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The winds of the hot massive first stars

We study dynamical aspects of circumstellar environment around massive zero-metallicity first stars. For this purpose we apply our NLTE wind models. We show that the hydrogen-helium stellar wind from stationary massive first generation (Population III) stars (driven either by the line (bound-bound) or continuum (bound-free and free-free) transitions) is unlikely. The possibility of expulsion of chemically homogeneous wind and the role of minor isotopes are also discussed. Finally, we estimate the importance of hydrogen and helium lines for shutting off the initial accretion onto first stars and its influence on initial mass function of first stars.


Introduction
The Big Bang nucleosynthesis has left our universe completely metal-free (e.g. Coc et al. 2004). Thus, at the dawn of the universe there existed a population of stars composed from hydrogen and helium only. Numerical simulations show that the physical properties and the evolution of these first stars (Population III stars) were very different from the stellar evolution and properties of Population I and II stars. Simulations of first star formation (cf. Bromm et al. 1999, Nakamura & Umemura 2002 show that due to the less efficient cooling the first stars were probably much more massive than the present stars. First stars with masses of the order 100 M ⊙ might have been formed. However, there are indications that the initial mass function of first stars was bimodal (Nakamura & Umemura 2002, Omukai & Yoshii 2003, and thus solar mass first stars may have been formed as well. There are no available observations of first stars up to now. Although the recently discovered extremely low metallicity star HE 0107-5240 (Christlieb et al. 2002) was claimed to be the first star (Shigeyama et al. 2003), there are some problems with this explanation. It is difficult to explain the present surface chemical composition of this star (especially that of carbon and nitrogen) on the basis of initial pure hydrogen and helium composition. For example, Shigeyama et al. (2003) explain the CN composition by the surface enrichment due to the central nucleosynthesis, however there is a question whether significant amount of CN elements may occur on the surface for this stellar type according to evolutionary calculations of Marigo et al. (2001). It is more likely that this star was formed from the ma-terial which was already enriched by the first stars (Bromm & Larson 2004) or by subsequent stellar populations.
Consequently, it is necessary to constraint ourselves only to indirect observation of the first stars. Probably the most reliable indirect observation of the first stars comes from the polarisation measurement of cosmic microwave background radiation. If the first stars ionized the matter in the universe significantly, we should observe its imprint on the cosmic microwave background due to the light scattering on free electrons. Recent cosmic microwave background polarisation measurement showed that the universe was reionized 180 × 10 6 years after the Big Bang (Kogut et al. 2003). Detailed numerical analysis shows that this effect can be attributed to the generation of massive zero-metallicity stars (Cen 2003, Wyithe & Loeb 2003, Sokasian et al. 2003. Moreover, the level of the sky infrared background radiation can also be explained by the generation of massive first-generation stars (Magliocchetti et al. 2003).
The detailed theoretical study of physical properties of the first generation stars is a challenging task. Especially appealing is the study of its stellar winds, since winds might have a significant impact on the first stars evolution. Moreover, the study of outflows of the first stars may be important for the estimate of the duration of supermassive stars formation. The possibility of supermassive stars formation depends critically on the metal abundance in the interstellar medium. The presence of a relative small amount of metals might inhibit subsequent formation of supermassive stars (see Bromm & Larson 2004 for a review). Note, for example, that there are indications that the initial formation of supermassive stars might have been inhibited by metals expelled due to the supernova explosions and that direct black hole formation by implosion is necessary for the explanation of the duration of massive star era (Ricotti &

Theoretical limits for line-driven winds
The necessary condition for launching the line-driven stellar wind is relatively simple. The radiative force should exceed the gravity force at some point. Thus, instead of presenting more detailed numerical models we start with a discussion of this condition and its application to simple situations. This helps to better understand more complicated behaviour of numerical models.

