Basic Principles of the Evolution of Binary Stars

3 Basic Principles of the Evolution of Binary Stars

Beautiful early general reviews of the topic can be found, e.g., in [45 , 795 *] and more later ones, e.g., in [755 , 171 , 799 ]. Here we restrict ourselves to recalling several facts concerning binary evolution that are most relevant to the formation and evolution of compact binaries. We will not discuss possible dynamical effects on binary evolution (like the Kozai–Lidov mechanism of eccentricity change in hierarchical triple systems, see e.g., [693 ]). Readers with experience in the field can skip this section.

3.1 Keplerian binary system and radiation back reaction

We start with some basic facts about Keplerian motion in a binary system and the simplest case of the evolution of two point masses due to gravitational radiation losses. The stars are highly condensed objects, so their treatment as point masses is usually adequate for the description of their interaction in the binary. Furthermore, Newtonian gravitation theory is sufficient for this purpose as long as the orbital velocities are small compared to the speed of light $c$ . The systematic change of the orbit caused by the emission of gravitational waves will be considered in a separate paragraph below.

3.1.1 Keplerian motion

Let us consider two point masses $M1$ and $M2$ orbiting each other under the force of gravity. It is well known (see [404 ]) that this problem is equivalent to the problem of a single body with mass $μ$ moving in an external gravitational potential. The value of the external potential is determined by the total mass of the system

The reduced mass $μ$ is

The body $μ$ moves in an elliptic orbit with eccentricity $e$ and major semi-axis $a$ . The orbital period $P$ and orbital frequency $Ω = 2π ∕P$ are related to $M$ and $a$ by Kepler’s third law

This relationship is valid for any eccentricity $e$ .

Individual bodies $M1$ and $M2$ move around the barycenter of the system in elliptic orbits with the same eccentricity $e$ . The major semi-axes $ai$ of the two ellipses are inversely proportional to the masses

and satisfy the relationship $a = a1 + a2$ . The position vectors of the bodies from the system’s barycenter are $⃗r1 = M2 ⃗r∕(M1 + M2 )$ and $⃗r2 = − M1 ⃗r∕(M1 + M2 )$ , where $⃗r = ⃗r1 − ⃗r2$ is the relative position vector. Therefore, the velocities of the bodies with respect to the system’s barycentre are related by

and the relative velocity is $⃗V = ⃗V1 − ⃗V2$ .

The total conserved energy of the binary system is

where $r$ is the distance between the bodies. The orbital angular momentum vector is perpendicular to the orbital plane and can be written as

The absolute value of the orbital angular momentum is

For circular binaries with $e = 0$ the distance between orbiting bodies does not depend on time,

r(t,e = 0) = a,

and is usually referred to as orbital separation. In this case, the velocities of the bodies, as well as their relative velocity, are also time-independent,

and the orbital angular momentum becomes

3.1.2 Gravitational radiation from a binary

The plane of the orbit is determined by the orbital angular momentum vector $⃗Jorb$ . The line of sight is defined by a unit vector $⃗n$ . The binary inclination angle $i$ is defined by the relation $⃗ cos i = (⃗n, Jorb∕Jorb)$ such that $∘ i = 90$ corresponds to a system visible edge-on.

Let us start from two point masses $M1$ and $M2$ in a circular orbit. In the quadrupole approximation [405 *], the two polarization amplitudes of GWs at a distance $r$ from the source are given by

G5∕31 2 2∕3 h+ = -c4-r-2(1 + cos i)(πf M ) μ cos(2πf t), (18 ) 5∕3 h = ± G---1-4cos i(πf M )2∕3μ sin(2πf t). (19 ) × c4 r

Here $f = Ω∕ π$ is the frequency of the emitted GWs (twice the orbital frequency). Note that for a fixed distance $r$ and a given frequency $f$ , the GW amplitudes are fully determined by $2∕3 5∕3 μM = ℳ$ , where the combination

ℳ ≡ μ3∕5M 2∕5

is called the “chirp mass” of the binary. After averaging over the orbital period (so that the squares of periodic functions are replaced by 1/2) and the orientations of the binary orbital plane, one arrives at the averaged (characteristic) GW amplitude

3.1.3 Energy and angular momentum loss

In the approximation and under the choice of coordinates that we are working with, it is sufficient to use the Landau–Lifshitz gravitational pseudo-tensor [405 *] when calculating the gravitational waves energy and flux. (This calculation can be justified with the help of a fully satisfactory gravitational energy-momentum tensor that can be derived in the field theory formulation of general relativity [17 ]). The energy $dE$ carried by a gravitational wave along its direction of propagation per area $dA$ per time $dt$ is given by

The energy output $dE ∕dt$ from a localized source in all directions is given by the integral

Replacing

( )2 ( )2 ∂h+- + ∂h-× = 4π2f2h2 (𝜃, ϕ) ∂t ∂t

and introducing

∫ 2 1-- 2 h = 4π h (𝜃,ϕ)dΩ,

we write Eq. (22 *) in the form

Specifically for a binary system in a circular orbit, one finds the energy loss from the system (sign minus) with the help of Eqs. (23 *) and (20 *):

This expression is exactly the same that can be obtained directly from the quadrupole formula [405 ],

rewritten using the definition of the chirp mass and Kepler’s law. Since energy and angular momentum are continuously carried away by gravitational radiation, the two masses in orbit spiral towards each other, thus increasing their orbital frequency $Ω$ . The GW frequency $f = Ω ∕π$ and the GW amplitude $h$ are also increasing functions of time. The rate of the frequency change is¹³

In spectral representation, the flux of energy per unit area per unit frequency interval is given by the right-hand-side of the expression

where we have introduced the spectral density $S2h (f )$ of the gravitational wave field $h$ . In the case of a binary system, the quantity $S h$ is calculable from Eqs. (18 * and 19 *):

3.1.4 Binary coalescence time

A binary system in a circular orbit loses energy according to Eq. (24 *). For orbits with non-zero eccentricity $e$ , the right-hand-side of this formula should be multiplied by the factor

( 73 37 ) f(e) = 1 + ---e2 + --e4 (1 − e2)−7∕2 24 96

(see [579 *]). The initial binary separation $a0$ decreases and, assuming Eq. (25 *) is always valid, the binary should vanish in a time

As noted above, gravitational radiation from the binary depends on the chirp mass $ℳ$ , which can also be written as $ℳ ≡ M η3∕5$ , where $η$ is the dimensionless ratio $η = μ∕M$ . Since $η ≤ 1∕4$ , one has $ℳ ≲ 0.435M$ . For example, for two NSs with equal masses $M1 = M2 = 1.4M ⊙$ , the chirp mass is $ℳ ≈ 1.22M ⊙$ . This explains the choice of normalization in Eq. (29 *).

Figure 3: The maximum initial orbital period (in hours) of two point masses that will coalesce due to gravitational wave emission in a time interval shorter than 10¹⁰ yr, as a function of the initial eccentricity $e0$ . The lines are calculated for $10 M ⊙ + 10 M ⊙$ (BH + BH), $10 M ⊙ + 1.4M ⊙$ (BH + NS), and $1.4 M ⊙ + 1.4M ⊙$ (NS + NS).

