Self-consistent Solutions for Line-driven Winds of Hot Massive Stars: The m-CAK Procedure

To calculate the line-acceleration and obtain a proper value of ${ \mathcal M }(t)$, Abbott (1982) established that it is necessary to sum the contribution of hundreds of thousands of spectral lines participating in the line-acceleration processes. Therefore, aiming to get the most accurate value of ${ \mathcal M }(t)$, we decided to employ around ∼900,000 line transitions. These atomic data were obtained (and modified in format) from the atomic database list used by the code CMFGEN⁶ (Hillier 1990; Hillier & Miller 1998). Specifically, we have extracted information related to energy levels, degeneracy levels, partition functions, and oscillator strengths f_l, which are necessary to calculate the absorption coefficient η_line of each line in terms of lower (l) and upper (u) level populations n_l and n_u, and their statistical weights g_l and g_u. The absorption coefficient η_line is given by:

$$\begin{eqnarray}&&{\eta }_{\mathrm{line}}=\displaystyle \frac{\pi {e}^{2}}{{mc}}{f}_{l}\displaystyle \frac{{n}_{l}}{\rho (r)}\left(1-\displaystyle \frac{{n}_{u}}{{n}_{l}}\displaystyle \frac{{g}_{l}}{{g}_{u}}\right).\end{eqnarray} \tag{ 9 }$$

Elements and ionization stages considered in this work are listed in Table 1.

Line-acceleration is calculated over the contribution of numerous transitions for every element at every ionization stage present in the wind. Abbott (1982) determined the ionization degrees using the Saha's equation for extended atmospheres (Mihalas 1978), namely:

$$\begin{eqnarray}&&{\left(\displaystyle \frac{{N}_{i+1}}{{N}_{i}}\right)}_{\mathrm{LTE}}=2{\left(\displaystyle \frac{2\pi {m}_{e}{k}_{B}}{{h}^{2}}\right)}^{3/2}\displaystyle \frac{{T}_{R}\sqrt{{T}_{e}}}{{N}_{e}/W}\displaystyle \frac{{U}_{i+1}}{{U}_{i}}{e}^{-\tfrac{{E}_{i}}{{k}_{B}{T}_{e}}},\end{eqnarray} \tag{ 10 }$$

where T_R, T_e are the radiation and electron temperatures, respectively, and E_i is the ionization energy from stage i to i + 1. More precise treatment called approximate NLTE (hereafter quasi-NLTE) has been used by, e.g., Mazzali & Lucy (1993) and Noebauer & Sim (2015). Here the ionization balance is determined by the application of the modified nebular approximation (Abbott & Lucy 1985). Following this treatment, the ratio of number densities for two consecutive ions can be expressed in term of its LTE value, multiplied by correction effects due to dilution of radiation field and recombinations:

$$\begin{eqnarray}&&\displaystyle \frac{{N}_{i+1}}{{N}_{i}}\approx {\left(\displaystyle \frac{{N}_{e}}{W}\right)}^{-1}[{\zeta }_{i}+W(1-{\zeta }_{i})]\sqrt{\displaystyle \frac{{T}_{e}}{{T}_{R}}}{\left(\displaystyle \frac{{N}_{i+1}{N}_{e}}{{N}_{i}}\right)}_{\mathrm{LTE}},\end{eqnarray} \tag{ 11 }$$

where ζ_i represents the fraction of recombination processes that go directly to the ground stage. Equation (11) is an alternative description to the one given by Puls et al. (2005), who included a different radiative temperature dependence in the wind, which is especially important in the far-UV region of the spectrum that is not optically thick.

Modifications in the treatment of atomic populations X_i, with i being the excitation level, are also based on the work of Abbott & Lucy (1985). It is necessary to make a distinction between metastable levels (with no permitted electromagnetic dipole transitions to lower energy levels) and all the other ones:

$$\begin{eqnarray*}\left(\displaystyle \frac{{X}_{i}}{{X}_{1}}\right)=\left\{\begin{array}{cc}{\left(\tfrac{{X}_{i}}{{X}_{1}}\right)}_{\mathrm{LTE}} & \mathrm{metastable}\ \mathrm{levels}\\ W(r){\left(\tfrac{{X}_{i}}{{X}_{1}}\right)}_{\mathrm{LTE}} & \mathrm{others}.\end{array}\right.\end{eqnarray*}$$