Minimum radiative force
The radiative acceleration due to the absorption or scattering of radiation is given by (Mihalas 1978) where ρ is the mass density, χ ν is the absorption coefficient, H ν is the radiative flux, and ν is the frequency. For the calculation of the radiative force in the optically thin environment, the radiative flux H c (ν) emerging from the stellar atmosphere can be directly inserted into Eq. (1). However, this is not the case of the stellar wind since the radiative flux H ν is influenced by the absorption and emission of the radiation in the sorrounding wind environment in such case. Thus, the self-shadowing in the given transition has to be accounted for, preferably by the solution of the radiative transfer equation. In the case of rapidly accelerating stellar wind a convenient approximation of the radiative transfer equation solution is given by the Sobolev approximation (see Sobolev 1947). From Eq. (1) follows the limit below which classical linedriven wind is not possible. In particular, neglecting continuum absorption, for a given set of lines the radiative force is maximum, if lines are not self-shadowed and if lines are optically thin, i.e. if the optical depths τ < 1 in lines (Gayley 1995). Thus, for given occupation numbers the line acceleration (i.e. line force per unit of mass) cannot be larger than where now H c (ν ij ) is the frequency dependent flux emerging from the star at the stellar radius R ⋆ , 1 − µ 2 c = R 2 ⋆ /r 2 is a correction due to the dilution of radiation, ν ij and g i f ij are the frequency and the oscillator strength of a given transition i ↔ j, n i , n j are the number densities, and g i , g j are the statistical weights of corresponding levels.
Note that g rad, max is related to the Gayley's parameterQ (Gayley 1995, Puls et al. 2000, namely where g rad e is the radiative acceleration on free electrons, where L is the stellar luminosity, s e is the Thomson scattering cross-section, and n e is the electron density. The line-driven wind is possible only if the total radiative force is greater than gravity at some point in the stellar atmosphere or circumstellar envelope. Thus, the necessary condition for line-driven wind reads (neglecting the gas-pressure) where g = GM /r 2 is the gravity acceleration. Using Eq.
(2) for maximal line acceleration and 1 − µ 2 c = R 2 ⋆ /r 2 we can obtain a condition for the lowest limit of line-driven wind as where M is stellar mass and Using the Gayley's parameter, which for our case reads Q = πe 2 m e cs e n e lines where H t (R * ) = L/(4πR * ) 2 is the total flux at the stellar surface R * , the condition (6) may be rewritten as Let us first assume that the stellar surface flux H c (ν ij ) is constant for a given transition i → j throughout the wind, i.e. that the frequency dependence of H c (ν ij ) due to the Doppler shift can be neglected. In such a case the only quantities in Eq. (7) which depend on radius are the fractions n i /ρ, n j /ρ. Thus, for a given star Eq. (7) gives a lower limit for n i /ρ below which the line-driven wind is not possible. Eq. (7) is a necessary condition for launching the stellar wind. Another formulation of the condition for launching the stellar wind may be obtained from the requirement that the work done by the radiative force (per unit mass) is greater than the absolute value of gravitational potential. It is noteworthy that in the case of wind driven purely by optically thin lines this condition can be easily obtained by the integration of Eq. (6) from r = R * to infinity if the wind ionization and excitation state is constant (note that in this case fraction n i /ρ is constant for homogeneous winds) (11) Apparently, this is an identical condition to Eq. (7) and thus Eq. (7) is also an energy condition for launching the stellar wind driven purely by optically thin lines. However, real linedriven stellar winds are accelerated not only by optically thin lines but by the mixture of optically thin and optically thick lines. Thus, a simple condition (11) cannot be used in this case. Moreover, some of the lines which accelerate the stellar wind may be optically thick in the photosphere. In such a case, the stellar flux H c depends on the wind velocity (and thus also on the radius) due to the Doppler effect and condition (11) cannot be applied.

Application to hydrogen
The necessary condition for the existence of line-driven winds (7) is especially simple in the case of the radiative force only due to the hydrogen lines. This is only a rough approximation even in the case of stars with zero metallicity for which also the radiative force due to the helium lines may be important. However, condition (7) at least provides a guess of occupation numbers for which the line-driven winds of stars at zero metallicity may exist. Moreover, adding other hydrogen-like ions is straightforward.
The oscillator strengths for hydrogen are given by the well known Kramers formula, which neglecting Gaunt factors reads (Mihalas 1978, Eq. (4.78) therein) For the resonance lines it may be assumed which in fact means neglecting the stimulated emission in all resonance transitions. Moreover, the contribution of other than resonance lines to the radiative force can be in many cases neglected. Thus, in the case of hydrogen, we can obtain from Eq. (7) relatively simple condition for the existence of line-driven wind in the form of The main contribution in the sum in condition (14) is due to the resonance lines and the lowest relative occupation number of the hydrogen ground level N H I,1 , for which the hydrogendriven wind is possible, is given by where m H is hydrogen atom mass, X = n H m H /ρ is the relative hydrogen abundance, n H is hydrogen number density and N min H I,1 = n min H I,1 /n H (where n min H I,1 is the minimum number density of ground state neutral hydrogen necessary to launch the stellar wind). Apparently, for stars close to the Eddington limit (Γ → 1) the minimum relative number density N H I,1 for launching the hydrogen line-driven wind is lower.

Application to helium
He II is the most important helium ion for radiative driving since helium is ionized in the stellar winds of hot stars. Thus, the radiative force due to helium can be calculated in a very similar way to that due to H I. Similarly to the case of H I, the resonance lines are the only important He II lines for radiative driving (assuming that helium is singly ionized ionized). Thus, we can write for He II (similarly as for H I) The condition of existence of wind driven by the absorption in He II lines is (see Eq. 14) and the lowest relative occupation number N He II,1 for which the helium-driven wind is possible is where m He is helium atom mass, Y = n He m He /ρ is the relative helium abundance, n He is helium number density and N min He II,1 = n min He II,1 /n He . The sums in (17) and (18) converge due to the asymptotic behaviour of hydrogen-like oscillator strengths, g i f ij ∼ 1/j 3 (see Eq. (12)).
In many cases the stellar wind is accelerated by lines which are optically thick in the photosphere and thus the stellar flux H c for frequencies close to the photospheric line center depends strongly on frequency. Since the radiative flux at a given point is shifted due to the Doppler effect, the optically thin radiative force depends on velocity (and consequently also on radius). Even in such a case the conditions (15,18) can also be used to estimate the minimum conditions for line driven winds. However, the actual radiative flux H c [ν(v r )], which depends on wind velocity (and thus also on radius) should be inserted in these equations.

