4 Some simple estimates

In this section we describe some simple estimates, based on general momentum and energy considerations of optically thick radiation driven winds.

   
4.1 The opacity at the sonic point

We can obtain a simple estimate of the opacity at the sonic point of an optically thick wind from the sonic point condition that the righthand side of the momentum equation should vanish, so $f_1(R_{\rm s})=0$ with f₁ defined by Eq. (21). The temperature derivative in an optically thick wind is given by Eq. (16). These equations predict that

$$\frac{\chi_{\rm s} L(R_{\rm s})} {4 \pi c} = \frac{G M_* - 2 ... ... \mbox{$\dot{M}$ })/( c_1 R_{\rm s}^2 T_{\rm s}^3 v_{\rm s})},$$ (39)

where we have used the mass continuity equation for the substitution of $\rho$. The second term of the numerator is much smaller than the first term, because the sound velocity at the sonic point (typically about 30 km s^-1) is much smaller than the escape velocity (typically about 2000 km s^-1). For reasonable values of $\dot{M}$ (between 10^-5 and 10^-4 $\mbox{$M_\odot$ }~{\rm yr}^{-1}$) and sonic point temperatures of order 10⁵ K (see below), the second term in the denominator is much smaller than unity. This is equivalent to the statement that the two dominant terms in the right hand side of the momentum equation are the gravity and the radiation pressure, with the gas pressure terms being negligible. So Eq. (39) reduces to

$$\chi_{\rm s}~\simeq~ \frac{4 \pi c G M_*}{L_*}\cdot$$ (40)

Using the mass-luminosity relation of H-poor WR-stars, Eq. (53), we find an estimate of the required opacity for radiation driven optically thick winds of $0.3 < \chi_{\rm s} < 0.9$ cm² g^-1. These are reasonable values for hot star winds.

   
4.2 The opacity gradient at the sonic point

We can derive an important condition for the opacity gradient at the sonic point of radiation driven winds. From de l'Hopital's rule at the sonic point (Eq. (26)), combined with the sonic point condition that f₁=0 and f₂=0, it is easy to show that the velocity gradient at the sonic point, $({\rm d}v/{\rm d}r)_{\rm s}$, can only be positive if

$$\biggl( \frac{{\rm d}\chi}{{\rm d}r} \biggr)_{\rm s} + \frac{... ...( \frac{{\rm d}^2 T}{{\rm d} r^2} \biggr)_{\rm s} \right) > 0.$$ (41)

The last term is due to the effect of the gradient of the gas pressure. In a radiation driven wind the force produced by the gas pressure is by definition negligible compared to the force produced by the radiation pressure. So we can neglect that term. Equation (41) thus shows that in a radiation driven wind with constant L(r)the gradient $({\rm d} \chi /{\rm d}r)_{\rm s}$ must be positive to allow a transonic flow. In the case of optically thick radiation driven winds ${\rm d}L(r)/{\rm d}r \leq 0$ (cf. with Quinn & Paczynski 1985) and so $({\rm d} \chi /{\rm d}r)_{\rm s}$ must certainly be positive.

We conclude that a stellar wind can only be driven by radiation pressure if ${\rm d}\chi/{\rm d}r>0$ at the sonic point! This implies that optically thick stellar winds must have their sonic point in the density and temperature regime where the opacity increases as a function of distance. This occurs for instance in the region of the iron opacity peak around $\log T \simeq 5.2$.

This condition for the gradient of the opacity of radiation driven winds is a special case of the general condition that a wind can only be accelerated through the sonic point if either energy or momentum is added at the sonic point (cf. Lamers & Cassinelli 1999, p. 100).

   
4.3 Mass-loss rates of optically thick winds

The energy equation near the sonic point (Eq. (19)) can be written as

$$\frac{G M \mbox{$\dot{M}$ }}{R_{\rm s}} \simeq \mbox{$L_{\rm adv}$ }- \left\{\dot{E} - L(R_{\rm s})\right\}\cdot$$ (42)

The test models and the optically thick wind models of the studied stars (described below in Sect. 5.1 and Table 1) reveal that $L(R_{\rm s})$ is very close to $\dot{E}$ and the absolute value of $L(R_{\rm s})-\dot{E}$ is much smaller than $\mbox{$L_{\rm adv}$ }(R_{\rm s})$. This means that we can estimate the mass-loss rate by equating the advection term with the potential energy term:

$$\frac{G M \mbox{$\dot{M}$ }}{R_{\rm s}} = \mbox{$L_{\rm adv}$ }(R_{\rm s})$$ (43)

and after some substitutions we find

$$\mbox{$\dot{M}$ }\simeq \frac{c_1 R_{\rm s}^3 T_{\rm s}^4 v_{... ...= \frac{c_1 a_1^{1/2} R_{\rm s}^{3} T_{\rm s}^{4.5}}{GM}\cdot$$ (44)