Atomic partition functions, U_i (necessary for Saha's equation and the calculation of atomic populations), are calculated following the formulation of Cardona et al. (2010), i.e.,

$$\begin{eqnarray}&&{U}_{i}={U}_{i,0}+{G}_{{jk}}{e}^{-{\varepsilon }_{{jk}}/T}+\displaystyle \frac{m}{3}({n}^{3}-343){e}^{-{\hat{E}}_{n* {jk}}/T},\end{eqnarray} \tag{ 12 }$$

where U_i,0 are the constant partition functions, ${\hat{E}}_{n* {jk}}$ is the mean excitation energy of the last level of the ion, n is the maximum excitation stage to be considered, while G_jk, ε_jk, and m are parameters tabulated by Cardona et al. (2010).

The advantage of this treatment is that it provides values for atomic partition functions explicitly as a function of temperature and implicitly of electron density, giving a more accurate ionization balance. Following Noebauer & Sim (2015), the temperature will be treated as a constant (T_R = T_e = T_eff). Then, for a specific value of (T_eff, N_e), the ratio between number densities of ionization stages j and i (for a specific Z-element) is calculated by a matrix (hereafter ionization matrix) given by:

$$\begin{eqnarray}&&{D}_{Z,i,j}=\displaystyle \frac{{N}_{j}}{{N}_{i}}=\displaystyle \prod _{i\leqslant k\lt j}\displaystyle \frac{{N}_{k+1}}{{N}_{k}}.\end{eqnarray} \tag{ 13 }$$

In reference to the abundances of the different chemical elements, these were adopted from the solar abundances given by Asplund et al. (2009). However, these can be easily modified to evaluate stars with nonsolar metallicity (see Section 4).

At this point, it is necessary to remark that previous authors (Abbott 1982; Noebauer & Sim 2015) have considered the diluted-electron density N_e/W as constant throughout the wind. Nevertheless, to calculate δ, ${ \mathcal M }(t)$ must be evaluated considering changes in the ionization stages, and therefore N_e(r)/W(r). Since, the calculation of electron density depends on the ionization stages of each species which in turn are functions of N_e, we deal with a coupled nonlinear problem. To obtain a solution, we use the following formula to calculate (as an initial value) the electron number density:

$$\begin{eqnarray}&&{N}_{e,0}=\displaystyle \frac{\rho (r)}{{m}_{{\rm{H}}}}\,\displaystyle \frac{{X}_{{\rm{H}}}+2{X}_{\mathrm{He}}}{{X}_{{\rm{H}}}+4{X}_{\mathrm{He}}},\end{eqnarray} \tag{ 14 }$$

with m_H being the hydrogen atom mass, and X_H and X_He the abundances of hydrogen and helium, respectively.

We used this initial electron density to start calculating the ionization matrix and to recalculate both atomic populations and electron density iteratively:

$$\begin{eqnarray}\begin{array}{rcl}{N}_{e}(r) & = & \left({X}_{{\rm{H}}}\displaystyle \frac{{D}_{\mathrm{1,1,2}}}{1+{D}_{\mathrm{1,1,2}}}+{X}_{\mathrm{He}}\displaystyle \frac{({D}_{\mathrm{2,1,2}}+2{D}_{\mathrm{2,1,3}})}{1+{D}_{\mathrm{2,1,2}}+{D}_{\mathrm{2,1,3}}}\right)\\ & & \times \ \displaystyle \frac{\rho (r)}{{X}_{{\rm{H}}}+4{X}_{\mathrm{He}}}.\end{array}\end{eqnarray} \tag{ 15 }$$

Convergence of N_e is easily obtained after just a few iterations (see Figure 1). It is important to remark that even when we use N_e,0 as a constant value (not described by Equation (14)), the final converged value for N_e is the same.

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Together with an accurate treatment of atomic populations and electron density, Equation (5) requires as an input the radiation field in the term F_ν/F.

Abbott (1982) used the radiation fields from Kurucz' models (Kurucz 1979), whereas Noebauer & Sim (2015) from a blackbody. In this work, we use the radiation field calculated by the NLTE line-blanketing plane-parallel code Tlusty (Hubeny & Lanz 1995; Lanz & Hubeny 2003).

The overlap effects among tens of thousands of spectral lines are not considered when we sum the contributions to the force multiplier ${ \mathcal M }(t)$ across the wind. However, line-blanketing effects are partially considered as we are using the Tlusty radiation field in the calculations of ${ \mathcal M }(t)$.