The coalescence time for a binary star with an initially eccentric orbit with $e0 ⁄= 0$ and initial separation $a0$ is shorter than the coalescence time for an object with a circular orbit and the same $a0$ [579 ]:

where the correction factor $f (e0)$ is

To merge in a time interval shorter than 10 Gyr the binary should have a small enough initial orbital period $P0 ≤ Pcr(e0,ℳ )$ and, accordingly, a small enough initial semi-major axis $a0 ≤ acr(e0,ℳ )$ . The critical orbital period is plotted as a function of the initial eccentricity $e0$ in Figure 3 *. The lines are plotted for three typical sets of masses: two neutron stars with equal masses ( $1.4M ⊙ + 1.4 M ⊙$ ), a black hole and a neutron star ( $10M ⊙ + 1.4 M ⊙$ ), and two black holes with equal masses ( $10 M ⊙ + 10M ⊙$ ). Note that in order to get a significantly shorter coalescence time, the initial binary eccentricity should be $e ≥ 0.6 0$ .

3.1.5 Magnetic stellar wind

In the case of low-mass binary evolution, there is another important physical mechanism responsible for the removal of orbital angular momentum, in addition to the GW emission discussed above. This is the magnetic stellar wind (MSW), or magnetic braking, which is thought to be effective for main-sequence G-M dwarfs with convective envelopes, i.e., approximately, in the mass interval $0.3– 1.2 M ⊙$ . The upper mass limit corresponds to the disappearance of a deep convective zone, while the lower mass limit stands for fully convective stars. In both cases a dynamo mechanism, responsible for enhanced magnetic activity, is thought to become ineffective. The idea behind angular momentum loss (AML) by magnetically-coupled stellar wind is that the stellar wind is compelled by magnetic field to corotate with the star to rather large distances, where it carries away large specific angular momentum [680 ]. Thus, it appears possible to take away substantial angular momentum without evolutionarily significant mass-loss in the wind. In the quantitative form, the concept of angular momentum loss by MSW as a driver of the evolution of compact binaries was introduced by Verbunt and Zwaan [810 *] when it became evident that momentum loss by GWs is unable to explain the observed mass-transfer rates in cataclysmic variables (CVs) and low-mass X-ray binaries, as well as the deficit of cataclysmic variables with orbital periods between 2 and 3 hr (the “period gap”).¹⁴ Verbunt and Zwaan based their reasoning on observations of the spin-down of rotation of single G-dwarfs in stellar clusters with age, which is expressed by the phenomenological dependence of the equatorial rotational velocity $V$ on age: $V = λt−1∕2$ (“Skumanich law” [718 ]). In the latter formula $λ$ is an empirically-derived coefficient $∼ 1$ . Applying this to a binary component and assuming tidal locking between the stellar axial rotation and orbital motion, one arrives at the rate of angular momentum loss via MSW

where $M2$ and $R2$ are the mass and radius of the optical component of the system, respectively, $ω = Ωorb$ is the spin frequency of the star’s rotation equal to the binary orbital frequency, and $k2 ∼ 0.1$ is the gyration radius of the optical component squared.

Radii of stars filling their Roche lobes should be proportional to binary separations, $Ro ∝ a$ , which means that the time scale of orbital angular momentum removal by MSW is $τMSW ≡ (J ˙MSW ∕Jorb)−1 ∝ a$ . This should be compared with AML by GWs with $τGW ∝ a4$ . Clearly, MSW (if it operates) is more efficient at removing angular momentum from a binary system at larger separations (orbital periods), and at small orbital periods GWs always dominate. Magnetic braking is especially important in CVs and in LMXBs with orbital periods exceeding several hours and is the driving mechanism for mass accretion onto the compact component. Indeed, the Doppler tomography reconstruction of the Roche-lobe–filling low-mass K7 secondary star in a well-studied LMXB Cen X-4 revealed the presence of cool spots on its surface. The latter provide evidence for the action of magnetic fields at the surface of the star, thus supporting magnetic braking as the driving mechanism of mass exchange in this binary [689 ].

Equation (32 *) with assumed $λ = 1$ and $2 k = 0.07$ [627 *] is often considered as a “standard”. However, Eqs. (25 *) and (32 *) do not allow one to reproduce some observed features of CVs, see, e.g., [627 , 600 , 434 , 771 , 368 ], but the reasons for this discrepant behavior are not clear as yet. Recently, Knigge et al. [368 ], based on the study of properties of donors in CVs, suggested that a better description of the evolution of CVs is provided by scaling Eq. (32 *) by a factor of $(0.66 ± 0.05)(R ∕ R )−1 2 ⊙$ above the period gap and by scaling Eq. (25 *) by a factor $2.47 ± 0.22$ below the gap (where AML by MSW is not acting). Note that these “semi-empirical”, as called by the their authors, numerical factors must be taken with some caution, since they are based on the fitting of the $Porb– M2$ relation for observed stars by a single evolutionary sequence with the initially-unevolved donor with $M2,0 = 1 M ⊙$ and $MWD,0 = 0.6 M ⊙$ .

The simplest reason for deviation of AML by MSW from Eq. (25 *) may be the unjustified extrapolation of stellar-rotation rates over several orders of magnitude – from slowly-rotating single field stars to rapidly-spinning components of close binaries. For the shortest orbital periods of binaries meant to evolve due to AML via GW, the presence of circumbinary discs may enhance their orbital angular momentum loss [841 ].

3.2 Mass exchange in close binaries

As mentioned in the introductory Section 1, all binaries may be considered either as “close” or as “wide”. In the former case, mass exchange between the components can occur. This process can be accompanied by mass and angular momentum loss from the system.

The shape of the stellar surface is determined by the shape of the equipotential level surface $Φ = const.$ Conventionally, the total potential, which includes gravitational and centrifugal forces is approximated by the Roche potential (see, e.g., [375 *]), which is defined under the following assumptions:

the gravitational field of two components is approximated by that of two point masses;

the binary orbit is circular;

the components of the system corotate with the binary orbital period.

Let us consider a Cartesian reference frame (x,y,z) rotating with the binary, with the origin at the primary $M1$ ; the x-axis is directed along the line of centers; the y-axis is aligned with the orbital motion of the primary component; the z-axis is perpendicular to the orbital plane. The total potential at a given point (x,y,z) is then

where $μ = M2 ∕(M1 + M2 )$ , $Ωorb = 2π∕Porb$ .

A 3-D representation of a dimensionless Roche potential in a co-rotating frame for a binary with mass ratio of components $q = 2$ is shown in Figure 4 *.

Figure 4: A 3-D representation of a dimensionless Roche potential in the co-rotating frame for a binary with a mass ratio of components $q = 2$ .

For close binary evolution the most important is the innermost level surface that encloses both components. It defines the “critical”, or “Roche” lobes of the components.¹⁵ Inside these lobes the matter is bound to the respective component. In the libration point $L1$ (the inner Lagrangian point) the net force exerted onto a test particle corotating with the binary vanishes, so the particle can escape from the surface of the star and be captured by the companion. The matter flows along the surface of the Roche-lobe filling companion in the direction of $L1$ and escapes from the surface of the contact component as a highly inhomogeneous stellar wind. Next, the wind forms a (supersonic) stream directed at a certain angle respective to the line connecting the centers of the components. Depending on the size of the second companion, the stream may hit the latter (called “direct impact”) or form a disk orbiting the companion [448 *].

