Reverberation mapping

A Primer on Active Galactic Nuclei

As far as we know, every galaxy has a supermassive black hole at its center, and the sizes of those black holes are strongly correlated with the properties of their host galaxies. At first this might seem obvious—black holes have really strong gravitational fields, so of course they're going to affect their hosts. However, the radius within which motions are dominated by a black hole's gravity, known as the "sphere of influence," is very small compared to the size of a galaxy. For example, the sphere of influence of the supermassive black hole at the center of our Milky Way galaxy is only about 3 parsecs (~10 light years) in radius, while the Milky Way itself is 5000 times larger at 15,000 parsecs in radius.

There are a number of theories that explain the black hole-galaxy connection, most related to the idea of co-evolution of the two bodies. Each theory predicts a slightly different behavior for how the relations change over time, so if we can measure the properties precisely enough, we can rule out some theories and support others. In nearby galaxies, we can look at the motions of stars and gas within the sphere of influence to measure the black hole mass. For distant galaxies required to study the early Universe, the sphere of influence is too small to resolve even with the most powerful telescopes. Instead, we can use a technique called reverberation mapping to measure the black hole mass indirectly.

Information Encoded in the Emission Lines

We don't know the exact geometry of the BLR, but the shape and width of a broad emission line can provide a lot of information. The wavelength of light we observe from a gas particle depends on the velocity of the gas along the line of sight to the observer. If the gas is moving towards us the light will be blue-shifted, and if the gas is moving away from us the light will be red-shifted. If the gas is moving tangentially to the line of sight, then there is no line-of-sight velocity component and no Doppler shift is observed. You can use the tool below to explore the effect of different geometries and velocities on the shape of the emission line.

If you change the velocity of the gas, you'll notice that the width of the emission line also changes: the faster the gas is moving, the broader the line. Assuming the motion of the gas is primarily dictated by the black hole's gravity^[?] , the orbital velocity of a gas particle is \begin{equation}\label{eq:orbital_velocity} v = \sqrt{\frac{GM_{\text{BH}}}{r}} \end{equation} where $M_{\text{BH}}$ is the mass of the black hole, $r$ is the distance from the gas particle to the black hole, and $G$ is the gravitational constant ($6.67 \times 10^{-8} \, \text{pc}\cdot M_\odot^{-1} \cdot (\text{km}/\text{s})^2$). Therefore, faster moving gas indicates either a larger black hole or a smaller radius of the BLR, but we cannot know which from the the shape of the line alone. Unfortunately, the BLRs of most AGNs only span a few microarcseconds (1 microarcsecond = 1/3,600,000,000th of a degree) on the sky, making it near-impossible to measure their size directly with telescopes^[?] .

Instead, to determine the physical size of the BLR, we can take advantage of the fact that the speed of light is finite and takes time to travel from point A to point B. If you imagine a pulse of light emitted from the accretion disk, that light will travel out in all directions. Some of the light will travel towards us, the observer, and we will see it after a time, $t$, that depends on the distance from us to the galaxy. From that same pulse, some light will also travel out to the BLR where it will be absorbed and re-emitted by the gas as an emission line. That re-emitted light will also travel towards us, but it will arrive later than the continuum light pulse with a time lag, $\tau$, that we can measure. This is illustrated in the animation below.

Of course, each point in the BLR will have a different time lag, depending on its distance from the accretion disk as well as its positioning along the $x$-axis. Points that are along our line of sight to the accretion disk will actually have no time lag. Regardless, multiplying the average time lags from all components of the BLR by the speed of light, $c$, gives a crude estimate of the size of the BLR: \begin{equation}\label{eq:blr_size} r_{\text{BLR}} \approx c \tau. \end{equation}

Measuring the Time Delay

Fortunately for us, AGN brightnesses are not constant over time, but rather increase and decrease on timescales of days to months. Therefore, if we continuously monitor the AGN and measure the brightness of the continuum and the broad emission lines, we should be able to see the emission lines respond to changes in the continuum brightness. This is precisely what's shown in the figure below. The black points, labeled $g$-band photometry, are measurements of the continuum brightness. The light blue points are measurements of a fit to the C IV broad emission line flux which follows the same fluctuations as the continuum, but delayed by $\tau = 139$ days^[?] . The dark blue points are the same C IV measurements, but shifted back in time by $\tau$ to show the overlap with the continuum.

Now that we have a measurement of the time delay, we can combine Equations \eqref{eq:orbital_velocity} and \eqref{eq:blr_size} and rearrange: \begin{equation}\label{eq:bh_mass} M_{\text{BH}} \approx \frac{c \tau v^2}{G}. \end{equation} This gives us a measurement of the black hole mass, $M_{\text{BH}}$, in terms of the time delay, $\tau$, and the velocity, $v$, of the gas in the BLR. There are a few different measurements of the velocity that can be used, but the most common is the Full Width at Half Maximum (FWHM) of the emission line. The FWHM is a measurement of the width of the emission line at half its maximum height, and is easy to measure from a spectrum. If we denote the FWHM as $\Delta V$ and add in a "fudge factor," $f$, that accounts for all of the approximations we've been making (e.g. the unknown BLR geometry, average time lag, etc.), we come to the most common form of the reverberation mapping equation: \begin{equation}\label{eq:bh_mass_fwhm} M_{\text{BH}} = f \cdot \frac{c \tau (\Delta V)^2}{G}. \end{equation} By far, the most uncertain part of this equation is the fudge factor, $f$, which introduces a factor of $\sim$$2.5$ uncertainty in individual black hole mass measurements.

Improvements: Velocity-Resolved Reverberation Mapping

Until now, we've been been condensing all the information from the broad line region into two measurements: the average time delay and the width of the emission line. These get us to a pretty good estimate of the black hole mass, but there's a lot more information in the emissinon line profile we're not utilizing. As we saw in the interactive tool above, the shape of the emission line depends not only on the velocity of the gas, but on the geometry and orientation of the BLR as well. If we can use the precise shape of the emission line to make inferences about the geometry of the BLR, we can limit the possible values of $f$ to reduce the black hole mass uncertainty.

The Gold Standard: Dynamical Modeling of the Broad Line Region