Previous studies by Abbott (1982) and Noebauer & Sim (2015) have considered a fixed range for the optical depth t to fit the force multiplier (Equation (8)).

However, given the definition of t (Equation (6)), it is clear that the optical depth range is constrained by the physical properties of the stellar wind (density and velocity profiles). For this reason, calculations presented in this work are constrained inside the wind, characterized by this range of t.

Because m-CAK theory is based upon Sobolev approximation (Sobolev 1960; Lamers & Cassinelli 1999) in this work we will use as upper and lower limits of t its values at the sonic point and at infinity (usually r ∼ 100 R_*), respectively. It is important to remark that although t decreases outward it never reaches zero. Therefore, it is always possible to define a proper range.

Velocity profile and $\dot{M}$ from hydrodynamics is required in order to calculate the line-acceleration g_line. At the same time, line-force parameters fitted from g_line, are necessary to solve the m-CAK hydrodynamic equations and obtain the mass-loss rate and velocity profile. Therefore, a self-consistent iterative procedure should be implemented to solve this coupled nonlinear problem.

Our procedure is the following: (i) using a β-law profile with a given mass-loss rate, initial values for the line-force parameters (k₀, α₀, δ₀) are calculated; (ii) a numerical solution of the equation of motion (Equation (3)) is obtained with HydWind,⁷ getting an improved hydrodynamics: v(r) and $\dot{M};$ (iii) a new force multiplier is calculated; (iv) new line-force parameters (k_i, α_i, δ_i) are fitted from ${ \mathcal M }(t)$; and (v) steps ii–iv are iterated. Convergence is usually obtained after ∼4–5 iterations (see Figure 2), independently on the initial values. Our criterion for convergence is when two consecutive iterations (i, i + 1) get a value for $\parallel {\rm{\Delta }}p\parallel =\parallel {p}^{i+1}-{p}^{i}\parallel \,\leqslant {10}^{-3}$, where p is a line-force parameter and this condition should be satisfied for each one of these parameters.

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Figure 3 shows the convergence of the mass-loss rate (top panel) and the terminal velocity (lower panel) as a function of the procedure's iterations. Both values depend nonlinearly on the stellar and line-force parameters.

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To compare our line-force parameters with the results obtained by Abbott (1982) and Noebauer & Sim (2015), we use one single-step only. Following these authors, δ and N_e/W are set as input and the optical depth range is fixed between $-6\lt \mathrm{log}\,t\lt -1$. The selection of a fixed interval of $\mathrm{log}\,t$ does not require any velocity field structure. Furthermore, we have considered Kurucz' and blackbody fluxes to reproduce Abbott (1982) and Noebauer & Sim (2015) calculations, respectively. Then, starting from a β-law and a $\dot{M}$, we calculate k₁ and α₁ (single-step). These results are shown in Table 2. The coefficients of determination, R-Squared, for α and k (respectively) between previous and our single-iteration results are (i) ${R}_{\alpha }^{2}=0.87$ and ${R}_{k}^{2}=0.93$ for T_eff ≥ 40,000 K; (ii) ${R}_{\alpha }^{2}=0.4$ and ${R}_{k}^{2}=0.81$ for T_eff ≥ 30,000 K. We conclude that our calculations reproduced previous results, now using a modern atomic database and abundances.

The following results are computed self-consistently with the methodology detailed in Section 3.

Self-consistent solutions for a grid of models are presented in Table 3. The effective temperature ranges from 32 to 45 kK and log g from 3.4 to 4.0 dex. This grid considers different stellar radii and two abundances: 1 and 1/5 of the solar value. This table shows the stellar parameters, the calculated t-range, and the fitted m-CAK line-force. In addition, we calculated the corresponding wind solution using HydWind, and their error margins were derived considering variations of ΔT_eff = ±500, ${\rm{\Delta }}\mathrm{log}\,g=\pm 0.05$, and ΔR_* = ±0.1 R_⊙ in the stellar radius, keeping constant the line-force parameters.

