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{\it Astronomy 45 problem set 2}\hfill {\it Spring 1997}\newline
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{\bf Astronomy 45.  Introduction to Astrophysics}
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\noindent\underline{Problem set 2} \hfill Due Friday, February 28
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\noindent {\bf 1.}  Since photons arrive statistically, when you expect to
observe $N$ photons on the average, you actually get a value drawn from 
a statistical distribution with a mean of $N$ and a standard deviation 
$\sqrt{N}$.
\begin{itemize}
\item[a)] How many photons do you need to collect to measure the magnitude
of a star to an accuracy of $\pm 0.02$ mag?
\item[b)] How long an exposure do you need, with a $1\,{\rm m}$ telescope,
to measure the $B$ magnitude of an apparent magnitude $m_B = 20$ star to 
$\pm 0.02$ mag. (Hint: use the table in \S2.2.1 in the notes.)
\end{itemize}

\noindent
For even fainter objects, the brightness of the night sky becomes a limiting
factor.  The darkest (ground-based) night sky has an equivalent brightness
in $B$ of one $m_B=22.5$ star per square arcsecond.

\begin{itemize}
\item[c)] What is the night sky brightness $I_{\lambda}$ in ${\rm erg/(s\,
cm^2\,ster\,\AA)}$?
\item[d)] How many photons do you need from an $m_B =24$ star to measure its
magnitude to $\pm 0.02$ mag? 
Assume that the starlight is spread out over exactly 1 square
arcsecond (this the the best resolution that can be achieved from the ground).
(Hint: photons from the star are competing with 
statistical fluctuations {\em from the night sky}, not just with their own 
fluctuations.)
\item[e)] What is the exposure time on a telescope of aperture $4\,{\rm m}$ 
required for the measurement in part (d)?
\end{itemize}

\vskip12pt\noindent{\bf 2.} A $1.2\times 10^6\,{\rm g}$ spacecraft is launched
from Earth and is to be accelerated radially away from the Sun using a circular
solar sail.  Assume a flat sail made from a very lightweight material (say 
mylar) $10\,{\rm \mu m}$ thick, with the density $2\, {\rm g/cm^3}$.
\begin{itemize}
\item[a)] Compute the initial acceleration of the sail {\em alone} if
\begin{enumerate}
\item it is black, so it absorbs the Sun's light;
\item it is shiny, so it reflects the Sun's light.
\end{enumerate}
Does your answer depend on the radius of the sail? Why?
\item[b)] Now let's assume that we want the initial acceleration of the sail
and the spacecraft {\em together} to be half the value you found in part (a).
What is the required radius of the sail?
\end{itemize}

\noindent
You can assume that the Sun is a blackbody of temperature $T_{\odot} = 5700\,
{\rm K}$ and radius $7\times 10^{10}\,{\rm cm}$.

\vskip12pt\noindent {\bf 3.} Telescopes.
\begin{itemize}
\item[a)] The atmosphere limits the angular resolution of ground based optical
telescopes to $\sim 1^{''}$.  What is the smallest diameter a telescope can
have for its angular resolution to be limited by the atmosphere and not by
the size of the mirror?  Why do you think astronomers keep building larger
and larger telescopes on the ground (Keck telescope in Hawaii has $D = 10\,
{\rm m}$)?
\item[b)] A CCD detector is mounted at the focus of an f/1.4 reflecting 
telescope with at $5\,{\rm m}$-diameter mirror.  The CCD chip contains 
$800 \times 800$ pixels, with each pixel being $15\,{\rm \mu m}$ square. 
What is the angular size of the sky which is imaged on each pixel?  What is
the total field of view?
\item[c)] The Hubble Space Telescope has the effective diameter of 
$2.4\,{\rm m}$ and the CCD chip as described in part (b).  What must be its
effective f-number to achieve diffraction limited resolution at 
$\lambda = 5000\,{\rm \AA}$?  
\end{itemize}

\vskip12pt\noindent {\bf 4.} The Universe is filled with blackbody radiation 
of temperature $2.7\,{\rm K}$.  It is called Cosmic Microwave Background
and has originated when our Universe was much denser and hotter.
\begin{itemize}
\item[a)]  How many microwave background photons are there per cubic centimeter
in the Universe today?
\item[b)] When our Universe was a factor of $\Lambda$ smaller in linear 
dimension, these same photons where in a volume $\Lambda^3$ smaller, and it 
turns out that their {\em temperature} was also larger by a factor of 
$\Lambda$.  Compute the radiation density of in the Universe now 
($\Lambda=0$), and in the past when $\Lambda=1000$.
\item[c)]  Assume that the mean density of matter in the Universe today
is $2\times 10^{-31}\,{\rm g\, cm^{-3}}$.  According to special relativity, 
each particle has a {\em rest energy} associated with it which is equal to 
$E=m c^2$, where $m$ is the mass of the particle.  Using this fact, compute
at what redshift factor $\Lambda$ the photon energy density was greater 
than the energy density in matter.
\end{itemize}
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