5 Test models of optically thick radiation driven winds

In this section we describe some of the results of the optically thick radiation driven winds of some test models. We discuss the properties of the models and investigate the dependence of the resulting mass-loss rates on the input parameters. We do this for the models calculated in Variant A0 (constant L(r)) and variant B0 (power-law dependence of $\chi$). These tests are useful for describing and understanding the models, and for comparing the results for different parameters.

For the stellar parameters of the test model we have chosen the values of a characteristic WN 5 star: WR139. This star is a member of a well studied binary system and its distance and stellar parameters are well determined (Nugis et al. 1998; Nugis & Lamers 2000). The adopted parameters are discussed below in Sect. 6 and listed in Table 4. The luminosity is $1.62 \times 10^5$ $L_\odot$, the observed mass-loss rate is $0.92 \times 10^{-5}$  $\mbox{$M_\odot$ }~{\rm yr}^{-1}$ and the terminal velocity of the wind is 1785 km s^-1. The adopted radius of the sonic point is $R_{\rm s}=2\mbox{$R_{\rm evol}$ }=1.648~\mbox{$R_\odot$ }$. This is much smaller than the empirically determined "core''-radius of the star, $R_{\rm c} \simeq 6$ $R_\odot$, derived from the empirical wind models with an adopted $\beta=1$ law by Hamann & Schwarz (1992). The reason for this smaller choice will be justified below in Sect. 6. Note that for the final set of optically thick wind models (Variants A1 and B1) we will determine the sonic point radius from the demand that ${\rm d}\chi/{\rm d}r$ is equal to the OPAL value for the particular density, temperature and chemical composition.

   
5.1 Test models with a fixed mass-loss rate

Here we describe the results of a set of models that were calculated with a fixed, pre-specified mass-loss rate. For these models $R_{\rm s}$ is fixed and the effective optical depth at the sonic point is a parameter that has to be solved. The results of the test models are listed in Table 1. The first part gives the results of Variant B0, i.e. for a power-law opacity of the type $\chi(r)=a \rho / T^{n}$, for various values of n. The last line gives the results for Variant A0, i.e. with a fixed value of L(r). We discuss some properties of the models.

 

 

Table 1: Test models for WR139 for fixed mass-loss rate.

n	$\taup_{\rm s}$	$T_{\rm s}$	$\left(\frac{{\rm d}\chi}{{\rm d}r}\right)_{\rm s}$	$v_{\rm s}$	$\frac{R_{\rm s}}{v_{\rm s}}\left(\frac{{\rm d}v}{{\rm d}r}\right)_{\rm s}$	$\frac{\dot{E}-L(R_{\rm s})}{\dot{E}}$	$\frac{\mbox{$L_{\rm adv}$ }(R_{\rm s})}{\dot{E}}$	$\chi_{\rm s}$	$\beta$
		105K	10-14 cm g-1	$\mbox{km~s$^{-1}$ }$		10-4	10-4	cm2 g-1
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
3.01	21.2	1.809	2.34	34.75	1.01	-0.233	101	0.756	5.09
3.10	20.7	1.799	3.72	34.65	1.18	-2.829	98.6	0.756	5.00
3.30	19.8	1.781	6.97	34.48	1.54	-7.128	94.3	0.756	4.95
3.50	19.2	1.768	10.45	34.35	1.88	-10.21	91.25	0.755	5.03
4.00	18.2	1.746	20.20	34.14	2.68	-15.23	86.25	0.755	5.85
4.50	17.6	1.732	31.36	34.00	3.44	-18.30	83.21	0.755	10.0
4.70	17.5	1.727	36.19	33.96	3.74	-19.20	82.25	0.755	80.0
-	22.6	1.837	1.45	35.02	0.805	+7.013	108.5	0.757	5.92

The upper part of the table is for Variant B0, the last line forVariant A0.

