Common Envelope Evolution Redux
Ronald F. Webbink
Department of Astronomy, University of Illinois, 1002 W. Green St., Urbana, IL
61801, USA
Summary. Common envelopes form in dynamical time scale mass exchange, when
the envelope of a donor star engulfs a much denser companion, and the core of
the donor plus the dense companion star spiral inward through this dissipative en-
velope. As conceived by Paczynski and Ostriker, this process must be responsible
for the creation of short-period binaries with degenerate components, and, indeed,
it has proven capable of accounting for short-period binaries containing one white
dwarf component. However, attempts to reconstruct the evolutionary histories of
close double white dwarfs have proven more problematic, and point to the need for
enhanced systemic mass loss, either during the close of the first, slow episode of
mass transfer that produced the first white dwarf, or during the detached phase
preceding the final, common envelope episode. The survival of long-period inter-
acting binaries with massive white dwarfs, such as the recurrent novae T CrB and
RS Oph, also presents interpretative difficulties for simple energetic treatments of
common envelope evolution. Their existence implies that major terms are missing
from usual formulations of the energy budget for common envelope evolution. The
most plausible missing energy term is the energy released by recombination in the
common envelope, and, indeed, a simple reformulation the energy budget explicitly
including recombination resolves this issue.
1 Introduction
From the realization [26, 27, et seq.] that all cataclysmic variables (CVs)
are interacting binary stars, their existence posed a dilemma for theories of
binary evolution. The notion that close binary stars might evolve in ways
fundamentally different from isolated stars was rooted in the famous ‘Algol
paradox’ (that the cooler, lobe-filling subgiant or giant components among
these well-known eclipsing binaries are less massive, but more highly evolved,
than their hotter main-sequence companions). The resolution of that para-
dox invoked large-scale mass transfer reversing the initial mass ratios of these
binaries [34]. Indeed, model calculations assuming conservation of total mass
and orbital angular momenum are qualitatively consistent with the main fea-
tures of Algol-type binaries. Even if quantitative consistency between models
http://arxiv.org/abs/0704.0280v1
2 Ronald F. Webbink
and observational data generally requires some losses of mass and angular
momentum among Algol binaries (e.g., [10, 21, 8]), the degree of those losses
is typically modest, and the remnant binary is expected to adhere closely to
an equilibrium core mass-radius relation for low-mass giant stars (see, e.g.,
the pioneering study of AS Eri by Refsdal, Roth & Weigert [46]). Those rem-
nant binaries are typically of long orbital period (days to weeks) in compari-
son with CVs, and furthermore typically contain helium white dwarfs of low
mass, especially in the short-period limit. In contrast, CVs evidently contain
relatively massive white dwarfs, in binary systems of much shorter orbital
periods (hours), that is, with much smaller total energies and orbital angular
momenta.
In an influential analysis of the Hyades eclipsing red dwarf/white dwarf
binary BD +16◦ 516 (= V471 Tau), Vauclair [57] derived a total system mass
less than the turnoff mass of the Hyades, and noted that the cooling age of
the white dwarf component was much smaller than the age of the cluster.
He speculated that V471 Tau in its present state was the recent product of
the ejection of a planetary nebula by the white dwarf. Paczynski [41] realized
that, immediately prior to that event, the white dwarf progenitor must have
been an asymptotic giant branch star of radius ∼600 R⊙, far exceeding its
current binary separation ∼3R⊙. He proposed that the dissipation of orbital
energy provided the means both for planetary nebula ejection and for the
severe orbital contraction between initial and final states, a process he labeled
‘common envelope evolution’ (not to be confused with the common envelopes
of contact binary stars). Discovery soon followed of the first ‘smoking gun’,
the short-period eclipsing nucleus of the planetary nebula Abell 63 [3].
Over the succeeding three decades, there have been a number of attempts
to build detailed physical models of common envelope evolution (see [55] for
a review). These efforts have grown significantly in sophistication, but this
phenomenon presents a daunting numerical challenge, as common envelope
evolution is inherently three-dimensional, and the range of spatial and tem-
poral scales needed to represent a common envelope binary late in its inspiral
can both easily exceed factors of 103. Determining the efficiency with which
orbital energy is utilized in envelope ejection requires such a code to conserve
energy over a similarly large number of dynamical time scales.
Theoretical models of common envelope evolution are not yet capable of
predicting the observable properties of objects in the process of inspiral. If
envelope ejection is to be efficient, then the bulk of dissipated orbital energy
must be deposited in the common envelope on a time scale short compared
with the thermal time scale of the envelope, else that energy be lost to ra-
diation. The duration of the common envelope phase thus probably does not
exceed ∼103 years. However, general considerations of the high initial orbital
angular momenta of systems such as the progenitor of V471 Tau, and the fact
that most of the orbital energy is released the envelope only very late in the
inspiral have led to a consensus view [60, 32, 33, 65, 50] that the planetary
nebulae they eject should be bipolar in structure, with dense equatorial rings
Common Envelope Evolution Redux 3
absorbing most of the initial angular momentum of the binary, and higher-
velocity polar jets powered by the late release of orbital energy. Indeed, this
appears to be a signature morphology of planetary nebulae with binary nuclei
(e.g., [2]), although it may not be unique to binary nuclei.
2 The Energetics of Common Envelope Evolution
Notwithstanding the difficulties in modeling common envelope evolution in
detail, it is possible to calculate with some confidence the initial total energy
and angular momentum of a binary at the onset of mass transfer, and the
corresponding orbital energy and angular momentum of any putative remnant
of common envelope evolution.
Consider an initial binary of component masses M1 and M2, with orbital
semimajor axis Ai. Its initial total orbital energy is
Eorb,i = −
GM1M2
. (1)
Let star 1 be the star that initiates interaction upon filling its Roche lobe. If
M1c is its core mass, and M1e = M1 − M1c its envelope mass, then we can
write the initial total energy of that envelope as
Ee = −
GM1M1e
λR1,L
, (2)
where R1,L is the Roche lobe radius of star 1 at the onset of mass transfer (the
orbit presumed circularized prior to this phase), and λ is a dimensionless pa-
rameter dependent on the detailed structure of the envelope, but presumably
of order unity. For very simplified models of red giants – condensed poly-
tropes [40, 14, 16] – λ is a function only of me ≡ Me/M = 1 − Mc/M , the
ratio of envelope mass to total mass for the donor, and is well-approximated
λ−1 ≈ 3.000− 3.816me + 1.041m
e + 0.067m
e + 0.136m
e , (3)
to within a relative error < 10−3.
For the final orbital energy of the binary we have
Eorb,f = −
GM1cM2
, (4)
where Af is of course the final orbital separation. If a fraction αCE of the
difference in orbital energy is consumed in unbinding the common envelope,
αCE ≡
orb − E
, (5)
4 Ronald F. Webbink
αCEλr1,L
M1 −M1c
, (6)
where r1,L ≡ R1,L/Ai is the dimensionless Roche lobe radius of the donor
at the start of mass transfer. In the classical Roche approximation, r1,L is a
function only of the mass ratio, q ≡ M1/M2 [7]:
r1,L ≈
0.49q2/3
0.6q2/3 + ln(1 + q1/3)
. (7)
Typically, the second term in brackets in (6) dominates the first term.
As formulated above, our treatment of the outcome of common envelope
evolution neglects any sources or sinks of energy beyond gravitational terms
and the thermal energy content of the initial envelope (incorporated in the
parameter λ). The justification for this assumption is again that common
envelope evolution must be rapid compared to the thermal time scale of the
envelope. This implies that radiative losses (or nuclear energy gains – see
below) are small. They, as well as terminal kinetic energy of the ejecta, are
presumably reflected in ejection efficiencies αCE < 1. We neglect also the
rotational energy of the common envelope (invariably small in magnitude
compared to its gravitational binding energy), and treat the core of the donor
star and the companion star as inert masses, which neither gain nor lose
mass or energy during the course of common envelope evolution. One might
imagine it possible that net accretion of mass by the companion during inspiral
might compromise this picture. However, the common envelope is typically
vastly less dense than the companion star. and may be heated to roughly
virial temperature on infall. A huge entropy barrier arises at the interface
between the initial photosphere of the companion and the common envelope
in which it is now embedded, with a difference in entropy per particle of order
(µmH/k)∆s ≈ 4–6. The rapid rise in temperature and decrease in density
through the interface effectively insulates the accreting companion thermally,
and strongly limits the fraction of the very rarified common envelope it can
retain upon exit from that phase [62, 15].
Common envelope evolution entails systemic angular momentum losses as
well as systemic mass and energy losses. Writing the orbital angular momen-
tum of the binary,
GM21M
2A(1 − e
M1 +M2
, (8)
in terms of the total orbital energy, E = −GM1M2/2A, we find immediately
that the ratio of final to initial orbital angular momentum is
)3/2 (
M1c +M2
M1 +M2
)−1/2 (
)1/2 (
1− e2f
1− e2i
. (9)
Since M1c < M1 and we expect the initial orbital eccentricity to be small (ei ≈
0), it follows that any final energy state lower than the initial state (|Ef | >
Common Envelope Evolution Redux 5
|Ei|) requires the loss of angular momentum. The reverse is not necessarily
true, so it is the energy budget that most strongly constrains possible outcomes
of common envelope evolution.
