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Suppose we have a Shakura and Sunyaev style accretion disk between $r_1$ and $r_2$. What is the average temperature? We can calculate it by integrating the local temperature $$T(r)=\left [ \frac{3GM\dot{m}}{8\pi\sigma r^3}\right ]^{1/4} \left (1-\sqrt{r_1/r}\right ) ^{1/4}$$ and dividing by the total disk area: $$\langle T\rangle=\frac{1}{\pi(r_2^2-r_1^2)}\int_{r_1}^{r_2} 2\pi r T(r) dr$$ If we assume the second factor in the temperature is $\approx 1$, then $T(r)=k r^{-3/4}$ and the integral becomes $2\pi k \int_{r_1}^{r_2} r^{1/4} dr = (8\pi k/5)[r^{5/4}]_{r_1}^{r_2} = (8\pi k/5)\left (r_2^{5/4}-r_1^{5/4}\right )$. So
$$\langle T\rangle=\frac{8k}{5}\frac{r_2^{5/4}-r_1^{5/4}}{r_2^2-r_1^2}.$$

As $r_2$ grows this declines, which is not unreasonable since we are averaging over more cool outer surface. However, I am interested in what happens as we move towards heavier central black holes with bigger disks with correspondingly larger inflows.

The disk size is sometimes estimated for AGNs using the self-gravity radius $r_{sg}$ where local gravity from the disk exceeds the vertical component of the black hole gravity and the disk becomes unstable. $$r_{sg}\approx 1680 \left[\frac{M}{10^9 M_\odot}\right]^{-2/9} \alpha^{2/9} \left[\frac{L_{AGN}}{L_{Edd}}\right]^{4/9} \left[\frac{\epsilon}{0.1}\right]^{-4/9} r_S$$ where $\alpha$ is the viscosity parameter $\approx 0.01-0.1$ and $\epsilon$ is the mass conversion efficiency. $r_s=2MG/c^2$. We can set $r_1=3r_s$ and $r_2=r_{sg}$.

If we set the luminosity to a fraction $f$ of the Eddington luminosity $3\times 10^4 (M/M_\odot)L_\odot$ the mass flow becomes $\dot{m}=L_{Edd}/\epsilon c^2$. Putting all of this into the formulas gives $r_2=r_{sg}\propto M^{7/9}$, $r_1\propto M$, and $k\propto M^{1/2}$ (because of the dependency on $M$ and $\dot{m}\propto M$). So $$\langle T\rangle \propto M^{1/2}\frac{M^{35/36}-M^{5/4}}{M^{14/9}-M^2}=\frac{M^{53/36}-M^{7/4}}{M^{14/9}-M^2}.$$ Assuming that I actually did not flub the algebra, this expression will be dominated by the second terms in the numerator and denominator as $M$ grows large. Together they suggest a scaling $\langle T\rangle \propto M^{-1/4}$.

Does this make sense? I have a feeling I have missed something elementary. While I can tell myself that very large accretion disks on average are pretty lukewarm and that they grow faster in radius and area than the mass inflow increases, it seems deeply counter-intuitive.

Hmm, it is the hot annulus with maximal temperature at $(49/36)r_1$ that dominates the spectrum, and it has $T\propto (M\dot{m}/r_1^3)^{1/4}=M^{-1/4}$. So this also tells us that we should expect really big black holes to have cool disks. I find that hard to swallow.

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