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The Astrophysical Journal, 754:49 (8pp), 2012 July 20 doi:10.1088/0004-637X/754/1/49
C
2012. The American Astronomical Society. All rights reserved. Printed in the U.S.A.
THE LICK AGN MONITORING PROJECT 2011: DYNAMICAL MODELING
OF THE BROAD-LINE REGION IN Mrk 50
Anna Pancoast1, Brendon J. Brewer1, Tommaso Treu1, Aaron J. Barth2, Vardha N. Bennert1,3, Gabriela Canalizo4,
Alexei V. Filippenko5, Elinor L. Gates6, Jenny E. Greene7, Weidong Li5, Matthew A. Malkan8, David J. Sand1,9,
Daniel Stern10, Jong-Hak Woo11 , Roberto J. Assef10,18 , Hyun-Jin Bae12, Tabitha Buehler13 , S. Bradley Cenko5,
Kelsey I. Clubb5, Michael C. Cooper2,19, Aleksandar M. Diamond-Stanic14 ,20, Kyle D. Hiner4, Sebastian F. H ¨
onig1,
Michael D. Joner13, Michael T. Kandrashoff5, C. David Laney13 , Mariana S. Lazarova4, A. M. Nierenberg1,
Dawoo Park11, Jeffrey M. Silverman5,15 , Donghoon Son11, Alessandro Sonnenfeld1, Shawn J. Thorman2,
Erik J. Tollerud2, Jonelle L. Walsh2,16, and Richard Walters17
1Department of Physics, University of California, Santa Barbara, CA 93106, USA; pancoast@physics.ucsb.edu
2Department of Physics and Astronomy, 4129 Frederick Reines Hall, University of California, Irvine, CA 92697-4575, USA
3Physics Department, California Polytechnic State University, San Luis Obispo, CA 93407, USA
4Department of Physics and Astronomy, University of California, Riverside, CA 92521, USA
5Department of Astronomy, University of California, Berkeley, CA 94720-3411, USA
6Lick Observatory, P.O. Box 85, Mount Hamilton, CA 95140, USA
7Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544, USA
8Department of Physics and Astronomy, University of California, Los Angeles, CA 90095-1547, USA
9Las Cumbres Observatory Global Telescope Network, 6740 Cortona Drive, Suite 102, Santa Barbara, CA 93117, USA
10 Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Boulevard, Pasadena, CA 91109, USA
11 Astronomy Program, Department of Physics and Astronomy, Seoul National University, Seoul 151-742, Republic of Korea
12 Department of Astronomy and Center for Galaxy Evolution Research, Yonsei University, Seoul 120-749, Republic of Korea
13 Department of Physics and Astronomy, N283 ESC, Brigham Young University, Provo, UT 84602-4360, USA
14 Center for Astrophysics and Space Sciences, University of California, San Diego, CA 92093-0424, USA
15 Physics Division, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA
16 Department of Astronomy, The University of Texas at Austin, Austin, TX 78712, USA
17 Caltech Optical Observatories, California Institute of Technology, Pasadena, CA 91125, USA
Received 2011 October 26; accepted 2012 May 11; published 2012 July 5
ABSTRACT
We present dynamical modeling of the broad-line region (BLR) in the Seyfert 1 galaxy Mrk 50 using reverberation
mapping data taken as part of the Lick AGN Monitoring Project (LAMP) 2011. We model the reverberation mapping
data directly, constraining the geometry and kinematics of the BLR, as well as deriving a black hole mass estimate
that does not depend on a normalizing factor or virial coefficient. We find that the geometry of the BLR in Mrk 50
is a nearly face-on thick disk, with a mean radius of 9.6+1.2
−0.9light days, a width of the BLR of 6.9+1.2
−1.1light days, and
a disk opening angle of 25 ±10 deg above the plane. We also constrain the inclination angle to be 9+7
−5deg, close to
face-on. Finally, the black hole mass of Mrk 50 is inferred to be log10(MBH/M)=7.57+0.44
−0.27. By comparison to the
virial black hole mass estimate from traditional reverberation mapping analysis, we find the normalizing constant
(virial coefficient) to be log10 f=0.78+0.44
−0.27, consistent with the commonly adopted mean value of 0.74 based on
aligning the MBH–σ* relation for active galactic nuclei and quiescent galaxies. While our dynamical model includes
the possibility of a net inflow or outflow in the BLR, we cannot distinguish between these two scenarios.