3 Does Common Envelope Evolution Work?
As an example of common envelope energetics, let us revisit the pre-CV
V471 Tau, applying the simple treatment outlined above. It is a member of
the Hyades, an intermediate-age metal-rich open cluster (t = 650 Myr, [Fe/H]
= +0.14) with turnoff mass MTO = 2.60 ± 0.06M⊙ [28]. The cooling age of
the white dwarf is much smaller than the age of the cluster (tcool,WD = 10
yr [39] – but see the discussion there of the paradoxical fact that this most
massive of Hyades white dwarfs is also the youngest). Allowing for the possi-
bility of significant mass loss in a stellar wind prior to the common envelope
phase, we may take MTO for an upper limit to the initial mass M1 of the
white dwarf component. The current masses for the white dwarf and its dK2
companion, as determined by O’Brien et al. [39] are MWD = 0.84± 0.05M⊙,
MK = 0.93±0.07M⊙, with orbital separation A = 3.30±0.08R⊙. A 2.60M⊙
star of Hyades metallicity with a 0.84M⊙ core lies on the thermally-pulsing
asymptotic giant branch, with radius (maximum in the thermal pulse cycle)
which we estimate at Ri = 680R⊙ = R1,L, making Ai = 1450R⊙. With this
combination of physical parameters, we derive an estimate of αCEλ = 0.057
for V471 Tau. Equation (3) then implies αCE = 0.054. This estimate of course
ignores any mass loss prior to common envelope evolution (which would drive
αCE to lower values), or orbital evolution since common envelope evolution
(which would drive αCE to higher values). In any event, the status of V471 Tau
would appear to demand only a very small efficiency of envelope ejection.1
The fact that V471 Tau is a double-lined eclipsing member of a well-
studied cluster provides an exceptionally complete set of constraints on its
prior evolution. In all other cases of short-period binaries with degenerate or
compact components, available data are inadequate to fix simultaneously both
the initial mass of the compact component and the initial binary separation,
for example. To validate the energetic arguments outlined above, one must
resort to consistency tests, whether demonstrating the existence of physically-
plausible initial conditions that could produce some individual system, or else
1 The anomalously small value of αCE deduced for V471 Tau may be connected to
its puzzlingly high white dwarf mass and luminosity: O’Brien et al. [39] suggest
that it began as a heirarchical triple star, in which a short-period inner binary
evolved into contact, merged (as a blue straggler), and later engulfed its lower-
mass companion in a common envelope. An overmassive donor at the onset of
common envelope evolution would then have a more massive core than produced
by its contemporaries among primordially single stars, and it would fill its Roche
lobe with a more massive envelope at somewhat shorter orbital period, factors all
consistent with a larger value of αCE having led to V471 Tau as now observed.
6 Ronald F. Webbink
following a plausible distribution of primordial binaries wholesale through the
energetics of common envelope evolution and showing that, after application
of appropriate observational selection effects, the post-common-envelope pop-
ulation is statistically consistent with the observed statistics of the selected
binary type. In the cases of interacting binaries, such as CVs, one should allow
further for post-common-envelope evolution. Nevertheless, within these limi-
tations, binary population synthesis models show broad consistency between
the outcomes of common envelope evolution and the statistical properties of
CVs and pre-CVs [4, 24, 44, 17, 63], as well as with most super-soft X-ray
sources [5], for assumed common envelope ejection efficiencies typically of or-
der αCE ≈ 0.3–0.5.
A useful tool in reconstructing the evolutionary history of a binary, used
implicitly above in analyzing V471 Tau, is the mass-radius diagram spanned
by single stars of the same composition as the binary. Figure 1 illustrates
such a diagram for solar-composition stars from 0.08 M⊙ to 50 M⊙. In it are
plotted various critical radii marking, as a function of mass, the transition
from one evolutionary phase to the next.2 Since the Roche lobe of a binary
component represents a dynamical limit to its size, its orbital period fixes the
mean density at which that star fills its Roche lobe,
logPorb(d) ≈
log(RL/R⊙)−
log(M/M⊙)− 0.455 , (10)
to within a very weak function of the binary mass ratio. The mass and radius of
any point in Fig. 1 therefore fixes the orbital period at which such a star would
fill its Roche lobe, just as the orbital period of a binary fixes the evolutionary
state at which such a star initiates mass transfer.
2 Not all evolutionary phases are represented here. In a binary, a donor initiates
mass transfer when it first fills its Roche lobe; if it would have done so at a prior
stage of evolution, then its present evolutionary state is ‘shadowed’, in the sense
that it only occurs by virtue of the binary not having filled its lobe previously.
Thus, for example, low- and intermediate-mass stars cannot in general initiate
mass transfer during core helium burning, because they would have filled their
Roche lobes on the initial ascent of the giant branch.
Fig. 1 (facing page). The mass-radius diagram for stars of solar metallicity, con-
structed from the parametric models of stellar evolution by Hurley, Pols, & Tout [19]
and models of thermally-pulsing asymptotic giant branch stars by Wagenhuber &
Weiss [58]. Also plotted in the locus of asymptotic giant branch stars at the onset
of the superwind, after Willson [64]; beyond this radius, systemic mass loss drives
orbital expansion faster than nuclear evolution drives stellar expansion, and a bi-
nary will no longer be able to initiate tidal mass transfer. The unlabeled dotted
line terminating at the junction between lines labeled ‘helium core flash’ and ‘core
helium ignition’ marks the division between those helium cores (at lower masses)
which evolve to degeneracy if stripped of their envelope, and those (at higher masses)
which ignite helium non-degenerately and become helium stars.
Common Envelope Evolution Redux 7
8 Ronald F. Webbink
In Fig. 2, the corresponding core masses of low- and intermediate-mass
stars are plotted in the mass-radius diagram. For a binary which is the imme-
diate product of common envelope evolution, the mass of the most recently
formed white dwarf (presumably the spectroscopic primary) equals the core
mass of the progenitor donor star. That donor (presuming it to be of solar
metallicity) must be located somewhere along the corresponding core mass
sequence in Fig. 2, with the radius at any point along that sequence corre-
sponding to the Roche lobe radius at the onset of the mass transfer, and the
mass a that point corresponding to the initial total mass of the donor. Thus,
if the mass of the most recently-formed white dwarf is known, it is possible
to identify a single-parameter (e.g., initial mass or initial radius of the donor)
family of possible common-envelope progenitors.
Using a mapping procedure similar to this, Nelemans & Tout [35] recently
explored possible progenitors for detached close binaries with white dwarf
components. Broadly speaking, they found solutions using (6) for almost all
systems containing only one white dwarf component. Only three putative
post-common-envelope systems failed to yield physically-plausible values of
αCEλ: AY Cet (G5 III + DA, Porb = 56.80 d [49]), Sanders 1040 (in M67:
G4 III + DA, Porb = 42.83 d [56]), and HD 185510 (=V1379 Aql: gK0 +
sdB, Porb = 20.66 d [22]). The first two of these systems are non-eclipsing,
but photometric masses for their white dwarf components are extremely low
(estimated at ∼0.25M⊙ and 0.22M⊙, respectively), with Roche lobe radii
consistent with the limiting radii of very low-mass giants as they leave the
giant branch (cf. Fig. 2, above). They are thus almost certainly post-Algol
binaries, and not post-common-envelope binaries. HD 185510 is an eclipsing
binary; a spectroscopic orbit exists only for the gK0 component [9]. The mass
(0.304 ± 0.015M⊙) and radius (0.052 ± 0.010R⊙) of the sdB component,
deduced from model atmosphere fitting of IUE spectra combined with solution
of the eclipse light curve, place it on a low-mass white dwarf cooling curve,
rather than among helium-burning subdwarfs [22]. Indeed, from fitting very
detailed evolutionary models to this system, Nelson & Eggleton [38] found a
Fig. 2 (facing page). The mass-radius diagram for low- and intermediate-mass
stars, as in Fig. 1, but with loci of constant core mass added. The solid lines added
correspond to core masses interior to the hydrogen-burning shell, dashed lines to
those interior to the helium-burning shell. Solid lines intersecting the base of the
giant branch (dash-dotted curve) correspond to helium core masses of to 0.15, 0.25,
0.35, 0.5, 0.7, 1.0, 1.4, and 2.0 M⊙; those between helium ignition and the initial
thermal pulse to 0.7, 1.0, 1.4, and 2.0 M⊙, and those beyond the initial thermal pulse
to 0.7, 1.0, and 1.4 M⊙. Dashed lines between helium ignition and initial thermal
pulse correspond to carbon-oxygen core masses of 0.35, 0.5, 0.7, 1.0, and 1.4 M⊙.
Beyond the initial thermal pulse, helium and carbon-oxygen core masses converge,
with the second dredge-up phase reducing helium core masses above ∼0.8 M⊙ to
the carbon-oxygen core.
Common Envelope Evolution Redux 9
10 Ronald F. Webbink
post-Algol solution they deemed acceptable. It thus appears that these three
problematic binaries are products of quasi-conservative mass transfer, and not
common envelope evolution.
The close double white dwarfs present a more difficult conundrum, how-
ever. Nelemans et al. [36, 37, 35] found it impossible using the energetic argu-
ments (6) outlined above to account for the existence of a most known close
double white dwarfs. Mass estimates can be derived for spectroscopically de-
tectable components of these systems from their surface gravities and effective
temperatures (determined from Balmer line fitting). The deduced masses are
weakly dependent on the white dwarf composition, and may be of relatively
modest accuracy, but they are independent of the uncertainties in orbital
inclination afflicting orbital solutions. These mass estimates place the great
majority of detectable components in close double white dwarf binaries below
∼0.46M⊙, the upper mass limit for pure helium white dwarfs (e.g., [51]). They
are therefore pure helium white dwarfs, or perhaps hybrid white dwarfs (low-
mass carbon-oxygen cores with thick helium envelopes). While reconstructions
of their evolutionary history yield physically-reasonable solutions for the final
common envelope phase, with values for 0 < αCEλ < 1, the preceding phase of
mass transfer, which gave rise to the first white dwarf, is more problematic. If
it also proceeded through common envelope evolution, the deduced values of
αCEλ ≤ −4 for that phase are unphysical. Nelemans & Tout [35] interpreted
this paradox as evidence that descriptions of common envelope evolution in
terms of orbital energetics, as described above, are fundamentally flawed.
4 An Alternative Approach to Common Envelope
Evolution?
Nelemans et al. [36] proposed instead parameterizing common envelope evo-
lution in terms of γ, the ratio of the fraction of angular momentum lost to
the fraction of mass lost:
Ji − Jf
M1 −M1,c
M1 +M2
. (11)
Both initial and final orbits are assumed circular, so the ratio of final to initial
orbital separations becomes
M1c +M2
M1 +M2
M1 −M1c
M1 +M2
. (12)
Among possible solutions leading to known close double white dwarfs, Nele-
mans & Tout [35] find values 1 < γ . 4 required for the second (final) common
envelope phase, and 0.5 . γ < 3 for the first (putative) common envelope
phase. They note that values in the range 1.5 < γ < 1.7 can be found among
possible solutions for all common envelope phases in their sample, not only
Common Envelope Evolution Redux 11
those leading to known double white dwarfs, but those leading to known pre-
CV and sdB binaries as well.
The significance of this finding is itself open to debate. At one extreme, it
would seem implausible for any mechanism to remove less angular momentum
per unit mass than the orbital angular momentum per unit mass of either
component in its orbit (so-called Jeans-mode mass loss). At the other extreme,
a firm upper limit to γ is set by vanishing final orbital angular momentum,
Jf . If M1c and M2 can be regarded as fixed, the corresponding limits on γ are
M1 +M2
M1 −M1c
> γ >
M1 +M2
M1 −M1c
M1 +M2
M1c +M2
. (13)
In a fairly typical example, M1c = M2 =
M1, γ is inevitably tightly con-
strainted for any conceivable outcome: 5
> γ > 5
. The ratio of final to initial
orbital separation, Af/Ai, is extremely sensitive to γ near the upper limit of
its range. It is therefore not surprising to find empirical estimates of γ clus-
tering as they do – their values merely affirm the fact that Af must typically
be much smaller than Ai.
