We write the logarithmic time derivative of the donor radius as:. We derive here a simple analytic approximation to the effective mass-radius exponent when the response of the donor is a combination of the adiabatic and thermal adjustments to mass loss.
As a consequence of mass loss, the donors radius will differ from the equilibrium radius corresponding to its instantaneous mass. With these definitions we write:. The secular evolution of the binary takes place on the mass transfer time scale. Differentiating Equation 46 with respect to time, we get:. Finally, setting in Equation 47 , and solving for , we obtain:. This expression shows that if the evolution is much slower than the thermal relaxation , the donor radius follows the equilibrium radius closely, whereas if mass transfer occurs rapidly, the donor reacts adiabatically.
Mass transfer will proceed on a time-scale which depends critically on the changes in the radius of the donor star and that of its Roche lobe in response to the mass loss; if the star expands faster than its Roche lobe or shrinks less rapidly than its Roche lobe for a prolonged time, mass transfer will be unstable and the donor star may disintegrate.
If the donor star expands less rapidly or shrinks faster than its Roche lobe, mass transfer will generally be stable and may continue for a long time. We now introduce the concept of the equilibrium mass transfer rate, which is the mass transfer rate that a stable, semi-detached binary undergoes for a given rate of driving.
The equilibrium mass transfer rate is a function of the driving rate, the consequential angular momentum loss mechanisms and the value of the mass ratio q. The mass transfer rate can be written generally as:. Under the assumption that is a function of the binary parameters, whilst f is a strong function of the depth of contact , we can write:. For equilibrium mass transfer, we let , i. Thus, using the evolution equations and Equation 46 we have:. For example, for a polytropic donor with which is representative of a white dwarf donor , and so.
On the other hand, when , we can see that the system cannot reach the equilibrium value and this implies that the mass transfer is unstable.
It is possible during the course of the evolution that a system that initially has can evolve into a system with.
In order to determine exactly when a binary with given total mass and mass ratio q becomes semi-detached, one needs to specify the radii of the component stars and their corresponding Roche lobes. The condition for stability can be expressed as:. The first factor we will consider is the change in the orbital separation resulting from a mass transfer event. The orbital angular momentum, for a binary composed of two point masses is given by Equation 2.
If we logarithmically differentiate Equation 57 with respect to time and from Equation 10 with , we obtain:. From Roche lobe geometry and following [10] , the Roche lobe radius can be crudely approximated as:. A more accurate approximation for the Roche lobe radius was determined by Eggleton and is given by.
Differentiating Equation 60 with respect to time, we obtain:. In the conservative case Equation 65 indicates that if the donor Roche lobe will be contract. Finally, we consider the response of the donors radius to change in its mass. For stars like the sun it is well known that the mass and radius are approximately proportional to one another. Using the expression of 66 , the mass radius relation for solar type stars implies. If we substitute Equation 65 and Equation 67 into Equation 55 we obtain a limiting stable mass ratio for the binary star.
If the mass ratio exceed this values the Roche lobe will be shrink faster than the star can contract and mass. If the star will, on a time scale set by the mass transfer. From Equation 64 any mechanism Schutz that removes orbital angular momentum from the binary will cause the Roche lobe of the donor to contract. Before we proceed upon studying the complete numerical solutions for the evolution of orbital parameters and the evolution equations we derived in Section 2 for different astrophysical scenarios, we obtain analytic solutions for the time evolution of the geometry of mass transfer rate.
We can do this only on the assumption that most of the parameters characterizing the binary remain constant or evolve slowly as compared to the evolution of the mass transfer rate. Analytic solutions are useful in providing physical insights into the expected behaviour of the binary system in the limit where the assumptions imposed to obtain the analytic solutions are valid approximately. To make Equations 60 , 61 and 68 , the two figures are plotted for different values of the mass ratio for comparison in Figure 1 and Figure 2.
