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doing report rewrites
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@ -30,28 +30,33 @@ $^{2}$Department of Physics, Western Michigan University\\
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\begin{abstract}
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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.
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(Should I just remove this?)
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Power spectral densities and time lags 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 emission is confirmed in the power spectral densities, and the time lags show the expected frequency dependence. The time lags 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.
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\end{abstract}
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\section{Introduction}
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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. Direct observation of active galactic nuclei (AGN) such as that thought to exist at the center of NGC 5548 is rarely possible. The astronomer may infer the properties of AGN from the dynamics of their variable spectra. \cite{2016ApJ...821...56F} published the most complete set of time-dependent 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. We now attempt to use frequency-domain analyses to map the reverberation in the observed light curves.
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Active galactic nuclei are powerful objects, both in luminosity and in the imaginations of modern astronomers.
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Seyfert galaxies are thought to have active galactic nuclei (AGN) at their center: very luminous and variable sources of electromagnetic radiation. The variability in these objects do not follow any pattern astronomers have been able to recognize. The Type-I Seyfert galaxy NGC 5548 is one of the most studied Seyfert galaxies, and yet remains an object of intense interest and study to modern astronomy.
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Direct observation of active galactic nuclei (AGN) such as that thought to exist at the center of NGC 5548 is rarely possible. The astronomer may infer the properties of AGN from the dynamics of their variable spectra. \cite{2016ApJ...821...56F} published the most complete set of time-dependent 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. We now attempt to use frequency-domain analyses to map the reverberation in the observed light curves.
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\section{Reverberation Mapping}
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One model for AGN suggests that a hot accretion disk is incident upon a central super-massive black hole (SMBH). Electromagnetic emission emergent from the gas surrounding the SMBH is reprocessed by the disk, resulting in observed response delays between emission peaks. If the temperature of the disk decreases radially from the SMBH, the observed time delays can be expected to increase with decreasing wavelength. Furthermore, as the emissions are reprocessed in the disk, they become blurred, so the variability can be expected to decrease with wavelength. A transfer function encodes the geometry of the system by describing the time-dependent response of each light curve against the others. Recovering the function from the observed light curves is a primary goal of reverberation mapping.
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One model for AGN suggests that a hot accretion disk is incident upon a central super-massive black hole (SMBH). Electromagnetic emission emergent from the gas surrounding the SMBH is reprocessed by the disk, resulting in observed time lags between emission peaks. If the temperature of the disk decreases radially from the SMBH, the time lags can be expected to increase with decreasing wavelength. Furthermore, as the emission is reprocessed in the disk, it becomes blurred, so the variability can be expected to decrease with wavelength. A transfer function encodes the geometry of the system by describing the time-dependent response of each light curve against the others. Recovering the function from the observed light curves is a primary goal of reverberation mapping.
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This technique has become a standard for calculating the black hole mass of AGN. It is well-described by \cite{2007MNRAS.380..669C} and \cite{2014A&ARv..22...72U} and many others. It continues to be refined, and may also become a tool to measure the black hole spin of these systems (\cite{2016arXiv160606736K}).
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\begin{figure}
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\centering
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\includegraphics[width=3.5in]{../img/basic_geometry.png}
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\caption{Simple geometry of reverberation in the accretion disk. Some continuum emissions are reprocessed before escaping toward the observer.}
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\caption{Simple geometry of reverberation in the accretion disk. Some continuum emission is reprocessed before escaping toward the observer.}
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\end{figure}
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\subsection{Frequency-domain Analysis}
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The transfer function $g(t)$ is related to two light curves as being convoluted against the reference light curve (\ref{time_transfunc}). The convolution theorem provides that convolution in the time domain is equivalent to pointwise multiplication in the frequency domain (\ref{freq_transfunc}). The transfer function therefore lends itself well to frequency-domain analyses.
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The transfer function $g(t)$ is related to two light curves as being convolved against the reference light curve (\ref{time_transfunc}). The convolution theorem provides that convolution in the time domain is equivalent to pointwise multiplication in the frequency domain (\ref{freq_transfunc}). The transfer function therefore lends itself well to frequency-domain analyses.