The unphysically large or, more commonly, negative values of αCEλ noted
above for the first mass transfer phase in the production of close white dwarf
binaries [35] implies that the orbital energies of these binaries have increased
through this phase (or, at any rate, decreased by significantly less than the
nominal binding energies of their common envelopes). Such an increase in
orbital energy is a hallmark of slow, quasi-conservative mass transfer, on a
thermal or nuclear time scale. Thermal time scale mass transfer is driven by
relaxation of the donor star toward thermal evolution; the re-expansion of
the donor following mass ratio reversal is powered by the (nuclear) energy
outflow from the core of the star. Likewise, the bulk expansion of the donor
star in nuclear time scale mass transfer draws energy from nuclear sources in
that star. It appears, therefore, that the first phase of mass transfer among
known close double white dwarfs cannot have been a common envelope phase,
but must instead have been a quasi-conservative phase, notwithstanding the
difficulties that conclusion presents, as we shall now see.
The dilemma that the close double white dwarfs present is illustrated in
Figs. 3 and 4. Figure 3 shows the distribution of immediate remnants of
mass transfer among solar-metallicity binaries of low and intermediate mass,
for a relatively moderate initial mass ratio. Conservation of total mass and
orbital angular momentum have been assumed. The remnants of the intial
primary include both degenerate helium white dwarfs, and nondegenerate he-
lium stars which have lost nearly all of their hydrogen envelopes. The helium
white dwarfs lie almost entirely along the left-hand boundary, the line labeled
‘envelope exhaustion’ in Fig. 1. (The extent of this sequence is more apparent
in the distribution of remnant secondaries.) Their progenitors have enough
angular momentum to accommodate core growth in the terminal phases of
mass transfer. In the calculation shown, the least massive cores grow from
12 Ronald F. Webbink
Common Envelope Evolution Redux 13
0.11M⊙ to ∼0.18M⊙ by the completion of mass transfer. In contrast, virtu-
ally all binaries leaving nondegenerate helium star remnants have too little
angular momentum to recover thermal equilibrium before they have lost their
hydrogen envelopes; for them, there is no slow nuclear time scale phase of
mass transfer, and core growth during mass transfer is negligible. The lowest-
mass helium star remnants have nuclear burning lifetimes comparable to their
hydrogen-rich binary companions, now grown through mass accretion. Those
more massive than ∼0.8M⊙ develop very extended envelopes during shell he-
lium burning, and will undergo a second phase of mass transfer from primary
to secondary, not reflected here; such massive white dwarfs are absent in the
Nelemans & Tout [35] sample, and so are omitted here.
In Fig. 4, the remnants of the first phase of conservative mass transfer
illustrated in Fig. 3 are followed through the second phase of mass transfer,
using (6). Because the remnants of the first phase have second-phase donors
much more massive their companions, and nearly all have deep convective en-
velopes, they are unstable to dynamical time scale mass transfer, and undergo
common envelope evolution. The systems labeled ‘Without Wind Mass Loss’
have been calculated assuming that no orbital evolution or mass loss occurs
between the end of the first phase of mass transfer and common envelope
evolution. It is assumed furthermore that αCE = 1, in principle marking the
most efficient envelope ejection energetically possible. Binary orbital periods
of 0.1, 1.0, and 10 days (assuming equal component masses) are indicated for
reference.
Observed close double white dwarfs, as summarized by Nelemans &
Tout [35], have a median orbital period of 1.4 d, and mass (spectroscopic pri-
mary) 0.39M⊙. Among double-lines systems, nearly equal white dwarf masses
are strongly favored, with the median q = 1.00. Clearly, most observed dou-
ble white dwarfs are too long in orbital period (have too much total energy
and angular momentum) to have evolved in the manner assumed here. Fur-
thermore, the computed binary mass ratios are typically more extreme than
observed, with the second-formed core typically 1.3-2.5 times as massive as
the first. The problem is that, while remnant white dwarfs or low-mass he-
Fig. 3 (facing page). Products of mass- and angular momentum-conservative mass
transfer for a typical initial mass ratio. The radii indicated refer to Roche lobe radii at
the onset or termination of mass transfer, as appropriate. To avoid common envelope
evolution, the donor stars (the region outlined in bold toward the lower right in the
diagram) must have radiative envelopes, and so arise between the terminal main
sequence and base of the giant branch. Their mass transfer remnants are outlined
in bold at the center-left of the diagram, with the remnant accretors at upper right.
The regions mapped are truncated in each case at a lower initial donor mass of
1.0 M⊙ and upper initial donor core mass of 0.7 M⊙. Lines of constant initial core
mass (with values as in Fig. 2) are indicated for the initial and remnant primaries.
Lines of constant remnant primary mass are indicated for the remnant secondaries.
14 Ronald F. Webbink
Common Envelope Evolution Redux 15
lium stars with suitable masses can be produced in the first, conservative mass
transfer phase, the remnant companions have envelope masses too large, and
too tightly bound, to survive the second (common envelope) phase of inter-
action at orbital separations and periods as large as observed. Evidently, the
progenitors of these double white dwarfs have lost a significant fraction of
their initial mass, while gaining in orbital energy, prior to the final common
envelope phase. These requirements can be fulfilled by a stellar wind, pro-
vided that the process is slow enough that energy losses in the wind can be
continuously replenished from nuclear energy sources.
The requisite mass loss and energy gain are possible with stellar wind
mass loss during the non-interactive phase between conservative and common
envelope evolution, or with stellar winds in nuclear time-scale mass transfer
or the terminal (recovery) phase of thermal time-scale mass transfer. Systemic
mass loss during or following conservative mass transfer will (in the absence
of angular momentum losses) shift the remnant regions to the left and upward
in Fig. 3 (subject to the limit posed by envelope exhaustion), while systemic
angular momentum losses shift them downwards. More extreme initial mass
ratios shift them downwards to the left.
The net effect of wind mass loss is illustrated by the regions labeled ‘With
Wind Mass Loss’ in Fig. 4. For simplicity, it is assumed here that half the
remnant mass of the original secondary was lost in a stellar wind prior to
the common envelope phase. Mass loss on this scale not only significantly re-
duces the mass of the second-formed white dwarf relative to the first, but the
concomitant orbital expansion produces wider remnant double white dwarfs,
bringing this snapshot model into good accord with the general properties of
real systems. Losses of this magnitude might be unprecedented among single
stars prior to their terminal superwind phase, but they have been a persis-
tent feature of evolutionary studies of Algol-type binaries [10, and references
Fig. 4 (facing page). Remnants of the second, common envelope, phase of mass
transfer of the systems shown in Fig. 3. Masses refer to the final remnants of the
original secondaries, and radii to their Roche lobe radii. Two groups of remnants are
shown. Those at lower right labeled ‘Without Wind Mass Loss’ follow directly from
the distributions of remnant primaries and secondaries shown in Fig. 3. Because
the remnant secondaries straddle the helium ignition line in Fig. 3, across which
core masses are discontinuous (see Fig. 2), the distribution of their post-common-
envelopes remnants is fragmented, some appearing as degenerate helium white dwarf
remnants (lower left), some as helium main sequence star remnants (lower center),
and the remainder as shell-burning helium star remnants (upper center). These latter
two groups overlap in the mass-radius diagram. The remnant distributions labeled
‘With Wind Mass Loss’ assume that the remnant secondaries of conservative mass
transfer lose half their mass in a stellar wind prior to common envelope evolution.
They too are fragmented, into degenerate helium white dwarf remnants (lower left)
and shell-burning helium stars (upper right). Within each group of remnants, lines
of constant remnant primary mass are shown, as in Fig. 3.
16 Ronald F. Webbink
therein] and, indeed, of earlier studies of close double white dwarf forma-
tion [11]. In the present context, their existence appears inescapable, if not
understood.
5 Long-Period Post-Common-Envelope Binaries and the
Missing Energy Problem
If the properties of short-period binaries with compact components can be
reconciled with the outcomes of common envelope evolution as expected from
simple energetics arguments, a challenge to this picture still comes from the
survival of symbiotic stars and recurrent novae at orbital separations too large
to have escaped tidal mass transfer earlier in their evolution. Notwithstanding
this author’s earlier hypothesis that the outbursting component in the recur-
rent nova T CrB (and its sister system RS Oph) might be a nondegenerate star
undergoing rapid accretion [59, 29, 61], it is now clear that the hot components
in both of these systems must indeed be hot, degenerate dwarfs [48, 6, 1, 18].
Furthermore, the short outburst recurrence times of these two binaries de-
mand that the degenerate dwarfs in each must have masses very close to the
Chandrasekhar limit.
The complexion of the problem posed by these systems can be illustrated
by a closer examination of T CrB itself. Its orbital period (P = 227.53 d)
and spectroscopic mass function (f(m) = 0.299M⊙) are well-established
from the orbit of the donor M3 III star [23]. The emission-line orbit for
the white dwarf [25] now appears very doubtful [18], but the system shows
very strong ellipsoidal variation (e.g., [1]), suggesting that the system is near
a grazing eclipse. Following Hric, et al. [18], I adopt MWD = 1.38 M⊙
amd MM3 = 1.2 M⊙. The Roche lobe radius of the white dwarf is then
RL,WD = 84 R⊙, nearly an order of magnitude larger than can be accom-
modated from the energetics arguments presented above, even for αCE = 1,
assuming solar metallicity for the system (see Fig. 5). A similar discrepancy
occurs for RS Oph.
It is evident that these long-period binaries are able to tap some en-
ergy source not reflected in the energy budget in (6). One possibility, dis-
cussed repeatedly in studies of planetary nebula ejection ([30, 42], more re-
cently [58, 12, 13]) is that the recombination energy of the envelope comes
Fig. 5 (facing page). Post-common-envelope masses and Roche lobe radii for bina-
ries consisting of a white dwarf or helium star plus a 1.2M⊙ companion, computed
with αCE = 1. Remnant systems inhabit the regions outlined in bold, and spanned
vertically by lines of constant white dwarf/helium star mass of 0.25, 0.35, 0.5, 0.7,
1.0, 1.4, and 2.0M⊙. Other initial sequences, encoded as in Fig. 2, have been mapped
through common envelope evolution. The location in this diagram of the white dwarf
in the recurrent nova T CrB is also indicated.
