Evolution of Low-Mass Helium Stars

Transcription

1 Bonner Zentrum für Lehrerbildung (BZL) Evolution of Low-Mass Helium Stars BACHELORARBEIT im Fach Physik für das Lehramt an Gymnasien und Gesamtschulen angefertigt am Argelander-Institut für Astronomie der Rheinischen Friedrich-Wilhelms-Universität Bonn vorgelegt von Frau Franziska Addy Heusgen betreut durch Priv. Doz. Dr. Thomas Tauris und Zweitgutachter Prof. Dr. Norbert Langer Sommersemester 2016 Bonn, den 12. September 2016

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3 Eigenständigkeitserklärung Ich versichere hiermit, dass die Bachelorarbeit mit dem Titel Evolution of low-mass helium stars von mir selbst und ohne jede unerlaubte Hilfe selbständig angefertigt wurde, dass sie noch an keiner anderen Hochschule zur Prüfung vorgelegen hat und dass sie weder ganz noch in Auszügen veröffentlicht worden ist. Die Stellen der Arbeit - einschließlich Tabellen, Karten, Abbildungen usw. -, die anderen Werken dem Wortlaut oder dem Sinn nach entnommen sind, habe ich in jedem einzelnen Fall kenntlich gemacht. Bonn, den - Unterschrift

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5 Abstract This bachelor thesis presents the evolution of low-mass helium stars with masses of M and metallicities between , as single stars and in a binary system with a 1.35 M neutron star companion. Low-mass helium stars are supposed to be ubiquitous in the Galaxy and, especially as part of a binary system, their evolution can give rise to various observable phenomena, such as supernovae or cataclysmic variable (CV) systems. Using BEC (binary evolution code), we find that carbon is ignited in all stars with a mass bigger than 1.2 M. In our binary systems we are able to find that oxygen-neon-magnesium white dwarf can be the remnants of helium stars with masses bigger than 2.0 M. In addition to that, we are able to present two binary systems resulting in electron-capture supernovae. Furthermore, we find a lower mass limit for which single helium stars experience helium shell flashes. These helium shell flashes are also noticed in several binary systems. We derive a relation between the cooling time and the initial mass of helium stars with masses between M. Additionally, we find that several binary systems evolve to become ultra-compact X-ray binaries. We present the correlations between the initial orbital separations of our binary systems and the duration of mass transfer, the mass accreted by the companion and the final mass of the resulting white dwarf. Binary systems where the helium star cool down without exhausting the central helium completely are found. Finally, we point out the different cases of Roche-lobe overflow and their consequences on the evolution of our helium stars.

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7 CONTENTS CONTENTS Contents 1 Introduction General classification Formation of He stars Evolution of He stars Binary stellar evolution Methods Binary evolution code - BEC Initial parameter setup General physical input and single star parameter setup Binary star setup Analysis Single stars General overview and special features of single He stars Observing He flash in low mass He stars Analysis of the inner structure and evolution of single He stars Observing the effect of different metallicities on the evolution and the inner structure Binary systems General overview Analyzing the different evolution and fates of the binary systems - Case BA RLO Analyzing the different evolution and fates of the binary systems - Case BB RLO Analyzing the different evolution and fates of the binary systems - Case BC RLO Ultra-compact X-ray binary systems Effects of a lower metallicity on binary evolution General trends in binary systems with a low-mass He star Discussion and outlook Discussion Outlook Appendices 35 -III-

8 List of Figures 1.1 HRD including hot subdwarfs positions Evolutionary track of a 2.5 M He star by Habets HRD for a 0.44 M He star Mass-accretion rate of NS in a binary system HRD with different single He stars Kippenhahn diagram of a single He star with M = 2.6 M with Z lower Analysis of the He envelope in stars with and without He shell flash Overview He shell flash t cool for He stars in the mass range of M Kippenhahn diagram of the final years of a single star with M = 1.9 M with Z lower Influence of different metallicities on single He stars Kippenhahn diagram of a single He star with M = 2.6 M with Z solar Kippenhahn diagram of a 0.46 M He star with Z solar and a 0.68 M star with Z lower Main-sequence lifetime of He stars as function of their mass and metallicity HRD of binary systems with a 2.4 M He star HRD of binary systems with a 0.52 M He star Kippenhahn diagrams of a 2.4 M He star with P orb,i = d and 2.2 M He star with P orb,i = d. Both have Z solar Mass-transfer rate of binary systems with a 2.4 M and a 2.2 M He star as function of the donor star s mass HRD of binary systems with a 2.2 M He stars HRD of binary systems with a 1.4 M He star HRD of binary systems with a 1.0 M He star and with a 0.7 M He star Chemical abundance structure of the last model of the 0.7 M He star in a binary system with P orb,i = d and its Kippenhahn diagram Chemical abundance structure of the last model of the 2.4 M He star in a binary system with P orb,i = d evolving Case BBB RLO HRD of binary systems with a 1.8 M and a 0.6 M He star Kippenhahn diagram of a 2.4 M He star with Z solar and with P orb,i = d Mass-transfer rate and orbital period of a binary system with a 1.2, M He star and P orb,i = d as function of the donor star s mass HRD of binary systems with a 2.6 M He star with Z solar and Z lower Chemical abundance structure of the last model of the 2.6 M He star in a binary system with P orb,i = d and its Kippenhahn diagram t mt against P i orb of the binary stars M NS against P i orb of the binary stars Mass of the WD against the P i orb IV

