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\BOOKMARK [1][-]{section.1}{Introduction}{}% 1
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\BOOKMARK [2][-]{subsection.1.1}{Reverberation Mapping}{section.1}% 2
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\BOOKMARK [2][-]{subsection.1.2}{Unevenly-Spaced Data}{section.1}% 3
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\BOOKMARK [1][-]{section.2}{Analysis}{}% 4
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\BOOKMARK [2][-]{subsection.2.1}{Dataset}{section.2}% 5
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\BOOKMARK [2][-]{subsection.2.2}{Error Analysis}{section.2}% 6
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\BOOKMARK [1][-]{section.3}{Results}{}% 7
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\BOOKMARK [2][-]{subsection.3.1}{Dubious Error Computations}{section.3}% 8
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\BOOKMARK [1][-]{section.4}{Discussion}{}% 9
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% mnras_template.tex
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%
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% LaTeX template for creating an MNRAS paper
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%
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% v3.0 released 14 May 2015
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% (version numbers match those of mnras.cls)
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%
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% Copyright (C) Royal Astronomical Society 2015
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% Authors:
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% Keith T. Smith (Royal Astronomical Society)
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% Change log
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%
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% v3.0 May 2015
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% Renamed to match the new package name
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% Version number matches mnras.cls
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% A few minor tweaks to wording
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% v1.0 September 2013
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% Beta testing only - never publicly released
|
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% First version: a simple (ish) template for creating an MNRAS paper
|
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
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% Basic setup. Most papers should leave these options alone.
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\documentclass[a4paper,fleqn,usenatbib]{article}
|
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% MNRAS is set in Times font. If you don't have this installed (most LaTeX
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% installations will be fine) or prefer the old Computer Modern fonts, comment
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% out the following line
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\usepackage{newtxtext,newtxmath}
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% Depending on your LaTeX fonts installation, you might get better results
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%\usepackage{mathptmx}
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%\usepackage{txfonts}
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% Use vector fonts, so it zooms properly in on-screen viewing software
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% Don't change these lines unless you know what you are doing
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\usepackage[T1]{fontenc}
|
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\usepackage{ae,aecompl}
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%%%%% AUTHORS - PLACE YOUR OWN PACKAGES HERE %%%%%
|
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% Only include extra packages if you really need them. Common packages are:
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\usepackage{graphicx} % Including figure files
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\usepackage{amsmath} % Advanced maths commands
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\usepackage{amssymb} % Extra maths symbols
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\begin{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%% AUTHORS - PLACE YOUR OWN COMMANDS HERE %%%%%
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% Please keep new commands to a minimum, and use \newcommand not \def to avoid
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% overwriting existing commands. Example:
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%\newcommand{\pcm}{\,cm$^{-2}$} % per cm-squared
|
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|
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
%%%%%%%%%%%%%%%%%%% TITLE PAGE %%%%%%%%%%%%%%%%%%%
|
||||
|
||||
% Title of the paper, and the short title which is used in the headers.
|
||||
% Keep the title short and informative.
|
||||
\title[Optical Reverberation with Max Likelihood; NGC 5548]{Optical
|
||||
Reverberation Mapping of NGC 5548 with Maximum Likelihood}
|
||||
|
||||
% The list of authors, and the short list which is used in the headers.
|
||||
% If you need two or more lines of authors, add an extra line using \newauthor
|
||||
\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\\
|
||||
}
|
||||
|
||||
% These dates will be filled out by the publisher
|
||||
%\date{Accepted XXX. Received YYY; in original form ZZZ}
|
||||
\date{August 8, 2016}
|
||||
|
||||
% Enter the current year, for the copyright statements etc.
|
||||
% \pubyear{2016}
|
||||
|
||||
% Don't change these lines
|
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\label{firstpage}
|
||||
%\pagerange{\pageref{firstpage}--\pageref{lastpage}}
|
||||
\maketitle
|
||||
|
||||
% Abstract of the paper
|
||||
\begin{abstract}
|
||||
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
|
||||
curves are made available in STORM III by \cite{2016ApJ...821...56F}. The
|
||||
uneven distribution of flux data in those curves necessitates the use of a
|
||||
maximum likelihood method in conjunction with Fourier transformations to
|
||||
produce the frequency-dependent values of interest. Variability in the
|
||||
emissions is confirmed in the power spectral densities, and the time delays
|
||||
show the expected frequency dependence. The time delays also appear to have
|
||||
wavelength dependence. There are issues computing accurate error estimates for
|
||||
both distributions that remain as yet unresolved. The transfer function should
|
||||
be recoverable once those and any additional computational issues are
|
||||
resolved.
|
||||
\end{abstract}
|
||||
|
||||
% Select between one and six entries from the list of approved keywords.
