Christer Watson edited sectionAnalysis__sub.tex  over 8 years ago

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where $\beta$ is assumed to be 2 \citep{Desert2008}. B$_0$ and T were taken as free parameters and a Levenberg{\textendash}Marquardt algorithm was used to find the best-fit. The fitting was done to the flux density in Jy, so B$_0$ carries these units. The total column density was calculated following Miettinen \& Harju (2010). Briefly, we used the following relations:  \begin{eqnarray*}  N_{tot} = \frac{I}{B_\nu \mu m_H \kappa R_d}\\ R_d}\nonumber\\  B_\nu = \frac{2 h \nu^3}{c^2 (e^\frac{h \nu}{kT}-1)}\\ \nu}{kT}-1)}\nonumber\\  I = 3.73\times 10^{-16} B_{mod} \left(\frac{1"}{\theta}\right)^2\\ \left(\frac{1"}{\theta}\right)^2\nonumber\\  \kappa = \kappa_{1.3mm} \left(\frac{\lambda}{1.3mm}\right)^{-\beta}\\ \left(\frac{\lambda}{1.3mm}\right)^{-\beta}\nonumber\\  \end{eqnarray*}  where {\bf m$_H$ is the mass of hydrogen} and 3.73 x 10$^{-16}$ converts the surface brightness from Jy/(1\arcsec beam) to SI units. We make the following assumptions: $\kappa_{1.3mm}$ = 0.11 $\frac{m^2}{kg}$, appropriate for ice-covered dust grains from \citet{1994A&A...291..943O}, $\theta$=15.0\arcsec , the beamsize of the GBT at 49 GHz, the mean molecular weight $\mu$ = 2.3 and dust to mass ratio $R_d$ = 1/100. Note that B$_\nu$, B$_{mod}$ and $\kappa$ all require a choice of frequency or wavelength. However, these dependencies cancel in the final calculation of N$_{tot}$. These results are summarized in Table \ref{mbb}, where we report the flux density at five wavelength bands for each CS detection, the best-fit temperature, the total column density and the CS abundance. We estimate the error in determining the extended flux to be dominated by defining the edge of the object. The GBT beam is formally tapered along the edges to minimize ringing in the side-lobes. We have used a simple cut-off at the edge of the beam. This difference in beam produces a 20\% uncertainty in the {\bf HiGal} fluxes reported below. To estimate the uncertainty in the model results, we fit the data after adjusting the fluxes either up or down by the 20\% uncertainty. The results indicate an uncertainty of 4 K in temperature and 20\% in column density.