5 Metallicity effects, thin winds and scaling relations

Having understood the origin of the specific shape of the line-strength distribution function and its impact on the force-multiplier parameters in detail, we are now able to consider the question raised at the beginning of this paper, namely in how far the situation changes for different wind conditions. We will concentrate here on principal effects which are valid under fairly general circumstances. In particular, let us firstly consider the consequences if the overall metallicity is changed.

5.1 The direct effect

Due to its definition (6), the line-strength scales with metallicity (under the realistic assumption that the ionization balance is not severely modified) as

$$k_{\rm L}(\epsilon) \sim z \frac{n_{\rm l}(\odot)}{\rho} \sim z k_{\rm L}(\epsilon_{\odot})$$ (78)

where z is the actual abundance $\epsilon$ relative to its solar value, $z=\epsilon / \epsilon_{\odot}$. Thus, the major effect of changing the metallicity is a shift of the according line-strength distribution functions (in the ${\rm log}-{\rm log}$ representation) to the "left'' (for z < 1) or to the "right'' (for z > 1).

Figure 26: Cumulative flux weighted line-strength distribution function for $T_{\rm eff}= 40\,000$ K (upper panel) and $10\,000$ K (lower panel), dilution factor W=0.5 and $n_{\rm e}/W = 10^{12}$. Asterisks: solar abundance; dashed: z = 0.1; dashed-dotted: z=3.0. Radiation field Planck

Figure 26 verifies this behaviour for some exemplaric atmospheric conditions ( $T_{\rm eff}= 40\,000$ K and $10\,000$ K, respectively) and three different metallicities, namely z=1 (solar), z=0.1 (roughly SMC) and z=3 (typical for the Galactic center). The shift to lower/higher line-strengths is clearly visible. Only for the largest line-strengths at $T_{\rm eff}= 10\,000$ K the distributions seem to be unaffected by metallicity, which is not surprising since the participating lines are transitions from the hydrogen Lyman series (cf. Sect. 4.2.6).

If we try to translate the shift in metallicity (affecting the independent variable $k_{\rm L}$) into the corresponding shift of the dependent variable $<N(k_{\rm L})>$, this relates to a modification of the vertical offset of the distribution, i.e., of the normalization constant, or, in other words, of the total number of contributing lines. In case of a perfect power-law then, the normalization varies according to

 

$${\rm d}N(\nu, k_{\rm L})(\epsilon) \,-\,N_{\rm o}(\epsilon) \,f_{\nu}(\nu)\, k_{\rm L}^{\alpha-2}\, {\rm d}\nu \,{\rm d}k_{\rm L}$$ =

$$N_{\rm o}(\epsilon) N_{\rm o}(\epsilon_{\odot}) z^{1-\alpha}.$$ = (79)

Since the force-multiplier parameter $k_{\rm CAK}$ is proportional to $N_{\rm o}$(Eq. 12), it should scale according to

$$k_{\rm CAK}(z) \sim z^{1-\alpha} \;\;\; \mbox{(perfect power-law)}.$$ (80)

With respect to $\bar Q$ and from its definition (37), obviously

$$\bar Q(z) \sim z$$ (81)

is predicted (cf. Gayley [1995]), (almost) independent from the specific shape of the line distribution.

 

 

Table 3: Force-multiplier parameters as a function of $T_{\rm eff}$ and metallicity z, for $n_{\rm e}/W = 10^{12}$ and W = 0.5. Values denoted with "(1)'' derived from regression in the range ${\rm log}\,t = -1 \ldots -6$ (as in Table 2); values with "(2)'' refer to a range ${\rm log}\,t = -1\ldots -5$ with optical depth parameter t

$T_{\rm eff}$	z	${\rm log}\,k_{\rm CAK}^{(1)}$	$\hat \alpha(1)$	${\rm log}\,k_{\rm CAK}^{(2)}$	$\hat \alpha(2)$	$\bar Q$
40000	3.0	-0.98	0.68	-0.98	0.69	5817
	1.0	-1.12	0.67	-1.15	0.69	1941
	0.1	-1.28	0.62	-1.42	0.67	196
10000	3.0	-0.41	0.47	-0.52	0.52	997
	1.0	-0.51	0.43	-0.64	0.47	767
	0.1	-0.87	0.36	-0.96	0.40	663

In Table 3 we have calculated the f.m. parameters for the same "models'' as in Fig. 26. The last column shows the validity of the linear dependence $\bar Q\sim z$ for the hotter atmospheres, whereas for the cooler ones $\bar Q$ remains much more constant. If we remember that $\bar Q$ is dominated by lines of maximum strength (Sect. 2.6 and Appendix C), this behaviour results from the fact that the strongest driving lines in this temperature domain are those from hydrogen and thus remain rather unaffected by a change of global metallicity. Again, the conceptual simplicity of the $\bar Q$-approach is hampered by additional effects becoming obvious only by means of detailed calculations.