Though it is evident that mass exchange is a complex 3D gas-dynamical process that must also take into account radiation transfer and, in some cases, even nuclear reactions, virtually all computations of the evolution of non-compact close binaries have been performed in 1-D approximations. The Roche-lobe overflow (RLOF) is conventionally considered to begin when the radius of an initially more massive and hence faster evolving star (primary component) becomes equal to the radius of a sphere with volume equal to that of the Roche lobe. For the latter, an expression precise to better than 1% for an arbitrary mass ratio of components $q$ was suggested by Eggleton [172 ]:

For practical purposes, such as analytical estimates, a more convenient expression is that suggested by Kopal [375 ] (usually called the “Paczyński formula”, who introduced it into close binary modeling):

which is accurate to $≲ 2%$ for $0 < q ≲ 0.8$ .

In close binaries, the zero-age main sequence (ZAMS) mass ceases to be the sole parameter determining stellar evolution. The nature of compact remnants of close binary components also depends on their evolutionary stage at RLOF, i.e., on the component separation and their mass-ratio $q$ . Evolution of a star may be considered as consumption of nuclear fuel accompanied by an increase of its radius. Following the pioneering work of Kippenhahn and his collaborators on the evolution of close binaries in the late 1960s, the following basic cases of mass exchange are usually considered: A – RLOF at the core hydrogen-burning stage; B – RLOF at the hydrogen-shell burning stage; C – RLOF after exhaustion of He in the stellar core. Also, more “fine” gradations exist: case AB – RLOF at the late stages of core H-burning, which continues as Case B after a short break upon exhaustion of H in the core; case BB – RLOF by the star, which first filled its Roche lobe in case B, contracted under the Roche lobe after the loss of the hydrogen envelope, and resumed the mass loss due to the envelope expansion at the helium-shell burning stage. Further, one may consider the modes of mass-exchange, depending, e.g., on the nature of the envelope of the donor (radiative vs. convective), its relative mass, reaction of accretor etc., see, e.g., [171 , 826 ] and Section 3.3.

In cases A and B of mass exchange the remnants of stars with initial masses lower than $(2.3 –2.8)M ⊙$ are degenerate He WD – a type of objects not produced by single stars.¹⁶

In cases A and B of mass exchange the remnants of stars with initial mass $≳ 2.5M ⊙$ are helium stars.

If the mass of the helium remnant star does not exceed $≃ 0.8 M ⊙$ , after exhaustion of the He in the core it does not expand, but transforms directly into a WD of the same mass [300 *]. Helium stars with masses between $≃ 0.8 M ⊙$ and $≃ (2.3– 2.8)M ⊙$ expand in the helium shell burning stage, re-fill their Roche lobes, lose the remnants of the helium envelopes and also transform into CO WD.

Stellar radius may be taken as a proxy for the evolutionary state of a star. We plot in Figure 5 * the types of stellar remnant as a function of both initial mass and the radius of a star at the instant of RLOF. In Figure 6 * initial-final mass relations for components of close binaries are shown. Clearly, relations presented in these two figures are only approximate, reflecting current uncertainty in the theory of stellar evolution (in this particular case, relations used in the population synthesis code IBiS [791 *] are shown.

Figure 5: Descendants of components of close binaries depending on the radius of the star at RLOF. The upper solid line separates close and wide binaries (after [293 ]). The boundary between progenitors of He- and CO-WDs is uncertain by several $0.1 M ⊙$ , the boundary between CO and ONe varieties of WDs and WD and NSs – by $∼ 2M ⊙$ . The boundary between progenitors of NS and BH is shown at $40 M ⊙$ after [643 ], while it may be possible that it really is between $20M ⊙$ and $50 M ⊙$ (see Section 1 for discussion and references.)

Figure 6: Relation between ZAMS masses of stars $Mi$ and their masses at TAMS (solid line), masses of helium stars (dashed line), masses of He WD (dash-dotted line), masses of CO and ONe WD (dotted line). For stars with $Mi ≲ 5 M ⊙$ we plot the upper limit of WD masses for case B of mass exchange. After [791 *].

3.3 Mass transfer modes and mass and angular momentum loss in binary systems

GW emission is the sole factor responsible for the change of orbital parameters of a detached pair of compact (degenerate) stars. However, it was recognized quite early on that in the stages of evolution preceding the formation of compact objects, the mode of mass transfer between the components and the loss of matter and orbital angular momentum by the system as a whole play a dominant dynamical role and define the observed features of the binaries, e.g., [556 , 781 , 631 *, 471 , 603 , 137 ].

Strictly speaking, as mentioned previously, these processes should be treated hydrodynamically and they require complicated numerical calculations. However, binary evolution can also be described semi-qualitatively, using a simplified description in terms of point-like bodies. The change of their integrated physical quantities, such as masses, orbital angular momentum, etc., governs the evolution of the orbit. This description turns out to be successful in reproducing the results of more rigorous numerical calculations (see, e.g., [705 ] for more details and references). In this approach, the key role is allocated to the total orbital angular momentum $Jorb$ of the binary.

Let star 2 lose matter at a rate $M˙ < 0 2$ and let $β$ $(0 ≤ β ≤ 1)$ be the fraction of the ejected matter that leaves the system (the rest falls on the first star), i.e., $˙ ˙ M1 = − (1 − β )M2 ≥ 0$ . Consider circular orbits with orbital angular momentum given by Eq. (17 *). Differentiate both parts of Eq. (17 *) by time $t$ and exclude $d Ω∕dt$ with the help of Kepler’s third law (10 *). This gives the rate of change of the orbital separation:

In Eq. (36 *) $a˙$ and $M˙$ are not independent variables if the donor fills its Roche lobe. One defines the mass transfer as conservative if both $β = 0$ and $J˙orb = 0$ . The mass transfer is called non-conservative if at least one of these conditions is violated.

It is important to distinguish some specific cases (modes) of mass transfer:

conservative mass transfer,
non-conservative Jeans mode of mass loss (or fast wind mode),
non-conservative isotropic re-emission,
sudden mass loss from one of the components during supernova explosion, and
common-envelope stage.

As specific cases of angular momentum loss we consider GW emission (see Section 3.1.3 and 3.1.4) and the magnetically-coupled stellar wind (see Section 3.1.5), which drive the orbital evolution for short-period binaries. For non-conservative modes, one can also consider some less important cases, such as, for instance, the formation of a circumbinary ring by the matter leaving the system (see, e.g., [722 *, 732 ]). Here, we will not go into details of such sub-cases.

3.3.1 Conservative accretion

In the case of conservative accretion, matter from $M2$ is fully deposited onto $M1$ . The transfer process preserves the total mass ( $β = 0$ ) and the orbital angular momentum of the system. It follows from Eq. (36 *) that

√ -- M1M2 a = const.,

so that the initial and final binary separations are related as

The orbit shrinks when the more massive component loses matter, and the orbit widens in the opposite situation. During such a mass exchange, the orbital separation passes through a minimum, if the masses become equal in the course of mass transfer.

3.3.2 The Jeans (fast wind) mode

In this mode the matter ejected by the donor completely escapes from the system, that is, $β = 1$ . Escape of the matter can occur either via fast isotropic spherically-symmetric wind or in the form of bipolar jets moving from the system at high velocity. Escaping matter does not interact with another component. The matter escapes with the specific angular momentum of the mass-losing star $J2 = (M1 ∕M )Jorb$ (we neglect a possible proper rotation of the star, see [795 ]). For the loss of orbital momentum $J˙ orb$ it is reasonable to take

In the case $β = 1$ , Eq. (36 *) can be written as

Then Eq. (39 *) in conjunction with Eq. (38 *) gives $Ωa2 = const$ , that is, $√ ------ GaM = const$ . Thus, as a result of the Jeans mode of mass loss, the change in orbital separation is

Since the total mass decreases, the orbit always widens.

