Mergers of massive main sequence binaries

Transcription

1 Mergers of massive main sequence binaries Bachelor research report February 2010 Adrian Hamers Under supervision of O.R. Pols S.E. de Mink Universiteit Utrecht Faculteit Bètawetenschappen Departement Natuur en sterrenkunde Sterrenkundig instituut

2 On the cover page: Image of the star-forming region N 11B in the Large Magellanic Cloud (LMC) taken by Hubble s Wide Field Planetary Camera 2 using filters that isolate light emitted by oxygen and hydrogen gas, taken in Abstract In this bachelor research project, we investigated mergers between close main-sequence binaries with the aim of determining whether they result in blue stragglers. We define blue stragglers to be hot and luminous stars that cannot be explained with canonical single star evolution models 2. Using a pre-calculated grid of binary models, we determined the composition of merger products for a wide range of binaries with the assumption of no mass loss and homogeneous mixing during the merging process. We find that there are two types of mergers which result from different phases of mass transfer. Both of these types have relatively low hydrogen content which leads to high luminosities. We then used the data of this analysis to simulate a cluster of a large number of stars with a binary percentage of 50% which formed instantaneously. The result of this simulation is that merger stars of close massive main-sequence binaries are dominant among the most luminous stars in the main-sequence band of the simulated cluster, however we have made assumptions that may greatly affect results. In addition, blue stragglers may also be formed in different processes that do not involve the merging of massive main-sequence binaries. We could improve on this by using stellar evolution models to model the merger products instead of using homology relations, assuming different cluster formation processes and including different blue straggler scenarios. Contents 1 Introduction 3 2 Mass transfer in binary systems Roche lobe overflow Sub-cases of case A evolution Mergers in binary systems Assumptions made Qualitative properties Quantitative properties: grid analysis Cluster simulation Method Cluster generation Cluster population Results Discussion Mass loss and angular momentum loss Homology relations Cluster simulation Conclusion 12 7 Acknowledgements 12 8 References 12 9 Appendix: flow diagram 14 1 Source: 2 By these we mean evolution models with no implementation of non-standard physical processes such as rotational mixing. 2

3 (L / L to as merger blue stragglers, to produce more luminous stars than the channel which leads to accretion blue stragglers. Also, as N 11 is a young star cluster, its stars are expected to have a relatively high binary fraction. Therefore, in this research we will focus on merger blue stragglers. More precisely, mergers between core-hydrogen burning stars in close massive binaries are investigated. The key questions are: Which binary configurations lead to merger stars on the main sequence? What is the relation between binary configuration and merger star composition and luminosity? What is the relative importance of mergers of close massive binaries among the massive stars in a stellar population, such as a (young) star cluster? (T eff / K Figure 1: Hertzsprung-Russel (HR) diagram of LMC cluster N 11 by Evans et al. (2006) (edited image), whose data originated from the VLT-Flames survey. N 11 is a young cluster with an age of a few Myr. The region of stars which we define to be blue stragglers has been indicated. Dotted lines indicate calculated evolution tracks. 1 Introduction Looking at regions of cluster N 11 in the Large Magellanic Cloud (LMC) (cover page), one immediately notices the presence of many bright blue- and white-colored stars. These stars are among the hottest and most massive stars known in the Universe. In the Hertzsprung-Russel (HR) diagram of this cluster (Fig. 1), they belong to the stars which lie in the region of stars which we will refer to as blue stragglers. In the context of low-mass stars, blue stragglers are known as stars which have luminosities and effective temperatures that are significantly greater for their age than predicted by standard single-star evolution models. In analogy, we define blue stragglers in the context of high-mass stars as very hot and luminous stars that cannot be explained with canonical single star evolution models, by which we mean star evolution models which do not incorporate effects such as rotational mixing. The origin of these blue stragglers is not yet properly understood as they could be very massive single stars which had an unusual evolutionary history or they could also have a binary origin. One of the binary explanations involves the collision of single or binary stars as pointed out by Lombardi et al. (2002), which can be modeled with hydrodynamic simulations. Other explanations are given by Han et al. (2009). One possibility is that a binary component accretes mass from its companion and gets rejuvenated, increasing the main-sequence life-time of the component. We will refer to the resulting blue stragglers as accretion blue stragglers. An other possibility is that a binary merges to a single star after a common envelope situation. We expect the latter channel, whose remnants we will refer The third question addresses the issue that in young clusters, blue stragglers are hard to distinguish from very massive, single (i.e. normal ) and young stars, as the latter appear in roughly the same position in the HR-diagram. Whereas the relative importance of mergers of close massive binaries with respect to single massive stars cannot easily be determined from observations, this determination can be done by a simulation of a cluster of stars, as we shall do in this research. We remark that this question relates to questions raised by Hunter et al. (2008) as to the possibly considerable binary fraction in the LMC needed to explain the observed low nitrogen surface abundances among massive main-sequence stars. In Sect. 2 we will begin with a brief overview