Common Envelope Evolution Redux 17
18 Ronald F. Webbink
into play. For solar composition material (and complete ionization), that re-
combination energy amounts to 15.4 eVamu−1, or 1.49 × 1013 erg g−1. For
tightly-bound envelopes on the initial giant branch of the donor, this term
is of little consequence; but near the tip of the low-mass giant branch, and
on the upper aymptotic giant branch of intermediate-mass stars, it can be-
come comparable with, or even exceed, the gravitational potential energy of
the envelope. In the model calculations of thermally-pulsing asymptotic giant
branch stars by Wagenhuber & Weiss [58], the threshold for spontaneous ejec-
tion by envelope recombination occurs consistently when the stellar surface
gravity at the peak thermal pulse luminosity falls to
log gHRI = −1.118± 0.042 . (14)
This threshold marks the presumed upper limit to the radii of lower-mass
asymptotic giant branch stars in Figs. 1 et seq. in the present paper. In fact,
the total energies of the envelopes of these stars formally becomes decidedly
positive even before the onset of the superwind phase, also shown in these
figures.
Whether single stars successfully tap this ionization energy in ejecting
planetary nebulae is still debated, but the circumstances of mass transfer in
binary systems would seem to provide a favorable environment for doing so.
In the envelopes of extended giants and asymptotic giant branch stars, pho-
tospheric electron densities and opacities are dominated by heavy elements;
the middle of the hydrogen ionization zone is buried at optical depths of order
τ ∼ 105. Adiabatic expansion of the envelope of the donor into the Roche lobe
of its companion can therefore trigger recombination even as the recombina-
tion radiation is itself trapped and reprocessed within the flow, much as the
same process occurs in rising convective cells.
Other possible energy terms exist that have been neglected in the energet-
ics arguments above: rotational energy, tidal contributions, coulomb energy,
magnetic fields, etc. But Virial arguments preclude most of these terms from
amounting to more than a minor fraction of the internal energy content of
the common envelope at the onset of mass transfer, when the energy budget
is established. The only plausible energy source of significance is the input
from nuclear reactions. In order for that input to be of consequence, it must
of course occur on a time scale short compared with the thermal time scale of
the common envelope. Taam [53] explored the possibility that shell burning
in an asymptotic giant branch core could be stimulated by mixing induced
dynamically in the common envelope (see also [54, 52]). Nothing came of
this hypothesis: mixing of fresh material into a burning shell required tak-
ing low-density, high-entropy material from the common envelope and mixing
it downward many pressure scale heights through a strongly stable entropy
gradient to the high-density, low-entropy burning region. In the face of strong
buoyancy forces, dynamical penetration is limited to scales of order a pressure
scale height.
Common Envelope Evolution Redux 19
6 Common Envelope Evolution with Recombination
The notion that recombination energy my be of importance specifically to
common envelope evolution is not new. It has been included, at least para-
metrically in earlier studies, for example, by Han et al. [13], who introduced
a second α-parameter, αth, characterizing the fraction of the initial thermal
energy content of the common envelope available for its ejection. The initial
energy kinetic/thermal content of the envelope is constrained by the Virial
Theorem, however, and it is not clear that there is a compelling reason for
treating it differently from, say, the orbital energy input from the inspiraling
cores. We choose below to formulate common envelope evolution in terms of
a single efficiency parameter, labeled here βCE to avoid confusion with αCE
as defined above.
By combining the standard stellar structure equations for hydrostatic equi-
librium and mass conservation, we can obtain an expression for the gravita-
tional potential energy, Ωe, of the common envelope:
Ωe ≡ −
dM = 3 PV
P dV , (15)
where subscripts c refer to the core-envelope boundary, and ∗ to the stellar
surface. This is, of course, the familiar Virial Theorem applied to a stellar
interior.
It is convenient to split the pressure in this integral into non-relativistic
(particle), Pg, and relativistic (photon), Pr, parts. The envelopes of giants un-
dergoing common envelope evolution are sufficiently cool and non-degenerate
to make the classical ideal gas approximation an excellent one for the particle
gas. One can then write
P = Pg + Pr =
ur , (16)
where ug and ur are kinetic energy densities of particle and radiation gases,
respectively. The total internal energy density of the gas is
u = ug + ur + uint , (17)
where the term uint now appearing represents non-kinetic contributions to
the total energy density of the gas, principally the dissociation and ionization
energies plus internal excitation energies of bound atoms and molecules. The
overwhelmingly dominant terms in uint are the ionization energies: uint ≈
ρχeff .
Integrating over the stellar envelope, we obtain for the total energy Ee of
the envelope:
Ee = Ωe + Ue
3 PU |
− 2Ug − Ur
+ (Ug + Ur + Uint)
= −4πR3cPc − Ug + Uint , (18)
20 Ronald F. Webbink
where we explicitly take P∗ → 0. In fact, experience shows that, for red-giant
like structures, Rc is so small that the first right-hand term in the last equality
can generally be neglected. In that case, we get the familiar Virial result, but
with the addition of a term involving the ionization/excitation/dissociation
energy available in the gas, Uint ≈ Meχeff , which becomes important for dif-
fuse, loosely-bound envelopes.
In the context of common envelope evolution, it is of course the dissipated
orbital energy, E
orb−E
orb, that must unbind the envelope. However, the inclu-
sion of Uint in Ee now opens the possibility that the common envelope began
with positive total energy; that is, in the usual αCE-prescription, it is possible
for λ−1 to be zero or even negative, which has the undesirable consequence
that αCE need not lie in the interval 0 ≤ αCE ≤ 1 for all physically-possible
outcomes. However, the gravitational potential energy of the envelope, Ωe, is
negative-definite, and by comparing it with all available energy sources (or-
bital energy released plus internal energy of the envelope), we can define an
ejection efficiency βCE that has the desired property, 0 ≤ βCE ≤ 1:
βCE ≡
orb − E
orb)− Ue
4πR3cPc + 2Ug + Ur
orb − E
orb + Ug + Ur + Uint
. (19)
By analogy to the form factor λ in the conventional αCE formalism above, we
can define separate form factors λΩ for the gravitational potential energy and
λP for the gas plus radiation contributions to the (kinetic) internal energy of
the envelope:
and Ug + Ur =
, (20)
In contrast, the recombination energy available can be written simply in terms
of an average ionization energy per unit mass,
Uint = Meχeff . (21)
The ratio of final to initial orbital separation then becomes
1 + 2
βCEλΩr1,L
λPr1,L
χeffAi
M1 −M1c
In the limit that radiation pressure Pr, ionization energy (Uint), and the
boundary term (4πR3cPc) are all negligible, then 2λΩ → λP → λ and
βCE → 2αCE/(1 + αCE).
Fig. 6 (facing page). Post-common-envelope masses and Roche lobe radii as in
Fig. 5, but with recombination energy included, computed from (22) with βCE = 1,
with the approximation 2λΩ = λP = λ from (3). At small separations, the differences
are inconsequential, but substantially larger final separations are allowed when Af &
10R⊙ (RL & 3R⊙).
Common Envelope Evolution Redux 21
22 Ronald F. Webbink
The ability to tap the recombination energy of the envelope has a profound
effect on the the final states of the longest-period intermediate-mass binaries,
those that enter common envelope evolution with relatively massive, degener-
ate carbon-oxygen (or oxygen-neon-magnesium) cores. As is evident in Fig. 6,
possible final states span a much broader range of final orbital separations.
Indeed, for the widest progenitor systems, the (positive) total energy of the
common envelope can exceed the (negative) orbital energy of the binary, mak-
ing arbitrarily large final semimajor axes energetically possible.3 The inclusion
of recombination energy brings both T CrB and RS Oph within energetically
accessible post-common-envelope states. It suffices as well to account for the
exceptionally long-period close double white binary PG 1115+166, as sug-
gested by Maxted, et al. [31].
7 Conclusions
Re-examination of global constraints on common envelope evolution leads to
the following conclusions:
Both energy and angular momentum conservation pose strict limits on the
outcome of common envelope evolution. Of these two constraints, however,
energy conservation is much the more demanding.
The recent study of close double white dwarf formation by Nelemans &
Tout [35] shows clearly that their progenitors can have lost little orbital en-
ergy through their first episodes of mass transfer. Since common envelope
ejection must be rapid if it is to be efficient, its energy budget is essentially
fixed at its onset by available thermal and gravitational terms. The preser-
vation of orbital energy through that first phase of mass transfer therefore
indicates that the observed close double white dwarfs escaped common en-
velope formation in that first mass transfer phase. They evidently evolved
through quasi-conservative mass transfer. However, strictly mass- and angular
momentum-conservative mass transfer leaves remnant accretors that are too
massive and compact to account for any but the shortest-period close double
white dwarfs. Significant mass loss and the input of orbital energy prior to the
onset of the second (common envelope) phase of mass transfer are required.
The requisite energy source must be of nuclear evolution, which is capable of
driving orbital expansion and stellar wind losses during the slower (thermal
recovery or nuclear time scale) phases of quasi-conservative mass transfer, or
during the interval between first and second episodes of mass transfer. Details
of this process remain obscure, however.
3 The final orbit remains constrained by the finite initial orbital angular momentum
of the binary. Final semimajor axes much in excess of the initial semimajor axis
may be energetically allowed, but the finite angular momentum available means
that they cannot be circular – see (9) – an effect which has been neglected in
Fig. 6.
Common Envelope Evolution Redux 23
Long-period cataclysmic variables such as T CrB and RS Oph pose a
more extreme test of common envelope energetics. With their massive white
dwarfs, the evident remnants of much more massive initial primaries, they
are nevertheless too low in total systemic mass to be plausible products of
quasi-conservative mass transfer, but too short in orbital period to have es-
caped tidal mass transfer altogether. They must be products of common en-
velope evolution, but to have survived at their large separations, they demand
the existence of a latent energy reservoir in addition to orbital energy to as-
sist in envelope ejection. It appears that these binaries efficiently tap ioniza-
tion/recombination energy in ejecting their common envelopes. That reservoir
is demonstrably adequate to account for the survival of these binaries. Its in-
clusion requires only a simple revision to the parameterization of common
envelope ejection efficiency.
Acknowledgement. This work owes its existence to both the encouragement and the
patience of Gene Milone, to whom I am most grateful. Thanks go as well to Ron
Taam for a useful discussion of possible loopholes in common envelope theory, and
to Jarrod Hurley for providing the source code described in Hurley, et al. (2000).
This work was supported in part by grant AST 0406726 to the University of Illinois,
Urbana-Champaign, from the US National Science Foundation.
