\documentstyle[12pt,aaspp]{article} \setlength{\textwidth}{4.5in} \setlength{\textheight}{3.25in} \setlength{\topmargin}{-1in} \setlength{\evensidemargin}{-0.5in} \setlength{\oddsidemargin}{\evensidemargin} \setlength{\parindent}{0pt} \setlength{\parskip}{12pt} \newcommand{\bc}{\begin{center}} \newcommand{\ec}{\end{center}} \newcommand{\bn}{\begin{enumerate}} \newcommand{\en}{\end{enumerate}} \newcommand{\bi}{\begin{itemize}} \newcommand{\ei}{\end{itemize}} \newcommand{\be}{\begin{equation}} \newcommand{\ee}{\end{equation}} \newcommand{\bea}{\begin{eqnarray}} \newcommand{\eea}{\end{eqnarray}} \newcommand{\vs}{{\it vs.}} \newcommand{\asec}{\mbox{$^{\prime\prime}$}} \newcommand{\etal}{{\it et al.}} \newcommand{\bfR}{\mbox{{\bf R}}} \newcommand{\bq}{\mbox{{\bf q}}} %\newcommand{\vr}{\vec{r}} \newcommand{\kms}{\mbox{km~s$^{-1}$}} \pagestyle{empty} \def\wisk#1{\ifmmode{#1}\else{$#1$}\fi} \def\percm {\wisk{{\rm cm}^{-1}}} \def\percmsq{\wisk{{\rm cm}^{-2}}} \def\percmcub{\wisk{{\rm cm}^{-3}}} \def\lt {\wisk{<}} \def\gt {\wisk{>}} \def\le {\wisk{_<\atop^=}} \def\ge {\wisk{_>\atop^=}} \def\lsim {\wisk{_<\atop^{\sim}}} \def\gsim {\wisk{_>\atop^{\sim}}} \def\Lsun {\wisk{{\rm L_\odot}}} \def\Msun {\wisk{{\rm M_\odot}}} \def\um {\wisk{{\rm \mu m}}} \def\sig {\wisk{\sigma}} \def\bsl {\wisk{\backslash}} \def\by {\wisk{\times}} \def\amin {\wisk{^\prime}} \def\asec {\wisk{^{\prime\prime}}} \def\cc {\wisk{{\rm cm^{-3}\ }}} \def\deg {\wisk{^\circ\ }} \def\ddeg {\wisk{{\rlap.}^\circ}} \def\damin {\wisk{{\rlap.}^\prime}} \def\dasec {\wisk{{\rlap.}^{\prime\prime}}} \def\approxeq{$\sim \over =$} \def\abouteq{$\sim \over -$} \def\perhz{Hz$^{-1}$} \def\perpc{$\rm pc^{-1}$} \def\persec{s$^{-1}$} \def\peryr{yr$^{-1}$} \def\te{$\rm T_e$} \def\tenup#1{10$^{#1}$} \def\to{\wisk{\rightarrow}} \def\thin{\thinspace} \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}