9 CHAPTER 1. INTRODUCTION Chapter 1 Introduction This thesis is about low-mass helium stars with focus on their evolution and their characteristics. Helium stars (He stars) are classified as stars which are burning helium in their cores and furthermore only have a real thin hydrogen envelope left (< 0.02M ) (Han et al. 2002). Astrophysicists are interested in He stars as they are connected to various observational phenomena, e.g. AM CVn stars, supernovae type Ia, Ib and Ic and the UV-upturn of early type galaxies (Tauris et al. 2015; Yungelson 2008; Heber 2009, and references therein). AM CVns are a relatively rare type of binary systems which consist of an accreting white dwarf (WD) and a compact companion star. The donor star is assumed to be a helium WD or a helium star. In opposite to other cataclysmic variables (CV), they don t have hydrogen lines in their spectra. Additionally, they have small orbital periods (Yungelson 2008). The UV-upturn, or UV excess, of early-type galaxies could be caused by hot subdwarf(sd) stars, as their radiation dominates the UV part in galaxies with mainly old red stars (Heber 2009). Supernovae of type Ia are important for cosmology and Galactic evolution. The progenitor of such a phenomena is a WD in a binary system, which grows to the Chandrasekhar mass limit due to mass transfer from its companion. The companion can be a helium star. Therefore, it is of scientific interest to study the evolution of He stars, as singles and their behavior in binary systems. Habets (1986) studied the evolution of He stars with masses between M as single ones and in binary systems up to neon ignition. In a more recent paper from Tauris et al. (2015), the evolution and fate of single and binary He stars of masses from M is studied. A special view on the very low-mass He stars is taken by Schindler et al. (2015), who evolved He stars with masses below 0.5M. The aim of this thesis is to evolve low-mass He stars between M in order to close the existing gap and to get an overview of the general behavior of He stars within the intermediate mass range. As these stars are supposed to play an important part in different binary systems, their evolution in binary systems with a neutron star (NS) companion is also carried out. We used the Binary Evolution Code (BEC) to evolve the single He stars and the binary systems, see (Tauris et al. 2015; Sanyal et al. 2015) for various applications and references of this code. 1.1 General classification There are two groups of observable non-degenerate He stars - He subdwarf and Wolf-Rayet stars, from here on WR stars. Hot subdwarfs are low mass He stars with masses around 0.5M (Han et al. 2002). Their expected mass distribution covers a range of M. They can be divided into sdb and sdo stars. Where B and O are the spectral class of the star. The first ones have hydrogendominated atmospheres, whereas the atmosphere of the sdos is often dominated by helium 1. There exists also a class in-between these two, the sdob stars (Yungelson and Tutukov 2005). The envelops of these subdwarfs are completely different to the ones from traditional subdwarfs. They are too thin to start hydrogen burning, see Heber (2009). Due to that, they evolve directly down the cooling track to become a carbon-oxygen WD (CO WD). The progenitors of these stars had to lose most part of their hydrogen envelope. We will present the different evolutionary scenarios later in this section. Subdwarfs can be found below the upper main sequence in the Hertzsprung-Russel diagram, from here on HRD, see Fig They are faint blue objects, which have been firstly observed 1 There are also sdo stars with a lack of helium. -1-

10 1.2. FORMATION OF HE STARS CHAPTER 1. INTRODUCTION Fig. 1.1: Hertzsprung-Russel diagram with the position of the sdb and sdo stars as well as the extreme horizontal branch (EHB). This Figure is taken from Heber (2009). by Humason and Zwicky in the late 1940s (Heber 2009). One can observe them in all Galactic stellar populations. The more massive He stars, which are observed, are the WR stars. They ve got masses from around 7M upwards (Nugis and Lamers 2000). Nowadays it is assumed, that the mass spectrum of He stars is continuous (Yungelson and Tutukov 2005). 1.2 Formation of He stars There are different ways in which He stars - in this case sdb stars - can be produce. The most important part in the evolution is the loss of the hydrogen envelope. The evolution in a binary system is one of the formation channels which has been studied by Han et al. (2002; 2003). As there has been found a high fraction of short-period sdb binary systems (Heber 2009), binary evolution seems to be very likely in the formation process of sdb stars. Han et al. (2003) presented three different channels for forming sdb stars in binary systems: There are the first and second Common Envelope (CE) ejection channel. In the first one the donor star, which is the more massive star, starts mass transfer during the first giant branch (FGB). After a CE phase, a close binary remains, and if the core of the giant is massive enough to ignite helium, an sdb star is produced. In the second one the companion is already a WD. Furthermore, we have the first and second stable Roche-lobe overflow (RLO) channel. In the first stable RLO channel, one assumes stable mass transfer which leads to a sdb star with a main sequence (MS) star as companion. In the second one, one expects the companion for the accretion to be very massive. -2-