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	Common Envelope Evolution Redux
	Ronald F. Webbink

  Common envelopes form in dynamical time scale mass exchange, when the
envelope of a donor star engulfs a much denser companion, and the core of the
donor plus the dense companion star spiral inward through this dissipative
envelope. As conceived by Paczynski and Ostriker, this process must be
responsible for the creation of short-period binaries with degenerate
components, and, indeed, it has proven capable of accounting for short-period
binaries containing one white dwarf component. However, attempts to reconstruct
the evolutionary histories of close double white dwarfs have proven more
problematic, and point to the need for enhanced systemic mass loss, either
during the close of the first, slow episode of mass transfer that produced the
first white dwarf, or during the detached phase preceding the final, common
envelope episode. The survival of long-period interacting binaries with massive
white dwarfs, such as the recurrent novae T CrB and RS Oph, also presents
interpretative difficulties for simple energetic treatments of common envelope
evolution. Their existence implies that major terms are missing from usual
formulations of the energy budget for common envelope evolution. The most
plausible missing energy term is the energy released by recombination in the
common envelope, and, indeed, a simple reformulation the energy budget
explicitly including recombination resolves this issue.

Introduction
From the realization [26, 27, et seq.] that all cataclysmic variables (CVs)
are interacting binary stars, their existence posed a dilemma for theories of
binary evolution. The notion that close binary stars might evolve in ways
fundamentally different from isolated stars was rooted in the famous ‘Algol
paradox’ (that the cooler, lobe-filling subgiant or giant components among
these well-known eclipsing binaries are less massive, but more highly evolved,
than their hotter main-sequence companions). The resolution of that para-
dox invoked large-scale mass transfer reversing the initial mass ratios of these
binaries [34]. Indeed, model calculations assuming conservation of total mass
and orbital angular momenum are qualitatively consistent with the main fea-
tures of Algol-type binaries. Even if quantitative consistency between models
http://arxiv.org/abs/0704.0280v1
2 Ronald F. Webbink
and observational data generally requires some losses of mass and angular
momentum among Algol binaries (e.g., [10, 21, 8]), the degree of those losses
is typically modest, and the remnant binary is expected to adhere closely to
an equilibrium core mass-radius relation for low-mass giant stars (see, e.g.,
the pioneering study of AS Eri by Refsdal, Roth & Weigert [46]). Those rem-
nant binaries are typically of long orbital period (days to weeks) in compari-
son with CVs, and furthermore typically contain helium white dwarfs of low
mass, especially in the short-period limit. In contrast, CVs evidently contain
relatively massive white dwarfs, in binary systems of much shorter orbital
periods (hours), that is, with much smaller total energies and orbital angular
momenta.
In an influential analysis of the Hyades eclipsing red dwarf/white dwarf
binary BD +16◦ 516 (= V471 Tau), Vauclair [57] derived a total system mass
less than the turnoff mass of the Hyades, and noted that the cooling age of
the white dwarf component was much smaller than the age of the cluster.
He speculated that V471 Tau in its present state was the recent product of
the ejection of a planetary nebula by the white dwarf. Paczynski [41] realized
that, immediately prior to that event, the white dwarf progenitor must have
been an asymptotic giant branch star of radius ∼600 R⊙, far exceeding its
current binary separation ∼3R⊙. He proposed that the dissipation of orbital
energy provided the means both for planetary nebula ejection and for the
severe orbital contraction between initial and final states, a process he labeled
‘common envelope evolution’ (not to be confused with the common envelopes
of contact binary stars). Discovery soon followed of the first ‘smoking gun’,
the short-period eclipsing nucleus of the planetary nebula Abell 63 [3].
Over the succeeding three decades, there have been a number of attempts
to build detailed physical models of common envelope evolution (see [55] for
a review). These efforts have grown significantly in sophistication, but this
phenomenon presents a daunting numerical challenge, as common envelope
evolution is inherently three-dimensional, and the range of spatial and tem-
poral scales needed to represent a common envelope binary late in its inspiral
can both easily exceed factors of 103. Determining the efficiency with which
orbital energy is utilized in envelope ejection requires such a code to conserve
energy over a similarly large number of dynamical time scales.
Theoretical models of common envelope evolution are not yet capable of
predicting the observable properties of objects in the process of inspiral. If
envelope ejection is to be efficient, then the bulk of dissipated orbital energy
must be deposited in the common envelope on a time scale short compared
with the thermal time scale of the envelope, else that energy be lost to ra-
diation. The duration of the common envelope phase thus probably does not
exceed ∼103 years. However, general considerations of the high initial orbital
angular momenta of systems such as the progenitor of V471 Tau, and the fact
that most of the orbital energy is released the envelope only very late in the
inspiral have led to a consensus view [60, 32, 33, 65, 50] that the planetary
nebulae they eject should be bipolar in structure, with dense equatorial rings
Common Envelope Evolution Redux 3
absorbing most of the initial angular momentum of the binary, and higher-
velocity polar jets powered by the late release of orbital energy. Indeed, this
appears to be a signature morphology of planetary nebulae with binary nuclei
(e.g., [2]), although it may not be unique to binary nuclei.
2 The Energetics of Common Envelope Evolution
Notwithstanding the difficulties in modeling common envelope evolution in
detail, it is possible to calculate with some confidence the initial total energy
and angular momentum of a binary at the onset of mass transfer, and the
corresponding orbital energy and angular momentum of any putative remnant
of common envelope evolution.
Consider an initial binary of component masses M1 and M2, with orbital
semimajor axis Ai. Its initial total orbital energy is
Eorb,i = −
GM1M2
. (1)
Let star 1 be the star that initiates interaction upon filling its Roche lobe. If
M1c is its core mass, and M1e = M1 − M1c its envelope mass, then we can
write the initial total energy of that envelope as
Ee = −
GM1M1e
λR1,L
, (2)
where R1,L is the Roche lobe radius of star 1 at the onset of mass transfer (the
orbit presumed circularized prior to this phase), and λ is a dimensionless pa-
rameter dependent on the detailed structure of the envelope, but presumably
of order unity. For very simplified models of red giants – condensed poly-
tropes [40, 14, 16] – λ is a function only of me ≡ Me/M = 1 − Mc/M , the
ratio of envelope mass to total mass for the donor, and is well-approximated
λ−1 ≈ 3.000− 3.816me + 1.041m
e + 0.067m
e + 0.136m
e , (3)
to within a relative error < 10−3.
For the final orbital energy of the binary we have
Eorb,f = −
GM1cM2
, (4)
where Af is of course the final orbital separation. If a fraction αCE of the
difference in orbital energy is consumed in unbinding the common envelope,
αCE ≡
orb − E
, (5)
4 Ronald F. Webbink
αCEλr1,L
M1 −M1c
, (6)
where r1,L ≡ R1,L/Ai is the dimensionless Roche lobe radius of the donor
at the start of mass transfer. In the classical Roche approximation, r1,L is a
function only of the mass ratio, q ≡ M1/M2 [7]:
r1,L ≈
0.49q2/3
0.6q2/3 + ln(1 + q1/3)
. (7)
Typically, the second term in brackets in (6) dominates the first term.
As formulated above, our treatment of the outcome of common envelope
evolution neglects any sources or sinks of energy beyond gravitational terms
and the thermal energy content of the initial envelope (incorporated in the
parameter λ). The justification for this assumption is again that common
envelope evolution must be rapid compared to the thermal time scale of the
envelope. This implies that radiative losses (or nuclear energy gains – see
below) are small. They, as well as terminal kinetic energy of the ejecta, are
presumably reflected in ejection efficiencies αCE < 1. We neglect also the
rotational energy of the common envelope (invariably small in magnitude
compared to its gravitational binding energy), and treat the core of the donor
star and the companion star as inert masses, which neither gain nor lose
mass or energy during the course of common envelope evolution. One might
imagine it possible that net accretion of mass by the companion during inspiral
might compromise this picture. However, the common envelope is typically
vastly less dense than the companion star. and may be heated to roughly
virial temperature on infall. A huge entropy barrier arises at the interface
between the initial photosphere of the companion and the common envelope
in which it is now embedded, with a difference in entropy per particle of order
(µmH/k)∆s ≈ 4–6. The rapid rise in temperature and decrease in density
through the interface effectively insulates the accreting companion thermally,
and strongly limits the fraction of the very rarified common envelope it can
retain upon exit from that phase [62, 15].
Common envelope evolution entails systemic angular momentum losses as
well as systemic mass and energy losses. Writing the orbital angular momen-
tum of the binary,
GM21M
2A(1 − e
M1 +M2
, (8)
in terms of the total orbital energy, E = −GM1M2/2A, we find immediately
that the ratio of final to initial orbital angular momentum is
)3/2 (
M1c +M2
M1 +M2
)−1/2 (
)1/2 (
1− e2f
1− e2i
. (9)
Since M1c < M1 and we expect the initial orbital eccentricity to be small (ei ≈
0), it follows that any final energy state lower than the initial state (|Ef | >
Common Envelope Evolution Redux 5
|Ei|) requires the loss of angular momentum. The reverse is not necessarily
true, so it is the energy budget that most strongly constrains possible outcomes
of common envelope evolution.
3 Does Common Envelope Evolution Work?
As an example of common envelope energetics, let us revisit the pre-CV
V471 Tau, applying the simple treatment outlined above. It is a member of
the Hyades, an intermediate-age metal-rich open cluster (t = 650 Myr, [Fe/H]
= +0.14) with turnoff mass MTO = 2.60 ± 0.06M⊙ [28]. The cooling age of
the white dwarf is much smaller than the age of the cluster (tcool,WD = 10
yr [39] – but see the discussion there of the paradoxical fact that this most
massive of Hyades white dwarfs is also the youngest). Allowing for the possi-
bility of significant mass loss in a stellar wind prior to the common envelope
phase, we may take MTO for an upper limit to the initial mass M1 of the
white dwarf component. The current masses for the white dwarf and its dK2
companion, as determined by O’Brien et al. [39] are MWD = 0.84± 0.05M⊙,
MK = 0.93±0.07M⊙, with orbital separation A = 3.30±0.08R⊙. A 2.60M⊙
star of Hyades metallicity with a 0.84M⊙ core lies on the thermally-pulsing
asymptotic giant branch, with radius (maximum in the thermal pulse cycle)
which we estimate at Ri = 680R⊙ = R1,L, making Ai = 1450R⊙. With this
combination of physical parameters, we derive an estimate of αCEλ = 0.057
for V471 Tau. Equation (3) then implies αCE = 0.054. This estimate of course
ignores any mass loss prior to common envelope evolution (which would drive
αCE to lower values), or orbital evolution since common envelope evolution
(which would drive αCE to higher values). In any event, the status of V471 Tau
would appear to demand only a very small efficiency of envelope ejection.1
The fact that V471 Tau is a double-lined eclipsing member of a well-
studied cluster provides an exceptionally complete set of constraints on its
prior evolution. In all other cases of short-period binaries with degenerate or
compact components, available data are inadequate to fix simultaneously both
the initial mass of the compact component and the initial binary separation,
for example. To validate the energetic arguments outlined above, one must
resort to consistency tests, whether demonstrating the existence of physically-
plausible initial conditions that could produce some individual system, or else
1 The anomalously small value of αCE deduced for V471 Tau may be connected to
its puzzlingly high white dwarf mass and luminosity: O’Brien et al. [39] suggest
that it began as a heirarchical triple star, in which a short-period inner binary
evolved into contact, merged (as a blue straggler), and later engulfed its lower-
mass companion in a common envelope. An overmassive donor at the onset of
common envelope evolution would then have a more massive core than produced
by its contemporaries among primordially single stars, and it would fill its Roche
lobe with a more massive envelope at somewhat shorter orbital period, factors all
consistent with a larger value of αCE having led to V471 Tau as now observed.