The simple form of Equation 60 is a reasonable approximation to the more accurate Eggleton [11] formula for mass ratios. Thus we see that the Roche lobe radius is directly proportional to the separation for a fixed mass ratio.
Given Roche geometry and the structure of the star, we can now specify the mass transfer rate. Figure 1. Comparison of [10] Eq. Figure 2. Change in the orbital angular momentum of the binary system with time. Thus if which implies that the orbit will be enlarge and if that the. This means the radius of a Roche lobe filling star depends only on the binary separation and the mass ratio. The blue curve shows that the stable conservative mass transfer while the green one is the unstable mass transfer.
We here produce analytical solutions for the non conservative mass transfer in close binary system taking the initial masses of the primary and secondary as and.
From Equation 18 , the change of orbital period is proportional to the change in. Prolonged, conservative mass transfer is obtained when the initial mass of donor star is more massive, and the period decreases; hence star move close together. It is worthwhile to describe the binary parameters used by [4] [5] in their pioneering analysis in some detail, since we are using it as a comparison to our numerical solutions in Section 4.
We shall thus obtain analytic as well as numerical solutions for a general polytropic index n to compare our results with those obtained from a full numerical evolution. Figure 3. Figure 4. The total rate of change of the orbital angular frequency of the binary Eq. As we go from top to bottom on the right side; the lines curves are calculated for magenta which corresponds to an initial donor whose central hydrogen abundance is 0. Raising both sides of the Equation 69 to the power and differentiating, we obtain:.
Defining a positive dimensionless mass transfer , and a characteristic time scale , Equation 70 becomes. Note that in the stable case, this value is positive; while it is negative in the unstable case. Before we attempt to solve the above differential equation, it is clear from its form and the signs just discussed, that it describes a stable solution in which when.
Since X diverges for the non driving case in a finite time, the driven case diverges even sooner. Considering Equation 70 again we define , and. Thus, for the stable case , while for the unstable case, and is defined positive.
The ge- neral analytic solution comprising both the stable and the unstable case can be given in terms of the hyper geometric function, as follows:. Type Ia supernovae are of great interest to astronomers in other areas of research. This type of supernova is brighter than supernovae produced by the collapse of a massive star. Thus, type Ia supernova e can be seen at very large distances, and they are found in all types of galaxies. The energy output from most type Ia supernovae is consistent, with little variation in their maximum luminosities, or in how their light output initially increases and then slowly decreases over time.
In contrast, type II supernovae are about 5 times less luminous than type Ia supernovae and are only seen in galaxies that have recent, massive star formation. Type II supernovae are also less consistent in their energy output during the explosion and can have a range a peak luminosity values.
It is possible that, under the right circumstances, a binary system can even survive the explosion of one of its members as a type II supernova. In that case, an ordinary star can eventually share a system with a neutron star. Such infalling gas will be compressed and heated to incredible temperatures.
It will quickly become so hot that it will experience an explosive burst of fusion. The energies involved are so great that we would expect much of the radiation from the burst to emerge as X-rays. If the neutron star and its companion are positioned the right way, a significant amount of material can be transferred to the neutron star and can set it spinning faster as spin energy is also transferred.
The radius of the neutron star would also decrease as more mass was added. Astronomers have found pulsars in binary systems that are spinning at a rate of more than times per second! These are sometimes called millisecond pulsars since the pulses are separated by a few thousandths of a second. Such a rapid spin could not have come from the birth of the neutron star; it must have been externally caused. View this short video to see Dr. Scott Ransom, of the National Radio Astronomy Observatory, explain how millisecond pulsars come about, with some nice animations.
Another binary neutron star system includes two pulsars that are orbiting each other every 2 hours and 25 minutes. As we discussed earlier, pulsars radiate away their energy, and these two pulsars are slowly moving toward one another, such that in about 85 million years, they will actually merge.
