Molecular Column Density Calculation Jeffrey G. Mangum National Radio Astronomy Observatory, 520 Edgemont Road, Charlottesville, VA 22903, USA
[email protected] and Yancy L. Shirley Steward Observatory, University of Arizona, 933 North Cherry Avenue, Tucson, AZ 85721, USA
[email protected] May 29, 2013 Received ; accepted – 2 – ABSTRACT We tell you how to calculate molecular column density. Subject headings: ISM: molecules – 3 – Contents 1 Introduction 5 2 Radiative and Collisional Excitation of Molecules 5 3 Radiative Transfer 8 3.1 The Physical Meaning of Excitation Temperature . 11 4 Column Density 11 5 Degeneracies 14 5.1 Rotational Degeneracy (gJ)........................... 14 5.2 K Degeneracy (gK)................................ 14 5.3 Nuclear Spin Degeneracy (gI ).......................... 15 5.3.1 H2CO and C3H2 ............................. 16 5.3.2 NH3 and CH3CN............................. 16 5.3.3 c–C3H and SO2 .............................. 18 6 Rotational Partition Functions (Qrot) 18 6.1 Linear Molecule Rotational Partition Function . 19 6.2 Symmetric and Asymmetric Rotor Molecule Rotational Partition Function . 21 2 7 Dipole Moment Matrix Elements (|µjk| ) and Line Strengths (S) 24 – 4 – 8 Linear and Symmetric Rotor Line Strengths 26 8.1 (J, K) → (J − 1, K) Transitions . 28 8.2 (J, K) → (J, K) Transitions . 29 9 Symmetry Considerations for Asymmetric Rotor Molecules 29 10 Hyperfine Structure and Relative Intensities 30 11 Approximations to the Column Density Equation 32 11.1 Rayleigh-Jeans Approximation . 32 11.2 Optically Thin Approximation . 34 11.3 Optically Thick Approximation . 34 12 Molecular Column Density Calculation Examples 35 12.1 C18O........................................ 35 12.2 C17O........................................ 37 + 12.3 N2H ....................................... 42 12.4 NH3 .......................................