First stars: evolution without mass loss

First Stars. I. Evolution without mass loss

Abstract

The first generation of stars was formed from primordial gas. Numerical simulations suggest that the first stars were predominantly very massive, with typical masses . These stars were responsible for the reionization of the universe, the initial enrichment of the intergalactic medium with heavy elements, and other cosmological consequences. In this work, we study the structure of Zero Age Main Sequence stars for a wide mass and metallicity range and the evolution of , , , and galactic and pregalactic Pop III very massive stars without mass loss, with metallicity and , respectively. Using a stellar evolution code, a system of 10 equations together with boundary conditions are solved simultaneously. For the change of chemical composition, which determines the evolution of a star, a diffusion treatment for convection and semiconvection is used. A set of 30 nuclear reactions are solved simultaneously with the stellar structure and evolution equations. Several results on the main sequence, and during the hydrogen and helium burning phases, are described. Low metallicity massive stars are hotter and more compact and luminous than their metal enriched counterparts. Due to their high temperatures, pregalactic stars activate sooner the triple alpha reaction self-producing their own heavy elements. Both galactic and pregalactic stars are radiation pressure dominated and evolve below the Eddington luminosity limit with short lifetimes. The physical characteristics of the first stars have an important influence in predictions of the ionizing photon yields from the first luminous objects; also they develop large convective cores with important helium core masses which are important for explosion calculations.

first stars, stars: models, evolution
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1 Introduction

A first generation of stars composed of primordial nearly pure H/He gas must have existed, since heavy elements can only be synthesized in the interior of the stars. These first stars, also called Population III (or Pop III), were responsible for the initial enrichment of the intergalactic medium (IGM) with heavy elements (Bond , 1981; Klapp , 1981, 1983, 1984; Cayrel , 1986, 1996; Carr , 1987, 1994; Bromm et al. , 2002).

UV photons radiated by the first stars, perhaps together with an early population of quasars, are expected to have contributed to the IGM reionization (Haiman and Loeb , 1997; Ferrara , 1998; Miralda-Escudé et al. , 2000; Tumlinson and Shull , 2000; Bromm et al. , 2001; Schaerer , 2002). The energy output from the first stars might have left a measurable imprint on the cosmic microwave background (CMB) on very small scales (Visniac , 1987; Dodelson and Jubas , 1995).

However, despite an intense observational effort, the discovery of a true Pop III remains elusive. To probe the time when star formation first started entails observing at very high redshifts .

The first stars were typically many times more massive and luminous than the Sun (Larson and Bromm , 2004). A review of theoretical results on the formation of the first stars has been made by (Bromm and Larson , 2004). The masses of the first star-forming clumps would have been about 500 to . Several numerical simulations suggest that the first stars were predominantly Very Massive Stars (VMS), with typical masses (Bromm et al. , 1999, 2002; Nakamura and Umemura , 2001; Abel et al. , 2000, 2002) and these stars had important effects on subsequent galaxy formation.

In another scenario, the hypothesis that first stars were VMS () has been strongly criticized because the pair-instability supernovae yield patterns are incompatible with the Fe-peak and r-process abundances found in very metal poor stars. Models including Type II supernova and/or hypernova from zero-metallicity progenitors with can better explain the observed trends (Tumlinson et al. , 2004). The same authors also pointed out that the sole generation of VMS ( cannot be possible and suggested that some VMS could be formed as companions of stars with masses . Their Initial Mass Function (IMF) proposition match quite well with the reionization and nucleosynthesis evidence. Tegmark et al. (1997) argued that the minimum baryonic mass is redshift dependent and lies in the range to , for and , respectively, and that a participation of of the whole baryonic matter in the generation of luminous stars is sufficient to reheating the universe by .

The Wilkinson Microwave Anisotropy Probe has observed the large-angle polarization anisotropy of the CMB (Cen , 2003; Kogut et al. , 2003; Sokasian et al. , 2003; Wyithe and Loeb , 2003). Some measurements have been interpreted as a signature of a substantial early activity of massive star (MS) formation at high redshifts .

The supernova explosions that ended the lives of the first stars were responsible for the initial enrichment of the intergalactic medium with heavy elements (Ostriker and Gnedin , 1996; Gnedin and Ostriker , 1997; Bromm et al. , 2003; Yoshida et al. , 2004). An interesting possibility unique to zero-metallicity massive stars is the complete disruption of their progenitors in pair-instability supernovae explosions, which are predicted to leave no remnant behind (Barkat et al. , 1967; Ober et al. , 1983; Bond et al. , 1984; Fryer et al. , 2001; Heger and Woosley , 2002; Heger et al. , 2003; Bromm and Larson , 2004). The later works consider that this peculiar explosion mode could have played an important role in quickly seeding the intergalactic medium with the first metals.

Related to the first stars, there are two very important unsolved questions: 1. What are their typical masses and Initial Mass Function (IMF)? and, 2. During their cuasi-static evolutionary phases, do they have radiation driven winds or mass loss due to other mechanisms?