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\begin{equation}
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\label{time_transfunc}
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@ -69,7 +74,7 @@ The local Type-I Seyfert galaxy NGC 5548, while perhaps the best-studied active
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|Y(\nu)|^2 = |G(\nu)|^2 |X(\nu)|^2
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\end{equation}
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Given two bands, a cross spectrum can also be computed. The cross spectrum is defined as $C(\nu) = X^*(\nu) Y(\nu)$. The argument $\phi$ of the cross spectrum is the phase lag between the two signals. The time delay $\tau$ can therefore be computed from the cross spectrum using
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Given two bands, a cross spectrum can also be computed. The cross spectrum is defined as $C(\nu) = X^*(\nu) Y(\nu)$. The argument $\phi$ of the cross spectrum is the phase lag between the two signals. The time lag $\tau$ can therefore be computed from the cross spectrum using
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\begin{equation}
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\tau(\nu) = \frac{\phi(\nu)}{2\pi\nu}
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@ -81,25 +86,25 @@ The local Type-I Seyfert galaxy NGC 5548, while perhaps the best-studied active
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C(\nu) = X^*(\nu) G(\nu) X(\nu) = G(\nu) |X(\nu)|^2
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\end{equation}
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The time delays are therefore trivially predicted as a function of frequency from the cross spectrum. The frequency dependence of these lags in turn relates directly to the transfer function. Very good explanations of these techniques and the associated mathematics are available from \cite{2014A&ARv..22...72U}.
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The time lags are therefore trivially predicted as a function of frequency from the cross spectrum. The frequency dependence of these lags in turn relates directly to the transfer function. Very good explanations of these techniques and the associated mathematics are available from \cite{2014A&ARv..22...72U}.
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\subsection{Tophat Transfer Function}
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A tophat 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 frequency. Once the time delays are extracted from the observational datasets, they might be fitted with a tophat function, however, a more complicated function such as a log-Gaussian function is probably more appropriate.
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A tophat 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 lag spectrum as a function of temporal frequency. This simple model provides a guideline for how the computed time lags are expected to be distributed as a function of frequency. Once the time lags are extracted from the observational datasets, they might be fitted with a tophat function, however, a more complicated function such as a log-Gaussian function is probably more appropriate.
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\begin{figure}
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\centering
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\begin{minipage}{.475\textwidth}
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\centering
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\includegraphics[width=1\linewidth]{../img/tophat_timedomain.pdf}
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\captionof{figure}{Tophat functions in the time domain show an average time delay of the reverberating curve and a constant distribution in time over an interval. An area of unity indicates no loss of signal in the response.}
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\captionof{figure}{Tophat functions in the time domain show an average time lag of the reverberating curve and a constant distribution in time over an interval. An area of unity indicates no loss of signal in the response.}
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\label{fig:th_time}
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\end{minipage}
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\hfill
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\begin{minipage}{.475\textwidth}
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\centering
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\includegraphics[width=1\linewidth]{../img/tophat_freqdomain.pdf}
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\captionof{figure}{The time delays associated with each tophat function. Distinct features related to the average time lag are present (maximum, value of $\nu$ at steepest change), and complicated relationships with higher frequency waves can be noted.}
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\captionof{figure}{The time lags associated with each tophat function. Distinct features related to the average time lag are present (maximum, value of $\nu$ at steepest change), and complicated relationships with higher frequency waves can be noted.}
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\label{fig:th_freq}
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\end{minipage}
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\end{figure}
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@ -113,9 +118,9 @@ The local Type-I Seyfert galaxy NGC 5548, while perhaps the best-studied active
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\section{Analysis}
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\cite{2016ApJ...821...56F} published the best dynamic data yet collected from NGC 5548 over a 260-day period, for 19 bands throughout the optical and into the UV domains. These data were collected from a variety of observatories, including both space and ground-based telescopes, and thus have significantly uneven and variable sampling rates. 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.
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\cite{2016ApJ...821...56F} published the best dynamic data yet collected from NGC 5548 over a 260-day period, for 19 bands throughout the optical and into the UV domains. These data were collected from a variety of observatories, including both space and ground-based telescopes, and thus have significantly uneven and variable sampling rates. 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 lags as a function of temporal frequency are computed for each band in the dataset -- 18 bands not including the reference band.
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In STORM III, a reverberation mapping analysis is performed using cross-correlation to find the average time delay for each wavelength. These results are compared to a number of possible models, however, the average time lag leaves a lot of uncertainty when trying to constrain an appropriate model. More information is contained in the light curves, and a frequency-domain analysis should provide better constraints.
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In STORM III, a reverberation mapping analysis is performed using cross-correlation to find the average time lag for each wavelength. These results are compared to a number of possible models, however, the average time lag leaves a lot of uncertainty when trying to constrain an appropriate model. More information is contained in the light curves, and a frequency-domain analysis should provide better constraints.
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\begin{figure}
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\centering
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@ -135,7 +140,7 @@ In STORM III, a reverberation mapping analysis is performed using cross-correlat
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\end{figure}
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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 analysing these data by producing the PSD and time delays. The latest version as of July, 2016, 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.
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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 analysing these data by producing the PSD and time lags. The latest version as of July, 2016, of the C++ program "psdlag" associated with that work is used to directly produce the PSD and cross spectra. The time lag spectrum is produced from the cross spectrum by dividing it by $2 \pi f$, with $f$ the mean frequency for a given bin.
