How to Calculate Molecular Column Density

The calculation of the molecular column density from molecular spectral (rotational or ro-vibrational) transition measurements is one of the most basic quantities derived from molecular spectroscopy. Starting from first principles where we describe the basic physics behind the radiative and collisional excitation of molecules and the radiative transfer of their emission, we derive a general expression for the molecular column density. As the calculation of the molecular column density involves a knowledge of the molecular energy level degeneracies, rotational partition functions, dipole moment matrix elements, and line strengths, we include generalized derivations of these molecule-specific quantities. Given that approximations to the column density equation are often useful, we explore the optically thin, optically thick, and low-frequency limits to our derived general molecular column density relation. We also evaluate the limitations of the common assumption that the molecular excitation temperature is constant, and address the distinction between beam- and source-averaged column densities. We conclude our discussion of the molecular column density with worked examples for C$^{18}$O, C$^{17}$O, N$_2$H$^+$, NH$_3$, and H$_2$CO. Ancillary information on some subtleties involving line profile functions, conversion between integrated flux and brightness temperature, the calculation of the uncertainty associated with an integrated intensity measurement, the calculation of spectral line optical depth using hyperfine or isotopologue measurements, the calculation of the kinetic temperature from a symmetric molecule excitation temperature measurement, and relative hyperfine intensity calculations for NH$_3$ are presented in appendices. The intent of this document is to provide a reference for researchers studying astrophysical molecular spectroscopic measurements.

Ammonia Thermometry of Star Forming Galaxies

With a goal toward deriving the physical conditions in external galaxies, we present a study of the ammonia (NH$_3$) emission and absorption in a sample of star forming systems. Using the unique sensitivities to kinetic temperature afforded by the excitation characteristics of several inversion transitions of NH$_3$, we have continued our characterization of the dense gas in star forming galaxies by measuring the kinetic temperature in a sample of 23 galaxies and one galaxy offset position selected for their high infrared luminosity. We derive kinetic temperatures toward 13 galaxies, 9 of which possess multiple kinetic temperature and/or velocity components. Eight of these galaxies exhibit kinetic temperatures $>100$ K, which are in many cases at least a factor of two larger than kinetic temperatures derived previously. Furthermore, the derived kinetic temperatures in our galaxy sample, which are in many cases at least a factor of two larger than derived dust temperatures, point to a problem with the common assumption that dust and gas kinetic temperatures are equivalent. As previously suggested, the use of dust emission at wavelengths greater than 160 $mu$m to derive dust temperatures, or dust heating from older stellar populations, may be skewing derived dust temperatures in these galaxies to lower values. We confirm the detection of high-excitation OH $^2Pi_{3/2}$ J=9/2 absorption toward Arp220 (Ott et. al. 2011). We also report the first detections of non-metastable NH$_3$ inversion transitions toward external galaxies in the (2,1) (NGC253, NGC660, IC342, and IC860), (3,1), (3,2), (4,3), (5,4) (all in NGC660) and (10,9) (Arp220) transitions.

Formaldehyde Densitometry of Galactic Star-Forming Regions Using the H2CO 3(12)-3(13) and 4(13)-4(14) Transitions

We present Green Bank Telescope (GBT) observations of the 3(12)-3(13) (29 GHz) and 4(13)-4(14) (48 GHz) transitions of the H2CO molecule toward a sample of 23 well-studied star-forming regions. Analysis of the relative intensities of these transitions can be used to reliably measure the densities of molecular cores. Adopting kinetic temperatures from the literature, we have employed a Large Velocity Gradient (LVG) model to derive the average hydrogen number density [n(H2)] within a 16 arcsecond beam toward each source. Densities in the range of 10^{5.5}--10^{6.5} cm^{-3} and ortho-formaldehyde column densities per unit line width between 10^{13.5} and 10^{14.5} cm^{-2} (km s^{-1})^{-1} are found for most objects, in general agreement with existing measurements. A detailed analysis of the advantages and limitations to this densitometry technique is also presented. We find that H2CO 3(12)-3(13)/4(13)-4(14) densitometry proves to be best suited to objects with T_K >~ 100 K, above which the H2CO LVG models become relatively independent of kinetic temperature. This study represents the first detection of these H2CO K-doublet transitions in all but one object in our sample. The ease with which these transitions were detected, coupled with their unique sensitivity to spatial density, make them excellent monitors of density in molecular clouds for future experiments. We also report the detection of the 9_2--8_1 A^- (29 GHz) transition of CH3OH toward 6 sources.

MUSTANG 3.3 Millimeter Continuum Observations of Class 0 Protostars

We present observations of six Class 0 protostars at 3.3 mm (90 GHz) using the 64-pixel MUSTANG bolometer camera on the 100-m Green Bank Telescope. The 3.3 mm photometry is analyzed along with shorter wavelength observations to derive spectral indices (S_nu ~ nu^alpha) of the measured emission. We utilize previously published dust continuum radiative transfer models to estimate the characteristic dust temperature within the central beam of our observations. We present constraints on the millimeter dust opacity index, beta, between 0.862 mm, 1.25 mm, and 3.3 mm. Beta_mm typically ranges from 1.0 to 2.4 for Class 0 sources. The relative contributions from disk emission and envelope emission are estimated at 3.3 mm. L483 is found to have negligible disk emission at 3.3 mm while L1527 is dominated by disk emission within the central beam. The beta_mm^disk <= 0.8 - 1.4 for L1527 indicates that grain growth is likely occurring in the disk. The photometry presented in this paper may be combined with future interferometric observations of Class 0 envelopes and disks.

Thioformaldehyde Emission from the Massive Star-Forming Region DR21(OH)

Using arguments parallel to those used in support of using H2CO as a sensitive probe of temperature and density in molecular clouds, we measured the J=7-6 and J=10-9 transitions of thioformaldehyde (H2CS) in several hot core sources. The goal here was to investigate more closely the conditions giving rise to H2CS emission in cloud cores containing young stars by modelling several transitions. The H2CS molecule is a slightly asymmetric rotor, a heavier analogue to H2CO. As in H2CO, transitions occur closely spaced in frequency, though they are substantially separated in energy. Transitions of H2CS originating from the K=0, 1, 2, 3, and 4 ladders in the 230 and 345 GHz windows can productively be used to constrain densities and temperatures. As a first step in developing the use of these transitions as thermometers and densitometers, we surveyed and modeled the emission from well known warm dense cores.