Star formation and accretion calculations suggest that the first stars were very massive (Omukai and Palla , 2003). On the other hand, by comparing the observed abundance patterns of Extremely Metal Poor (EMP) stars with supernova explosion calculations, some authors have concluded that the first stars are more likely to have masses in the range , but not more massive that (Umeda and Nomoto , 2002; Heger and Woosley , 2002).

We have suggested that a possible solution to this inconsistency is that first stars are born very massive but that during their cuasi-static evolutionary phases, lose mass and reach the pre-supernova stage with the masses required from supernova calculations to reproduce the EMP abundance pattern (Klapp et al. , 2005; Bahena , 2006). It is then very important to estimate the amount of first stars mass loss during the cuasi-static evolutionary phases.

Kudritzki (2000, 2002) calculated wind models of massive stars between 100 and and metallicities in the range 0.0001 to 1.0 solar, in an effective temperature range from to K, with the objective of predicting mass-loss rates at very low metallicities applicable to the first generation of massive stars. It was found that for very low metallicities, the line driven mechanism becomes very inefficient and wind solutions can only be obtained very close to the Eddington limit. He also pointed out that very massive stars are pulsationally unstable, which might contribute to stellar mass loss, in particular at low metallicity when the contribution of the radiative driving to the wind decreases. However, the critical mass for the onset of the nuclear pulsational instability is uncertain.

Other mechanisms could induce first stars mass loss, for example, the low metallicity rotation models of Meynet and Maeder (2002) show fast rotating cores that lose significant amounts of mass and thus angular momentum. Rotation by enhancing the luminosity and lowering the effective gravity increases the mass loss rate. Then, it is possible that pulsation, rotation and Luminous Blue Variables (LBV) type phenomenae could induce significant amounts of mass loss during the first stars cuasi-static evolutionary phases.

Motivated by the above arguments, in a series of papers we will study the structure, evolution and nucleosynthesis of the first stars with and without mass loss and rotation. In this Paper I we present the evolutionary results without mass loss and with no-rotation.

This work is organized as follows: In Sect. 2 we describe the initial conditions of the stellar models and the way in which the main physical variables are computed. Then, in Sect. 3 we describe the main results, and in Sect. 4 we discuss our results and compare them with other authors. Finally, in Sect. 5 we outline our conclusions.

2 Numerical modelling and input physics

We present Zero Age Main Sequence (ZAMS) models for stars in the mass range with compositions for Pop I, for Pop II, for galactic Pop III, and and for pregalactic Pop III.

In this paper, evolutionary models for Pop III stars have been calculated without mass loss and with no-rotation. The chemical composition of the models is and , for galactic and pregalactic stars, respectively. For the evolution the chosen stellar masses were , , , and .

The main difference between galactic and pregalactic stars is their initial metallicity. According to Castellani (2000) we consider a Pop III metallicity range from to . The transition from pregalactic to galactic stars occur at the critical metallicity (Bromm and Larson , 2004). Pregalactic stars are characterized by self-producing their own heavy elements; galactic stars correspond to the next generation of stars which have been previously enriched with metals.

The computer program used for the calculations has been described by Klapp (1981, 1983) and Bahena (2006, 2007), with updated input physics.

For the nuclear reaction rates we use the Nuclear Astrophysics Compilation of Reaction Rates (NACRE) by Angulo et al. (1999). A diffusion treatment for convection and semiconvection is used. For the opacity we have adopted the OPAL radiative opacities (Rogers and Iglesias , 1992; Iglesias and Rogers , 1993).

3 Results

3.1 Zero Age Main Sequence

For our Pop I, II and III models, in Fig. 1 we show their ZAMS Hertzsprung-Russell (HR) diagram for the mass range . The objective is to understand the ZAMS structure differences as function of mass and metallicity. A large number of very detailed ZAMS models with different masses and metallicities have been calculated.

The luminosity and effective temperature increases with mass. Low mass stars are located at the right lower part of the diagram, while massive and very massive stars are found in the left upper part, because they are the most luminous and hotter. Pop III stars are hotter than their Pop I and II counterparts, and so their locus on the HR diagram is shifted to the left upper part. Pregalactic stars are bluer than galactic stars. This is shown in Fig. 1 and also in Fig. 2 that is an amplification for the range. In Fig. 3 we show a log - log diagram for a large mass and metallicity range.

Figure 1: Hertzsprung-Russell (HR) diagram for ZAMS Pop I, II and III stars for the mass range and metallicities and .
Figure 2: HR-diagram for ZAMS Pop I, II and III stars for the mass range . Depending on metallicity the Pop III ZAMS is systematically shifted to higher effective temperature.
Figure 3: ZAMS Pop I, II, and III Log -Log plane for the mass range . With increasing density and temperature, radiation pressure becomes more important.

In Figs. 4 to 9 we show the main physical variables for the ZAMS models. All quantities are plotted as function of the mass. As the mass increases, ZAMS stars become bigger, brighter and less dense. With decreasing metallicity, Pop III stars get very hot and compact. All massive and very massive stars are dominated by radiation pressure and develop a large convective core. In all cases, however, their luminosity is below the Eddington upper luminosity limit.

