The Structure and Evolution of Massive Rotating Single and Binary Population III Stars

Figure 1(a) shows the convective core mass for nonrotational single stars as a function of evolutionary age. We can find that using different convection criteria contributes to no significant difference for the mass of convective cores at zero-age main sequence (ZAMS). When evolution proceeds, Schwarzschild convection models can attain much larger mass of convective cores than their Ledoux convection counterparts. This difference is because of the presence of a molecular weight gradient outside of the burning cores. An interesting thing occurs in the Population III stars. Normally, the lower the metallicity, the bigger the mass of the convective core (and thus the higher the luminosity). This results from the fact that the star with low metallicity has higher central temperature. However, Population III stars obviously violate this trend. For example, convective cores for model S2 are smaller than the ones for model S5. This can be attributed to the fact that the central temperature is much higher but the released energy is much lower in Population III stars.

Zoom In Zoom Out Reset image size

Figure 2 illustrates the evolution of the hydrogen profile in a nonrotational 130 M_⊙ Population III star with Schwarzschild convection and Ledoux convection criteria. It is displayed that the Ledoux convection model S1 can attain the helium core at the central hydrogen exhaustion and favor the dredge-up. When convective motion attains the hot layers that are rich in carbon and oxygen, energy generation in the H-burning shell increases dramatically and triggers the expansion of the envelope to large radii. In model S1 using the Ledoux criterion, convection shifts outward after the end of core hydrogen burning because the molecular weight gradient prevents inward movement and mixing of hydrogen into carbon-rich layers. In model S2 using the Schwarzschild criterion, convection moves inward during the core He-burning phase. However, there appears to be a significant distance between convection front and central helium cores. For instance, mass coordinates for helium cores are 67.27 M_⊙, whereas they are 71.92 M_⊙ for convection fronts at a central helium mass fraction of 0.9. This fact implies that the model with the Schwarzschild criterion does not easily form the deep dredge-up.

Zoom In Zoom Out Reset image size

Figure 3(a) displays the evolution of nonrotational single stars in the Hertzsprung–Russell (H-R) diagram. It is found that models using the Schwarzschild convection criterion can widen the main-sequence width and lengthen the main sequence because of the larger extension of the convective core. A larger amount of hydrogen is available for helium production in the core and leads to a larger helium core (panel (a1) in Figure 10). The star displays a lower effective temperature and a higher luminosity at the end of the main sequence. This implies that stars with a Schwarzschild convection criterion have larger radius.

Zoom In Zoom Out Reset image size

Figure 4(a) illustrates the evolution of stellar mass for nonrotational single stars as a function of evolutionary age. As expected, the main sequence is barely affected for Population III stars in panel (a). The reason is the extremely low value of radiative winds during this phase. The mass-loss rate for 130 Population III stars with a Schwarzschild convection criterion is larger during the post–main-sequence phase compared to the one during the main sequence. The nonrotational model S2 suffers a huge amount of mass loss during the post–main-sequence because it approaches Eddington luminosity. Its surface effective temperature falls down to $\mathrm{log}{T}_{\mathrm{eff}}\sim 3.8\mbox{--}4.0$ (see Figure 3). The star turns back blueward in the H-R diagram owing to mass removal. If the stellar evolution timescale is long compared to the growth time of the instability, which is of order the thermal timescale of the convective element, then the temperature gradient from the Schwarzschild convection model is appropriate (Lawlor et al. 2015). In the following sequences, we use the Schwarzschild convection criterion to model the evolution of rotating single and binary Population III stars.

Zoom In Zoom Out Reset image size

Figure 5 shows the relationship between the equatorial rotational speeds and evolutionary ages for single stars and binarities. For the nonrotating model S2, the rotational velocity maintains zero until the end of the calculation. We find that rotational velocities in rotating models increase toward critical velocities, except for model S11. Actually, from the beginning of evolution, spin angular velocities go down slightly owing to the combined effect of mass loss via stellar winds and stellar expansion. Therefore, the increase of rotational velocities mainly originates from the expansion of stellar radius. The growth of the ratio of angular velocities $\tfrac{{\rm{\Omega }}}{{{\rm{\Omega }}}_{\mathrm{crit}}}$ in panel (c) can be explained by three reasons. First, the angular velocity Ω decreases less steeply with the stellar radius than Ω_crit (∝R^−3/2) for Population III stars. Second, the removal of angular momentum via stellar winds during evolution is much weaker than the counterparts with high metallicity. Third, there is some transport by shear induced by the angular velocity gradient that forms between the contracting core and the expanding envelope, instead of meridional circulations that might be negligible for Population III stars. The results indicate that the internal transport of angular momentum from the stellar interior to the surface is sufficient to bring the surface to critical rotation during the main sequence. This process effectively compensates the decreasing of angular velocities thanks to stellar expansion and wind braking. Model S11 has a higher initial value of $\tfrac{{\rm{\Omega }}}{{{\rm{\Omega }}}_{\mathrm{crit}}}$ than other models because the quantity $\tfrac{{\rm{\Omega }}}{{{\rm{\Omega }}}_{\mathrm{crit}}}$ is proportional to the quantity of R^1/2. Moreover, Ω_crit decreases with the increasing of metallicity Z because the star with higher metallicity Z has a larger radius.