Stellar wind driven by optically thin lines
We have derived the necessary condition for launching the linedriven stellar wind. However, there is another question which should be answered. Is this condition also the sufficient one? In other words, are the optically thin lines only able to drive stellar wind?

Constant ionization fraction
Let us first assume that the wind ionization and excitation state does not depend on wind density and on radius and that the stellar surface flux H c does not significantly vary with frequency. In such a case the optically thin radiative acceleration (2) does not depend on wind density. Thus, also the wind momentum equation (where the optically thin radiative acceleration g thin = g rad, max , a is sound speed, and where we assumed isothermal flow) does not depend on wind density. It may seem that the optically thin line-force may launch the stellar wind with an arbitrary mass-loss rate. However, this is not true, since in such a case either for higher wind densities some of the lines would become optically thick or the photon tiring effect (Owocki & Gayley 1997) would impose constrains on the mass-loss rate.

Variable ionization fraction
In the case when the wind ionization and excitation state depends on wind density, the value of wind mass-loss rate can be derived from the hydrodynamic equations. The optically thin momentum equation (19) has a critical point for v r = a. To obtain a smooth wind solution, the regularity condition must be fulfilled at the critical point. Since wind ionization and excitation state depends on wind density now, also the optically thin radiative acceleration depends on wind density. Thus, the critical condition (20) depends implicitly on the wind density and using this equation the mass-loss rate of thin winds can be calculated. Note that in the common case when the gas pressure term is negligible, the condition (6) will be just fulfilled at the critical point. This again shows the importance of the condition (6). However, this analysis does not tell us anything about the possibility to drive the wind material out from the stellar gravitational potential well. This condition has to be tested using detailed numerical modelling. On the other hand, more wind properties can be unveiled using a simple analysis. The stellar photosphere deep below the wind region shall be quasi-static. Thus, to obtain a smooth flow, the radiative force has to be smaller than the gravity in this region and the condition (6) has not to be fulfilled below the critical point. The change of wind ionization or excitation state then has to assure that the equality in (6) is fulfilled at the critical point. Finally, in the outer parts of the wind the inequality in condition (6) has to hold, i.e. the radiative force has to be greater than the gravity in the outer wind parts.

Frequency dependent stellar flux
The frequency variable stellar flux gives probably the most attractive possibility to obtain consistent radiative force to drive an optically thin wind, i.e. radiative force which is lower than the gravity force below the critical point and higher outwards. This is caused by the fact that some lines, which accelerate the stellar wind, may be optically thick in the photosphere and optically thin in the stellar wind. In such a case the stellar flux H c strongly depends on frequency and, consequently, due to the Doppler effect, also on the wind velocity. The stellar wind near the star has low velocity and a given line is accelerated mainly by the weak flux from the line core. On the other hand, stellar wind with higher velocities is accelerated mainly by the stronger flux from the line wings and thus the line force of the outer wind may be higher than that of inner part of the wind close to the star. However, this is not as simple as it looks like since higher flux may also cause higher excitation and ionization and consequently lower the radiative force. Clearly, detailed numerical calculations are necessary to test this possibility.

Numerical calculation of thin winds
For the numerical tests we used NLTE wind models described by Krtička & Kubát (2004). The computer program allows to solve hydrodynamic, simplified radiative transfer, and statistical equilibrium equations in a radiatively driven stellar wind. However, for the present purpose we slightly changed our code. Since we do not know in advance whether the stellar wind is possible or not, the hydrodynamical variables, i.e. velocities, temperatures, and densities of all wind components (with an exception of the electron density) are kept fixed. This enables us to calculate the model occupation numbers and the radiative force regardless the existence of the wind and, consequently, to test the wind existence.
The method itself was slightly improved with respect to the published one. We included accelerated lambda iterations (cf. Rybicki & Hummer 1992) and Ng acceleration (Ng 1974). These improvements will be described in detail in a separate paper (Krtička & Kubát, in preparation). Moreover, to obtain a correct surface flux we accounted for the Doppler effect during the calculation of radiative transfer in lines, i.e. we shifted the stellar surface flux H c according to the actual wind velocity (we accounted for the detailed profiles of photospheric lines, cf. Babel 1996). Finally, to assure a smooth transitions between stellar atmosphere and wind, the model atmosphere fluxes are taken at the point where the density is equal to the boundary wind density.
First, we have performed tests whether the radiative force is large enough to drive stellar wind in optically thin lines. Although this gives only the necessary condition for stellar wind and not the sufficient one (and thus further calculations are necessary to test whether the stellar wind is possible or not), this approach enables us to better understand more detailed models.