Figure 4: Central density for ZAMS models in the mass range .
Figure 5: Ibidem. Effective temperature.
Figure 6: Ibidem. Luminosity.
Figure 7: Ibidem. Central temperature.
Figure 8: Ibidem. Radius.
Figure 9: Ibidem. Gamma factor which is the ratio of the luminosity to the Eddington Luminosity.

Then, we present ZAMS models in the metallicity range from to for , , , and , which are shown in Figs. 10 to 15. The central density increases with decreasing metallicity. The most massive stars are less dense. MS and VMS develop a large convective core, but its size decreases slowly with decreasing metallicity. Stellar radius is lower for low-metallicity. The radius also decrease with decreasing mass, however, lower-metallicity stars are more compact. Central temperature, and effective temperature, increase with decreasing metallicity, and the most massive stars are hotter. Luminosity does not depend on metallicity but on the mass. The most massive stars are the most luminous stars. The Eddington luminosity factor , which is the ratio of the luminosity to the Eddington luminosity does not depend on metallicity. The most massive stars are the ones closest to the Eddington luminosity, and the upper luminosity limit.

Figure 10: Central density in the metallicity range from to for , , , and .
Figure 11: Ibidem. Effective temperature.
Figure 12: Ibidem. Luminosity.
Figure 13: Ibidem. Central temperature.
Figure 14: Ibidem. Radius.
Figure 15: Ibidem. Gamma factor.

3.2 Stellar structure

We have calculated the evolution of , , , and stars with metallicities , , , , and . Pop III stars have higher density, temperature and pressure than their Pop I and II counterparts, and are radiation pressure dominated and very luminous.

Pregalactic stars have central temperatures of about log and effective temperatures of log . A direct consequence of higher central temperatures is that they have higher energy generation rates. On the main sequence, lower metallicity stars produce slightly less energy. However, these stars are hotter than the others and so require higher temperatures to produce the same amount of energy.

The most striking feature of the low metallicity stellar models is their atmospheric high temperature they are able to maintain. During hydrogen burning, these stars derive their nuclear energy from the inefficient pp-chains and the CNO-cycles. This is possible because a small fraction of carbon is produced during the pre-main sequence phase (Castellani et al. , 1983). Some carbon is also generated by the triple- process before the star reaches the main sequence (Bromm et al. , 2001). That is, in the absence of metals nuclear burning proceeds in a non-standard way. First, the hydrogen burning occurs via the pp-chain. However, metal-free stars are hotter and very luminous reaching high central temperatures which are high enough for the simultaneous occurrence of helium burning via the triple- reaction. After a brief initial period of this process, a trace amount of heavy metals are formed and this makes that most of the energy generation rate during hydrogen burning comes from the CNO-cycles.

Lower metallicity stars have a higher central temperature to support the star against gravity. This is a consequence of their high central density and temperature, and because this type of stars are very compact.

3.3 Stellar evolution without mass loss

Physical variables

For the present work stellar evolution models without mass loss have been computed during the hydrogen and helium burning phases, for very massive galactic and pregalactic Pop III and stars with metallicities and , respectively.

As an example of our evolution models, we list in Tables 1 and 2 some properties of galactic and pregalactic Pop III stars.

The first two columns give the lifetime , given in units of years, and the helium mass fraction during the hydrogen and helium burning phases. Several physical variables in logarithmic units are included such as the central density , the central temperature , the luminosity (in solar units), the effective temperature , the radius , and the nuclear energy generation rate . Then, the other quantities listed are the convective core size , which is the ratio of the mass of the convective core to the total mass of the star, the radiation factor in the centre of the star, defined by , where is the radiation pressure and the total pressure, and the Eddington luminosity factor .

