# Radiation-Driven Wind Hydrodynamics of Massive Stars: A Review

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Radiation Force

#### 2.1. Radiative Force Due to Electron Scattering

#### 2.2. Radiative Force due to Lines

#### 2.2.1. The Sobolev Approximation

#### 2.2.2. The Line Force due to a Single Line

- (i)
- $({F}_{\nu}\phantom{\rule{0.166667em}{0ex}}\Delta {\nu}_{d}/c)$ is the rate of momentum emitted by the star per unit area at frequency $\nu $ with bandwidth $\Delta {\nu}_{d}$;
- (ii)
- $({\tau}_{l}/{k}_{l})=\rho \phantom{\rule{0.166667em}{0ex}}{v}_{\mathrm{th}}/(dv/dr)$ represents the amount of mass that can absorb this momentum;
- (iii)
- $(1-{e}^{-{\tau}_{l}})$ is the probability that such an absorption occurs.

#### 2.2.3. The Line Force due to a Statistical Distribution of Line Strength

#### 2.2.4. The Correction Factor

#### 2.3. The Ionization Balance

## 3. The m-CAK Hydrodynamic Model

## 4. Topological Analysis

#### The Critical Point Function R

- (i)
- Set the density at the stellar surface to a specific value,$$\rho \left({R}_{*}\right)={\rho}_{*}$$Usually this base density is in the range ${10}^{-8}\phantom{\rule{0.166667em}{0ex}}\mathrm{g}\phantom{\rule{0.166667em}{0ex}}{\mathrm{cm}}^{-3}$ to ${10}^{-13}\phantom{\rule{0.166667em}{0ex}}\mathrm{g}\phantom{\rule{0.166667em}{0ex}}{\mathrm{cm}}^{-3}$. For some examples, see the works of de Araujo and de Freitas Pacheco [47], Friend and MacGregor [48], Madura et al. [49], Curé [50] and Araya et al. [51].
- (ii)
- Set the optical depth integral to a specific value, i.e.,$${\tau}_{*}={\int}_{{R}_{*}}^{\infty}{\sigma}_{E}\phantom{\rule{0.166667em}{0ex}}\rho \left(r\right)dr=\frac{2}{3}.$$

## 5. Types of Solutions

#### 5.1. Fast Solution

#### 5.2. $\mathsf{\Omega}$-Slow Solution

#### 5.3. $\delta $-Slow Solution

#### 5.4. The $\beta $-Law Approximation

## 6. Analytical Wind Solutions

#### 6.1. Solution of the Dimensionless Equation of Motion

#### 6.2. The Fast Regime

#### 6.3. The $\delta $-Slow Regime

## 7. Summary and Discussion

- The self-consistent CAK procedure [44], based on the m-CAK model. Here, we iteratively calculate the line force parameters using the atomic line database from CMFGEN, coupled with the m-CAK hydrodynamic until convergence. We obtain the line force parameters and, therefore, the velocity profile and the mass loss rate. Thus, none of the input parameters are ’free’, but self-consistently calculated.
- The LambertW procedure [45]. In this procedure, we start using a $\beta $-law and a value for $\dot{M}$ in CMFGEN. After convergence, we calculate the line acceleration as a function of r, and, using the LambertW function, we obtain a new velocity profile. This is inserted in CMFGEN and the cycle is repeated until convergence. In this LambertW procedure, the only input parameter is the mass loss rate.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

ZAMS | Zero-Age Main Sequence |

EoM | Equation of Motion |

CAK | Castor et al. [16] |

RHS | Right Hand Side |

MMR | Multivariate Multiple Regression |

WLR | Wind momentum–Luminosity Relationship |

VDD | Viscous Decretion Disc |

## Appendix A

## Appendix B

## Notes

1 | The surface gravity g is given in CGS units, i.e., cm/s${}^{2}$. The quantity $logg$ is dimensionless, see Matta et al. [19]. |

2 | Electrons with a velocity distribution function given by the Maxwellian distribution. |

3 | This equation is for the direct radiation force as no scattering contributions are included within the Sobolev approximation. |

4 | The subscript s means at the singular point. |

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**Figure 1.**Function $R(u,Z)$ for a typical O5 V star without rotation. The (

**left panel**) shows only the function $R(u,Z)$, while the (

**right panel**) is similar to the (

**left panel**), but the plane $R(u,Z)=0$ is also plotted in light grey. Furthermore, the intersection of both curves (black solid lines) shows two loci of singular points.