The other columns display the $k_{\rm CAK}$ and $\hat \alpha$ values derived by linear regressions to the calculated force-multipliers, both in the range of ${\rm log}\,t = -1 \ldots -6$ ("1'') as well as in the range of ${\rm log}\,t = -1\ldots -5$ ("2'') with optical depth parameter $t = k_{\rm 1}^{-1}$. From the differences, it is immediately clear that the assumption of a more or less perfect power-law is only valid for the hotter atmosphere and low to intermediate line-strengths, consistent with the run of $\hat \alpha(k_{\rm 1})$shown in Fig. 27. Thus, the predicted scaling of $k_{\rm CAK}$(Eq. 80) is only verified for case "2'' at 40000 K, whereas in all other cases the reaction of $k_{\rm CAK}$ is much weaker.

One should note, however, that the primary influence of $k_{\rm CAK}$ regards the definition of the mass-loss rate. Thus, $k_{\rm CAK}$ is most important in the subcritical region of the wind, where $k_{\rm 1}$ is low (and t is large), and the contributing range in $k_{\rm 1}$ is also small (typically 2 dex). Under those conditions, however, a power-law distribution with $\hat \alpha\approx \alpha(k_{\rm 1})$ can be always justified (cf. Sect. 2.3.2), so that the effective $k_{\rm CAK}$-value controlling the mass-loss rate should actually scale with (80), provided we compare winds of similar density. In so far, the variations displayed in Table 3 are an artefact of the much larger range of regression applied.

5.2 The indirect " $\hat \alpha$''-effect for low metallicity and thin winds

Besides the obvious direct effect that the cumulative number of lines varies in concert with z, we have to account for an additional complication: By comparison of the various $\hat \alpha$ values derived by linear regression, we find that also $\hat \alpha$ is a function of metallicity, especially for cooler temperatures. Regarding the difference between actual (Fig. 27) and "power-law'' fitted values, the depth-dependent values of $\hat \alpha$ are typically smaller than the mean for large $k_{\rm 1}$, whereas they are larger or similar at low $k_{\rm 1}$-values. (Due to the dominance of hydrogen lines with their $\alpha = 2/3$ statistics, for the cooler atmosphere we even encounter an - otherwise untypical - steep increase of $\hat \alpha$ towards maximum $k_{\rm 1}$).

The reason for the outlined behaviour is, again, the steep decline of the line-strength distribution at its upper end, due to the excitation effects discussed extensively in Sect. 4.2, and the horizontal shift of the distribution as a function of metallicity. Thus, for lower z the steeper end of the distribution becomes visible at lower values of $k_{\rm 1}$, and $\hat \alpha$ can be roughly expressed as

$$\hat \alpha({\rm log}\,k_{\rm 1}, z) \approx \hat \alpha({\rm log}\,k_{\rm 1}- {\rm log}\,z,z=1),$$ (82)

neglecting certain subtleties arising from non-metallic lines.

Figure 27: As Fig. 26, however for $\hat \alpha$ (derived from force-multiplier) as function of $k_{\rm 1}$. Solar abundances: fully drawn

Whereas $k_{\rm L}$ is an (almost) density independent quantity, $k_{\rm 1}= {\rm d}v/{\rm d}r/\rho \propto v_{\infty}^2 R_{\ast}/{\dot M}$ scales with the inverse of the mean wind density (times $v_{\infty}/R_{\ast}$). Thus, in addition to the metallicity shift (argument of rhs in Eq. (82)), the range of ${\rm log}\, k_{\rm 1}$ present in the wind is also shifted compared to solar conditions. Since a reduced metallicity yields a reduced wind density, this shift is towards higher $k_{\rm 1}$, i.e., a low-metallicity wind plasma "doubles'' the effect of lowering $\hat \alpha$. In contrast, enhanced metallicities have almost no effect on $\hat \alpha$, since the corresponding shift is towards lower $k_{\rm 1}$, where the line-strength distribution function has a more constant slope.

Of course, the described process is also present if the wind-density is low for other reasons, e.g. because the luminosity is low. Compared to supergiant winds then, the $k_{\rm 1}$ range to be considered is shifted towards higher values, and $\hat \alpha$ is accordingly lower.

In conclusion, thin and fast winds as well as low metallicity winds tend to have lower $\hat \alpha$-values than high density or high metallicity winds, both on the average as well as locally. Once more, the reason for this effect is the curvature of the line-strength distribution function, especially at highest line-strengths, which is also the answer to the problem raised at the end of Sect. 2.5 concerning the origin of the lower $\hat \alpha$-values calculated in a low metallicity environment. If, on the other hand, a perfect power-law were present, $\hat \alpha\approx \alpha \approx {\rm const}$, independent on wind density and metallicity.