Key words: galaxies: active – galaxies: individual (Mrk 50) – galaxies: nuclei
Online-only material: color figures
1. INTRODUCTION
The standard model of active galactic nuclei (AGNs;
Antonucci 1993; Urry & Padovani 1995) explains their broad
emission lines as being produced in a broad emission line re-
gion (BLR) situated on the order of light days from the black
hole (Wandel et al. 1999; Kaspi et al. 2000; Bentz et al. 2006).
The distance of the BLR from the black hole can be measured
using reverberation mapping, in which the average delay time is
measured between a timeseries of the variable AGN continuum
luminosity and a timeseries of the variable broad line emis-
sion (Blandford & McKee 1982; Peterson 1993; Peterson et al.
2004). Standard reverberation mapping analysis also provides
18 NASA Postdoctoral Program Fellow.
19 Hubble Fellow.
20 Southern California Center for Galaxy Evolution Fellow.
estimates of the black hole mass, MBH, in AGNs from a nor-
malized virial product. The virial product, Mvir=fv
2cτ/G,
is derived from the average light travel time lag of the BLR,
τ, and the typical velocity of the BLR gas, v, measured from
the width of the broad lines. The small sample of ∼50 rever-
beration mapped AGNs is then used to determine single-epoch
MBH estimates for much larger samples of AGNs using the
BLR-size-to-luminosity relation (Vestergaard & Peterson 2006;
McGill et al. 2008; Vestergaard 2011).
However, there are certain limitations to the standard rever-
beration mapping techniques. The object-to-object scatter of the
normalization factor fis believed to be of the order of ∼0.4dex
(Onken et al. 2004; Collin et al. 2006; Woo et al. 2010; Greene
et al. 2010b; Graham et al. 2011) based on assuming the same
MBH–σ* relation (e.g., Bennert et al. 2011) as for quiescent
galaxies. It would be desirable to avoid this assumption and es-
timate MBH from reverberation mapping data alone. The details
1
The Astrophysical Journal, 754:49 (8pp), 2012 July 20 Pancoast et al.
Figure 1. Integrated Hβbroad line and V-band continuum light curves. The Hβ
light curve has flux units of 10−15 erg cm−2s−1.TheV-band light curve is in
arbitrary flux units.
(A color version of this figure is available in the online journal.)
of the BLR geometry and dynamics are also poorly constrained
by standard reverberation mapping analysis. Measuring the time
lag as a function of line-of-sight velocity has shown that while
some BLRs are consistent with virial motion in a Keplerian
potential (Peterson & Wandel 1999; Bentz et al. 2009; Denney
et al. 2010), some show suggestions of inflowing gas (Bentz
et al. 2010; Denney et al. 2010). In addition to the mean radius
of the BLR as obtained in the standard analysis, we would like
to constrain the overall geometry of the BLR in more detail.
Recent improvements in reverberation mapping data and anal-
ysis are starting to provide better constraints on the geometry
and dynamics of the BLR. Velocity-resolved transfer functions
(VRTFs) have been measured using high-quality reverberation
mapping data from the Lick AGN Monitoring Project in 2008
(LAMP 2008; Walsh et al. 2009; Bentz et al. 2009) and from
the 2007 MDM Observatory reverberation mapping campaign
(Denney et al. 2010), showing signatures consistent with inflow,
outflow, and virialized motion for different AGNs. However,
a clear interpretation of VRTFs requires additional modeling
steps, since they are functions of time lag instead of position
within the BLR. The traditional reverberation mapping analy-
sis has also been recently improved by Zu et al. (2011), who
model the AGN continuum and line light curve using an imple-
mentation equivalent to Gaussian Processes (Kelly et al. 2009;
Kozłowski et al. 2010; MacLeod et al. 2010;Zuetal.2012).
Members of our team have developed a method for determining
the geometry and dynamics of the BLR by directly modeling
reverberation mapping data (Pancoast et al. 2011;Breweretal.