Zoom In Zoom Out Reset image size

As shown by the comparison of model S8 with model S11, rotational velocities for model S11 with Z = 0.0021 decline rather rapidly, while they go up for model S8 at Z = 10⁻¹⁴. The reason is that the mass-loss rate by stellar winds is expected to depend on the chemical composition of the atmosphere as $\dot{M}\sim {\left(\tfrac{Z}{{Z}_{\odot }}\right)}^{m}$ with m = 0.85 for Vink et al. (2001). This indicates that at high metallicity Z there will be larger mass loss than at lower metallicity Z, so less angular momentum will be kept in the star with high metallicity Z at the surface. Actually, meridional circulations are much more efficient at high metallicity Z at feeding the surface with angular momentum than the ones at lower metallicity Z. This fact implies that wind mass loss, which removes mass and thus spin angular momentum, has exceeded the outward transportation of angular momentum induced by meridional circulations in model S11. It is also displayed in Figure 5(a) that model S12 with a lower stellar mass reaches the critical velocity later. There are three main reasons. First, meridional circulations and shears that can transfer angular momentum are weaker in low-mass stars than in high-mass stars. Second, the critical velocity goes up for decreasing mass, which is due to the decreasing contribution of radiation pressure. Third, the expansion of low-mass stars is slower than that of high-mass stars.

Although the initial rotational velocities for S10 are highest, they attain the critical velocities later (see Figures 5(a) and (c)). There are two main reasons. First, this physical process can be explained by the fact that Population III stars do not experience significant mass loss at the beginning of evolution because lines by hydrogen and helium are too weak to trigger radiation-driven winds (Krticka et al. 2006). However, when core temperature attains T_c ∼ 10⁸ K, helium fusion via 3α-reaction starts to generate ¹²C. When the mass fraction of ¹²C attains the value of ${X}_{{}^{12}{\rm{C}}}\simeq {10}^{-10}$, the CNO cycle is activated and the C, N, O elements are diffused from the stellar core to the surface by meridional circulations. Vink & de Koter (2005) have noticed that most of the opacity driving of the radiative winds is due to iron (70%, and 15% to CNO) at solar metallicity Z. However, CNO contributes to 60% of the driving and Fe to 10% at Z_⊙/30. CNO (to some extent extended to H and He) lines govern the wind-driving mechanism instead of only H and He lines, and then this results in stronger winds that remarkably reduce rotational velocities. Second, the radius expands slowly owing to quasi-chemically homogeneous evolution (hereafter, quasi-CHE) for S10, and this process can lead to a higher critical rotation (see Figure 7).

It takes a significant fraction of evolutionary time for models S8 and S10 to maintain critical velocity because critical velocity declines with stellar expansion. Rotational velocities for S2 and S10 can attain the critical velocity again during the Hertzsprung gap. The reason is that violent expansion results in strong differential rotation, which can efficiently transfer spin angular momentum outward. Moreover, rapid contractions of the convective envelope easily bring the star to critical rotation (Heger & Langer 1998). Rotation velocities for models S8 and S10 fall with the rapid expansion of envelopes during helium burning because of the conservation of spin angular momentum. This fact also indicates that large amounts of mass may be lost at this advanced stage.

Figure 4(b) shows stellar mass for single stars as a function of evolutionary age. Comparing models S8 and S10, one can find that the amount of mass lost is higher for rapidly rotating stars during the main sequence. For comparison, one can notice that mass loss also increases with the increase of mass and metallicity. Because W-R stars generate a much larger amount of carbon and other elements than stars that do not go through the W-R phase, the mass-loss rate for model S11 goes up rapidly during the main sequence. A bi-stability jump can occur in model S8. The effect of mass loss can bring about lower luminosity because the luminosity is directly proportional to the power law of stellar mass (Kippenhahn & Weigert 1990).

It is shown in Figure 1(b) that the convective core for model S8 is smaller than the one for model S2 at or near ZAMS. The reason is that centrifugal forces partly sustain the star against its own gravity. Therefore, model S8 behaves like a lower-mass nonrotating one. This leads to a smaller luminosity in the H-R diagram. In fact, the size of the convective core is governed by radiative pressure. Therefore, convective cores in low-mass model S12 are smaller than those in model S8.

Away from ZAMS, convective cores appear to be larger in the star with high velocity because rotational mixing becomes efficient. The larger core induced by rotational mixing gives rise to higher central temperature and lower opacity. Meridional circulation governs rotational mixing above the convective core because the gradients of chemical elements decrease the shear turbulence (Song et al. 2018a). The core for the star with Z = 10⁻¹⁴ has a smaller value than the one for the star with Z = 10⁻⁸ because of the weak released luminosity. The maximum size of the convective core occurs for model S11 with CHE because rotational mixing is so efficient that hydrogen and helium diffuse freely without the restrictions of μ-gradients.

Close to the hydrogen exhaustion, the convective core for S2 has a value of $\sim 61.85\,{M}_{\odot }$, whereas it has a value of ∼72.74 M_⊙ for S8. Therefore, convective cores can be remarkably enlarged by rotational mixing. In convective regions, the density gradient is smaller; thus, more matter is contained in a given volume. This enables local gravity to be larger, and thus the star is better counteracted against the centrifugal force. This implies that rotational mixing gives rise to a larger mean density and a more compact star. Furthermore, the rotationally induced mixing refuels the core in fresh hydrogen and contributes to higher luminosity. The corresponding lifetime of main sequence is significantly prolonged. By comparing models S2 and S10, one finds that rotational mixing also delays the shrinkage of the convective core. The decreasing of the core size may occur just when the model starts to lose a huge amount of matter. When evolution proceeds, model S11 has a smaller convective core owing to mass removal via W-R winds.