log log log log log log
0.04407 0.23500 0.73760 7.87567 6.55521 4.87844 11.88977 5.76058 0.89348 0.42004 0.73286
0.34457 0.30006 0.74533 7.88046 6.57138 4.87399 11.90675 5.77809 0.89744 0.40450 0.74736
0.76005 0.40003 0.76022 7.88834 6.59460 4.86358 11.93919 5.80563 0.90758 0.38060 0.77037
1.13125 0.50000 0.77986 7.89717 6.61619 4.84854 11.98007 5.83358 0.91887 0.35667 0.78979
1.46357 0.60006 0.80543 7.90722 6.63618 4.82777 12.03159 5.86349 0.92563 0.33274 0.81376
1.76063 0.70001 0.84086 7.91962 6.65458 4.80000 12.09633 5.89558 0.93542 0.30890 0.83259
2.02594 0.80008 0.89235 7.93622 6.67161 4.76299 12.17888 5.93093 0.94607 0.28509 0.85120
2.26415 0.90014 0.97789 7.96280 6.68784 4.71413 12.28471 5.96946 0.95284 0.26120 0.87436
2.46034 0.99983 1.34155 8.07854 6.70443 4.67744 12.36639 5.97202 0.46767 0.23943 0.87764
2.46037 0.99996 1.34222 8.07873 6.70444 4.67745 12.36637 5.97430 0.46771 0.23942 0.87766
2.46045 0.99986 1.34278 8.07892 6.70445 4.67745 12.36638 5.97639 0.46776 0.23942 0.87784
2.50534 0.90010 2.45172 8.42220 6.73086 4.43326 12.86797 6.41240 0.30804 0.26984 0.99198
2.52450 0.80004 2.40005 8.40679 6.73326 4.32939 13.07691 6.39498 0.40993 0.26280 0.91068
2.54783 0.70015 2.37961 8.40012 6.73469 4.24234 13.25172 6.39498 0.43042 0.25803 0.89312
2.57361 0.60009 2.37056 8.39699 6.73547 4.17638 13.38405 6.39740 0.43887 0.25356 0.88887
2.60105 0.50017 2.36860 8.39610 6.73588 4.14063 13.45575 6.40088 0.44543 0.24923 0.89512
2.62969 0.40017 2.37269 8.39710 6.73603 4.13471 13.46766 6.40352 0.45009 0.24500 0.89164
2.65913 0.30001 2.38339 8.40015 6.73596 4.14619 13.44467 6.40307 0.45347 0.24090 0.89055
2.68881 0.20014 2.40316 8.40603 6.73581 4.15927 13.41843 6.40012 0.45335 0.23697 0.88681
2.71795 0.10005 2.44120 8.41767 6.73569 4.16384 13.40923 6.36450 0.45693 0.23337 0.88916
2.73912 0.01070 2.51746 8.44134 6.73582 4.15837 13.42023 5.91299 0.45401 0.23109 0.88956
2.74053 0.00123 2.53190 8.44585 6.73588 4.15804 13.42092 5.50297 0.45446 0.23099 0.89017
Table 1: Physical variables and quantities for galactic Pop III stars with initial metallicity , without mass loss and with no-rotation, during the hydrogen and helium burning phases.
log log log log log log
0.04407 0.23500 1.46281 8.11724 6.56930 5.00039 11.65293 5.68780 0.87527 0.42049 0.73962
0.32884 0.30008 1.47318 8.12283 6.58481 4.99569 11.67006 5.70573 0.88415 0.40503 0.75446
0.72025 0.40000 1.49258 8.13204 6.60693 4.98487 11.70278 5.73226 0.75495 0.38138 0.77484
1.06814 0.50003 1.51732 8.14230 6.62713 4.96938 11.74386 5.75997 0.90139 0.35781 0.79531
1.37810 0.60005 1.54912 8.15406 6.64545 4.94843 11.79492 5.78962 0.64234 0.33438 0.81714
1.65450 0.70003 1.59211 8.16853 6.66204 4.92085 11.85836 5.82162 0.91933 0.31113 0.83462
1.90105 0.80005 1.65303 8.18772 6.67699 4.88498 11.93759 5.85683 0.93180 0.28799 0.85101
2.11975 0.90016 1.75252 8.21821 6.69054 4.83938 12.03556 5.89503 0.93966 0.26493 0.87007
2.30096 0.99992 2.19813 8.35947 6.70693 4.80493 12.11265 5.73874 0.41750 0.24530 0.85123
2.30098 1.00006 2.19971 8.35997 6.70695 4.80505 12.11242 5.74387 0.41763 0.24533 0.85064
2.30103 0.99994 2.20122 8.36047 6.70697 4.80517 12.11221 5.74887 0.41776 0.24534 0.85011
2.32034 0.90054 2.40066 8.42261 6.71089 4.80396 12.11658 6.36513 0.41280 0.24675 0.88169
2.33776 0.80019 2.35674 8.40641 6.71097 4.78656 12.15143 6.34291 0.42242 0.24439 0.87961
2.35934 0.70013 2.34010 8.39941 6.71137 4.77235 12.18005 6.34426 0.42915 0.24187 0.87765
2.38348 0.60006 2.33432 8.39608 6.71175 4.76042 12.20409 6.34729 0.43179 0.23922 0.87840
2.40920 0.50011 2.33553 8.39510 6.71207 4.75027 12.22454 6.35226 0.43465 0.23647 0.88157
2.43607 0.40027 2.34233 8.39598 6.71239 4.74112 12.24300 6.35622 0.43754 0.23367 0.88169
2.46378 0.30015 2.35559 8.39895 6.71276 4.73206 12.26131 6.35760 0.44051 0.23084 0.88270
2.49173 0.20016 2.37737 8.40467 6.71329 4.72292 12.27985 6.34892 0.43961 0.22803 0.88389
2.51901 0.10025 2.41608 8.41584 6.71408 4.71410 12.29789 6.31307 0.43980 0.22538 0.88633
2.53894 0.01019 2.48863 8.43781 6.71513 4.70969 12.30723 5.81421 0.43147 0.22366 0.88707
2.53997 0.00157 2.49814 8.44068 6.71524 4.70981 12.30705 5.86620 0.43179 0.22360 0.88798
Table 2: Physical variables and quantities for pregalactic Pop III stars with initial metallicity , without mass loss and with no-rotation, during the hydrogen and helium burning phases.

These tables summarize the most representative data of the models, and they include hydrogen and helium burning from the initial chemical composition, in % intervals of the helium mass fraction, to the end of helium burning when the helium mass fraction is approximately .

