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scaled to size 8.0pt on input line 89. -LaTeX Font Info: Font shape `OT1/ntxtlf/m/n' will be -(Font) scaled to size 6.2pt on input line 89. -LaTeX Font Info: Try loading font information for TS1+aer on inp \ No newline at end of file diff --git a/press/report/report.out b/press/report/report.out deleted file mode 100644 index 327d0c2..0000000 --- a/press/report/report.out +++ /dev/null @@ -1,9 +0,0 @@ -\BOOKMARK [1][-]{section.1}{Introduction}{}% 1 -\BOOKMARK [2][-]{subsection.1.1}{Reverberation Mapping}{section.1}% 2 -\BOOKMARK [2][-]{subsection.1.2}{Unevenly-Spaced Data}{section.1}% 3 -\BOOKMARK [1][-]{section.2}{Analysis}{}% 4 -\BOOKMARK [2][-]{subsection.2.1}{Dataset}{section.2}% 5 -\BOOKMARK [2][-]{subsection.2.2}{Error Analysis}{section.2}% 6 -\BOOKMARK [1][-]{section.3}{Results}{}% 7 -\BOOKMARK [2][-]{subsection.3.1}{Dubious Error Computations}{section.3}% 8 -\BOOKMARK [1][-]{section.4}{Discussion}{}% 9 diff --git a/press/report/report.tex b/press/report/report.tex index 660ec3d..f8fa776 100644 --- a/press/report/report.tex +++ b/press/report/report.tex @@ -1,320 +1,232 @@ -% mnras_template.tex -% -% LaTeX template for creating an MNRAS paper -% -% v3.0 released 14 May 2015 -% (version numbers match those of mnras.cls) -% -% Copyright (C) Royal Astronomical Society 2015 -% Authors: -% Keith T. Smith (Royal Astronomical Society) - -% Change log -% -% v3.0 May 2015 -% Renamed to match the new package name -% Version number matches mnras.cls -% A few minor tweaks to wording -% v1.0 September 2013 -% Beta testing only - never publicly released -% First version: a simple (ish) template for creating an MNRAS paper - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -% Basic setup. Most papers should leave these options alone. -\documentclass[a4paper,fleqn,usenatbib]{article} - -% MNRAS is set in Times font. If you don't have this installed (most LaTeX -% installations will be fine) or prefer the old Computer Modern fonts, comment -% out the following line -\usepackage{newtxtext,newtxmath} -% Depending on your LaTeX fonts installation, you might get better results -% with one of these: -%\usepackage{mathptmx} -%\usepackage{txfonts} - -% Use vector fonts, so it zooms properly in on-screen viewing software -% Don't change these lines unless you know what you are doing -\usepackage[T1]{fontenc} -\usepackage{ae,aecompl} - - -%%%%% AUTHORS - PLACE YOUR OWN PACKAGES HERE %%%%% - -% Only include extra packages if you really need them. Common packages are: -\usepackage{graphicx} % Including figure files -\usepackage{amsmath} % Advanced maths commands -\usepackage{amssymb} % Extra maths symbols - -\begin{document} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -%%%%% AUTHORS - PLACE YOUR OWN COMMANDS HERE %%%%% - -% Please keep new commands to a minimum, and use \newcommand not \def to avoid -% overwriting existing commands. Example: -%\newcommand{\pcm}{\,cm$^{-2}$} % per cm-squared - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -%%%%%%%%%%%%%%%%%%% 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 -\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} -keyword1 -- keyword2 -- keyword3 -\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 \ No newline at end of file +\documentclass[11pt,letterpaper]{article} + +\usepackage{natbib} +\usepackage{graphicx} +\usepackage[margin=1.in,centering]{geometry} + +\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: + + +\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} + + + + +\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. + + + + + + + + + +\begin{equation} +y(t) = \int_{-\infty}^{\infty} g(\tau) x(t-\tau) {\rm d}\tau +\end{equation} + +So, $y(t)$ is a delayed and blurred version of $x(t)$, with the amount of delay and blurring encoded in $g(\tau)$. + +The power spectral density (PSD) of $x(t)$ is calculated from the Fourier transform of $x(t)$, which we denote $X(\nu)$. The PSD is $|X(\nu)|^2 = X^*(\nu)X(\nu)$, where the $^*$ denotes the complex conjugate. From the convolution theorem of Fourier transforms we can write: + +\begin{equation} +Y(\nu) = G(\nu) X(\nu) +\end{equation} + +This means it is easy to relate the PSD of the reprocessed lightcurve to the PSD of the driving lightcurve and the impulse response function: + +\begin{equation} +|Y(\nu)|^2 = |G(\nu)|^2 |X(\nu)|^2 +\end{equation} + +The cross spectrum is defined as +\begin{equation} +C(\nu) = X^*(\nu) Y(\nu) +\end{equation} +the phase, $\phi$, of which gives the phase lag between X and Y at each Fourier frequency, $\nu$. This can be converted to a time lag through: +\begin{equation} +\tau(\nu) = \frac{\phi(\nu)}{2\pi\nu} +\end{equation} +Since $Y(\nu) = G(\nu) X(\nu)$, the cross spectrum can be written as: +\begin{equation} +C(\nu) = X^*(\nu) G(\nu) X(\nu) = G(\nu) |X(\nu)|^2 +\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. + +\end{document} \ No newline at end of file diff --git a/scripts/tophat_fft.pl b/scripts/tophat_fft.pl index e4c8f76..f3fef57 100755 --- a/scripts/tophat_fft.pl +++ b/scripts/tophat_fft.pl @@ -63,7 +63,9 @@ foreach (@tophat_list) { # This currently multiplies the imaginary component by -1, # and I really need to figure out why this is necessary for # proper output. - my $φdiff = atan2(-$V,$U); + my $φdiff = atan2($V,$U); + + wcols $f,$φdiff,"analyses/tables/tophat_φdiff" my $offset = $φdiff/($_2π*$f); # Output frequency-domain time delay for given tophat