However, a high rate of mass overflow onto a compact star from a normal star is always expected when the normal star goes off the main sequence and develops a deep convective envelope. The physical reason for this is that convection tends to make entropy constant along the radius, so the radial structure of convective stellar envelopes is well described by a polytrope i. Removing mass from a star with a negative power of the mass-radius relation increases its radius.
On the other hand, the Roche lobe of the more massive star should shrink in response to the conservative mass exchange between the components. This further increases the mass loss rate from the Roche-lobe filling star leading to a continuation of an unstable mass loss and eventual formation of a common envelope. The common envelope stage is, usually, treated in the following simplified way [ , 72 ].
The orbital evolution of the compact star m inside the envelope of the normal star M 1 is driven by the dynamical friction drag. This leads to a gradual spiral-in process of the compact star. What remains of the normal star M 1 is its stellar core M c. The above energy condition reads. R L is the Roche lobe radius of the normal star that can be approximated as [ 89 ]. From Equation 50 one derives.
The mass M c of a helium core of a massive star may be approximated as [ ]. There are debates in the literature as to should additional sources of energy e. In the case of systems with at least one white-dwarf component one can try to reconstruct the evolution of double compact binaries with known masses of both components, since there is a unique relation between the mass of a white dwarf and the radius of its red giant progenitor. Close binary white dwarfs should definitely result from the spiral-in phase in the common envelope that appears inevitable during the second mass transfer i.
Such an analysis [ ], extended in [ , ], suggests that the standard energy prescription for the treatment of the common envelope stage cannot be applied to the first mass transfer episode. The applicability of this algorithm should be investigated further. Note also, that formulations of the common envelope equation different from Equation 50 are met in the literature see, e. The evolutionary scenario of massive binaries was elaborated shortly after the discovery of binary X-ray sources [ , , ] and is depicted in Figure 4.
Evolutionary scenario for the formation of neutron stars or black holes in close binaries. A massive X-ray binary is an inevitable stage preceding the formation of a double compact system after the second supernova explosion of the helium-rich companion in such a stellar system. In fact, the scenario for binary pulsars was proposed even earlier in [ ], but because no binary pulsars were known at that time, it was suggested that all pairs of NS are disrupted at the second NS formation.
It is convenient to separate the evolution of a massive binary into several stages according to the physical state of the binary components, including phases of mass exchange between them. The simplest evolutionary scenario can be schematically described as follows see Figure 4. Initially, two high-mass OB main-sequence stars are separated and are inside their Roche lobes. Tidal interaction is very effective so that a possible initial eccentricity vanishes before the primary star M 1 fills its Roche lobe.
The expected number of such binaries in the Galaxy is around 10 4. After core hydrogen exhaustion, the primary leaves the main sequence and starts to expand rapidly. When its radius approaches the Roche lobe see Equation 51 , mass transfer onto the secondary, less massive star which still resides on the main sequence begins. While the mass of the primary star reduces, the mass of the secondary star increases, since the mass transfer at this stage is thought to be quasi-conservative.
The secondary star acquires large angular momentum due to the infalling material, so that its outer envelope can be spun up to an angular velocity close to the limiting Kepler orbit value. Such massive rapidly rotating stars are observed as Be-stars. During the conservative stage of mass transfer, the semimajor axis of the orbit first decreases, reaches a minimum when the masses of the binary components become equal to each other, and then increases.
This behavior is dictated by the angular momentum conservation law After the completion of the conservative mass transfer, the initially more massive star becomes less massive than its initially lighter companion. For the typical parameters the duration of the first RLOF is rather short, of the order of 10 4 yr, so only several dozens of such binaries are expected to be in the Galaxy.
The duration of the WR stage is about several 10 5 yr, so the Galactic number of such binaries should be several hundreds. At this stage the disruption of the binary is possible e. Some runaway Galactic OB-stars must have been formed in this way. If the system survives the first SN explosion, a rapidly rotating Be star in pair with a young NS appears. Orbital evolution following the SN explosion is described above by Equations 40— The orbital eccentricity after the SN explosion is high, so enhanced accretion onto the NS occurs at the periastron passages.