11 CHAPTER 1. INTRODUCTION 1.3. EVOLUTION OF HE STARS Finally, we have the helium WD merger channel. If two WD merge and produce a star massive enough to ignite helium this channel could lead to single sdb stars. However, there are also a lot of suggestions of formation channels for sdb stars with a single star as progenitor. One example, is an enhanced stellar wind taking place close to the tip of the first giant branch (FGB) and thus production of an sdb star. Another one is the hot-flasher scenario, see Heber (2009) and references therein. As they are likely ubiquitous in the Galaxy, it is important to know how such low-mass He stars evolve from the zero-age helium main sequence (ZAHeMS) on. WR stars progenitors are stars with zero-age main sequence (ZAMS) masses greater than 50M. Their evolution to He stars is caused by stellar wind mass lost (Yungelson and Tutukov 2005, and references therein). 1.3 Evolution of He stars Of particular interest for this thesis is the evolution of low-mass He stars. The following information are based on lecture notes from Prof. Norbert Langer see (Pols 2009) based on (Kippenhahn and Weigert 1990). Their evolution depends on the hydrostatic equilibrium, a state of equilibrium where the inward gravitational force and the outward pressure gradient in the star balance each other. Therefore a star has to be hot and thus lose energy by radiation. As the star radiates, it has to contract and gets even hotter. If the star is hot enough in its inner region, nuclear reactions can take place and interrupt the contraction for a certain amount of time. One calls the state of a star, when the loss of energy at the surface is equal to its production by nuclear reactions, the thermal equilibrium state. As soon as the nuclear fuel is exhausted in the interior, the star contracts again and its central temperature rises. The nuclear reactions can lead up to production of iron (Fe) for massive stars. These processes can only take place at well-defined temperatures. There is a minimum core mass for a star to ignite a specific kind of nuclear fuel. The heavier the elements, the more massive the core must be. The evolution of stars can be divided into two categories. If the metal core of the star is lighter than the Chandrasekhar mass, it can become a WD. But if the metal core mass is bigger the star s evolution will end when its core consists of Fe. As Fe fusion requires energy the core will collapse and become either a black hole or a NS. When a star is born it evolves towards the ZAMS where it reaches thermal equilibrium. Similar to the ZAMS of hydrogen stars, one can define a ZAHeMS. The luminosity caused by He burning depends strongly on the core mass. The mass-luminosity relation which describes approximately hydrogen stars is defined by: L 1 κ µ4 M 3 (1.1) in which µ is the mean molecular weight, κ the opacity and M the mass of the star. For ZAHeMS stars one can adjust this relation by changing the value of µ. As noted before, the evolution of He stars (from 2.2 M upwards) up to neon ignition is described in the thesis from Habets (1986). He also took a deeper look at the evolution up to the off-centre ignition of carbon of a 2.0 M He star. The evolution of these stars in an HRD is shown in Figure 1.2. While on the ZAHeMS the dominant reaction is the so called 3α-reaction. Effectively the reaction can be written as: 3α 12 C + γ (1.2) But as there is more and more carbon in the core, the 12 C +α reaction becomes more important. The reaction equation is: 12 C + α 16 O + γ (1.3) -3-

12 1.3. EVOLUTION OF HE STARS CHAPTER 1. INTRODUCTION Fig. 1.2: The evolutionary track of a 2.5 M He star in a Hertzsprung-Russel diagram. The figure is a modified version from (Habets 1986). During this burning phase, Habet noticed a growth of the mass of the core of around 50%, depending on the mass of the He star. The growth of the convective core is contrary to the behavior of the core during hydrogen burning. Like hydrogen burning stars, the radii and luminosities of the stars increase during the central He burning phase. The temperature, however, stays nearly constant. The reason is the strong temperature dependence of the nuclear reactions, which keeps the central temperature nearly constant. This leads to the expansion of the outer layers, so that the radius increases. Simultaneously, the luminosity increases as it s strongly dependent on µ, as one can see in equation 1.1. µ increase as the nearly pure He core converts into a mixture of carbon and oxygen. Before He is exhausted in the core, the central temperature increases in order to maintain the energy production. This is necessary as the number of nuclear reactions decreases. When He is exhausted, the star loses more energy due to radiation than is produced in the core. The star is no longer in thermal equilibrium and the contraction starts again. This heats up the core so that He can ignite in a shell around the carbon-oxygen (CO) core. As the core contracts, the radius of the star starts to increase due to the mirror principle. The core grows in mass and the luminosity increases too, which can be seen in Fig Habets (1986) stars are able to ignite carbon in the center, which interrupts the contraction again. Now carbon burning is the main energy source of the star. The main reactions are 12 C + 12 C 20 Ne + 4 He and 12 C + 12 C 23 Na + 1 H. At point D in Fig. 1.2, carbon shell burning takes place and produces the energy which is necessary to maintain the luminosity from the prior phase. When the energy production diminishes, the core contracts again and the temperature rises once more. A second carbon burning layer arise so that contraction is discontinued again. This behavior occurs multiple times throughout the evolution. For the stars more massive than 2.2 M, Ne can be ignited non-violently because of their weakly degenerated oxygen-neon-magnesium (ONeMg) cores. -4-