VMS are hot and luminous and so are located in the left upper part of the Hertzprung-Russell (HR) diagram. Because Pop III stars are hotter than their enriched counterparts, their locus in the HR-diagram is shifted to the left upper part; pregalactic stars are bluer than galactic ones.

For a given stellar mass, the evolution of a massive star, i.e., its location in the HR-diagram, depends strongly on metallicity. Metal-free stars have unique physical characteristics and they exhibit high effective temperatures and small radii. In relationship with their cosmological consequences, metal-free models are important for predicting the ionizing photon production of the first generation of stars.

Pregalactic stars were denser and hotter than galactic Pop III stars. Some of their main physical variables during the hydrogen and helium burning phases are shown in Figs.15 to 21.

Figure 16: Central density for (solid line), (dash-dot-dot-dot-dash), (dashes), (dash-dot-dash) and (dots) pregalactic Pop III stars with metallicity , without mass loss, during the hydrogen burning.
Figure 17: Ibidem. Effective temperature.
Figure 18: Ibidem. Radius.
Figure 19: Central density for (solid line), (dash-dot-dot-dot-dash), (dashes), (dash-dot-dash) and (dots) pregalactic Pop III stars with metallicity , without mass loss, during the helium burning phase.
Figure 20: Ibidem. Effective temperature.
Figure 21: Ibidem. Radius.

The following are the main properties of the studied evolutionary models for galactic and pregalactic very massive Pop III stars.

  a) Central density

Pop III stars with settle down on the main sequence with central densities of , and g cm for , and , respectively. For the same stellar masses, and , central densities are , and g cm, respectively. That is, the central density is higher for lower metallicity stars, and so, Pop III pregalactic stars are denser than galactic stars. However, the central density decreases with increasing stellar mass.

During hydrogen burning, the central density increases slowly but then by the end of this burning phase its increase rate grow significantly. The central density increases during the transition from hydrogen to helium burning and keeps increasing during the whole helium burning phase.

  b) Central temperature

The VMS high energy requirements, demand high central temperatures in order to be able to maintain their structure and energy output. That is, the central temperature is high for more massive stars, but with lower metallicity the stars have an even higher temperature. At the beginning of hydrogen burning, the central temperature with is , and , for , and stars, respectively, where K. For , the corresponding temperatures are , and , respectively. During hydrogen burning, slowly increases until the end of this burning phase when it begins to increase to high values and continues to do so during the whole helium burning phase.

  c) Nuclear energy generation

During hydrogen burning the main energy source is given by the CNO-cycles because of their strong temperature dependence. In lower metallicity stars these cycles are activated just after a short helium burning phase that produces enough CNO elements for the CNO-cycles to operate.

With increasing stellar mass, the central temperature is higher and so the nuclear energy generation. However, for pregalactic stars it increases with decreasing metallicity. As a consequence of the larger central temperatures, CNO-cycles carbon production is enhanced at an earlier evolutionary phase than in less massive stars. The nuclear energy generation increases during hydrogen burning. But, for galactic Pop III stars with metallicity , the energy generation rate decreases when hydrogen tends to be exhausted. When the existing hydrogen mass fraction is about , and the helium mass fraction reaches , the nuclear energy generation decreases. For pregalactic stars with , during the transition from hydrogen to helium burning there is only a small decrease in the nuclear energy generation rate, i.e. the transition from hydrogen to helium burning is smooth, quite different than for galactic stars.

There are different nuclear generation rates at the end of hydrogen burning because the central temperature in galactic Pop III stars is not high enough for helium ignition. This situation does not take place in the case of pregalactic stars precisely because they have a higher central temperature at the end of hydrogen burning. Then, the transition to the next burning phase occurs in a smoother form. In the other case, a strong explosive helium flash takes place during the transition to helium burning. At this moment, the star is contracting, consequently, it is heated and reaches an appropriate central temperature to ignite helium.

When helium is ignited in the core of the star, the nuclear energy generation rate increases rapidly during the transition to helium burning, then reaches a maximum and decreases towards the end of this burning phase. This occurs for galactic stars with metallicity . In the case of pregalactic stars, with metallicity , this transition takes place very smoothly because the stars are hot enough to ignite helium immediately after exhausting hydrogen. Then the generation rate increases towards a maximum and decreases by the end of helium burning.

  d) Luminosity

Very massive Pop III stars are very luminous. For , and with metallicity , at the beginning of the main sequence, their luminosity are , 3.59 and , respectively. For stars with , the corresponding figures are , and . During hydrogen burning the stellar luminosity increases slightly and varies smoothly during the transition from hydrogen to helium burning. Then it remains practically constant during helium burning.

  e) Effective temperature

Pregalactic Pop III stars are hotter than galactic stars. At the beginning of the main sequence, galactic stars with have an effective temperature , and K for , and , respectively. For the same stellar masses, pregalactic stars with have , and K, respectively. These different values are due to their higher central temperatures and the different mechanisms to drive nuclear burning.