Figure 6(a) displays surface nitrogen abundances for the single star as a function of evolutionary age. Nitrogen enrichment for S2 and S3 can be ascribed to mass removal of hydrogen envelopes via stellar winds after main sequence. Markova et al. (2018) have noticed that the envelope is really stripped in the most luminous supergiants by the strong winds ($\mathrm{log}L/{L}_{\odot }\geqslant 5.8$ and $\mathrm{log}\dot{M}[{M}_{\odot }\,{\mathrm{yr}}^{-1}]\geqslant -5.4$). Maeder et al. (2009) pointed out that the behavior of the excess of nitrogen abundances is a multivariate function (i.e., stellar mass, evolutionary age, projected rotational velocity, metallicity) for a single rotating star. As expected, we find that nitrogen enrichment goes up with the increase of initial velocity and evolutionary age during the main sequence when comparing models S8 and S10. Moreover, nitrogen enrichment for model S9 is higher than the one for model S8. Surface nitrogen abundance reaches quite high values for model S11 with CHE because more nitrogen can be synthesized via CNO cycles in the star with high values of initial CNO. After helium burning, surface nitrogen abundances increase rapidly because W-R winds expose nitrogen at the position of hydrogen-burning shells.

Zoom In Zoom Out Reset image size

One can find that nitrogen enrichments in less massive stars S12 proceed slower than the ones in massive stars S8 at early stages of main sequence. However, less massive Population III stars may be responsible for the overabundance of nitrogen at the subsequent stage. There are three main reasons. First, a high rotational velocity for model S12 favors efficient carbon mixing between helium-burning cores and hydrogen-burning shells. Second, more primary nitrogen can be produced at the position of hydrogen-burning shells. Third, the surface nitrogen enrichment factor can attain a value of 1.23 during the dredge-up. These lower-mass Population III stars contribute through their winds to the enrichment in newly synthesized elements of the interstellar medium. This may be closely related to unusually high nitrogen abundances found in extremely metal-poor objects (CS 22949–037). This star was discovered to have an extreme overabundance of nitrogen relative to iron (Norris et al. 2001).

Figure 7(a) shows surface helium mass faction Y_s versus central helium mass fraction Y_c from ZAMS up to the end of the helium burning for single stars. Surface helium abundance for model S1 goes up at the end of helium burning because stellar winds expose the productions of the hydrogen-burning shell. Note that surface helium enrichment for model S2 appears after main sequence owing to the dredge-up. In contrast to model S11, other rotational sequences do not undergo CHE, and their surface becomes less enriched with helium. Model S10 represents a transition between CHE and normal evolution and is marked as quasi-CHE. It is difficult for Population III stars to form CHE because meridional circulations are too weak to give rise to rapid mixing. Model S11 displays the CHE, and its surface helium abundance goes up as fast as the central He abundance during the main sequence. At core hydrogen exhaustion, surface helium mass fraction has attained a value close to 1. The star can turn into a massive helium star with a very thin radiative envelope. After main sequence, surface helium abundance decreases steeply in the second half of helium burning. This can be ascribed by the fact that helium is mixed from radiative envelopes to convective cores, whereas both carbon and oxygen in the core are transferred to the envelope. Surface helium abundance for other Population III stars remains approximately the constant value because hydrogen envelopes of these stars are very thick.

Zoom In Zoom Out Reset image size

Panels (a) and (b) in Figure 8 show the evolution of the mass fraction of hydrogen and effective temperature as a function of the central helium mass fraction. From panel (a), one can find that models S2 and S8 cannot turn into W-R stars, which are defined as $\mathrm{log}{T}_{\mathrm{eff}}\gt 4.0$ and X_H < 0.4 because hydrogen envelopes are not removed significantly by weak stellar winds. One can also find that surface mass fraction of hydrogen goes down rapidly with the increase of rotational velocities and metallicity. Meanwhile, the star shifts toward high temperature during the main sequence in panel (b). There are small round excursions for models S9, S10, and S12 at central hydrogen exhaustion because of the first dredge-up. Surface hydrogen drops rapidly during the first dredge-up. For instance, surface hydrogen mass fraction for S12 falls down from 0.441 to 0.254, whereas the corresponding effective temperature goes up from 4.538 to 4.817. The first dredge-up can cause the star to shift toward higher effective temperature and can favor the formation of W-R stars. Model S11 is first transformed into a W-R star because it has experienced CHE. Therefore, it will take more time for model S11 to evolve at the stage of W-R stars, and stellar mass is smaller at the end of helium burning. Model S12 turns into W-R stars later because mass loss induced by rotation is weak for lower-mass stars.

Zoom In Zoom Out Reset image size

Figure 9 shows mass fraction of various chemical elements as a function of the mass coordinate for models S2 and S8. The abundances show a step profile above the convective core at the end of hydrogen burning, representing outer convective regions. One can notice that the included masses for outer convective regions in model S2 are smaller than the ones for S8. For example, we have the following mass sizes for the two convective layers: between 86.35 and 100.33 M_⊙ for model S2 (i.e., 14 M_⊙), and between 102 and 123 M_⊙ for model S8 (i.e., 21 M_⊙). Therefore, rotation favors convection in stellar envelopes and thus leads to the deep dredge-up.

Zoom In Zoom Out Reset image size

Surface abundances for ¹H in model S10 are reduced, whereas surface abundances of He, C, O, N, Ne, Mg, and ²⁶Al are increased by rotational mixing at hydrogen exhaustion in comparison with model S2 without rotation (see Table 3). One can find that during the main sequence surface carbon and oxygen abundances decrease in S11, whereas they go up in S8. This fact implies that these chemical elements are transferred inward in S11 and are transported outward in Population III stars in S8. We have noticed that the high surface abundances of carbon, nitrogen, and oxygen for model S10 make the metallicity Z up to 1.26 × 10⁻⁷, whereas these abundances are less than 10⁻¹² in the nonrotating model S2. This results in a stronger mass-loss rate for stellar winds. Note that helium burning via the 3α-reaction may produce C and O in the core, and carbon is transported to the envelope by rotational mixing. After that, some amounts of C and O are brought from the core to the hydrogen-burning shell, which has a much higher temperature and is closer to the core for Population III stars. These two elements are processed to not only nitrogen but also all the other CNO nuclei by CNO cycles. Rotational velocities and central temperature are higher for Population III stars, and they have created favorable conditions that can reproduce the occurrence of primary nitrogen.