of the physical processes that occur in binary systems before a possible merging event. To tackle the second question quantitatively, we analyzed a grid of binary SMC models computed by De Mink et al. (2007) which allowed us to calculate the hydrogen fraction of the merger star, which we shall discuss in Sect We considered the third question by simulating a cluster of stars with SMC metallicity (Sect. 4) which formed instantaneously, i.e. in a starburst, where we generated single stars and binaries. We processed the binaries with the potential to merge on the main sequence and thus we could distinguish between single stars, binaries that quite certainly do not merge on their main sequence (which we will refer to as unevolved binaries) and mergers that originate from close main-sequence binaries. This allowed us to make statements of the importance of these mergers among other stars with comparable luminosity, i.e. massive single stars and massive unevolved binaries. 2 Mass transfer in binary systems Since interacting binaries are the focus of this research, we discuss some relevant aspects of mass transfer in this section: Roche lobe overflow (Sect. 2.1) and two important binary evolution sub-cases (Sect. 2.2). 3

4 z z Field point Roche Lobe lobe surface Figure 3: Depiction of the possible binary configurations 5. r r p r r s r s s L 1 Eggleton (1983): y r p p p a x x R L,p = 0.49 a q 2/3 ln(1 + q 1/3 ), (2) Axis of rotation Figure 2: Contour plot of surfaces of constant Roche potential for a typical binary system 4 with mass ratio 2. For the Roche-lobe surface, which corresponds to the inner thick solid line, the point of intersection L 1 has been indicated. 2.1 Roche lobe overflow The theory of Roche lobe overflow describes mass transfer in binary systems as explained by Verbunt (2008). For a binary system with primary 3 mass M p, secondary mass M s and period P, consider the geometry of surfaces of constant potential (Fig. 2), which is determined by the mass distribution and the period. The potential per unit mass, Φ(r p, r s, r ω), is given by: Φ(r p, r s, r ω) = GMp r p GMs r s ω2 r 2 ω 2, (1) with r p and r s the distances from the field point (the point of interest) to the centers of mass of the primary and secondary star respectively, r ω the (shortest) distance of the field point to the axis of rotation, ω = 2π the angular frequency of P rotation and G the gravitational constant. Thus the forces we consider are the gravitational attraction of both stars and the outward centrifugal repulsion. There exists a certain equipotential surface which consists of two surfaces that each enclose one of the stars. These two surfaces intersect at a point called the inner Lagrangian point L 1 and each enclose a certain volume; the radius of the sphere with the same corresponding volume is called the Roche-lobe radius R L. Thus in a binary system there exist two Roche-lobe radii, one for the primary and one for the secondary star. In general these radii depend on the separation a of the centers of mass of the stars (Fig. 2) and on the mass ratio q, here defined as the ratio of the mass of the primary and secondary star, i.e. q Mp M s. An approximate equation for all mass ratios accurate to better than 1% for the primary star was given by 3 We define the primary star as the initially most massive star and the secondary star as the initially least massive star. 4 Unedited image extracted on from Internet page p5.html. which is an increasing function of q. The Roche-lobe radius of the secondary star can be obtained by using Eq. 2 with q 1. It is useful to mention the relation between the q separation a and period P, which is Kepler s Third Law: P 2 a 3 = 4π 2 G(M p + M s). (3) A particle within the Roche lobe of a star is attached to this particular star because the gravitational attraction of this star exceeds the gravitational attraction of the other star and the centrifugal force. At the inner Lagrangian point L 1, the forces exactly balance such that a particle around this point can migrate to the other star. Hence if the radius of a star in a binary exceeds the Roche-lobe radius, in other words, if a star fills its Roche lobe, then mass transfer can occur through the nozzle around the inner Lagrangian point L 1, where hydrostatic equilibrium is no longer possible. This leads to three possible binary configurations which are also observed (Fig. 3): Detached binary: both stars do not fill their Roche lobe and thus there is no mass transfer. Binary evolution closely resembles single star evolution. Semi-detached binary: one of the stars fills its Roche lobe and may transfer mass to its companion. Contact binary: both stars fill their Roche lobes and may transfer mass and heat to each other (common-envelope situation). This may result in a merger star. Mass transfer greatly affects binary evolution. It is of importance to be able to specify the evolutionary state at which it occurs, hence the following evolutionary cases are discerned: Case A: mass transfer occurs during core-hydrogen burning. Case B: mass transfer occurs during hydrogen shell burning. Case C: mass transfer occurs after central helium depletion. 5 Unedited image extracted on from Internet page imamura/122/lecture- 9/typeIaSN.html. 4

