Bolocam Mass Calculations

Adopt an effective Bolocam center frequency, f = 271.1 GHz

Use Ossenkopf, V. & Henning, Th. 1994, A&A, 291, 943 Table 1. [http://adsabs.harvard.edu/abs/1994A%26A...291..943O] Interpolate last two entries (lambda = 1.0 mm and 1.3 mm to f)

This gives \(\kappa = 0.01166 cm^2 g^{-1}\) (or round to 0.0117) for gas-to-dust = 100. But the slope between these 2 points is only 1.61!

The points in OH94 table 1 jump around and locally give a variety of slopes around 1.8. A fit to kappa longward of 100 microns gives a slope of 1.8.

Interpolating kappa from 1.00 mm where OH5 gives \(\kappa(1.00) = 0.0137\) (for a gas-to-dust ratio = 100) to 271.1 GHz (= 1.1058 mm) using a slope of 1.8 gives:

\(\kappa = 0.01143 cm^{2} g^{-1}\)

This is the same as the value used by Enoch, so let's adopt it (round to 0.0114), but note that this was calculated for 271.1 GHz.

For the v1.0 data, the BGPS has the following properties:

effective beam FWHM = 33"

gaussian beam \(\sigma = 14.01\)" (= FWHM / \(\sqrt{(8*\ln(2))}\))

effective beam radius = 19.82" (= FWHM / \(\sqrt{(4*\ln(2))}\))

This is the radius of a circular top-hat function that has the same solid angle as a gaussian. At a distance of D = 1 kpc, this radius corresponds to a length = \(2.97 x 10^{17}\) cm. The effective beam solid angle is \(\Omega = 2.902 x 10^{-8}\) sr.

Adopt a fiducial dust temperature for mass calculations, \(T_d = 20K\)

The total mass per Jy at \(D_{kpc}\) = 1 kpc is given by

\begin{equation*} M = 1.0 x 10^{-23} * S(Jy) * D^2 / \kappa * B_{\nu}(T) \end{equation*}
\begin{equation*} = 14.30 * [e^{13.01/T(K) - 1}] * S(Jy) * D^2_{kpc} M_{\odot} \end{equation*}

For \(T = 20 K\):

\begin{equation*} M = 13.07 * S(Jy) * D^2_{kpc} M_{\odot} \end{equation*}

This corresponds to a beam-filling column density of \(H_2\),

\begin{equation*} N(H2) = 2.19e+22 * [e^{13.01/T(K) - 1}] * S(Jy) (cm^{-2}) \end{equation*}

For \(T = 20 K\):

\begin{equation*} N(H2) = 2.01 x 10^{22} * S(Jy) (cm^{-2}) A_V = 21.1 \mathrm{magnitudes} Jy^{-1} \end{equation*}

where we use the Bohlin & Savage calibration

\begin{equation*} A_V = 1 mag. <=> N(H) = 1.9 x 10^{20} cm^{-2} \mathrm{and} N(H_2) = N(H) / 2 \end{equation*}

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