6 Ronald F. Webbink
following a plausible distribution of primordial binaries wholesale through the
energetics of common envelope evolution and showing that, after application
of appropriate observational selection effects, the post-common-envelope pop-
ulation is statistically consistent with the observed statistics of the selected
binary type. In the cases of interacting binaries, such as CVs, one should allow
further for post-common-envelope evolution. Nevertheless, within these limi-
tations, binary population synthesis models show broad consistency between
the outcomes of common envelope evolution and the statistical properties of
CVs and pre-CVs [4, 24, 44, 17, 63], as well as with most super-soft X-ray
sources [5], for assumed common envelope ejection efficiencies typically of or-
der αCE ≈ 0.3–0.5.
A useful tool in reconstructing the evolutionary history of a binary, used
implicitly above in analyzing V471 Tau, is the mass-radius diagram spanned
by single stars of the same composition as the binary. Figure 1 illustrates
such a diagram for solar-composition stars from 0.08 M⊙ to 50 M⊙. In it are
plotted various critical radii marking, as a function of mass, the transition
from one evolutionary phase to the next.2 Since the Roche lobe of a binary
component represents a dynamical limit to its size, its orbital period fixes the
mean density at which that star fills its Roche lobe,
logPorb(d) ≈
log(RL/R⊙)−
log(M/M⊙)− 0.455 , (10)
to within a very weak function of the binary mass ratio. The mass and radius of
any point in Fig. 1 therefore fixes the orbital period at which such a star would
fill its Roche lobe, just as the orbital period of a binary fixes the evolutionary
state at which such a star initiates mass transfer.
2 Not all evolutionary phases are represented here. In a binary, a donor initiates
mass transfer when it first fills its Roche lobe; if it would have done so at a prior
stage of evolution, then its present evolutionary state is ‘shadowed’, in the sense
that it only occurs by virtue of the binary not having filled its lobe previously.
Thus, for example, low- and intermediate-mass stars cannot in general initiate
mass transfer during core helium burning, because they would have filled their
Roche lobes on the initial ascent of the giant branch.
Fig. 1 (facing page). The mass-radius diagram for stars of solar metallicity, con-
structed from the parametric models of stellar evolution by Hurley, Pols, & Tout [19]
and models of thermally-pulsing asymptotic giant branch stars by Wagenhuber &
Weiss [58]. Also plotted in the locus of asymptotic giant branch stars at the onset
of the superwind, after Willson [64]; beyond this radius, systemic mass loss drives
orbital expansion faster than nuclear evolution drives stellar expansion, and a bi-
nary will no longer be able to initiate tidal mass transfer. The unlabeled dotted
line terminating at the junction between lines labeled ‘helium core flash’ and ‘core
helium ignition’ marks the division between those helium cores (at lower masses)
which evolve to degeneracy if stripped of their envelope, and those (at higher masses)
which ignite helium non-degenerately and become helium stars.
Common Envelope Evolution Redux 7
8 Ronald F. Webbink
In Fig. 2, the corresponding core masses of low- and intermediate-mass
stars are plotted in the mass-radius diagram. For a binary which is the imme-
diate product of common envelope evolution, the mass of the most recently
formed white dwarf (presumably the spectroscopic primary) equals the core
mass of the progenitor donor star. That donor (presuming it to be of solar
metallicity) must be located somewhere along the corresponding core mass
sequence in Fig. 2, with the radius at any point along that sequence corre-
sponding to the Roche lobe radius at the onset of the mass transfer, and the
mass a that point corresponding to the initial total mass of the donor. Thus,
if the mass of the most recently-formed white dwarf is known, it is possible
to identify a single-parameter (e.g., initial mass or initial radius of the donor)
family of possible common-envelope progenitors.
Using a mapping procedure similar to this, Nelemans & Tout [35] recently
explored possible progenitors for detached close binaries with white dwarf
components. Broadly speaking, they found solutions using (6) for almost all
systems containing only one white dwarf component. Only three putative
post-common-envelope systems failed to yield physically-plausible values of
αCEλ: AY Cet (G5 III + DA, Porb = 56.80 d [49]), Sanders 1040 (in M67:
G4 III + DA, Porb = 42.83 d [56]), and HD 185510 (=V1379 Aql: gK0 +
sdB, Porb = 20.66 d [22]). The first two of these systems are non-eclipsing,
but photometric masses for their white dwarf components are extremely low
(estimated at ∼0.25M⊙ and 0.22M⊙, respectively), with Roche lobe radii
consistent with the limiting radii of very low-mass giants as they leave the
giant branch (cf. Fig. 2, above). They are thus almost certainly post-Algol
binaries, and not post-common-envelope binaries. HD 185510 is an eclipsing
binary; a spectroscopic orbit exists only for the gK0 component [9]. The mass
(0.304 ± 0.015M⊙) and radius (0.052 ± 0.010R⊙) of the sdB component,
deduced from model atmosphere fitting of IUE spectra combined with solution
of the eclipse light curve, place it on a low-mass white dwarf cooling curve,
rather than among helium-burning subdwarfs [22]. Indeed, from fitting very
detailed evolutionary models to this system, Nelson & Eggleton [38] found a
Fig. 2 (facing page). The mass-radius diagram for low- and intermediate-mass
stars, as in Fig. 1, but with loci of constant core mass added. The solid lines added
correspond to core masses interior to the hydrogen-burning shell, dashed lines to
those interior to the helium-burning shell. Solid lines intersecting the base of the
giant branch (dash-dotted curve) correspond to helium core masses of to 0.15, 0.25,
0.35, 0.5, 0.7, 1.0, 1.4, and 2.0 M⊙; those between helium ignition and the initial
thermal pulse to 0.7, 1.0, 1.4, and 2.0 M⊙, and those beyond the initial thermal pulse
to 0.7, 1.0, and 1.4 M⊙. Dashed lines between helium ignition and initial thermal
pulse correspond to carbon-oxygen core masses of 0.35, 0.5, 0.7, 1.0, and 1.4 M⊙.
Beyond the initial thermal pulse, helium and carbon-oxygen core masses converge,
with the second dredge-up phase reducing helium core masses above ∼0.8 M⊙ to
the carbon-oxygen core.
Common Envelope Evolution Redux 9
10 Ronald F. Webbink
post-Algol solution they deemed acceptable. It thus appears that these three
problematic binaries are products of quasi-conservative mass transfer, and not
common envelope evolution.
The close double white dwarfs present a more difficult conundrum, how-
ever. Nelemans et al. [36, 37, 35] found it impossible using the energetic argu-
ments (6) outlined above to account for the existence of a most known close
double white dwarfs. Mass estimates can be derived for spectroscopically de-
tectable components of these systems from their surface gravities and effective
temperatures (determined from Balmer line fitting). The deduced masses are
weakly dependent on the white dwarf composition, and may be of relatively
modest accuracy, but they are independent of the uncertainties in orbital
inclination afflicting orbital solutions. These mass estimates place the great
majority of detectable components in close double white dwarf binaries below
∼0.46M⊙, the upper mass limit for pure helium white dwarfs (e.g., [51]). They
are therefore pure helium white dwarfs, or perhaps hybrid white dwarfs (low-
mass carbon-oxygen cores with thick helium envelopes). While reconstructions
of their evolutionary history yield physically-reasonable solutions for the final
common envelope phase, with values for 0 < αCEλ < 1, the preceding phase of
mass transfer, which gave rise to the first white dwarf, is more problematic. If
it also proceeded through common envelope evolution, the deduced values of
αCEλ ≤ −4 for that phase are unphysical. Nelemans & Tout [35] interpreted
this paradox as evidence that descriptions of common envelope evolution in
terms of orbital energetics, as described above, are fundamentally flawed.
4 An Alternative Approach to Common Envelope
Evolution?
Nelemans et al. [36] proposed instead parameterizing common envelope evo-
lution in terms of γ, the ratio of the fraction of angular momentum lost to
the fraction of mass lost:
Ji − Jf
M1 −M1,c
M1 +M2
. (11)
Both initial and final orbits are assumed circular, so the ratio of final to initial
orbital separations becomes
M1c +M2
M1 +M2
M1 −M1c
M1 +M2
. (12)
Among possible solutions leading to known close double white dwarfs, Nele-
mans & Tout [35] find values 1 < γ . 4 required for the second (final) common
envelope phase, and 0.5 . γ < 3 for the first (putative) common envelope
phase. They note that values in the range 1.5 < γ < 1.7 can be found among
possible solutions for all common envelope phases in their sample, not only
Common Envelope Evolution Redux 11
those leading to known double white dwarfs, but those leading to known pre-
CV and sdB binaries as well.
The significance of this finding is itself open to debate. At one extreme, it
would seem implausible for any mechanism to remove less angular momentum
per unit mass than the orbital angular momentum per unit mass of either
component in its orbit (so-called Jeans-mode mass loss). At the other extreme,
a firm upper limit to γ is set by vanishing final orbital angular momentum,
Jf . If M1c and M2 can be regarded as fixed, the corresponding limits on γ are
M1 +M2
M1 −M1c
> γ >
M1 +M2
M1 −M1c
M1 +M2
M1c +M2
. (13)
In a fairly typical example, M1c = M2 =
M1, γ is inevitably tightly con-
strainted for any conceivable outcome: 5
> γ > 5
. The ratio of final to initial
orbital separation, Af/Ai, is extremely sensitive to γ near the upper limit of
its range. It is therefore not surprising to find empirical estimates of γ clus-
tering as they do – their values merely affirm the fact that Af must typically
be much smaller than Ai.