The duration of this stage depends on the binary parameters, but in all cases it is limited by the time left for the now more massive secondary to burn hydrogen in its core. An important parameter of NS evolution is the surface magnetic field strength. In binary systems, magnetic field, in combination with NS spin period and accretion rate onto the NS surface, determines the observational manifestation of the neutron star see [ ] for more detail.
Accretion of matter onto the NS can reduce the surface magnetic field and spin-up the NS rotation pulsar recycling [ 37 , , , 36 ]. The secondary expands to engulf the NS. The fate of TZ stars remains unclear see [ 17 ] for the recent study.
Single possibly, massive NS or BH should descend from them. A note should be made concerning the phase when a common envelope engulfs the first-formed NS and the core of the secondary.
Chevalier [ 55 ] suggested that this may be the case for the accretion in common envelopes. An essential caveat is that the accretion in the hyper-Eddington regime may be prevented by the angular momentum of the captured matter.
The magnetic field of the NS may also be a complication. The possibility of hyper-critical accretion still has to be studied. Nevertheless, implications of this hypothesis for different types of relativistic binaries were explored in great detail by H. Bethe and G. Brown and their coauthors see, e.
Even for a symmetric SN explosion the disruption of binaries after the second SN explosion could result in the observed high average velocities of radiopulsars see Section 3. In the surviving close binary NS system, the older NS is expected to have faster rotation velocity and possibly higher mass than the younger one because of the recycling at the preceding accretion stage. The subsequent orbital evolution of such double NS systems is entirely due to GW emission see Section 3.
Detailed studies of possible evolutionary channels which produce merging binary NS can be found in the literature see, e. We emphasize that this scenario applies only to initially massive binaries. A scenario similar to the one presented in Figure 4 may be sketched for them too, with the difference that the secondary component stably transfers mass onto the companion see, e.
This scenario is similar to the one for low- and intermediate-mass binaries considered in Section 7 , with the WD replaced by a NS or a BH. Compact low-mass binaries with NSs may be dynamically formed in dense stellar environments, for example in globular clusters.
The dynamical evolution of binaries in globular clusters is beyond the scope of this review; see [ 26 ] and [ 36 ] for more detail and further references. So far, we have considered the formation of NSs and binaries with NSs. It is believed that very massive stars end up their evolution with the formation of stellar mass black holes.
We will discuss now their formation. In the analysis of BH formation, new important parameters appear. The first one is the threshold mass M cr beginning from which a main-sequence star, after the completion of its nuclear evolution, can collapse into a BH. The upper mass limit for BH formation with the caveat that the role of magnetic-field effects is not considered is, predominantly, a function of stellar-wind mass loss in the core-hydrogen, hydrogen-shell, and core-helium burning stages.
For a specific combination of winds in different evolutionary stages and assumptions on metallicity it is possible to find the types of stellar remnants as a function of initial mass see, for instance [ ]. The recent discovery of the possible magnetar in the young stellar cluster Westerlund 1 [ ] hints to the reality of such a scenario.
Current reassessment of the role of clumping generally results in the reduction of previous mass-loss estimates. Other factors that have to be taken into account in the estimates of the masses of progenitors of BHs are rotation and magnetic fields.
There are various studies as for what the mass of the BH should be see, e. The parameter k BH can vary in a wide range. The third parameter, similar to the case of NS formation, is the possible kick velocity w BH imparted to the newly formed BH see the end of Section 3. In general, one expects that the BH should acquire a smaller kick velocity than a NS, as black holes are more massive than neutron stars.
A possible relation as adopted, e. The allowance for a quite moderate w BH can increases the coalescence rate of binary BH [ ].
The possible kick velocity imparted to newly born black holes makes the orbits of survived systems highly eccentric. It is important to stress that some fraction of such binary BH can retain their large eccentricities up to the late stages of their coalescence. This signature should be reflected in their emitted waveforms and should be modeled in templates.
