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\documentclass[11pt,letterpaper]{article} \documentclass[11pt,letterpaper,fleqn]{article}
\usepackage{natbib} \usepackage{natbib}
%\usepackage{cite}
\usepackage{graphicx} \usepackage{graphicx}
\usepackage[margin=1.in,centering]{geometry} \usepackage[margin=1.in,centering]{geometry}
\usepackage{hyperref}
\begin{document} \begin{document}
Consider two lightcurves $x(t)$ and $y(t)$, where $x(t)$ is the driving lightcurve and $y(t)$ is the reprocessed lightcurve. If they are related by a linear impulse response, $g(\tau)$, then: \title{Optical/UV Band
Reverberation Mapping of NGC 5548 with Frequency-Resolved Techniques}
\author[Ulrich et al.]{
Otho A. Ulrich,$^{2}$\thanks{E-mail: otho.a.ulrich@wmich.edu}
Edward M. Cackett,$^{1}$
\\
% List of institutions
$^{1}$Department of Physics and Astronomy, Wayne State University, 666 W.
Hancock St., Detroit, MI 48201, USA\\
$^{2}$Department of Physics, Western Michigan University, Kalamazoo, MI
49008-5252, USA\\
}
\date{August 8, 2016}
\maketitle
\begin{abstract} \begin{abstract}
Power spectral densities and time delays of 19 wavelength bands are recovered Power spectral densities and time delays of 19 wavelength bands are recovered
as part of a reverberation mapping of NGC 5548. The latest time-variable light as part of a reverberation mapping of NGC 5548. The latest time-variable light
curves are made available in STORM III by \cite{2016ApJ...821...56F}. The curves are made available in STORM III by \citet{2016ApJ...821...56F}. The
uneven distribution of flux data in those curves necessitates the use of a uneven distribution of flux data in those curves necessitates the use of a
maximum likelihood method in conjunction with Fourier transformations to maximum likelihood method in conjunction with Fourier transformations to
produce the frequency-dependent values of interest. Variability in the produce the frequency-dependent values of interest. Variability in the
@ -22,34 +39,36 @@ wavelength dependence. There are issues computing accurate error estimates for
both distributions that remain as yet unresolved. The transfer function should both distributions that remain as yet unresolved. The transfer function should
be recoverable once those and any additional computational issues are be recoverable once those and any additional computational issues are
resolved. resolved.
\end{abstract} \end{abstract}
\section{Introduction} \section{Introduction}
The local Type-I Seyfert galaxy NGC 5548, while perhaps the best-studied The local Type-I Seyfert galaxy NGC 5548, while perhaps the best-studied
active galaxy, remains an object of intense interest and study to modern active galaxy, remains an object of intense interest and study to modern
astronomy. An extensive observational campaign has been carried out on this astronomy. Direct observation of active galactic nuclei (AGN) such as that
object, producing the most complete set of time-dependent light curves yet thought to exist at the center of NGC 5548 is rarely possible. The astronomer
collected from an active galactic nucleus (AGN). The physics underlying the may infer the properties of AGN from the dynamics of their variable spectra.
nature of these light curves is not completely understood, and so remains a \citet{2016ApJ...821...56F} published the most complete set of time-dependent
topic of debate and great interest. light curves yet collected from an active galactic nucleus as part III of
STORM, an extensive optical/UV observational campaign carried out on NGC 5548.
\subsection{Reverberation Mapping} \subsection{Reverberation Mapping}
A primary model of AGN suggests that an accretion disk is incident upon a A primary model of AGN suggests that an accretion disk is incident upon a
central super-massive black hole (SMBH). Electromagnetic emission emergent central super-massive black hole (SMBH). Electromagnetic emission emergent
from the accreting gases close to the SMBH is reprocessed by the from a corona surrounding the SMBH is reprocessed by the
surrounding gas clouds, resulting in observed response delays between surrounding gas clouds, resulting in observed response delays between
emission peaks that are dependent on the geometry of the system. The emission peaks that are dependent on the geometry of the system. The
impulse response encodes this geometry, and astronomers have combined impulse response encodes this geometry and other interactions, so recovering it from observed emission allows astronomers to probe the properties of
and astronomers have combined
models for the orbiting gas velocities and ionization states with these models for the orbiting gas velocities and ionization states with these
observed time delays to calculate it for some known systems. This observed time delays to calculate it for some known systems. This
technique has become a standard for calculating the black hole mass of technique has become a standard for calculating the black hole mass of
AGN, and is well-described by \cite{2007MNRAS.380..669C} and AGN, and is well-described by \citet{2007MNRAS.380..669C} and
\cite{2014A&ARv..22...72U}. It continues to be refined, and may also \citet{2014A&ARv..22...72U}. It continues to be refined, and may also
become a tool to measure the black hole spin of these systems become a tool to measure the black hole spin of these systems
\citep{2016arXiv160606736K}. \citet{2016arXiv160606736K}.
(Probably would be good to put a picture here describing simple (Probably would be good to put a picture here describing simple
reverberation.) reverberation.)