**Figure 2.**Velocity profile for a typical O5 V star without rotation. The velocity profile is plotted as a function of $log(r/{R}_{*}-1)$ (

**left panel**) and as a function of u (

**right panel**). The location of the singular point is shown with a red dot, while the sonic point is in black.

**Figure 3.**The radiative acceleration, ${g}^{\mathrm{line}}$, for a typical O5 V star without rotation as function of $r/{R}_{*}$ for $r<10\phantom{\rule{0.166667em}{0ex}}{R}_{*}$.

**Figure 4.**Dependence of the wind parameters as a function of the line force parameter $\alpha $. Terminal velocity (

**left panel**) and mass loss rate (

**right panel**). The values obtained for our typical O5 V star without rotation are shown in red.

**Figure 5.**Dependence of the wind parameters as a function of the line force parameter k. Terminal velocity (

**left panel**) and mass loss rate (

**right panel**). The values obtained for our typical O5 V star without rotation are shown in red.

**Figure 6.**Dependence of the wind parameters as a function of the line force parameter $\delta $. Upper panels are for $k=0.124$ and lower panels are for $k=0.0124$. The values obtained for our typical O5 V star without rotation are shown in red.

**Figure 7.**The function $R(u,Z)$ (topology) of the m-CAK theory as function of $\mathsf{\Omega}$: $\mathsf{\Omega}=0.3$ (

**upper left panel**), $\mathsf{\Omega}=0.5$ (

**upper right panel**), $\mathsf{\Omega}=0.7$ (

**lower left panel**) and $\mathsf{\Omega}=0.9$ (

**lower right panel**). The plane $R(u,Z)=0$ is shown in light grey, and its intersection with the surface $R(u,Z)$ (locus of singular points) is plotted with a black line.

**Figure 8.**Velocity profiles, $v\left(u\right)$, as function of the rotational rate $\mathsf{\Omega}$. (

**Left panel**): fast solutions ($\mathsf{\Omega}\lesssim 0.74$) are plotted in grey lines, while the $\mathsf{\Omega}$-slow solutions are in coloured lines: $\mathsf{\Omega}=0.74$ (red line), $\mathsf{\Omega}=0.76$ (blue line), $\mathsf{\Omega}=0.80$ (cyan line), $\mathsf{\Omega}=0.82$ (magenta line), $\mathsf{\Omega}=0.85$ (green line), $\mathsf{\Omega}=0.90$ (black line) and $\mathsf{\Omega}=0.95$ (orange line). (

**Right panel**): The same $\mathsf{\Omega}$-slow solutions, but zoomed and including the location of the singular points (red dots).

**Figure 9.**Wind density profiles, $\rho \left(u\right)$ (in gr/cm${}^{3}$) versus u, for fast and $\mathsf{\Omega}$-slow solutions. The colour scheme is the same as the one used in Figure 8.

**Figure 10.**The topological function $R(u,Z)$ of the m-CAK theory as function of $\delta $. (

**Upper left panel**): $\delta =0.1$. (

**Upper right panel**): $\delta =0.12$. (

**Lower left panel**): $\delta =0.24$. (

**Lower right panel**): $\delta =0.3$. The plane $R(u,Z)=0$ is shown in light grey, and its intersection with the surface $R(u,Z)$ (locus of singular points) is plotted with black lines.