5 Close binaries, i.e. binaries with short periods (lower than about 8 days), interact at a relatively early stage of their evolution, i.e. on the main sequence. This is because short periods imply small separations a (Eq. 3), which imply small Roche-lobe radii (Eq. 2). Thus the binary components fill their Roche lobes at a relatively early stage in their evolution because stars expand on the main sequence. Hence case A mass transfer applies in close binaries. It must be noted that three assumptions are made in the Roche geometry displayed in Fig. 2. First, the gravitational fields of both stars are assumed to be those of point masses. This is a reasonable assumption, even when stars are filling their Roche lobes, because in most stars the mass is very centrally located. Second, the binary orbit is assumed to be circular. This assumption is not generally valid, but since the circularization timescale depends on (R/a) 8 as shown by Zahn (1977), it is small compared to the stellar expansion timescale when R is close to R L, i.e. when the binary components are filling their Roche lobes. Last, the stellar rotation is assumed to be synchronized with the orbital motion, since the Roche geometry only applies to matter that co-rotates with the orbit. As with the second assumption, this assumption is not generally valid, but since the synchronization timescale depends on (R/a) 6, also shown by Zahn (1977), it is again small compared to the stellar expansion timescale if R is close to R L. 2.2 Sub-cases of case A evolution Further distinctions of case A binary evolution can be made as illustrated in detail by Nelson & Eggleton (2001). We are only interested in sub-cases that lead to contact binaries with both components still on the main sequence, hence we only consider the two sub-cases which Nelson & Eggleton (2001) refer to as case AR and case AS. For each of these sub-cases a contact binary is eventually formed, but in different phases of mass transfer in case A systems. These phases are referred to as the rapid and slow phase, which are driven by expansion on the thermal timescale of the primary and expansion on the nuclear timescale of the primary respectively. In the rapid phase of mass transfer, the initial phase of mass transfer in case A systems, mass is transferred from the more massive primary to the less massive secondary, which causes the orbit to shrink. The shrinking orbit implies a decrease of the Roche-lobe radius (Eq. 2), such that the equilibrium radius of the primary exceeds the primary Roche-lobe radius and the primary is brought out of thermal equilibrium. Hence mass transfer occurs at the relatively short thermal timescale of the primary. If the mass ratio is moderate or large - these corresponding binaries are referred to as case AR -, then the secondary has, by definition, a (much) smaller mass than the primary, which implies that its thermal timescale is (much) longer. This means that for these mass ratios, the secondary is also brought out of thermal equilibrium and quickly expands (point B in Fig. 4(a)). Soon a contact binary is formed (point C in Fig. 4(a)). Binaries with small mass ratios and small periods - which are referred to as case AS binaries as discussed below (R/R (R/R M p,i = 10 M ; M s,i = 5.6 M ; P i = 1.7 d M/M (a) Case AR mass transfer. M p,i = 10 M ; M s,i = 8.9 M ; P i = 1.1 d M/M (b) Case AS mass transfer. Primary Secondary Roche Lobe Figure 4: Radius as function of mass for (a) typical AR systems and (b) typical AS systems 6. The index i stands for initial; d stands for day. - remain semi-detached during this phase. This is because for these binary systems, the secondary is able to maintain its thermal equilibrium since the mass and hence thermal timescale of the secondary are comparable to that of the primary. After the primary has transferred so much mass that it has become the least massive component, the orbit widens again and hence the primary restores its thermal equilibrium. Mass transfer is continued in the slow phase, where the primary transfers mass to the secondary on its own nuclear timescale. 6 Images extracted on from lecture notes used for a binary stars course by O.R. Pols, p. 22 Fig. 8.4, which can be found at pols/education/binaries/lnotes/binaries 2007.pdf. 5

6 If the initial mass ratio and initial period are small (case AS systems), the secondary remains in thermal equilibrium and expands as it accretes mass from the primary (point B in Fig. 4(b)). Then the secondary overtakes the primary (point C in Fig. 4(b)) and eventually a contact binary is formed. From this it is clear that in case AR systems, mass transfer can quickly lead to a contact binary, i.e. on the thermal timescale of the primary (hence the name AR, where R stands for rapid), while in case AS systems it can take considerably more time before a contact binary is formed, i.e. on the nuclear timescale of the primary (hence the name AS, where S stands for slow). It will turn out that this difference leads to different compositions of the merger product if the contact binary were to merge to a single star, as we shall discuss below in Sect Mergers in binary systems 3.1 Assumptions made There still exists much uncertainty about the merging process of a binary system as merging binaries are only rarely observed in practice. Therefore we are forced to make the necessary assumptions. First, we assume that main-sequence mergers of close binary systems can only originate from contact binaries whose components are both still main-sequence stars. This, in conjunction with Sect. 2.2, rules out all subcases of case A evolution except case AR and case AS systems. Furthermore, we assume that the merging process is quick, i.e. occurs on the dynamical timescale of one of the components and that the stars merge to a single star without any loss of mass into the interstellar medium. Also, we assume that the merger star has a homogeneous composition, constant opacity, ideal gas pressure and that it is in radiative equilibrium. These quite severe assumptions for the merger star are necessary to justify the use of a homology relation (Eq. 6), which greatly reduces computing time in the cluster simulation in Sect. 4 compared to the use of single stellar evolution models with modified composition. 3.2 Qualitative properties Using the assumptions made in Sect. 3.1, we can make statements about the merger product composition based on the properties of case AR and case AS binary systems discussed in Sect Case AS binaries merge at a much later moment in their evolution than case AR binaries. This implies that the components of AS systems have lower core-hydrogen abundances at the time of merging than the components of AR systems, because stars steadily burn core-hydrogen on the main sequence. With the assumption of no mass loss, this implies that the case AS mergers, i.e. mergers that originate from case AS binary systems, have (much) lower hydrogen abundances than case AR mergers at the moment of their formation. This in turn implies (much) larger helium abundances for AS mergers compared to AR mergers and thus higher mean molecular weights. These higher mean molecular weights imply higher luminosities, for which we refer to Eq. 6. Thus we expect case AS mergers to be more luminous than case AR mergers. In general, case AS and case AR mergers have lower hydrogen abundances than components in binaries that have not merged. We therefore also expect that both case AS and case AR mergers are more luminous than single stars with the same mass and binary stars with comparable mass that do not merge on the main sequence. 