Minimum number density for line-driven winds
We performed two different tests of the possibility of a stellar wind of zero-metallicity stars driven by optically thin lines. First, we calculated the ionization structure of a possible extended stellar atmosphere and compared this ionization structure with minimum number densities for launching a linedriven wind obtained using Eqs. (15) and (18). Table 1. Stellar and wind parameters used to calculate discussed wind models taken from Kudritzki (2002). For our detailed discussion we selected only those stars from the Kudritzki (2002) list, which have the largest luminosity, i.e. those which are close to the Eddington limit. These stars have the largest radiative force due to the light scattering on free electrons in their atmospheres and thus also the highest chance to launch the stellar wind. Detailed models were calculated also for other stars from the Kudritzki's list, however for these stars we describe the final results only. Parameters of all studied model stars are given in Tab. 1.
We compared the minimum number densities necessary for launching the stellar wind with actual number densities calculated by our NLTE code. Since we do not know in advance whether the radiative force is strong enough to drive a wind, the velocity structure of our models is given by an artificial velocity law v(r) = 10 −3 5 3 This velocity law corresponds to the Taylor expansion of the wind velocity around the sonic point. According to our experience with wind models of O stars this expression is valid in the wind region close to the star with r 1.2R * within the accuracy of 20%. This is the region of rapid wind acceleration where the stellar wind attains velocities of the order of 100 km s −1 . Note, however, that obtained results do not significantly depend on the detailed form of the velocity law. The density structure is obtained from the equation of continuity. Wind temperature in these models was equal to T wind and electron density was consistently calculated from hydrogen and helium ionization balance. For each of these stars with different effective temperatures three models with different values of mass-loss rates were calculated, namely 10 −6 M ⊙ year −1 , Comparison of model number densities and minimum number densities necessary to launch the wind is given in Figs. 1 -3. Near the stellar surface number densities of ground levels of hydrogen-like ions are relatively high and they decrease with radius due to increasing ionization. The ionization is higher in the outer regions due to both lower density and higher radiative flux at frequencies outside the line core (due to the Doppler effect the line frequency shifts from the saturated core of photospheric line-profile, consequently larger flux absorbed in the lines increases populations of higher levels, for which the ionization rate is much higher than for lower levels). Minimum number densities also decrease. This is caused by increasing stellar flux as the wind lines move out from the centre of photospheric line due to the Doppler effect.
The relative number density of the ground level of H I is very low, mostly much lower than the minimum number density necessary to launch the stellar wind by H I lines. Thus, we conclude that H I lines are not important for line-driven winds of first stars. On the other hand, for specific regions of some models the actual number density of ground level of He II is higher than the minimum number density necessary to launch the stellar wind. In these regions the total radiative force may be higher than the gravity force and the sufficient condition for launching the stellar wind may be fulfilled there.

Maximum radiative force
The preceding simple analysis can be extended using the detailed calculation of the radiative force. For stars in Tab. 1 we have calculated the relative difference g thin + g rad e − g /g (i.e. the net acceleration in terms of gravity acceleration) as a function of radius. Corresponding graphs are plotted in Figs. 4 -6.
We came to the same conclusion as in the previous paragraph. The maximum possible radiative force is greater than the gravity in some regions of some models. However, the functional dependence of the difference g thin + g rad e − g in most cases does not have a correct radial dependence necessary to launch the stellar stellar wind, i.e. the total radiative force in these models is higher than the gravity near the star and lower than the gravity in the outer regions.
Moreover, we have to keep in mind that the optically thin radiative force is maximum for a given set of lines and clearly the actual radiative force will be lower due to the selfshadowing effect. Moreover, for many of the models the total radiative force is not greater than the gravity. Thus, we conclude that winds of zero metallicity stars driven purely by the optically thin lines are unlikely. However, these results shall be later firmed up by a more detailed models with the possibility that some lines are optically thick. Inspecting Figs. 1 -6 note that Eqs. (15) and (18) are able to reliably estimate the minimum occupation number necessary to launch a line-driven stellar wind because they correctly predict the location of region where the optically thin radiative force is greater than gravity.

