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% Homework #1

\title{Astronomy 545: Homework 1}


\begin{document}

$ $

\begin{enumerate}

\item Imagine that we are searching for the following planetary
  systems.  Specify the detection technique you would use for each,
  and discuss the justifcation for your choice.  If more than one
  technique is suitable (and can provide better constraints than one
  technique one its own), feel free to discuss.  Also describe what
  planetary properties can be characterized with your chosen technique(s).
  \begin{enumerate}
    \item An A star at 10 pc distance, with a Jupiter-mass planet with
      a 10-year period.
    \item A G star at 10 pc distance, with a Jupiter-mass planet with
      a 1-year period
    \item A G star at 10 pc distance, with an Earth-mass planet with a
      10-day period.
    \item An M star at 10 pc distance with an Earth-mass planet in the
      Habitable zone.
    \item An F star at 300 pc distance with a Netune-mass planet with
      a 4-year period.
    \end{enumerate}
      
\item Estimate how
  long the Sun could shine if it was powered entirely by chemical reactions.  The
  ionization potential of Hydrogen is 13.6 eV, and thus this is a
  reasonable guess as to how
  much energy could be involved in a given chemical reaction.
  Assuming that every atom in the Sun could release 13.6 eV through
  such a reaction, and that the Sun is made entirely of Hydrogen, how
  long could the Sun shine?

\item At what rate is the mass of the Sun decreasing due to 
nuclear reactions?  Has the mass of the Sun changed significantly
(due to nuclear reactions) over its $\sim 5$ billion years of life?

\item Show that the Hydrostatic Equilibrium equation can be written as 
\begin{equation}
\frac{dP}{d\tau} = \frac{g}{\kappa}.
\end{equation}
Here, $g=GM/r^2$ is the local gravitational acceleration.

\item Show that $P_c < (\pi/6)^{1/3}  GM^{2/3} \rho_c^{4/3}$.

\item Assume a star of mass $M_{\odot}$ and radius $R_{\odot}$ with a
  density profile $\rho = \rho_c \left[ 1 - (r/R) \right] .$
\begin{enumerate}
\item Derive an expression for $\rho_c$ in terms of the stellar mass
  and radius, and compute a numerical value.
\item Use the Virial theorem to derive the average temperature of the 
star. Express your answer 
in terms of $\mu$, $M$, $R$, and the relevant physical constants, and
evaluate to obtain a numerical value.
\end{enumerate}

\end{enumerate}

\end{document}