Most of the CEMP-no stars (i.e., carbon-enriched extremely metal-poor stars without r- and s-process elements) display not only an enriched C abundance but also excesses in nitrogen and oxygen. Maeder & Meynet (2015) have presented that the variety of abundances found at the surface of such stars can be explained by various degrees of back-and-forth mixing between the hydrogen- and helium-burning regions. The released energy in the hydrogen shell goes up dramatically and drives the expansion of the hydrogen envelope to larger radii. Therefore, the dredge-up is easy to develop (see Figure 14(a)). The production of primary nitrogen can be greatly influenced by mass loss because H-burning shells (i.e., the engine for the production of primary nitrogen) for model S11 can be extinguished by a strong W-R wind at central helium of 0.8. In this case, CEMP-no stars might also be explained by the convective dredge-up.

One can find that the nitrogen peak for S8 at the position of the H shell increases remarkably and diffusion spreads nitrogen further outward and inward. Nitrogen in the hydrogen-burning shell that is diffused back to the center is quickly converted into ²²Ne first and then into ^25,26Mg, turning into an efficient primary neutron source. Then, central abundances for ¹⁴N and ²²Ne are increased by this interplay (Meynet & Maeder 2002; Limongi & Chieffi 2018). More importantly, the helium that is produced in H-burning shells is also transferred towards the helium-burning core, and it supports the conversion of ¹²C into ¹⁶O, reducing therefore the final ¹²C/¹⁶O ratio in the CO core. Therefore, surface abundances of O, ²⁰Ne, and Al go up remarkably for model S8 during central helium burning.

Panels (a1)–(a4) in Figure 10 display the evolution of helium cores for single stars. We have noticed that the helium core for model S1, which has a value of 54.65 M_⊙, disappears at the evolutionary time of 2.541 Myr and displays a larger one of 57.38 M_⊙ again at the evolutionary time of 2.69 Myr. The reason is that the first deep dredge-up can mix the hydrogen from the envelope to the core at hydrogen exhaustion. Hydrogen burning may be reignited and can generate a larger helium core. We can notice that nonrotational models S1 and S4 with the Ledoux convection criterion have experienced two episodes of deep dredge-up, whereas models S2 and S5 with the Schwarzschild convection criterion have not gone through the deep dredge-up. During the dredge-up, both the central temperature and mean molecular weight are decreased, while the opacity is increased owing to the engulfed hydrogen. Although model S12 has experienced the dredge-up, helium cores in S8 are larger than that in S12 in panel (a2). The reason is that helium cores scale with the size of the hydrogen convective core and increase with the initial mass of the star. Radiative pressure is larger for more massive stars and results in larger helium cores.

Zoom In Zoom Out Reset image size

From panel (3a), we can find that model S8 has a helium core with a mass of 77.33 M_⊙ at 2.685 Myr, whereas model S2 has a helium core with a mass of 67.31 M_⊙ at 2.685 Myr. Therefore, rotational mixing can remarkably increase the mass of the helium core during the main sequence. In order to obtain such mixing, large overshooting is often introduced in the literature (Chin & Stothers 1991). Both rotational mixing and larger overshooting can result in bigger CO core masses and therefore tracks in the ρ_c–T_c diagram that are moved toward the pair-instability region (i.e., ${\rm{\Gamma }}\lt \tfrac{4}{3}$). Helium cores for S8 increase slightly because the H-burning shell continuously adds new helium mass to the He core. It is noticed that model S10 goes through the dredge-up owing to higher initial rotational velocities (see Appendix B). Maeder & Meynet (2008) have presented that the effects of rotation on the thermal gradient are much larger and that rotation triggers convection in stellar envelopes. We have noticed that the convective dredge-up might prefer the lower-mass star with higher rotational velocities and metallicities. From panel (a3), helium cores for model S9 with high metallicities are larger compared to model S8. However, helium cores for model S11 reduce substantially because of the strong stellar wind expected at the phase for W-R stars.

Panel (3b) in Figure 3 shows the evolution of single stars in the H-R diagram. On or close to ZAMS, effective temperature $\mathrm{log}{T}_{\mathrm{eff}}$ and luminosity $\mathrm{log}{L}_{1}/{L}_{\odot }$ of model S2 are 5.004 and 6.319, respectively, whereas their values in rotational model S10 are 4.972 and 6.301, respectively (see Table 2). Therefore, rapid rotation may enable the star to shift toward low temperature and luminosity. There are two main reasons. First, the centrifugal force gives rise to a weaker effective gravity, which can contribute to lower central temperature and smaller convective core. For example, at the age of ∼0.0489 Myr, the mass of the convective core of model S2 is 106.99 M_⊙, whereas it is 105.71 M_⊙ for model S10 (see Figure 1(b)). Second, low effective gravity can result in low surface temperature and luminosity according to gravity darkening (von Zeipel 1924). Generally, the star with low metallicity Z is more compact, and density profiles are flatter in the interior of this model owing to larger cores. Therefore, the luminosities for the star with low Z can be more influenced by rotation.