The unphysically large or, more commonly, negative values of αCEλ noted
above for the first mass transfer phase in the production of close white dwarf
binaries [35] implies that the orbital energies of these binaries have increased
through this phase (or, at any rate, decreased by significantly less than the
nominal binding energies of their common envelopes). Such an increase in
orbital energy is a hallmark of slow, quasi-conservative mass transfer, on a
thermal or nuclear time scale. Thermal time scale mass transfer is driven by
relaxation of the donor star toward thermal evolution; the re-expansion of
the donor following mass ratio reversal is powered by the (nuclear) energy
outflow from the core of the star. Likewise, the bulk expansion of the donor
star in nuclear time scale mass transfer draws energy from nuclear sources in
that star. It appears, therefore, that the first phase of mass transfer among
known close double white dwarfs cannot have been a common envelope phase,
but must instead have been a quasi-conservative phase, notwithstanding the
difficulties that conclusion presents, as we shall now see.
The dilemma that the close double white dwarfs present is illustrated in
Figs. 3 and 4. Figure 3 shows the distribution of immediate remnants of
mass transfer among solar-metallicity binaries of low and intermediate mass,
for a relatively moderate initial mass ratio. Conservation of total mass and
orbital angular momentum have been assumed. The remnants of the intial
primary include both degenerate helium white dwarfs, and nondegenerate he-
lium stars which have lost nearly all of their hydrogen envelopes. The helium
white dwarfs lie almost entirely along the left-hand boundary, the line labeled
‘envelope exhaustion’ in Fig. 1. (The extent of this sequence is more apparent
in the distribution of remnant secondaries.) Their progenitors have enough
angular momentum to accommodate core growth in the terminal phases of
mass transfer. In the calculation shown, the least massive cores grow from
12 Ronald F. Webbink
Common Envelope Evolution Redux 13
0.11M⊙ to ∼0.18M⊙ by the completion of mass transfer. In contrast, virtu-
ally all binaries leaving nondegenerate helium star remnants have too little
angular momentum to recover thermal equilibrium before they have lost their
hydrogen envelopes; for them, there is no slow nuclear time scale phase of
mass transfer, and core growth during mass transfer is negligible. The lowest-
mass helium star remnants have nuclear burning lifetimes comparable to their
hydrogen-rich binary companions, now grown through mass accretion. Those
more massive than ∼0.8M⊙ develop very extended envelopes during shell he-
lium burning, and will undergo a second phase of mass transfer from primary
to secondary, not reflected here; such massive white dwarfs are absent in the
Nelemans & Tout [35] sample, and so are omitted here.
In Fig. 4, the remnants of the first phase of conservative mass transfer
illustrated in Fig. 3 are followed through the second phase of mass transfer,
using (6). Because the remnants of the first phase have second-phase donors
much more massive their companions, and nearly all have deep convective en-
velopes, they are unstable to dynamical time scale mass transfer, and undergo
common envelope evolution. The systems labeled ‘Without Wind Mass Loss’
have been calculated assuming that no orbital evolution or mass loss occurs
between the end of the first phase of mass transfer and common envelope
evolution. It is assumed furthermore that αCE = 1, in principle marking the
most efficient envelope ejection energetically possible. Binary orbital periods
of 0.1, 1.0, and 10 days (assuming equal component masses) are indicated for
reference.
Observed close double white dwarfs, as summarized by Nelemans &
Tout [35], have a median orbital period of 1.4 d, and mass (spectroscopic pri-
mary) 0.39M⊙. Among double-lines systems, nearly equal white dwarf masses
are strongly favored, with the median q = 1.00. Clearly, most observed dou-
ble white dwarfs are too long in orbital period (have too much total energy
and angular momentum) to have evolved in the manner assumed here. Fur-
thermore, the computed binary mass ratios are typically more extreme than
observed, with the second-formed core typically 1.3-2.5 times as massive as
the first. The problem is that, while remnant white dwarfs or low-mass he-
Fig. 3 (facing page). Products of mass- and angular momentum-conservative mass
transfer for a typical initial mass ratio. The radii indicated refer to Roche lobe radii at
the onset or termination of mass transfer, as appropriate. To avoid common envelope
evolution, the donor stars (the region outlined in bold toward the lower right in the
diagram) must have radiative envelopes, and so arise between the terminal main
sequence and base of the giant branch. Their mass transfer remnants are outlined
in bold at the center-left of the diagram, with the remnant accretors at upper right.
The regions mapped are truncated in each case at a lower initial donor mass of
1.0 M⊙ and upper initial donor core mass of 0.7 M⊙. Lines of constant initial core
mass (with values as in Fig. 2) are indicated for the initial and remnant primaries.
Lines of constant remnant primary mass are indicated for the remnant secondaries.
14 Ronald F. Webbink
Common Envelope Evolution Redux 15
lium stars with suitable masses can be produced in the first, conservative mass
transfer phase, the remnant companions have envelope masses too large, and
too tightly bound, to survive the second (common envelope) phase of inter-
action at orbital separations and periods as large as observed. Evidently, the
progenitors of these double white dwarfs have lost a significant fraction of
their initial mass, while gaining in orbital energy, prior to the final common
envelope phase. These requirements can be fulfilled by a stellar wind, pro-
vided that the process is slow enough that energy losses in the wind can be
continuously replenished from nuclear energy sources.
The requisite mass loss and energy gain are possible with stellar wind
mass loss during the non-interactive phase between conservative and common
envelope evolution, or with stellar winds in nuclear time-scale mass transfer
or the terminal (recovery) phase of thermal time-scale mass transfer. Systemic
mass loss during or following conservative mass transfer will (in the absence
of angular momentum losses) shift the remnant regions to the left and upward
in Fig. 3 (subject to the limit posed by envelope exhaustion), while systemic
angular momentum losses shift them downwards. More extreme initial mass
ratios shift them downwards to the left.
The net effect of wind mass loss is illustrated by the regions labeled ‘With
Wind Mass Loss’ in Fig. 4. For simplicity, it is assumed here that half the
remnant mass of the original secondary was lost in a stellar wind prior to
the common envelope phase. Mass loss on this scale not only significantly re-
duces the mass of the second-formed white dwarf relative to the first, but the
concomitant orbital expansion produces wider remnant double white dwarfs,
bringing this snapshot model into good accord with the general properties of
real systems. Losses of this magnitude might be unprecedented among single
stars prior to their terminal superwind phase, but they have been a persis-
tent feature of evolutionary studies of Algol-type binaries [10, and references
Fig. 4 (facing page). Remnants of the second, common envelope, phase of mass
transfer of the systems shown in Fig. 3. Masses refer to the final remnants of the
original secondaries, and radii to their Roche lobe radii. Two groups of remnants are
shown. Those at lower right labeled ‘Without Wind Mass Loss’ follow directly from
the distributions of remnant primaries and secondaries shown in Fig. 3. Because
the remnant secondaries straddle the helium ignition line in Fig. 3, across which
core masses are discontinuous (see Fig. 2), the distribution of their post-common-
envelopes remnants is fragmented, some appearing as degenerate helium white dwarf
remnants (lower left), some as helium main sequence star remnants (lower center),
and the remainder as shell-burning helium star remnants (upper center). These latter
two groups overlap in the mass-radius diagram. The remnant distributions labeled
‘With Wind Mass Loss’ assume that the remnant secondaries of conservative mass
transfer lose half their mass in a stellar wind prior to common envelope evolution.
They too are fragmented, into degenerate helium white dwarf remnants (lower left)
and shell-burning helium stars (upper right). Within each group of remnants, lines
of constant remnant primary mass are shown, as in Fig. 3.
16 Ronald F. Webbink
therein] and, indeed, of earlier studies of close double white dwarf forma-
tion [11]. In the present context, their existence appears inescapable, if not
understood.
5 Long-Period Post-Common-Envelope Binaries and the
Missing Energy Problem
If the properties of short-period binaries with compact components can be
reconciled with the outcomes of common envelope evolution as expected from
simple energetics arguments, a challenge to this picture still comes from the
survival of symbiotic stars and recurrent novae at orbital separations too large
to have escaped tidal mass transfer earlier in their evolution. Notwithstanding
this author’s earlier hypothesis that the outbursting component in the recur-
rent nova T CrB (and its sister system RS Oph) might be a nondegenerate star
undergoing rapid accretion [59, 29, 61], it is now clear that the hot components
in both of these systems must indeed be hot, degenerate dwarfs [48, 6, 1, 18].
Furthermore, the short outburst recurrence times of these two binaries de-
mand that the degenerate dwarfs in each must have masses very close to the
Chandrasekhar limit.
The complexion of the problem posed by these systems can be illustrated
by a closer examination of T CrB itself. Its orbital period (P = 227.53 d)
and spectroscopic mass function (f(m) = 0.299M⊙) are well-established
from the orbit of the donor M3 III star [23]. The emission-line orbit for
the white dwarf [25] now appears very doubtful [18], but the system shows
very strong ellipsoidal variation (e.g., [1]), suggesting that the system is near
a grazing eclipse. Following Hric, et al. [18], I adopt MWD = 1.38 M⊙
amd MM3 = 1.2 M⊙. The Roche lobe radius of the white dwarf is then
RL,WD = 84 R⊙, nearly an order of magnitude larger than can be accom-
modated from the energetics arguments presented above, even for αCE = 1,
assuming solar metallicity for the system (see Fig. 5). A similar discrepancy
occurs for RS Oph.
It is evident that these long-period binaries are able to tap some en-
ergy source not reflected in the energy budget in (6). One possibility, dis-
cussed repeatedly in studies of planetary nebula ejection ([30, 42], more re-
cently [58, 12, 13]) is that the recombination energy of the envelope comes
Fig. 5 (facing page). Post-common-envelope masses and Roche lobe radii for bina-
ries consisting of a white dwarf or helium star plus a 1.2M⊙ companion, computed
with αCE = 1. Remnant systems inhabit the regions outlined in bold, and spanned
vertically by lines of constant white dwarf/helium star mass of 0.25, 0.35, 0.5, 0.7,
1.0, 1.4, and 2.0M⊙. Other initial sequences, encoded as in Fig. 2, have been mapped
through common envelope evolution. The location in this diagram of the white dwarf
in the recurrent nova T CrB is also indicated.