2011b, hereafter P11 and B11, respectively), estimating the un-
certainties in the framework of Bayesian statistics. Our model-
ing method constrains MBH without requiring a normalization
constant f. We also constrain the geometry of the BLR, its ori-
entation with respect to the line of sight, and the possibility of
net inflowing or outflowing gas in the BLR. We have previously
demonstrated our method on LAMP 2008 data for Arp 151
and estimated MBH with smaller uncertainties than traditional
reverberation mapping analysis (B11).
What is now needed to make further progress is large sam-
ples of high quality velocity resolved reverberation mapping
data. For this purpose we carried out an 11 week spectro-
scopic observing campaign at Lick Observatory, the Lick AGN
Figure 2. Top: Hβspectra in velocity units for each epoch in the light curve for
data (left panel) and model (right panel). Dark red corresponds to the highest
levels of flux and dark blue corresponds to the lowest levels, where the same
color scale is used for the data and model. Middle: integrated Hβflux for
each epoch in the light curve for the data (blue solid line with error bars) and
model (red dashed line). As an illustration of the range of solutions, we show
light curves for five acceptable models as dotted gray lines. For the correct
time separation between light curve epochs, see Figure 1. The model is able to
reproduce the major features of the data. Bottom: two examples of Hβspectra
fit by the model, with data shown by blue and green error bars and model fits
shown by red lines.
(A color version of this figure is available in the online journal.)
Monitoring Project 2011. The project focused on nearby AGNs
with bright Hβlines, which are good candidates for dynami-
cal modeling. Here, we present the first results of dynamical
modeling for the project, focusing on one of the most variable
objects in the sample, Mrk 50. The average time lag and virial
MBH estimates from traditional reverberation mapping analysis
are presented by Barth et al. (2011b). Here, we present an al-
ternative analysis based on our direct modeling technique. The
Hβand V-band continuum light curve data are briefly described
in Section 2, the dynamical model for the BLR is described in
Section 3, and our results and conclusions are given in Section 4.
2. DATA
We observed Mrk 50 in the spring of 2011. The data, shown
in Figure 1and the top left panel of Figure 2, include a light
curve of V-band continuum flux and a time series of the broad
Hβline spectral profile. More observational details, as well as
details about the measurement of V-band and Hβlight curves,
are described by Barth et al. (2011b). We model all 156 epochs
of the V-band light curve and 55 of the 59 epochs of the Hβ
line profile, ignoring those epochs with low signal-to-noise ratio
(S/N) or other problems. The median S/NfortheHβline profile
throughout the campaign is 75 per pixel.
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The Astrophysical Journal, 754:49 (8pp), 2012 July 20 Pancoast et al.
AGNs and stellar continuum lines can significantly alter the
measured broad line widths in AGNs, affecting single-epoch
MBH estimates (Denney et al. 2009;Parketal.2012). In order to
reduce contamination from other lines when modeling the Hβ
line profiles, the Mrk 50 spectra have been fitted with AGNs
and stellar continuum components and the He ii λ4686 line just
blueward of Hβ, and then these components were subtracted to
yield the “pure” Hβprofile (Barth et al. 2011b).
3. THE DYNAMICAL MODEL OF THE BLR
We now give a brief description of our method for directly
modeling reverberation mapping data. The motivation for our
approach is developed in P11 and further implementation details
are described in B11. We model the BLR as a large number
of point-like clouds, each with a given position and velocity.
Several parameters describe the overall spatial distribution of the
clouds and the prescription for assigning velocities to the clouds,
given their positions. Our goal is to estimate these parameters.
The continuum emission from the central ionizing source is
absorbed by these clouds and re-emitted as broad line emission,
allowing us to predict the line flux and shape as a function
of time, i.e., to produce mock data sets of the form shown in
Figure 2.
The full set of model parameters includes the geometry
and dynamics parameters for the BLR clouds corrected to the
rest frame of Mrk 50, as well as a continuous version of the
continuum light curve, since the continuum light curve must
be evaluated at arbitrary times in order to compute mock data
for comparison with the actual data. The observed continuum
light curve is interpolated using Gaussian Processes to create
a continuous light curve and to account for the uncertainty in
the interpolation. Gaussian Processes have been found to be a
good model for larger samples of AGN light curves (Kelly et al.
2009;Kozłowskietal.2010; MacLeod et al. 2010;Zuetal.
2011,2012).
