Commit 0f1a3f8e by Parameswaran Ajith

minor edit.

parent 04aa6282
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......@@ -49,8 +49,9 @@ P(r) = K \, \rho(r) ^{\gamma}.
We also need to specify initial conditions for the variables $m, P, \Phi$. The following conditions can be used
\begin{equation}
\label{eq:tov_boundary}
m ( r = 0) = 0, ~~~~ P(r = 0) = P_c = P(\rho_c), ~~~~ \Phi(r_\star) = \frac{1}{2} \ln \left(1 - \frac{2Gm_\star}{r_\star c^2} \right)
m ( r = 0) = 0, ~~~~ P(r = 0) = P_c = P(\rho_c), ~~~~ \Phi(r_\star) = \frac{1}{2} \ln \left(1 - \frac{2Gm_\star}{r_\star c^2} \right).
\end{equation}
For the problems in this section, you may use Scipy's high-level interface to various ODE solvers \href{https://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.solve_ivp.html}{\texttt{solve\_ivp}}.
\subsubsection*{Problems:}
\begin{enumerate}
......@@ -91,7 +92,3 @@ where $\eta(t)$ is Gaussian noise with zero mean and unit variance. Plot $x(t)$
\item Using the \href{http://matplotlib.org/api/pyplot_api.html#matplotlib.pyplot.hist}{\texttt{hist}} function, plot the probability distribution $P(x)$ of $x(t)$ at $t = 10, 50, 100$.
\end{enumerate}
\subsection{Two-point boundary value problems: The shooting method}
Solve the TOV equations described in Sec.~\ref{sec:TOV} as a two-point boundary value problem using the Shooting method. Use the Newton-Raphson method for root finding. The boundary conditions are given in Eq.~(\ref{eq:tov_boundary}).
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