Asymmetric explosions accompanied by a kick change the space orientation of the orbital angular momentum. As a result, some distribution of the angles between the BH spins and the orbital angular momentum denoted by J will be established [ ]. This means that in these binaries the orbital angular momentum vector is oriented almost oppositely to the black hole spins. This is one more signature of imparted kicks that can be tested observationally.
These effects are also discussed in [ ]. A rough estimate of the formation rate of double compact binaries can be obtained ignoring many details of binary evolution. Initial binary distributions. According to [ ], the present birth rate of binaries in our Galaxy can be written in factorized form as.
An almost flat logarithmic distribution of semimajor axes was also found in [ 4 ]. Constraints from conservative evolution. Equation 56 says that the formation rate of such binaries is about 1 per 50 years. We shall restrict ourselves by considering only close binaries, in which mass transfer onto the secondary is possible.
The mass ratio q should not be very small to make the formation of the second NS possible. This yields. An upper limit for the mass ratio is obtained from the requirement that the binary system remains bound after the sudden mass loss in the second supernova explosion Footnote 5. From Equation 45 we obtain. Of course, this is a very crude upper limit — we have not taken into account the evolution of the binary separation, ignored initial binary eccentricities, non-conservative mass loss, etc.
However, it is not easy to treat all these factors without additional knowledge of numerous details and parameters of binary evolution such as the physical state of the star at the moment of the Roche lobe overflow, the common envelope efficiency, etc. All these factors should decrease the formation rate of double NS. The model-dependent distribution of NS kick velocities provides another strong complication. We also stress that this upper limit was obtained assuming a constant Galactic star-formation rate and normalization of the binary formation by Equation Further semi- analytical investigations of the parameter space of binaries leading to the formation of coalescing binary NSs are still possible but technically very difficult, and we shall not reproduce them here.
The detailed semi-analytical approach to the problem of formation of NSs in binaries and evolution of compact binaries has been developed by Tutukov and Yungelson [ , ]. A distinct approach to the analysis of binary star evolution is based on the population synthesis method — a Monte-Carlo simulation of the evolution of a sample of binaries with different initial parameters.
This approach was first applied to model various observational manifestations of magnetized NSs in massive binary systems [ , , 78 ] and generalized to binary systems of arbitrary mass in [ ] The Scenario Machine code.
To achieve a sufficient statistical significance, such simulations usually involve a large number of binaries, typically of the order of a million. The total number of stars in the Galaxy is still four orders of magnitude larger, so this approach cannot guarantee that rare stages of the binary evolution will be adequately reproduced Footnote 6.
Presently, there are several population synthesis codes used for massive binary system studies, which take into account with different degree of completeness various aspects of binary stellar evolution e.
A review of applications of the population synthesis method to various types of astrophysical sources and further references can be found in [ , ]. Some results of population synthesis calculations of compact binary mergers carried out by different groups are presented in Table 4.
Actually, the authors of the studies mentioned in Table 4 make their simulations for a range of parameters. There are two clear outliers, [ ] and [ ]. The high rate in [ ] is due to the assumption that kicks to nascent neutron stars are absent.
A considerable scatter in the rates of mergers of systems with BH companions is due, mainly, to uncertainties in stellar wind mass loss for the most massive stars. A word of caution should be said here. It is hardly possible to trace a detailed evolution of each binary, so one usually invokes the approximate approach to describe the change of evolutionary stages of the binary components the so-called evolutionary track , their interaction, effects of supernovae, etc.
Thus, fundamental uncertainties of stellar evolution mentioned above are complemented with i uncertainties of the scenario and ii uncertainties in the normalization of the calculations to the real galaxy such as the fraction of binaries among allstars, the star formation history, etc. The intrinsic uncertainties in the population synthesis results for example, in the computed event rates of binary mergers etc.