Wind with optically thick lines
Our final test of the possibility of the existence of line-driven stellar wind of first stars should include the natural possibility that some lines are optically thick. Thus, we now turn our atten- tion to more realistic case of the stellar wind driven partially by the optically thick lines. This is a much more complicated task than the wind driven purely by optically thin lines since now we cannot use any simple criterion to decide whether the wind is possible or not. We have to use detailed modelling. Moreover, the nonlinear character of wind equations (especially equations of statistical equilibrium) does not guarantee that the obtained solutions are unique. This is especially the case of zero metallicity stars when the wind may be driven just by one ionization state. Simply, for some ionization structures the wind may be possible and for some other not.
To make this problem more tractable we calculate radiative force in the Sobolev approximation for the same models, for which we studied optically thin radiative force in Sect. 3. The radiative force in the Sobolev approximation is given by the sum of partial radiative forces due to the individual lines (Castor 1974, Abbott 1982 where ρ is the wind density, v r is the radial velocity, H c is the stellar flux, µ c = 1 − R 2 * /r 2 is the direction cosine of the photospheric limb, R * is stellar radius, σ = d ln v r /d ln r − 1 and the Sobolev optical depth τ µ is If some of the lines are optically thick, then the actual radiative acceleration given in the Sobolev approximation Eq. (22) is lower than the maximum radiative acceleration (2). This is caused by the effect of self-shadowing, i.e. that some part of the stellar flux which is supposed to accelerate the wind in a line was already absorbed by a given line in the region closer to the star. Thus, here we present only results for models for which optically thin radiative force can drive a stellar wind, since it is clear that there will not be any wind in the opposite case.
The radiative force calculated using Eq. (22) in the selected models is given in Fig. 7. Clearly, since most of the strongest lines are optically thick, the radiative force due to these lines is much lower than the optically thin radiative force. More realistic radiative force given by Eq. (22) is generally by several orders of magnitude lower than the gravity force. Thus it is unlikely that some change of wind density or velocity may give significantly stronger radiative force which would drive a stellar wind. This is also supported by our models, because we were not able to find sufficiently large radiative force for any reasonable wind models. We conclude that it is unlikely that stationary massive H-He stars have any H-He stellar wind driven by the line transitions.
We have also calculated similar models for other stars from Kudritzki (2002) list with lower masses and lower luminosities (with parameters given in Tab. 1). These stars are not so close to the Eddington limit as stars discussed sofar. Consequently, the radiative force due to light scattering on free electrons is lower for these additional stars and the minimum radiative force necessary to launch the stellar wind is higher in this case (see Eqs. 15,18). Clearly, the possibility for launching of the stellar wind is lower for stars which are not so close to the Eddington limit.

Influence of continuum radiative force
Another force which can potentially accelerate the stellar wind of first stars is the radiative force due to bound-free and free-  Fig. 7. The plot of the net radiative acceleration g rad + g rad e − g and the net radiative acceleration of the whole gas due to free electrons only g rad e − g relative to the gravitational acceleration g. Clearly, the contribution of the radiative acceleration due to the line transitions g rad calculated after Eq. (22) is much smaller than the gravity acceleration and is not able to drive a stellar wind. free transitions. Numerical calculation showed that the contribution of free-free transitions to the radiative force in the circumstellar environment is much lower than that of bound-free transitions. Thus, we shall discuss the bound-free transitions only, although the free-free transitions are also included in our models. The radiative acceleration has in this case the form (Mihalas 1978) where χ ν is the opacity due to the bound-free transitions, where summation goes over all levels i for which bound-free transitions are accounted for and the asterisk refers to the LTE value. Let us first briefly discuss an interesting possibility that the stellar wind is driven purely by the bound-free absorption. Combining Eqs. (24) and (25), the radiative acceleration is given by As has been already mentioned in Sect. 2.4, in order to obtain smooth wind solution, the radiative force has to be smaller than the gravity force in the stellar atmosphere below the critical point and larger than the gravity force above the critical point. It is very difficult to achieve this in this case, since the radiative flux H ν decreases with radius rapidly, especially in the case when strong bound-free absorption is present. Moreover, the critical condition is not a sufficient one to drive the stellar wind, since it does not tell us anything whether the radiative force in the outer wind regions is strong enough to lift the wind material out from the stellar gravitational potential well. The decrease of the stellar flux with radius due to the bound-free absorption (see also the photon tiring effect, Owocki & Gayley 1997) can complicate this. Note that the situation with the line-driven wind is simpler, because the lines are usually illuminated by the unattenuated stellar flux thanks to the Doppler effect.
We included the radiative force due to the bound-free and free-free transitions Eq. (24) into our models. The numerical calculations (for stars with parameters given in Tab. 1) showed that the radiative force due to the bound-free and free-free transitions does not significantly contribute to the acceleration of stellar wind of first stars because it is typically at least two orders of magnitude lower than the gravity force.
Finally, we calculated also static spherically symmetric NLTE model H-He atmospheres (for the description of the code see Kubát 2003), where the radiative force was calculated without any simplification. It turned out that all of the models with parameters from the Table 1 are sub-Eddington i.e. the total radiative force is always lower than gravity. However, for the most luminous models the value of Γ reached the value of 0.99.