The model for the BLR geometry is simple yet flexible,
allowing for disk-like or spherical geometries with asymmetric
illumination of the gas. Examples of possible BLR geometries
are shown in Figure 3. The model for the spatial distribution
of the BLR gas is first generated from an axisymmetric two-
dimensional configuration in the x–yplane, with a parameterized
radial profile. The radius rof a cloud from the origin is generated
as follows. First, a variable gis drawn from a Γdistribution with
shape parameter αand scale parameter 1:
g∼Γ(α, 1).(1)
Then, the radius rof the cloud is computed by applying the
following linear transformation to g:
r=Fμ+μ(1 −F)
αg. (2)
The parameters {μ, F, α }control the radial profile of the BLR.
μis the overall mean radius of the BLR (this can be verified by
taking the expectation value of rin Equation (2)). The parameter
F∈[0,1] allows for the possible existence of a hard lower
limit Fμ on radius, because there may be some radius interior
to which the BLR gas would all be ionized and thus unable to
respond to changes in the continuum emission (Korista & Goad
2004). αcontrols the shape of the Γdistribution: a value of α
close to 1 imposes an exponential distribution (allowing for disk
or ball configurations), whereas large values of αcreate a narrow
Figure 3. Geometry of the BLR for three models, with the x-, y-, and z-axis
scales in light days and the observer’s line of sight along the x-axis. The top
panel BLR distribution is a close-to-face-on torus of clouds, the middle BLR
distribution is a close-to-face-on disk of clouds similar to the geometry inferred
for Mrk 50, and the bottom BLR distribution is a dense sphere of clouds.
(A color version of this figure is available in the online journal.)
3
The Astrophysical Journal, 754:49 (8pp), 2012 July 20 Pancoast et al.
normal distribution (allowing for shell or ring configurations).
In the implementation, and in the description of the same model
in B11, we parameterize the shape by β=1/√αinstead of α
because βhas a simple interpretation as the standard deviation
of gin units of its mean. The radial width of the BLR can be
defined as the standard deviation of r:
σr=μβ (1 −F).(3)
In order to assign velocities to the BLR gas clouds, the
model uses probabilistic perturbations about circular orbits. The
solution for the radial velocity of a BLR cloud given its position
r, energy E, and angular momentum Lis
vr=±2E+GMBH
r−L2
r21/2
.(4)
If we wished to impose circular orbits, the values for Eand L
would be fully determined by the radius rof the cloud:
Ecirc =−1
2
GMBH
r(5)
Lcirc =±r2E+GMBH
r1/2
.(6)
To obtain elliptical orbits, we generate values for Eand Lproba-
bilistically, given r. The probability distributions for energy and
angular momentum are parameterized by the parameter λand
are given by
E=1
1+exp(−χ)Ecirc (7)
p(L)∝exp |L|
λ,(8)
where χ∼N(0,λ
2) and |L|<|Lcirc|.Forλ→0 we recover
circular orbits and increasing λcreates more elliptical orbits.
Since there are two solutions for the sign of vr, the model
also includes a parameter for the fraction of outflowing versus
inflowing gas. The inflowing and outflowing gas is bound to
the gravitational potential of the black hole, but an inequality
in the fraction of inflowing and outflowing gas has the desired
effect of modeling asymmetries in the Hβspectral line profile
as observed in Arp 151 (B11) when an asymmetric illumination
model is included.
Once a two-dimensional configuration of clouds in the x–y
plane has been generated, and velocities have been assigned
to the clouds, rotations are applied to “puff up” the two-
dimensional configuration into a three-dimensional configura-
tion. We first rotate the cloud positions about the y-axis by a
small random angle; the typical size of these angles determines
the opening angle of the cloud distribution. The opening angle
is defined as the angle above the midplane of the disk or sphere.
We then rotate around the z-axis by random angles to restore the
axisymmetry of the model. Finally, we rotate again about the
y-axis, by the inclination angle (common to all of the clouds) to
model the inclination of the system with respect to the line of
sight. The inclination angle is defined so that 0 deg corresponds
to a face-on configuration and 90 deg corresponds to an edge-on
configuration.
