Quasar mass and accretion rates

This page on Wikipedia -- Quasars mentions that the "The largest known [quasar] is estimated to consume matter equivalent to 600 Earths per minute". However, there is no citation for this comment. How can I find out where this information came from? I've commented in the Talk section for the page.

Tricky to say for sure, but I would imagine it comes about from measurements of the luminosity and inference of the black hole mass in such systems.

The most extreme objects radiate at the Eddington luminosity, where gravitational forces on matter falling into the black hole are balanced by radiation pressure from the heated material closer in.

If infalling mass is converted to luminosity at a rate of $$L = \epsilon \dot{M} c^2,$$ where $\dot{M}$ is the mass accretion rate, $L$ is the luminosity and $\epsilon$ is an efficiency factor, which should be of order 0.1; then the mass accretion rate at the Eddington limit is given by $$\dot{M} = \frac{4\pi G M m_p}{\epsilon c \sigma_T} \simeq 1.4\times 10^{15}\frac{M}{M_{\odot}}\ {\rm kg/s},$$ where $M$ is the black hole mass, $m_p$ the mass of a proton and $\sigma_T$ is the Thomson scattering cross-section for free electrons (the major source of opacity in the infalling hot gas).

The biggest supermassive black holes in the universe have $M \simeq 10^{10}M_{\odot}$ and thus the Eddington accretion rate for such objects is about $1.4\times 10^{25}$ kg/s or about 2.3 Earths/second or 140 Earths per minute. The difference between this estimate and the one on the wikipedia page could be what is assumed for the biggest $M$ or that $\epsilon$ is a bit smaller than 0.1 or indeed that the luminosity could exceed the Eddington luminosity (because the accretion isn't spherical).

Perhaps a simpler way to get the answer is to find the most luminous quasar and divide by $\epsilon c^2$. The most luminous quasar ever seen is probably something like 3C 454.3, which reaches $\sim 5\times 10^{40}$ Watts in its highest state. Using $\epsilon = 0.1$ yields about an Earth mass per second for the accretion rate.

So perhaps the number on the wikipedia page is a little exaggerated.

• Great answer! Your link to 3C 454.3 (mdpi.com/2075-4434/5/1/3/pdf) doesn't work. EDIT: Just found a chached copy at webcache.googleusercontent.com/… – Jim421616 Jul 4 '18 at 2:29
• Using your equation for M-dot, and the mass of 3C273 of 886 million solar masses I get 1.24E18 kg/s. Does that sound right? For the Thomson scattering cross-section I used 6.65E-29 m^2 – Jim421616 Jul 4 '18 at 3:33
• @Jim421616 No, you forgot the factor of a million! – Rob Jeffries Jul 4 '18 at 3:47
• Oh yes, I just saw that I used the wrong mass for solar :) – Jim421616 Jul 4 '18 at 3:51

Here is a study from 2012 for the largest recorded quasar which quotes an output of 400 times the mass of the sun per year, which is 253 earth masses per minute (133178400 M ⊕ / 525600 mins) at 2.5 percent the speed of light, located 1 billion light years away.

https://vtnews.vt.edu/articles/2012/11/112912-science-quasar.html

It's the largest recorded quasar, I don't know the figure for the largest theoretical quasar, there are apparently hundreds of people theorizing and debating the theoretical maximum.

• The question asks about the mass accretion rate, not the size of any outflow. – Rob Jeffries Jul 6 '18 at 7:21