Stellar Evolution in Close Binaries: Processes and Outcomes

Stellar Evolution in Close Binaries: Processes and Outcomes

We discuss some aspects of stellar evolution in binary systems. While single stars can swell following the chemical evolution of their interior, stars belonging to binary systems cannot overflow the size of the Roche lobe and hydrostatic equilibrium is strictly impossible. The system is forced to exchange mass between its members through the inner Lagrangian point. In the first part of the paper, we discuss the standard evolution of binaries that have a non-degenerate donor star and a compact companion. We show that the model fails when to account for the occurrence of binary pulsars when they predict a long-standing mass transfer episode. Models including irradiation feedback and evaporation in close binaries are examined next. Following these sections, we discuss the case of systems with a black hole. We show that if black holes are born non-rotating, binary interaction seems insufficient to speed them up, an indication that black holes' rotation is a feature present at birth. Finally, we discuss Blue Straggler Stars detected in open and globular clusters. Since they cannot be understood as single-born stars, we evaluate one of the proposed channels is mass transfer in close binaries, and discuss its viability and the limitations of the present models.

Since the beginning of the past century, it became clear that stars are stable nuclear fusion cauldrons in hydrostatic equilibrium that play a central role in the production of most elements in Nature. This is one of the most remarkable achievements of Science. Stellar evolution is a direct consequence of the joint action of several physical ingredients, e.g., nuclear reaction rates, radiative and conductive opacities, an equation of state (including partial ionizations, electron degeneracy, Coulomb interactions, etc.), non-adiabatic convection, and wind mass loss, among others. Although the present understanding of these ingredients is still far from complete, it allows us to state that the evolution of single stars is well understood. For a description of these topics, see e.g., Kippenhahn and Weigert. Despite this fact, the modeling of single-star evolution is still not fully predictive. Among the uncertainties that affect the models, perhaps the most serious one is related to the still poor treatment of convection. But it is important to stress that the enormous development of Physics rendered as a bonus and a consistent picture of Stellar Evolution as a prime application field. When the more complex context of binary systems is considered, not surprisingly, Stellar Evolution is not so well understood. The interplay of physical ingredients has to be disentangled, and accurate observations play a major role in modeling and understanding different classes of binaries.

This paper aims to review the research work carried out in our working group GESBI and some general issues. Here, we describe some of the main ingredients that determine Stellar Evolution in binary systems and related astrophysical objects. Firstly, in Section Two, we describe the usual approximations to make binary evolution tractable with simple one-D simulations. Then, we present the case in which one of the components of the system is a neutron star. In Section Three, we describe the usual approximations employed to treat these systems that are responsible for the occurrence of binary millisecond pulsars and X-ray sources. Since these models have severe difficulties in accounting for the existence of some objects, it is necessary to include non-standard ingredients, such as irradiation and evaporation (which are strongly suggested by observations). This is addressed in Section Four. Section Five is devoted to the discussion of the case in which the compact companion is a black hole. The case in which both stars are normal, the low-mass star case is presented in Section Six in connection with the existence of blue straggler stars. Lack of space prevents us from discussing other astrophysical systems related to binary evolution, e.g., cataclysmic variables (in which one of the stars is a white dwarf), and progenitors of supernovae.

Two THE GENERAL CONTEXT OF BINARY EVOLUTION

It is usual to study binary evolution considering the circular, restricted three-body problem, assuming that stellar masses are concentrated in points. This is a very good approximation in this context; see, e.g., Hilditch. The shape of the stars has to be that of the equipotential surfaces. As it is well-known, these surfaces can be described analytically. In a corotating reference frame, there are five Lagrangian points at which the net force on any test particle is zero. Particularly relevant is the one located in between the stars, usually called L one. The equipotential surfaces that surround each star and are connected at L one define the Roche Lobes.

Binary configurations are generally classified into different types. Binaries may be detached if both stars accommodate inside their respective Roche lobes. If only one of them fills its lobe, this is called semi-detached. If both stars fill their lobes, these are referred to as contact binaries. While isolated stars may swell freely, in binary systems there is a limit imposed by the Roche lobes.

The size of the Roche lobes is proportional to the orbital separation A. Usually, in computing binary evolution the departure from strict sphericity is ignored. The radius of a sphere R R I corresponding to star one, with a volume equal to that of the lobe is usually approximated by the expression due to Eggleton

R R L equals A times zero point four nine q to the power of two thirds divided by zero point six q to the power of two thirds plus the natural logarithm of one plus q to the power of one third, zero is less than q is less than infinity.

where q equals M one divided by M two is the mass ratio. The orbital period P or b is related to A by the Third Kepler's Law as P or b equals two pi times A to the power of three divided by the square root of G times the sum of M one and M two to the power of one half where M i, i equals one, two are the masses of the components and G is the Gravitational Constant.

