Explosive Common-Envelope Ejection: Implications for Gamma-Ray Bursts and Low-Mass Black-Hole Binaries

Abstract

We present a new mechanism for the ejection of a common envelope in a massive binary, where the energy source is nuclear energy rather than orbital energy. This can occur during the slow merger of a massive primary with a secondary of 1-3 Msun when the primary has already completed helium core burning. We show that, in the final merging phase, hydrogen-rich material from the secondary can be injected into the helium-burning shell of the primary. This leads to a nuclear runaway and the explosive ejection of both the hydrogen and the helium layer, producing a close binary containing a CO star and a low-mass companion. We argue that this presents a viable scenario to produce short-period black-hole binaries and long-duration gamma-ray bursts (LGRBs). We estimate a LGRB rate of about 1.e-6 per year at solar metallicity, which implies that this may account for a significant fraction of all LGRBs, and that this rate should be higher at lower metallicity. Comment: 9 pages, 4 figures, accepted for publication in MNRAS

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Mon. Not. R. Astron. Soc. 000, 1–8 (2010)Printed 5 April 2010(MN LATEX style file v2.2)
Explosive Common-Envelope Ejection: Implications for
Gamma-Ray Bursts and Low-Mass Black-Hole Binaries
Philipp Podsiadlowski1?, Natasha Ivanova2,3, Stephen Justham1,4, Saul Rappaport5
1Dept. of Astronomy, Oxford University, Oxford, OX1 3RH, UK
2CITA, University of Toronto, 60 St. George, Toronto, ON M5S 3H8, Canada
3Department of Physics, University of Alberta, Edmonton, AB T6G 2G7, Canada
4Kavli Institute for Astronomy and Astrophysics, Peking University, Beijing 100871, China
5Department of Physics and Kavli Institute for Astrophysical Research, Massachusetts Institute of Technology, Cambridge, MA 02139
5 April 2010
ABSTRACT
We present a new mechanism for the ejection of a common envelope in a massive
binary, where the energy source is nuclear energy rather than orbital energy. This
can occur during the slow merger of a massive primary with a secondary of 1 − 3M?
when the primary has already completed helium core burning. We show that, in the
final merging phase, hydrogen-rich material from the secondary can be injected into
the helium-burning shell of the primary. This leads to a nuclear runaway and the
explosive ejection of both the hydrogen and the helium layer, producing a close binary
containing a CO star and a low-mass companion. We argue that this presents a viable
scenario to produce short-period black-hole binaries and long-duration gamma-ray
bursts (LGRBs). We estimate a LGRB rate of ∼ 10−6yr−1at solar metallicity, which
implies that this may account for a significant fraction of all LGRBs, and that this
rate should be higher at lower metallicity.
Key words: black holes – binaries: general – gamma-rays: bursts – stars: individual:
X-ray Nova Sco – X-rays: stars.
1 INTRODUCTION
Roughly half of the known short-period black-hole binaries
have low-mass companions (Lee, Brown & Wijers 2002).
However, it has been realized for more than a decade now
that such systems are difficult to form (see the discussion in
Podsiadlowski, Rappaport & Han 2003 [PRH] and further
references therein). The problem is that, in order to produce
a short-period system (i.e.,<
∼15 hrs) with a low-mass donor
star, the progenitor system has to pass through a common-
envelope (CE) phase where the binary’s period is reduced
from a typical period of several years to less than a few days.
In the standard model for CE evolution (Paczy´ nski 1976),
this requires that the orbital energy released in the spiral-
in process is sufficient to eject the massive envelope of the
primary. For a low-mass companion (i.e.,<
energetically difficult, if not impossible, in particular con-
sidering the large binding energies of the massive envelope
of the primaries (Dewi & Tauris 2001; PRH).1
∼2 M?), this is
?E-mail: podsi@astro.ox.ac.uk
1In a recent study by Yungelson et al. (2006), as well as in some
other studies, this problem did not seem to arise. However, these
authors used a prescription for the binding energy of the envelope
that underestimates the binding energy by up to a factor of 10
This problem has led to the suggestion of a number of
more exotic formation scenarios for low-mass black-hole bi-
naries, e.g., involving a triple scenario (Eggleton & Verbunt
1986) or the formation of the companion from a disrupted
envelope (Podsiadlowski, Cannon & Rees 1995; PRH). Al-
ternatively, the low-mass black-hole binaries could descend
from intermediate-mass systems (Justham, Rappaport &
Podsiadlowski 2006), as is the case for the majority of low-
mass neutron-star binaries (Pfahl, Rappaport & Podsiad-
lowski 2003). In this paper, we present a new mechanism for
the ejection of the common envelope: “explosive common-
envelope ejection”, involving nuclear rather than orbital en-
ergy, which can be highly efficient in ejecting a massive en-
velope even if the companion is a relatively low-mass star.
