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\title{Optical/UV Band
Reverberation Mapping of NGC 5548 with Frequency-Resolved Techniques}

\date{August 20, 2016}

\maketitle

\begin{abstract}

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 sampling of the 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. This is the first time frequency-resolved time lags have been measured from UV/optical light curves in AGN. Variability in the emission is confirmed in the power spectral densities, and the time lags show a complex frequency dependence. The total power seen in each PSD decreases with wavelength, and the average time lag increases with wavelength, both in support of thermal reprocessing by an accretion disk. There are computational issues yet to be resolved regarding accurate error estimates, so errors presented here can only be considered a lower limit. The transfer function should be recoverable once those and any additional computational issues are resolved.

\end{abstract}


\section{Introduction}
\label{sec:intro}

Alpha-partcicle spectroscopy is a method for testing and measuring the properties of any alpha emitter, which is a class of radioactive particles that emit alpha particles, a bound collection of two protons and two neutrons. This is one of the primary classes of radiation, along with beta and gamma radiation, and of these the only hadronic form of radiation. An alpha particle is emitted during alpha decay of a nucleus, when the nucleus gives off 2 protons and 2 neutrons. The isotope would thus lose 2 from its atomic number, decaying into an isotope of a new element. The spectrum measured from the radium sample exemplifies this, as the radium decays into daughter elements further down the periodic table. Alpha particles created in this process often have kinetic energy near 5 MeV and are highly ionizing, but with a small penetration depth. They are therefore not considered a dangerous form of radiation unless ingested.





\section{Reverberation Mapping}
\label{sec:reverbmap}

    \subsection{Continuum Reverberation}
    \label{sec:cont_reverb}
    
    \begin{equation}
        \label{eq:timelag}
        \tau\left(\lambda\right) = 
            \left(3.9 \textrm{d}\right) 
            \left(\frac{T_0}{10^4\mathrm{K}}\right)^{4/3}
            \left(\frac{\lambda}{10^4\mathrm{\AA}}\right)^{4/3}
            \left(\frac{X}{4}\right)^{4/3}
    \end{equation

    \begin{figure}
        \centering
        \includegraphics[width=2.5in]{../img/basic_geometry.png}
        \caption{Simple geometry of reverberation in the accretion disk. Some continuum emission is reprocessed before escaping toward the observer.}
        \label{fig:disk_reverb}
    \end{figure}


    \begin{figure}
        \centering
        \begin{minipage}{.475\textwidth}
            \centering
            \includegraphics[width=1\linewidth]{../img/tophat_timedomain.pdf}
            \captionof{figure}{Tophat transfer 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.}
            \label{fig:th_time}
        \end{minipage}
        \hfill
        \begin{minipage}{.475\textwidth}
            \centering
            \includegraphics[width=1\linewidth]{../img/tophat_freqdomain.pdf}
            \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.}
            \label{fig:th_freq}
        \end{minipage}
    \end{figure}



\section{Discussion}

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