3.3 Quantitative properties: grid analysis In order to quantify the effect of case AR and case AS mergers, we analyzed a grid of binary models computed by De Mink et al. (2007). The code used for this grid is an updated version of the STARS/TWIN evolution code which implements convective mixing and overshooting and spin-orbit interaction by tides. Heat transfer between the binary components is not treated and mass transfer is assumed to be fully conservative, i.e. no mass is lost during mass transfer. For more details we refer to Sect. 2 of De Mink s paper. The binaries in this grid have SMC metallicity, i.e. Z = and are evolved up to the end of their main sequence. In the grid there are three parameters: initial primary mass M p,i, initial mass ratio q i = M p,i M s,i at equal logarithmic intervals: 8 < : and initial period P i which are spaced (M p,i/m ) = 0.05, 0.10, ; (q i) = 0.050, 0.075, ; (P i/p ZAMS) = 0.050, 0.10, , hence in total there are over 6000 models. The unit P ZAMS is an approximation of the orbital period at which the initially more massive component would fill its Roche lobe on the zeroage main sequence for a system with equal masses and was given approximately by Nelson & Eggleton (2001): P ZAMS/d = (4) 0.19 (Mp,i/M ) (Mp,i/M ) (M p,i/m ) 2, (5) with day abbreviated as d. For the given mass ranges in Eq. 4, the minimum and maximum values of P ZAMS are d and d respectively. Using a self-written code in Fortran 77, we determined which binary configurations lead to contact binaries on the main sequence. This was done by comparing the radii of both binary components to their Roche-lobe radii; we defined the time of contact of both stars to be the maximum of the times at which both stars fill their Roche lobes. Since we assumed that a binary merges if it is a contact binary and that this happens on a short, i.e. dynamical timescale (Sect. 3.1), we let the time of merging be equal to the time of contact. We determined the amount of hydrogen in the merger product by adding the amounts of hydrogen of both components at the time of contact, which required our assumption of no mass loss during the merging process. Thus we could calculate the hydrogen fraction X of the merger product using X = M hydrogen,merger /M total, merger. We show results of this grid analysis in Fig. 5(a) for M p,i = 12.6M. In general the hydrogen fraction decreases 6

7 (P i / P zams ) (P i / P zams ) (a) (b) AS AS M p, i = 12.6 M (q i ) M p, i = 50.1 M (q i ) Figure 5: Plot of results of the grid analysis for (a) M p,i = 12.6M and (b) M p,i = 50.1M. Each rectangle corresponds to a binary evolution model. On the x-axis is the initial mass ratio q i in log scale and on the y-axis the initial period P i, also in log scale. The period has been plotted in units of P ZAMS, which is defined in Eq. 5. The color shading indicates the hydrogen fraction of the resulting merger product; black signifies that this particular binary does not merge on the main sequence and green signifies that this particular binary model was missing from the binary grid. The regions of binary cases AR and AS have been indicated. with initial period. This is because as the initial period increases, the initial separation a increases (Eq. 3), the initial Roche-lobe radii increase (Eq. 2) and the time of merger increases, hence the binary components have burnt more hydrogen before a contact situation. This results in a low hydrogen fraction in the merger product. Furthermore, it is clear that case AS mergers indeed contain significantly less hydrogen and thus more helium than their case AR counterparts. For some AS mergers the hydrogen fraction can be as low as about 0.4 while the AR mergers do not have a fraction lower than about 0.5. Next, we show a comparable diagram in Fig. 5(b) for M p,i = 50.1M. Here the hydrogen fraction for higher initial periods is lower compared to the same situation at lower initial primary masses (Fig. 5(a)). This is because AR AR Hydrogen fraction Hydrogen fraction the relative size of the convective core is greater for higher masses, hence a high-mass binary component can burn a larger fraction of its hydrogen fuel before a merging event than a low-mass binary component. We see in Fig. 5(b) that the case AS mergers have a lower hydrogen fraction than their AR counterparts, as is the case for lower initial primary masses. The difference between the lowest hydrogen fractions for both cases is about 0.2. Thus, as we have shown qualitatively in Sect. 3.2, we have also shown quantitatively that mergers of close binaries can lead to merger stars with rich helium content. More specifically, case AS mergers contain significantly more helium than case AR mergers as expected. In order to relate these helium abundances to luminosity quantitatively, we recall the following homology relation given by Kippenhahn & Weigert (1990), L M 3 µ 4, (6) with L luminosity, M mass and µ the mean molecular weight. This relation is an analytic scaling relation which applies to main-sequence stars and only gives rough estimates. We note that our assumptions for the merger product of homogeneous composition, constant opacity, ideal gas pressure and radiative equilibrium (which we made in Sect. 3.1), are necessary to justify the use of this homology relation. The mean molecular weight µ is also given by Kippenhahn & Weigert (1990): µ = 1 2X Y Z, (7) with X the hydrogen fraction, Y the helium fraction and Z the metallicity. Taking the metallicity Z constant, we can rewrite Eq. 7 as follows with the relation X + Y + Z = 1 (which is true by the definitions of X, Y and Z): µ = Y 3 2 Z. (8) Hence we see that, taking the metallicity Z constant, greater helium fractions Y imply greater mean molecular weights µ according to Eq. 8, which in turn imply higher luminosities L according to Eq. 6. Thus case AS mergers are more luminous than case AR mergers when comparing stars of the same mass, a property that is thus fully attributed to their greater mean molecular weights. Also we see that merger stars are more luminous than single stars and binaries that do not merge on the main sequence with comparable mass, as their mean molecular weights are always greater. In Sect. 4 we will apply the results of this grid analysis to a cluster simulation to be able to make statements about the relative importance of these mergers among other massive stars (stars with high luminosity). 