A particularly interesting black-hole binary is the well
studied system GRO J1655–40 (Nova Scorpii 1994). It has
an orbital period of 2.6d and a black hole with a mass
∼ 5.4M? (Beer & Podsiadlowski 2002). Israelian et al.
(1999) claimed that the secondary in this system has been
highly enriched with the products of explosive nucleosynthe-
compared to the actual binding energies calculated from realistic
models for the structure of massive supergiants (Dewi & Tauris
2001; PRH).
arXiv:1004.0249v1 [astro-ph.HE] 1 Apr 2010

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2Podsiadlowski et al.
sis (e.g., Mg, Si, S, Ti) produced in the supernova explosion
that produced the black hole.2Based on the actual abun-
dance ratios, Podsiadlowski et al. (2002) found some ten-
tative evidence that these are better explained by an ener-
getic supernova explosion, a hypernova, with a typical ejecta
energy of>
∼1052ergs (i.e., 10 times the energy in a ‘typi-
cal’ supernova). This may suggest that the formation of the
black hole in this system could have been accompanied by
a long-duration gamma-ray burst (LGRB). However, in this
case it is puzzling why the black-hole progenitor would have
been rapidly rotating as required in the collapsar model for
LGRBs (Woosley 1993; MacFadyen & Woosley 1999) since
the core of the primary should have been spun down rather
than have been spun up during its evolution (Heger, Woosley
& Spruit 2005). As we will show later in this paper, in the
case of explosive CE ejection, this problem does not arise,
possibly linking the formation of compact black-hole bina-
ries having low-mass secondaries to LGRBs.3
This process of explosive CE was discovered in a sys-
tematic study of the slow merger of massive stars (Ivanova
2002; Ivanova & Podsiadlowski 2003; Ivanova & Podsiad-
lowski 2010), where it was found that, in some cases in
the late stage of the spiral-in process, hydrogen-rich mate-
rial could be mixed into the helium-burning shell leading to
a thermonuclear runaway which released enough energy to
eject both the hydrogen and the helium envelope (see Fig-
ure 1 for a schematic representation), possibly explaining
why to date all supernovae associated with LGRBs appear
to be Type Ic supernovae (i.e., supernovae without hydro-
gen and helium in their spectrum; cf. Podsiadlowski et al.
2004).4
In this paper, we will first discuss the numerical method
in Section 2, and the physics of the process of explosive
CE ejection in Section 3. In Section 4 we apply it to the
formation of low-mass black-hole binaries and LGRBs, and
end with a broader discussion in Section 5.
2 THE NUMERICAL CODE
To model the merging of the two stars, we used a modified
one-dimensional Henyey-type stellar evolution code (based
on Kippenhahn, Weigert & Hofmeister 1967), which has re-
cently been updated (Podsiadlowski, Rappaport & Pfahl
2002). It uses OPAL opacities (Rogers & Iglesias 1992), sup-
plemented by molecular opacities from Alexander & Fergu-
2In this context, we refer the reader to two related studies: one
by Foellmi, Dall & Depagne (2007) challenging the original claim
of these overabundances, and one by Gonz´ alez Hern´ andez, Rebolo
& Israelian (2008) re-affirming them.
3Brown et al. (2000) were the first to suggest that the formation
of the black hole in GRO J1655-40 was associated with an LGRB
and proposed a general link between low-mass black-hole binaries
and LGRBs (see, in particular, Brown, Lee & Moreno M´ endez
2007). Similar to the present study, they suggest late Case C mass
transfer for the progenitor systems, but argue for the spin-up of
the cores rather than spin-down during the CE phase, invoking
tidal locking, an assumption that remains to be proven.