The value of the optical depth $\chi_{\rm s}$ at the sonic point is similar for all models. This was already predicted in the previous section where we showed that the gravity force and the radiative force should cancel each other at the sonic point (Eqs. (39) and (40)). All models show that the temperature at the sonic point is about $1.8 \times 10^5$ K. The value of $\taup_{\rm s}$ needed to reach this temperature is about 20. The high sonic temperature is needed because the mass-loss rate is basically determined by the energy conservation at the sonic point. The velocity gradient implies a gain in potential and kinetic energy that has to be provided under the constraint of constant total energy. For the pre-chosen mass-loss rate this can only be achieved at high temperature, and hence at high effective optical depth. The optical depth of order 20 at the sonic point, with the given mass-loss rate and terminal velocity, requires a high column density above the sonic point, which implies a high value of the velocity law exponent $\beta$ of about 5 in the supersonic part of the wind for most models. The models with high values of $n \ge 4$ require a larger value of $\beta$ because the opacity drops steeply outwards and so the density $\rho \sim (r^2 v)^{-1}$ has to drop slowly outwards, implying a high value of $\beta$. In fact there is an upper limit for the opacity exponent nin our models. If $n \ge 4.7$ the opacity decreases so rapidly outwards that the required value of $\taup_{\rm s} \simeq 20$ cannot be reached by any $\beta$-type velocity law. The slowest velocity law with $\beta \rightarrow \infty$ still has a density decrease of $\rho \sim r^{-2}$ and so the maximum column density above the sonic point is $\mbox{$\dot{M}$ }/ 4 \pi R_{\rm s} v_{\rm s}$. We also found that there is a lower limit for the exponent n of the opacity law that can produce optically thick radiation driven wind models. If n < 3 then the opacity does not increase through the sonic point, so the wind cannot be accelerated through the sonic point. We conclude that optically thick radiation driven winds for WR-stars with an opacity of the type $\chi \sim \rho / T^n$ near the sonic point can only exist for a small range of n of approximately 3 < n < 5.

The data in Table 1 show that the radiative luminosity in the comoving frame is very close to the total luminosity L_*. The difference is less than about a factor 10^-2. This justifies our assumption that in the calculation of the radiative acceleration (Eq. (17)) we can substitute L_*for L(r). Note that in models with variant B0 the value of $L(R_{\rm s})$, that is calculated from the energy Eq. (19) is larger than the value of $\dot{E}$, but less than L_*. In the case of Variant A0, which has a constant L(r), the comoving luminosity is slightly smaller than $\dot{E}$ but again the difference is very small. It is interesting to compare the advected luminosity with the difference between $\dot{E}$ and $L(R_{\rm s})$. We see that in all these models the advective luminosity is much larger than this difference. So at the critical point, the energy balance (Eq. (19)) is practically reduced to $\mbox{$L_{\rm adv}$ }= GM\mbox{$\dot{M}$ }/R_{\rm s}$. (This property was already used to derive an estimate for mass-loss rate in Eq. (44).)

   
5.2 Test models with different mass-loss rates

For this set of test models we have solved the full set of equations described above, for calculating the mass-loss rate of optically thick radiation driven wind models. In these calculations the mass-loss rate is determined basically from the predicted velocity gradient near the sonic point which is required by the sonic point conditions. The velocity gradient implies a potential and kinetic energy gain that has to be provided under the constraint of constant total energy. Table 2 gives the results for Variant B0, i.e. with an opacity-law of the form $\chi=a \rho / T^{n}$ for n=3.5 and with fixed $R_{\rm s}=2\mbox{$R_{\rm evol}$ }$. In this case we have adopted a pre-chosen set of values for $\taup_{\rm s}$and we derive the mass-loss rate.

 

 

Table 2: Test models for WR139 with variable mass-loss rate.

$\taup_{\rm s}$	$\dot{M}$	$T_{\rm s}$	$\beta$
	$10^{-5}~\mbox{$\mbox{$M_\odot$ }~{\rm yr}^{-1}$ }$	105 K
2.0	0.098	1.074	2.89
3.0	0.140	1.163	3.32
5.0	0.228	1.296	3.87
10.0	0.462	1.516	4.56
20.0	0.960	1.784	5.05
30.0	1.477	1.964	5.25
50.0	2.531	2.214	5.38

Models with $\chi \sim \rho / T^{3.5}$.

We see that both the mass-loss rate and the temperature at the sonic point increase with increasing effective optical depth. This is in agreement with the predictions described in the approximate formulae Eqs. (46) and (47). For higher optical depth the velocity laws in the supersonic part of the wind have to become "softer'', i.e. higher values of $\beta$, to have a sufficient column density. We see that the observed mass-loss rate of $\mbox{$\dot{M}$ }= 0.92 \times 10^{-5}$  $\mbox{$M_\odot$ }~{\rm yr}^{-1}$ requires an effective optical depth at the sonic point of about 20, in agreement with the results of Table 1.

Figure 1 shows the mass-loss rates and the values of $\beta$for a series of test models for WR139, for different values of n. We see that the mass-loss rate scales almost linearly with the adopted value of $\taup_{\rm s}$ as it is expected according to the approximate formula (46). The mass-loss rates are not very sensitive to the values of n.