4 Cluster simulation In the introduction we have given a motivation for a simulation of a cluster of stars to see whether the most luminous main-sequence stars originate from mergers of close binary systems. In Sect. 4.1 we clarify the methods used in this simulation and in Sect. 4.2 we present our results. 7

8 4.1 Method Cluster generation The cluster simulation consists of stars with a binary percentage of 50% which all formed at the same time, an assumption known as a starburst. We assumed the current age of the cluster t 0 to be known. Furthermore, we let the ranges of initial parameters M p,i, q i and P i in this simulation be the same as in the binary grid we analyzed (Eq. 4), with the exception that we extended the period to about 10 3 days (more precisely: (P i/p ZAMS) = 3.000). This is to also account for binaries that do not interact on the main sequence. We generated the initial star parameters using Monte Carlo methods and assumed the following distribution functions. For the initial mass distribution, we assumed that the probability density is proportional to the mass to the power -2.3 as found to be valid by Kroupa (2001) for stars with masses greater than 1.0 M. Furthermore, for the initial mass ratio distribution (applicable only to binaries), we assumed that the probability density is constant with respect to the inverse of the mass ratio, which Goldberg et al. (2003) showed to be approximately valid for 1 < q < 2.2. Lastly, for the initial period distribution (binaries only), we assumed that the probability density is proportional to the log of the period as found by Mazeh et al. (2007). In symbols: 8 >< >: dn dm i m 2.3 dn = C; dq i dn dp i i ; (P i), with N the number of stars, m i the initial primary mass, q i = 1/q i and C a constant. In the remainder of this section, we shall elaborate on the technical details of the initial parameter generation. For the mass distribution, we let 7 m 1 and m 2 be the lowest and highest occurring initial masses respectively. The normalized probability density is then given by dn dm ρ(m) = (1 α)m α m 1 α 2 m 1 α 1 (9), (10) with α = 2.3, such that R m 2 m 1 ρ(m) dm = 1. Note that m 1 α 2 m 1 α 1 < 0 and that (1 α) < 0 such that ρ(m) > 0 for all m 1 < m < m 2. Now we define the function f(m) to be the integral of ρ(m) of m 1 to a value m where m 1 < m < m 2: f(m) Z m 1 α 1 ρ(m ) dm = m1 α m m 1 m 1 α 2 m 1 α. (11) 1 We have sketched the function f(m) in Fig. 6. Note that 0 f(m) 1 for all m 1 < m < m 2. Now we associate a random number x between 0 and 1 with f(m), i.e. we obtain m from the inverse function of f(m), f 1 (m), where now 7 To avoid cumbersome notation, we shall omit the index i for initial in the rest of this section. 1 f (m O m 1 m m 2 Figure 6: Sketch of the function f(m) as function of m, defined in Eq. 11. f(m) x. This inversion yields: m = 1 α ˆm1 (1 x) + xm 1 α 1 1 α 2. (12) Thus if a number of N initial masses must be generated with probability density ρ(m) and range m 1 to m 2, we can execute a simple do-loop N times to generate a random number x which we can use in conjunction with Eq. 12 to generate a mass m(x). Hence we refer to Eq. 12 as a generating function. For the initial mass ratio distribution, let the range of q be determined by q 1 and q 2 where q 2 > q 1. The normalized probability density is given by ρ(q) = dn dq = dn dq = dq dq = 1 dn = C q 2 dq q 2 q 1q 2 1 q 2 q 1 q, (13) 2 with q = 1/q; we took the absolute value of the derivative dq /dq to ensure that the probability density is always positive. Hence the function f(q), defined similarly as with the initial mass distribution, is given by: f(q) = q2 q q q 1 q 2 q 1. (14) Inversion of this function (defining x f 1 (q)) yields the generating function: q = q 1q 2 q 1x + q 2(1 x). (15) For the initial period distribution, we generated the periods in units of P = (P/P ZAMS) where P ZAMS was defined in Eq. 5. Hence the probability density with respect to P is linear, i.e. ρ( P ) = C, (16) with C a constant. This leads to the generating function P = P 1(1 x) + x P 2, (17) where P 1 and P 2 refer to the lowest and highest periods respectively. 8

9 P i > 8 d Binaries P i < 8 d Cluster population Merging on MS No merging on MS Mergers 8592 Unevolved binaries Single stars 0 t 0 birth of cluster t m merging event current cluster age nuc nuc + t m t t single stars; unresolved binaries mergers Figure 8: Timeline to illustrate the criteria for the main-sequence band. Figure 7: Overview of the cluster population. We indicated the absolute numbers of objects within the groups Cluster population We divided the simulated cluster population into three groups: binary stars with initial periods lower than about 8 days (more precisely: (P i/p ZAMS) = 0.750), binary stars with initial periods greater than about 8 days and single stars. We analyzed the first group, which can lead to mergers on the main sequence, with De Mink s grid as discussed in Sect For this group we used Eq. 7 to obtain the mean molecular weights and neglected the metals, i.e. we put 8 Z = 0 and used Eq. 6 to obtain the luminosity of the merger product 9. The second group, which we refer to as pre-interacting main-sequence binaries or unevolved binaries, we did not analyze