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\begin{document}



\title{Lab 4: Nuclear Reactions and Nucleosynthesis}
\author{Otho Ulrich, Eugene Kopf, Asghar Kayani, Mike Pirkola, }
\maketitle

\begin{abstract}
\end{abstract}

%─────────────
\section{Nuclear Reactions Induced by a Proton Beam}
    \label{sec:beamreactions}


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


%─────────────
\section{Analysis of Reverberating Lightcurves}
    \label{sec:analysis}

    \subsection{Time Delay and Transfer Function Theory}
        \label{subsec:transferfunc}


        \begin{equation}
            \label{eq:time_transfunc}
            y(t) = \int_{-\infty}^{\infty} g(\tau) x(t-\tau)  {\rm d}\tau.
        \end{equation}


        \begin{equation}
            \label{eq:freq_transfunc}
            Y(\nu) = G(\nu) X(\nu).
        \end{equation}


       \begin{equation}
            |Y(\nu)|^2 = |G(\nu)|^2 |X(\nu)|^2.
        \end{equation}

        We also define a cross spectrum between two lightcurves as $C(\nu) = X^*(\nu) Y(\nu)$. The argument $\phi$ of the cross spectrum is the phase between those two signals. The time lag can be computed from the phase using equation \ref{eq:timelag}. An empirical fit of the time delay in frequency space can be produced, and then transformed to the time domain. The average time delay can be obtained from this new function, but this function describes more fully the time-dependent response of the reprocessed lightcurve. Composing the many specific time-dependent responses into a single function $g(t,\lambda)$ provides the desired description of the reveberation in the system.

        \begin{equation}
            \tau(\nu) = \frac{\phi(\nu)}{2\pi\nu}.
            \label{eq:timelag}
        \end{equation}


    \subsection{Computational Details}
        \label{subsec:computation}

        \begin{figure}
            \centering
            \includegraphics[width=0.5\linewidth]{tophat_timedomain.pdf}[A]
            \includegraphics[width=0.5\linewidth]{tophat_freqdomain.pdf}[B]
            \caption{[A] 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. [B] The frequency-dependent time lags associated with each tophat function. Important features related to the average time lag are observable (maximum, corresponding to the average time delay; value of $\nu$ at steepest change), and complicated relationships with higher frequency waves can be noted.}
            \label{fig:tophat_theory}
        \end{figure}





%─────────────
\section{Results}
\label{sec:results}
    \begin{figure}
        \centering
        \includegraphics[width=4in]{psd1367A.pdf}
        \caption{The power spectral density for the chosen reference lightcurve. It is strongly variable across all computed Fourier frequencies.}
        \label{fig:PSD_ref}
    \end{figure}

    \begin{figure}
        \includegraphics[width=.46\textwidth]{psd_atlas.pdf}[A]
        \includegraphics[width=.46\textwidth]{timelag_atlas.pdf}[B]
        \caption{[A] Power spectral densities for all observed light curves, excluding the reference curve. A decrease in variability is observed with increasing wavelength. [B] Time lags as a function of Fourier frequency. Computational issues make many of these results dubious, but ten curves provide at least a lower-limit estimate of the uncertainty. It can nevertheless be noted that time delay increases with wavelength.}
        \label{fig:grids}
    \end{figure}


    \begin{figure}
        \centering
        \includegraphics[width=.45\textwidth]{freq_3465A.png}
        \includegraphics[width=.45\textwidth]{time_3465A.png}[3465\AA]
        \centering
        \includegraphics[width=.45\textwidth]{freq_3471A.png}
        \includegraphics[width=.45\textwidth]{time_3471A.png}[3471\AA]
        \centering
        \includegraphics[width=.45\textwidth]{freq_6175A.png}
        \includegraphics[width=.45\textwidth]{time_6175A.png}[6175\AA]
        \caption{The shorter wavelengths, corresponding to emission from the inner region of the accretion disk, do not show strong agreement with the average time lags reported by Fausnaugh et al. This may be due to difficulty fitting the function because the computer is unable to distinguish between emission from the source and reprocessing.}
        \label{fig:results1}
    \end{figure}
    
    \begin{figure}
        \centering
        \includegraphics[width=.45\textwidth]{freq_6439A.png}
        \includegraphics[width=.45\textwidth]{time_6439A.png}[6439\AA]
        \centering
        \includegraphics[width=.45\textwidth]{freq_7657A.png}
        \includegraphics[width=.45\textwidth]{time_7657A.png}[7657\AA]
        \centering
        \includegraphics[width=.45\textwidth]{freq_9157A.png}
        \includegraphics[width=.45\textwidth]{time_9157A.png}[9157\AA]
        \caption{The empirical fits in these wavelengths show a strong agreement with the average time lags reported by Fausnaugh et al. The strong response trending downward from time 0 may be from the source emission, while the bump is from thermal emission further out in the accretion disk. These components appear to overlap in 6439\AA.}
        \label{fig:results2}
    \end{figure}

\section{Conclusion}
    \label{sec:conclusion}

\printbibliography


\end{document}