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\begin{figure}
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\centering
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@ -149,21 +154,23 @@ The light curves analysed here are unevenly distributed along the time axis, whi
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\begin{minipage}{.475\textwidth}
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\centering
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\includegraphics[width=1\linewidth]{../img/timelag_1367Å_≺_7647Å_{σ∊CM}.pdf}
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\captionof{figure}{The time delay computed from the cross spectrum of 7647\AA and 1367\AA.}
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\captionof{figure}{The time lag computed from the cross spectrum of 7647\AA and 1367\AA.}
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\label{fig:timelag_7647}
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\end{minipage}
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\end{figure}
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\subsection{Errors}
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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}.
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The standard errors reported for the power spectral densities and time lags are taken from the covariance matrix. This method assumes that the errors between frequency bins are not correlated, so these values represent a lower limit of the true variance. 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}.
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An error analysis by scanning the likelihood function was attempted, but this analysis was plagued by computational issues that prevented some models from properly optimizing. Monte Carlo simulations were also attempted as a way of estimating the variability of the resultant values. Many errors obtained from this method were larger than the expected accurate values. Therefore, this analysis was also excluded. The ultimate goal is to use one of these more accurate methods, and the errors currently reported should be considered temporary and unreliable.
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An error analysis by scanning the likelihood function was attempted, but some computational issues have yet to be resolved. Monte Carlo simulations were also attempted as a way of estimating the variability of the resultant values. Some errors obtained from this method are larger than the expected accurate values, so this analysis was also excluded. Moving forward, one of these methods will provide more accurate estimates of the variance, but the errors currently reported should be considered only a lower limit.
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The
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\subsection{Results}
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\label{results}
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In figure \ref{psd_atlas}, an atlas of the power spectral densities for all 18 reverberated bands is provided. Figure \ref{timelag_atlas} provides the time delay spectra for each band. The reference band PSD is provided separately. Producing the time delay map is a significant step toward recovering the transfer function.
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In figure \ref{psd_atlas}, an atlas of the power spectral densities for all 18 reverberated bands is provided. Figure \ref{timelag_atlas} provides the time lag spectra for each band. The reference band PSD is provided separately. Producing the time lag map is a significant step toward recovering the transfer function.
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\begin{figure}
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\centering
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@ -175,36 +182,36 @@ The light curves analysed here are unevenly distributed along the time axis, whi
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\begin{figure}
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\centering
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\includegraphics[width=.9\textwidth]{../img/timelag_atlas.pdf}
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\caption{Time delays for all observed light curves relative to 1367\AA. The delay increases with wavelength, as predicted by the accretion disk model. These functions can ultimately be used to reconstruct the transfer function.}
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\caption{Time lags for all observed light curves relative to 1367\AA. The lag increases with wavelength, as predicted by the accretion disk model. These functions can ultimately be used to reconstruct the transfer function.}
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\label{timelag_atlas}
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\end{figure}
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\section{Discussion}
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Frequency-dependent power spectral densities confirm time-dependent variability in the emission strengths for each band. This was apparent from inspection of the time-domain light curves, and is confirmed by the analysis. That behaviour is expected for any active galactic nucleus and has been long-confirmed in NGC 5548. The power spectral densities also show a decrease in variability with increasing wavelength. That behaviour is expected due to the blurring of reverberated emissions reprocessed by the accretion disk, and was one of the primary hypotheses of this reverberation mapping analysis. Better error sampling is preferred, but with the trend as clear as it is, even with low-estimate errors, it appears safe to say that this predicted trend is observed in these data.
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Frequency-dependent power spectral densities confirm time-dependent variability in the emission strengths for each band. This was apparent from inspection of the time-domain light curves, and is confirmed by the analysis. That behaviour is expected for any active galactic nucleus and has been long-confirmed in NGC 5548. The power spectral densities also show a decrease in variability with increasing wavelength. That behaviour is expected due to the blurring of reverberated emission reprocessed by the accretion disk, and was one of the primary hypotheses of this reverberation mapping analysis. Better error sampling is preferred, but with the trend as clear as it is, even with low-estimate errors, it appears safe to say that this predicted trend is observed in these data.
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Analysis of the top-hat impulse response model predicted frequency-dependent time delays, which have been recovered from the light curves in this analysis. The time delays extracted from the light curves do not mimic very closely the saw-tooth character seen in the tophat model, so a more complex model may be a better choice for fitting the time delays and ultimately recovering the transfer function. A log-Gaussian distribution is likely a good function to try next.
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Analysis of the top-hat impulse response model predicted frequency-dependent time lags, which have been recovered from the light curves in this analysis. The time lags extracted from the light curves do not mimic very closely the saw-tooth character seen in the tophat model, so a more complex model may be a better choice for fitting the time lags and ultimately recovering the transfer function. A log-Gaussian distribution is likely a good function to try next.