with De Mink s grid. Instead, we treated its binaries essentially as single stars. We defined the luminosity of an unevolved binary to be the sum of luminosities of both components calculated using Eq. 6. We also used Eq. 6 for the third group, which consists of single stars. The reader might wonder why we did not use single stellar evolution models for the single stars and unevolved binaries to obtain more accurate results. This is because we needed to ensure that our comparisons are reliable in the sense that we used the same method to model all three groups. We note that we included the binaries of the first group that do not merge on the main sequence in the second group. In Fig. 7 we illustrate the cluster population, where we also include the absolute numbers of objects within the groups. We calculated a nuclear timescale τ nuc for the stars of all groups. The physical meaning of the formula we used is that the (main sequence) nuclear timescale is the amount of available nuclear fuel for hydrogen burning divided by the luminosity. Thus: τ nuc Xµ 4 τ nuc, SMC(M), (18) 8 We neglected metals because it proved to be complicated to obtain compositions of the merger product other than hydrogen and helium abundances. We expect the error made in neglecting metals to be only small however. 9 The reference value of µ used in Eq. 6 was taken to be about 0.6, which is approximately the current solar value. For the purposes of this research however (comparing the different groups), the actual value of this number is not important. with X the hydrogen fraction, µ the mean molecular weight and τ nuc, SMC(M) the nuclear timescale for a single star with SMC metallicity for particular mass M. We calculated τ nuc, SMC(M) for a few values of M using the Window to the Stars interface developed by Izzard & Glebbeek (2006). To obtain τ nuc, SMC(M) for other masses, we used a linear interpolation. We used this nuclear timescale to determine which stars in the cluster lie in the main-sequence band. The required criterion depends on the group type; as an illustration, consider the following time line in Fig. 8. At t = 0 the cluster is formed in an infinitesimal burst of stars. For a single star or unevolved binary, if the current age of the cluster is t 0 and if the nuclear timescale τ nuc is larger than t 0, then this star is on the main sequence. For a merger star, we must modify this criterion: the merger star merged at a certain time, say t m. Thus it is only on the main sequence if t 0 lies between t m and τ nuc + t m. The reason why we added t m to τ nuc is that the nuclear timescale must be relative to t = 0 and not to the time of merger t m. Therefore, expressing these criteria in inequalities, we have that for main-sequence stars: j t 0 < τ nuc, (G 2, G 3); t m < t 0 < τ nuc + t m, (G 1), (19) with G 1 the group of merger stars, G 2 the group of unevolved binaries and G 3 the group of single stars. 4.2 Results Using the procedure discussed in Sect , we could calculate a luminosity for each group member. In Fig. 9, we show for all three group types the luminosity function, i.e. the number of objects formed in a small interval of luminosity log L/L = 0.1 as function of luminosity. It is important to note that this figure does not represent data which could be observed in practice because it does not contain any time information, i.e. in reality, all the mergers shown would never exist simultaneously. The aim of the figure however is to give an impression of the cluster population and demonstrate the potential importance of mergers. It is clear from Fig. 9 that for all groups, the bulk of the stars have low luminosities. This is a result of the mass distribution: there is a relatively high probability for a star being a low-mass star (Eq. 9) and low-mass stars have low luminosities (Eq. 6). There are relatively few 9

10 (a) t 0 = 5 Myr Single stars Unevolved binaries Merger stars Single stars Unevolved binaries Merger stars (N (N (L/L (L/L Figure 9: Luminosity function, i.e. number of objects formed as function of luminosity, of the simulated cluster. All objects have been counted in bins of bin width log L/L = 0.1. (b) t 0 = 500 Myr Single stars Unevolved binaries Merger stars merger stars, this is because of the many criteria for the existence of these objects: they must originate from a binary with periods less than about 8 days and with the right combination of mass ratio and period, i.e. they may not lie in the black areas in figures like Fig. 5(a) and Fig. 5(b). At luminosities greater than about L, only the merger stars remain. This is firstly due to the fact that compared to single stars and unevolved binaries, the merger stars have a larger mean molecular weight and thus higher luminosity (Eq. 6). Secondly, when comparing single stars and mergers of the same mass, the mergers always have larger mass due to the companion star and hence also larger luminosity. (N (L/L To be able to say more about the relative importance of the merger stars in the main-sequence band of the cluster population, we must consider the nuclear timescales, i.e. crudely evolve the simulated stars. In Fig. 10(a) we plotted a luminosity function for all three groups for a cluster age of t 0 = 5 Myr, i.e., we only plotted the objects that still lie in the main-sequence band 5 Myr after the formation of the cluster, which formed in a starburst by assumption. In contrast to Fig. 9, this is a figure we expect to be observable in princible. In Fig. 10(a), there are no merger stars at luminosities lower than about L. This is due to the fact that for this very young cluster age, only the binaries with very massive components, with high luminosities, merge. This is because the more massive a binary, the faster its (main sequence) evolution and hence the faster a merger may be formed and appear in the main-sequence band. From the figure it is clear that the merger stars are dominant among the high luminosities. All stars with luminosities greater than about L are expected to be merger stars, although they are not as abundant in quantity as the lower-mass stars. We show the same diagram