If P sub orbit is long enough, a binary system will always remain detached and each star evolves as if they were isolated. On the contrary, if P sub orbit is short enough, the most massive star evolves faster and will fill its Roche lobe. Since at L sub one, there is no net force capable of balancing the pressure gradient of the sub-photospheric layers, hydrostatic equilibrium is no longer possible and the system undergoes a mass transfer episode. Hereafter, we will refer as donor star to the one that is transferring mass and accretor to its companion. The components of the system may go through configurations unreachable for isolated stars. These systems are usually called Close Binary Systems. We denote the quantities related to the donor star with subindex two meanwhile for the companion star, we employ subindexes one (for normal stars), NS or BH, depending on the context.

This paper will focus on binaries that sometimes reach a semi-detached configuration during their evolution. Of course it is possible that, as a consequence of the mass exchange, the companion star evolves and swells enough to fill its Roche lobe, reaching a contact configuration. The evolution during and after contact is too complex to be treated with one-D simulations, and therefore, this evolutionary stage is beyond the scope of this work and will not be further discussed.

As stated above, understanding stellar evolution in binary systems is essential for accounting for a variety of astrophysical objects. A very important type of Close Binary System is that formed by a normal star together with a NS. NSs are extremely compact objects with radii of tens of kilometers and masses up to approximately two M sub sun. Their interior densities are approximately ten to the power of fifteen grams per cubic centimeter. One of the most remarkable characteristics of magnetized NSs is that they support pulsar phenomenology, at least in zeroth order.

There are two well-known families of eclipsing binary millisecond pulsars in which one of the components is a normal star. Both have P sub arcsec less than or approximately one day. Black widows have very light donor stars, with masses of M sub two much less than zero point one M sub sun whereas Redbacks are around ten times more massive, with zero point one M sub sun less than or approximately M sub two less than or approximately zero point four M sub sun. Because of the association with the names given by the discoverers, they are now usually collectively termed as spider pulsars. It is widely believed that the donor stars first transfer mass to the NS, recycling their spin, and later the fastly spinning NS irradiates and evaporates the donor (see below for further details). This resembles the behaviour of the above-cited black widow spiders that, after mating, the female eats the male, but on a cosmic scale. Australians identified the higher M two group, and gave them the "red back" name corresponding to the Australian spider relative to the former black widow, an American one.

If a binary system including a NS is detached, it may be detected as a binary pulsar (in the sense of having a normal star companion). On the contrary, if it is semi-detached, it will be an X-ray source. If the companion star is a low (high) mass star, it is called Low (High) Mass X-ray Binary, LMXB and HMXB, respectively. Here, we are interested in LMXBs, and the class of spiders related to them.

One of the fundamental problems of binary evolution is related to the reaction of the donor star to the onset of mass transfer. Mass transfer may be a stable or unstable process depending on the ability of the donor to adjust itself to the change of the Roche lobe size due to mass exchange. Usually, stability is favoured if donors have a radiative envelope since they tend to contract. On the contrary, if the envelope is convective, the star swells as it transfers mass, which tends to destabilize the former process. It is also important that in the cases of interest, the donor star is not much more massive than its companion. If this were not the case, the size of the lobe would change too fast for the donor to be able to adjust to the lobe's size. Moreover, if the timescale of mass accretion is shorter than its thermal timescale, the companion star cannot release the gravitational energy fast enough; thus, the system gets in contact. If the mass transfer rate evolves stably, it is possible to compute the evolution of the system. However, if it is unstable, the mass transfer will grow, and the system will enter the common envelope stage. Common envelope evolution is a very difficult issue, beyond the scope of the present paper. Some progress has been recently achieved in this problem.

A mass transfer process may be conservative, in which the mass and angular momentum of the binary remain constant. But non-conservative situations may occur. This mainly determines the evolution of the orbit of the system.

It is usually considered that tidal effects are so efficient that synchronization and circularization are instantaneous, and therefore the orbit remains circular over the entire evolution of the system. Relaxing these simple assumptions makes the numerical models largely increase their complexity.

Usually, and in the simplest form, the mass transfer process is described by two free parameters. Beta is the fraction of mass transferred by the donor, and retained by the companion (zero less than or equal to beta less than or equal to one), and alpha is the specific angular momentum of the matter lost from the system in units of the specific angular momentum of the companion. Often, it is assumed that alpha equals zero point five, and beta equals one; see Podsiadlowski, Rappaport, and Pfahl for further details. A NS can accrete up to a rate known as the Eddington rate M Edd equals two times ten to the negative eight M sub sun per year. The mass transfer rate M is usually expressed as

M two equals negative M sub one exp [(R two minus R RL) divided by H P] (two)

where M sub one is the mass transfer rate for a donor star with radius R two that just fills its Roche lobe, and H P is the pressure scale height at the donor's photosphere.

Another ingredient of key relevance is magnetic braking. Suppose the donor star has a magnetic field attached to a convective envelope and suffers mass loss. In that case, this material corotates with the star along the field lines up to distances of a few stellar radii, slowing down its rotation. Since rotation is coupled to the orbit by tidal effects, the system's orbit is affected by this process. Remarkably, the strength and the functional dependence of braking remain considerably uncertain. The most used, "standard law" has been presented in Verbunt and Zwaan. However, other prescriptions have been suggested by Van, Ivanova, and Heinke and Van and Ivanova. The results presented and discussed below are based on the standard prescription.