Table 3.  Self-consistent Line-force Parameters (k, α, δ) for Adopted Standard Stellar Parameters, Together with the Resulting Terminal Velocities and Mass-loss Rates (${\dot{M}}_{\mathrm{SC}}$)

Teff	$\mathrm{log}\,g$	R*/R⊙	Z/Z⊙	$\mathrm{log}\,{t}_{\mathrm{in}}$	$\mathrm{log}\,{t}_{\mathrm{out}}$	k	α	δ	${v}_{\infty }^{\mathrm{SC}}$	${\dot{M}}_{\mathrm{SC}}$	${\dot{M}}_{\mathrm{SC}}/{\dot{M}}_{\mathrm{Vink}}$
(kK)									(km s−1)	(10−6 M⊙ yr−1)
45	4.0	12.0	1.0	−0.31	−4.53	0.167	0.600	0.021	3 432 ± 240	$2.0{\pm }_{0.5}^{0.65}$	1.00
45	4.0	12.0	0.2	−0.77	−4.85	0.142	0.493	0.017	2 329 ± 210	$0.38{\pm }_{0.11}^{0.15}$	0.74
45	3.8	16.0	1.0	0.28	−4.07	0.135	0.648	0.022	3 250 ± 300	$6.4{\pm }_{1.3}^{1.6}$	0.84
45	3.8	16.0	0.2	−0.06	−4.28	0.114	0.545	0.014	2 221 ± 230	$1.7{\pm }_{0.45}^{0.6}$	0.88
42	3.8	16.0	1.0	−0.10	−4.36	0.137	0.629	0.027	3 235 ± 300	$3.4{\pm }_{0.7}^{0.9}$	0.94
42	3.8	16.0	0.2	−0.55	−4.73	0.108	0.534	0.019	2 313 ± 230	$0.73{\pm }_{0.21}^{0.3}$	0.79
42	3.6	20.4	1.0	0.70	−3.80	0.122	0.671	0.039	2 738 ± 230	$11{\pm }_{2.5}^{3.5}$	0.74
42	3.6	20.4	0.2	0.37	−4.09	0.091	0.586	0.022	2 043 ± 200	$3.1{\pm }_{0.75}^{1.2}$	0.82
40	4.0	12.0	1.0	−0.88	−4.97	0.164	0.581	0.027	3 300 ± 220	$0.66{\pm }_{0.15}^{0.19}$	1.17
40	4.0	12.0	0.2	−1.43	−5.44	0.133	0.492	0.038	2 329 ± 160	$0.11{\pm }_{0.03}^{0.05}$	0.76
40	3.6	20.4	1.0	0.42	−3.96	0.118	0.659	0.044	2 813 ± 290	$6.6{\pm }_{1.4}^{1.8}$	0.89
40	3.6	20.4	0.2	−0.05	−4.40	0.091	0.572	0.025	2 116 ± 230	$1.7{\pm }_{0.4}^{0.5}$	0.90
40	3.4	18.0	1.0	1.30	−3.14	0.099	0.715	0.094	1 548 ± 240	$14.5{\pm }_{3.5}^{5.0}$	0.73
40	3.4	18.0	0.2	1.90	−3.50	0.073	0.650	0.047	1 334 ± 230	$4.7{\pm }_{1.3}^{2.4}$	0.92
38	3.8	16.0	1.0	−0.63	−4.79	0.130	0.610	0.036	3 153 ± 240	$1.2{\pm }_{0.25}^{0.3}$	1.10
38	3.8	16.0	0.2	−1.18	−5.28	0.091	0.542	0.033	2 473 ± 300	$0.25{\pm }_{0.06}^{0.08}$	0.89
36	4.0	12.0	1.0	−1.45	−5.50	0.132	0.580	0.036	3 314 ± 200	$0.21{\pm }_{0.05}^{0.065}$	1.17
36	4.0	12.0	0.2	−1.97	−5.97	0.101	0.517	0.068	2 402 ± 140	$0.036{\pm }_{0.01}^{0.014}$	0.78
36	3.6	20.4	1.0	−0.29	−4.55	0.104	0.644	0.062	2 809 ± 240	$2.2{\pm }_{0.5}^{0.7}$	1.12
36	3.6	20.4	0.2	−0.87	−5.09	0.071	0.581	0.033	2 534 ± 220	$0.5{\pm }_{0.13}^{0.17}$	1.00
36	3.4	18.0	1.0	1.78	−3.77	0.091	0.686	0.116	1 708 ± 170	$4.4{\pm }_{1.0}^{1.6}$	1.13
36	3.4	18.0	0.2	0.41	−4.21	0.072	0.607	0.048	1 566 ± 160	$1.0{\pm }_{0.25}^{0.4}$	1.01
34	3.8	16.0	1.0	−1.27	−5.37	0.103	0.604	0.043	3 093 ± 210	$0.34{\pm }_{0.07}^{0.1}$	1.12
34	3.8	16.0	0.2	−1.93	−5.94	0.069	0.555	0.028	2 791 ± 180	$0.074{\pm }_{0.018}^{0.025}$	0.95
34	3.6	20.4	1.0	−0.61	−4.82	0.095	0.637	0.074	2 732 ± 180	$1.2{\pm }_{0.3}^{0.4}$	1.25
34	3.6	20.4	0.2	−1.29	−5.46	0.058	0.590	0.031	2 642 ± 180	$0.25{\pm }_{0.05}^{0.07}$	1.03
32	3.4	18.0	1.0	0.37	−4.30	0.078	0.675	0.159	1 653 ± 190	$1.3{\pm }_{0.3}^{0.5}$	1.67
32	3.4	18.0	0.2	−0.70	−4.15	0.053	0.610	0.052	1 847 ± 140	$0.23{\pm }_{0.05}^{0.075}$	1.16