Moreover, one can find that at ZAMS the effective temperature $\mathrm{log}{T}_{\mathrm{eff}}$ and luminosity $\mathrm{log}{L}_{1}/{L}_{\odot }$ for model S11 are 4.746 and 6.264, respectively whereas their values for model S8 are 4.995 and 6.307, respectively. This indicates that low-metallicity model S8 gives rise to lower opacities, which generate a bigger outgoing luminosity. The radiative gradient ${{\rm{\nabla }}}_{\mathrm{rad}}\propto \tfrac{\kappa P}{{T}^{4}}$ is smaller because of both the smaller opacity and the higher internal temperature. This leads to larger surface effective temperature for Population III stars.

Nonrotational model S2 shows a quite large span of temperature at almost constant luminosity after main sequence, which is caused by a rapid expansion of the envelope during evolution. It evolves redward to the temperature of $\mathrm{log}{T}_{\mathrm{eff}}\simeq 4.0$, implying that they become blue supergiant stars. One can find that the more rapidly rotating stars in S10 remain bluer and are more luminous than the nonrotational stars in S2. The reason is that the increase of helium in outer envelopes leads to a declining of the opacity and a growing in mean molecular weight, resulting in more luminous and more compact stars. Actually, stellar contraction and expansion in the outer envelope are governed by the photon luminosity and therefore by the opacities. The luminosity available for stellar expansion is decreased by rotational mixing because the envelope becomes transparent. Moreover, the decreased specific entropy may originate from the increasing of mean molecular weight in the outer envelope. At the same time, fresh hydrogen is mixed into the core and thus contributes to a higher specific entropy.

The central compactness, which is defined by the ratio of central density to mean density (i.e., $\tfrac{{\rho }_{c}}{\bar{\rho }}$), increases with the initial rotational velocity on or close to ZAMS. Rapid rotation enables stars to turn into a central compact one. In order to offset the centrifugal force, more matter tends to deposit toward the core for increasing the local gravitational force (see Table 2). Therefore, rapid rotation is not in favor of fast apsidal motion, which has been found by Claret (1999). However, the central compactness is decreased by rotational mixing at the subsequent stage. The reason is that rotating mixing brings about a lower central density ρ_c, which originates from inward hydrogen mixing in the core and a higher mean density $\bar{\rho }$ due to a smaller radius. For instance, model S2 inflates to a value of $\mathrm{log}R/{R}_{\odot }=1.739$ at hydrogen exhaustion, whereas model S10 expands to a radius of $\mathrm{log}R/{R}_{\odot }=1.091$ (see Table 2).

The model S11 can undergo CHE because the chemical mixing timescale induced by Eddington–Sweet circulations can be shorter than the nuclear burning time of hydrogen. Model S11 shifts toward blueward instead of redward on the H-R diagram. The reason is that effective temperature for a homogeneous star has a power law of 1.5 with mean molecular weight (i.e., T_eff ∝ μ^1.5M^0.75). Because almost all the hydrogen in the star takes part in nuclear reaction in this way, the star gradually turns into a massive helium star on main sequence. The central temperature and specific entropy are lower for model S11 in contrast to Population III stars. Except for the surface layer, there appears to be a flat entropy profile inside the star with CHE. The luminosity for model S11 with CHE falls down rapidly at the stage for W-R stars because the decreasing of stellar mass exceeds the increasing of mean molecular weight (i.e., $L\propto \tfrac{{\mu }^{7.5}{M}^{5.5}}{{R}^{0.5}}$). CHE is an important evolutionary channel that can generate massive He stars at low metallicities, but it is difficult for Population III stars to produce CHE. For a given mass, meridional circulations for Population III stars proceed much slower than the ones in stars with high metallicity because weak angular momentum loss via stellar winds restricts the velocity of meridional circulations. The rotational mixing timescale will be longer than the nuclear timescale. Model S10 goes through a quasi-CHE, and its effective temperature lies between CHE and normal evolution.

Figures 5(b) and (d) illustrate that rotational velocities and corresponding angular velocities of the primary star (more massive star; solid lines) and the secondary star (less massive star; dashed lines) vary with evolutionary age. It is shown that equatorial velocities of both components in binaries drop from ZAMS as a result of tidal interactions. The reason is that tidal torques can spin the star down when the spin angular velocity is larger than the orbital one. The primary star for models B2, B6, and B7 attains a synchronous state (i.e., Ω₁ = Ω_orb) at an age of 1.607, 0.0181, and 0.00215 Myr, respectively. The secondary star in B2 achieves this state at 1.944 Myr. The fact indicates that synchronous timescales become larger for less massive components with low metallicity owing to a smaller radius. The main reason is that synchronous timescales are sensitive to the factor ${\left(\tfrac{a}{R}\right)}^{8.5}$ according to Zahn (1977). The quantity R is stellar radius, and the value a is the orbital separation. Furthermore, tidal torques are proportional to the factor of E₂ (i.e., ${E}_{2}=1.592\times {10}^{-9}\tfrac{M}{{M}_{\odot }}$), which is smaller for less massive stars. Population III stars can maintain higher rotational speeds in a longer evolutionary time than the counterparts with high metallicity, and rapid rotation for Population III stars can slow down the production of mean molecular weight gradients, which can efficiently inhibit rotational mixing in the stellar interior.

As expected, the synchronous timescale also has a bigger value in the system with a longer orbital period. Rotational velocities for two components in model B4 behave as the ones for single stars owing to a larger initial orbital period. Therefore, a single star can maintain a high rotational velocity, whereas rotational velocities are braked by tidal torques in the binary system, except for the system with a very large orbital period.

