\documentstyle[11pt]{article}
%
\textheight 9.5in
\textwidth  6.5in
%
\topmargin  0.0cm
\headheight 0.0cm
\headsep    0.0cm
\oddsidemargin  0.0cm
\evensidemargin 0.0cm
\marginparwidth 2.5cm
\marginparsep   0.0cm
%
\parskip 14pt
\parindent 0pt
% --------------
\newcommand{\kms}{\mbox{$km \ s^{-1}$ \ }} 
\newcommand{\Msun}{\mbox{M$_{\odot}$}}
\newcommand{\Rsun}{\mbox{R$_{\odot}$}}
\newcommand{\Lsun}{\mbox{L$_{\odot}$}}
% Define commands for `less than or approximately equal to' \ltsimeq
%                     `greater than or approximately equal to' \gtsimeq
\newcommand{\ltsimeq}{\raisebox{-0.6ex}{$\,\stackrel
        {\raisebox{-.2ex}{$\textstyle <$}}{\sim}\,$}}
\newcommand{\gtsimeq}{\raisebox{-0.6ex}{$\,\stackrel
        {\raisebox{-.2ex}{$\textstyle >$}}{\sim}\,$}}
\newcommand{\prpsimeq}{\raisebox{-0.6ex}{$\,\stackrel
        {\raisebox{-.2ex}{$\textstyle \propto $}}{\sim}\,$}}
%--------------------------------------------------------------
\begin{document}
\vspace*{-2cm}
\begin{center}
{\LARGE
{\bf 
\underline{Astronomy 680 Homework \#1
}}}
\end{center}

{\Large
Fall 2004      \hfill 2004 Aug 31\\
Due Tues Sep 7 \hfill Prof. Welsh}

[50 points total]

0. Learn unix! Suggestions: See the class WWWeb pages for help, ask your
classmates for help, borrow books on unix/linux, etc..


1. Send me an e-mail message from your sciences account. \hfill [5]\\
(send to wfw@sciences.sdsu.edu, not wwelsh@mail.sdsu.edu)


2. Show that the full-width-at-half-max (FWHM) of a Gaussian distribution
equals 2.355 $\sigma$, where $\sigma$ is the standard deviation of the 
Gaussian. \hfill [5]


3. Show (using the propagation of errors formula) that\\
a) the uncertainty in the mean $\sigma_{\bar{x}}$ is
      $\frac{\sigma}{\sqrt{N}}$, where $N$ is the number
      of measurements and $\sigma$ is the uncertainty in each 
      measurement and is a constant. \hfill [5]\\
b) the uncertainty in the inverse-variance weighted mean is 
      $\sigma_{\bar{x}_{wgt}} =$
      $\sqrt{ 1 / \sum{\frac{1} {\sigma^{2}_{i} } }  }$\\
      where the individual $\sigma_{i}$ are no longer the same.
\hfill [10] \\
Note: In this context, the symbol $\sigma$ stands for the uncertainty in 
a measurement. This is not necessarily the same as the standard
deviation of a Gaussian!


4. A ``magnitude'' in astronomy is defined as m= --2.5 log (flux) + const.
Note the base--10 log, not the natural ln. The constant is defined so as
to give historical consistency and it also depends strongly on the
wavelength of the light.\\
a) What is the uncertainty in magnitude ($\sigma_m$) given a
   relative uncertainty in flux ($\sigma_f/f$)? \hfill [10]\\
b) Show that if a star changes brightness by 2\%, the change in
   magnitude is roughly 0.02. \hfill [5]


5. Do problem 2.13 in Bevington \& Robinson. \hfill [10] \\
Hint: (i) Use the Poisson distribution. (ii) Part b is tricky - don't
waste too much time on it if you don't see the trick.\\

\hrule

Headstart for next homework:\\
-- Learn scientific programming (e.g. fortran)\\
-- Using the files$^\dagger$ hw2--a.dat, hw2--b.dat, and hw2--c.dat,
determine the following for each file:\\
the mean; standard deviation; error in the mean;
weighted mean; error in the weighted mean; the median

$^\dagger$ The files are located in
/iraftmp/iraftmp2/wfw/ASTR680/ \\
$>$cp the files to your own directory and work with them there.
The files are in ASCII format and have 3 columns:
the time of observation, the observation, and the error estimate of
the observation. In other words, the measured values are in column 2 and
the 1-$\sigma$ error bars are in column 3.\\


%---------------------------------------------------------------------
\end{document}