Figure 1: The results of models B0, i.e. for $\chi \sim \rho / T^n$, for the star WR139 (WN5). The temperature $T_{\rm s}$ at the sonic point (upper), the mass-loss rate (middle) and the value of $\beta$ (lower) are plotted as a function of the effective optical depth $\tau ^{\prime }_{\rm s}$ at the sonic point. The curves are for n=3.01, 3.30, 3.50, 4.0 and 4.5. In all plots the lowest curve is for n=3.01 and the highest one is for n=4.50. Notice that the results are insensitive to the value of n, except the value of $\beta$ for n>3.5. The dotted line is for model A0, i.e. for L(r)=constant in the transonic region. The results for models A0 and B0 are very similar.

   
5.3 The influence of the sonic radius

In this section we study the influence of the sonic radius on the mass-loss rates of WR-stars. We also investigate the influence of the choices for determining $\chi(r)$ (Variants A0 or B0) on the mass-loss rates, predicted for the models. Table 3 gives the results for a series of test models for WR139, for various values of the ratio $R_{\rm s}/\mbox{$R_{\rm evol}$ }$ and for different values of $\taup_{\rm s}$. We give the results for both variants (A0 and B0). The values of the sonic point temperature $T_{\rm s}$ are slightly different for Variants A0 and B0. However, this difference is so small, less than a factor 10^-3, that we only listed the values of $T_{\rm s}$ for variant A0. We see that the results of the two variants are quite similar. The difference in the mass-loss rates and in $\beta$ is less than about 25 percent, with mass-loss rates of Variant B0 being slightly higher than those of Variant A0. The values of $\beta$ of Variant B0 are slightly smaller than those of Variant A0. This is because Variant B0 has higher mass-loss rates, so a slightly steeper velocity law is needed to produce the same value of $\taup_{\rm s}$. The mass-loss rates increase with increasing adopted location of the sonic point. In the range of $1 < R_{\rm s}/\mbox{$R_{\rm evol}$ }< 4$ the mass-loss rate for a fixed value of $\taup_{\rm s}$ increases almost linearly with $R_{\rm s}$. This is in accord with the prediction of the approximate formula (46).

 

 

Table 3: Test models for WR139 with variable sonic point radius.

$R_{\rm s}$	$R_{\rm hc}$	$\taup_{\rm s}$	$T_{\rm s}$	$\mbox{$\dot{M}$ }^{1}$	$\mbox{$\dot{M}$ }^{1}$	$\beta$	$\beta$
$\mbox{$R_{\rm evol}$ }$	$\mbox{$R_{\rm evol}$ }$		K
			A0	A0	B0	A0	B0
1.0	1.0	2.0	152 000	0.049	0.058	3.22	2.81
1.0	1.0	3.0	164 500	0.070	0.083	3.72	3.23
1.0	1.0	5.0	183 400	0.114	0.136	4.35	3.77
1.0	1.0	10.0	214 700	0.232	0.276	5.12	4.45
1.0	1.0	20.0	252 900	0.485	0.576	5.67	4.94
1.0	1.0	30.0	278 700	0.750	0.891	5.88	5.14
1.0	1.0	50.0	314 900	1.301	1.543	6.05	5.28
2.0	1.0	2.0	107 400	0.082	0.098	3.32	2.89
2.0	1.0	3.0	116 300	0.118	0.140	3.84	3.32
2.0	1.0	5.0	129 600	0.192	0.228	4.49	3.87
2.0	1.0	10.0	151 700	0.389	0.462	5.28	4.56
2.0	1.0	20.0	178 500	0.809	0.960	5.85	5.05
2.0	1.0	30.0	196 500	1.247	1.477	6.06	5.25
2.0	1.0	50.0	221 700	2.143	2.531	6.23	5.38
4.0	2.0	2.0	76 000	0.138	0.165	3.40	2.94
4.0	2.0	3.0	82 300	0.198	0.235	3.93	3.37
4.0	2.0	5.0	91 700	0.323	0.384	4.59	3.93
4.0	2.0	10.0	107 300	0.654	0.778	5.40	4.61
4.0	2.0	20.0	126 300	1.365	1.620	5.97	5.08
4.0	2.0	30.0	139 100	2.108	2.499	6.19	5.26
4.0	2.0	50.0	157 000	3.635	4.298	6.35	5.37
4.0	1.0	2.0	75 900	0.138	0.164	3.40	2.94
4.0	1.0	3.0	82 200	0.198	0.235	3.93	3.37
4.0	1.0	5.0	91 600	0.321	0.382	4.59	3.93
4.0	1.0	10.0	107 100	0.649	0.770	5.40	4.61
4.0	1.0	20.0	125 900	1.343	1.589	5.97	5.09
4.0	1.0	30.0	138 300	2.055	2.425	6.19	5.27
4.0	1.0	50.0	155 500	3.481	4.086	6.36	5.39

¹ $\dot{M}$ is in units of $10^{-5} \mbox{$\mbox{$M_\odot$ }~{\rm yr}^{-1}$ }$.

