Source-linked AI summary
A computer program for fast non-LTE analysis of interstellar line spectra
Floris van der Tak, John Black, Fredrik Schoeier, David Jansen, Ewine van Dishoeck
TL;DR
Modern radio and infrared line observations require efficient models to infer physical and chemical conditions when simple equilibrium assumptions are inadequate. The paper presents RADEX, a statistical-equilibrium code using collisional and radiative processes, background radiation, and escape probabilities, and compares it with other methods. RADEX is intended for rapid analysis of large datasets and supports line-ratio diagnostics and synthetic spectra within its stated scope.
Problem
Efficient modeling is needed to infer temperature, density, and molecular abundances from modern radio and infrared line observations.
Method
RADEX calculates line intensities in a uniform medium by iteratively solving statistical equilibrium with collisional and radiative processes, background radiation, and escape-probability optical-depth treatment.
Results
RADEX provides rapid analysis of large datasets, supports line-ratio estimates of density and kinetic temperature, and can generate synthetic spectra.
Takeaways & Limitations
RADEX can analyze molecules with available collisional data and compare observed line ratios with diagnostic plots to estimate physical parameters.
Takeaways & Limitations
RADEX omits continuous opacity and treats only one molecule at a time, limiting applications involving infrared pumping or line overlap.
Abstract
from arXiv · showhide
The large quantity and high quality of modern radio and infrared line observations require efficient modeling techniques to infer physical and chemical parameters such as temperature, density, and molecular abundances. We present a computer program to calculate the intensities of atomic and molecular lines produced in a uniform medium, based on statistical equilibrium calculations involving collisional and radiative processes and including radiation from background sources. Optical depth effects are treated with an escape probability method. The program is available on the World Wide Web at http://www.sron.rug.nl/~vdtak/radex/index.shtml . The program makes use of molecular data files maintained in the Leiden Atomic and Molecular Database (LAMDA), which will continue to be improved and expanded. The performance of the program is compared with more approximate and with more sophisticated methods. An Appendix provides diagnostic plots to estimate physical parameters from line intensity ratios of commonly observed molecules. This program should form an important tool in analyzing observations from current and future radio and infrared telescopes.
1. Introduction
Spectral-line observations probe physical and chemical conditions in dilute astronomical gas, where thermodynamic equilibrium is often a poor approximation. When only limited lines are available, excitation must be inferred indirectly, while rotation diagrams provide a common multi-line approach under restricted conditions.
- Rotation diagrams fit line intensities versus upper-level energy with a single excitation temperature to estimate excitation temperature and molecular column density.These estimates require similar beam sizes and low optical depths, or appropriate corrections.
- Spectral lines at radio, (sub)millimeter, and infrared wavelengths probe physical and chemical conditions in dilute astronomical gas.
- When only one or two molecular lines are observed, excitation must be inferred from other species or theoretical considerations.
- Assuming excitation temperature equals kinetic temperature defines Local Thermodynamic Equilibrium, which holds at high densities.
2. Radiative transfer and molecular excitation
Radiative-transfer modeling couples molecular level populations, radiation, and collisions through statistical equilibrium and transport equations. Escape probability simplifies this coupled problem by representing photon escape with a geometry- and optical-depth-dependent average.
- Molecular excitation: Intermediate-level methods such as statistical equilibrium and LVG use collisional data to constrain column density, kinetic temperature, and volume density.Their use is limited when accurate collision rates are unavailable.
- Radiative transfer: Specific intensity describes radiation energy flow per area, time, bandwidth, and solid angle, while the transfer equation tracks emission and extinction along a path.
- Molecular excitation: Molecular excitation is determined by radiative and collisional rates, with collisional rates based on collision-partner density and rate coefficients.Upward rates are obtained through detailed balance.
- Radiative transfer: The formalism assumes complete angular and frequency redistribution, an approximation stated to be strictly valid when collisional excitation dominates.
- Molecular excitation: Statistical-equilibrium equations include non-local radiation, while the presented treatment sets state-specific formation and destruction rates to zero.Formation and destruction should be included explicitly when chemical timescales or radiative lifetimes make them important.
- Escape probability: Escape probability β is the geometry-averaged probability that a photon escapes the medium from its creation site, depending on optical depth τ.It simplifies the interdependence between molecular level populations and the local radiation field when global cloud properties are sufficient.