Expulsion of individual elements by a wind
Whereas first stars probably fail to drive H-He outflow, there is a possibility that hydrogen or helium are expelled separately, i.e. that there exists a pure hydrogen or helium wind. To show whether such outflow of chemically pristine elements is possible or not, we recalculated models with the same stellar parameters presented in the Sect. 3 assuming that one element may form the wind while the other not and compared the radiative force in such chemically homogenous wind with the gravity force acting against such wind. To avoid charging of the star we assume that a corresponding number of electrons follow the escaping element making the wind electrically neutral. We assume that such wind exists above the H-He photosphere.
Since these models are chemically homogenous, the electron density in such models is calculated only from the contribution of the element that remained in the wind (i.e. either hydrogen or helium). This difference is crucial for the final results, at least in the case of helium. If we assume that the helium gas consists of only the isotope 4 He, we may say that helium has two times less electrons per nucleon (proton or neutron) than hydrogen. Hence, the radiative force per unit of mass (i.e. the radiative acceleration) due to free electrons is lower in the case of helium. Consequently, although the number of free electrons per atomic nucleus is higher in pure He than in H-He models, the total radiative force is lower. Surprisingly, although helium lines significantly contribute to the radiative force in the H-He models, the possibility of pure He outflow is lower than that of H-He.
On the other hand, also the situation in pure hydrogen models is different. Hydrogen plasma has higher number of free electrons per nucleon than H-He plasma. Consequently, H-He stars which are close to the Eddington limit may have slightly higher radiative force than gravity force for the same stellar parameters (mass, radius, luminosity) in a pure hydrogen wind, whereas the radiative force in hydrogen-helium atmosphere is (due to the contribution of heavier helium atom) lower than gravity force. These considerations are supported by our numerical models. The total radiative force in helium models is lower than gravitational force in all models considered (for stars with parameters given in Tab. 1). On the other hand, for model stars which are very close to the Eddington limit, the radiative force in purely hydrogen models may be greater than gravity, thus potentially enabling hydrogen wind.
Condition for the existence of such hydrogen wind is relatively simple. If the wind exists, radiative acceleration due to the free electrons in a pure hydrogen plasma is greater than the gravitational acceleration, or using Eddington parameter Γ H in a pure hydrogen plasma (cf. Eq. 8) where s e is the Thomson scattering cross-section, and n H e is the electron number density in hydrogen plasma with the density ρ H . Condition (28) can be rewritten in a more convenient form. Now we compare conditions in pure hydrogen wind with that in hydrogen-helium atmosphere. Using Eqs. (8) and (5) for the hydrogen-helium atmosphere, we can express the fraction L/ (4πcGM ) in (28) using the Eddington parameter in the stellar atmosphere Γ and we obtain which yields the condition (ρ = ρ H + ρ He is the total density in the atmosphere) For a chemical composition given by primordial nucleosynthesis and assuming a fully ionized gas both in the atmosphere and in the wind we obtain simple condition Γ 0.859.
Clearly, pure hydrogen-helium stars close to the Eddington limit (with 0.859 Γ < 1) may have a pure hydrogen wind, but not the H-He wind.
The only stars from our sample for which Γ 0.859 and which thus can have a pure hydrogen wind are the most massive stars with M = 300 M ⊙ (for all considered values of effective temperature) and star with M = 250 M ⊙ and T eff = 40 000 K.
However, despite this behaviour of the radiative force in pure hydrogen models, pure hydrogen winds need not necesserily exist in all stars for which they are theoretically possible. Radiative force in these models is not due to the hydrogen line transitions, but mostly due to the free electrons. Thus, in order to hydrogen wind exists, there should be some proccess which is able to separate hydrogen and helium atoms. The mass-loss rate of such a theoretical wind depends mainly on the rate of this separating process. The only possible process, which comes into consideration, is the gravitational settling due to the different atomic masses (see Michaud 2005 for a recent review). However, the typical rate of such a process is very small, thus mass-loss rates of a pure hydrogen wind would be probably also very small (see also Appendix A). Moreover, any macroscopic motion in the stellar atmosphere may inhibit this gravitational settling. Thus, we conclude that any possible pure hydrogen wind has either a very low mass-loss rate or is even missing.

Contribution to the radiative force
Hydrogen and helium isotopes 2 1 H and 3 2 He were produced in a nonnegligible amount during the era of the Big-Bang nucleosynthesis. The positions of their lines are slightly shifted with respect to those of 1 1 H and 4 2 He and thus they can potentially be exposed to a slightly higher flux than their much more abundant counterparts.
To test their importance we included 2 1 H and 3 2 He into our atomic list. We used basically the same atomic data for the calculation of model atoms, however with modified energy levels due to the isotopic shift. The isotopic abundances were taken from the simulation of primordial nucleosynthesis by Coc et al. (2004), N ( 1 1 H)/N ( 2 1 H) = 2.6 × 10 −5 , N ( 1 1 H)/N ( 3 2 He) = 1.04 × 10 −5 , which are in a relatively good agreement with the observed values.
The importance of other hydrogen and helium isotopes was tested using optically thin models discussed in Sect. 3. The calculated excitation and ionization states of 2 1 H and 3 2 He are very similar to those of 1 1 H and 4 2 He. Moreover, due to the relatively small isotopic shifts and due to the relatively broad photospheric lines the flux at the positions of lines of less abundant isotopes is similar to that of more abundant isotopes. Finally, the abundance of these isotopes is several orders of magnitude lower than the ordinary ones, which significantly diminishes the optically thin radiative force. Consequently, our models showed that the contribution of less abundant hydrogen and helium isotopes 2 1 H and 3 2 He to the radiative force is negligible. Similarly, we have also tested the contribution of lithium to the total radiative force. To this end, we have selected data on excitation and ionization of lithium ions from the Opacity Project database (Seaton 1987, Peach et al. 1988, Fernley et al. 1987. The primordial lithium abundance N ( 1 1 H)/N ( 7 3 Li) = 4.15 × 10 −10 was taken from Coc et al. (2004). Numerical calculation showed that due to the extremely low lithium abundance its contribution to the radiative force can be also neglected.

