# Chapter 4. Spherically Symmetric Flow

## Basics

Let us consider systems in steady state (**Parker's wind & Bondi accretion**)

The EoC requires that $\dot M=4\pi r^2\rho u$ is a constant, which gives out the relation

$$
\frac1\rho\frac{\text d\rho}{\text dr}+\frac2r+\frac1v\frac{\text dv}{\text dr}=0
$$

The EoM gives

$$
\begin{align\*}
v\frac{\text dv}{\text dr}&=-\frac1\rho\frac{\text dP}{\text dr}-\frac{GM\_*}{r^2}\\
&=-\frac{c\_s^2}\rho\frac{\text d\rho}{\text dr}-\frac{GM\_*}{r^2}
\end{align\*}
$$

Combine EoC and EoM to eliminate the density gradient $\text d\rho/\text dr$, we find

$$
v\frac{\text dv}{\text dr}=\frac{2c\_s^2}r+\frac{c\_s^2}v\frac{\text dv}{\text dr}-\frac{GM\_\*}{r^2}
$$

$$
\Rightarrow \frac12\left(1-\frac{c\_s^2}{u^2}\right)\frac{\text dv^2}{\text dr}=-\frac{GM\_*}{r^2}\left(1-\frac{2c\_s^2r}{GM\_*}\right)
$$

To solve this ODE, we apply the so-called **trans-sonic condition**, that is, $v=c*s$ at $r=r*\text c\equiv GM\_\*/2c\_s^2$.

### Isothermal Solution

In isothermal flow, $c\_s$ is a constant, the solution is thus

$$
\mathcal M^2-\ln \mathcal M^2=4\ln\xi+\frac4\xi-3
$$

where $\mathcal M$ is the Mach number, and $\xi\equiv r/r\_\text c$. The solutions are shown as below.

![](https://1509032923-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-MPMxe8Bu9WDT3p-DA_8%2Fsync%2F9b3600a5527693768cce9e351962ab6b335e732c.gif?generation=1608873190445126\&alt=media)

* I & II - unphysical
* III - supersonic
* IV - subsonic
* V - **wind solution (Parker's wind)**
* The other trans-sonic solution - **accretion solution (Bondi accretion)**

### Accretion Solution

If the flow is not isothermal, we assume a polytropic equation of state $P=K\rho^\gamma$. Bernoulli's theorem for steady flow ensures that $\frac12u^2+h+\phi$ is a constant along each streamline, where $h$ is the specific enthalpy

$$
h(p)\equiv\int\frac{\text dp}{\rho(p)}=\frac K{\gamma-1}\gamma\rho^{\gamma-1}=\frac{c\_s^2}{\gamma-1}
$$

and $\phi$ is the gravitational potential. Thus at radius $r$,

$$
\frac12v^2+\frac{c\_s^2}{\gamma-1}-\frac{GM\_\*}{r}=\frac{c\_\infty^2}{\gamma-1}
$$

where $c\_\infty$ is the adiabatic sound speed at infinity. Here we have applied the infinite boundary condition, that is, $v\to0$ as $r\to\infty$.

Setting $v=c\_s$ at

$$
r=r\_c\equiv\frac{GM\_\*}{2c\_{s,\text c}},\quad c\_{s,\text c}\equiv c\_s(r\_\text c)
$$

as we are only interested in the trans-sonic solution. Consequently,

$$
\begin{align\*}
r\_\text c&=\frac{5-3\gamma}2\frac{GM\_*}{2c\_\infty^2}\\
c\_{s,\text c}&=\left(\frac2{5-3\gamma}\right)^{1/2}c\_\infty\\
\rho\_\text c&=\left(\frac2{5-3\gamma}\right)^{1/(\gamma-1)}\rho\_\infty
\end{align*}
$$

$\rho\_\text c$ is calculated from the EoS.

Obviously, for $\gamma\ge5/3$, $r\_\text c\le0$, so there is no trans-sonic solution for adiabatic (no radiative cooling) EoS. $\gamma=5/3$ is critical for spherical, point-mass accretion.

### Mass Accretion Rate

$$
\begin{align\*}
\dot M\_B&=4\pi\rho r^2v=4\pi\rho\_\text c r\_\text c^2v\_{s,\text c}\\
&=4\pi\left(\frac2{5-3\gamma}\right)^{\frac{5-3\gamma}{2(\gamma-1)}}\left(\frac{GM\_*}{2c\_\infty^2}\right)^2\rho\_\infty c\_\infty\quad\left(\propto \rho M\_*^2T^{-3/2}\right)\\
&\equiv4\pi q(\gamma)R\_B^2\rho\_\infty c\_\infty
\end{align\*}
$$

where we define

$$
q(\gamma)\equiv\frac14\left(\frac2{5-3\gamma}\right)^{\frac{5-3\gamma}{2(\gamma-1)}}
$$

and the **Bondi radius**

$$
R\_B\equiv\frac{GM\_\*}{c\_\infty^2}
$$

* For diffusive ISM accretion onto a sun-like object

  $$
  \rho\_\infty\sim10^{-24}\text{ g/cm}^3,\quad T\sim10^4\text{ K},\quad \gamma\sim1.4
  $$

  $$
  \Rightarrow \dot M\_B\sim10^{-15}\left(\frac{M\_\*}{M\_\odot}\right)^2M\_\odot/\text{yr}
  $$

