Modeling absorbers
Astrocook includes several tools to model the absorption features observed in quasar spectra. Absorbers are modeled with composite Voigt profiles, which are fitted to the data using the Levenberg-Marquardt algorithm for least-squares minimization, as implemented by the Lmfit package.
To model absorbers, the spectrum must be first normalized to continuum. 🚧
Table of contents
Voigt-profile modeling
The Voigt profile is the convolution of a Gaussian and a Cauchy-Lorentz distribution. In the context of line fitting, it is typically parametrized as a function of three quantities that describe the absorbing medium: its redshift \(z\), its column density \(N\), and its Doppler broadening \(b\) (due to the thermal and/or turbulent motions of the particles). The profile of an absorption line as a function of the wavelength \(\lambda\) is thus \(e^{-\tau_\lambda}\), where the opacity \(\tau_\lambda\) is computed as
\[\tau_\lambda=N\frac{\sqrt{\pi}e^2}{m_ec}\frac{f}{\Delta\nu_b}V(a,u_\lambda),\]with \(e\) the electron charge, \(m_e\) the electron mass, \(c\) the speed of light, and \(f\) the oscillator strength of the transition producing the line. The Voigt function \(V(a,u)\) id defined as
\[V(a,u_\lambda)=\frac{a}{\pi}\int_\infty^\infty\frac{e^{-y^2}}{a^2+(u_\lambda-y)^2}dy,\]where \(a\) and \(u_\lambda\) depend on \(z\) and \(b\) as follows:
\[a=\frac{\Gamma}{4\pi b}\frac{\lambda_\mathrm{obs}}{1+z},\] \[u_\lambda=\frac{c}{b}\left(\frac{\lambda}{\lambda_\mathrm{obs}}-1\right),\]with \(\Gamma\) the transition damping constant and \(\lambda_\mathrm{obs}\) the observed wavelength of the line.
Voigt-profile modeling assumes that an arbitrarily complex absorbers can be analyzed as a superposition of finitely-many discrete components, each one characterized by a single set of parameters \((z,N,b)\). This is of course an oversimplification. The decomposition is not always unique, as the parameters may be degenerate across different components and even within a single component (\(N\) and \(b\) are degenerate when a line is saturated). It is generally possible to find a composite Voigt profile which is fitting to the data and complies with a chi-squared test or a given information criterion, but its physical interpretation is not always straightforward.
In Astrocook, each component in a Voigt-profile fit is called an absorption system (or simply a system) and appears as a separate entry in the corresponding list. In the list, the column series specifies if each system is a single line (e.g. Ly_a for Lyman-alpha) or a multiplet of lines (e.g. Ly, including all transitions in the Lyman series, or CIV, including the two members of the C iv doublet at 154.8204 and 155.0781 nm). When a system is a multiplet, it includes several Voigt profiles centered at different wavelengths, and with different equivalent widths depending on the oscillator strength of the multiplet members. All these profiles are nevertheless described by a single set of parameters \((z,N,b)\).
Here you can find a complete list of the ionic transitions used to model absorption features.