Note. Ratios between self-consistent mass-loss rates and Vink's recipe values (rescaled to match metallicity from Asplund et al. 2009) using ${v}_{\infty }/{v}_{\mathrm{esc}}=2.6$ are shown in the last column. Error margins for mass-loss rates and terminal velocities are derived over a variation of ±500 for effective temperature, ±0.05 for logarithm of surface gravity, and ±0.1 for stellar radius.

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Convergence has been checked for each solution. Figure 4 shows the last iteration of ${ \mathcal M }(t)$ for four models from Table 3 with different ranges of t. Due to the quasilinear behavior of the logarithm of the force multiplier, parameters k and α are easily fitted and their values can be considered constant throughout the wind (see Section 4.2). To fit δ in the ${ \mathcal M }(t)$–N_e/W plane, it is necessary to perform an extra calculation of ${ \mathcal M }(t)$ using a slightly different value for the diluted-electron density. Last column of this table shows the ratio between our mass-loss rate and the one calculated using Vink's recipe (Vink et al. 2001), with ${v}_{\infty }/{v}_{\mathrm{esc}}=2.6$ and rescaled to current abundances (Asplund et al. 2009). The mean value of ${\dot{M}}_{\mathrm{SC}}/{\dot{M}}_{\mathrm{Vink}}=0.98\pm 0.2$. As we have not included in our procedure multi-line nor line-overlapping processes, we support Puls' (1987) conclusion that these effects are somewhat canceled, because we do not observe relevant discrepancies in the mass-loss rates when a comparison with Vink's recipe is performed.

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In Figure 5, we observe clear trends for the behavior of the (k, α, δ) parameters with T_eff, $\mathrm{log}\,g$, and Z. While k increases and δ decreases as a function of the effective temperature, for both metallicities. It is interesting to remark the influence of the surface gravity on the resulting line-force parameters, values for k and δ decrease as the gravity decreases. Notice that globally our line-force parameter results are similar to the values obtained in previous works (Puls et al. 2000; Kudritzki 2002; Noebauer & Sim 2015). However, we found an important dependence on log g as a result of the hydrodynamic coupling in the self-consistent procedure.

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On the other hand, the behavior of α depends on the metallicity, it increases with effective temperature for solar abundance, but for low abundance and low gravities, it slowly decreases with temperature. Moreover, the change in α is more significant for log g than for T_eff: a difference in ${\rm{\Delta }}\mathrm{log}\,g\,\pm 0.2$ dex produces a Δα ∼ 0.04, whereas variations on ΔT_eff = ±2000 K, might produce Δα ∼ 0.02.

Figure 6 shows the results for the mass-loss rates as a function of the effective temperature, for different gravities and metallicities. The upper panel shows the results from our self-consistent procedure and the bottom panel shows the result using Abbott's methodology (a single iteration) to calculate line-force parameters and apply them in our hydrodynamic code HydWind (hereafter Abbott's procedure). We found that $\dot{M}$ increases with effective temperature and metallicity and decreases with gravity. This behavior is similar to the one obtained using Abbott's procedure, but the self-consistent calculated mass-loss rates are about 30% larger.

