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\begin{document}

\title{Far Infrared Observations of Distant Galaxies with SPECS}

\author{Edward L. (Ned) Wright (UCLA)}

\bc
wright@astro.ucla.edu\\
http://www.astro.ucla.edu/\raisebox{-0.5em}{{\large\~{}}}wright/intro.html\\
\ec

\bc
COSMOLOGY TUTORIAL:\\
http://www.astro.ucla.edu/\raisebox{-0.5em}{{\large\~{}}}wright/cosmolog.htm
\ec

\noindent
ABSTRACT: Observations of galaxies with background limited far
infrared single dish instruments in space are confusion
limited after short integration times. One way to use this
sensitivity is to scan the sky rapidly to survey a large
solid angle with short integration times, as is planned by
the IRIS experiment on the ASTRO-F satellite.  But SPECS
will use this sensitivity to perform interferometric
observations that will spread the confusing forest in three
dimensions: two spatial angles and the redshift.  This will
allow detailed studies of individual objects at large
distances and redshifts. The Guiderdoni et al. model counts
flatten to a slope well below Euclidean at $\log(N) = 8.7$
with $N$ in sources per steradian at a 175 $\mu$m flux of
160 $\mu$Jy.  But an aperture greater than 10 meters is
needed to study sources at this density, so counting out the
cosmic infrared background detected by DIRBE will require an
interferometer like SPECS.

\clearpage

The intensity from unresolved sources fainter than $S_{lim}$ is given by
$$
I_{S_{lim}} = \int_0^{S_{lim}} S dN
$$
where $N(>S)$ is the number of sources brighter than $S$ per unit solid
angle.  For source counts steeper than $N \propto S^{-1}$ this integral 
diverges at low fluxes, while for counts less steep than $N \propto S^{-1}$ 
it diverges for high fluxes.  Thus the largest contribution comes in the flux
range where $NS \approx \mbox{const}$.

The flux variance due to unresolved sources is
$$
\sigma(S)^2 = \Omega_{eff} \int_0^{S_{lim}} S^2 dN
$$
where $\Omega_{eff}$ is the noise effective solid angle, given by
$$
\Omega_{eff} = \frac{[\int \phi(\Omega) d\Omega]^2}{\int \phi(\Omega)^2
d\Omega}
$$
for a beam profile $\phi(\Omega)$.  This solid angle is $4(\lambda/D)^2$
for a diffraction-limited circular aperture with a 32\% linear obscuration.

The BLIP SNR for a given observation
is then given by
$$
\mbox{SNR} = \sqrt{\frac{A\eta t}{h\Omega_{eff}} \int \frac{S_\nu^2}{B_\nu}
d\ln\nu}
$$
where $A$ is the collecting area, $\eta$ is the DQE, $t$ is the integration
time, $B_\nu$ is the specific intensity of the background and $S_\nu$ is the
spectrum of the source.

The scaling for the integration time at fixed SNR is determined using
$$\Omega_{eff} \propto (\lambda/D)^2 \propto N^{-1}$$
which then gives the integration time going like
$$t \propto  D^{-2}\Omega B S^{-2} \propto \lambda^{-2}B (NS)^{-2}$$
This time is shortest when the product $NS$ is highest, which is roughly the
same flux range that gives the largest contribution to the integrated
intensity.

The wavelength factor, $B_\nu/(\lambda S_\nu)^2$, when evaluated using the
DIRBE darkest pixel for $B$ and the DIRBE measured flux of the Milky Way for
$S_\nu$, is {\bf 1,000,000 times smaller} for 240 $\mu$m than it is for
2.2 $\mu$m.  The far-IR is the best wavelength range to use for detecting
distant faint galaxies.