13 CHAPTER 1. INTRODUCTION 1.4. BINARY STELLAR EVOLUTION Tauris et al. (2015) analyzed the progenitor evolution of helium stars that lead to ultra-stripped supernovae. Therefore, they evolved tight binary systems composed of a NS and a He star. The He star s mass range was chosen from M. For He stars with masses between M binary evolution lead to electron-capture supernovae (EC SNe). Furthermore, only stars which do not ignite oxygen, and which have a core mass greater than 1.37 M result in an EC SN. For more massive stars in such systems, normally the fate is an iron core-collapse SN (Fe CCSN). The approximated mass limit for the metal core to undergo an Fe CCSN is M core,f = 1.43 M. The fate of the lighter stars is to become an ONeMg or a CO WD. Tauris et al. (2015) also evolved these stars as singles. Here the 2.5 M stars evolve to ONeMg WDs and the stars with masses of 2.6 and 2.7 M undergo a EC SNe. All more massive single stars end in Fe CCSN. Low-mass He stars become red giants, so their radii increase to a great extent after the main sequence (Tauris and van den Heuvel 2006). This can lead to mass transfer in wider binary systems. A recent paper focusing on sdb stars is from Schindler et al. (2015). They analyzed the internal structure of the modeled stars and compared these to asteroseismological observations. They evolved their He stars out of hydrogen stars in binary systems. Their He stars have masses between M. For these sdb stars they found lifetimes of about Myr. Additionally, they had a look at the influence of the overshooting parameter on the lifetime and the convective core mass of the star. Analyzing the lifetimes of stars resulting from different diffusion models, they found variations from Myr. 1.4 Binary stellar evolution The evolution of stars in binary systems is based on parameters as e.g. the separation of the stars and their stellar masses. One has to understand the evolution of the single stars in order to understand how two stars interact in a binary system. As mentioned earlier, the evolution of one star up to the red giant branch can cause a phase of mass transfer in the late evolution. Important for the fate of the binary is the so called Roche-lobe (Tauris and van den Heuvel 2006). It is defined by the gravitational potential in the rotating system, more precisely by the equipotential surface which crosses at the first Lagrangian point, L1. The two stars are classified as a donor and an accretor. A donor star loses mass that can be accreted onto the accretor. If the donor star fills its Roche-lobe during its evolution, mass transfer occurs. It is called Rochelobe overflow, RLO. As the effective gravitational potential depends strongly on the masses and the centrifugal force between the two stars, the radius of the Roche-lobe also depends on the mass ratio, which is defined as q = M donor /M accretor. Furthermore, it is also a function of the orbital separation a. Hence, the mass transfer can be caused by an increasing radius due to the evolutionary state of the donor star, or it can be caused by a shrinking orbital separation. When RLO has started, the two stars separate again after the donor star does no longer fill its Roche-lobe. In general, the donor star looses over 70% of its total mass (Tauris and van den Heuvel 2006). To describe the different phases of RLO, Kippenhan and Weigert introduced three types of RLO (Tauris and van den Heuvel 2006). The first case, Case A, occurs during the core hydrogen burning phase of the donor star. The second case, Case B, describes RLO after the main sequence but before the star ignites helium. And the last case, Case C, refers to RLO during or after helium shell burning. Regarding of He stars, one talks of Case BA, BB or BC RLO. Then we talk about He core burning, He shell burning and carbon burning respectively. -5-

14 1.4. BINARY STELLAR EVOLUTION CHAPTER 1. INTRODUCTION The angular momentum of a binary system can change not only because of mass loss, which is usually the dominant part, but also by gravitational-wave radiation (Tauris and van den Heuvel 2006, and references therein). In a binary consisting of two stars with different masses, the asymmetric distribution of the mass related to the rotation axis leads to a quadrupol momentum which varies in time. This variation causes the emission of gravitational waves. Especially in tight systems, this radiation leads to a shrinking orbital separation as the system loses angular momentum. This is of particular interest for CVs and low-mass X-ray binaries with small orbital periods, where gravitational wave radiation has an observable influence. Also stellar wind-mass loss can influence the orbital separation of stars, especially if they re in a wide binary system. The consequence of the mass loss is an increasing orbital period. During the mass transfer, the accretion efficiency of the accretor, is not 100%. Mass is lost from the donor in form of direct fast winds or e.g. as mass ejection from the vicinity of the accretor. This process disturbs the donor star s hydrostatic and thermal equilibrium. It needs to get back into equilibrium and hence the radius has to either increase or decrease. During the mass transfer, the Roche-lobe radius also varies as the angular momentum of the system is modified. It increases when the donor star is less massive than the accretor, hence the orbit expands. If the mass ratio is the other way round, the Roche-lobe decreases and the orbit shrinks. But due to the mass, which is lost by the ejection in the environment of the accretor - given by the parameter β - the orbit can expand, even if he donor star is a bit heavier than the accretor, for 1 < q < The stability of the mass transfer depends on the reactions of the donor star and the Roche-lobe (Tauris et al. 2011). If the accretor of a binary system is a black hole or a NS, and if there is an epoch of mass transfer, one can observe this system as an X-ray binary (Tauris et al. 2012). Most of the observed strong Galactic X-ray sources belong to either low-mass X-ray binaries (LMXBs) or high-mass X-ray binaries (HMXBs). A long accretion phase onto a pulsar can result in a millisecond pulsar. The NS spins up and one speaks of recycled pulsars. Observed X-ray binary systems show mass accretion rates for the neutron star of about M yr 1 (Tauris and van den Heuvel 2006). This is below the Eddington accretion limit, Ṁ Edd M yr 1. The accurate value depends on the composition of the accreted material. If the rate is higher than the limit, the excess can t be accreted by the NS. The excess matter accumulates and forms a cloud which is optically thick to X-rays. Low-mass He stars can be a companion of a NS and hence could cause such X-ray binaries and recycled pulsars, which is another reason to study them in more depth. -6-