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The time delays show an increase in overall magnitude as wavelength increases. This was predicted by the assumed geometry of the accretion disk coupled with a decreasing temperature distribution. The error computed for the time delay at 3465\AA is extremely large, but is probably no more suspicious than the values whose errors are extremely small. While, again, better error calculations are preferred, the trend is very strong, and this analysis appears protective of the accretion disk hypothesis.
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The time lags show an increase in overall magnitude as wavelength increases. This was predicted by the assumed geometry of the accretion disk coupled with a decreasing temperature distribution. The error computed for the time lag at 3465\AA is extremely large, but is probably no more suspicious than the values whose errors are extremely small. While, again, better error calculations are preferred, the trend is very strong, and this analysis appears protective of the accretion disk hypothesis.
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\begin{figure}
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\centering
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\begin{minipage}{.475\textwidth}
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\centering
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\centeringQ
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\includegraphics[width=1\linewidth]{../img/tophat_freqdomain.pdf}
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\captionof{figure}{Time delays modeled from tophat impulse responses.}
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\captionof{figure}{Time lags modeled from tophat impulse responses.}
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\label{fig:top_freq_comp}
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\end{minipage}
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\hfill
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\begin{minipage}{.475\textwidth}
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\centering
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\includegraphics[width=1\linewidth]{../img/timelag_1367Å_≺_7647Å_{σ∊CM}.pdf}
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\captionof{figure}{The time delay computed from the cross spectrum of 7647\AA and 1367\AA.}
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\captionof{figure}{The time lag computed from the cross spectrum of 7647\AA and 1367\AA.}
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\label{fig:timelag_comp}
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\end{minipage}
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\end{figure}
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The analytical method developed by \cite{2013ApJ...777...24Z} applies well to the quality of data available for optical reverberation mapping. 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. The transfer function is within the reach of this analysis, and should be recovered in the next few steps. The error computation issues must be remedied so that any conclusions made from this analysis may be judged valid. It is our hope that this mode of analysis will be judged valid so it can be applied to datasets across the landscape of optical reverberation mapping, where consider information awaits discovery.
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The analytical method developed by \cite{2013ApJ...777...24Z} applies well to the quality of data available for optical reverberation mapping. The analyses performed on these data have elucidated clear trends in the PSD and time lags. With reverberation mapping, the goal is to recover the transfer function, which encodes the geometry of the system. Recovering the time lags is a significant step toward that goal. The transfer function is within the reach of this analysis, and should be recovered in the next few steps. The error computation issues must be remedied so that any conclusions made from this analysis may be judged valid. It is our hope that this mode of analysis will be judged valid so it can be applied to datasets across the landscape of optical reverberation mapping, where consider information awaits discovery.
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%\bsp
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\bibliographystyle{plainnat}
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@ -1,43 +0,0 @@
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\documentclass[11pt,letterpaper]{article}
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\usepackage{natbib}
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\usepackage{graphicx}
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\usepackage[margin=1.in,centering]{geometry}
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\begin{document}
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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:
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\begin{equation}
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y(t) = \int_{-\infty}^{\infty} g(\tau) x(t-\tau) {\rm d}\tau
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\end{equation}
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So, $y(t)$ is a delayed and blurred version of $x(t)$, with the amount of delay and blurring encoded in $g(\tau)$.
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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:
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\begin{equation}
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Y(\nu) = G(\nu) X(\nu)
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\end{equation}
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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:
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\begin{equation}
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|Y(\nu)|^2 = |G(\nu)|^2 |X(\nu)|^2
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\end{equation}
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The cross spectrum is defined as
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\begin{equation}
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C(\nu) = X^*(\nu) Y(\nu)
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\end{equation}
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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:
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\begin{equation}
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\tau(\nu) = \frac{\phi(\nu)}{2\pi\nu}
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\end{equation}
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Since $Y(\nu) = G(\nu) X(\nu)$, the cross spectrum can be written as:
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\begin{equation}
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C(\nu) = X^*(\nu) G(\nu) X(\nu) = G(\nu) |X(\nu)|^2
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\end{equation}
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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.
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\end{document}
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@ -63,7 +63,7 @@ foreach (@tophat_list) {
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# This currently multiplies the imaginary component by -1,
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# and I really need to figure out why this is necessary for
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# proper output.
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my $φdiff = atan2($V,$U);
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my $φdiff = atan2(-1*$V,$U);
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wcols $f,$φdiff,"analyses/tables/tophat_φdiff${tophat_count}.tab";
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@ -71,7 +71,6 @@ foreach (@tophat_list) {
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# Output frequency-domain time delay for given tophat
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wcols $f,$offset,"analyses/tables/tophat_fft${tophat_count}.tab";
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}
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sub tophat {
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