3.5 Common envelope stage

This is a very important stage in the evolution of binaries of all masses. In different contexts, the evolution of binary components in an envelope engulfing both of them was considered, for instance, in [3453846], but the importance of common envelopes was really recognized after Paczyński [302Jump To The Next Citation Point] and Ostriker [299] applied them to explain the formation of cataclysmic variables and massive X-ray binaries, respectively. A detailed review of problems related to common envelopes may be found, e.g., in [160Jump To The Next Citation Point393].

Generally, common envelopes form in binary systems where the mass transfer from the mass-losing star is high, and the companion cannot accommodate all the accreting matter. The common envelope stage appears unavoidable on observational grounds. The evidence for a dramatic orbital angular momentum decrease in some preceding evolutionary stage follows from observations of certain types of close binary stars. They include cataclysmic variables, in which a white dwarf accretes matter from a small red dwarf main-sequence companion, planetary nebulae with double cores, low-mass X-ray binaries and X-ray transients (neutron stars and black holes accreting matter from low-mass main-sequence dwarfs). The radii of progenitors of compact stars in these binaries typically should have been 100 – 1000 solar radii, that is, much larger than the currently observed binary separations. This testifies of some dramatic reduction of the orbital momentum in the earlier stages of evolution and eventual removal of the common envelope. Additional indirect evidence for reality of the common envelope stage in the typical pre-cataclysmic binary V471 Tau has recently been obtained from X-ray Chandra observations [87] showing anomalous C/N contamination of the K-dwarf companion. Recent studies also indicate that many planetary nebulae are actually binaries, which may suggest that most of them result from common envelope interaction [47474].

Exact criteria for the formation of a common envelope are absent. 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.e. the equation of state can be written as P = K ρ1+1∕n) 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 well known inverse mass-radius relation, −1∕3 R ∝ M, first measured 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 a continuation of an unstable mass loss and eventual formation of a common envelope. The critical mass ratio for the unstable Roche lobe overflow depends on specifics of the stellar structure and mass ratio of components; typically, mass loss is unstable for stars with convective envelopes, stars with radiative envelopes if q ≳ 2, and white dwarfs if q ≳ 2∕3.

As other examples for the formation of a common envelope one may consider, for instance, direct penetration of a compact star into the dense outer layers of the companion, which can happen as a result of the Darwin tidal orbital instability in binaries [6612]; it is 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 stellar radius of the companion; a common envelope enshrouding both components of a binary may form due to unstable thermonuclear burning in the surface layers of an accreting WD.

The common envelope stage is, usually, treated in the following simplified way [444Jump To The Next Citation Point73]. The orbital evolution of the compact star m inside the envelope of the normal star M1 is driven by the dynamical friction drag. This leads to a gradual spiral-in process of the compact star. The released orbital energy ΔEorb, or a fraction of it, can become numerically equal to the binding energy E bind of the envelope with the rest of the binary system. It is generally assumed that the orbital energy of the binary is used to expel the envelope of the donor with an efficiency αce:

Ebind = αceΔEorb,

where Ebind is the total binding energy of the envelope and ΔEorb is the orbital energy released in the spiral-in. What remains of the normal star M1 is its stellar core Mc. The above energy condition reads

GM (M − M ) ( GmM GM m ) ----1---1-----c-= αce ------c− ----1-- , (50 ) λRL 2af 2ai
where ai and af are the initial and the final orbital separations, and λ is a numerical coefficient that depends on the structure of the donor’s envelope. RL is the Roche lobe radius of the normal star that can be approximated as [91]
RL 0.49 ---= -------2∕3--------1∕3- (51 ) ai 0.6 + q ln(1 + q )
and q ≡ M1 ∕m. From Equation (50View Equation) one derives
af Mc ( 2ai M1 − Mc )− 1 Mc Mc -- = ---- 1 + ----------------- ≲ --------, (52 ) ai M1 λαceRL m M1 ΔM
where ΔM = M1 − Mc is the mass of the ejected envelope. The mass Mc of a helium core of a massive star may be approximated as [415Jump To The Next Citation Point]
1.4 MHe ≈ 0.1(M1 ∕M ⊙) , (53 )
so the orbital separation during the common envelope stage may decrease as much as by factor 100 and more.

The above formalism for the common envelope stage depends in fact on the product of two parameters: λ, which is the measure of the binding energy of the envelope to the core prior to mass transfer in a binary system, and αce, which is the common envelope efficiency itself. Numerical calculations of evolved giant stars with masses 3– 10M ⊙ [81] showed that the value of the λ-parameter is typically between 0.2 and 0.8; however, it can be as high as 5 on the asymptotic giant branch. For more massive primaries (> 20 M ⊙), which are appropriate for the formation of BH binaries, the λ-parameter was found to depend on the mass of the star and vary within a wide range 0.01 – 0.5 [322Jump To The Next Citation Point]. Some hydrodynamical simulations [342] indicated that αce ≃ 1, while in others [368Jump To The Next Citation Point] a wider range for values of αce was obtained. There are debates in the literature as to should additional sources of energy (e.g., ionization energy in the envelope [136]) should be included in the ejection criterion of common envelopes [382].

There is another approach, different from the standard Webbink formalism, which is used to estimate the common envelope efficiency αce. 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.e. from the red giant to the white dwarf remnant of the original primary in the system). Such an analysis [285Jump To The Next Citation Point], extended in [284Jump To The Next Citation Point431], suggests that the standard energy prescription for the treatment of the common envelope stage cannot be applied to the first mass transfer episode. Instead, the authors proposed to apply the so-called γ-formalism for the common envelope, in which not the energy but the angular momentum J is balanced and conservation of energy is implicitly implied:

δJ-= γ ΔM---, (54 ) J Mtot
where ΔM is the mass of the common envelope, Mtot is the total mass of the binary system before the common envelope, and γ is a numerical coefficient. For all binary systems considered, the parameter γ was found to be within the range 1.1 – 4, with the mean value around 1.5. (Notice that γ = 1 corresponds to loss of the angular momentum through stellar wind, as considered above, which always increases the orbital separation of the binary.) The applicability of this algorithm should be investigated further.

Note also, that formulations of the common envelope equation different from Equation (50View Equation) are met in the literature (see, e.g., [416160]); a ∕a f i similar to the values produced by Equation (50View Equation) are then obtained for different αceλ values.

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