Mallory Molina The Ohio State Univeristy
REU programSummer 2011

Introduction Reverberation Mapping Empirical Method Fitting Process Analysis Results Future Work Sources Links 
Quasars (or QuasiStellar Objects) are one of the least understood objects in our Universe. Their complete physical structure, evolution and many other properties still elude scientists. Many are very luminous and very distant from us. Quasars are the dense central core of an Active Galactic Nuclei, or AGN. Therefore, sometimes quasars are referred to as an active galactic nucleus. This central core surrounds the central supermassive black hole and radiates more than the entire galaxy, where matter actively accretes on the central black hole (10^{6} to 10^{8} solar mass) in a structure called an accretion disk.
Radiation from accretion disk excites gas farther out. This material is accelerated by the gravitational force of the black hole, thus giving the mass high Doppler motions, and making the lines broad. The region is called the broadline region (BLR).
With these concepts in mind, scientists are working out the mechanics of calculating basic physical measurements, such as the central black hole mass. This is complicated because of the distance and chaotic area. However, knowing the properties of the BLR, where the material is dominated by gravity, we can use those gas motions to help find out properties of the central black hole. Therefore, to calculate the central black hole mass, we need to look at spectra from this region.
The most direct method of measuring black hole masses is called reverberation mapping. In order to calculate this directly, we assume that the gas motions around the central source are virialized (or gravity dominated). Given that the BLR is a region in which the gas motions are gravity dominated, this assumption holds. Therefore, we can use a simple physical formula called the virial theorem, which states:
M =  fcτΔV^{2} G 
Where cτ is the radius of the BLR, ΔV is the line width, G is the gravitational constant, and f is a constant of order unity that accounts for the geometry of the BLR. Here the M represents the black hole mass, but in general the virial theorem holds true for any central mass for which another object is rotating with a given radius and velocity.
The calculation of the radius of the BLR is the actual "reverberation" part of reverberation mapping. By looking at changes in the continuum flux from the accretion disk, and then tracking those changes in the BLR flux, we can measure a light distance travel time between the accretion disk and the BLR. When multiplied by c, this gives us the radius of the BLR. The physical interpretation of this method is shown below.
The ΔV, or line width, is the velocity dispersion of the BLR. The reason we cannot measure the actual velocity of the BLR is that the clouds are not moving at one consistent velocity. Yet, they are gravity dominated, and affected only by the central black hole, so they are ideal for measuring the velocity component of the virial theorem. The line can be parametrized by two main parameters: full width at half maximum (FWHM) or the line dispersion, which is σ, or the second moment of the line profile. The line Hβ, which is the line we studied this summer, is a well accepted line to use for this measurement.
While reverberation mapping is the best and most direct way of measuring central black hole mass, it is very time and resource consuming. For instance, the wait time for the light reverberation can be up to days and weeks. Therefore it is not a feasible solution for all projects. However, this does not mean that only those who can afford this amount of time and resources can calculate black hole masses.
The basis behind the empirical formula is that v^{2}R∝FWHM^{2}L^{γ}, or in other words, the radius and the velocity (from the virial theorem) is proportional to the line width of the line used to calculate the velocity dispersion and the luminosity. The luminosity can be calculated from many areas in the spectrum (5100 Å, Hβ, etc.), and the γ in the relation corresponds to the slope of the RL relationship.
This summer, we used the FWHM(Hβ) and L(5100 Å), which gives us a γ of 0.5. Therefore the equation we use is:
Where λ is 5100 Å, and μ is the unscaled mass, that relates to the reverberation black hole mass with only one degree of freedom, the scaling factor, a, as in the following equation:
In our case, a is 6.91. The M_{BH}(RM) is given in solar masses. Therefore, given this simple relation, we should easily be able to take a single epoch spectrum (taken at one time) to the more precise reverberation mapping mass. While this seems rather straight forward, the problem is the paremetrization of Hβ is complicated, as the line is contaminated with narrow components and blending with Fe II lines and sometimes [OIII] lines. Therefore, finding the FWHM(Hβ) is a process in itself.
All equations in this section are from Vestergaard and Peterson, 2006.
Each spectrum took many iterations of the fitting program when each Gaussian was added, starting with three or four Gaussians and then working into more complicated profiles. The precision on the guesstimates used had to be very high. Shown below is the fitting process for one of the twelve spectra used, PG 1444+407:
As is evident we have fit the three main curves in the spectrum as best we could. We identified the first curve as Hβ, the second as a shelf that we sometimes see on Hβ, and the third a broad curve (possible an Fe II line) around the [OIII]λ5007 region. From this fit we will add more Gaussians in order to minimize the size and systematic trends of the residuals.
This fit better constrains the Hβ, and fits the narrow line component on Hβ more accurately. The blue component on Hβ as well as at the [OIII]λ5007 region are poorly fit. The shelf on Hβ is also better constained.
This fit has sufficiently constrained the [OIII]λ5007 region and the Hβ shelf. The blue wing on Hβ is still not sufficiently fit. Therefore one more Gaussian is needed to eliminate the large systematic trends in the residuals.
This is the finalized fit of the spectrum for PG 1444+407. As is obvious, the residuals are minimal and there are no real systematic trends.This is the best fit for this spectrum.
This process was repeated for all twelve spectra. After completing all the fits, we had to go through and identify all the components that we fit. This included finding components that contribute to the Hβ broad line, and what the broader components in the [OIII]λ5007 region are.
The next step after fitting all of spectra is to figure out what components contribute to the FWHM of Hβ. We decided to emulate a paper done by Kovačević et. al. (2010). They claim that Hβ can be divided into three components: the narrow line region (NLR), intermediate line region (ILR) and the very broad line region (VBLR). The narrow line region comes from further out in the galaxy and is therefore subtracted from the width of Hβ, as we only care about the BLR. Then what should be done is add the functions of the two remaining components, the ILR and VBLR, and calculated the FWHM from that. This is shown below:
While this seems rather simplistic in essence, there were problems that arose. One was that in our spectra, the Hβ region was not as simple as the paper proposed. There were red and blue shifted components in the line. We therefore are working out which parts contribute to the Hβ broad line, and which do not. There were also shelves on Hβ lines in certain spectra.
In the [OIII]λ5007 region, we saw what appeared to be broadening of the line [OIII]λ5007. Given that this is a forbidden line, the line should not be broadened. This region is also riddled with Fe II lines, and Kovačević et. al. (2010) have an Fe II template that they used to subtract out these lines. Based on the relative shift from the expected center of the line, we are trying to identify the components as either Fe II lines, or actual components of one of the quasar lines in that region. The template is shown below:
Comparing these to currently calculated empirical and reverberation black hole masses, they seem in the right size range, and seem to agree. A better calculation of these masses may come when further data is reduced and ready for analysis.
We will not only do this with the current data we have, but also examine more data that has higher resolution to better answer these questions.
Vestergaard and Peterson, 2006
SIMBAD (Stellar/Galactic database)
NASA Astrophysics Data Service