AGN structure and emission

Let's talk about the structure of an active galactic nucleus like a quasar, and the types of emission we see from it. The classic unified model of an AGN consists of a supermassive black hole (of perhaps $\sim10^8\text{-}10^9M_{\odot}$) surrounded by an accretion disk about $\sim10^{13}\text{-}10^{14}$ meters in radius. The disk has a radial temperature distribution of $$T(r)\approx3\times10^5\dot{m}\left(\frac{M}{10^8M_{\odot}}\right)^{1/4}\left(\frac{r}{r_{\text{Sch}}}\right)^{-3/4}\text{ K}$$ where $r_{\text{Sch}}$ is the Schwarzschild radius and $\dot{m}$ is the ratio of the accretion rate to the maximum accretion rate specified by the Eddington limit. If we assume that $\dot{m}\approx1$ and $M\sim10^8M_{\odot}$, the luminosity should be about $L\sim10^{39}\text{ W}$. The disk itself should emit most strongly in the x-ray and ultraviolet portion of the spectrum, with UV emission beginning at $\sim10^{13}\text{ m}$.

Beyond the disk lies the broad-line region, where high-velocity gas clouds produce secondary emission. This area should have an outer radius of maybe $\sim10^{14}\text{-}10^{15}$ meters. Beyond the broad-line region lies the narrow-line region, (radius $\sim1000$ light-years) which includes the obscuring torus (radius $\sim100$ light-years), the latter being a structure of gas and dust that may be fed by a wind from the accretion disk. The narrow-line region contains slower-moving gas clouds; the low speeds produces less Doppler broadening - hence the name.

I think that what you've been considering is only the jets that arise from the accretion disk. Matter from the disk travels along magnetic field lines; electrons are accelerated, producing the synchotron emission observed from many AGN. This is indeed strong, but keep in mind that the jets are narrow and usually perpendicular to the plane of the galaxy, meaning that most objects in the galaxy are far away from the jet. If your planet exists in the equatorial plane of the quasar, it won't be hit by the jets, although it could experience radiation from the accretion disk.

Different radii

We could attempt to compute the flux the planet would receive from the disk via the inverse square law if the disk was a point source and emitting isotropically. This is decidedly not the case. If we want to look at the best-case scenario, where the black hole is accreting below the Eddington limit, we could attempt to model the disk as a thin disk and use the Shakura-Sunyaev model, where the flux is given by $$F(r)=\frac{3GM\dot{M}}{8\pi r^3}\left(1-\sqrt{\frac{r_0}{r}}\right)$$ where $r_0$ is the inner radius of the disk. If we assume the accretion disk somehow extends to the black hole's surface, we get that at a distance of $1.17\times10^{16}$ meters (1.24 light-years), the flux from the disk is about 10% of that from the Sun.

There's another radius we may want to consider, which is $r_{\text{blr}}$, the outer radius of the broad-line region. It's calculated by $$r_{\text{blr}}\approx0.26\times10^{15}\left(\frac{L}{10^{37}\text{ W}}\right)^{1/2}\text{ m}\approx0.27\text{ light-years}$$ This radius is quite similar to the dust sublimation radius, inside which dust will be vaporized. This occurs when $T\approx1500\text{ K}$. I then calculate that for our supermassive black hole, $r_{\text{sub}}\approx0.037$ light-years. The order-of-magnitude difference is because of my assumption that the black hole is accreting as fast as it can - a sort of worst-case scenario for our quasar. Finally, consider that the largest accretion disks may be around $0.01$ light-years in radius.

Here's a summary of the various length scales:

• $2\text{ AU}$: the Schwarzschild radius of the black hole
• $0.001\text{ ly}$ ($64\text{ AU}$): the outermost point of UV emission
• $0.01\text{ ly}$: the outer radius of the accretion disk
• $0.037\text{ ly}$: the dust sublimation radius
• $0.27\text{ ly}$: the outer radius of the broad-line region
• $1.24\text{ ly}$: the distance at which the jet's flux becomes 10% of that from the Sun
• $\sim100\text{ ly}$: the outer edge of the obscuring torus
• $\sim1000\text{ ly}$: the outer edge of the narrow-line region

The answer to your question, then, depends on the innermost zone you're comfortable with having your planets in, assuming you're good with the systems being in the plane. If not, you can be around 80 light-years away, for a direct hit from the jet.

posted about 6 years ago

HDE 226868‭

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