This should always be born in mind when using the population synthesis calculations. A pedagogical derivation of the signal-to-noise ratio and its discussion for different detectors is given, for example, in Section 8 of the review [ ]. In this section we focus of two particular points: the plausible enhancement of the detection of merging binary black holes with respect to binary neutron stars and the way how absolute detection rates of binary mergings can be calculated.
Coalescing binaries emit gravitational wave signals with a well known time-dependence waveform see Section 3. This allows one to use the technique of matched filtering [ ].
However, the BH mass can be significantly larger than the NS mass. Hence, the registration volume for such bright binaries is significantly larger than the registration volume for relatively weak binaries.
If we assign some characteristic mean chirp mass to different types of double NS and BH systems, the expected ratio of their detection rates by a given detector is. Here, we discuss the ratio of the detection rates, rather than their absolute values. The derivation of absolute values requires detailed evolutionary calculations, as we discussed above. This estimate is, of course, very rough, but it can serve as an indication of what one can expect from detailed calculations.
Now we shall briefly discuss how the detection rates of binary mergings can be calculated for a given gravitational wave detector. This requirement determines the maximum distance from which an event can be detected by a given interferometer. So to assess the merger rate from a large volume based on the Galactic values, the best one can do at present appears to be using formulas like Equation 4 given earlier in Section 2.
This, however, adds another factor two of uncertainty in the estimates. Clearly, a more accurate treatment of the transition from Galactic rates to larger volumes with an account of the galaxy distribution is required. Routine increase of the statistics of binary pulsars, especially with low radio luminosity.
More indirectly, a larger sample of NS parameters in binary pulsars would be useful for constraining the range of parameters of scenarios of formation for double NSs and, hence, a better understanding their origin see, for example, a recent attempt of such an analysis in [ ]. Measurements of its parameters would be crucial for models of formation and evolution of BHs in binary systems in general. Current estimates of the number of such binaries in the Galaxy, obtained by the population synthesis method, range from one per several thousand ordinary pulsars [ , ] to much smaller values of about 0.
Search for unusual observational manifestations of relativistic binaries e. Stellar physics: post-helium burning evolution of massive stars, supernova explosion mechanism, masses of compact stars formed in the collapse, mechanism s of kick velocity imparted to nascent compact remnants neutron stars and black holes , stellar winds from hydrogen-and helium-rich stars.
Binary evolution: treatment of the common envelope stage, magnetic braking for low-mass binaries, observational constraints on the initial distributions of orbital parameters of binary stars masses, semimajor axes, eccentricities.
Binary systems with white dwarf components that are interesting for general relativity and cosmology come in several flavours:. Cataclysmic variables CVs — a class of semidetached binary stars containing a white dwarf and a companion star that is usually a red dwarf or a slightly evolved star, a subgiant.
They appear to be important LISA verification sources. Faulkner et al. The origin of all above mentioned classes of short-period binaries was understood after the notion of common envelopes and the formalism for their treatment were suggested in s see Section 3. We recall, however, that most studies of the formation of compact objects through common envelopes are based on a simple formalism of comparison of binding energy of the envelope with the orbital energy of the binary, thought to be the sole source of energy for the loss of the envelope as described in Section 3.
Though full-scale hydrodynamic calculations of a common-envelope evolution exist, for instance a series of papers by Taam and coauthors published over more than two decades see [ , ] and references therein , the process is still very far from comprehension. We recall also that the stability and timescale of mass-exchange in a binary depends on the mass ratio of components q , the structure of the envelope of Roche-lobe filling star, and possible stabilizing effects of mass and momentum loss from the system [ , , , , , , , 47 ].
Mass loss occurs on a dynamical time scale if the donor has a deep convective envelope or if it is degenerate and conditions for stable mass exchange are not satisfied. It is currently commonly accepted, despite a firm observational proof is lacking, that the distribution of binaries over q is even or rises to small q see Section 5. Formation of compact binaries with WDs. A flowchart schematically presenting the typical scenario for formation of low-mass compact binaries with white-dwarf components and some endpoints of evolution is shown in Figure 5.