**Figure 11.**Velocity profiles, $v\left(u\right)$, for different values of the line force parameter $\delta $. (

**Left panel**): fast solutions are plotted in grey lines for $\delta =0.0,0.1,0.2,0.24$, while $\delta $-slow solutions are in coloured lines. The red line corresponds to $\delta =0.3$, the blue line to $\delta =0.31$, the cyan line to $\delta =0.32$ and the magenta line to $\delta =0.33$. The (

**right panel**) shows only $\delta $-slow solutions, and the location of the singular points for each solution is shown with a red dot.

**Figure 12.**Fast solution velocity profile (solid blue), $v\left(u\right)$ vs. u. Six different $\beta $-law velocity profiles are also plotted. It is clearly seen that the $\beta $-law approximation is a good one for $0.7\lesssim \beta \lesssim 1.2$. See text for details.

**Figure 13.**$\delta $-slow solution velocity profile (solid blue), $v\left(u\right)$ vs. u. Six different $\beta $-law velocity profiles are also plotted. For values around $\beta =0.7$, the profiles can be considered similar, but it can be clearly concluded that for $\beta >0.7$, the $\beta $-law profile cannot fit the m-CAK hydrodynamical $\delta $-slow solution.

**Figure 14.**$\mathsf{\Omega}$-slow solution velocity profile (solid blue), $v\left(u\right)$ vs. u. Six different $\beta $-law velocity profiles are also plotted. It can be clearly concluded that no $\beta $-law profile can fit the m-CAK hydrodynamical $\mathsf{\Omega}$-slow solution.

**Figure 15.**Velocity profiles as a function of the inverse radial coordinate $u=-{R}_{*}/r=-1/\widehat{r}$ for four models. The hydrodynamic results from Hydwind are shown in solid blue lines and the analytical solutions are shown by dashed lines. The stellar and line force parameters for the models are given in Araya et al. [60].

**Figure 16.**Velocity profiles of $\u03f5$ Ori as a function of $log(r/{R}_{*}-1)$ in a region near to the stellar surface. The solid blue line shows the numerical hydrodynamic result and the analytical solution is shown by a dashed line. The dot symbol indicates the position of the sonic (or critical) point. The difference between both curves is around one thermal speed.

**Figure 17.**Real and complex regions where the line acceleration expression given by Villata [58] can be found. These regions are delimited by the values of the line force parameters $\alpha $ and $\delta $.

**Figure 18.**Hydrodynamic models in the ${T}_{\mathrm{eff}}$-$log\phantom{\rule{0.166667em}{0ex}}g$ plane. Blue dots represent the converged solutions. Grey solid lines are the evolutionary tracks for stars of $7{M}_{\odot}$ to $60{M}_{\odot}$ without rotation [72], and black lines represent the zero-age main sequence (ZAMS) and the terminal age main sequence (TAMS).

**Figure 19.**Velocity profiles as a function of the inverse radial coordinate $u=-{R}_{*}/r=-1/\widehat{r}$ for the model R24 from Curé et al. [27]. The hydrodynamic result from Hydwind is shown in the solid blue line and the analytical solution is a dashed black line.

Parameter | Range |
---|---|

$\alpha $ | 0.45–0.69 (step size of 0.02) |

k | 0.05–1.00 (step size of 0.05) |

$\delta $ | 0.26–0.35 (step size of 0.01) |

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## Share and Cite

**MDPI and ACS Style**

Curé, M.; Araya, I. Radiation-Driven Wind Hydrodynamics of Massive Stars: A Review. *Galaxies* **2023**, *11*, 68.
https://doi.org/10.3390/galaxies11030068

**AMA Style**

Curé M, Araya I. Radiation-Driven Wind Hydrodynamics of Massive Stars: A Review. *Galaxies*. 2023; 11(3):68.
https://doi.org/10.3390/galaxies11030068

**Chicago/Turabian Style**

Curé, Michel, and Ignacio Araya. 2023. "Radiation-Driven Wind Hydrodynamics of Massive Stars: A Review" *Galaxies* 11, no. 3: 68.
https://doi.org/10.3390/galaxies11030068