At the synchronous state, tidal torque is zero according to Equation (20) in Paxton et al. (2015). Both components can be spun down further by stellar winds. Spin velocities of two components finally attain the equilibrium velocity, which is always slightly inferior to the orbital one (Song et al. 2018a, 2018b). For example, at the evolutionary age of 1.793 Myr, the equilibrium velocity for the donor star in model B2 is 112.18 km s⁻¹, whereas the orbital velocity is 115.85 km s⁻¹. We have derived a theoretical equation for the equilibrium velocity in Appendix A. For the donor star in model B2, we obtain the quantities τ_mc = 0.493 Myr, τ_mi = 2.703 Myr, τ_wb = 220.32 Myr, and τ_tide = 0.0150 Myr, respectively. The theoretical value for equilibrium velocities is 111.82 km s⁻¹, which is very close to the numerical solution of 112.18 km s⁻¹. The fact indicates that the equilibrium velocity strongly depends on the timescales of stellar expansion, tidal torques, wind torques, and meridional circulations. Tides tend to spin surface layers of the star up, whereas other mechanisms (i.e., meridional circulations, wind braking, and stellar expansion) are inclined to spin the star down at the equilibrium state.

After synchronization, equatorial velocities of both components in models B2 and B6 increase owing to stellar expansion because spin angular velocities are locked to approach the orbital one. The donor star in models B2 and B6 suffers from heavy mass and angular momentum loss during the first episode of RLOF. This process can efficiently spin the donor star down. Mass and angular momentum are transferred to the companion star. Generally, the spin-up torque due to RLOF is 3–4 orders of magnitude larger than tidal torques (Song et al. 2017). Packet (1981) has found that only about 10% of the stellar material is able to spin the companion star up significantly. The accreting star attains critical rotation once for models B2 and B6 during RLOF.

The accreting star for B7 is not spun up to critical velocities because of the deficiency of RLOF, in comparison with binary Population III stars. W-R winds for model B7 take away the specific orbital angular momentum of the mass-losing star and broaden the orbital separation evidently. Rotational velocities of two components go down progressively because two components undergo CHE.

Figure 11 shows the evolution of the orbital period for the binaries. One may find that the orbital period for B2 is slightly larger than the one for B1 before RLOF. The reason is that tidal torques spin the star down when the spin angular velocity of two components is larger than the orbital one. More spin angular momenta can be converted into the orbit, and thus the orbital period is increased. Therefore, model B1 starts RLOF at the time of 2.640 Myr, whereas model B2 starts RLOF at the time of 2.518 Myr. The orbital period for models B3 and B4 remains unchanged during the main sequence because the tidal torques are very weak owing to the larger initial orbital period. This implies that there is no or very tiny angular momentum exchange between spin angular momentum and the orbital one. Furthermore, stellar winds for models B3 and B4, which can widen the orbital separation, are also very negligible.

Zoom In Zoom Out Reset image size

The orbital period for B6 increases faster than the one for B5 before RLOF. There are two main reasons. First, during synchronization (i.e., before the age of 0.0286 Myr), the efficiency of exchange of angular momentum is much higher. Second, stellar winds for model B6 are much stronger. The orbital period for all binaries goes up continuously before mass reverse because the accretion efficiency is so low that the inefficient mass transfer can not cause the orbit to shrink (Song et al. 2018b). Also, critical rotation for the accreting star induced by RLOF can enlarge the orbital period rapidly because most transferred matter is thrown away from the system in a form of winds. For instance, the orbital period for B2 increases from 4.16 to 11.48 days, whereas it increases from 4.00 to 9.73 days for the B1 system. The mass transfer phase generally terminates when the donor star has been stripped of most of its envelope. The hydrogen-burning shell can be extinguished, and the corresponding engine that produces primary nitrogen is turned off. For instance, hydrogen-burning shells for the donor star in B6 and B7 extinguish at central helium Y_c = 0.639 and 0.818, respectively. However, the donor star still has some hydrogen left in its envelope (typically a few 0.1 M_⊙). Strong mass loss via W-R winds in model B7 tends to widen the orbit dramatically in comparison with model B4.

Figure 12 displays the mass transfer rate due to RLOF as a function of evolutionary age in binaries. It is found in panel (a) that the peak of mass transfer rate during the first episode of RLOF increases with the orbital period because the donor star evolves fully. The mass transfer rate in Model B4 can reach 3.7 × 10⁻³ M_⊙ yr⁻¹ because the donor star has developed a convective envelope. Although the orbit for high-metallicity model B6 is enlarged by strong stellar winds, it begins mass transfer early at the time of 2.186 Myr. The reason is that the radius for model B6 is much larger than the one for model B2. During the first episode of RLOF, the peak of the mass transfer rate is high for Population III stars. For example, the mass transfer rate for B2 is 9.98 × 10⁻⁴ M_⊙ yr⁻¹, whereas it has the value of 1.684 × 10⁻⁴ M_⊙ yr⁻¹ in B6. The reason is that less mass is lost by stellar winds in B2 and more mass can be transferred to the companion star correspondingly. Wider systems induced by strong stellar winds also do not favor RLOF because their Roche lobe is larger (see Table 4). For example, at the beginning of RLOF, the orbital period for B2 is 4.16 days, whereas it can arrive at a value of 6.26 days for B6. During the two episodes of RLOF, the donor star loses 54.65 M_⊙ for B2, whereas it loses 17.19 M_⊙ for B6.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Panels (c) and (d) of Figure 1 show the evolution of the convective core mass for the binaries. It is clear that the convective core for model B6 shrinks faster than the one for model S11 from ZAMS because rotational velocities can be braked by tidal interactions during synchronization. This implies that rotational mixing effectively mixes the hydrogen from the envelope to the core and slows down the reduction speed of Thomson scattering opacity (i.e., κ_Th = 0.2(1 + X_H)), which originates from the consumed core hydrogen. Like model S11, the donor star for model B7, which has gone through CHE, has the largest convective core during the main sequence. The convective core for the donor star in B7 does not undergo rapid contraction at core hydrogen exhaustion because of CHE.