Approximately the same formula can be derived quite generally from the following simple considerations. Let us start with the formulae for $T_{\rm s}$ and $\tau ^{\prime }_{\rm s}$ (Eqs. (34) and (35)). Using Eq. (4) we can express $\tau ^{\prime }_{\rm s}$in the form

$$\tau_{\rm s}^{\prime} = \int_{R_{\rm s}}^{\infty} \frac{\chi \dot{M} R_{\rm s}^{2}} {4 \pi v r^{4}} {\rm d}r.$$ (45)

The ratio $\chi/v$ is changing very little in the inner part of the wind (this is the specific property of optically thick winds of WR-stars!) and the ratio $\chi/v$ at infinity is only about two times smaller than at the sonic point. This property and the steep drop of the integral kernel in Eq. (45) with increasing r, due to the factor v^-1r^-4, implies that we can estimate $\tau_{\rm s}^{\prime}$as

$$\tau_{\rm s}^{\prime} \approx \frac{ c_{\rm m} \chi_{\rm s}}{v_{\rm s}} \frac{\dot{M}} {4 \pi R_{\rm s}},$$ (46)

where $c_{\rm m}$ is a multiplier of the order of unity. The temperature at the sonic point can be expressed approximately as

$$T_{\rm s}^{4} \approx \frac{0.75 L_{\rm s}} {4 \pi \sigma R_{\rm s}^{2}} \tau_{\rm s}^{\prime},$$ (47)

because $\tau_{\rm s}^{\prime} \gg 2/3$ for WR-winds. Using Eqs. (47) and (46) and $\chi_{\rm s}$ from Eq. (40), we obtain

$$\mbox{$\dot{M}$ }\simeq \frac{a_1^{1/2}c_1 R_{\rm s}^{3} T_{\rm s}^{4.5}}{GM} \frac{0.75}{c_{\rm m}}\cdot$$ (48)

Note that this formula differs from the Eq. (44) only by the constant $0.75/c_{\rm m}$ which is very close to unity.

Therefore we can conclude that formula (44) is a very good approximation formula for deriving the mass-loss rates of WR-stars.

   
4.4 The minimum mass-loss rate for optically thick winds

We can find the minimum estimated mass-loss rate for the particular WR-star by using the formula (44) with the minimim estimate of the temperature at the sonic point derived from formula Eq. (34) by adopting $\tau_{\rm s}=0$. This gives a minimum value of the mass-loss rate of

$$\mbox{$\dot{M}$ }_{\rm min} \simeq \frac{a_1^{1/2}c_1} {(8 \pi)^{9/8}}\frac{R_{\rm s}^{3/4} L_*^{9/8}}{GM \sigma^{9/8}}\cdot$$ (49)

Applying this equation to a typical hydrogen-free WNE-star with $M_*\simeq 10~ \mbox{$M_\odot$ }$, $R_{\rm s} \simeq 2~ \mbox{$R_\odot$ }$ and with the luminosity given by the M-L law (Eq. (53): $L_*= 1.85 \times 10^5~\mbox{$L_\odot$ }$), we find a minimum mass-loss rate of about $0.2 \times 10^{-6}$  $\mbox{$M_\odot$ }~{\rm yr}^{-1}$.

   
4.5 The maximum mass-loss rate

An absolute upper limit for the mass-loss rate of radiation driven winds is set by the condition that all the energy generated in the nucleus is used to drive the wind. This gives

$$\mbox{$\dot{M}$ }< L / (\mbox{$v_\infty$ }^{2}/2 + GM/R_{\rm hc}).$$ (50)

For a typical hydrogen-free WNE-star with $M_*\simeq 10~ \mbox{$M_\odot$ }$, $v_{\infty}=2000$ km s^-1 and with the luminosity given by the M-L law (Eq. (53): $L_*= 1.85 \times 10^5~\mbox{$L_\odot$ }$) and the hydrostatic core radius given by the evolutionary models (Eq. (54): $R_{\rm hc} \simeq 0.86~ \mbox{$R_\odot$ }$) we derive an upper limit of $2.7 \times 10^{-4}$  $\mbox{$M_\odot$ }~{\rm yr}^{-1}$.

A more realistic upper limit can be found from the formula Eq. (43) by using for $\mbox{$L_{\rm adv}$ }(R_{\rm s})$ the advective luminosity at infinity which ought to be higher than $L_{\rm adv}(R_{\rm s})$. In the case of thin winds we know that near the star $L_{\rm adv} \approx v/c L(r)$, which is much lower than $L_{\rm adv}(\infty) = 2v_{\infty} L(\infty) / c$. In the case of thick winds we know that $L_{\rm adv}(\infty) \ge L_{\rm adv}(R_{\rm s})$. So the upper limit for the mass-loss rate of optically thick winds can be obtained from the formula

$$\mbox{$\dot{M}$ }_{\rm max} \simeq \frac{2 L(\infty) \mbox{$v_\infty$ }R_{\rm s}} {c G M}\cdot$$ (51)

For a typical WNE-star with the parameters given above we find that $\mbox{$\dot{M}$ }_{\rm max}= 1.6 \times 10^{-5}$ $\mbox{$M_\odot$ }~{\rm yr}^{-1}$. The observed mass-loss rates of WNE stars are indeed in between the minimum and maximum values derived here.