3.3.3 Isotropic re-emission

The matter lost by star 2 can first accrete onto star 1, and then, a fraction $β$ of the accreted matter, can be expelled from the system. This happens, for instance, when a massive star transfers matter onto a compact star on the thermal timescale (usually $<$ 10⁶ years). Accretion luminosity may exceed the Eddington luminosity limit, and the radiation pressure pushes the infalling matter away from the system, in a manner similar to the spectacular example of the SS 433 binary system. Other examples may be systems with helium stars transferring mass onto relativistic objects [442 *, 238 ]. In this mode of mass-transfer, the binary orbital momentum carried away by the expelled matter is determined by the orbital momentum of the accreting star rather than by the orbital momentum of the mass-losing star, since mass loss happens in the vicinity of the accretor. The assumption that all matter in excess of accretion rate can be expelled from the system, thus avoiding the formation of a common envelope, will only hold if the liberated accretion energy of the matter falling from the Roche lobe radius of the accretor star to its surface is sufficient to expel the matter from the Roche-lobe surface around the accretor, i.e., $M˙d ≲ M˙dmax = M˙Edd (rL,a∕ra)$ , where $ra$ is the radius of the accretor [364 ].

The orbital momentum loss can be written as

where $J = (M ∕M )J 1 2 orb$ is the orbital momentum of the star $M 1$ . In the limiting case when all the mass initially accreted by $M1$ is later expelled from the system, $β = 1$ , Eq. (41 *) simplifies to

After substitution of this formula into Eq. (36 *) and integration over time, one arrives at

The exponential term makes this mode of mass transfer very sensitive to the components’ mass ratio. If $M1 ∕M2 ≪ 1$ , the separation $a$ between the stars may decrease so much that the approximation of point masses becomes invalid. Tidal orbital instability (Darwin instability) may set in, and the compact star’s may start spiraling toward the companion star centre (the common envelope stage; see Section 3.6). On the other hand, “isotropic reemission” may stabilize mass-exchange if $M ∕M > 1 1 2$ [877 *].

Mass loss may be considered as occurring in the “isotropic re-emission” mode in situations in which hot white dwarf components of cataclysmic variables lose mass by optically-thick winds [346 *] or when time-averaged mass loss from novae is considered [867 ].

3.4 Supernova explosion

Supernovae explosions in binary systems occur on a timescale much shorter than the orbital period, so the loss of mass is practically instantaneous. This case can be treated analytically (see, e.g., [55 , 59 , 753 ]).

Clearly, even a spherically-symmetric sudden mass loss due to an SN explosion will be asymmetric in the reference frame of the center of mass of the binary system, leading to system recoil (‘Blaauw–Boersma’ recoil). In general, the loss of matter and radiation is non-spherical, so that the remnant of the supernova explosion (neutron star or black hole) acquires some recoil velocity called kick velocity $w⃗$ . In a binary, the kick velocity should be added to the orbital velocity of the pre-supernova star.

The usual treatment proceeds as follows. Let us consider a pre-SN binary with initial masses $M 1$ and $M2$ . The stars move in a circular orbit with orbital separation $ai$ and relative velocity $⃗Vi$ . The star $M1$ explodes leaving a compact remnant of mass $Mc$ . The total mass of the binary decreases by $ΔM = M1 − Mc$ . Unless the binary is disrupted, it will end up in a new orbit with eccentricity $e$ , semi-major axis $af$ , and angle $𝜃$ between the orbital planes before and after the explosion. In general, the new barycenter will also receive some velocity, but we neglect this motion. The goal is to evaluate the parameters $af$ , $e$ , and $𝜃$ .

It is convenient to use an instantaneous reference frame centered on $M2$ right at the time of explosion. The $x$ -axis is the line from $M 2$ to $M 1$ , the $y$ -axis points in the direction of $⃗V i$ , and the $z$ -axis is perpendicular to the orbital plane. In this frame, the pre-SN relative velocity is $⃗ Vi = (0,Vi,0)$ , where $∘ ---------------- Vi = G (M1 + M2 )∕ai$ (see Eq. (16 *)). The initial total orbital momentum is $J⃗i = μiai(0, 0,− Vi)$ . The explosion is considered to be instantaneous. Right after the explosion, the position vector of the exploded star $M 1$ has not changed: $⃗r = (a,0, 0) i$ . However, other quantities have changed: $⃗ Vf = (wx, Vi + wy, wz)$ and $⃗ Jf = μfai(0,wz, − (Vi + wy ))$ , where $⃗w = (wx, wy,wz )$ is the kick velocity and $μf = McM2 ∕(Mc + M2 )$ is the reduced mass of the system after explosion. The parameters $af$ and $e$ are found by equating the total energy and the absolute value of the orbital momentum of the initial circular orbit to those of the resulting elliptical orbit (see Eqs. (13 *, 17 *, 15 *)):

2 μfV-f − GMcM2--- = − GMcM2---, (44 ) ∘ --2-------ai---- 2af 2 2 ∘ --------------------2- μfai w z + (Vi + wy ) = μf G(Mc + M2 )af(1 − e ). (45 )

For the resulting $af$ and $e$ one finds

and

where $χ ≡ (M1 + M2 )∕(Mc + M2 ) ≥ 1$ . The angle $𝜃$ is defined by

J⃗f ⋅ ⃗Ji cos𝜃 = --------, |⃗Jf| |⃗Ji|

which results in

The condition for disruption of the binary system depends on the absolute value $Vf$ of the final velocity, and on the parameter $χ$ . The binary disrupts if its total energy defined by the left-hand-side of Eq. (44 *) becomes non-negative or, equivalently, if its eccentricity defined by Eq. (47 *) becomes $e ≥ 1$ . From either of these requirements one derives the condition for disruption:

The system remains bound if the opposite inequality is satisfied. Eq. (49 *) can also be written in terms of the escape (parabolic) velocity $Ve$ defined by the requirement

V 2e GMcM2 μf--- − -------- = 0. 2 ai

Since $χ = M ∕(M − ΔM )$ and $2 2 V e = 2G (M − ΔM )∕ai = 2Vi ∕χ$ , one can write Eq. (49 *) in the form

The condition of disruption simplifies in the case of a spherically-symmetric SN explosion, that is, when there is no kick velocity, $⃗w = 0$ , and, therefore, $Vf = Vi$ . In this case, Eq. (49 *) reads $χ ≥ 2$ , which is equivalent to $ΔM ≥ M ∕2$ . Thus, the system unbinds if more than half of the mass of the binary is lost. In other words, the resulting eccentricity

following from Eqs. (46 *) and (47 *), and $⃗w = 0$ becomes larger than 1, if $ΔM > M ∕2$ .

So far, we have considered an originally circular orbit. If the pre-SN star moves in an originally eccentric orbit, the condition of disruption of the system under symmetric explosion reads

1-r- ΔM = M1 − Mc > 2 a , i

where $r$ is the distance between the components at the moment of explosion.