  This rate is extremely small. Even if we neglect the accretion feedback, a sun-like object can only accrete $\sim10^{-5}M\_\odot$ in Hubble time.
* For denser and colder molecular cloud

  $$
  \rho\_\infty\sim10^{-20}\text{ g/cm}^3,\quad T\sim10\text{ K},\quad \gamma\sim1.4
  $$

  $$
  \Rightarrow \dot M\_B\sim3\times10^{-7}\left(\frac{M\_\*}{M\_\odot}\right)^2M\_\odot/\text{yr}
  $$

  In this way, the accretion timescale is quite interesting, as it only takes several million years for a sun-like object to double its mass. For a SMBH $\sim10^6M\_\odot$, the timescale to reach double mass is only 0.3 yr.

We note that in reality, BH feedback suppresses the accretion somehow. Thus, $\dot M\_B$ is the upper limit of the actual accretion rate. In this case,

$$
\frac{\text dM\_*}{\text dt}=AM\_*^2
$$

$$
\Rightarrow\frac1{M\_*}-\frac1{M\_0}=-A(t-t\_0)\Rightarrow M\_*=\frac{M\_0}{1-AM\_0(t-t\_0)}
$$

The growth of $M*\*$ is so rapid that before $t$ goes to infinity, $M$ diverges within \*Bondi timescale* $t\_B$

$$
t\_B\equiv\frac1{AM\_0}
$$

This is known as the **super-exponential growth**.

## Accretion Flows

### Radiative Luminosity

The luminosity caused by radiation can be estimated as

$$
L\_\text{acc}\sim\frac{\text dE\_\text{g}}{\text dt}\sim\frac{\text d}{\text dt}\left(\frac{GM^2}{R}\right)\sim\frac{GM\dot M}{R}-\frac{GM^2}{R^2}\dot R
$$

The second term is important only when there is significant orbital shrink in the system, such as binary spiral-in or stellar contraction. For steady accretion onto a point mass, it is negligable, so we can focus on the first term only.

* Neutron star

  $$
  M\_*\sim M\_\odot,\quad R\_*\sim10\text{ km}\Rightarrow L\_\text{acc}\sim\frac{GM\_*}{R\_*}\dot M\sim10^{38}\left(\frac{\dot M}{10^{-8}M\_\odot/\text{yr}}\right)\text{ erg/s}
  $$
* Black hole

  $$
  L\_\text{acc}\sim\frac{GM}{R\_\text{ISCO}}\dot M
  $$

  where $R\_\text{ISCO}$ is the innermost stable circular orbit. For a Schwarzschild black hole,

  $$
  R\_\text{ISCO}=3R\_{Sch}=\frac{6GM}{c^2}
  $$

  so the accretion rate is approximately

  $$
  L\_\text{acc}\sim\frac16\dot Mc^2
  $$

We define the **radiation efficiency** $\eta$ so that

$$
L\_\text{acc}\equiv\eta\dot Mc^2
$$

$\eta$ can be of the order $10^{-1}$ for black holes, but is negligible for less compact objects, such as normal stars.

### Eddington Accretion Rate

For any astrophysical system, for materials to accrete, the gravity should exceed the radiation force, that is

$$
\frac{GM}{r^2}\ge\frac{L}{4\pi r^2}\frac{\rho\kappa}{c}
$$

Immediately, we notice that $L$ cannot exceed a critical luminosity $L\_{Edd}$,

$$
L\le L\_{Edd}\equiv\frac{4\pi cGM}{\kappa}
$$

$L\_{Edd}$ is the famous **Eddington luminosity**. Assuming $\kappa$ is solely given by the electron scattering, the typical Eddington luminosity is

$$
L\_{Edd}\sim1.3\times10^{38}\left(\frac{M}{M\_\odot}\right)\text{ erg/s}
$$

Since $\dot M=Mc^2$, there is also a critical $\dot M\_{Edd}$, known as the **Eddington accretion rate**, given as

$$
\dot M\le\dot M\_{Edd}\equiv\frac{4\pi GM}{\eta c\kappa}\propto M
$$

Note that

$$
\frac{\text dM}{\text dt}\propto BM\Rightarrow M=M\_0e^{B(t-t\_0)}
$$

So Eddington accretion undergoes **exponential growth**.