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From the mass-loss results tabulated in Table 3, a simple relationship for solar-like metallicities (with a coefficient of determination or R-squared, R² = 0.999) reads:

$$\begin{eqnarray}\begin{array}{rcl}\mathrm{log}{\dot{M}}_{Z=1.0} & = & 10.443\times \mathrm{log}\left(\displaystyle \frac{{T}_{\mathrm{eff}}}{1000\ {\rm{K}}}\right)\\ & & -\ 1.96\times \mathrm{log}\,g\\ & & +\ 0.0314\times ({R}_{* }/{R}_{\odot })\\ & & -\ 15.49,\end{array}\end{eqnarray} \tag{ 16 }$$

and for metallicity Z/Z_⊙ = 0.2 the relationship reads (with R² = 0.999):

$$\begin{eqnarray}\begin{array}{rcl}\mathrm{log}{\dot{M}}_{Z=0.2} & = & 11.668\times \mathrm{log}\left(\displaystyle \frac{{T}_{\mathrm{eff}}}{1000\ {\rm{K}}}\right)\\ & & -\ 2.126\times \mathrm{log}\,g\\ & & +\ 0.04\times ({R}_{* }/{R}_{\odot })\\ & & -\ 17.63,\end{array}\end{eqnarray} \tag{ 17 }$$

where $\dot{M}$ is given in ${10}^{-6}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$. These relationships could be considered analogous to that given by Vink et al. (2000) to obtain theoretical mass-loss rates for solar-like metallicities. However, the advantage of our description is that it depends only on stellar parameters and we do not need to consider the value of ${v}_{\infty }/{v}_{\mathrm{esc}}$. It is important to remark, however, that this formula has been derived for the following ranges:

• 1.  
  T_eff = 32–45 kK
• 2.  
  $\mathrm{log}\,g$ = 3.4–4.25
• 3.  
  M_*/M_⊙ ≥ 25.0.

Concerning terminal velocities, see Figure 7, self-consistent calculations (top panel) show that ${v}_{\infty }$ is almost constant with respect to the effective temperature, but it decreases as a function of $\mathrm{log}\,g$ and Z. On the other hand, Abbott's procedure results do not show the same behavior and exhibit a maximum in the T_eff interval.

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It is important to remember that the range of optical depths used to calculate our self-consistent line-force parameters is defined along almost all the atmosphere of the star, i.e., downstream from the sonic point. This procedure improves the criterion used by Abbott (1982), who determined the parameters at t = 10⁻⁴. This value sometimes lays outside the optical depth range here defined, as shown in Figure 4.

To analyze the change on the line-force parameters due to the selection of the t-range, we define four different intervals inside the whole range of t, and compute these parameters in each range. Table 4 summarizes these calculations. Regarding the uncertainties of our procedure in the terminal velocities, these are of the same order as the uncertainties owed to the errors in the determination of the stellar parameters in the range 32,000 K < T_eff < 40,000 K, while, the uncertainties in $\dot{M}$ are much lower than the ones produced by variations of stellar parameters. These small uncertainties indicate that it is a good approximation to consider line-force parameters as constants throughout the wind. Due to the fact that the entire t-range represents the physical conditions of almost all the wind, we recommend using the complete optical depth range to derive the line-force parameters.

Table 4.  Influence of the Optical Depth Interval on the Line-force Parameters for Some Reference Models Given in Table 3

Teff	$\mathrm{log}\,g$	$\mathrm{log}\,{t}_{\mathrm{in}}$	$\mathrm{log}\,{t}_{\mathrm{out}}$	k	α	δ	$| {\rm{\Delta }}{v}_{\infty }|$	$| {\rm{\Delta }}\dot{M}|$
							(km s−1)	(10−6 M⊙ yr−1)
45,000	4.0	−0.31	−2.03	0.099	0.686	0.037	780	0.23
		−0.31	−2.87	0.107	0.650	0.029	600	0.30
		−0.31	−3.71	0.120	0.638	0.027	420	0.21
		−0.31	−4.55	0.167	0.600	0.021	0	0
40,000	4.0	−0.87	−2.50	0.099	0.633	0.040	521	0.09
		−0.87	−3.32	0.099	0.634	0.036	610	0.07
		−0.87	−4.14	0.107	0.621	0.026	594	0.07
		−0.87	−4.96	0.164	0.581	0.027	0	0
40,000	3.6	0.08	−1.44	0.095	0.666	0.090	247	0.58
		0.08	−2.28	0.098	0.680	0.075	75	0.13
		0.08	−3.12	0.101	0.692	0.067	323	0.92
		0.08	−3.96	0.118	0.659	0.044	0	0
36,000	3.6	−0.29	−2.00	0.084	0.637	0.112	520	0.58
		−0.29	−2.85	0.092	0.648	0.078	114	0.15
		−0.29	−3.70	0.089	0.668	0.075	267	0.01
		−0.29	−4.55	0.104	0.644	0.062	0	0
32,000	3.4	0.37	−1.49	0.066	0.630	0.251	631	0.77
		0.37	−2.43	0.075	0.636	0.221	457	0.57
		0.37	−3.37	0.079	0.662	0.179	168	0.11
		0.37	−4.31	0.078	0.675	0.159	0	0