As a consequence of the variations of $\hat \alpha$ as a function of $k_{\rm 1}$, $\hat \alpha$ varies throughout the wind, since $k_{\rm 1}$changes by typically three dex from inside to outside. Hence, any exact hydrodynamic solution requires depth dependent force-multipliers (cf. Kudritzki et al. [1998])

At this point of reasoning, we like to reiterate our findings in a somewhat different context. From our experience, the behaviour of the line-force in a low-density environment is frequently misinterpreted. E.g., after having calculated the according f.m. parameters - with the result of $\alpha \rightarrow0$ in the outermost wind part -, there seems to be a common concern whether this is not only an artefact of an incomplete line-list at lower gf-values. Actually, however, almost the opposite effect is present! If, e.g. in B-dwarf winds, the density becomes so low that (corresponding to ), all lines contribute to the line-acceleration at their optically thin limit, $g_{\rm rad}\propto \bar Q$. Thus, the strongest (however optically thin) lines have the largest influence and the numerical value of the total force does not depend on any incompleteness of the line list at low gf-values. The fact that $\alpha$tends to zero in this case is, as explained already in Sect. 2.3, given by the independence of the line-force on any variation of t (or $k_{\rm 1}$).

With respect to the horizontal shift "to the left'' in a low-metallicity environment, this independence on $k_{\rm 1}$ can start even earlier, i.e., the line-force becomes saturated ( $\alpha \rightarrow0$) at lower values of $k_{\rm 1}$. Even in cases of a higher metallicity (where the effects of an incomplete line-list may become obvious at least in principle), the actual range of contributing $k_{\rm L}$ values is normally much too small that this might become a real problem.

5.3 Scaling relations

Including now the aforementioned finite disk correction factor and accounting for ionization effects $N_{\rm o}\propto (n_{\rm e11}/W)^{\delta}$, we can summarize the resulting scaling relations for ${\dot M}$ and $v_{\infty}$ as function of metallicity, which arise if a metal dependent line-force is used to solve the hydrodynamic equations (for actual solution methods, cf. PPK and Kudritzki et al. [1989]) and the f.m. parameters were constant throughout the wind:

$${\dot M} \sim (z^{1-\hat \alpha})^\frac{1}{\alpha'} \left(\frac{L}{L_{\odot}}\r... ...'} \left(\frac{M}{M_{\odot}} (1 - \Gamma)\right)^{1 - \frac{1}{\alpha'}} \times$$

$$\times g(k_{\rm CAK}(z=1),\hat \alpha,\alpha')$$ (83)

$$v_{\infty} \frac{\hat \alpha}{1 - \hat \alpha} \,v_{\rm esc} \,f(\hat \alpha, \hat \delta)$$ = (84)

$$\alpha' \hat \alpha- \hat \delta$$ = (85)

$$\approx \left(\frac{\hat \alpha}{1-\hat \alpha}\right)^{\frac{\hat \alpha... ...'}}\, \bigl(3~10^{-5}\,k_{\rm CAK}(1-\hat \alpha)\bigr)^{\frac{1}{\alpha'}} \,.$$ g (86)

M is the stellar mass and $v_{\rm esc}$ the escape velocity, corrected for the Eddington factor $\Gamma$. $f(\hat \alpha, \hat \delta)$ is a decreasing function of $\hat \delta^{-1}$, and has a value of roughly 2.2 if $\hat \delta$is small (cf. Kudritzki et al. [1989]). The function g finally accounts for the (moderate) dependence on terms of order $\hat \alpha$, on the proportionality to $k_{\rm CAK}^{1/\alpha'}$, and, most important (and frequently forgotten), on the scaling factor $\Gamma \approx 3~10^{-5} (L/L_{\odot})/(M/M_{\odot})$ , since the mass-loss rate actually depends on the Eddington factor $\Gamma^{1/\alpha'}$ and not on $L^{1/\alpha'}$ itself. Note, that the variation of g has to be considered in any comparison where $\hat \alpha$ is different (e.g., A-star vs. O-star winds, see below).

In case of depth dependent parameters, ${\dot M}$ relates to the conditions at the critical point ( ${\rm log}\,k_{\rm 1}= 2\ldots 3$ for not too thin winds), where $\hat \alpha$ and $\hat \delta$ do not vary heavily. The terminal velocity, however, is dependent on some average value of $(\hat \alpha, \hat \delta)$between the location of the critical point and large values of $k_{\rm 1}$, and will be typically smaller compared to using the $\hat \alpha$ values present at the critical point, because of the reasons outlined above.