3. The program
RADEX is a public non-LTE radiative-transfer program for homogeneous, isothermal media that uses escape probabilities and iterative statistical-equilibrium calculations. It supports multiple geometries, background radiation, molecular collision partners, and practical line-analysis workflows, while having explicit limitations for continuum opacity, line overlap, masers, and self-absorption.
- Basic capabilities: RADEX uses escape-probability radiative transfer for an isothermal, homogeneous medium without large-scale velocity fields.Users can select expanding-spherical/LVG, static-spherical, or slab escape-probability expressions; the online version uses the uniform-sphere formula.
- Basic capabilities: The program can handle up to seven collision partners and uses molecular collisional data to model excitation in environments including dense clouds, diffuse clouds, PDRs, and comets.Applications include varying molecular column density to match line intensities or deriving source-averaged temperature and density from same-molecule line ratios.
- Background radiation field: RADEX includes external and internal continuum radiation and outputs background-subtracted line intensities in equivalent Rayleigh–Jeans radiation temperature.Its adopted Galactic background includes the CMB, Galactic starlight, thermal dust emission, and low-frequency non-thermal radiation.
- Calculation: The program iteratively solves statistical equilibrium, recalculating line optical depths and excitation until lines with τ > 10^-2 converge to a specified tolerance.Calculations begin with optically thin populations and then include internally produced radiation consistently with the background field.
- Results: Single line ratios cannot constrain both temperature and density, while multi-line observations are preferably compared with models using χ2.The paper illustrates abundance estimation from inferred column density and discusses differing density estimates from HCO+ line ratios in the Orion Bar.
- Limitations of the program: RADEX is limited when continuous opacity, line overlap, maser amplification, or strong self-absorption materially affects the modeled emission.Continuous-opacity omissions particularly restrict applications involving infrared pumping, and self-absorbed lines cannot be modeled satisfactorily.
4. Comparison with other methods
RADEX was compared with rotation-diagram, multi-zone escape-probability, and Monte Carlo methods under constant and varying physical conditions. Agreement is generally good in suitable regimes, while optical depth and geometry increasingly affect excitation and line intensities.
- RADEX was benchmarked against optically thin LTE, multi-zone escape-probability, and Monte Carlo calculations using LAMDA molecular data.
- At N(HCO+) ≈ 10^12 cm−2, excitation is nearly radius-independent and geometrical calculations agree to ≈10%.
- At N(HCO+) > 10^15 cm−2, excitation curvature becomes significant and the corresponding optical depth is ≈100, with ≈20% spread between estimates.Photon trapping thermalizes the cloud center, while emission escapes more readily at the edge.
- At high optical depth, RADEX is not recommended because excitation may not represent the emitting region and velocity-field assumptions affect line flux.Higher-lying H2O transitions can nevertheless remain suitable when they are less optically thick.
- For p-H2CO, non-LTE fitting yields a lower minimum χ2 than LTE and substantially different temperature and column-density estimates.The reported temperatures are 50 versus 150 K, and the authors prefer the non-LTE results because they involve fewer assumptions about cloud state.
5. Conclusions
The paper presents RADEX as a fast non-LTE line-analysis program based on escape probabilities and LAMDA molecular data. It supports broad line-analysis applications while identifying extensions needed for more complex radiative-transfer conditions.
- RADEX analyzes radio and infrared spectral-line observations using the escape-probability approximation.
- The program supports any molecule with collisional data available in the required LAMDA format.
- RADEX handles optical depths from approximately −0.1 to 100 and uses few free parameters for rapid analysis of large datasets.
- Diagnostic plots and Python scripts allow line-ratio estimates of density and kinetic temperature and calculations for other lines and molecules.
Appendix A: Program input and output
Appendix A defines key quantities used in RADEX outputs, including excitation temperature, line optical depth, line intensity, and integrated line flux.
- The excitation temperature Tex is defined in Eq. (10), and different transitions generally have different excitation temperatures.Lines are thermalized when Tex = Tkin; in LTE, all lines are thermalized.
- Line optical depth is defined as the optical depth of an equivalent rectangular line shape.
- Line intensity is reported as the Rayleigh–Jeans equivalent temperature TR in kelvin.
- Line flux is the velocity-integrated intensity, reported in K km s−1.
