Due to their high values of luminosity and effective temperature, massive hot stars have strong stellar winds that are driven by the scattering of continuum radiation in spectral lines. CMFGEN (Hillier & Miller 1998) is a state of the art stellar atmosphere code for modeling observables of hot stars such as OB-type, LBV, and Wolf-Rayet stars. This code provides unified photosphere-wind models from solving the radiative transfer problem in non-LTE (local thermodynamic equilibrium) conditions assuming a symmetric spherical wind. Further information on the code CMFGEN can be found at:
The main physical parameters of stellar winds are the mass-loss rate (Ṁ) and the terminal velocity (v∞). Considering a spherically symmetrical stationary wind, the mass-loss rate and the wind velocity structure are related to each other by the mass conservation equation as shows in the following. Here, the wind velocity v(r) is parameterized, in its simplest form, by the so-called β-law approximation, where the v(r) is given by the terminal velocity and the wind velocity law exponent (β).
In the following, I show the emergent spectrum in the wavelegnth region of∼6530-6600 Å, around the Hα line line profile, from a test CMFGEN calculated by myself for studying the archetype Luminous Blue Variable (LBV) star P Cygni. Further details on these spectroscopic and interferometric studies on P Cygni are found in Rivet et al. (2020) and de Almeida et al. (2022). Here, by "test model" I mean that the the model parameters are not tuned from the fit to P Cygni's observations. Nevertheless, this model has a high enough value of mass-loss rate (1.0 x 10⁻⁵ M⊙ yr⁻¹) to produce a strong P Cygni profile in the Hα line. Note that the C II λλ6580,6585 lines also form P Cygni profiles, but very weak in comparison with Hα.
From the solution of the radiative transfer problem in the photosphere+wind system, CMFGEN provides a description of the specific intensity as a function of the impact parameter (p) in the (p,z) coordinate system that is related to the a radial coordenate r. Further details on the (p,z) coordinate system can be found in Fig. 7-29 of Mihalas 1978. Thus, higher values of impact parameter mean larger distances from e stellar photosphere throughout the wind region.
From the same CMFGEN model previously described, figure below shows the intensity profiles for each value of impact parameter as function of wavelength around the Hα line. For the sake of clarity, intensity profiles shown in yellow/orange come closer to the photosphere, while one the one shown in purple/violet are calculated at larger distances towards the wind region of the model.