   
5.4 Conclusions from the tests

We conclude from these tests that the mass-loss rates predicted by optically thick radiation driven wind models show the following properties:

1.

The high mass-loss rates of the star WR139 requires a high temperature at the sonic point of order 10⁵ K, depending on the adopted radius of the sonic point. This high temperature then implies a considerable effective optical depth, $\taup_{\rm s} \simeq 20$. The large optical depth requires a slow velocity law in the supersonic part of the wind with $\beta \approx 5$. In the discussion, Sect. 9, we will show that there is indeed observational evidence for such a high value of $\beta$.

2.

The mass-loss rate of the WR139 models (or more generally of models with fixed L, M, $v_\infty$ and chemical composition) scales almost linearly with the adopted value of the sonic radius, with the optical depth and with the square root of the sonic point temperature, i.e. $\mbox{$\dot{M}$ }\sim R_{\rm s} \taup_{\rm s} T_{\rm s}^{1/2}$. The temperature at the sonic point scales as $T_{\rm s} \sim R_{\rm s}^{-1/2} \taup_{\rm s}^{1/4}$. Combining these two relations we find that

$$\mbox{$\dot{M}$ }\sim T_{\rm s}^{4.5} R_{\rm s}^{3}.$$ (52)

This is in accord with the prediction of Eq. (44). We can conclude that optically thick wind models (A0 and B0) for a particular WR-star lead to the same mass-loss rate for different combinations of parameters $T_{\rm s}$ and $R_{\rm s}$ that satisfy the formula (52). This conclusion means that for a particular star we can get the estimate of $R_{\rm s}$ by demanding that $T_{\rm s}$ must be such that the OPAL opacity gradient from OPAL Tables is positive. We will show below in Sect. 8 that this occurs at $T_{\rm s} \ge 156~000$ K or $37~000 <T_{\rm s} < 71~000$ K.

3.

The models with the two variants A0 and B0 for the opacity give about the same mass-loss rates and the same values of $\beta$ with a difference less than about 25 percent. This gives confidence in the results that we will obtain below for the modeling of the individual WR-stars.

4.

The results are insensitive to the choice of the parameter $R_{\rm hc}$, as can bee seen by comparing the models of $R_{\rm s}/\mbox{$R_{\rm evol}$ }=4$with $R_{\rm hc}/\mbox{$R_{\rm evol}$ }=2.0$ or 1.0 (and confirmed with other test calculations). So a possible uncertainty in the value of the parameter $R_{\rm hc}$ has only a minor effect on the resulting model.

 

 

Table 4: Parameters of WR-stars used for modeling the optically thick winds.

Star	Sp. Type	$\log L$	$M_{\rm WR}$	$R_{\rm evol}$	Y	Z	$\dot{M}$(obs)	$v_\infty$	$\frac{\dot{M} v_{\infty}} {L/c}$
		($L_\odot$)	($M_\odot$)	($R_\odot$)			(10-5)	(km s-1)	$\eta$
							( $\mbox{$M_\odot$ }~{\rm yr}^{-1}$)
2	WN 2b	5.27	10.0	0.863	0.983	0.0172	0.40	3100	3.29
139	WN 5+O 6V	5.21	9.3	0.824	0.936	0.0172	0.92	1785	4.96
136	WN 6b	5.73	19.1	1.235	0.866	0.0173	6.25	1605	9.20
22	WN 7+OB	6.08	55.3	1.622	0.546	0.0176	4.20	1790	3.06
105	WN 9	5.81	21.8	1.315	0.624	0.0176	2.80	1200	2.54
111	WC 5	5.31	10.6	0.891	0.381	0.619	1.00	2415	5.82
42	WC 7+O 7V	5.23	9.5	0.837	0.497	0.503	1.28	1645	6.11
103	WC 9	5.20	9.2	0.818	0.585	0.415	2.40	1190	8.75