A.1. Program input
RADEX input specifies molecular data, output selection, cloud conditions, collision partners, and background radiation. Its line-profile treatment limits the interpretation of integrated emission at high optical depth.
- RADEX requires a molecular data file, an output file, and a frequency range for reported transitions.All transitions in the molecular data file enter the calculation, although output may be restricted to selected lines.
- The kinetic temperature and number of collision partners are user-specified inputs, with H2 typically used as the sole partner.
- The adopted rectangular profile can be converted to a Gaussian with FWHM ΔV, but integrated profiles have limited meaning at high optical depth.Optically thick lines require programs resolving the source spectrally and spatially.
- A positive background-temperature input uses a blackbody at that temperature, while zero selects the average interstellar radiation field.
Appendix B: Coding standards
The coding standards emphasize clear, maintainable source organization and robust input handling. They also document radiation-field and line-parameter inputs used by the program.
- Inputs: Input radiation fields may use an interstellar radiation spectrum or user-defined frequency, intensity, and dilution-factor values with spline interpolation and extrapolation.The interstellar spectrum is not adjustable by a scale factor.
- Inputs: The program accepts molecular column density in cm−2 and FWHM line width in km s−1 as model inputs.
- Organization: RADEX organizes functionality in subroutines grouped across files, with compilation instructions provided through a Makefile.The main program primarily illustrates the program structure.
- Documentation: Comments are maintained at approximately a 1:1 ratio with program text, including descriptions of each subroutine’s inputs, outputs, and call relationships.
A.2. Program output
RADEX output begins by repeating the input parameters and then reports line-specific molecular and spectroscopic quantities. These include state energies, frequencies, and wavelengths copied from the molecular data file.
- Output structure: RADEX first replicates the input parameters in its output file before listing quantities for each spectral line.
- Line data: For each line, the output lists quantum numbers, upper-state energy in K, frequency in GHz, and wavelength in µm.These values are copied from the molecular data file, usually sourced from LAMDA.
Appendix C: Diagnostic plots of molecular line ratios
The diagnostic plots use RADEX and LAMDA calculations to relate molecular line ratios to kinetic temperature and H2 density. Their interpretation depends on molecule type, transition selection, optical thinness, and observational beam assumptions.
- Purpose: RADEX and LAMDA were used to calculate line ratios for commonly observed molecules across kinetic temperatures and H2 densities.The plots are intended to help observers estimate physical conditions from observations.
- Assumptions: Under the illustrative assumptions, the lines are optically thin, so their ratios do not depend on molecular column density.The calculations use N=10^12 cm−2, a 1.0 km s−1 line width, and a 2.73 K blackbody background.
- Observational use: Line ratios are less sensitive to calibration errors than absolute line strengths, especially when both lines use the same telescope, receiver, and spectrometer.
- Molecular tracers: CO traces density at low densities, where collisions compete with radiative decay, while HCO+ and HCN show trends similar to CS.
- Molecular tracers: H2CO line ratios can probe temperature and density within one frequency range: different J-states primarily trace density, while different K-states primarily probe temperature.The lines are often strong, supporting H2CO’s use as a temperature and density tracer.
Appendix D: The Python scripts
The Python scripts automate RADEX grid calculations and column-density fitting from observed line data. Users configure physical, molecular, observational, and numerical parameters, then inspect tabulated or fitted results.
- Execution: The scripts are written in Python and run from the Unix shell after users manually edit the parameters.
- grid.py: grid.py produces a tabular radex.out file containing temperature, log density, and line ratio for plotting.The output may be plotted with the user’s preferred plotting program.
- line.py: line.py calculates molecular column density from an observed line intensity given estimates of kinetic temperature and H2 volume density.Inputs include the molecule, transition frequency, observed intensity, background temperature, and observed line width.
- Numerical settings: The fitting accuracy defaults to 10%, corresponding to typical telescope calibration uncertainty, while the free spectral range defaults to 10%.The free spectral range may need adjustment for molecules with many nearby lines.
- line.py: line.py iterates over column density until modeled and observed line fluxes agree within the requested accuracy, then reports the best-fit value.The radex.out file provides details of the best-fit model.
- grid.py: The scripts include grid.py for estimating kinetic temperature and/or volume density from observed line ratios.Users specify temperature and H2-density grid boundaries, background temperature, molecular column density, and line width.