In order to produce asymmetric broad line profiles, we
include a simple prescription for asymmetric illumination of
the BLR clouds. We assign a weight wto each cloud, given by
w=0.5+κcos φ, where φis the azimuthal position of the
cloud in spherical polar coordinates. The parameter κranges
from 0.5, corresponding to illuminating the near side of the
BLR, to −0.5, corresponding to illuminating the far side of the
BLR. Physically, the near side of the BLR could be preferentially
illuminated if the far side of the BLR were obscured by gas, and
the far side of the BLR could be preferentially illuminated if the
BLR clouds only reradiate the continuum emission toward the
central ionizing source due to self-shielding within the cloud.
Inflowing gas with the near side of the BLR illuminated can, in
principle, be distinguished from outflowing gas with the far side
of the BLR illuminated by the VRTF, since the lags for these
two cases are different.
In addition, we allow for a scaling factor to describe the
percentage variability of the emission line compared to that
of the continuum. While for Arp 151 the variability of the
continuum was approximately equal to that of the Hβflux,
in the case of Mrk 50 we find that the continuum variability is
less than that of the line. This is consistent with the amplitude
of variability of the ionizing continuum responsible for Hβ
being larger than that of the V-band (Meusinger et al. 2011 and
references therein).
Once the dynamical model has been defined, we are able to
compute simulated data that are then blurred with a Gaussian
kernel to model the instrumental resolution. The simulated data
are then compared with the actual data. For the likelihood
function, we use the standard Gaussian assumption:
P(data|model) ∝exp −1
2χ2(model,data).(9)
With the likelihood function defined, the modeling problem
is reduced to computing the inferences on all of the model pa-
rameters. The likelihood function, P(data|model), is combined
with the prior distribution for the parameters using Bayes’ theo-
rem: P(model|data) ∝P(model) ×P(data|model). The poste-
rior probability distribution for the parameters is sampled using
the Diffusive Nested Sampling algorithm (Brewer et al. 2011a).
Nested Sampling algorithms initially sample the prior distri-
bution, and subsequently create and sample more constrained
distributions, climbing higher in likelihood. In the specific case
of Diffusive Nested Sampling, uphill and downhill moves are
allowed, allowing the exploring particles to return to the prior,
take large steps, and then climb the likelihood function again.
We assigned uniform priors to most parameters except for the
mean radius and MBH, which have log uniform priors to de-
scribe initial uncertainty about the order of magnitude of the
parameter.
By computational necessity, our model is relatively simple.
While it is still rather flexible and can reproduce the large-scale
features of the reverberation mapping data, it is unable to model
every detail of the Hβlight curve. The large-scale features of the
variability in the Hβlight curve are well modeled, for example,
but the small epoch-to-epoch fluctuations in the light curve are
not (see Figure 2). In addition, the error bars reported on the
data are very small, and our model is not able to fit the data set
to within these small error bars (i.e., we cannot achieve reduced
χ2∼1). If we did not take this into account our uncertainties
would be unrealistically small. This issue is a generic feature of
the fitting of simply parameterized models to informative data
sets and will be discussed in depth in a forthcoming contribution
4
The Astrophysical Journal, 754:49 (8pp), 2012 July 20 Pancoast et al.
Figure 4. Inferred posterior PDFs for model parameters, including MBH, inclination angle (0 deg is face-on), and opening angle of the BLR disk. Joint posterior PDFs
are also shown to illustrate the major degeneracies.
(A color version of this figure is available in the online journal.)
(B. J. Brewer et al., in preparation). In order to account for this
effect and to obtain realistic and conservative uncertainties, we
explore the effect of inflating the error bars on the spectrum data
by a factor H, or equivalently, choosing to form the posterior
distribution from different chunks of the Nested Sampling run
(i.e., different ranges of allowed likelihood values). For each
value of Htested, we inspect the posterior distribution over
simulated data (top right panel in Figure 2) to ensure that the
major features of the data are reproduced. We find that, as long
as His low enough that the models fit the major features in the
data, the resulting posterior distributions on the parameters are
insensitive to the exact choice of the value for H.