The effective temperature continuously decrease during hydrogen burning, until the transition to helium burning that it increases. Then, the effective temperature start to decrease but then remains practically constant until the end of this burning phase when the effective temperature decreases even more.

  f) Radius

Very massive Pop III stars are compact. On the main sequence, galactic stars with metallicity have initial radii , and for , and , respectively. For pregalactic stars with , their radii are , and , respectively.

As the effective temperature decreases, the radii increases. At the end of hydrogen burning, for galactic , and stars with initial metallicity their radii are , and , respectively; and for pregalactic stars with initial metallicity , , and , respectively.

On the contrary to effective temperatures, during the transition from hydrogen to helium burning, radii slowly increase and remain almost constant during helium burning until the end when they increase again. This is, pregalactic lower metallicity stars are more compact because they are hotter than their galactic counterparts.

  g) Convective core

A convective core is always present from the ZAMS to the end of the helium burning phase. Very massive Pop III stars develop a large convective core. For , and stars with , the convective core size at the beginning of the main sequence is , and , respectively. For the same stellar masses and , , and , respectively. The core is larger for higher metallicity.

The convective core size is larger for higher masses and increases during hydrogen burning. In fact, the studied stars are almost fully convective during hydrogen burning. At the end of this burning phase the stars contract while forming a helium core. For the masses above mentioned and metallicity , at the end of hydrogen burning, the convective core size , and for , and , respectively. For , , and , respectively. Then, stars with form a helium core of , and ; with , the core masses are , and , respectively.

Because for galactic stars with metallicity the transition from hydrogen to helium burning is explosive, they suddenly contract, affecting momentarily their core size which rapidly increases, but then decreases while forming a helium core. That is, the exhaustion of hydrogen in the centre of the star causes a progressive contraction of the star and the shrinking of the convective core which finally vanishes when . During the transition, the energy released by the star is supplied by the gravitational contraction. In the pregalactic case with the burning transition is very smooth and the stars contract immediately forming a helium core.

During helium burning, for , and stars with , the convective core is , and , respectively. For , , and , respectively. That is, a carbon core mass of , and , respectively, is formed for stars with , while for , , and , respectively.

  h) Radiation pressure

Very massive Pop III stars are dominated by radiation pressure. At the centre of the stars we have , and for , and galactic stars with metallicity . That is, the contribution of radiation pressure to the total pressure is considerable. For pregalactic stars with metallicity we have that , and , respectively.

The contribution of the central radiation pressure increases during hydrogen burning. After the transition from hydrogen to helium burning, the central radiation pressure contribution increases and then remains almost constant.

  i) The Eddington luminosity

Very massive Pop III stars evolve during hydrogen burning below the Eddington upper luminosity limit. For , at the beginning of hydrogen burning, the ratio , and for , and , respectively; and for , and , respectively.

During hydrogen burning, this ratio increases and has a maximum close to the end of this burning phase, approximately when the helium mass fraction is and then decreases slightly as the star reaches helium ignition. After the transition from hydrogen to helium burning, increases when the helium mass fraction is approximately . Furthermore, this ratio decreases slightly but then remains almost constant during the helium burning phase.

Figure 22: Evolutionary tracks in the HR-diagram for , , , and Pop III stars with metallicity and , respectively and without mass loss during the hydrogen and helium burning phases.

Evolutionary tracks

Fig. 22 shows evolutionary tracks in the HR-diagram for , , , and galactic and pregalactic Pop III stars with metallicity and , respectively. The most massive stars are the hotter and most luminous. Luminosity decreases with lower metallicity. For the same stellar mass, galactic and pregalactic Pop III stars have similar luminosities. All stars settle down on the main-sequence with a high effective temperature and luminosity. During hydrogen burning all stars increase their luminosity while the helium mass fraction increase and both the central density and temperature increase.

For stars with metallicity , the transition between nuclear burnings is explosive while the star contracts. At this moment, the effective temperature and luminosity increase. Then, the luminosity remains almost constant while the effective temperature decreases. However, for different stars their luminosity and effective temperature decrease with decreasing mass and metallicity. For pregalactic stars the hydrogen to helium burning transition occurs very smoothly because they are hotter and their central temperature is high enough to ignite helium promptly.

Galactic and pregalactic stars evolve with different lifetimes. For , and galactic Pop III stars, their lifetimes during hydrogen burning are , and megayears, respectively. For pregalactic stars, their lifetimes for the same masses are , and megayears, respectively. During helium burning, stellar lifetimes are shorter than during hydrogen burning.

The initial hydrogen and helium burning phases take place at the blue side of the HR-diagram. When the stars evolve they move toward the red. However, the presently studied galactic and pregalactic very massive Pop III stars, evolving without mass loss and no-rotation, do not experience the asymptotic giant branch (AGB) phase. The most massive zero-metal stars tend to be cooler but the temperature remains above K.

The HR-diagram evolution of massive lower metallicity Pop III stars is different than for Pop I and II stars, because they do not evolve to red-giants before core collapse (Umeda et al. , 2000). In the quoted range of masses used by (Castellani et al. , 1983), massive zero-metal stars fail to reach the red giant region either at the hydrogen or helium burning phases.