3.5 Kick velocity of neutron stars

The kick imparted to a NS at birth is one of the major problems in the theory of stellar evolution. By itself, it is an additional parameter, the introduction of which has been motivated first of all by high space velocities of radio pulsars inferred from the measurements of their proper motions and distances. Pulsars were recognized as a high-velocity Galactic population soon after their discovery in 1968 [256 ]. Shklovskii [704 ] put forward the idea that high pulsar velocities may result from asymmetric supernova explosions. Since then this hypothesis has been tested by pulsar observations, but no definite conclusions on its magnitude and direction have as yet been obtained.

Indeed, the distance to a pulsar is usually derived from the dispersion measure evaluation and crucially depends on the assumed model of electron density distribution in the Galaxy. In the middle of the 1990s, Lyne and Lorimer [449 ] derived a very high mean space velocity of pulsars with known proper motion of about 450 km s^–1. This value was difficult to adopt without invoking an additional natal kick velocity of NSs.

The high mean space velocity of pulsars, consistent with earlier results by Lyne and Lorimer, was confirmed by the analysis of a larger sample of pulsars [284 ]. The recovered distribution of 3D velocities is well fit by a Maxwellian distribution with the mean value $w = 400 ± 40 km s−1 0$ and a 1D rms $− 1 σ = 265 km s$ .

Possible physical reasons for natal NS kicks due to hydrodynamic effects in core-collapse supernovae are summarized in [402 , 401 ]. Large kick velocities ( $∼$ 500 km s^–1 and even more) imparted to nascent NSs are generally confirmed by detailed numerical simulations (see, e.g., [536 , 846 ]). Neutrino effects in the strong magnetic field of a young NS may also be essential in explaining kicks up to $∼$ 100 km s^–1 [109 , 160 , 398 ]. Astrophysical arguments favoring a kick velocity are also summarized in [754 ]. To get around the theoretical difficulty of insufficient rotation of pre-supernova cores in single stars to produce rapidly-spinning young pulsars, Spruit and Phinney [731 ] proposed that random off-center kicks can lead to a net spin-up of proto-NSs. In this model, correlations between pulsar space velocity and rotation are possible and can be tested in further observations.

Here we should note that the existence of some kick follows not only from the measurements of radio pulsar space velocities, but also from the analysis of binary systems with NSs. The impact of a kick velocity $∼$ 100 km s^–1 explains the precessing pulsar binary orbit in PSR J0045–7319 [345 ]. The evidence of the kick velocity is seen in the inclined, with respect to the orbital plane, circumstellar disk around the Be star SS 2883 – an optical component of a binary PSR B1259–63 [617 ]. Evidence for $∼$ 150 km s^–1 natal kicks has also been inferred from the statistics of the observed short GRB distributions relative to their host galaxies [31 ].

Long-term pulse profile changes interpreted as geodetic precession are observed in the relativistic pulsar binaries PSR 1913+16 [831 ], PSR B1534+12 [734 ], PSR J1141–6545 [289 ], and PSR J0737–3039B [78 ]. These observations indicate that in order to produce the misalignment between the orbital angular momentum and the neutron star spin, a component of the kick velocity perpendicular to the orbital plane is required [835 , 839 *, 840 ]. This idea seems to gain observational support from recent thorough polarization measurements [324 ] suggesting alignment of the rotational axes with the pulsar’s space velocity. Detailed discussion of the spin-velocity alignment in young pulsars and implications for the SN kick mechanisms can be found in Noutsos et al. [538 ]. Such an alignment acquired at birth may indicate the kick velocity directed preferably along the rotation of the proto-NS. For the first SN explosion in a close binary system this would imply that the kick is mostly perpendicular to the orbital plane. Implications of this effect for the formation and coalescence rates of NS binaries were discussed by Kuranov et al. [396 ].

It is worth noting that the analysis of the formation of the relativistic pulsar binaryPSR J0737–3039 [595 ] may suggest, from the observed low eccentricity of the system $e ≃ 0.09$ , that a small (if any) kick velocity may be acquired if the formation of the second NS in the system is associated with the collapse of an ONeMg WD due to electron captures. The symmetric nature of electron-capture supernovae was discussed in [597 ] and seems to be an interesting issue requiring further studies (see, e.g., [582 , 397 ] for the analysis of the formation of NSs in globular clusters in the frame of this hypothesis). Note that electron-capture SNe are expected to be weak events, irrespective of whether they are associated with the core collapse of a star that retained some original envelope or with the AIC of a WD [644 , 366 , 146 ]. In the case of AIC, rapid rotation of a collapsing object along with flux freezing and dynamo action can grow the WD’s magnetic field to magnetar strengths during collapse. Further, magnetar generates outflow of the matter and formation of a pulsar wind nebula, which may be observed as a radio-source for a few month [592 ].

We also note the hypothesis of Pfahl et al. [583 *], based on observations of high-mass X-ray binaries with long orbital periods ( $≳$ 30 d) and low eccentricities ( $e < 0.2$ ), that rapidly rotating precollapse cores may produce neutron stars with relatively small kicks, and vice versa for slowly rotating cores. Then, large kicks would be a feature of stars that retained deep convective envelopes long enough to allow a strong magnetic torque, generated by differential rotation between the core and the envelope, to spin the core down to the very slow rotation rate of the envelope. A low kick velocity imparted to the second (younger) neutron star ( $<$ 50 km s^–1) was inferred from the analysis of large-eccentricity pulsar binary PSR J1811–1736 [119 ]. The large orbital period of this pulsar binary (18.8 d) may then suggest an evolutionary scenario with inefficient (if any) common envelope stage [148 ], i.e., the absence of a deep convective shell in the supernova progenitor (a He-star). This conclusion can be regarded as supportive to ideas put forward by Pfahl et al. [583 *]. A careful investigation of radio profiles of double PSR J0737–3039A/B [198 *] and Fermi detection of gamma-ray emission from the recycled 22-ms pulsar [254 ] imply its spin axis to be almost aligned with the orbital angular momentum, which lends further credence to the hypothesis that the second supernova explosion in this system was very symmetric.

Small kicks in the case of $e$ -capture in the O-Ne core leading to NS formation are justified by hydrodynamical considerations. Indeed, already in 1996, Burrows and Hayes [80 ] noted that large scale convective motions in O and Si burning stages preceding the formation of a Fe core may produce inhomogeneities in the envelope of the protostellar core. They in turn may result in asymmetric neutrino transport, which impart kicks up to 500 km s^–1. Such a violent burning does not precede the formation of O-Ne cores and then small or even zero kicks can be expected.

In the case of asymmetric core-collapse supernova explosion, it is natural to expect some kick during BH formation as well [430 *, 214 , 607 *, 609 *, 518 *, 34 *, 872 *]. The similarity of NS and BH distribution in the Galaxy suggesting BH kicks was noted in [327 ]. Evidence for a moderate (100 – 200 km s^–1) BH kick has been derived from the kinematics of several BH X-ray transients (microquasars): XTE J1118+180 [208 ], GRO 1655–40 [838 ], MAXI J1659–152 [400 ]. However, no kicks or only small ones seem to be required to explain the formation of other BH candidates, such as Cyg X-1 [485 ], [845 ], X-Nova Sco [514 ], V404 Cyg [484 ]. Population synthesis modeling of the Galactic distribution of BH binaries supports the need for (possibly, bimodal) natal BH kicks [634 *]. Janka [321 ] argued that the similarity of BH kick distribution with NS kick distributions as inferred from the analysis by Repetto et al. [634 *] favors BH kicks being due to gravitational interaction with asymmetric mass ejection (the “gravitational tug-boat mechanism”), and disfavors neutrino-induced kicks (in the last case, by momentum conservation, BH kicks are expected to be reduced by the NS to the BH mass ratio relative to the NS kicks). Facing current uncertainties in SN explosions and BH formation mechanisms, it is not excluded that low-kick BHs can be formed without associated SN explosions due to neutrino asymmetry, while high-velocity Galactic BHs in LMXBs analyzed by Repetto et al. [634 ] can be formed by the gravitational tug-boat mechanism suggested by Janka [321 ].

