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converting report to basic letter and working with tophat program to find -1 problem
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\BOOKMARK [2][-]{subsection.1.1}{Reverberation Mapping}{section.1}% 2
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%%%%% AUTHORS - PLACE YOUR OWN COMMANDS HERE %%%%%
|
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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\title[Optical Reverberation with Max Likelihood; NGC 5548]{Optical
|
|
||||||
Reverberation Mapping of NGC 5548 with Maximum Likelihood}
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Edward M. Cackett,$^{1}$
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$^{1}$Department of Physics and Astronomy, Wayne State University, 666 W.
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||||||
Hancock St., Detroit, MI 48201, USA\\
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$^{2}$Department of Physics, Western Michigan University, Kalamazoo, MI
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|
||||||
Power spectral densities and time delays of 19 wavelength bands are recovered
|
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
|
as part of a reverberation mapping of NGC 5548. The latest time-variable light
|
||||||
@ -104,15 +24,8 @@ be recoverable once those and any additional computational issues are
|
|||||||
resolved.
|
resolved.
|
||||||
\end{abstract}
|
\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}
|
\section{Introduction}
|
||||||
The local Type-I Seyfert galaxy NGC 5548, while perhaps the best-studied
|
The local Type-I Seyfert galaxy NGC 5548, while perhaps the best-studied
|
||||||
@ -278,43 +191,42 @@ 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}
|
\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}
|
\end{document}
|
||||||
|
|
||||||
% End of mnras_template.tex
|
|
||||||
@ -63,7 +63,9 @@ foreach (@tophat_list) {
|
|||||||
# This currently multiplies the imaginary component by -1,
|
# This currently multiplies the imaginary component by -1,
|
||||||
# and I really need to figure out why this is necessary for
|
# and I really need to figure out why this is necessary for
|
||||||
# proper output.
|
# proper output.
|
||||||
my $φdiff = atan2(-$V,$U);
|
my $φdiff = atan2($V,$U);
|
||||||
|
|
||||||
|
wcols $f,$φdiff,"analyses/tables/tophat_φdiff"
|
||||||
my $offset = $φdiff/($_2π*$f);
|
my $offset = $φdiff/($_2π*$f);
|
||||||
|
|
||||||
# Output frequency-domain time delay for given tophat
|
# Output frequency-domain time delay for given tophat
|
||||||
|
|||||||
Loading…
Reference in New Issue
Block a user