Common Envelope Evolution Redux 17
18 Ronald F. Webbink
into play. For solar composition material (and complete ionization), that re-
combination energy amounts to 15.4 eVamu−1, or 1.49 × 1013 erg g−1. For
tightly-bound envelopes on the initial giant branch of the donor, this term
is of little consequence; but near the tip of the low-mass giant branch, and
on the upper aymptotic giant branch of intermediate-mass stars, it can be-
come comparable with, or even exceed, the gravitational potential energy of
the envelope. In the model calculations of thermally-pulsing asymptotic giant
branch stars by Wagenhuber & Weiss [58], the threshold for spontaneous ejec-
tion by envelope recombination occurs consistently when the stellar surface
gravity at the peak thermal pulse luminosity falls to
log gHRI = −1.118± 0.042 . (14)
This threshold marks the presumed upper limit to the radii of lower-mass
asymptotic giant branch stars in Figs. 1 et seq. in the present paper. In fact,
the total energies of the envelopes of these stars formally becomes decidedly
positive even before the onset of the superwind phase, also shown in these
figures.
Whether single stars successfully tap this ionization energy in ejecting
planetary nebulae is still debated, but the circumstances of mass transfer in
binary systems would seem to provide a favorable environment for doing so.
In the envelopes of extended giants and asymptotic giant branch stars, pho-
tospheric electron densities and opacities are dominated by heavy elements;
the middle of the hydrogen ionization zone is buried at optical depths of order
τ ∼ 105. Adiabatic expansion of the envelope of the donor into the Roche lobe
of its companion can therefore trigger recombination even as the recombina-
tion radiation is itself trapped and reprocessed within the flow, much as the
same process occurs in rising convective cells.
Other possible energy terms exist that have been neglected in the energet-
ics arguments above: rotational energy, tidal contributions, coulomb energy,
magnetic fields, etc. But Virial arguments preclude most of these terms from
amounting to more than a minor fraction of the internal energy content of
the common envelope at the onset of mass transfer, when the energy budget
is established. The only plausible energy source of significance is the input
from nuclear reactions. In order for that input to be of consequence, it must
of course occur on a time scale short compared with the thermal time scale of
the common envelope. Taam [53] explored the possibility that shell burning
in an asymptotic giant branch core could be stimulated by mixing induced
dynamically in the common envelope (see also [54, 52]). Nothing came of
this hypothesis: mixing of fresh material into a burning shell required tak-
ing low-density, high-entropy material from the common envelope and mixing
it downward many pressure scale heights through a strongly stable entropy
gradient to the high-density, low-entropy burning region. In the face of strong
buoyancy forces, dynamical penetration is limited to scales of order a pressure
scale height.
Common Envelope Evolution Redux 19
6 Common Envelope Evolution with Recombination
The notion that recombination energy my be of importance specifically to
common envelope evolution is not new. It has been included, at least para-
metrically in earlier studies, for example, by Han et al. [13], who introduced
a second α-parameter, αth, characterizing the fraction of the initial thermal
energy content of the common envelope available for its ejection. The initial
energy kinetic/thermal content of the envelope is constrained by the Virial
Theorem, however, and it is not clear that there is a compelling reason for
treating it differently from, say, the orbital energy input from the inspiraling
cores. We choose below to formulate common envelope evolution in terms of
a single efficiency parameter, labeled here βCE to avoid confusion with αCE
as defined above.
By combining the standard stellar structure equations for hydrostatic equi-
librium and mass conservation, we can obtain an expression for the gravita-
tional potential energy, Ωe, of the common envelope:
Ωe ≡ −
dM = 3 PV
P dV , (15)
where subscripts c refer to the core-envelope boundary, and ∗ to the stellar
surface. This is, of course, the familiar Virial Theorem applied to a stellar
interior.
It is convenient to split the pressure in this integral into non-relativistic
(particle), Pg, and relativistic (photon), Pr, parts. The envelopes of giants un-
dergoing common envelope evolution are sufficiently cool and non-degenerate
to make the classical ideal gas approximation an excellent one for the particle
gas. One can then write
P = Pg + Pr =
ur , (16)
where ug and ur are kinetic energy densities of particle and radiation gases,
respectively. The total internal energy density of the gas is
u = ug + ur + uint , (17)
where the term uint now appearing represents non-kinetic contributions to
the total energy density of the gas, principally the dissociation and ionization
energies plus internal excitation energies of bound atoms and molecules. The
overwhelmingly dominant terms in uint are the ionization energies: uint ≈
ρχeff .
Integrating over the stellar envelope, we obtain for the total energy Ee of
the envelope:
Ee = Ωe + Ue
3 PU |
− 2Ug − Ur
+ (Ug + Ur + Uint)
= −4πR3cPc − Ug + Uint , (18)
20 Ronald F. Webbink
where we explicitly take P∗ → 0. In fact, experience shows that, for red-giant
like structures, Rc is so small that the first right-hand term in the last equality
can generally be neglected. In that case, we get the familiar Virial result, but
with the addition of a term involving the ionization/excitation/dissociation
energy available in the gas, Uint ≈ Meχeff , which becomes important for dif-
fuse, loosely-bound envelopes.
In the context of common envelope evolution, it is of course the dissipated
orbital energy, E
orb−E
orb, that must unbind the envelope. However, the inclu-
sion of Uint in Ee now opens the possibility that the common envelope began
with positive total energy; that is, in the usual αCE-prescription, it is possible
for λ−1 to be zero or even negative, which has the undesirable consequence
that αCE need not lie in the interval 0 ≤ αCE ≤ 1 for all physically-possible
outcomes. However, the gravitational potential energy of the envelope, Ωe, is
negative-definite, and by comparing it with all available energy sources (or-
bital energy released plus internal energy of the envelope), we can define an
ejection efficiency βCE that has the desired property, 0 ≤ βCE ≤ 1:
βCE ≡
orb − E
orb)− Ue
4πR3cPc + 2Ug + Ur
orb − E
orb + Ug + Ur + Uint
. (19)
By analogy to the form factor λ in the conventional αCE formalism above, we
can define separate form factors λΩ for the gravitational potential energy and
λP for the gas plus radiation contributions to the (kinetic) internal energy of
the envelope:
and Ug + Ur =
, (20)
In contrast, the recombination energy available can be written simply in terms
of an average ionization energy per unit mass,
Uint = Meχeff . (21)
The ratio of final to initial orbital separation then becomes
1 + 2
βCEλΩr1,L
λPr1,L
χeffAi
M1 −M1c
In the limit that radiation pressure Pr, ionization energy (Uint), and the
boundary term (4πR3cPc) are all negligible, then 2λΩ → λP → λ and
βCE → 2αCE/(1 + αCE).
Fig. 6 (facing page). Post-common-envelope masses and Roche lobe radii as in
Fig. 5, but with recombination energy included, computed from (22) with βCE = 1,
with the approximation 2λΩ = λP = λ from (3). At small separations, the differences
are inconsequential, but substantially larger final separations are allowed when Af &
10R⊙ (RL & 3R⊙).
Common Envelope Evolution Redux 21
22 Ronald F. Webbink
The ability to tap the recombination energy of the envelope has a profound
effect on the the final states of the longest-period intermediate-mass binaries,
those that enter common envelope evolution with relatively massive, degener-
ate carbon-oxygen (or oxygen-neon-magnesium) cores. As is evident in Fig. 6,
possible final states span a much broader range of final orbital separations.
Indeed, for the widest progenitor systems, the (positive) total energy of the
common envelope can exceed the (negative) orbital energy of the binary, mak-
ing arbitrarily large final semimajor axes energetically possible.3 The inclusion
of recombination energy brings both T CrB and RS Oph within energetically
accessible post-common-envelope states. It suffices as well to account for the
exceptionally long-period close double white binary PG 1115+166, as sug-
gested by Maxted, et al. [31].
7 Conclusions
Re-examination of global constraints on common envelope evolution leads to
the following conclusions:
Both energy and angular momentum conservation pose strict limits on the
outcome of common envelope evolution. Of these two constraints, however,
energy conservation is much the more demanding.
The recent study of close double white dwarf formation by Nelemans &
Tout [35] shows clearly that their progenitors can have lost little orbital en-
ergy through their first episodes of mass transfer. Since common envelope
ejection must be rapid if it is to be efficient, its energy budget is essentially
fixed at its onset by available thermal and gravitational terms. The preser-
vation of orbital energy through that first phase of mass transfer therefore
indicates that the observed close double white dwarfs escaped common en-
velope formation in that first mass transfer phase. They evidently evolved
through quasi-conservative mass transfer. However, strictly mass- and angular
momentum-conservative mass transfer leaves remnant accretors that are too
massive and compact to account for any but the shortest-period close double
white dwarfs. Significant mass loss and the input of orbital energy prior to the
onset of the second (common envelope) phase of mass transfer are required.
The requisite energy source must be of nuclear evolution, which is capable of
driving orbital expansion and stellar wind losses during the slower (thermal
recovery or nuclear time scale) phases of quasi-conservative mass transfer, or
during the interval between first and second episodes of mass transfer. Details
of this process remain obscure, however.
3 The final orbit remains constrained by the finite initial orbital angular momentum
of the binary. Final semimajor axes much in excess of the initial semimajor axis
may be energetically allowed, but the finite angular momentum available means
that they cannot be circular – see (9) – an effect which has been neglected in
Fig. 6.
Common Envelope Evolution Redux 23
Long-period cataclysmic variables such as T CrB and RS Oph pose a
more extreme test of common envelope energetics. With their massive white
dwarfs, the evident remnants of much more massive initial primaries, they
are nevertheless too low in total systemic mass to be plausible products of
quasi-conservative mass transfer, but too short in orbital period to have es-
caped tidal mass transfer altogether. They must be products of common en-
velope evolution, but to have survived at their large separations, they demand
the existence of a latent energy reservoir in addition to orbital energy to as-
sist in envelope ejection. It appears that these binaries efficiently tap ioniza-
tion/recombination energy in ejecting their common envelopes. That reservoir
is demonstrably adequate to account for the survival of these binaries. Its in-
clusion requires only a simple revision to the parameterization of common
envelope ejection efficiency.
Acknowledgement. This work owes its existence to both the encouragement and the
patience of Gene Milone, to whom I am most grateful. Thanks go as well to Ron
Taam for a useful discussion of possible loopholes in common envelope theory, and
to Jarrod Hurley for providing the source code described in Hurley, et al. (2000).
This work was supported in part by grant AST 0406726 to the University of Illinois,
Urbana-Champaign, from the US National Science Foundation.
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	Common Envelope Evolution Redux
	Ronald F. Webbink