15 CHAPTER 2. METHODS Chapter 2 Methods In this thesis, the modeled He stars were created and evolved with the Binary Evolution Code BEC, see Section 2.1. The calculation in this work, handling and visualization of the data was carried out with Python and its packages SciPy (Jones et al ), NumPy (Van Der Walt et al. 2011) and matplotlib (Hunter 2007). Besides that, we used IDL to create the Kippenhan diagrams shown in this thesis. 2.1 Binary evolution code - BEC First of all, we want to explain the creation and evolution of the He star models. For the stellar evolution calculation of single stars and binary systems we used BEC. Originally the code was developed by Wellstein et al. (2001), which is based on a single-star code (Langer 1998; Braun 1997, and references therein). It makes usage of a one-dimensional implicit Lagrangian code that solves the hydrodynamic stellar structure and evolution equations. For binary evolution, the donor star s mass-transfer rate and the orbital separation are computed simultaneously (Wellstein and Langer 1999). It uses the Roche approximation in the formulation of Eggleton (1983). The stellar model s nuclear network for this code was developed by Heger et al. (2000). For more detailed information see e.g. Tauris et al. (2015) and Yoon et al. (2010) and references therein. 2.2 Initial parameter setup Handling the parameters of BEC is of utter importance to create scientific usable data. Finding the right parameter setup has been very time consuming and thus we explain it in more detail. Although BEC offers a variety of parameters, the most important ones for the He stars in this thesis are presented in the following section General physical input and single star parameter setup To close the gap in the existing modeled He star mass range we evolved stars with masses from M, as single stars and most of them also as a donor star in a binary system with a neutron star companion. The models were calculated with a mixing-length parameter of α = l/h p = 2.0. We assumed a core convective overshooting parameter of δ ov = The mass loss due to a stellar wind can be neglected for most of the stars, especially in binary systems. Nevertheless, all stars are calculated including stellar wind-mass loss. For the lower mass stars, He is burned at relatively low temperatures. This resulted in wrong calculations of the timestep termination. Most of the central helium was burned within an insufficient number of calculated models. The parameter which controls the timestep termination by the change of central He abundances, did not respond as expected as the inner temperature of these stars was too low. Hence, the temperature criterion in the timestep subroutine of the BEC had to be adjusted. The temperature threshold had to be changed from K down to K. This is from now on called the new timestep subroutine. We controlled whether the threshold was correct and the termination of the timestep due to abundance changes of He worked well. Therefore, we calculated stars with the new timestep subroutine and we created two other setups. In the second setup, we restricted the maximum timestep to low values, so that we were able to control that He wasn t burned too fast. Finally, -7-

16 2.2. INITIAL PARAMETER SETUP CHAPTER 2. METHODS R 1 R 2.0 log(l/l ) R ECHEB = 0.02 ECHEB = log(t eff /K) ECHEB = 0.02, t max red t max red Fig. 2.1: HRD for a 0.44 M He star. The evolutionary tracks of stars with four different setups are shown. The different parameters are explained in the text. The circle indicates the ZAHeMS. the third setup was a combination of restricting the maximum timestep and applying the new timestep subroutine. The parameter which controls the amount of helium burned in one timestep, ECHEB, was set to 0.02 for all three setups. We found out that for the new threshold all three calculations results in similar lifetimes. For our 0.6 M star, the maximum discrepancy was 3.91% with the new timestep subroutine. Furthermore, it was only 0.2% for the 1.2 M star. Besides that test, we also made another calculation to see whether the code correctly computes the He usage caused by the burning process. Therefore, we firstly worked out the amount of helium, carbon and oxygen before and after a certain timestep. For that, we evolved a special model which only had two nuclear reactions turned on: the 3α and the α + 12 C. We calculated the number of reactions for both processes. With this information we were able to determine the energy released based on the two reactions. As a crosscheck for our computed values, we used the luminosity calculated by the BEC. With it we calculated the time it takes to produce the amount of energy, we gained before. For a 1.2 M He star the calculated timestep and the timestep given by the code are in agreement. The small differences between the code and our calculations varied between 1.4% and 8.7%. For the lightest models ( M < 0.6 M ), we obtained bigger differences in the lifetime than for heavier stars. The lifetimes of the stars with our new timestep routine were always bigger than the ones from the other two setups. We tried to solve the problem by setting the threshold even lower without success. In another attempt, we changed the He abundance parameter to lower values, namely we set ECHEB = and ECHEB = The difference between the and was negligible and so we decided to set ECHEB = For the stars up to 0.46 M - the evolutionary tracks in the HRD are similar. For least massive stars, one can see in Fig. 2.1 that the models start doing loops due to shell flashes for some setups. Thus their evolution is very dependent on the initial parameter setup. The importance of the initial parameter choice on the evolution and especially on the lifetime was -8-