To summarize, the kick velocity remains to be one of the important unknown parameters of binary evolution with NSs and BHs, and further phenomenological input here is of great importance. Large kick velocities will significantly affect the spatial distribution of coalescing compact binaries (e.g., [352 ]) and BH kicks are extremely important for BH spin misalignment in coalescing BH-BH binary systems (e.g., Gerosa et al. [237 ] and references therein). Further constraining this parameter with various observations and a theoretical understanding of the possible asymmetry of core-collapse supernovae seem to be of paramount importance for the formation and evolution of close compact binaries.

3.5.1 Effect of the kick velocity on the evolution of a binary system

The collapse of a star to a BH, or its explosion leading to the formation of a NS, are normally considered as instantaneous. This assumption is well justified in binary systems, since typical orbital velocities before the explosion do not exceed a few hundred km/s, while most of the mass is expelled with velocities of about several thousand km/s. The exploding star $M1$ leaves the remnant $Mc$ , and the binary loses a part of its mass: $ΔM = M1 − Mc$ . The relative velocity of stars before the event is

Right after the event, the relative velocity is

Depending on the direction of the kick velocity vector $w⃗$ , the absolute value of $⃗Vf$ varies in the interval from the smallest $Vf = |Vi − w |$ to the largest $Vf = Vi + w$ . The system gets disrupted if $Vf$ satisfies the condition (see Section 3.4)

where $χ ≡ (M1 + M2 )∕(Mc + M2 )$ .

Let us start from the limiting case when the mass loss is practically zero ( $ΔM = 0$ , $χ = 1$ ), while a non-zero kick velocity can still be present. This situation can be relevant to BH formation. It follows from Eq. (54 *) that, for relatively small kicks, $√ -- w < ( 2 − 1)Vi$ , the system always (independent of the direction of $⃗w$ ) remains bound, while for $√ -- w > ( 2 + 1)Vi$ the system always decays. By averaging over equally probable orientations of $⃗w$ with a fixed amplitude $w$ , one can show that in the particular case $w = Vi$ the system disrupts or survives with equal probabilities. If $V < V f i$ , the semi-major axis of the system becomes smaller than the original binary separation, $af < ai$ (see Eq. (46 *)). This means that the system becomes harder than before, i.e. it has a greater negative total energy than the original binary. If $√ -- Vi < Vf < 2Vi$ , the system remains bound, but $af > ai$ . For small and moderate kicks $w ≳ Vi$ , the probabilities for the system to become more or less bound are approximately equal.

In general, the binary system loses some fraction of its mass $ΔM$ . In the absence of the kick, the system remains bound if $ΔM < M ∕2$ and gets disrupted if $ΔM ≥ M ∕2$ (see Section 3.4). Clearly, a “properly” oriented kick velocity (directed against the vector $⃗V i$ ) can keep the system bound, even if it would have been disrupted without the kick. And, on the other hand, an “unfortunate” direction of $⃗w$ can disrupt the system, which otherwise would stay bound.

Consider, first, the case $ΔM < M ∕2$ . The parameter $χ$ varies in the interval from 1 to 2, and the escape velocity $V e$ varies in the interval from $√2V- i$ to $V i$ . It follows from Eq. (50 *) that the binary always remains bound if $w < Ve − Vi$ , and always unbinds if $w > Ve + Vi$ . This is a generalization of the formulas derived above for the limiting case $ΔM = 0$ . Obviously, for a given $w$ , the probability for the system to disrupt or become softer increases when $ΔM$ becomes larger. Now turn to the case $ΔM > M ∕2$ . The escape velocity of the compact star becomes $V < V e i$ . The binary is always disrupted if the kick velocity is too large or too small: $w > Vi + Ve$ or $w < Vi − Ve$ . However, for all intermediate values of $w$ , the system can remain bound, and sometimes even more hard than before, if the direction of $⃗w$ happened to be approximately opposite to $⃗Vi$ . A detailed calculation of the probabilities for the binary survival or disruption requires integration over the kick velocity distribution function $f(⃗w )$ (see, e.g., [68 ]).

3.6 Common envelope stage

3.6.1 Formation of the common envelope

Common envelopes (CE) are definitely the most important (and not as yet solved) problem in the evolution of close binaries. In the theory of stellar evolution, it was recognized quite early that there are several situations when formation of an envelope that engulfs an entire system seems to be inevitable. It can happen when the mass transfer rate from the mass-losing star is so high that the companion cannot accommodate all the accreting matter [41 , 868 , 631 *, 523 , 823 , 614 , 590 ]. Another instance is encountered when it is impossible to keep synchronous rotation of a red giant and orbital revolution of a compact companion [730 *, 8 ]. Because of tidal drag forces, the revolution period decreases, while the rotation period increases in order to reach synchronism (Darwin instability). If the total orbital momentum of the binary $Jorb < 3JRG$ , the synchronism cannot be reached and the companion (low-mass star, white dwarf or neutron star) spirals into the envelope of the red giant. Yet another situation is the formation of an extended envelope, which enshrouds the system due to unstable nuclear burning at the surface of an accreting WD in a compact binary [738 , 559 , 534 ]. It is also possible that a compact remnant of a supernova explosion with “appropriately” directed kick velocity finds itself in an elliptic orbit whose minimum periastron distance $af(1 − e)$ is smaller than the radius of the companion.

The common-envelope stage appears to be unavoidable on observational grounds. The evidence for a dramatic orbital angular momentum decrease in the course of evolution follows immediately from observations of cataclysmic variables, in which a white dwarf accretes matter from a small red dwarf main-sequence companion, close binary nuclei of planetary nebulae, low-mass X-ray binaries, and X-ray transients (neutron stars and black holes accreting matter from low-mass main-sequence dwarfs). At present, the typical separation of components in these systems is $∼ R ⊙$ , while the formation of compact stars requires progenitors with radii $∼ (100 – 1000)R ⊙$ . ¹⁷ Indirect evidence for the common envelope stage comes, for instance, from X-ray and FUV observations of a prototypical pre-cataclysmic binary V471 Tau showing anomalous C/N contamination of the K-dwarf companion to white dwarf [164 , 714 ].

Let us consider the following mass-radius exponents that describe the response of a star to the mass loss in a binary system:

where $ζL$ is the response of the Roche lobe to the mass loss, $ζth$ – the thermal-equilibrium response, and $ζad$ – the adiabatic hydrostatic response. If $ζad > ζL > ζth$ , the star retains hydrostatic equilibrium, but does not remain in thermal equilibrium and mass loss occurs in the thermal timescale of the star. If $ζL > ζad$ , the star cannot retain hydrostatic equilibrium and mass loss proceeds in the dynamical time scale [558 ]. If both $ζad$ and $ζth$ exceed $ζL$ , mass loss occurs due to expansion of the star caused by nuclear burning or due to the shrinkage of the Roche lobe owed to the angular momentum loss.

