Tut 5 added

tuturials/gw_tutorials.tex

@@ -161,6 +161,22 @@ A_{\mu\nu} \equiv A \, e_{\mu\nu}, ~~ e^{\mu\nu} e_{\mu\nu}^* = 1, ~~ k_\nu \equ
 
 \end{enumerate}
 
+\subsection{Tutorial 5}
+\begin{itemize}
+	\item In the class we have derived 
+	\begin{equation}\label{hij_quad}
+		\left[h_{ij}^{TT}\right]_{quad} = \frac{1}{r}\frac{2G}{c^4}\Lambda_{ij,kl}(\hat{n})\ddot{M}^{kl}\left(t-\frac{r}{c}\right)
+	\end{equation}
+    where $$M^{ij} = \frac{1}{c^2}\int \rho\left(t,\vec{x}\right) x^i x^j d^3\vec{x}$$
+\end{itemize}
+\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 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.  
+\end{enumerate}
+
+
+
 
 \bibliography{Lab}
 \end{document}