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{\it Astronomy 45 problem set 1}\hfill {\it Spring 1997}\newline
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{\bf Astronomy 45.  Introduction to Astrophysics}
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\noindent\underline{Problem set 1} \hfill Due Friday, February 14
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\noindent {\bf 1.} The synodic period of a planet ($S$) is defined as the 
time interval between successive oppositions or conjunctions, whereas the 
sidereal period ($P$) is the amount of time necessary to make a complete 
orbit relative to the background stars. 
\begin{itemize}
\item[a)] Derive the following relationships between the synodic and sidereal 
periods
\[\frac{1}{S} = \frac{1}{P} - \frac{1}{P_{\oplus}},\ \ \ 
{\rm for\ inferior\ planets} \] 
\[\frac{1}{S} = \frac{1}{P_{\oplus}} - \frac{1}{P},\ \ \ 
{\rm for\ superior\ planets},\]
where $P_{\oplus}$ is the sidereal period of Earth.  (Hint: it might be easier
to work in the rotating frame of reference in which Earth is stationary.)
\item[b)] Which planet in the Solar system has the shortest synodic period 
and why? Compute this period (in days) using the fact that 
$P_{\oplus}=365.26\,{\rm days}$, and the following table.
\end{itemize}

\begin{table}[h]
%\caption{Sidereal Orbital Periods for the Solar System Planets} 
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\begin{tabular}{l r|l r}
\hline \\
Planet & Period (yr) & Planet & Period (yr) \\
\hline \\
Mercury & 0.2408 & Saturn & 29.4577 \\
Venus & 0.6152 & Uranus & 84.0139 \\ 
Earth & 1.0000 & Neptune & 164.793 \\
Mars & 1.8809  & Pluto & 248.54 \\
Jupiter & 11.8622 & &\\ 
\hline 
\end{tabular}
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\end{table}


\vskip12pt\noindent{\bf 2.}  Using the understanding of synodic periods
and/or rotating frames from the previous problem: 
\begin{itemize}
\item[a)] Calculate the time it takes Venus to transit (cross) the Sun, if it
crosses at an apparent latitude on the Sun's disk of $42^{\circ}$N.  You can
take the angular diameter of the Sun, as seen from Earth, to be 32
arcminutes.  The period of Venus' orbit is 0.615 years.
\item[b)] Two observers on Earth are separated by a North-South distance
of 8000 km.  How different are the apparent latitudes on the Sun's disk at
which they see Venus cross?  That is, if the N observer sees
$42.0000^{\circ}$N, what does the S observer see.
\item[c)] What is the difference in absolute time between when the N
observer sees Venus first touch the Sun, and when the S observer sees
this.  That is, if it occurs for the N observer at 11:00:00.000, when does it
occur for the S observer (in the same time zone?)  (You can assume that
Venus is a point for this estimate, if you want.)
\end{itemize}

\vskip12pt\noindent {\bf 3.} In 1672, an international effort was made to 
measure the parallax angle of 
Mars at the time of the opposition, when it was closest to Earth.
\begin{itemize}
\item[a)] Consider two observers who are separated by a baseline equal to 
Earth's
diameter.  If the difference in their measurements of Mars' angular position 
is $33.6^{\prime \prime}$, what is the distance between Earth and Mars at
opposition?
\item[b)] If the distance to Mars is to be measured to within $10 \%$, how 
closely 
must the clocks used by the two observers be synchronized?  (Hint: Use the
average orbital velocities of Earth and Mars, you should have enough 
information in this problem set to compute them.
Can the Earth's rotation be ignored in this problem?)
\end{itemize}


\vskip12pt\noindent {\bf 4.} Proxima Centauri is the closest star to the Sun and is a part of a triple
star system.  It has the epoch 1950.0 coordinates $(\alpha, \delta) = 
(14^h 26.3^m, -62^{\circ} 28^{\prime})$, while the brightest star in the 
system, $\alpha$ Centauri is located at $(\alpha, \delta) = (14^h 39.6^m, 
-60^{\circ} 49.8^{\prime})$.
\begin{itemize}
\item[a)] What is the angular separation between the two stars?
\item[b)] If the distance to Proxima Centauri is $4.3\,{\rm lyr}$, what is the 
projected distance between the two stars?  The distance to $\alpha$ Centauri
is roughly $4.4\,{\rm lyr}$.  What is the distance between the two stars?
\item[c)] The apparent visual magnitudes of the three stars in a triple system
are 0.0, 1.3, and 11.0 ($\alpha$ Centauri being the brightest and Proxima 
Centauri the dimmest).  What is the total apparent magnitude of the system?  
What is the total flux observed in the visual band?
\item[d)] Compute the absolute magnitudes of the three stars (it is necessary 
to make some assumption about the distance to the third star).  Based on this 
result, what are their approximate spectral types and masses?  
\end{itemize}

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