4. RESULTS AND CONCLUSIONS
Our inferred geometry and dynamics parameters of the BLR
in Mrk 50 are shown in Figures 4and 5. The shape of the
BLR gas radial profile is constrained to be closer to exponential
(α1), with a mean radius of μ=9.6+1.2
−0.9light days and a
width of σr=6.9+1.2
−1.1light days (the uncertainties quoted are
symmetric 68% confidence limits). Even though the mean radius
is not simply ctimes the mean lag in the general asymmetric
case, we expect our mean radius to roughly correspond to the
lag measurements using cross-correlation analysis by Barth et al.
(2011b), which are τpeak =9.75+0.50
−1.00 and τcen =10.64+0.82
−0.93 light
days. Our mean radius agrees more closely with τpeak, although
τcen is more commonly used for black hole mass estimation. We
infer the inner radius of the BLR distribution to be Fμ =2.0+1.3
−1.1
light days. The opening angle of the BLR disk, defined between
0 and 90 deg, is 25 ±10 deg, closer to a thin disk than to
a sphere. The inclination angle of the thick BLR disk with
respect to the line of sight is constrained to be 9+7
−5deg, closer to
face-on, consistent with the standard model of broad-line AGNs
(Antonucci 1993; Urry & Padovani 1995).
The dynamical modeling results constrain Mrk 50 to have
39% probability of net inflowing gas and 61% probability of net
outflowing gas, with equal amounts of inflowing and outflowing
gas ruled out (inflow fraction =0.5), as shown in Figure 5.This
result suggests only a slight preference for outflow while the
need for either outflow or inflow is quite robust, suggesting
that a more physical model for inflow and outflow is needed in
order to distinguish between them for the case of Mrk 50. Equal
amounts of inflowing and outflowing gas are ruled out because
net inflowing or outflowing gas, along with the illumination
model, creates the asymmetry in the Hβline profile observed in
the data.
In addition to constraining the geometry of the BLR, our dy-
namical model also places an independent estimate on MBH,
inferred to be log10(MBH /M)=7.57+0.44
−0.27. Part of the un-
certainty in this estimate comes from the range in possible MBH
5
The Astrophysical Journal, 754:49 (8pp), 2012 July 20 Pancoast et al.
Figure 5. Inferred posterior PDFs for model parameters, including the mean
radius of the BLR, radial width of the BLR, and the inflow fraction of BLR gas.
(A color version of this figure is available in the online journal.)
values at nearly face-on inclinations (close to 0 deg), as shown in
Figure 4. Recent cross-correlation reverberation mapping results
quote statistical uncertainties of the order of 0.15 dex (Bentz
et al. 2009; Denney et al. 2010; Barth et al. 2011a,2011b),
but this neglects the uncertainty in the normalization factor, f,
that is believed to have an object to object scatter of 0.44 dex
(Woo et al. 2010; Greene et al. 2010a). Thus, our uncertainty in
MBH for Mrk 50 is smaller than that achieved by traditional
reverberation mapping estimates. Our independent measure-
ment of MBH can be used to estimate the appropriate value
of ffor Mrk 50 by comparing it to the virial estimate by Barth
et al. (2011b), Mvir =fv
2cτ/G, where τand vare obtained
from the cross-correlation of the continuum and broad line light
curves and from the width of the broad line, respectively. We
find log10 f=0.78+0.44
−0.27, which agrees to within the errors with
the commonly used mean values of log10 f=0.74+0.12
−0.17 from
Onken et al. (2004), log10 f=0.72+0.09
−0.10 from Woo et al.
(2010), and log10 f=0.45+0.09
−0.09 from Graham et al. (2011).
We have used fto denote a normalization factor derived from
large samples of reverberation mapped AGN MBH estimates as
distinct from the fvalue we measure individually for Mrk 50. A
sample of 10 independent black hole mass and fmeasurements
with comparable uncertainties to Mrk 50 and Arp 151 would
allow us to calculate a mean fvalue to ∼0.3/√10 0.1dex
uncertainty and to distinguish between these commonly used
mean values.
An additional interesting feature of Figure 4is the complex
structure in the joint posterior distribution for the inclination
angle and opening angle, a feature that was not seen in Arp
151. The joint posterior appears to have two distinct families of
solutions, although one has almost four times as much weight as
the other. In an attempt to understand the origin of this structure,
we separated the posterior samples in the two modes in order to
test whether they are correlated with any other parameters (such
as the inflow fraction); however, we were unable to find any such
correlations. Future improvements to the flexibility and realism
of the model may enable us to rule out one of these modes, and
hence constrain the parameters more tightly and further reduce
the uncertainties.