4We note that Siess & Livio (1999a,b) were the first to point out
the potential importance of nuclear energy on the CE ejection
process in their modelling of the dissolution of planets/brown
dwarfs in red-giant stars.
son (1994) at low temperatures and a mixing-lenth param-
eter α = 2. Following the calibration by Schr¨ oder, Pols &
Eggleton (1997) and Pols et al. (1997), we generally assume
0.25 pressure scale heights of convective overshooting, unless
stated otherwise.
In the calculation of the spiral-in and merger phase we
followed the angular-momentum transport in the envelope
of the primary, using prescriptions similar to those given by
Heger & Langer (1998), where the source of the angular mo-
mentum is the orbital decay of the immersed binary. This is
important for determining the averaged distribution of the
heating sources inside the common envelope. The envelope
heating is provided by the dissipation of the kinetic energy
of the differentially rotating shells due to viscous friction.
Note that this energy is not necessarily released in the same
place where the angular momentum has been deposited ini-
tially. For modelling the latter, we included a prescription
that simulates the frictional energy input during the spiral-
in phase of the companion star similar to the prescription
given by Meyer & Meyer-Hofmeister (1979) (also see Podsi-
adlowski 2001).
Once the immersed secondary fills its own critical po-
tential surface (i.e., its equivalent “Roche lobe” defined by
the effective potential of the core-secondary pair) within the
common envelope, we simulate the mass transfer guided by
the detailed stream–core simulations presented in Ivanova,
Podsiadlowski & Spruit (2002). We use their equation (28)
to determine the depth to which the stream penetrates the
hydrogen-exhausted core of the primary and vary the param-
eter k to take into account uncertainties in the modelling of
the stream–core impact.5
In each time step, we determine how much mass is lost
from the secondary and deposit this material with the ap-
propriate entropy at the bottom of the stream. In this phase,
the mass-transfer rate from the secondary is still determined
by the frictional angular-momentum loss the immersed bi-
nary experiences, and reaches up to 100M?yr−1in the final
phase.
Since hydrogen is deposited in layers with temperatures
T > 108K, an extended nuclear reaction network, including
all important weak interactions, is included using the REA-
CLIB library (Thielemann, Truran & Arnould 1986). This
library provides the data for reaction rates which can be ap-
plied for a wide range of temperatures (T = 107− 1010K)
for all elements up to
brary has been updated for neutron-capture cross-sections
and half-lifes of beta-decays for the s-elements from
up to
from the work of Beer, Voss and Winters (1992) and have
been interpolated using the method of minimal squares to fit
the formulae for reaction rates used in REACLIB. The data
for the half lives of the beta-decays were taken from New-
man (1978). We note that, in explosive phases, the nuclear
timescales of many reactions are shorter than the convec-
tive turnover timescales in convective regions, and thus the
84Kr. For heavier elements, the li-
85Rb
209Bi. The data for these cross-sections were taken
5
The parameter k defines the stream penetration efficiency
and depends on the amount of entropy that is generated in the
stream–core impact: a larger k corresponds to the case where the
ambient medium near the core has a larger pressure and temper-
ature gradient resulting in the generation of stronger shocks and
faster dissipation of the stream.

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Explosive Common-Envelope Ejection3
Figure 1. Schematic illustration of the process of explosive
common-envelope ejection. The H-rich stream from the Roche-
lobe-filling immersed companion penetrates deep into the core of
the primary, mixing hydrogen into the helium-burning shell. This
leads to a thermonuclear runaway ejecting both the helium shell
and the hydrogen-rich envelope and leaving a bare CO core.
assumption of homogeneous mixing in the convective layers
is not valid. To treat the time-dependent nuclear burning in
convective zones, we therefore use a modified version of the
two-stream formalism developed by Cannon (1993).6
3 EXPLOSIVE MERGING AND
COMMON-ENVELOPE EJECTION
The case of an 18M? primary
To explore the conditions for explosive common-envelope
ejection, we performed a detailed numerical study of the
merger of a 18M? primary with secondaries in the range of
1–5M?. In all cases, we assumed that the spiral-in phase
started when the primary had completed helium core burn-
ing and was ascending the red-supergiant branch for the sec-
ond time (so-called Case C mass transfer; Lauterborn 1970).
