What's the size of a Quasar's black hole? Take a look at light curves!
Estimating Black Hole Masses in Hundreds of Quasars from Reverberation Mapping
Determining black hole (BH) masses is important for understanding the AGN phenomenon, cosmological evolution of BHs, and the coevolution of AGNs and their containing host galaxies. Direct BH mass measurements have been established only in very nearby galaxies from stellar and gas dynamics near their center.
In AGN, the accretion disk's continuum radiation excites gas in the so-called broad line region (BLR), where it is moving at ~ 1,000 to 10,000 km/s, leading to Doppler broadening of the emission lines. Changes in the accretion disk luminosity will lead to changes in the BLR luminosity, modulated by a time-delay τdelay due to finite light travel time. This τdelay contains information about BLR size and structure, and when combined with the BLR velocities, also the BH mass. This method is called reverberation mapping.
Traditional reverberation mapping requires spectroscopic monitoring of individual objects which is observationally very expensive, and has been carried out only for a few tens of objects.
This work entails the development and application of a rigorous approach for stochastic reverberation mapping of existing sparsely sampled broad-band flux measurements. It is applied to SDSS broad-band observations in 'Stripe 82' (S82) having flux measurements precise to 2 % at ~ 60 epochs, where BLR line flux contributes up to 15 % of the flux in some bands, further constrained by spectroscopy at one epoch.
The AGN continuum is modeled as a stochastic Gaussian process. A flux model describes variations of the observed flux having emission line contribution as a scaled version of the pure continuum band plus a scaled, smoothed and delayed version of the continuum. This method is capable not only to interpolate between data points, but also to make and include self-consistently estimates and determine statistical confidence limits. Despite the effectiveness of this approach in handling uneven time sampling, the S82 temporal sampling proves a serious limitation. Also, suitable redshift ranges must be identified. This makes pre-selection of sufficient light curves necessary.
Through generating and evaluating problem-specific mock data, we verify that SDSS S82-like data can constrain τdelay. In application to SDSS S82 data, we estimated τdelay for a well-defined sample of 323 objects spanning redshifts from z=0.225-0.846. This results in ensemble-scaling relationships for the BLR size and BH mass as a function of luminosity and redshift. Comparing our findings with earlier results by [Kaspi2000] and [Vestergaard2002], we find tentative evidence that the BLR is ~ 1.7 larger than proposed. The formalism developed here should also be useful for application to data sets from upcoming surveys.
principle of reverberation mapping:
i) continuum radiation from accretion disk excites BLR clouds ⇒ broad emission lines
ii) changes in BLR excitation and luminosities
ii) due to finite light travel time, BLR luminosity variations are delayed with respect to accretion disk luminosity variation
BH mass is calculated using equation
where f: proportional factor (BLR geometry and kinematics) [Kaspi2000]
modeling a light curve as stochastic Gaussian process
The main advantage of this approach is that it not only does interpolation between data points, but also estimates the uncertainties in the interpolation. If we generated individual realizations of light curves each constrained by the data, they would track the mean light curve and statistically stay inside the "error snake" defined by the variances but they would show much more structure on short time scales and excursions outside the "error snakes" consistent with the estimated variances.
fig. 1: example of a light curve model; first image: measured r band light curve, second image: fitted light curve.
The light curve is from a spectroscopically confirmed quasar in the SDSS Stripe 82 in a redshift region where r band is only continuum. For fitting the light curves, outliers are excluded. The solid lines in the right figure panel represent the best fit mean model light curves. The area between the grey lines represents the "error snake", the 1σ range of possible stochastic models. The "error snakes" bound the reconstructed light curve are thinner than the data points because of the additional measurement error on the data.
methodology of stochastic reverberation mapping:
the following flow chart shows the methodology of stochastic reverberation mapping illustrated using the example of a r band continnum light curve and a g band contiuum+emission line light curve
Master thesis and source code are available on request.