Analytical Solutions for Radiation-Driven Winds in Massive Stars II: The $\delta$-slow Regime

TL;DR

This paper develops an analytical velocity profile for the delta-slow wind regime in massive stars, enabling easier calculation of mass-loss rates and linking new line-force parameters with established models.

Contribution

It introduces an approximate analytical expression for the line-force and a closed-form velocity profile for the delta-slow regime, expanding previous work on fast winds.

Findings

Analytical velocity profile derived using Lambert W function.

Good agreement between analytical and numerical models.

Established relation between new and known line-force parameters.

Abstract

Accurate mass-loss rates and terminal velocities from massive stars winds are essential to obtain synthetic spectra from radiative transfer calculations and to determine the evolutionary path of massive stars. From a theoretical point of view, analytical expressions for the wind parameters and velocity profile would have many advantages over numerical calculations that solve the complex non-linear set of hydrodynamic equations. In a previous work, we obtained an analytical description for the fast wind regime. Now, we propose an approximate expression for the line-force in terms of new parameters and obtain a velocity profile closed-form solution (in terms of the Lambert $W$ function) for the $\delta$-slow regime. Using this analytical velocity profile, we were able to obtain the mass-loss rates based on the m-CAK theory. Moreover, we established a relation between this new set of line-force parameters with the known stellar and m-CAK line-force parameters. To this purpose, we calculated a grid of numerical hydrodynamical models and performed a multivariate multiple regression. The numerical and our descriptions lead to good agreement between their values.