Note. Absolute values of the differences in the resulting Wind parameters with respect to the reference solution are presented.

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For T_eff < 30,000 K, we found that $\mathrm{log}\,{ \mathcal M }(t)$ is no longer linear with respect to $\mathrm{log}\,t$ and the corresponding line-force parameters can be approximated to a linear piecewise description. Due to this reason, we establish that our set of self-consistent solutions describes stellar winds for effective temperatures and $\mathrm{log}\,g$ in the range 32,000–45,000 K and 3.4–4.0 dex, respectively.

It is well known that the scaling relation for the terminal velocity in the frame of the line-driven wind theory. This relation (Puls et al. 2008) reads:

$$\begin{eqnarray}&&{v}_{\infty }\approx 2.25\,\sqrt{\displaystyle \frac{\alpha }{1-\alpha }}\,{v}_{\mathrm{esc}}.\end{eqnarray} \tag{ 18 }$$

This is an approximation of the formula found by Kudritzki et al. (1989, their Equations (62) to (65)).

In Figure 12 we have plotted ${v}_{\infty }/{v}_{\mathrm{esc}}$ versus $\sqrt{\alpha /(1-\alpha )}$ using the results from Table 3. Contrary to the expected result (Equation (18)) for solar abundances, we find a different linear behavior that strongly depends on the value of $\mathrm{log}\,g$. This is a new result that comes from applying our self-consistent procedure. The m-CAK equation of motion shows an interplay between the gravity ($\mathrm{log}\,g$) and the line-force term. This balance of forces defines the location of the singular point and therefore fixes the value of $\dot{M}$. As a consequence, the velocity profile depends also on the value of $\mathrm{log}\,g$. This result cannot be obtained from Equation (18) which is an oversimplification of this nonlinear coupling. However, Equation (18) presents a fair fit when Z = Z_⊙/5, where the dependence of the slope on $\mathrm{log}\,g$ is weak because the radiation force is driven by fewer ions.

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The dependence of ${v}_{\infty }/{v}_{\mathrm{esc}}$ on $\mathrm{log}\,g$ yield that stars with solar abundances present an intrinsic variations of ${v}_{\infty }/{v}_{\mathrm{esc}}$ in the range of 2.4–3.7, as shown in Figure 12. This range might explain the scatter observed on the hot side of the bistability jump shown by Markova & Puls (2008, in their Figure 12).

In this section we want to compare our theoretical results with the ones obtained from line-profile fittings for homogeneous (unclumped) winds with a β-law, and the mass-loss (recipe) from Vink et al. (2000).

Table 7 shows our results for the only two O-type star reported by Bouret et al. (2005): HD 96715, T_eff = 43.5 kK, $\mathrm{log}\,g=4.0$, and HD 1904290A, T_eff = 39 kK, $\mathrm{log}\,g=3.6$. These results were obtained for the self-consistent solution together with the ones after just one iteration starting from a β-law. It is observed that models starting from a β-law largely overestimate the terminal velocity and slightly underestimate the mass-loss rate. Self-consistent calculations find a fairly good agreement to both: the observed mass-loss rate and terminal velocity. For the mass-loss rate in this figure, we have included the result calculated using Vink et al.'s (2000) recipe. It is clear that our self-consistent method gives values of $\dot{M}$ much closer to the observed ones.