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.e., the equation of state can be written as $1+1∕n P = K ρ$ ) with an index $n = 3∕2$ . The polytropic approximation with $n = 3∕2$ is also valid for degenerate white dwarfs with masses not too close to the Chandrasekhar limit. For a star in hydrostatic equilibrium, this results in the inverse mass-radius relation, $R ∝ M −1∕3$ , first found for white dwarfs. 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 dynamical mass loss and eventual formation of a common envelope. If the star is completely convective or completely degenerate, dynamically-unstable mass loss occurs if the mass ratio of components (donor to accretor) is $≳ 2∕3$ .

In a more realistic case when the star is not completely convective and has a radiative core, certain insight may be gained by the analyses of composite polytropic models. Conditions for the onset of dynamical mass loss become less rigorous [555 , 283 *, 722 ]: the contraction of a star replaces expansion if the relative mass of the core $mc > 0.214$ and in a binary with a mass ratio of components close to 1, a Roche-lobe filling star with a deep convective envelope may remain dynamically stable if $mc > 0.458$ . As well, a stabilizing effect upon mass loss may have mass and momentum loss from the system, if it happens in a mode that results in an increase of the specific angular momentum of the binary (e.g., in the case of isotropic reemission by a more massive component of the system in CVs or UCXBs).

Criteria for thermal or dynamical mass loss upon RLOF and the formation of a common envelope need systematic exploration of the response of stars to mass removal in different evolutionary stages and at different rates. The response also depends on the mass of the star. While computational methods for such an analyses are elaborated [224 , 225 ], calculations of respective grids of models with full-fledged evolutionary codes at the time of writing of this review were not completed.

3.6.2 “Alpha”-formalism

Formation of the common envelope and evolution of a binary inside the former is a 3D hydrodynamic process that may also include nuclear reactions; and current understanding of the process as a whole, as well as computer power, are not sufficient for a solution of the problem (see, for detailed discussion, [319 ]). Therefore, the outcome of this stage is, most commonly, evaluated in a simplified way, based on the balance of the binding energy of the stellar envelope and the binary orbital energy, following independent suggestions of van den Heuvel [794 ], Tutukov and Yungelson [783 *], and Webbink [825 *] and commonly named after the later author as “Webbink’s” or “ $α$ ”-formalism. The orbital evolution of the compact star $m$ inside the envelope of the donor-star $M1$ is driven by dynamic friction drag [553 *]. This leads to a gradual spiral-in process of the compact star. The above-mentioned energy condition may be written as

where $ai$ and $af$ are the initial and final orbital separations, $αce$ is the common envelope parameter [783 , 438 ] that describes the efficiency of expenditure of orbital energy on expulsion of the envelope and $λ$ is a numerical coefficient that depends on the structure of the donor’s envelope, introduced by de Kool et al. [138 ], and $RL$ is the Roche lobe radius of the normal star that can be approximated, e.g., by Eqs. (34 *) or (35 *). From Eq. (56 *) one derives

where $ΔM = M1 − Mc$ is the mass of the ejected envelope. For instance, the mass $Mc$ of the helium core of a massive star can be approximated as [782 *]

Then, $ΔM ∼ M$ and, e.g., in the case of $m ∼ M ⊙$ the orbital separation during the common envelope stage can decrease by as much as a factor of 40 or even more.

This treatment does not take into account possible transformations of the binary components during the CE-stage. In addition, the pre-CE evolution of the binary components may be important for the onset and the outcome of the CE, as calculations of synchronization of red giant stars in close binaries carried out by Bear and Soker [29 ] indicate. The outcome of the common envelope stage is considered as a merger of components if $af$ is such that the donor core comes into contact with the companion. Otherwise, it is assumed that the core and companion form a detached system with the orbital separation $af$ . Note an important issue raised recently by Kashi and Soker [341 ]: while Eq. (57 *) can formally imply that the system becomes detached at the end of the CE stage, in fact, some matter of the envelope can not reach the escape velocity and remains bound to the system and forms a circumbinary disk. Angular momentum loss due to interaction with the disk may result in a further reduction of binary separation and merger of components. This may influence the features of such stellar populations as a close WD binary, hot subdwarfs, cataclysmic variables) and the rate of SN Ia (see Section 7).

In the above equation for the common envelope, the outcome of the CE stage depends on the product of two parameters: $λ$ , which is the measure of the binding energy of the envelope to the core prior to the mass transfer in a binary system, and $αCE$ , which is the common envelope efficiency itself. Evaluation of both parameters suffers from large physical uncertainties. For example, for $λ$ , the most debatable issues are the accounting of the internal energy in the binding energy of the envelope and the definition of the core/envelope boundary itself. Some authors argue (see e.g., [315 ]) that enthalpy rather than internal energy should be included in the calculation of $λ$ , which seems physically justifiable for convective envelopes¹⁸ . We refer the interested reader to the detailed discussion in [139 , 319 ].

There exist several sets of fitting formulas for $λ$ [856 , 855 ] or binding energy of the envelopes [446 ] based on detailed evolutionary computations for a range of stellar models at different evolutionary stages. But we note, that in both studies the same specific evolutionary code ev [601 ] was used and the core/envelope interface at location in the star with hydrogen abundance in the hydrogen-burning shell $X = 0.1$ was assumed; in our opinion, the latter assumption is not justified neither by any physical assumptions nor by evolutionary computations for Roche-lobe filling stars. Even if the inaccuracy of the latter assumption is neglected, it is possible to use these formulas only in population synthesis codes based on the evolutionary tracks obtained by ev code.

Our test calculations do not confirm the recent claim [313 ] that core/envelope interface, which also defines the masses of the remnants in all cases of mass-exchange, is close to the radius of the sonic velocity maximum in the hydrogen-burning shell of the mass-losing star at RLOF. As concerns parameter $αCE$ , the most critical issue is whether the sources other than orbital energy can contribute to the energy balance, which include, in particular, recombination energy [724 ], nuclear energy [299 ], or energy released by accretion onto the compact star. It should then be noted that sometimes imposed to Eq. (56 *) restriction $αCE ≤ 1$ can not reflect the whole complexity of the processes occurring in common envelopes. Although full-scale hydrodynamic calculations of a common-envelope evolution exist, (see, e.g., [747 , 140 , 636 , 637 , 571 ] and references therein), we stress again that the process is still very far from comprehension, especially, at the final stages.

Actually, the state of the common envelope problem is currently such that it is possible only to estimate the product $αCE λ$ by modeling specific systems or well defined samples of objects corrected for observational selection effects, like it is done in [765 ].