Expulsion of particular isotopes and abundance stratification
Slightly different radiative force acting on different isotopes and different isotopic mass may cause radiative levitation of some isotopes and/or gravitation settling of others, the effect also referred to as the light induced drift (Aret & Sapar 2002). Thus, atmospheres of first generation stars may be chemically stratified. However, the discussion of these issues is beyond the scope of the present paper which is mainly intended to the study of an outflow from first generation stars. There appears a more interesting question in this context. Although the radiation force due to the less abundant isotopes is very small and has only marginal influence on the stellar atmospheres, it might be sufficient to expel these elements out from the stellar surface into the interstellar medium. However, numerical calculations showed that this is probably not the case. Similar considerations already discussed in Sect. 6 for ordinary hydrogen and helium are valid also for deuterium and the 3 2 He isotope. Radiative force acting on the deuterium is too low to allow a pure deuterium wind. Moreover, the radiative force due to free electrons is lower in 3 2 He wind due to a lower number of free electrons per one nucleon compared to the H-He wind.
Similarly, numerical calculations showed that pure lithium wind is also unlikely. Although the radiative force acting on on lithium may be higher than the gravity force in deeper layers of some models, in the outer regions is always too low to drive the lithium wind.

Influence of line transitions on accretion physics
Up to now, the initial mass function of first stars is not known (see Bromm & Larson 2004). One of the problems with the determination of initial mass function of first stars is connected with the process of termination of accretion onto a protostar. Dust grains are not present there, so the radiation pressure on dust grains, which operates in contemporary stars is not apparently available there. Thus, other mechanisms may be important for metal-free stars (e.g. Omukai & Inutsuka 2002, Tan & McKee 2004).
Here we do not intend to study accretion physics in detail, but only to depict such processes which may be important for termination of accretion, namely we would like to test whether line transitions due to hydrogen and helium are able to influence the accretion.
Thus, we selected a fixed velocity law which can be obtained from the solution of the momentum equation for isothermal (T = T wind ) spherical accretion (Mihalas 1978 The boundary condition of Eq. (33) was artificially selected to be v r (100R * ) = 5 3 a and the density structure was calculated from the continuity equation whereṀ is the accretion rate. We selected three different values of accretion rates, namely 10 −3 M ⊙ year −1 , 10 −5 M ⊙ year −1 , 10 −7 M ⊙ year −1 . The highest value of accretion rate roughly corresponds to the critical accretion rate over which protostellar core reaches the Eddington limit and no supermassive star forms (Omukai & Palla 2003). For the velocity and density structure mentioned above we solved statistical equilibrium equations for hydrogen and helium inflow. We neglected any influence of possible shock at the stellar surface and assumed that the protostar is in hydrostatic equilibrium. In such a case for our numerical calculations we may use the same stellar parameters as we used for wind tests, namely the stellar parameters taken from Kudritzki (2002). Our numerical calculations showed that the sum of the radiative force in the Sobolev approximation calculated after Eq. (22) and the radiative force acting on free electrons can together exceed the gravitational force only in stars close to the Eddington limit and only for relatively small accretion rates (of the order 10 −5 M ⊙ year −1 and smaller).
The principal difference between possible stellar wind and accretion flow is that important regions of accretion occur at much larger distances from the star than the acceleration region of the stellar wind. Since the ionizing stellar flux at larger distances is lower, the inflow (especially helium) is less ionized and the line radiative force due to He II becomes important relative to the gravity force.
Clearly, the importance of the radiative force rises for more massive stars closer to the Eddington limit. Thus, there is a possibility that for even more massive stars than those considered here the radiative force both due to line transitions and free electrons is able to overcome gravity and set the initial mass function of supermassive stars. These considerations have to be, however, tested against more advanced accretion models.

Radiative transfer
There are several problems which can influence our final results. Probably the most serious one is the simplified radiative transfer in lines. This can influence the calculated occupation numbers and hence also the radiative force. However, this is probably not the case, since the results obtained are relatively robust, i.e. in many cases even the optically thin radiative force is not sufficient to drive a stellar wind.

