Why Are These Accretion Disk Concepts Challenging?

[Shan K]

I was studying about Accretion disks and found some difficulties regarding some concepts. My questions are

1. For an accretion disk, $v_r<<c_s$ Why?
2. The conservation of mass equation is derived as,


The mass in an annulus of radius $\Delta r$ is $2\pi r \Delta r\Sigma$ and it is changing due to the mismatch of mass outflow and inflow which is written as,

$$ \frac{\partial}{\partial t}(2πr\Delta r\Sigma) = v_r(r,t)2πr\Sigma(r,t)−v_r(r+\Delta r,t)2π(r+\Delta r)\Sigma(r+\Delta r,t)\\ \approx-2\pi\Delta r\frac{\partial (r\Sigma v_r)}{\partial r} $$

up to this I have no problem but it further says that this equation in the $\Delta r\rightarrow0$ limit gives,

$$r\frac{\partial\Sigma}{\partial t}+\frac{\partial (r\Sigma v_r)}{\partial r}=0$$

and I can't able to derive how that limit gives that equation.

3. The angular momentum conservation equation is derived from the Navier-Stokes equation as,
$$ \frac{\partial \vec{v}}{\partial t}+(\vec{v}.\vec{\nabla})\vec{v}=-\frac{1}{\rho}(\vec{\nabla}P-\vec{\nabla}.\sigma)-\vec{\nabla}\Phi $$

In the axisymmetry assumption and no momentum loss or gain from the $z$ direction one gets,

$$\Sigma\Big(\frac{\partial v_\phi}{\partial t}+\frac{v_rv_\phi}{r}+v_r\frac{\partial v_\phi}{\partial r}\Big)=\frac{1}{r^2}\frac{\partial}{\partial r}(r^2 T_{r\phi}) $$

Now my question is, shouldn't the RHS be $\frac{1}{r}\frac{\partial}{\partial r}(r T_{r\phi})$ instead of $\frac{1}{r^2}\frac{\partial}{\partial r}(r^2 T_{r\phi})$ because the $r$ part of $\vec{\nabla}.\vec{f}$ in the cylindrical coordinate is $\frac{1}{r}\frac{\partial}{\partial r}(r f_r)$.

Thanks in advance..

 

[stefan r]

Can you post the source. I like to check the authors definition of each symbol.

up to this I have no problem but it further says that this equation in the $\Delta r\rightarrow0$ limit gives,

$$r\frac{\partial\Sigma}{\partial t}+\frac{\partial (r\Sigma v_r)}{\partial r}=0$$

and I can't able to derive how that limit gives that equation.

∇r is the "size" of the annulus?? The physical manifestation of that limit could be stated as "if the accretion disc becomes more like a ring or planet". There is no viscous laminar flow if the annulus has no area. could conclude that the mass loss is 0.

$$ \frac{\partial}{\partial t}(2πr\Delta r\Sigma) = v_r(r,t)2πr\Sigma(r,t)−v_r(r+\Delta r,t)2π(r+\Delta r)\Sigma(r+\Delta r,t)\\ \approx-2\pi\Delta r\frac{\partial (r\Sigma v_r)}{\partial r} $$

up to this I have no problem but it further says that this equation in the $\Delta r\rightarrow0$ l

For a generic f(x) = a-b
limit_b→a f(x)⇒0

1. For an accretion disk, $v_r<<c_s$ Why?

Is that saying the "turbulent velocity is much lower than the speed of sound"? Would be scary if that were no true. Storms on Earth would be quite intense.

 

[Shan K]

Can you post the source. I like to check the authors definition of each symbol.

See the attachment I can not understand how to derive equation (2.4) from (2.3).

Is that saying the "turbulent velocity is much lower than the speed of sound"? Would be scary if that were no true. Storms on Earth would be quite intense.

$v_r$ is the radial velocity of the gas in the accretion disk and $c_s$ is the sound speed.