Introduction

Inspired by the work of Neufeld 1990 for a plane-parallel slab, Dijkstra et al. 2006 present an analytical approximation for the radiation spectrum emerging from a static, uniform neutral hydrogen sphere with a central point source emitting at the Lyman-alpha line center. The approximation becomes more accurate for higher optical depths at lower gas temperatures.

References and downloads

Publications	Neufeld 1990 [ADS] Dijkstra et al. 2006 [ADS] Camps et al. 2021 [ADS]
Ski file	dijkstra_template.ski

SKIRT results

The figure below is copied from Camps et al. 2021. It shows the spectrum emerging from the sphere as a function of the dimensionless frequency \(x\) defined as

\[ x = \frac{\nu - \nu_\alpha}{\nu_\alpha} \,\frac{c}{v_\mathrm{th}} \qquad \mathrm{with} \qquad v_\mathrm{th} = \sqrt{\frac{2 k_\mathrm{B} T}{m_\mathrm{p}}}, \]

where \(\nu\) is the regular frequency, \(\nu_\alpha\) is the frequency at the Lyman-alpha line center, and \(v_\mathrm{th}\) is the thermal velocity of the neutral hydrogen gas corresponding to its temperature \(T\).

The figure compares the SKIRT output (solid lines) with the analytical approximation (dotted lines) for a gas temperature of 10 K and the three optical depth values shown by Dijkstra et al. 2006. It includes a fourth combination with a higher gas temperature to verify that SKIRT handles other temperature values as well. It is apparent from the figure that the numerical and analytical solutions indeed converge for higher optical depths at constant gas temperature. Also, the discrepancies between the numerical and analytical solutions are essentially identical to those shown by Dijkstra et al. 2006.

Performing this benchmark

To perform this benchmark, download the ski file provided above (References and downloads). Open the ski file in a text editor to adjust the following parameter values to a particular benchmark configuration:

Parameter	XML element	XML attribute
Gas temperature	`LyaNeutralHydrogenGasMix`	defaultTemperature
Optical depth	`OpticalDepthMaterialNormalization`	opticalDepth
Wavelength range	`LinWavelengthGrid`	minWavelength and maxWavelength

The benchmark specifies radial optical depth values at the Lyman-alpha line center (integrated from the origin to infinity) while the values in the ski file normalize the line-center optical depth along the complete X-axis (integrated from negative to positive infinity). The value in the ski file therefore must be set to twice the value specified for the corresponding benchmark.

The wavelength range for the instrument recording the output spectrum must be adjusted to the expected spectral dispersion for the specified temperature/optical depth combination. The appropriate values for the four combinations shown in the figure above are:

Gas temperature (K)	Radial optical depth	Minimum wavelength (micron)	Maximum wavelength (micron)
10	1e5	0.121562881	0.121571119
10	1e6	0.121558762	0.121575238
10	1e7	0.121550524	0.121583476
10000	1e7	0.1213846437	0.1217493563

Then pass the (name of) the ski file to SKIRT as a single command line argument. Higher optical depths and lower temperatures lead to longer simulation run times. At the end of the simulation run, SKIRT outputs a spectrum that can be compared to the analytical approximation.