|
||||
% Don't make up new ones.
|
||||
\begin{keywords}
|
||||
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|
||||
\end{keywords}
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
%%%%%%%%%%%%%%%%% BODY OF PAPER %%%%%%%%%%%%%%%%%%
|
||||
|
||||
\section{Introduction}
|
||||
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
|
||||
astronomy. An extensive observational campaign has been carried out on this
|
||||
object, producing the most complete set of time-dependent light curves yet
|
||||
collected from an active galactic nucleus (AGN). The physics underlying the
|
||||
nature of these light curves is not completely understood, and so remains a
|
||||
topic of debate and great interest.
|
||||
|
||||
\subsection{Reverberation Mapping}
|
||||
A primary model of AGN suggests that an accretion disk is incident upon a
|
||||
central super-massive black hole (SMBH). Electromagnetic emission emergent
|
||||
from the accreting gases close to the SMBH is reprocessed by the
|
||||
surrounding gas clouds, resulting in observed response delays between
|
||||
emission peaks that are dependent on the geometry of the system. The
|
||||
impulse response encodes this geometry, and astronomers have combined
|
||||
models for the orbiting gas velocities and ionization states with these
|
||||
observed time delays to calculate it for some known systems. This
|
||||
technique has become a standard for calculating the black hole mass of
|
||||
AGN, and is well-described by \cite{2007MNRAS.380..669C} and
|
||||
\cite{2014A&ARv..22...72U}. It continues to be refined, and may also
|
||||
become a tool to measure the black hole spin of these systems
|
||||
\citep{2016arXiv160606736K}.
|
||||
|
||||
(Probably would be good to put a picture here describing simple
|
||||
reverberation.)
|
||||
|
||||
Many reverberation mapping techniques involve time-domain analyses, such
|
||||
as cross-correlation. Time-domain techniques have limitations:
|
||||
cross-correlation, for instance, provides only the average time delay
|
||||
between two light curves; they also require data that is evenly-sampled
|
||||
across the time domain. X-ray reverberation mapping in particular has
|
||||
developed a body of techniques based on frequency-domain techniques,
|
||||
primarily Fourier analysis; these techniques still require evenly-sampled
|
||||
data and have been enabled by the relatively good data coverage in X-ray
|
||||
bands. They provide the astronomer with more detailed information about
|
||||
the variability and response delay within the system compared to
|
||||
time-domain techniques.
|
||||
|
||||
The power spectral density (PSD) as a function of temporal frequency for a
|
||||
given emission band can be produced using Fourier transforms, providing a
|
||||
measure of the time-scale of variability in that band. Given two bands,
|
||||
typically a reference or "driving" band and a delayed or "response" band,
|
||||
a cross spectrum can also be constructed. From the complex
|
||||
argument of the cross-correlation function, one can derive the
|
||||
frequency-dependent time delay between those bands; an important step
|
||||
toward
|
||||
constituting the transfer function of a system. Very good explanations of
|
||||
these techniques and the associated mathematics are available from
|
||||
\cite{2014A&ARv..22...72U}.
|
||||
|
||||
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
|
||||
response provides the time delay spectrum as a function of temporal
|
||||
frequency. This simple model provides a guideline for how the computed
|
||||
time delays are expected to be distributed as a function of temporal
|
||||
frequency.
|
||||
|
||||
|
||||
(Side-by-side graphic of top-hat impulse response function and FFT of
|
||||
top-hat giving time delays.)
|
||||
|
||||
\subsection{Unevenly-Spaced Data}
|
||||
|
||||
Some X-ray datasets contain gaps due to orbital mechanics, which motivated
|
||||
the work in \cite{2013ApJ...777...24Z}, where a maximum likelihood method
|
||||
is used to perform Fourier analysis on light curves with gaps. Since its
|
||||
development, this technique has found success among studies of
|
||||
observations captured by low-orbit X-ray telescopes that exceed the
|
||||
telescopes' orbital periods, such as the analysis performed by
|
||||
\cite{2016arXiv160606736K}. Until now, reverberation mapping in the
|
||||
optical bands has been limited to time-domain techniques. Many datasets
|
||||
available for these bands have uneven sampling across the time domain,
|
||||
however, and so do not lend themselves well to time-domain or traditional
|
||||
frequency-domain analyses. The maximum likelihood method is well-suited
|
||||
to extracting useful information from the data available in those
|
||||
datasets.
|
||||
|
||||
|
||||
|
||||
\section{Analysis}
|
||||
The 1367\AA$ $ light curve, obtained from observations made with the Hubble
|
||||
Space Telescope, is chosen as the reference curve. The power spectral
|
||||
densities and time delays as a function of temporal frequency are computed for
|
||||
each band in the dataset -- 18 bands not including the reference band.
