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The SKIRT project
advanced radiative transfer for astrophysics
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The SKIRT project
advanced radiative transfer for astrophysics
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When particles in a medium (atoms, molecules, electrons) emit or absorb photons, their random thermal motion causes Doppler shifts that result in broadening of the intrinsic emission and absorption lines. This page presents the formulas involved in this process as they are used in the SKIRT implementation. This is especially useful because the notation and definitions sometimes differ between authors.
The normalized Gaussian probability distribution of a variable \(x\) with center \(x_0\) and dispersion \(x_\mathrm{s}\) can be written as
\[ g(x;x_0,x_\mathrm{s}) = \frac{1}{x_\mathrm{s}\sqrt{2\pi}}\,\exp\!\left(-\frac{1}{2}\frac{(x-x_0)^2}{x_\mathrm{s}^2} \right) \qquad \mathrm{with} \quad \int_{-\infty}^\infty g(x;x_0,x_\mathrm{s}) \,\mathrm{d}x = 1, \]
where \(x\), \(x_0\), \(x_\mathrm{s}\) and \(1/g(x;x_0,x_\mathrm{s})\) all have the same units.
A Gaussian line profile can be written as a function of frequency, \(g(\nu;\nu_0,\nu_\mathrm{s})\), or as a function of wavelength, \(g(\lambda;\lambda_0,\lambda_\mathrm{s})\). Given that \(\nu\lambda=c\) and \(\nu_0\lambda_0=c\), and assuming that the profile is narrow ( \(\nu_\mathrm{s}\ll\nu_0\) or \(\lambda_\mathrm{s}\ll\lambda_0\)), the dispersions are related as
\[ \nu_\mathrm{s} = \frac{c}{\lambda_0^2}\lambda_\mathrm{s} \qquad \mathrm{and} \qquad \lambda_\mathrm{s} = \frac{c}{\nu_0^2}\nu_\mathrm{s}. \]
We now consider the Doppler shift velocity \(v\) corresponding to the frequency or wavelength displacement. For narrow profiles and thus \(v\ll c\), this velocity is defined through
\[ \lambda = \left( 1+\frac{v}{c}\right)\lambda_0 \qquad \mathrm{and} \qquad \nu = \left( 1-\frac{v}{c}\right)\nu_0, \]
so that the line profile can also be written as \(g(v;0,v_\mathrm{s})\) with
\[ v_\mathrm{s} = \frac{c}{\lambda_0}\lambda_\mathrm{s} \qquad \mathrm{and} \qquad v_\mathrm{s} = \frac{c}{\nu_0}\nu_\mathrm{s}. \]
The velocity probability distribution of particles with mass \(m\) in a medium at temperature \(T\) is given by the Maxwell-Boltzmann law,
\[ f(v)\,\mathrm{d}^3v = \left(\frac{m}{2\pi kT}\right)^{3/2} \, \exp\left(-\frac{mv^2}{2kT}\right)\,\mathrm{d}^3v = g(v_x;0,v_\mathrm{s})\,g(v_y;0,v_\mathrm{s})\,g(v_z;0,v_\mathrm{s})\,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z, \]
where \(k\) is the Boltzmann constant and we derived the velocity dispersion to be
\[ v_\mathrm{s} = \sqrt{\frac{kT}{m}}. \]
In other words, each of the velocity vector components is distributed as a Gaussian with dispersion \(v_\mathrm{s}\). Now consider observing a medium where the particles, in their local rest frame, isotropically emit photons at wavelength \(\lambda_0\) (or frequency \(\nu_0\)). The thermal motion is isotropic as well, so we select the \(x\)-axis as the line of sight without loss of generality. Because the particle velocity projected on the \(x\)-axis is distributed as \(g(v_x;0,v_\mathrm{s})\), it follows from the equations in the previous section that the observed line profile is given by \(g(\lambda;\lambda_0,\lambda_0 v_\mathrm{s}/c)\) or \(g(\nu;\nu_0,\nu_0 v_\mathrm{s}/c)\).
We further define the thermal velocity in a medium of particles with mass \(m\) at temperature \(T\) as
\[ v_\mathrm{th} = \sqrt{\frac{2kT}{m}} = \sqrt{2}\,v_\mathrm{s}. \]
This value corresponds to the most probable particle speed, i.e. the point where the probability distribution of the velocity vector norm reaches its maximum value. This result is easily derived, see e.g. Wikipedia.
One often considers an additional source of line broadening caused by subgrid processes other than those corresponding to the macroscopic temperature. This motion is characterized by the turbulent velocity \(v_\mathrm{turb}\), and the total velocity dispersion is then defined as
\[ \sqrt{v_\mathrm{th}^2 + v_\mathrm{turb}^2} = \sqrt{2}\,v_\mathrm{s}. \]
We can also represent subgrid turbulence as an artifical second term in an effective temperature, \(T_\mathrm{eff}\), that can be plugged into the regular equations for macroscopic thermal motion. Indeed, according to our treatment above, we have
\[ v_\mathrm{s}^2 = \frac{1}{2} (v_\mathrm{th}^2 + v_\mathrm{turb}^2) = \frac{1}{2} (\frac{2kT}{m} + v_\mathrm{turb}^2), \]
which leads to
\[ T_\mathrm{eff} = T + \frac{m v_\mathrm{turb}^2}{2k}. \]
The SKIRT implementation is consistent with the analysis in the previous sections. This includes the following areas in the code:
Many authors in this field omit the factor \(1/2\) in the exponent of the Gaussian-like distribution (adjusting the normalization accordingly) and use \(v_\mathrm{th}\) as the dispersion-like quantity instead of \(v_\mathrm{s}\). While this is equivalent to our treatment, it can be confusing, especially in combination with variations in the definitions or names of the involved variables.