@ -76,7 +95,7 @@ topic of debate and great interest.
toward toward
constituting the transfer function of a system. Very good explanations of constituting the transfer function of a system. Very good explanations of
these techniques and the associated mathematics are available from these techniques and the associated mathematics are available from
\cite{2014A&ARv..22...72U}. \citet{2014A&ARv..22...72U}.
A top-hat function provides a simple model of the impulse response of a A top-hat function provides a simple model of the impulse response of a
delayed light curve. A fast Fourier transform method of this impulse delayed light curve. A fast Fourier transform method of this impulse
@ -92,12 +111,12 @@ topic of debate and great interest.
\subsection{Unevenly-Spaced Data} \subsection{Unevenly-Spaced Data}
Some X-ray datasets contain gaps due to orbital mechanics, which motivated Some X-ray datasets contain gaps due to orbital mechanics, which motivated
the work in \cite{2013ApJ...777...24Z}, where a maximum likelihood method the work in \citet{2013ApJ...777...24Z}, where a maximum likelihood method
is used to perform Fourier analysis on light curves with gaps. Since its is used to perform Fourier analysis on light curves with gaps. Since its
development, this technique has found success among studies of development, this technique has found success among studies of
observations captured by low-orbit X-ray telescopes that exceed the observations captured by low-orbit X-ray telescopes that exceed the
telescopes' orbital periods, such as the analysis performed by telescopes' orbital periods, such as the analysis performed by
\cite{2016arXiv160606736K}. Until now, reverberation mapping in the \citet{2016arXiv160606736K}. Until now, reverberation mapping in the
optical bands has been limited to time-domain techniques. Many datasets optical bands has been limited to time-domain techniques. Many datasets
available for these bands have uneven sampling across the time domain, available for these bands have uneven sampling across the time domain,
however, and so do not lend themselves well to time-domain or traditional however, and so do not lend themselves well to time-domain or traditional
@ -115,7 +134,7 @@ each band in the dataset -- 18 bands not including the reference band.
The light curves analysed here are unevenly distributed along the time axis, The light curves analysed here are unevenly distributed along the time axis,
which suggests that the maximum likelihood method developed by which suggests that the maximum likelihood method developed by
\cite{2013ApJ...777...24Z} is a reasonable candidate for producing the PSD and \citet{2013ApJ...777...24Z} is a reasonable candidate for producing the PSD and
time delays in the frequency domain. The latest version (CHECK THIS) of the time delays in the frequency domain. The latest version (CHECK THIS) of the
C++ program psdlag associated with that work is used to directly produce the C++ program psdlag associated with that work is used to directly produce the
PSD and cross spectra. The time delay spectrum is produced from the cross PSD and cross spectra. The time delay spectrum is produced from the cross
@ -123,7 +142,7 @@ spectrum by dividing it by $2 \pi f$, with $f$ the mean frequency for a given
bin. bin.
\subsection{Dataset} \subsection{Dataset}
\cite{2016ApJ...821...56F} published the best dynamic data yet collected \citet{2016ApJ...821...56F} published the best dynamic data yet collected
from NGC 5548 over a 200-day (CHECK THIS) period, for 19 bands throughout from NGC 5548 over a 200-day (CHECK THIS) period, for 19 bands throughout
the optical and into the UV spectra. These data were collected from a the optical and into the UV spectra. These data were collected from a
variety of observatories, including both space and ground-based variety of observatories, including both space and ground-based
@ -138,10 +157,11 @@ bin.
limit of the true variability. Scanning the likelihood function can limit of the true variability. Scanning the likelihood function can
provide better error estimates at the cost of computation time, as can provide better error estimates at the cost of computation time, as can
running Monte Carlo simulations. All of these methods are built into the running Monte Carlo simulations. All of these methods are built into the
psdlag program provided by \cite{2013ApJ...777...24Z}, however, some psdlag program provided by \citet{2013ApJ...777...24Z}, however, some
issues have prevented proper error analysis using the latter two methods. issues have prevented proper error analysis using the latter two methods.
This is discussed in more detail in section \ref{results}. This is discussed in more detail in section \ref{results}.
\section{Results} \section{Results}
\label{results} \label{results}
@ -189,11 +209,19 @@ and time delays. With reverberation mapping, the goal is to recover the transfer
function, which encodes the geometry of the system. Recovering the time delays function, which encodes the geometry of the system. Recovering the time delays
is a significant step toward that goal. is a significant step toward that goal.
%\bsp
\bibliographystyle{plainnat}
\bibliography{wsu_reu}
Consider two lightcurves $x(t)$ and $y(t)$, where $x(t)$ is the driving lightcurve and $y(t)$ is the reprocessed lightcurve. If they are related by a linear impulse response, $g(\tau)$, then:
@ -229,4 +257,10 @@ C(\nu) = X^*(\nu) G(\nu) X(\nu) = G(\nu) |X(\nu)|^2
\end{equation} \end{equation}
thus, for a given impulse response function, one can trivially predict the time lags as a function of frequency, $\tau(\nu)$, by calculating the phase of $G(\nu)$, and the frequency dependence of the lags directly relates to the shape of the response function. thus, for a given impulse response function, one can trivially predict the time lags as a function of frequency, $\tau(\nu)$, by calculating the phase of $G(\nu)$, and the frequency dependence of the lags directly relates to the shape of the response function.
\end{document} \end{document}