In any case, to first order we find the metallicity effect as

$${\dot M}\sim z^\frac{1 - \hat \alpha}{\alpha'};\, v_{\infty}\sim \frac{\hat \alpha}{1 - \hat \alpha}(z),$$ (87)

which, in case of O-star winds (small $\hat \delta$) yields the often quoted scaling relation for the mass-loss rate ${\dot M}\sim \sqrt z$ since $\hat \alpha\approx 0.6$.

Due to the metallicity dependent factor $\hat \alpha/(1 -\hat \alpha)$, one can expect lower terminal velocities in a metallicity-deficient environment. This is just what has been found by comparing O-star terminal velocities in the Galaxy and the Clouds, cf. Fig. 28. For a detailed discussion, we refer the reader to the papers by Garmany & Conti ([1984]), Kudritzki et al. ([1987]), Haser et al. ([1993]) and Walborn et al. ([1995]).

Figure 28: Terminal velocities of O-type stars in the Galaxy and the Magellanic clouds. Data from Haser ([1995]) and Puls et al. ([1996])

Figure 29: Wind momentum (in cgs units) and luminosity of galactic and SMC supergiants and two A-supergiants in M 33. Open square: M 33 A-supergiant with galactic metallicity. Cross: Extremely metal poor A-supergiant in the outskirts of M 33. (From McCarthy et al. [1995])

Finally and with respect to the wind-momentum luminosity relation (Kudritzki et al. [1995]; Puls et al. [1996]), our findings imply (leading terms only)

 

$${{\rm log}D_{\rm mom}\,= \,{\rm log}({\dot M}v_{\infty}(R_{\ast}/R_{\odot})^{{1 \over 2}}) \sim}$$

$$\sim \frac{1}{\alpha'} \,{\rm log}L/L_{\odot}\,+ \, \frac{1 - \hat \al... ... log}z \,+\, 2\,{\rm log}\,\left(\frac{\hat \alpha}{1 - \hat \alpha}\right)\, +$$

$$\frac{1}{\alpha'}\,{\rm log}\bigl(3~10^{-5}\,k_{\rm CAK}(z=1)\,(1-\hat \alpha)\bigr) + \ldots ,$$ + (88)

where, of course, in case of $\hat \alpha(z!) \ne 2/3$ an additional correction for mass effects might be necessary.

From the presently available data, it is clear that at least in the SMC a different offset is visible (due to the second term in the above equation, resulting from the "direct'' effect (cf. Fig. 29, and also Puls et al. [1996]; Kudritzki [1997]). Whether there is actually a different slope (as a consequence of a reduced $\alpha'$), is not certain due to the small number statistics for SMC O-stars. To clarify the situation, more objects have to be analyzed. This work is well under way in our group.

Contrasted to the above uncertainty concerning the reduction of $\hat \alpha$in a metal poor environment, the observational status quo with respect to the difference of A-star vs. O-star winds is much more promising. From the latest results by Kudritzki et al. ([1999]), the WLR for Galactic A-Supergiants (in a temperature range $T_{\rm eff}= 8\,400 \ldots ~9\,400$ K) reads ${\rm log}D_{\rm mom}= 0.38^{-1} {\rm log}L/L_{\odot}+ 14.22$, to be compared with the relation valid for Galactic O-Supergiants (from Puls et al. [1996]), ${\rm log}D_{\rm mom}= 0.65^{-1} {\rm log}L/L_{\odot}+ 20.40$.

Note at first that the observed slope (interpreted as $\alpha' = \hat \alpha-\hat \delta= 0.38$) lies exactly in the range to be expected for A-type winds, cf. Tables  2 and 3. Second, from the difference in the offset compared to O-stars ( $\Delta {\rm log}D_{\rm o} = 6.18$), we can calculate the average value of the parameter $k_{\rm CAK}$(A-SG) using Eq. (88), the values found for $\alpha'$ from the WLRs and the appropriate values for $\hat \alpha$and $k_{\rm CAK}(40\,000$ K) from Table 3. In result, we find ${\rm log}\, k_{\rm CAK}$(A-SG) = -0.84. This number is reasonable when compared to our theoretical prediction ${\rm log}\,k_{\rm CAK}(10\,000$ K) = -0.51 (Table 3) at $n_{\rm e}/W = 10^{12}$, accouting for the fact that A-Supergiants have lower densities at the critical point (smaller ${\dot M}$and larger radii) than their O-type counterparts. In so far, the difference found is a consequence of the $\delta$-term, which is implicitely included in the value of $k_{\rm CAK}$ derived in our comparison.

Thus, we conclude that our theoretical predictions concerning the run of $\hat \alpha$ (and $k_{\rm CAK}$) with respect to temperature are correct, and, additionally, in Sect. 4 we have explained the reason for this behaviour.