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 [474, 74].
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 ) with an index
. The
polytropic approximation with
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,
, 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
, and white dwarfs if
.
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 [66, 12]; 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 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 [444, 73]. The orbital
evolution of the compact star
inside the envelope of the normal star
is driven by the
dynamical friction drag. This leads to a gradual spiral-in process of the compact star. The
released orbital energy
, or a fraction of it, can become numerically equal to the binding
energy
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
:
where is the total binding energy of the envelope and
is the orbital energy released in the
spiral-in. What remains of the normal star
is its stellar core
. The above energy condition reads
The above formalism for the common envelope stage depends in fact on the product of two parameters:
, which is the measure of the binding energy of the envelope to the core prior to mass transfer in a
binary system, and
, which is the common envelope efficiency itself. Numerical calculations of evolved
giant stars with masses
[81] showed that the value of the
-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
(
), which are appropriate for the formation of BH binaries, the
-parameter was found to
depend on the mass of the star and vary within a wide range 0.01 – 0.5 [322
]. Some hydrodynamical
simulations [342] indicated that
, while in others [368
] a wider range for values of
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 . 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 [285
],
extended in [284
, 431], 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
-formalism for the common envelope, in which not the energy
but the angular momentum
is balanced and conservation of energy is implicitly implied:
Note also, that formulations of the common envelope equation different from Equation (50) are met in
the literature (see, e.g., [416, 160]);
similar to the values produced by Equation (50
) are then
obtained for different
values.
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