Figure 2 shows the results of a typical calculation which il-
lustrates the evolution from the initial spiral-in phase to
the point at which the envelope is ejected. To test the ef-
fects of convective overshooting, we used models with 0.25
pressure scale heights of convective overshooting and models
without convective overshooting, where the former produces
a H-exhausted core of 6.8M?, 1.3M? larger than the lat-
ter. To examine the uncertainties due the modelling of the
stream–core interaction, we use two different parameters for
the entropy generation in equation (28) of Ivanova et al.
(2002), k = 0.2 for low-entropy generation and k = 0.4 for
high-entropy generation. We note that, in the present study,
we did not model how rotation of the core affects the pene-
tration depth (see section 5.3.2 of Ivanova et al. [2002]) and
how this changes during the stream–core interaction phase.
We do not expect that this would change our conclusions
significantly.
6Further details of the code can be found in Ivanova (2002) and
in Ivanova & Podsiadlowski (2010).
The initial set-up at the start of the merger
At the end of the spiral-in, when the secondary starts to fill
its Roche lobe within the common envelope, the secondary
itself is located partially within the outer convective zone
(OCZ) of the primary, although the inner Lagrangian point,
L1, is within the radiative zone. The layer of envelope mate-
rial at the position of the secondary has expanded because of
the frictional energy input from the spiral-in and has a den-
sity roughly two orders of magnitude less than before the
spiral-in, with a typical range from about 10−7gcm−3to
about 10−5gcm−3. The spiral-in does not affect the struc-
ture of the primary’s core (the CO core plus the surrounding
He shell), it only affects the structure of the radiative hy-
drogen zone between the OCZ and the He shell (∆MH,rad),
since the upper part of this zone has been heated and may
have become convective. For example, in the case of a 5M?
donor inside an 18M? primary, the total mass of the OCZ
is increased by 0.4M?, while, in the case of a 2M? donor, it
is only increased by 0.04M?. It should be noted that, in the
first case, the initial ∆MH,rad is 0.13M? for our standard
case with convective overshooting. Without convective over-
shooting, it would be 0.7M?. The corresponding radial sizes
of this zone (∆RH,rad) are 1.6 × 1011cm and 2.5 × 1011cm
for the case with and without convective overshooting, re-
spectively.
The stream behavior and the surrounding medium
Once the stream leaves the secondary, the depth of its pen-
etration depends on the initial entropy of the stream ma-
terial and on how much this entropy is modified during
the stream’s penetration into the primary’s core by shocks
(Ivanova et al. 2002). The latter is a function of the density
contrast between the stream and the surrounding matter
and depends on how strongly the medium itself is stratified.
A larger density gradient in the primary’s material near the
stream leads to more entropy generation in the stream. The
stream will stop its fall when the ambient pressure becomes
comparable to the stream pressure.
In the test cases, the penetration depth is two times
smaller in the model with large entropy generation (with
k = 0.4) than in the model with k = 0.2. The distance to the
He shell from L1 is also larger. For a secondary that is more
massive than ∼ 3M?, the stream is unable to penetrate into
the He shell since the density contrast between the stream
and the ambient medium is lower and the stream entropy
higher.
The response of the He shell to the injection of hydrogen
In the layer where the hydrogen-rich stream penetrates into
the helium-dominated layer (see the middle and right panels
of Figure 2), the stream entropy is higher than that of the
surrounding material. As a consequence, both the entropy
and the hydrogen mass fraction X have negative gradients.
This leads to the formation of a local convective zone that
rapidly expands outwards. As the stream continues to hit
the core, it penetrates progressively deeper into the core for
several reasons: first, the entropy of the donor material de-
creases as material from deeper inside the star is transferred.

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4 Podsiadlowski et al.
Figure 2. Simulations of the spiral-in of a 2M? star inside the envelope of a 18M? red supergiant (after He core burning) up to
the phase where the H/He envelope is being ejected. The left and medium panels show the time-evolution of the specific entropy
profile as a function of radius; left panel: from the start of the spiral-in up to the beginning of the He shell explosion phase; middle
panel: during the final year around the He shell explosion. The right panel shows the time-evolution of the helium profile as a function
of mass during the last year. The black solid curve in the left and medium panels shows the orbital position of the secondary, the
solid red curves show the depth of the stream penetration. The dashed curves are curves of constant mass and correspond to M =
14,9,7.30,6.96,6.92,6.88,6.84,6.80,6.72,4.457M?(from the top). The dashed curves in the right panel are curves of constant radii and
correspond to logR/cm = 13.6,13,12,11,10 (from the top). The stream penetration efficiency parameter for this model is k = 0.2.