In the Hertzprung-Russell diagram the locus of very massive Pop III stars is in the left upper part. These stars are hotter and very luminous. Pregalactic stars are hotter than galactic Pop III stars. Then, stars with lower metallicity are shifted to the left because they are bluer than the others.

  k) plane

The plane describes the state of the gas in the innermost stellar regions and it is important for the diagnosis of the stellar structure and evolution. The evolution of the central conditions determine the boundaries in which the equation of state is dominated by different pressure components, i.e., ideal gas, radiation pressure or electron pressure.

According to the zones of the equation of state of an electron gas, the studied stars occupy the upper loci of a non-degenerate and non-relativistic gas. In this zone there is a boundary at which pair production could become important. Regarding the gas in thermodynamic equilibrium, these stars are dominated by radiation pressure.

The central conditions varies in the range log and log for galactic stars with . For pregalactic stars with , both the central temperature and density are higher. The evolution of the central conditions can be described by . This behaviour depends on the equation of state and it is similar for models of any metallicity.

We distinguish two regions in the plane in correspondence to the stages of gravitational contraction of stellar cores between nuclear burnings. In the second region, corresponding to helium burning, the central density and temperature increases more than during hydrogen burning.

4 Discussion

The evolutionary tracks of our massive Pop III stars are shifted to the left upper part of the HR-diagram as in Tumlinson et al. (2003) models. Then, stars evolve to the red with increasing luminosity and decreasing effective temperature. In both cases, luminosities are similar but, in the present case, final effective temperatures are slightly higher. This is probably due to the different parameters used, chemical composition and specific implementation of physical processes, e.g., convection.

  a) Nuclear lifetimes

Our models are hot and luminous and have short lifetimes. For stars with and , their lifetimes during hydrogen burning are and megayears, respectively. Lifetimes during helium burning are of the order of of their lifetime during hydrogen burning. Our results for galactic and pregalactic hydrogen burning Pop III stars are in good agreement with other authors.

For , even metal-free stars evolve toward K and eventually become red supergiants, as the hydrogen burning shell becomes more active with increasing stellar mass (Baraffe et al. , 2001). In the present work, the most massive metal-free stars tend to be cooler and likely could become red supergiants. However, for a helium mass fraction equal to , they maintain high effective temperatures. In all cases, stars do not reach low effective temperatures. Evolving stars without mass loss and with no-rotation, fail to reach the AGB phase. At the end of helium burning, the studied galactic stars are hotter than log and pregalactic stars are hotter than .

  b) Nuclear energy generation

The peculiar behaviour of low metallicity stars was first pointed out by Ezer (1961) and their structure during hydrogen and helium burning has been investigated by several authors.

According to the results presented here the stars begin to settle down on the main-sequence with higher initial central temperatures of order log . Then, the onset of the reaction occurs at earlier stages of hydrogen burning. This is because the reaction requires a much higher temperature than the pp-chains and ignites at the beginning of the hydrogen burning phase. In the present models, after a brief initial period of burning, a trace amount of heavy elements has been formed. Then, the stars expand and follow the CNO-cycles.

In Fig. 23 the nuclear energy generation as function of the temperature is shown, and Fig. 24 shows this energy generation during hydrogen burning for both galactic and pregalactic Pop III stars.

Figure 23: Main sequence nuclear energy generation rate as function of the temperature for galactic (left panel), and pregalactic (right panel) Pop III stars, with metallicity and , respectively.

  c) Convective core size

In their models, Marigo et al. (2001) found that if the convective core grows, it eventually reaches the H-shell and engulfs some hydrogen-rich material, which is rapidly burnt via the CNO-cycles. This causes a flash that expands the core, so that central helium burning weakens and the convective core recedes temporarily (in mass). After the flash has occurred, the convective core starts growing again.

In the present models, the same picture takes place for galactic Pop III stars but it does not in the pregalactic case. In this case, the central temperature at the transition of nuclear burnings is higher than in the first one.

Hydrogen exhaustion in the centre of the star causes a progressive contraction of the star and the shrinking of the convective core which finally vanishes when , so that no standard overall contraction phase is found (Castellani et al. , 1983). A maximum in the energy released by gravitation occurs during this burning phase, when , and of the energy released by the star is supplied by contraction. Ignition of the full triple- cycle is only slightly delayed with respect to hydrogen burning ignition in a shell, and once again the gravitational contraction supplies energy to the star. Once the chain has become fully efficient, a convective core is again developed.

For the present work, pregalactic Pop III stars have higher temperatures than galactic stars allowing a soon ignition of the reactions. Then the transition from hydrogen to helium burning is smooth because the convective core does not vanish.

Figure 24: Main sequence nuclear energy generation rate as function of the hydrogen mass fraction for galactic (left panel), and pregalactic (right panel) Pop III stars, with metallicity and , respectively.

  d) Eddington luminosity

Stellar models may become gravitationally unbound, i.e., the rate of energy outflow at the surface exceeds the corresponding Eddington luminosity.

