tutorial 3 and 5 modified

tuturials/gw_tutorials.tex

@@ -102,6 +102,9 @@ and that the corresponding Ricci tensor can be written as
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 \emph{Hint:} Use geodesic deviation equation in the proper frame at the centre of the circle. For a GW propagating in the z direction, we can write $$\frac{D^2n^j}{D\tau^2} + R_{0j0k}n^k=0$$ where $n^j = x_B^j-x_A^j$ and use $h_{jk}^{TT} = \mathrm{Re} [A_0 e^{i(kz-\omega t)}{\mathrm{e}_{\boldmath{p}}}_{jk}]$, $\mathrm{e}_{\boldmath{p}}$ is the polarization tensor. Initially $x_B^j = x_B^j(0)$, $h_{ij} = 0$.
 \emph{Hint:} Use geodesic deviation equation in the proper frame at the centre of the circle. For a GW propagating in the z direction, we can write $$\frac{D^2n^j}{D\tau^2} + R_{0j0k}n^k=0$$ where $n^j = x_B^j-x_A^j$ and use $h_{jk}^{TT} = \mathrm{Re} [A_0 e^{i(kz-\omega t)}{\mathrm{e}_{\boldmath{p}}}_{jk}]$, $\mathrm{e}_{\boldmath{p}}$ is the polarization tensor. Initially $x_B^j = x_B^j(0)$, $h_{ij} = 0$.
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+\emph{N.B:} You need derive the equation of the curve (i.e ellipse or rotated ellipse if it is rotated ellipse what is the rotation etc.) on which the particles will lie. Plot that curve at different times. [Using a python script or mathematica]
 
 
 \item For GWs at large distances from the source ($T_{\mu\nu} = 0$) show that 
 \item For GWs at large distances from the source ($T_{\mu\nu} = 0$) show that 
 \begin{equation}
 \begin{equation}
@@ -112,7 +115,7 @@ Also, show that constructed energy-momentum tensor $t_{\mu\nu}$ of GWs is invari
 \item Derive that momentum transported (per unit time) by outward propagating GW at a large distance $r$ from the source, over a solid angle $d\Omega$ is 
 \item Derive that momentum transported (per unit time) by outward propagating GW at a large distance $r$ from the source, over a solid angle $d\Omega$ is 
 $$\frac{dP_k}{dt} = -\frac{c^3}{32\pi G} r^2\int d\Omega \left<\dot{h}^{TT}_{ij}\partial^k h^{TT}_{IJ}\right>.$$
 $$\frac{dP_k}{dt} = -\frac{c^3}{32\pi G} r^2\int d\Omega \left<\dot{h}^{TT}_{ij}\partial^k h^{TT}_{IJ}\right>.$$
 
 
-\item Estimate the total energy energy radiated by the binary black hole merger that produced the  GW event GW150914, assuming that the distance to the source is $r \simeq 400\;\text{Mpc}$ and assuming isotropic emission of GWs. \emph{Hint:} Download the whitened data $h(t)$ from the LIGO Hanford detector at the time of the GW150914 event (you may use \href{https://www.gw-openscience.org/s/events/GW150914/GW150914_tutorial.html}{this} notebook). Compute the Fourier transform $\tilde{h}(f)$ using \texttt{numpy.fft}.  The radiated energy is: 
+\item Estimate the total energy energy radiated by the binary black hole merger that produced the  GW event GW150914, assuming that the distance to the source is $r \simeq 400\;\text{Mpc}$ and assuming isotropic emission of GWs. \emph{Hint:} Download the strain data $h(t)$ from the LIGO Hanford detector at the time of the GW150914 event (you may use \href{https://www.gw-openscience.org/s/events/GW150914/GW150914_tutorial.html}{this} notebook). Compute the Fourier transform $\tilde{h}(f)$ using \texttt{numpy.fft}.  The radiated energy is: 
 \begin{equation}
 \begin{equation}
 E = \int_0^\infty \frac{dE}{df} \simeq \frac{4 \pi r^2 c^3}{2G} \int_{10 \mathrm{Hz}}^{300 \mathrm{Hz}} f^2 \tilde{h}(f)^2 df.  
 E = \int_0^\infty \frac{dE}{df} \simeq \frac{4 \pi r^2 c^3}{2G} \int_{10 \mathrm{Hz}}^{300 \mathrm{Hz}} f^2 \tilde{h}(f)^2 df.  
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@@ -173,6 +176,7 @@ A_{\mu\nu} \equiv A \, e_{\mu\nu}, ~~ e^{\mu\nu} e_{\mu\nu}^* = 1, ~~ k_\nu \equ
 	\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 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 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.
 \end{enumerate}
 \end{enumerate}