Second, the outer layer of the He shell starts to expand, re-
ducing the density in the outer part of the He shell. Third,
the mass transfer rate steadily goes up, and, fourth, the L1
point moves closer to the helium core. In the case where the
initial stream does not penetrate deeply enough into the He-
rich layer and is unable to create a negative entropy gradient
in the ambient medium, a convective zone still develops in
the region where the stream is being dissolved, but more
slowly (the stream’s entropy is still higher than the ambi-
ent medium and stream energy is being deposited, further
increasing the entropy). This convective zone propagates in-
wards as the stream penetrates deeper. We found that in
some models this zone can connect with the OCZ; however,
it either never penetrates deeply enough to connect to the
convective helium-burning shell, or it starts efficient steady
hydrogen burning, leading to a drop of the temperature in
the He convective shell.
The conditions for explosive CE ejection
Based on these simulations, it seems that the main condi-
tion for experiencing explosive CE ejection (ECEE) is that
the stream is able to penetrate deeply enough into the He-
rich zone from the very beginning such that both a negative
entropy and a negative composition gradient are created.
This favours models with convective overshooting and low-
entropy generation. It also favours lower-mass secondaries
with lower entropies. For example, for M2 = 2M?, after
leaving L1, the stream can penetrate as much as 2.5R? of
the core, while for M2 = 5M? it stops further from the
primary center, passing through only 1R?. An alternative
criterion is that ∆MH,rad is small: ∆MH,rad≤ 0.2M?.
Based on our test calculations, we conclude that the
conditions for a successful explosive common-envelope ejec-
tion are satisfied for secondary masses up to 3M? in the
models with convective overshooting. A lower limit on the
secondary mass is roughly given by 1M? where the energy
released in the spiral-in does not provide enough energy to
lead to an expansion of the envelope (i.e. lead to a “slow”
merger); in that case the merger is expected to occur on a
dynamical timescale.
With these constraints to guide us, we find that the
ECEE conditions are found for primaries with masses up
to 40M?. If the minimum mass for black-hole formation is
25M?, this leads to CO cores in the range of 6.5–13M?
(using our standard convective overshooting parameter).
The amount of mass transferred from the secondary be-
fore the CE ejection also depends on the secondary’s mass:
it is about 1M? for a 3M? secondary, 0.8M? for a 2M?
secondary and 0.2–0.3M? for a 1M? secondary (but this
depends somewhat on the stream entropy parameter).
The final stage
While the initial spiral-in and the early merger phase take
place on a timescale of ∼ 100yr, once the H-rich stream con-
nects to the helium-core-burning shell, the evolution acceler-
ates. The overall duration of the explosive phase is ∼ 1/4yr,
although 90% of the energy is released in the last few days.
At the end of our calculations, the outer parts of the core
expand with a velocity exceeding the local escape velocity,

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Explosive Common-Envelope Ejection5
and typically 0.03–0.06M? of stream material has been
burned explosively. The nuclear energy that has been re-
leased (>
∼2 × 1050ergs) exceeds the binding energy of both
the He-rich layer and the common envelope by about a fac-
tor of 2.
By the time the envelope is ejected, we find that the
CO core has been moderately spun-up by the accretion of
angular momentum and transport into the core during the
pre-explosive core-accretion phase. The characteristic spe-
cific angular momentum of the core is ∼ 1016cm2s−1, at
the low end of what is required in the collapsar model for
LGRBs. However, further spin-up is possible, since even af-
ter the ejection of the envelope, mass transfer is likely to
continue and the system is close enough for tidal spin-up to
operate.7
Indeed, despite the ejection of the helium shell, the sys-
tem does not widen appreciably, if at all, because there is
still enough frictional angular-momentum loss to keep the
secondary in Roche-lobe contact. This suggests that the sys-
tem will continue to transfer mass after the CE ejection.8
This mass transfer occurs on a thermal timescale since the
secondary is highly out of thermal equilibrium, in fact has
a radius smaller than its equilibrium radius, and will try to
expand towards a new thermal-equilibrium stage.