|
||||
|
||||
The light curves analysed here are unevenly distributed along the time axis,
|
||||
which suggests that the maximum likelihood method developed by
|
||||
\cite{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
|
||||
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
|
||||
spectrum by dividing it by $2 \pi f$, with $f$ the mean frequency for a given
|
||||
bin.
|
||||
|
||||
\subsection{Dataset}
|
||||
\cite{2016ApJ...821...56F} published the best dynamic data yet collected
|
||||
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
|
||||
variety of observatories, including both space and ground-based
|
||||
telescopes, and thus have significantly variable sampling rates.
|
||||
|
||||
(Include picture of Fausnaugh data here)
|
||||
|
||||
\subsection{Error Analysis}
|
||||
For the presented set of resultant data, the error estimates are extracted
|
||||
from the covariance matrix. This method assumes that the errors between
|
||||
frequency bins are not correlated, so these values only represent a lower
|
||||
limit of the true variability. Scanning the likelihood function 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
|
||||
psdlag program provided by \cite{2013ApJ...777...24Z}, however, some
|
||||
issues have prevented proper error analysis using the latter two methods.
|
||||
This is discussed in more detail in section \ref{results}.
|
||||
|
||||
\section{Results}
|
||||
\label{results}
|
||||
|
||||
An atlas of the power spectral densities as functions
|
||||
of temporal frequency for all 18 delayed bands is provided in this section.
|
||||
One is also provided of the time delay spectra for each band. The reference
|
||||
band PSD is also provided separately. Errors presented in these atlases
|
||||
are obtained from the covariance matrix.
|
||||
|
||||
(Atlas of PSD)
|
||||
|
||||
(Atlas of Time Delays)
|
||||
|
||||
\subsection{Dubious Error Computations}
|
||||
The errors obtained from the covariance matrix are only a lower estimate
|
||||
of the true error. An error analysis by scanning the likelihood function
|
||||
was attempted, but dubious values led to their exclusion from these results.
|
||||
In the case of
|
||||
|
||||
(Example of bad LF error)
|
||||
|
||||
Monte Carlo simulations were also attempted as a way of estimating the
|
||||
variability of the resultant values. Many errors obtained from this method
|
||||
were much larger than the expected accurate values. Therefore, this analysis
|
||||
was also excluded.
|
||||
|
||||
(Example of bad MC error)
|
||||
|
||||
|
||||
\section{Discussion}
|
||||
Frequency-dependent power spectral densities confirm time-dependent variability
|
||||
in the emission strengths for each band. This behaviour is expected for
|
||||
any active galactic nucleus and has been long-confirmed in NGC 5548, so
|
||||
it comes as no surprise to find those results here.
|
||||
|
||||
Analysis of the top-hat impulse response model predicted frequency-dependent
|
||||
time delays, which have been recovered from the light curves in this analysis.
|
||||
Furthermore, the distribution of time delays indicates a wavelength-dependent
|
||||
nature. This warrants further study and analysis.
|
||||
|
||||
(Maybe a graph comparing the top-hat time delays to one band's time delays.)
|
||||
|
||||
The analyses performed on these data have elucidated clear trends in the PSD
|
||||
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
|
||||
is a significant step toward that goal.
|
||||
|
||||
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%% REFERENCES %%%%%%%%%%%%%%%%%%
|
||||
|
||||
% The best way to enter references is to use BibTeX:
|
||||
|
||||
\bibliographystyle{mnras}
|
||||
\bibliography{wsu_reu} % if your bibtex file is called example.bib
|
||||
|
||||
|
||||
% Alternatively you could enter them by hand, like this:
|
||||
% This method is tedious and prone to error if you have lots of references
|
||||
% \begin{thebibliography}{99}
|
||||
% \bibitem[\protect\citeauthoryear{Author}{2012}]{Author2012}
|
||||
% Author A.~N., 2013, Journal of Improbable Astronomy, 1, 1
|
||||
% \bibitem[\protect\citeauthoryear{Others}{2013}]{Others2013}
|
||||
% Others S., 2012, Journal of Interesting Stuff, 17, 198
|
||||
% \end{thebibliography}
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
%%%%%%%%%%%%%%%%% APPENDICES %%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
% \appendix
|
||||
%
|
||||
% \section{Some extra material}
|
||||
%
|
||||
% If you want to present additional material which would interrupt the flow of
|
||||
% the main paper, it can be placed in an Appendix which appears after the list
|
||||
% of references.
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
|
||||
% Don't change these lines
|
||||
\bsp % typesetting comment
|
||||
\label{lastpage}
|
||||
\end{document}
|
||||
|
||||
% End of mnras_template.tex
|
||||
Loading…
Reference in New Issue
Block a user