for a cluster age of 500 Myr in Fig. 10(b). There is a shift for all groups towards lower lumi- Figure 10: Luminosity function, i.e. number of objects as function of luminosity of the simulated cluster for a cluster age of (a) 5 Myr and (b) 500 Myr. All objects have been counted in bins of bin width log L/L = 0.1. nosities. This is because a high cluster age implies that only older stars remain in the main-sequence band, which are stars with lower masses and hence lower luminosities. We see that interestingly at t 0 = 500 Myr the merger-band consists of two bumps. The reason for lies in the difference between case AR and case AS mergers. The first bump, centered around a luminosity of about L, is due to case AR mergers; their number decreases with increasing luminosities. This is because a higher luminosity corresponds to a greater mean molecular weight which corresponds to a lower hydrogen fraction. This, according to Figs. 5(a) and 5(b), corresponds to a higher period (for case AR mergers). Thus there is a decrease in number with increasing luminosities as there are less stars with high periods than there are stars with low periods (Eq. 9). The second bump appears at luminosities greater than about L. For these luminosities, the case AS mergers, stars with higher luminosities than case AR mergers as we have shown before in Sect. 3.3, appear and ensure that the 10

11 number of mergers again increases with increasing luminosity. As the binaries that produce AS mergers do not have a very wide range of periods (Figs. 5(a) and 5(b)), there is no visible decrease of number with increasing luminosity in Fig. 10(b) until no stars remain at all. We see that in Fig. 10(b), as in Fig. 10(a), merger stars are expected to be dominant among the brightest stars in the main-sequence band. The overlap between the other groups in luminosity is even smaller which makes their presence even more pronounced. 5 Discussion It is clear that in this research we have made many assumptions and choices that could potentially greatly influence the results we obtained. In the following section we shall elaborate on these. 5.1 Mass loss and angular momentum loss By analyzing a grid of binary models that implements conservative mass transfer (Sect. 3.3), we have implicitly made the assumption that no mass is lost before a contact binary is formed. From observations it is known that this is not necessarily the case however (as shown by e.g. De Mink et al. (2007)). Grids with non-conservative mass transfer were also available but we did not consider them. Furthermore, we assumed the merging process itself to be short-lasting and without mass loss. Although the assumption of no mass loss does appear to be severe, hydrodynamic simulations of stellar collisions have shown that only a modest fraction of mass is lost as pointed out by Lombardi et al. (2002). It is reasonable to assume that mass loss in a merging event of binary components is less severe than mass loss in a collision of stars, as the latter situation is much more violent. Furthermore, in the hydrodynamic simulations there remains the issue of angular momentum. It appears that a substantial amount of angular momentum would need to be lost for the remnant star to rotate at a rotational velocity smaller than or equal to the critical rotational velocity. Mass loss is the main mechanism to achieve this loss of angular momentum. This issue also exists in the merging within a binary system, which we confirmed using data from the binary grid. It appeared that up to 50% of angular momentum would need to be lost for noncritical rotation of the merger product. Thus it would not be unreasonable to implement mass loss in the calculation of merger compositions. The uncertainties involved however have forced us to implement the assumption of no mass loss during the merging event in this research. 5.2 Homology relations We frequently used homology relation Eq. 6 to calculate luminosities. Although the dependence of mean molecular weight in this relation is quite accurate, even for inhomogeneous stars, as shown by Poelarends (2002), the mass dependence is not. As an illustration, consider Fig. 11, which shows the luminosity as function of mass. The thick red curve shows the relation according to accurate stellar (L/L (M/M Figure 11: Thick red curve: plot of luminosity as function of mass, made using Window to the Stars, developed by Izzard & Glebbeek (2006). Dashed line: luminosity as function of mass according to Eq. 6. evolution models. The dashed line shows the relation according to Eq. 6, i.e. L M 3. We see that the assumed M 3 -dependence is only approximate: it fails for masses greater than about 25 M. This is because in high-mass stars, radiative pressure is considerable, which is not taken into account in the homology relation since ideal gas pressure is assumed. We could correct this problem in future research by interpolating data from stellar evolution models. An other important assumption that we made, is that the luminosity remains constant during the main sequence. In reality however, the luminosity increases slightly on the main sequence because of the slight increase of mean molecular weight. As this increase of mean molecular weight during the main sequence is relatively greater for more massive stars, this implies that high-mass stars of all group types in Figs. 10(a) and 10(b) are shifted towards higher luminosities. This would mean that mergers are less important among the massive stars than is suggested by these figures. To improve on this, we could assume the luminosity to increase linearly with time on the main sequence. A better solution would be to use stellar evolution models for the merger stars (and hence also the single stars and unresolved binaries to keep comparisons fair), although unfortunately this is quite time consuming for a large cluster population. The latter solution would of course also remove the objection that homology relations are not accurate for very high masses and it would remove the need to interpolate data from existing models as we did e.g. in Eq Cluster simulation In this research we have assumed that the simulated cluster formed instantaneously, i.e. in a starburst. It must be said however that this assumption is not very realistic. In reality, the stars in a cluster are formed in bursts that can be short-lasting, i.e. 5 Myr, or also long-lasting, i.e. 100 Myr, as pointed out by McQuinn et al. (2009). As a result, 11