3.6.3 “Gamma”-formalism

Nelemans et. al. [517 *] noted that the $α$ -formalism failed to reproduce parameters of several close white dwarf binaries known circa 2000. In particular, for close helium white dwarf binaries with known masses of both components, one can reconstruct the evolution of the system “back” to the pair of main-sequence progenitors of components, since there is a unique relationship between the mass of a white dwarf and the radius of its red giant progenitor [631 ], which is almost independent of the total mass of the star. Formation of close white dwarf binaries should definitely involve a spiral-in phase in the common envelope during the second episode of mass loss (i.e., from the red giant to the white dwarf remnant of the original primary in the system). In observed systems, mass ratios of components tend to concentrate to $Mbright∕Mdim ≃ 1$ . This means, that, if the first stage of mass transfer occurred through a common envelope, the separation of components did not reduce much, contrary to what is expected from Eq. (57 *). The values of $αCEλ$ , which appeared necessary for reproducing the observed systems, turned out to be negative, which means that the simple energy conservation law (57 *) is violated in this case. Instead, Nelemans et al. suggested that the first stage of mass exchange, between a giant and a main-sequence star with comparable masses ( $q ≥ 0.5$ ), can be described by what they called “ $γ$ -formalism”, in which not the energy but the angular momentum $J$ is balanced and conservation of energy is implicitly implied, though formally this requires $αCE λ > 1$ :

Here $ΔM$ is the mass lost by the donor, $Mtot$ is the total mass of the binary system before formation of the common envelope, and $γ$ is a numerical coefficient. Similar conclusions were later reached by Nelemans and Tout [516 *] and van der Sluys et al. [803 ] who analyzed larger samples of binaries and used detailed fits to evolutionary models instead of simple “core-mass – stellar radius” relation applied by Nelemans et al. [517 *]. It turned out that combination of $γ = 1.75$ for the first stage of mass exchange with $αCE λ = 2$ for the second episode of mass exchange enables, after taking into account selection effects, a satisfactory model for population of close binary WD with known masses. In fact, findings of Nelemans et al. and van der Sluys et al. confirmed the need for a loss of mass and momentum from the system to explain observations of binaries mentioned in Section 3.3. (Note that $γ = 1$ corresponds to the loss of the angular momentum by a fast stellar wind, which always increases the orbital separation of the binary.) We stress that no physical process behind $γ$ -formalism has been suggested as yet, and the model remains purely phenomenological and should be further investigated.¹⁹

We should note that “ $γ$ -formalism” was introduced under the assumption that stars with deep convective envelopes (giants) always lose mass unstably, i.e., Ṁ is high. As mentioned above, if the mass exchange is nonconservative, the associated angular momentum loss can increase the specific orbital angular momentum of the system $∝ √a--$ and hence the orbital separation $a$ . Then the increasing Roche lobe can accommodate the expanding donor, and the formation of a common envelope can be avoided. In fact, this happens if part of the energy released by accretion is used to expel the matter from the vicinity of the accretor (see, e.g., computations presented by King and Ritter [365 ], Beer et al. [30 ], Woods et al. [849 ] who assumed that reemission occurs). It is not excluded that such nonconservative mass-exchange with mass and momentum loss from the system may be the process underlying the “ $γ$ -formalism”.

Nelemans and Tout [516 ], Zorotovic et al. [886 ] and De Marco et al. [139 ] attempted to estimate $αCE$ using samples of close WD + M-star binaries. It turned out that their formation can be explained both if $αCEλ > 0$ in Eq. (56 *) or if $γ ≈ 1.5$ in Eq. (59 *). This is sometimes considered as an argument against the $γ$ -formalism. But we note that in progenitors of WD + M-star binaries the companions to giants are low-mass small-radius objects, which are quite different from $M ≥ M ⊙$ companions to WD in progenitors of WD + WD binaries and, therefore, the energetics of the evolution in CE in precursors of two types of systems can differ. In a recent population synthesis study of the post-common-envelope binaries aimed at comparing to SDSS stars and taking into account selection effects, Toonen and Nelemans [765 ] concluded that the best fit to the observations can be obtained for small universal $αCE λ = 0.25$ without the need to invoke the $γ$ -formalism. However, they also note that for almost equal mass precursors of WD binaries the widening of the orbit in the course of mass transfer is needed (like in $γ$ -formalism), while the formation of low mass-ratio WD+M-star systems requires diminishing of the orbital separation (as suggested by $α$ -formalism).

3.7 Other notes on the CE problem

A particular case of common envelopes (“double spiral-in”) occurs when both components are evolved. It was suggested [520 *] that a common envelope is formed by the envelopes of both components of the system and the binding energy of them both should be taken into account, i.e., the outcome of CE must be found from a solution of an equation

Formulations of the common envelope equation different from Eq. (56 *) are found in the literature (see, e.g., [139 ] for a review); $a ∕a f i$ similar to the values produced by Eq. (56 *) are then obtained for different $αceλ$ values.

Common envelope events are expected to be rare ( $≲ 0.1 yr−1$ [791 *]) and short-lasting ( $∼$ years). Thus, the binaries at the CE-stage itself are difficult to observe, despite energy being released at this stage comparable to the binding energy of a star, and the evidence for them comes from the very existence of compact object binaries, as described above in Section 3.6.1. Recently, it was suggested that luminous red transients – the objects with peak luminosity intermediate between the brightest Novae and SN Ia (see, e.g., [342 ]) – can be associated with CE [316 , 792 ].

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 the Roche-lobe filling star, and possible stabilizing effects of mass and momentum loss from the system [780 , 283 , 873 *, 261 , 268 *, 318 *, 196 , 76 , 849 ]. We note especially recent claims [848 , 570 ] that the local thermal timescale of the super-adiabatic layer existing over a convective envelope of giants may be shorter than the donor’s dynamic timescale. As a result, giants may adjust to a high mass loss and retain their radii or even contract (instead of dramatically expanding, as expected in the adiabatic approximation [224 , 225 ]). As well, the response of stars to mass loss evolves in the course of the latter depending on whether the super-adiabatic layer may be removed. The recent discovery of young helium WD companions to blue stragglers in wide binaries ( $Porb$ from 120 to 3010 days) in open cluster NGC 188 [248 ] supports the idea that in certain cases RLOF in wide systems may be stable. These considerations, which need further systematic exploration, may change the “standard” paradigm of the stability of mass exchange formulated in this section.

	Abstract
1	Introduction
	1.1	Formation of stars and end products of their evolution
	1.2	Binary stars
2	Observations of Double Compact Stars
	2.1	Compact binaries with neutron stars
	2.2	How frequent are NS binary coalescences?
	2.3	Black holes in binary systems
	2.4	A model-independent upper limit on the BH-BH/BH-NS coalescence rate
3	Basic Principles of the Evolution of Binary Stars
	3.1	Keplerian binary system and radiation back reaction
	3.2	Mass exchange in close binaries
	3.3	Mass transfer modes and mass and angular momentum loss in binary systems
	3.4	Supernova explosion
	3.5	Kick velocity of neutron stars
	3.6	Common envelope stage
	3.7	Other notes on the CE problem
4	Evolutionary Scenario for Compact Binaries with Neutron Star or Black Hole Components
	4.1	Compact binaries with neutron stars
	4.2	Black-hole–formation parameters
5	Formation of Double Compact Binaries
	5.1	Analytical estimates
	5.2	Population synthesis results
6	Detection Rates
7	Short-Period Binaries with White-Dwarf Components
	7.1	Formation of compact binaries with white dwarfs
	7.2	White-dwarf binaries
	7.3	Type Ia supernovae
	7.4	Ultra-compact X-ray binaries
8	Observations of Double-Degenerate Systems
	8.1	Detached white dwarf and subdwarf binaries
9	Evolution of Interacting Double-Degenerate Systems
	9.1	“Double-degenerate family” of AM CVn stars
	9.2	“Helium-star family” of AM CVn stars
	9.3	Final stages of evolution of interacting double-degenerate systems
10	Gravitational Waves from Compact Binaries with White-Dwarf Components
11	AM CVn-Type Stars as Sources of Optical and X-Ray Emission
12	Conclusions
	Acknowledgments
	References
	Footnotes
	Updates
	Figures
	Tables