Of course, not all possible scenarios are plotted, but only the most probable routes to SNe Ia and systems that may emit potentially detectable gravitational waves. For simplicity, we consider only the most general case when the first RLOF results in the formation of a common envelope.
Formation of close binary dwarfs and their descendants scale and colour-coding are arbitrary. The overwhelming majority of stars overflow their Roche lobes when they have He- or CO-cores. In stars with a mass below 2. If, additionally, the mass-ratio of components is favourable for stable mass transfer a cataclysmic variable may form.
If the WD belongs to the CO-variety and accreted hydrogen burns at the surface of the WD stably, the WD may accumulate enough mass to explode as a type Ia supernova; the same may happen if in the recurrent outbursts less mass is ejected than accreted the so-called SD scenario for SNe Ia originally suggested by Whelan and Iben [ ]; see, e. Some CV systems that burn hydrogen stably or are in the stage of residual hydrogen burning after an outburst may be also observed as supersoft X-ray sources see, e.
The outcome of the evolution of a CV is not completely clear. Matter flowing in from the companion circularises onto unstable orbits. This can efficiently prevent mass being transferred onto the compact object. These endpoints of the evolution of binaries with low-mass donors, were, in fact, never studied.
The second common envelope may form when the companion to the WD overfills its Roche lobe. If the system avoids merger and the donor had a degenerate core, a close binary WD or double-degenerate, DD is formed. The outcome of the contact depends on the chemical composition of the stars and their masses. For total masses lower than M Ch the formation of a single WD is expected.
It is important in this respect that binary white dwarfs at birth have a wide range of separations and merger of them may occur gigayears after formation.
Formation of helium stars via merger may be at least partially responsible for the ultraviolet flux from the giant elliptical galaxies, where all hot stars finished their evolution long ago. This is illustrated by Figure 6 which shows the occurrence rate of mergers of pairs of He-WDs vs. The age of merging pairs of helium WDs. Two components of the distribution correspond to the systems that experienced in the course of formation two or one common envelope episodes, respectively.
Type Ia supernovae. Table 5 summarizes order of magnitude model estimates of the occurrence rate of SNe Ia produced via different channels. For comparison, the rate of SNe Ia from wide binaries symbiotic stars is also given.
The estimates are obtained by a population synthesis code used before in, e. The differences in the assumptions with other population synthesis codes or in the assumed parameters of the models result in numbers that vary by a factor of several; this is the reason for giving only order of magnitude estimates. Both populations have a mass comparable to the mass of the Galactic disk. We also list in the table the types of observed systems associated with a certain channel and the mode of mass transfer.
Table 5 shows that, say, for elliptical galaxies where star formation occurred in a burst, the DD scenario is the only one able to respond to the occurrence of SNe Ia, while in giant disk galaxies with continuing star formation other scenarios may contribute as well.
Currently, it is likely that this problem is resolved see Section 8. The merger of pairs of WDs occurs via an intermediate stage in which the lighter of the two dwarfs transforms into a disc [ , 28 , , ] from which the matter accretes onto the central object.
The latter will collapse without a SN Ia [ ]. The latter authors find that the critical accretion rate for the off-center ignition is hardly changed by the effect of rotation. The problem has to be considered as unsettled until a better understanding of redistribution of angular momentum during the merger process will become available. Because of a long absence of apparent candidates for the DD scenario and its theoretical problems, the SD scenario is often considered as the most promising one.
However, it also encounters severe problems. Even stably burning white dwarfs must have radiatively driven winds. On the other hand, it was noted that the flashes become less violent and more effective accumulation of matter may occur if mass is transferred on a rate close to the thermal one or the dwarf is rapidly rotating [ , , , , , , ].
If the diversity of SNe Ia is associated with the spread of mass of the exploding objects, it would be more easily explained in a SD scenario, since the latter allows white dwarfs to grow efficiently in mass by shell burning, which is stabilized by accretion-induced spin-up.