Matter is transferred to the companion star owing to RLOF. The donor star responds to the removal in mass by decreasing its convective cores. The accreting star responds to the increase in mass by increasing convective cores. The size of its convective core goes up, and fresh hydrogen is mixed toward the center, effectively rejuvenating the star. After accretion, the properties of the star resemble the properties of a younger single star of the same mass. The accreting star becomes brighter but remains hot, appearing as an O-type star (see panel (c) of Figure 3). An exception appears in the donor star in B4, and convective cores do not decrease owing to mass removal. This implies that the donor star with central helium burning does not respond to the decreasing in convective core mass rapidly by adapting its internal structure. It is found that in wider systems convective cores for two components can grow larger before RLOF starts. The reason is that tidal braking is too weak to reduce rotational mixing. The shorter the initial orbital period, the smaller the mass of convective cores for the donor star after mass transfer. This can be understood by the fact that more mass can be transferred to the companion star in the system with shorter orbital period (see Figure 4(c)). One can find that the accreting star with higher metallicity can obtain less mass, except for model B7, which goes through CHE.

Figure 6(b) illustrates that surface nitrogen abundances of two components in binaries vary with evolutionary age. Nitrogen enrichment for the donor star in B1 occurs (i.e., t = 2.66 Myr) early in comparison with the one in model S2 (i.e., t = 2.90 Myr) because RLOF can quickly expose the inner hydrogen-burning region, which is rich in nitrogen. Nitrogen enrichment for the nonrotating accreting star in B1 can also reach a high value as a result of the transferred matter. Nitrogen abundances for model S8 reach 2.11 dex at 2.0 Myr, whereas it arrives at 1.34 dex for model B2 owing to tidal braking. The result indicates that nitrogen enrichment is weaker in binarities than the one in single stars. A very interesting case occurs in model B5. Nitrogen enrichment for the less massive accreting star can outweigh the more massive donor star during synchronization. The reason is that the accreting star can maintain a higher rotational speed owing to a longer timescale for synchronization (see Figure 5(b)). It is difficult for two components in binarities to produce CHE unless the orbital period is extremely short (de Mink et al. 2009; Song et al. 2016). After synchronization, rotational velocities can attain high values in an extremely tight binary system, which are beneficial to mixing chemical elements (see Figure 7(b)). This case may happen in model B7. These facts indicate that rotational mixing in binaries does not heavily depend on the initial rotational velocity but is strongly related to the orbital period.

In contrast with rotational models for single stars, nitrogen enrichment in binaries does not depend on stellar mass. The low-mass components may show the high surface nitrogen abundance because of the mass reverse during RLOF. There is only a mild or no nitrogen enrichment for the accreting star in rotational sequences during RLOF in comparison with nonrotational sequence B1. There are three main reasons (Song et al. 2018b). First, mass gainers stop accretion when its outer layers rotate critically. The transferred matter, which is rich in nitrogen, will deposit in an outer disk surrounding the accreting star. This matter might be dissipated by high radiative pressures. Second, thermohaline mixing swiftly dilutes the accreted nitrogen within the thermal timescale, which is very short for massive stars. We can notice that four peaks for model B6 in Figure 6(b) indicate that mass transfer due to RLOF occurs at these episodes. The decline of nitrogen after peaks implies that the thermohaline instability allows mixing downward of the accreted nitrogen-rich matter to occur. Third, the accretor has been spun down by tidal torques during previous synchronization, and this process easily develops a high gradient of mean molecular weight at the boundary of the convective core, which can drastically slow down rotational mixing. The transferred matter that is enriched in CNO elements produces strong stellar winds for the accreting star because the reduced opacity in the outer envelope also contributes to high temperature and luminosity.

Figure 7(b) displays surface helium mass faction Y_s versus central helium mass fraction Y_c from ZAMS up to the end of the helium burning for the donor star in binaries. Surface helium abundance for B6 reaches a value of ∼0.566 at the terminal of the main sequence, while it has a value of ∼0.97 for S4. Therefore, CHE can be inhibited in binarities except for the system B7 with very short orbital period (de Mink et al. 2009). At the end of central helium burning, surface helium mass fraction from model B1 to model B4 can reach the values of 0.715, 0.787, 0.757, and 0.500, respectively. The result indicates that rotation can expose more helium and less helium enrichment happens in wider systems. At the same time, the surface helium mass fraction for models B5 and B6 can reach values of 0.719 and 0.1497, respectively. Therefore, the system with high metallicities appears to have a lower surface helium mass fraction. System B7 does not experience RLOF at all, and no mass transfer occurs in this system. Surface helium abundance for the donor star in models B6 and B7 progressively increases after the main sequence because of the combined effect of RLOF and W-R winds. Then, it falls down rapidly because of the removal of helium envelopes and the exposure of CO cores. After the main sequence, the surface helium abundance for the donor star in systems B1–B5 increases gradually with the exposing of the products for hydrogen-burning shells.

Panels (c)–(d) of Figure 8 illustrate the evolution of mass fraction of hydrogen and effective temperature as a function of the central helium mass fraction in binaries. Before RLOF, the surface hydrogen fraction decreases slowly for model B2 because mass loss induced by rotation is decreased by tidal braking in comparison with the one for B4. Furthermore, the surface hydrogen fraction for model B7 is lower in comparison with B2 because stars with high metallicity have strong winds. Rapid decreasing of hydrogen in B7 can cause the star to shift toward high effective temperature, whereas model B2 evolves toward low effective temperature. Surface hydrogen fraction falls down steeply for close binaries because hydrogen envelopes are removed remarkably by RLOF. Therefore, there is no dredge-up that occurs in binaries. There is also some indication that the total hydrogen content may be a function of the initial orbital period. Larger orbital periods give rise to a larger hydrogen content. The surface hydrogen fraction disappears early for model B7 with CHE. The surface hydrogen mass fraction for B2 vanishes earlier than the counterpart for B3 because the donor star for model B2 turns into W-R stars early. One can find that high-metallicity stars B6 display a higher effective temperature at the terminal of helium burning because of the removal of a huge amount of hydrogen envelopes.