17 CHAPTER 2. METHODS 2.2. INITIAL PARAMETER SETUP also observed by Schindler et al. (2015). He studied the interplay of different opacities, atomic diffusion and overshooting parameters for a 0.48 M star. It resulted in different behavior and also differences in the lifetime (mostly between Myr). So he also concluded, that the lifetimes are quite influenced by the initial input parameters, which agrees with our testings. Due to this, we decided to evolve stars from M with the ECHEB parameter set to As the evolution code takes less time with a higher value, we chose 0.02 for all stars from 0.6 M upwards. With these setups, we evolved stars with metallicities in the range of Z = Binary star setup We evolved the same stars in binary systems with a 1.35 M NS as companion, which is treated as a point mass. To get a good overview of the evolution of low-mass He stars in binary systems, we calculated the evolution of He stars in a mass range of M, with the same parameter setups as described above. Furthermore, we chose two different metallicities: Z = 0.02 or Z = In order to find the range of initial orbital periods, we calculated the biggest and smallest Roche-lobe radius that can be filled by the He star. Due to that, we looked up the maximum and minimum radius, R max He and R min He, of the donor stars. If we want the star to fill its Rochelobe, and starting mass transfer that way, its radius R He should be equal to the Roche-lobe radius R L. The Roche-lobe radius is a function of the orbital separation, a, and the masses of the stars of the system. For the calculation we use Eggleton (1983) approximation given as: R l a = 0.49q 2/3 0.6q 2. (2.1) /3 + ln 1 + q1/3 In this equation q is the mass ratio. Hence, we calculated with this equation the minimum and maximum initial orbital separation by setting R L equal to R max He and R min He. We chose a few orbital separations in between these two to get a general view on the behavior of the helium stars in binary systems. However, this treatment didn t work for the lightest stars. As mentioned earlier, the orbital angular momentum of a binary system can shrink due to gravitational-wave radiation. This causes the orbit to shrink. As the smallest stars radii do not grow much during their evolution, all orbits are tight, and so gravitational-wave radiation always leads to Case BA RLO. Hence, we had to choose larger initial orbital separations than calculated by R L = Rmax. He As described in Section 1.4 the NS is limited in accreting material in a certain time. If the calculated mass-transfer rate is higher than the Eddington accretion limit, the NS can t accrete more mass. In these cases, we assumed that only 30% is accreted onto the NS. Additionally we assumed that 15% of the gravitational rest mass is binding energy. The results on the mass accretion onto the NS of these calculations are shown in Fig.2.2. The light blue line indicates that 100% is accreted by the NS star. The darker blue line correspond to an accretion of 30% and the black ones to the case including binding energy calculation. The star is a 1.2 M with three different initial orbital separations. We can see that the accretion rates for the closest orbits lay parallel to the black line. Hence, this is the maximum mass-accretion rate for our models. -9-

18 2.2. INITIAL PARAMETER SETUP CHAPTER 2. METHODS 1.48 E gravbind 1.46 Mass of the NS/M % 30% d d d Mass of the primary/m Fig. 2.2: Mass of the accretor, a NS with an initial mass of 1.35 M, against the mass of the donor star, a 1.2 M He star with different initial orbital separations, colored lines. The dotted blue and black lines denote the different assumptions on mass accretion on the NS as explained in the text. -10-

19 CHAPTER 3. ANALYSIS Chapter 3 Analysis In this chapter, we explain the results of our calculated models. First we analyze our single stars with their different masses, metallicities and overshooting parameters. Afterwards in the second part of this analysis, we investigate the binary systems regarding the different initial orbital separations, the influence of the metallicity and their fate. 3.1 Single stars We will start with a general overview of the evolutionary tracks in the HRD. Afterwards, we will present several characteristics of these stars General overview and special features of single He stars In Figure 3.1 we plotted a selection of our single stars in an HRD. The displayed He stars cover the general evolution of all developed stars with a metallicity of Z = This is solar metallicity, Z solar. For a better overview, we also plotted the lines of constant radii (dashed, grey lines). All stars begin at the ZAHeMS. During the core-helium burning phase, they all behave similar: they burn He and their convective cores grow during that time. An example of the evolution of the inner structure of a 2.6 M star with Z = 0.001, from now on called Z lower, can be seen in a Kippenhahn diagram in Fig It is equal to the stars with solar metallicity. First the radius increases, but as soon as He is exhausted, the radius shrinks. Simultaneously to this contraction, the star gets hotter and so He is ignited in a shell, marked by an arrow in Fig This causes the radius to increase again. Also the luminosity increases for all evolved stars, as expected. For all stars heavier than 0.8 M the code crashes before the stars reach their final fate. A more detailed view on the inner evolution can be read in section These stars evolve to red giants, which makes them interesting concerning their behavior in a binary system. Furthermore, we were able to see stars making loops in the evolutionary tracks, e.g. the 0.8 M mass star in Fig This is caused by a He shell flash, see Next, we had a look at the effect of different metallicities on the evolution of our stars. The tracks of stars with Z lower can also be found in Fig. 3.1, as dashed colored lines. One can see that their tracks differ at some points to the stars with Z solar. The ZAHeMS of the stars with lower metallicity is shifted to higher temperatures but also smaller radii. A more detailed analysis on the different impacts of the metallicity can be found in section Observing He flash in low mass He stars As one can see in Fig. 3.1, the 0.8 M mass star undergoes a loop in the HRD. We assume that this loop is a He shell flash occurring in the evolution of stars more massive than 0.6 M. To analyze this further, we calculated the mass of the He envelope of the star (M He,env ) before and after this loop. We chose the time in the evolution of the star where its temperature is at its maximum for the first time and then we looked at M He,env when log (L He /L ) = 3 is fulfilled. The data is shown in Fig Here we can see that the mass of He in the envelope has decreased more for the 0.8 M star in comparison to a non-looping star as e.g. 0.6 M. Additionally, we compared the Kippenhahn diagrams of these two stars. The 0.8 M He star shows a higher energy-production rate than the lower mass star with no flash. In order to find the mass limit for flash occurrence, we developed more stars in smaller steps -11-