12 high-mass stars could also lie in the main-sequence band for high cluster ages as they need not have formed at t = 0, as we assumed here. This would make the difference in luminosity between mergers, i.e. potential blue stragglers and massive single stars or unevolved binaries much smaller than suggested in this research. Furthermore, we based the choice of initial parameter ranges in the cluster simulation largely on the initial parameter ranges of the binary grid we used, which were given in Eq. 4. Although we increased the maximum initial period from about 8 to about 10 3 days, in reality larger initial periods of course do occur in binary systems. Also it must be noted that the assumed period distribution is only valid for 2 to about 10 days as shown by Mazeh et al. (2007). According to Goldberg et al. (2003) the distribution does vary from being flat in log P for periods higher than about 10 days. The same objection of limited parameter ranges applies to the initial masses and initial mass ratios. In this research, we could easily extend the maximum initial period because high-initial-period binaries are not expected to merge on the main sequence and hence do not need to be analyzed with help of the binary grid. In other words, they will certainly not become merger stars as defined in Sect This does not apply to the initial mass and initial mass ratio however, as binaries with these parameters outside the grid could very well still merge on the main sequence and hence become merger stars. It is possible however to make a crude assumption for the mass ratios. In the grid used, we considered mass ratios smaller than about 2.2, i.e. only about 50% of all binaries with respect to the mass ratio. As it is reasonable to state from Figs. 5(a) and 5(b) that binaries with these higher mass ratios and sufficiently low periods are also case AR binaries, we expect these binaries also to become contact binaries and thus merge and have properties comparable to the case AR mergers present in the grid. Finally, as we have already mentioned in the introduction, mergers of massive binaries are only one of the plausible explanations for the existence of blue stragglers. Other scenarios, such as accretion blue stragglers (mass transfer in binaries without a merging event) or stellar collisions, may also play a crucial role. To examine the relative importance of these groups, we would also have to model these in the cluster simulation. Then by experimenting with different assumptions for all groups, i.e. changing their relative frequency and comparing the resulting luminosity distribution with observational data, we could in principle obtain more information of the actual relative importance of mergers of massive main-sequence binaries. 6 Conclusion From Sect. 2.2, it has become clear that mass transfer in relatively close main-sequence binaries may lead to contact binary situations which may lead to merger stars. We have identified two types of binaries that may lead to contact binaries on the main sequence: case AR and case AS, which become contact binaries quickly and slowly respectively after formation of the binary. As the result of merging after an appreciable amount of hydrogen has been burnt, case AR and case AS mergers contain significantly lower hydrogen abundances than single stars on the main sequence. Furthermore, AS mergers contain less hydrogen than their AR counterparts. Homology relations then imply that these lower hydrogen abundances, i.e. higher mean molecular weights, lead to more luminous stars. Furthermore, from a simulation of a cluster of stars in Sect. 4, we have shown that these AR and AS mergers are very important if compared with single stars and unevolved binaries. This is because they dominate the brightest regions in the main-sequence band for young clusters, as well as older clusters. For young clusters, we expect that all stars that are on the main sequence with luminosities greater than about 10 5 L (Fig. 10(a)), originate from mergers of close massive binaries. This would imply that all blue stragglers in the HR-diagram of N 11 (Fig. 1), which is an example of a young cluster, are mergers of these binaries. This rather bold statement depends severely on our assumptions made however. One of the main objections is that we have ignored other possible binary interactions that do not lead to a merging event but could produce blue stragglers, e.g. accretion blue stragglers, or explanations that do not even require a binary explanation, such as stellar collisions. Furthermore we have made many other assumptions, such as instantaneous star formation (starburst) which may significantly change our conclusions as well. To finish up, although in this research we have gained more insight in the possible consequences of mergers of massive main-sequence binaries, more effort needs to be made before the results from this research can be compared to empirical data. 7 Acknowledgements I would like to thank my supervisors Onno Pols and Selma de Mink for their excellent guidance in this bachelor research project. I hope that they will be able to make use of the results of this research in the future. Furthermore I would like to thank Joke Claeys for taking me to the Stellar Mergers workshop in Leiden, her assistance in using Window to the Stars and the calculation of nuclear timescales from stellar evolution models. Last but not least, I would like to thank the participants of the stellar evolution and hydrodynamics weekly group meetings who provided valuable comments. 8 References Eggleton, P.P. ApJ, 1983, 268, 368 Evans, C. J., Lennon, D. J., Smartt, S., J. & Trundle, C. 2006, A&A, 456, 635 Goldberg, D., Mazeh, T., Latham, D. 2003, ApJ, 591, 399,

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