) by stating simplifying assumptions for a first model of a star: gaseous, non-magnetic, non-rotating and single star. Altogether, this implies spherical symmetry.
My concern here lies with the continuity equation (which is promptly derived in the first few pages). This equation describes the transport of some physical quantity and in the concerning case it's the density field, $ \rho = \rho(r,t)$. It is also an equation that implies that a physical quantity is conserved, which in this case is the mass. This equation for the density field is as follows:
$$ \frac{\partial\rho(r,t)}{\partial t} = -\nabla \cdot (\rho(r,t)\mathbf{v}_{r}(t)) \quad \quad (0)$$
I can easily derive it one way and it goes like this (bear with me please):
Since it's being assumed spherical symmetry, the total derivative of the mass enclosed within a sphere of radius $r$ is:
$$dm(r,t) = \frac{\partial m(r,t)}{\partial r}dr + \frac{\partial m(r,t)}{\partial t}dt \quad\quad (1) $$
Since I want to enforce the law of conservation of mass, which is translated as:
$$ \frac{dm(r,t)}{dt} = 0 \quad \quad (2)$$
In $(1)$ I divide both sides by $dt$ and equal the whole expression to zero.
$$\frac{dm(r,t)}{dt} = \frac{\partial m(r,t)}{\partial r}\frac{dr}{dt} + \frac{\partial m(r,t)}{\partial t} = 0 \quad \quad (3)$$
Now, I know relation between mass and density, which is:
$$ m(r,t) = \int_{V} \rho(r,t) dV(r) = \int_{V} \rho(r,t) 4\pi r^{2} dr \quad \quad (4)$$
Partial differentiating $(4)$ with respect to $r$, I get:
$$ \frac{\partial m(r,t)}{\partial r} = \frac{\partial}{\partial r} \bigg[ \int_{V} \rho(r,t) 4\pi r^{2} dr \bigg] = \rho(r,t)4\pi r^{2} \quad \quad (5)$$
Inputing this result into $(3)$, I get:
$$ \frac{\partial m(r,t)}{\partial t} = - 4\pi r^{2}\rho(r,t)\frac{dr}{dt} \quad \quad (6) $$
And equation $(1)$ can now be rewritten as:
$$dm(r,t) = 4\pi r^{2}\rho(r,t)dr - 4\pi r^{2}\rho(r,t)v_{r}(t)dt \quad\quad (7) $$
The crucial step comes next, which is to partially differentiate $\frac{\partial m(r,t)}{\partial r}$ with respect to $t$ and partially differentiate $\frac{\partial m(r,t)}{\partial t}$ with respect to $r$ and equate them, like so:
$$\frac{\partial}{\partial r}\bigg[\frac{\partial m(r,t)}{\partial t}\bigg] = -8\pi r \rho(r,t)v_{r}(t) - 4\pi r^{2} \frac{\partial \rho(r,t)}{\partial r}v_{r}(t) \quad\quad (8.1)$$
$$\frac{\partial}{\partial t}\bigg[\frac{\partial m(r,t)}{\partial r}\bigg] = 4\pi r^{2} \frac{\partial \rho(r,t)}{\partial t} \quad\quad (8.2)$$
Equating them gives:
$$-8\pi r \rho(r,t)v_{r}(t) - 4\pi r^{2} \frac{\partial \rho(r,t)}{\partial r}v_{r}(t) = 4\pi r^{2} \frac{\partial \rho(r,t)}{\partial t} \quad\quad (9)$$
Diving everything by $4\pi r^{2}$ and ackowledging the divergence operator in spherical coordinates (with spherical symmetry), this becomes:
$$ \frac{\partial \rho(r,t)}{\partial t} = -\frac{2}{r}\rho(r,t)v_{r}(t) - \frac{\partial \rho(r,t)}{\partial r}v_{r}(t) = - \frac{1}{r^{2}}\frac{\partial}{\partial r}\bigg[r^{2}\rho(r,t)v_{r}(t)\bigg] = - \nabla \cdot \bigg[\rho(r,t)\mathbf{v}_{r}(t)\bigg] \quad\quad (10) $$
$$ \frac{\partial \rho(r,t)}{\partial t} = - \nabla \cdot \bigg[\rho(r,t)\mathbf{v}_{r}(t)\bigg] \quad \quad (11)$$
And we got back the continuity equation in $(0)$.
Now, sorry for the long post but my problem and the reason I posted is this. From what I derived, the continuity equation implies that some physical quantity is conserved, but in other derivations they do not assume the mass conservation law, that is, $\frac{dm(r,t)}{dt} = 0$. Another derivation is given in:
But in this post nothing is said of the sort:
$$ \frac{dm(r,t)}{dt} = 0 $$
In fact, if they did both terms on equation $(11)$ would be equal to zero, but they're not right?
Am I missing something critical here?
Please, help me out in any way you can.
Thanks in advance!