20 3.1. SINGLE STARS CHAPTER 3. ANALYSIS R 1 R 10 R 100 R 4 log(l/l ) R M 0.8 M 2.0 M 0.52 M 0.9 M 2.6 M M 1.5 M log(t eff /K) Fig. 3.1: HRD with an overview of the simulated single stars for M He = ( ) M. The solid lines belong to the stars with Z solar, the dashed lines are related to the same mass but with Z lower. The circle indicates the ZAHeMS and the arrow the onset of He shell burning, both for our 0.46 M He star. Fig. 3.2: Kippenhahn diagram of a single He star with M = 2.6 M and Z lower. It shows the cross section of the star in mass coordinates from the inner to the outer part of the star on the y-axis. Along the x-axis we ve got the age of the star, with (t t)/yr giving the remaining time of our calculation. The intensity of the blue, respectively the purple color is related to the net energy production rate. Convection is indicated by green hatched zones, semi-convection is marked by the red colored regions. -12-

21 CHAPTER 3. ANALYSIS 3.1. SINGLE STARS MHe,env/M Z = 0.02 Z = M init /M Fig. 3.3: Helium envelope mass when the star reaches its maximum temperature for the first time. The filled stars indicate stars with Z solar and filled circles indicate stars with Z lower. The open symbols denote the helium envelope mass when log (L He /L ) = 3. The grey shaded area indicates the occurrence of a He shell flash for the stars with Z solar. The limit for the low-metallicity stars is given in the text. between M. In addition to that, we evolved more stars between M to find the mass for which the code can t calculate the evolution up to the WD cooling track. The result of this more detailed survey is presented in Fig We also developed the stars with the same mass but with Z lower to see whether this has an effect on the shown behavior. They re plotted in Fig. 3.4 as well. As shown in Fig. 3.4, for solar metallicity the lightest star having a helium shell flash has a mass of 0.68 M. The evolution of the 0.66 M He star crashes during the He shell flash. For stars with Z lower the mass limit for evolving directly in a white dwarf seems to be 0.66 M. In addition to the calculation above, we also analyzed the mass change of the He envelope for all these stars as shown in Fig We can see that the more massive a star is, the less massive is its He envelope. This figure points out that the stars lose much more He due to the flash compared to the stars without flash. Especially the lightest stars undergoing a He flash lose more He than the more massive ones. Last but not least, the mass of the He envelope is lighter for stars with Z solar. For a further analysis, we calculated the time the stars need to cool down. This approach is comparable to the inspection of hydrogen stars with low metallicity during hydrogen shell flashes by Istrate et al. (2016). We chose the same points as described above for the mass of the He envelope and determined the time, t cool, it took the star to cool down. The results are shown in Fig As one can see, the cooling age time seems to be a linear function of the mass of the star. Hence, we fitted a linear function to these data points and found the following relations: t Z=0.02 cool = ( ± 0.266) 10 8 M/M + (2.031 ± 0.176) 10 8 yr (3.1) t Z=0.001 cool = ( ± 0.244) 10 8 M/M + (1.865 ± 0.165) 10 8 yr (3.2) In addition to that, we were able to infer the mass limit for which a star could be evolved up -13-

22 3.1. SINGLE STARS CHAPTER 3. ANALYSIS 0.1 R 1 R 10 R 100 R R log(l/l ) M 0.66 M 0.72 M 0.8 M 0.84 M 0.86 M 0.68 M log(t eff /K) Fig. 3.4: HRD with He stars in the mass range of M. The solid lines belong to stars with Z solar, the dashed lines are related to the same mass but with Z lower tcool/yr Z = Z = Linear Fit Linear Fit M init /M Fig. 3.5: t cool for stars in the mass range of M. The red symbols indicates stars with Z solar, the blue ones stars with Z lower. The calculated linear fits are also shown in the corresponding color. The grey shaded area indicates the occurrence of a helium shell flash for the stars with solar metallicity. to the cooling track. In Fig. 3.3 we can see that the 0.86 M star is the first one not evolving to a WD in the HRD. -14-