Level dissolution
There is another more subtle problem which is related to the neglect of the so-called level dissolution (Hummer & Mihalas 1988). It is the statistical treatment of the fact that the highest atomic levels cease to exist under the influence of surrounding particles. Usually this problem is treated approximately using the lowering of ionization potential. However, it can be treated in a consistent way using the so-called occupation probability formalism (Hubeny et al. 1994, Kubát 1997). Possibly, neglect of level dissolution can influence the calculated radiative force, specially in the case of hydrogen. However, test calculations with different number of hydrogen and helium lines showed this is not the case. Although total number densities of individual ionization states may differ, number densities of lower excited levels (which are important for line driving) are nearly the same. Since there is a relatively low radiative flux in the wavelength regions of lines originating from higher excited levels, their contribution to the radiative force is negligible (see also Sect. 2.2). Thus, we conclude that the influence of level dissolution on our results is only marginal.

Pulsations
Since we assume stationary flow here, the effect of possible stellar pulsations is neglected. According to numerical simulations of Baraffe et al. (2001), primordial stars may actually lose mass via pulsational instability. However, such effect is connected mostly with the stellar interior, so it is beyond the scope of the present paper.

Stars with chemical composition given by primordial nucleosynthesis
We have studied the dynamical aspects of a possible circumstellar envelope around massive first stars using our NLTE models. Although many of studied stars are very close to the Eddington limit and the maximum possible radiative force (calculated assuming that all lines are optically thin) is high in some cases, the stellar winds driven by line transitions are unlikely, because the line radiative force is very small. The influence of other isotopes and elements, which had been produced during primordial nucleosynthesis, was found to be marginal. Also the effect of the continuum radiative force is negligible.
We have also tested whether first stars can expel individual elements, either the dominant ones (i.e. hydrogen and helium) or the trace ones (i.e. hydrogen and helium isotopes or lithium). Calculations showed that the only theoretically possible chemically homogeneous outflow is that of hydrogen (for stars very close to the Eddington limit with Γ 0.859), since other elements yield a small number of free electrons per nucleon (and the line-radiative force is always negligible). However, even the case of a pure hydrogen outflow is unlikely since the radiative force in such a wind would be predominantly due to the free electrons. Hence, another process which would be able to separate hydrogen and helium atoms in the atmosphere should exists (e.g. gravitational settling) to enable a pure hydrogen wind. Without such a process a pure hydrogen wind is not possible.
Finally, we have tested whether the radiative force due to line transitions is able to terminate the initial accretion onto first stars. We have found that this is only the case of stars relatively close to the Eddington limit and with relatively small accretion rates. Consequently, it is possible that also for even more massive stars this process can influence the accretion physics and the initial mass-function of first stars.
Our results do not imply that the first stars are completely without line driven winds during all of their life time. For example, cooler stars (which were not inlcuded into our present study) could have wind due to the hydrogen or helium lines. Moreover, if the accretion terminates, then as soon as sufficient amount of heavier elements is synthesised in the stellar core and transported to the surface layers, the line-driven wind is initiated (Kudritzki 2002). If this happens at a sufficiently low metallicity, such a stellar wind is multicomponent, allowing for element separation and frictional heating (Krtička et al. 2003).

General picture of stellar wind of massive low-metallicity stars
From this study and previous ones (Kudritzki 2002, Krtička et al. 2003 a general picture of the properties of stellar wind of massive stars at low metallicities emerges. i Zero-metallicity stars with Γ 0.859 likely do not have any stellar wind. If their atmospheres are relatively quiet (i.e. in the absence of violent pulsations or fast rotation) the chemical peculiarity in their atmospheres may develop. On the other hand, zero-metallicity H-He stars with Γ 0.859 may have pure-hydrogen wind (probably very thin with mass-loss rates of order 10 −14 M ⊙ year −1 , see Appendix B), driven only by the light scattering on free electrons. ii With increasing metallicity the line-driven wind may develop. For extremely low metallicities such a stellar wind is likely pure metallic (see Babel 1996). iii For higher metallicities also hydrogen and helium are expelled from the atmosphere and multicomponent effects become important in such a stellar wind (Krtička et al. 2003). The multicomponent structure does not significantly influence the wind mass-loss rate, which may be approximated by formulas obtained for one-component flow by Kudritzki (2002). iv For higher metallicities, a line-driven wind which is similar to the stellar wind of present hot stars exists and its massloss rate was calculated by Kudritzki (2002).
Thus, in our opinion, the best choice for the evolutionary calculation now is to assume zero mass-loss rate for zero and low metallicity stars mentioned in items i-ii and to use Kudritzki (2002) prescription for other stars.  Fig. 4. The plot of the relative difference g thin + g rad e − g /g for model T40. Note that the points where this difference is zero match the points where the actual occupation number N He II,1 is equal to N min He II,1 relatively well. Thus, the simple conditions Eqs. (15) and (18) for the minimum occupation number necessary to launch the stellar wind are able to predict relatively correctly the regions where the total radiative force is greater than gravity.