Hydrogen may be discovered both in very early and late optical spectra of SNe and in radio- and X-ray ranges [ 87 , , ]. Panagia et al. The SD scenario also predicts the existence of many more supersoft X-ray sources than are expected from observations, even considering severe problems in estimating incompleteness of the samples of the latter see for instance [ 82 ].
To summarize, the problem of progenitors of SNe Ia is still unsettled. Large uncertainties in the model parameters involved in the computation of the evolution leading to a SN Ia and in computations of the explosions themselves, do not allow to exclude any type of progenitors. The existence of at least two families of progenitors is suggested by observations see, e. As shown in the flowchart in Figure 5 , there are configurations for which it is expected that stable accretion of He onto a CO-WD occurs: in AM CVn systems in the double-degenerate formation channel and in precursors of AM CVn systems in the helium-star channel.
Thus, until recently, the real identification of these events remained a problem. However, it was shown recently by Yoon and Langer [ ], who considered angular-momentum accretion effects, that the helium envelope is heated efficiently by friction in the differentially rotating spun-up layers.
As a result, helium ignites much earlier and under much less degenerate conditions compared to the corresponding non-rotating case. If the efficiency of energy dissipation is high enough, detonation may be avoided and, instead of a SN, recurrent helium novae may occur. The outburst, typically, happens after accumulation of 0. Currently, there is one object known, identified as He-nova — V Pup [ , , 9 , 10 ].
As we mentioned above, intermediate mass donors, before becoming white dwarfs, pass through the stage of a helium star. For the range of mass-accretion rates expected for these stars, both the conditions for stable and unstable helium burning may be fulfilled.
Ultra-compact X-ray binaries. In progenitors of these systems, the primary becomes a neutron star, while the secondary is not massive enough. Then, several scenarios similar to the scenarios for the systems with the first-formed white dwarf are open.
A low-mass companion to a neutron star may overflow the Roche lobe at the end of the main sequence and become a low-mass He-rich donor. An additional scenario is provided by the formation of a neutron-star component by AIC of an accreting white dwarf. We refer the reader to the pioneering papers [ , , , , , , ] and to more recent studies [ , , , , 25 , , , , , ]. An analysis of the chemical composition of donors in these systems seems to be a promising way for discrimination between systems of different origin [ , , , ]: Helium dwarf donors should display products of H-burning, while He-star descendants should display products of He-burning products.
In globular clusters, UCXBs are formed most probably by dynamical interactions, as first suggested by Fabian et al. The state of interrelations between observations and theoretical interpretations are different for different groups of compact binaries. Such cataclysmic variables as novae stars have been observed for centuries, their lower-amplitude cousins including AM CVn-type stars for decades, and their origin and evolution found their theoretical explanation after the role of common envelopes and gravitational waves radiation and magnetic braking were recognized [ , , , ].
At present, about 2, CVs are known; see the online catalogue by Downes et al. More than of them have measured orbital periods; see the online catalogue by Kolb and Ritter [ ] at [ ]. In particular, there are at present 17 confirmed AM CVn-stars with measured or estimated periods and two more candidate systems; see the lists and references in [ , ] and [ ].
So only a small fraction of binaries can be seen to eclipse. In , John Goodricke found that Algol dimmed periodically to about a third of its normal brightness. So eclipsing binaries also provide an important piece in the puzzle of understanding the attributes of stars.
Formation of Binaries -- wide binaries: tidal capture or conucleation models have been suggested -- close binaries: fragmentation or fission for very close binaries. Close Binaries and Stellar Evolution -- Algol is an interesting system. There are actually more than just two stars, but we'll focus on the close binary the third star is well separated from these two.
Calculations show that the two stars are a hot dwarf and a cool giant. The hot dwarf looks just like a 3. What's wrong with this picture? You need a different evolutionary path. The evolution is affected by mass exchange in the binary.
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