23 CHAPTER 3. ANALYSIS 3.1. SINGLE STARS Fig. 3.6: Kippenhahn diagram of the final years of a single He star with M = 1.9 M with Z lower. Off-centre carbon ignition is seen at t = 3.85 Myr. See Fig. 3.2 for further explanations Analysis of the inner structure and evolution of single He stars For our massive He star with 2.6 M and Z lower we see the ignition of carbon in its center. Later in the evolution, we can investigate carbon shell burning phases in these massive stars, too. One can see the inner structure of the 2.6 M star in Fig The plot shows the cross-section of the He star as function of the remaining lifetime. The blue/purple regions indicate the net energy production and the green, respectively the red zones denotes convection/semi-convetion. For He stars with initial masses M 2.2 M and M 1.9 M, for Z solar and Z lower, respectively, the stellar code crashed prior carbon ignition. We tried to find the mass limit for carbon to be ignited in stars with Z lower as stars with Z solar do not evolve so far. We recognized that the lightest star for which we could observe carbon burning, even though it is ignited in one of the last models - i.e. we also have a small nuclear reaction rate - is the 1.2 M at the age of yr. The mass of the metal core in the final model, shortly after carbon is ignited, is M. This is consistent with the known mass limit of 1.0 M for a stars core mass to ignite carbon. So we can assume that this star and stars with higher masses become an ONeMg WD instead of a CO WD. The lightest star for which we can see this nuclear reaction in the Kippenhahn diagram is the 1.9 M one. We can also see phases of carbon shell burning occurring in this star. In Fig. 3.6 one can also see another effect occurring during the evolution of the low-mass He stars: the stars show convective envelopes during He shell burning and carbon burning. It is observable for all stars with M 0.9 M. The more massive the initial mass of the star, the larger is the convective zone. Additionally, the He burning shell moves outwards close to the surface of the star. Doing so, the convective envelope seems to link the surface with this shell, see e.g. the last convective zone in Fig The convective envelope becomes larger as the mass increases for the stars lighter than 1.9 M. Besides that, we can see in Fig. 3.6 that for this star the convective envelope appears and disappears alternating. This behavior can also be seen for the 2.0 and the 2.3 M star. The more massive stars have no or only a small convection zone in their envelope, e.g. see Fig

24 3.1. SINGLE STARS CHAPTER 3. ANALYSIS In addition to that, the described behavior of the He burning shell reaching close to the surface of the star can also be seen for the stars lighter than 0.9 M. They don t develop deep convective envelopes, but their helium burning zone moves close to the surface. For the stars having a He flash, we can observe at the closest point to the surface a small convective zone in the He burning shell. Furthermore, the stars with a He flash show a much higher net energy production in the shell. If we compare the maximum of the last star not having a He flash with the first one having one, we noticed a difference: the maximum net energy production of the 0.64 M star is around 10 5 erg/g/s whereas the one from the 0.68 M star is about 10 9 erg/g/s. Only the star with a mass of 0.6 M has such a high net energy production without undergoing a He shell flash. During the He core burning phase, the mass of the convective cores of all He stars increase. One can observe a bigger growth for the lighter stars. E.g. the core of the 2.6 M He star just rises from to M (a growth of 16.98%), the one s from the 0.8 M He star starts with a core mass of M and it rises up to M i.e. a growth of 75% Observing the effect of different metallicities on the evolution and the inner structure Next, we analyzed the effect of the metallicity on the evolution of He star in more depth. In Fig. 3.7 we ve plotted the evolutionary track of three different stars in a HRD. The tracks of the four different metallcities are indicated by different line-types explained in the caption of the figure. The lower the metallcity of the star the hotter they are. Additionally the maximum radii of the stars with a lower metallicity tend to be smaller. Furthermore, the stars with lower metallicities show higher luminosities. Besides that, some of the lower metallicity stars (e.g. the 1.5 M star in Fig. 3.7) show a steep increase of their luminosity at the end of their evolution. A detailed look at the He flash of the low mass stars in Fig. 3.4 shows that the mass limit for the flash moves to higher masses, when the metallicity gets lower. The flash itself seems not to be affected by the metallicity. Nevertheless, we can see that the stars with higher metallicity become colder during the helium flash than the ones with smaller metallicity. For a better understanding of some of these evolutions, we took a look at the inner structure of these stars. If we compare the evolution of the 2.6 M star with Z solar, see Fig. 3.8, with the one with Z lower, Fig. 3.2, we can notice that with Z lower the star has started carbon shell burning while for the other one the code crashed during core carbon burning. So a lower metallicity leads to a further evolution, which is one reason for the different tracks of the stars. The internal abundances of the last models of these two stars underline this difference. In the star with lower metallicity, more neon is produced as result of the carbon burning and there is almost no carbon left in the core region. Furthermore, we observe that the mass loss due to winds is bigger for the stars with an higher metallicity. As the mass loss is proportional to the opacity, which on the other hand depends on the metallicity, one expects that a higher metallicity causes a higher mass loss by winds. In addition to that, the more massive stars loses more mass too. Nevertheless, one can also observe wind-mass loss in stars with lower metallicity. Note, we can t observe mass loss in stars with Z solar as they aren t evolving far enough. Besides its proportionality to the opacity, the wind-mass loss depends also on the luminosity. Lower metallicity stars evolve further in BEC and have higher energy-production rates compared to the higher metallicity stars. This leads to more wind-mass loss in the intermediate-mass stars compared to the similar massive ones with higher metallicity. The higher the initial mass of the star, the less efficient is this effect. In addition to these effects, the metallicity also affects the shape of the convective cores of the lower mass stars. In Fig. 3.9 we plotted the Kippenhahn diagrams of a 0.46 M He star with Z solar and a 0.68 M He star with Z lower. One can see that their convective cores aren t -16-