In the present models, the Eddington factor is calculated by estimating the ratio between the current stellar luminosity and the Eddington luminosity. In both the hydrogen and helium burning phases, the studied cases of , and galactic and pregalactic Pop III stars evolve below the upper Eddington luminosity limit. During hydrogen burning, the factor increases, reaches a maximum toward the end of this burning phase and then decreases. The most massive stars evolve during the helium burning phase closest to the Eddington limit.

5 Conclusions

For the present work a large number of ZAMS models have been calculated showing the main physical variables as function of the mass, and the metallicity. As the mass increases, ZAMS stars get bigger, brighter, and less dense. Pop III stars get very hot and compact as metallicity decreases.

When metal-free stars settles down on the main-sequence, they have smaller radii, hotter cores, and higher effective temperatures than metal-enriched stars. Like their counterparts, lower metallicity stars become systematically cooler, larger and more luminous over their hydrogen burning lifetimes. The most massive stars have shorter lifetimes than less massive stars.

To study their properties and internal structure, stellar structure models on the main sequence have been calculated, emphasizing the case of metal-free stars which are compared with their Pop I and II counterparts. Pop III stars are more centrally condensed, denser and hotter. On the main sequence their different internal regions are always below the upper Eddington luminosity limit.

The most important feature of the metal-free models is the high temperature they maintain in their photosphere. These stars have a high ionizing photon production rate. The ionization caused by these stars is a direct result of their high effective temperatures. For , the studied metal-free stars have effective temperatures K. Consequently, they are very efficient at producing photons capable of ionizing hydrogen and helium. According to Bromm, Kudritzki and Loeb (2001), very massive Pop III stars might have played a significant role in the IGM reionization of hydrogen and helium at high redshifts.

Nuclear burning proceeds in a non-standard way. For stars with no metals, the pp-chains and the CNO-cycles do not provide enough energy to support the star, so when it reaches the main sequence keeps contracting until the process starts to operate, halts the contraction and in a very small time produces enough CNO elements so that the energy from the CNO-cycle is capable of supporting the star, which expands and settles down to the main sequence. The stars maintain core temperatures in excess of K which are high enough for the simultaneous occurrence of the -chains, the CNO-cycles and helium burning via the process.

For these stars, the radiative opacity in their envelopes is reduced, and their core temperature is high, then the first stars are hotter and smaller than metal-enriched stars. On the other hand, the opacity of stellar matter is reduced at low metallicity, permitting steeper temperature gradients and more compact configurations at the same mass.

Very massive () galactic and pregalactic Pop III stars develop large convective cores with important helium core masses . These quantities are important for explosion calculations (Umeda et al. , 2000) and synthetic derivation of the SN yields (Portinari et al. , 1998). Semiconvection does not greatly affect the stellar structure during the main-sequence phase. However, during the shell hydrogen burning and helium burning phases, it plays a significant role, and the evolutionary results depends on the adopted criterion and the input physics of the models (Chiosi and Maeder , 1986).

We have calculated evolutionary models for lower metallicity very massive Pop III stars. The evolution of these stars is similar to that of metal enriched stars but now the evolutionary tracks are shifted to the left of the HR-diagram, i.e. the models are bluer than their metal enriched counterparts. This is because massive Pop III stars are hotter and very luminous. Pregalacic stars are denser and hotter than galactic Pop III stars. However, during their evolution these stars are more luminous than the first ones.

During the hydrogen burning phase, very massive galactic and pregalactic Pop III stars evolve almost fully convective. They form a large core with a different structure depending on metallicity. Very massive stars are dominated by radiation pressure and electron scattering. As a consequence of the high radiation pressure, the convective core tends to be larger at higher masses and to expand as the star evolves.

Because the stars radiate near the Eddington limit, radiation pressure due to electron scattering opacity can become substantial. For pregalactic Pop III stars, with the metallicity considered in this work, they develop a helium core , and , corresponding to initial stellar masses of , and , respectively. Then, according to Fryer et al. (2001) and Heger and Woosley (2002), these stars will likely explode by pair-instability supernovae for the and cases or collapse into black holes for the case. However, stars could explode like hypernovae.

The present evolutionary models have been calculated without mass loss and with ro-rotation, in accordance with other existing points of view. However, the uncertain role of mass loss can be viewed as one of the major systematic uncertainties remaining in the study of metal-free stars. In subsequent papers we will extend the present study to mass losing and rotating models.

Acknowledgements: This work has been partially supported by the Mexican Consejo Nacional de Ciencia y Tecnología (CONACyT), Project CB-2007-84133-F, and the German Deutscher Akademischer Austauschdienst. DB also acknowledges CONACyT for a Ph. D. grant.

References

Footnotes

  1. affiliation: Astronomical Institute of the Academy of Sciences,
    Boční II 1401, 14131 Praha 4, Czech Republic.
    e-mail: bahen@universo.com
  2. affiliation: Instituto Nacional de Investigaciones Nucleares,
    Km 36.5 Carretera México-Toluca s/n, Salazar, Ocoyoacac,
    52750 Estado de México, Mexico.
    e-mail: jaime.klapp@inin.gob.mx
  3. slugcomment:
  4. Accepted for publication in Astrophysics & Space Science

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