While MBH is well constrained, there are many ways to
successfully model the large-scale structure of the reverberation
mapping data. This is illustrated by the degeneracies in the
posterior distributions plotted in Figure 4. The quality of the
model fits to the data are illustrated in Figure 2, including
six model integrated Hβflux light curves, an example of a
model data set of spectra for each epoch in the light curve,
and two data spectra with the model spectra overplotted. The
smoothness of the models compared to the data is illustrated
in the spectral data sets of the data and model shown in the
top panels of Figure 2.TheMrk50Hβspectral profile did not
change in shape drastically over the course of the LAMP 2011
reverberation mapping campaign, and the model spectral profile
is likewise very similar for all epochs. Even though the shape of
the individual spectral profiles can be well modeled, more
sophisticated models will be required to match the detail of
the small-scale variability of the integrated Hβdata light curve.
Note that the uncertainties quoted throughout this paper
are determined from a Monte Carlo method and are therefore
subject to error themselves. As we are interested in reducing the
uncertainties on black hole mass estimates from reverberation
mapping data, it is important to quantify the uncertainty on our
estimate of the uncertainty on black hole mass. To investigate
this, we estimated the effective number of independent samples
produced by our Diffusive Nested Sampling runs by counting
the number of times the exploring particles returned to the prior
(allowing large steps to be taken) before climbing the likelihood
peak again. Our effective number of independent samples was
found to be ∼180. We then generated samples of size 180
from our full posterior sample and determined the scatter in
the resulting log10(MBH ) uncertainties to be 0.02. Thus, the
uncertainty on the black hole mass for Mrk50 is +0.44
−0.27 ±0.02
dex.
Previous attempts to understand the geometry and dynam-
ics of the BLR have focused on reconstructing the VRTF
(Kollatschny & Bischoff 2002; Bentz et al. 2010; Denney et al.
2010). In the interests of comparing future transfer function
studies to our physically motivated model of the BLR, we show
three inferred VRTFs for Mrk 50 in Figure 6. These three trans-
fer functions were chosen out of the many inferred possible
models for Mrk 50 to show some of the variety in allowed trans-
fer function shapes. The top left VRTF has a fairly typical shape
and level of asymmetry, while the top right VRTF is more asym-
metric than average. One measurement of the VRTF asymmetry
is to compare the integral of the mean lag per velocity bin on ei-
ther side of line center, corresponding to the zero velocity point
in the middle right panel of Figure 6. By this measurement of
asymmetry, 43% of the possible models inferred for Mrk 50
have VRTFs that are less asymmetric than the top left VRTF,
while only 8% of the possible models have VRTFs that are more
6
The Astrophysical Journal, 754:49 (8pp), 2012 July 20 Pancoast et al.
Figure 6. Examples of acceptable transfer functions for Mrk 50. The top two and
middle left panels show examples of VRTFs drawn from the model parameter
posterior PDFs, illustrating the range in inferred transfer function shape. In the
color code of the VRTFs, red corresponds to the highest levels of response and
dark blue corresponds to the lowest levels. The middle right panel shows the
mean lag for each of the VRTFs. The mean lag in seven velocity bins from
Barth et al. (2011b) is shown by red error bars, which were measured by cross-
correlation analysis. We calculate the mean lag in the seven velocity bins of
Barth et al. (2011b)for∼200 VRTFs made using model parameters drawn
randomly from their posterior PDFs, shown in light blue. The bottom panel
shows the velocity-integrated transfer functions for the VRTFs shown in the
first three panels.
(A color version of this figure is available in the online journal.)
asymmetric than the top right VRTF. The middle left transfer
function illustrates the extent to which our inferred model for
Mrk 50 can agree with the velocity-resolved cross-correlation
measurements by Barth et al. (2011b), shown by red error bars in
the middle right panel of Figure 6. This VRTF has the smallest
χ2distance from the cross-correlation measurements by Barth
et al. (2011b) and models with this level of agreement (or better)
have a probability of ∼0.3%.