Table 7.  Comparison of Self-consistent with β-law (Single-step) Models for the Two Stars Analyzed By Bouret et al. (2005)

Model	Teff	$\mathrm{log}\,g$	R*/R⊙	k	α	δ	${v}_{\infty }$	$\dot{M}$
	(kK)						(km s−1)	(10−6 M⊙ yr−1)
Self-Consistent	43.5	4.0	11.9	0.159	0.603	0.032	3 342 ± 240	$1.55{\pm }_{0.3}^{0.45}$
β = 1.0	43.5	4.0	11.9	0.118	0.647	0.021	4 187 ± 290	$1.45{\pm }_{0.25}^{0.35}$
Self-Consistent	39	3.6	19.45	0.116	0.657	0.079	2 412 ± 210	$5.8{\pm }_{1.3}^{2.0}$
β = 0.8	39	3.6	19.45	0.039	0.815	0.062	6 789 ± 570	$4.2{\pm }_{0.7}^{0.9}$

Note. Self-consistent models reproduce better the line-fitted Wind parameters obtained by these authors (β = 1: ${v}_{\infty }=3000$ km s⁻¹, $\dot{M}=1.8\times {10}^{-6}$ M_⊙ yr⁻¹, and β = 0.8: ${v}_{\infty }=2300$ km s⁻¹, $\dot{M}=6\times {10}^{-6}$ M_⊙ yr⁻¹).

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We also apply our self-consistent procedure to objects analyzed by means of FASTWIND adopting unclumped winds. For that purpose, we also examine some field Galactic O-type stars from Markova et al. (2018). Table 8 summarizes our results. We found a fair agreement between observed and calculated mass-loss rates (see Figure 13). These results confirm that our methodology delivers the proper mass-loss rate for the ranges in T_eff and $\mathrm{log}\,g$ given above. Below these thresholds, mass-loss rates present larger values compared with both: observational and Vink's theoretical values. This is probably due to the fact that the line-force multiplier is no longer a linear function of t (in the $\mathrm{log}$-$\mathrm{log}$ plane, see Figure 4), and the line-force parameters are not constant throughout the wind.

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Table 8.  Resulting Self-consistent Wind Parameters (${v}_{\infty }^{\mathrm{SC}}$ and ${\dot{M}}_{\mathrm{SC}}$) Calculated for Stars Analyzed by Markova et al. (2018)

Field Star	Teff	$\mathrm{log}\,g$	R*/R⊙	k	α	δ	${v}_{\infty }^{{SC}}$	$\dot{M}$	${\dot{M}}_{\mathrm{SC}}/{\dot{M}}_{\mathrm{obs}}$	${\dot{M}}_{\mathrm{SC}}/{\dot{M}}_{\mathrm{Vink}}$
	(kK)						(km s−1)	(10−6 M⊙ yr−1)
HD 169582	37	3.5	27.2	0.102	0.668	0.063	3017 ± 700	$7.1{\pm }_{2.4}^{3.6}$	1.10	1.26
CD-43 4690	37	3.61	14.1	0.105	0.653	0.058	2 310 ± 540	$1.5{\pm }_{0.55}^{0.9}$	1.22	1.16
HD 97848	36.5	3.9	8.2	0.123	0.601	0.034	2532 ± 470	$0.17{\pm }_{0.06}^{0.09}$	0.89	0.95
HD 69464	36	3.51	20.0	0.099	0.664	0.076	2412 ± 580	$3.2{\pm }_{1.2}^{1.9}$	1.14	1.30
HD 302505	34	3.6	14.1	0.092	0.643	0.077	2331 ± 460	$0.68{\pm }_{0.26}^{0.42}$	1.24	0.98
HD 148546	31	3.22	24.4	0.073	0.718	0.243	1300 ± 350	$5.3{\pm }_{2.5}^{4.7}$	0.94	2.24
HD 76968a	31	3.25	21.3	0.071	0.711	0.248	1212 ± 300	$3.5{\pm }_{1.7}^{3.3}$	1.43	2.11
HD 69106	30	3.55	14.2	0.068	0.644	0.149	1455 ± 300	$0.21{\pm }_{0.09}^{0.16}$	1.48	1.78

Note. Error margins presented here for Wind parameters are undergone from uncertainties of ±1000 for T_eff and ±0.1 for log g. The last two columns show the ratio between self-consistent and observed mass-loss rates and the ratio between self-consistent and Vink's mass-loss rates.

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However, it is important to remark that uncertainties of ΔT_eff ∼ ±1000 K and ${\rm{\Delta }}\mathrm{log}\,g\sim \pm 0.1$ dex, produce uncertainties in the mass-loss rates up to a factor of 2 (see blue error bar in the top panel of Figure 13), which can be considered as the upper threshold for the mass-loss rate. Hence, even though our self-consistent hydrodynamics gives confident values for $\dot{M}$, these good results are strongly dependent on the assumed stellar parameters.