Figure 13 illustrates the mass fraction of H, He, C, N, O, and Ne as a function of the mass coordinate for two components in model B2. After the second episode of RLOF, surface abundances for H decline while surface abundances for He, C, N, O, and ²²Ne increase for two components in B2. This indicates that chemical elements in the envelope for the donor star can be transferred to the accreting stars owing to RLOF. Moreover, convective cores for the donor star are reduced, whereas convective cores for the accreting star are increased correspondingly. The central abundance of H increases for the accreting star, whereas central He, C, N, O, and Ne decline owing to the expansion of convective cores. This implies that the chemical structure of the accreting star can be significantly modified by RLOF.

Before RLOF, one can find that surface abundances for H, ²²Ne, and Mg go down sightly while surface abundances for He, C, N, O, and Al go up owing to rotational mixing, by comparing the donor star in B1 with the one in B2 (see Table 5). Surface abundances for He, C, N, O, ²⁰Ne, and Al go up in B2, while surface abundances for H, ²²Ne, and Mg go down sightly owing to RLOF. At hydrogen exhaustion, surface Mg abundance for the donor star in model B3 increases remarkably, and less Al is enriched in contrast to B2. The reason is that RLOF does not occur in model B3 during the main sequence and a higher temperature might remain inside the star (see Table 4). During helium burning, surface abundances for H, He, N, and Al go down remarkably while surface abundances for C, O, Ne, and ²⁵Mg go up in models B6 and B7. This can be ascribed to the fact that hydrogen envelopes have been removed for the star with high metallicities. Stellar mass for the donor star in B1 has a value of 79.37 M_⊙, whereas it is 75.09 M_⊙ for the one in B2 at the end of the second episode of RLOF. Therefore, rotation tends to produce stellar expansion and greatly favors RLOF in the binary system. At the end of central helium burning, surface abundances for He, C, and ²²Ne go up for B7 with CHE, while surface abundances for N, O, ²⁰Ne, Mg, and ²⁶Al go down sightly compared to B6. The main reason is that the donor star in model B7 with CHE has a minimum value of mass (see Table 5).

Panel (b) of Figure 10 displays the evolution of helium cores for the donor star in the binarities. The mass of helium cores for rotating stars in close binaries will often be smaller than the counterpart in single stars or wide binaries. For example, helium cores for S8 have a value of 76.89 M_⊙ at 2.836 Myr, whereas they have a value of 65.42 M_⊙ for B2 at 2.701 Myr. The reason is that helium cores are decreased by RLOF. A donor star that loses its hydrogen envelope will generate a smaller helium core compared to a single star. Furthermore, tidal braking can reduce rotational velocities and thus smaller helium cores. The donor star from model B1 to model B4 has values of 64.91, 65.42, 70.58, and 71.28 M_⊙, respectively. The results reveal that the shorter the initial orbital period, the smaller the mass of helium cores. This fact implies that more matter can be transferred toward the companion star in a system with shorter orbital period. Moreover, helium cores for models B5 and B6 have values of 67.31 and 69.38 M_⊙, respectively. This can be understood by the fact that the orbital separation can be widened via strong winds in the system with high metallicities and contribute to larger Roche lobe. Mass transfer rates due to RLOF are lower in system B6. We can notice that there is no deep dredge-up for binaries compared to single stars because of RLOF. However, helium core mass is 55.60 M_⊙ for model B7 with CHE, and it is the smallest one. This is expected because the star has turned into a W-R star during the main sequence. Strong W-R winds can reduce remarkably the mass of helium core.

Panels (c) and (d) of Figure 3 display the evolution of two components in the H-R diagram. At the terminal of helium burning, the temperature and luminosity for model S2 are 4.389 and 6.565, respectively, whereas they are 4.596 and 6.526 for model B1 (see Tables 2 and 4). The effective temperature is increased, whereas the luminosity is decreased in binarities because the hydrogen envelopes are stripped owing to RLOF. However, the temperature and luminosity for model B2 are 4.581 and 6.543, respectively. Rotation can cause the star to shift toward lower temperature and high luminosity because less mass is transferred to the companion star. The donor star in model B6 makes a small blue loop at hydrogen exhaustion, and they turn into W-R stars later in contrast to the one in model B7. The evolution of the donor star in B7 has the same trend as model S11 because both of them go through CHE.

The secondary star in B1 is able to accrete the matter coming from the donor star efficiently, and then it may contract appreciably, attaining high effective temperature. Podsiadlowski et al. (1992) have presented that an important factor in determining the time of transition to the low temperature is the fractional core mass ${\xi }_{c}=\tfrac{{m}_{c}}{M}$, where m_c is the mass of the hydrogen-exhausted core and M is the total mass of the star. Therefore, accretion of matter from a close binary tends to increase the m_c, which can cause the star to prevent the transition to the red supergiant stage. The accreting star in B4 shifts toward low effective temperature because it merely obtains the least matter from the donor star (see Figure 4(c)). From Figure 3(d), one can also notice that the accreting star in B6 shifts toward low effective temperature, whereas the one in B5 evolves toward high effective temperature. The main reason is that the high-metallicity star B6 loses a large amount of mass via stellar winds, and thus less matter can be transferred toward the companion star.