The average shape of the VRTF is also shown in Figure 6,
with the same velocity bins as used by Barth et al. (2011b)for
their cross-correlation based measurement. This average VRTF
is fairly symmetric, but the higher velocity bins have larger error
bars as a result of averaging over transfer functions that have
asymmetries from either net inflowing or outflowing gas (see
the dashed line in the middle right panel of Figure 6).
Note that the average VRTF we infer and the results obtained
from cross-correlation measurements by Barth et al. (2011b)
do not all agree to within the 1σerror bars. In order to
understand the differences between the time-lags as measured
in our dynamical model and those measured through the cross-
correlation procedure, we consider the ideal continuous noise-
free case. In this case, the cross-correlation function (CCF)
between the line and continuum light curves is the transfer
function convolved with the autocorrelation function (ACF) of
the continuum light curve, which is the CCF of the light curve
with itself. While the ACF is symmetric, the transfer function
may be asymmetric, as we find for Mrk 50, so the CCF may
also be asymmetric. One measurement of the cross-correlation
time-lag often used to measure black hole mass is the CCF-
weighted mean lag, τcen, which is by definition affected by the
asymmetry in the CCF. Therefore, in the case of asymmetric
transfer functions, τcen may not correspond to the mean lag
of our dynamical model of the BLR. For the non-ideal case,
a direct comparison between cross-correlation measurements
and the results of our dynamical modeling approach is not
straightforward, since the peak (or mean) of the CCF does
not measure the true mean lag but only a noisy version of the
convolution between the ACF and the transfer function.
We explored this issue by running the cross-correlation
technique as implemented by Barth et al. (2011b) on light
curves generated by models drawn from the posterior prob-
ability distribution function (PDF) for Mrk 50. For sim-
plicity we considered noise-free light curves sampled in
the same way as our data. We find that the peak and
CCF-weighted mean (τpeak and τcen) of the CCF can be sys-
tematically off by ∼1–2 light days with respect to the true mean
lag of the model. The amount of the offset varies as a function
of the actual shape of the transfer function as well as the details
of the implementation of the cross-correlation algorithm. Thus,
it is not surprising that we find systematic differences of this
order between our estimates of the mean lag and τcen. Clarifying
and quantifying systematically the relationship between these
two approaches as a function of BLR structure and data quality
is an important topic that goes beyond the scope of this paper
and is left for future work.
In conclusion, the analysis presented here provides new
and unique insights into the geometry and kinematics of the
BLR, and an MBH estimate that is competitive with the most
accurate methods. However, since our modeling uncertainties
are greater than data uncertainties, more physical models that
take into account the complex processes occurring in the BLR
should allow for even better constraints. In the future, we plan
to develop such models and apply them to large samples of
reverberation mapping data.
We thank the Lick Observatory staff for their exceptional
support during our observing campaign. In addition, we thank
Brandon Kelly for suggesting changes to our code that yielded
significant improvements. The referee also provided valuable
feedback that enabled us to improve the paper. The Lick AGN
Monitoring Project 2011 is supported by NSF grants AST-
1107812, 1107865, 1108665, and 1108835. The West Moun-
tain Observatory is supported by NSF grant AST-0618209. A.P.
acknowledges support from the NSF through the Graduate Re-
search Fellowship Program. B.J.B. and T.T. acknowledge sup-
port from the Packard Foundation through a Packard Fellowship
to T.T. A.D. acknowledges support from the Southern Califor-
nia Center for Galaxy Evolution, a multi-campus research pro-
gram funded by the University of California Office of Research.
A.V.F. and his group at UC Berkeley acknowledge generous
financial assistance from Gary & Cynthia Bengier, the Richard
& Rhoda Goldman Fund, NASA/Swift grants NNX10AI21G
and GO-7100028, the TABASGO Foundation, and NSF
grant AST-0908886. S.H. acknowledges support by Deutsche
Forschungsgemeinschaft (DFG) in the framework of a research
7
The Astrophysical Journal, 754:49 (8pp), 2012 July 20 Pancoast et al.
fellowship (“Auslandsstipendium”). The work of D.S. and R.A.
was carried out at the Jet Propulsion Laboratory, California
Institute of Technology, under a contract with NASA. J.H.W.
acknowledges support by Basic Science Research Program
through the National Research Foundation of Korea funded
by the Ministry of Education, Science and Technology (2010-
0021558).
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