To evaluate these quantities, I take the specific prediction of source counts 
at 175 $\mu$m in Guiderdoni \etal, astro-ph/9707134.  I also take 10 MJy/sr
as the galactic plus zodiacal background intensity at 175 $\mu$m, 
a DQE $\eta = 0.25$ and a
fractional bandwidth $d\ln\nu = 0.25$.
I find the $\Omega_{eff}$ for a
given $S$ that gives $\sigma(S) = 0.2 S_{lim}$, for a confusion-limited survey
with $\mbox{SNR} > 5$.  I find the telescope diameter $D$ needed to give this
$\Omega_{eff}$ using $D = 2\lambda/\sqrt{\Omega_{eff}}$.

\begin{table}[p]
\begin{center}
\begin{tabular}{rrrrrr}
\hline
  \multicolumn{1}{c}{S[Jy]} &   
\multicolumn{1}{c}{N} & 
\multicolumn{2}{c}{$(\Omega N)^{-1}$ \hspace{16pt}  D[m]}
& $I_S/I_\infty$ & 
$t\;[\mu\mbox{sec}]$ \\
\hline
1.25$\times10^{-5}$ & 1.88$\times10^{+9}$ &   6 & 37.571 & 0.019 & 132893 \\
1.26$\times10^{-4}$ & 5.95$\times10^{+8}$ &  10 & 26.339 & 0.107 &   5415 \\
5.55$\times10^{-4}$ & 1.88$\times10^{+8}$ &  16 & 19.198 & 0.276 &    987 \\
1.66$\times10^{-3}$ & 5.95$\times10^{+7}$ &  24 & 13.307 & 0.465 &    479 \\
4.06$\times10^{-3}$ & 1.88$\times10^{+7}$ &  35 &  8.934 & 0.625 &    395 \\
9.51$\times10^{-3}$ & 5.95$\times10^{+6}$ &  42 &  5.540 & 0.745 &    486 \\
2.16$\times10^{-2}$ & 1.88$\times10^{+6}$ &  49 &  3.348 & 0.833 &    708 \\
4.70$\times10^{-2}$ & 5.95$\times10^{+5}$ &  56 &  2.027 & 0.895 &   1111 \\
9.71$\times10^{-2}$ & 1.88$\times10^{+5}$ &  67 &  1.243 & 0.937 &   1838 \\
1.90$\times10^{-1}$ & 5.95$\times10^{+4}$ &  82 &  0.773 & 0.963 &   3210 \\
3.49$\times10^{-1}$ & 1.88$\times10^{+4}$ & 105 &  0.491 & 0.979 &   5817 \\
6.53$\times10^{-1}$ & 5.95$\times10^{+3}$ & 120 &  0.296 & 0.988 &  12581 \\
1.06$\times10^{+0}$ & 1.88$\times10^{+3}$ & 176 &  0.202 & 0.993 &  22503 \\
% 1.88$\times10^{+0}$ & 5.95$\times10^{+2}$ & 200 &  0.121 & 0.995 &  55355 \\
% 3.36$\times10^{+0}$ & 1.88$\times10^{+2}$ & 227 &  0.072 & 0.997 & 134776 \\
\hline
\end{tabular}
\end{center}
\end{table}

{\large In order to match the Kawara \etal\ ISO point with the ISO diameter
of 0.6 m, I have to use SNR = 2.5 instead of 5.}

For a wide range of fluxes, the integration times required for a
background-limited, diffraction-limited, confusion-limited far infrared
continuum survey are less than one-fifth of a second, and are as small as
0.4 millisec for a survey done with a 9 meter single dish to a depth of
4 mJy.  

\clearpage

{\large
Thus
\begin{itemize}
\item
efficient use of a single dish to do a DL\&CL\&BLIP 
far-IR continuum survey requires a very fast, agile telescope and a very
fast scan strategy.  
\item
Counting out a significant fraction of the far-IR
background requires a large telescope, with a diameter of ten meters or more.
\item
Large and agile seldom go together in space missions.
\end{itemize}
}

\clearpage

What should be done with this embarrassment of riches?
{\large
\begin{list}{$\bullet$}{\itemsep 2pt \parsep 2pt}
\item
Waste it with a warm telescope? - NO!
\item
Add agility as SIRTF does with the MIPS scan mirror
\item
Build an interferometer
\end{list}
}
If we take the diameters given earlier as baselines $L$, and use smaller 
dishes of diameter $D$ to collect photons, the integration time needed to 
reach a given flux varies like ${\cal O}(L/D)^4$.  Even though this is a very
steep dependence, the integration times for $L = D$ are so short that
working with $L/D$ as large as 30 is very practical.  This plan gives
us\\
\begin{center}
{\Huge SPECS}
\end{center}

\end{document}