The remaining evolution of the primary’s CO core to
core collapse is expected to be of order a few 103to at most
104yr (in the 18M?simulation it was 3000yr). We note that
this time is significantly longer than it would have taken the
core to core collapse without the ejection of the He layer
because of the signifiant core expansion and cooling.
4 APPLICATION TO LGRBS AND
SHORT-PERIOD BLACK-HOLE BINARIES
In order to assess the importance of ECEE events for LGRBs
and low-mass black-hole binaries, we first estimate in this
section the expected rate of these events and then present
detailed binary population synthesis results that simulate
the properties of the resulting black-hole (BH) binaries.
4.1 The ECEE Rate
Using the results discussed in the previous section, it is
straightforward to estimate the fraction of binaries that ex-
perience explosive CE ejection. This depends mainly on the
secondary mass and the condition that the systems experi-
ence Case C mass transfer.
Figure 3 shows the cumulative rate for ECEE events
7
binary (with Porb<
on a timescale of ∼ 104yr (also see Brown et al. 2007). However,
unlike the case considered by Detmers et al. (2008), the remaining
lifetime of the primary after case C mass transfer is short enough
that wind mass loss will not cause significant widening of the
system which would then tidally spin down the star again.
8We note, however, that there are enough uncertainties in our
calculations that we cannot rule out that the secondary will be out
of Roche-lobe contact after the CE ejection. This will ultimately
require full three-dimensional stellar-structure calculations.
Detmers et al. (2008) found that, in a sufficiently close He-star
∼10h), the core can be spun up significantly
Figure 3. The cumulative rate for ECEE events as a function
of the minimum primary mass required. The black curves assume
a maximum primary mass of 40M? whilst the grey curves take
a maximum primary mass of 30M?. The solid curves assume a
flat initial mass-ratio distribution (p(q) = constant), the dotted
curves use p(q) ∝ q. The light grey shading indicates a plausible
range for the minimum mass of a single star which can form a
black hole at solar metallicity. For these estimates we assumed
that 1% of binaries have an orbital period that leads to Case
C mass transfer. The actual rate could be higher by an order
of magnitude at low metallicity, and the minimum mass for BH
formation could be lower.
as a function of the minimum primary mass for different as-
sumptions about the maximum mass of the primary and dif-
ferent mass-ratio distributions. Here, we assume that 1% of
all binaries experience Case C mass transfer at solar metal-
licity. The primary masses M1 are chosen from an initial
mass function p(M1) ∝ M−2.7
1993). The rates are nomalised to a core-collapse SN rate of
1 per century (i.e., assume that one star more massive than
9M? is formed per century in a typical galaxy). This prob-
ably underestimates the core-collapse supernova rate in the
Milky Way (Cappellaro, Evans & Turatto 1999). We also
assumed that all massive stars form in binaries (Kobulnicky
& Fryer 2007), with a distribution of initial separations a
that is flat in loga between 3 and 104R?.
A significant source of uncertainty in the ECEE rates
is the initial mass-ratio distribution p(q), where q = M2/M1
is the mass ratio and M2 is the mass of the secondary. In
Figure 3 we consider a distribution where p(q) = constant
and one where p(q) ∝ q.
In using the ECEE mechanism to produce short-period
black-hole X-ray binaries, we require the primary to form a
BH. The light grey shading in Figure 3 indicates the ex-
pected minimum mass for black-hole formation, which is
generally believed to be in the range of 20–25M? (see,
e.g., Fryer & Kalogera 2001 and further references in PRH).
Black-hole formation is also generally assumed to be a re-
quirement for the formation of LGRBs, though we note that
rapid rotation at core-collapse for lower mass cores may
lead to the formation of magnetars (e.g., Thompson & Dun-
can 1993; Akiyama et al. 2003). Magnetar formation has
also been proposed as a channel for LGRBs (e.g., Wheeler,
H¨ oflich & Wang 2000; Burrows et al. 2007; Uzdensky & Mac-
Fadyen 2007; Bucciantini et al. 2008), which would increase
1
(Kroupa, Tout & Gilmore