where $$M^{ij}=\frac{1}{c^2}\int\rho\left(t,\vec{x}\right) x^i x^j d^3\vec{x}$$
where $$M^{ij}=\frac{1}{c^2}\int\rho\left(t,\mathbf{x}\right) x^i x^j d^3\mathbf{x}$$
\end{itemize}
\end{itemize}
\begin{enumerate}
\begin{enumerate}
\item\label{p1-5} Consider GW travelling in a generic direction $\hat{n}$ where $n_i =\left(sin\theta cos\phi,sin\theta sin\phi, cos\theta\right)$. Compute the angular distribution for the quadrupole radiation (i.e $h_{+}(\theta,\phi)$ and $h_{\times}(\theta,\phi)$) for given ${M_{ij}}$
\item\label{p1-5} Consider GW travelling in a direction $\hat{\mathbf{n}}$ where $n_i =\left(\sin\theta\cos\phi, \sin\theta\sin\phi, \cos\theta\right)$. Compute the angular distribution for the quadrupole radiation (i.e $h_{+}(\theta,\phi)$ and $h_{\times}(\theta,\phi)$) for given ${M_{ij}}$
\item Consider a binary system of mass $m_1$ and $m_2$, and assume circular motion in relative coordinate (neglect back radiation on motion due to GW). Compute $h_+$ and $h_{\times}$ (use result of \ref{p1-5}), and radiated power. Plot the angular distribution of the radiated power.
\item Consider a binary system of mass $m_1$ and $m_2$, and assume circular motion in relative coordinate (neglect back radiation on motion due to GW). Compute $h_+$ and $h_{\times}$ (use result of \ref{p1-5}), and radiated power. Plot the angular distribution of the radiated power.
\item Consider a mass $m$ is performing simple harmonic motion along $z$ axis. Compute $h_{+}$ and $h_{\times}$ and power radiated. Plot the angular distribution of emitted power.
\item Consider a mass $m$ is performing simple harmonic motion along $z$ axis. Compute $h_{+}$ and $h_{\times}$ and power radiated. Plot the angular distribution of emitted power.
\item Discuss the physical differences in the two radiation pattern obtained in Problem 2 and 3. Discuss the symmetry of the radiation pattern for both cases.
\item Discuss the physical differences in the two radiation pattern obtained in Problem 2 and 3. Discuss the symmetry of the radiation pattern for both cases.
\item link to mathematica notebook \href{http://gitlab.icts.res.in/ajith/gwcourse2023/tree/master/tuturials}{http://gitlab.icts.res.in/ajith/gwcourse2023/tree/master/tuturials}
\item link to mathematica notebook \href{http://gitlab.icts.res.in/ajith/gwcourse2023/tree/master/tuturials}{http://gitlab.icts.res.in/ajith/gwcourse2023/tree/master/tuturials}
\end{enumerate}
\end{enumerate}
\subsection{Tutorial 7}
\subsection{Tutorial 7: Astrophysics of GW sources}
\begin{enumerate}
\begin{enumerate}
\item Assume that all the spin down of pulsar is due to GW emissions, compute the strain amplitude of all pulsars in the \href{https://www.atnf.csiro.au/research/pulsar/psrcat/}{ATNF} catalog, using the $f$ and $\dot{f}$ values from the catalog.
\item Assume that all the spin down of pulsar is due to GW emissions, compute the strain amplitude of all pulsars in the \href{https://www.atnf.csiro.au/research/pulsar/psrcat/}{ATNF} catalog, using the $f$ and $\dot{f}$ values from the catalog.
\item Plot the amplitude of the GW signals as a function of the Fourier frequency for the following continuous sources of GWs.
\item Plot the amplitude of the GW signals as a function of the Fourier frequency for the following continuous sources of GWs.
In the case a known signal $h(t)$ buried in stationary Gaussian noise, the optimal technique for
signal extraction is the \emph{matched filtering}, which is a noise-weighted correlation of the data
with a signal {template}. If the noise is white, this is just a simple cross correlation. The correlation function between two real valued time series $x(t)$ and $\hat h(t)$ is:
Above, $\hat{h(t)} := h(t)/ ||h||$, where the norm $||h||$
of the template is defined by $||h||^2=\int\, |h(t)|^2/\sigma^2\, dt,$ where
$\sigma^2$ is the variance of the noise.
%The optimal signal-to-noise ratio (SNR) is obtained when the template exactly matches with the signal. i.e., $\mathrm{SNR}_\mathrm{opt} = ||h||$. If the SNR is greater than a predetermined threshold (which corresponds to an acceptably small false alarm probability), a detection can be claimed.
\subsubsection{Problems}
\begin{enumerate}
\item You are given a time-series data $d(t)$. This contains a simulated gravitational-wave signal from a black hole binary buried in zero-mean, Gaussian white noise $n(t)$ with standard deviation $\sigma=10^{-21}$. i.e., $d(t)= n(t)+ h(t)$, where $h(t)$ is the leading order PN waveform from a binary with (unknown masses) $m_1$ and $m_2$. Detect the location of the signal in the data and values of $m_1$ and $m_2$ by maximising the correlation of the waveform templates with the data.:
