Common envelope: on the mass and the fate of the remnant

0.3 MB·English

by  Natalia Ivanova

Draftversion January17,2011
PreprinttypesetusingLATEXstyleemulateapjv.11/10/09
COMMON ENVELOPE:ON THE MASS AND THE FATE OF THE REMNANT.
N. Ivanova1
Draft version January 17, 2011
ABSTRACT
One of the most important and uncertain stages in the binary evolution is the common envelope (CE) event.
Significant attention has been devoted in the literature so far to the energy balance during the CE event, expected
1 to determine the outcome. However this question is intrinsically coupled with the problem of what is left from the
1 donor star after the CE and its immediate evolution. In this paper we argue that an important stage has been
0 overlooked: post-CE remnant thermal readjustment phase. We propose a methodology for unambiguously defining
2 thepost-CEremnantmassafterithasbeenthermallyreadjusted,namelybycallingthecoreboundarytheradiusinthe
n hydrogen shell corresponding to the local maximum of the sonic velocity. We argue that the important consequences
a of the thermal readjustment phase are: (i) a change in the energy budget requirement for the CE binaries and (ii) a
J companion spin-up and chemical enrichment, as a result of the mass transfer that occurs during the remnant thermal
4 readjustment (TR). More CE binaries are expected to merge. If the companion is a neutron star, it will be mildly
1 recycledduringtheTRphase. ThemasstransferduringtheTRphaseismuchstrongerthantheaccretionrateduring
the commonenvelope,andthereforesatisfiesthe conditionfor ahypercriticalaccretionbetter. We alsoarguethat the

] TR phase is responsible for a production of mildly recycled pulsars in double neutron stars.
R
Subject headings: binaries: close — stars: evolution — X-rays: binaries — pulsars: general
S
.
h
p 1. UNCERTAINTYINTHECOMMONENVELOPETHEORY We anticipatethat introductionofthe two parameters
introduced accordingly two uncertainties. It is common
- In the standard treatment of common envelope (CE)
o to remove these uncertainties at the same time, consid-
outcomes via the “energy formalism” (Webbink 1984),
r ering the product of α and λ, by means of compari-
t thefinalseparationofthebinaryisdeterminedbyequat- CE
s son of observations with the binary population synthe-
ing the binding energy of the (shunned) envelope E
a bind sis calculations, where the product of the two parame-
[ to the decrease in the orbital energy Eorb: ters is varied to match the observations. However, this
1 approach has shown inconsistencies with the observa-
v E =E −E =−Gm1m2 + Gmcm2 (1) tions, especially large for the formation rates of black
3 bind orb,i orb,f 2a 2a hole LMXBs (Podsiadlowskiet al. 2003; Justham et al.
i f
6 2006). In particular, for LMXBs this required αCEλ&2
8 Hereaiandaf aretheinitialandfinalbinaryseparations, (Yungelson et al. 2006), although in massive giants λ≪
2 m1 andm2 aretheinitialstarmassesandmc isthefinal 0.1 (Podsiadlowski et al. 2003), and αCE is bound to be
. mass of the star that lost its envelope. ≤1.
1
0 Ebind is considered to be the sum of the potential en- The other way to reduce uncertainties is to consider
ergy of the envelope and its internal energy, and can them separately, e.g. one can try to determine an ‘ac-
1
1 be found directly from stellar structure for any accepted curate’ value of λ from stellar structure calculations.
: core mass (there are also modifications for Ebind, where It is then crucial to be precise about the definition of
v ionization energy or enhanced winds are taken into ac- the core – should only the hydrogen envelope be re-
Xi count, e.g. Han et al. 1995, 2002; Soker 2004): moved, or together with the H-burning shell, and so
on (Tauris & Dewi 2001). Without knowing what ex-
ar surface Gm1me actly counts as the core and which material ought to be
E = ǫ(m)dm= (2)
bind Z λR ejected, the inferred λ can vary by a factor of several
core 1 from this uncertainty alone.
Here λ is a parameter introduced to fit E ; it charac- The physical reason for this variation is that in gi-
bind
terises the donor envelope central concentration. m is ants, within the hydrogenshell, the potential is strongly
e
themassoftheremovedgiantenvelopeandiscommonly increasing towards the core. The uncertainty increases
assumed to be m = m −m , R is the radius of the as the mass of the donor increases, and changes from a
e 1 c 1
giant star at the onset of CE, and ǫ is the sum of the about a factor of 2 in intermediate mass stars at early
specific internal and potential energies. giant stage to a factor of 20 and more for well-evolved
For the final balance of energy, one more parameter massive stars (Tauris & Dewi 2001).
is introduced, α , to measure the energy transfer effi- We stressthatneither observationsnortheoryprovide
CE
ciency from the orbital energy into envelope expansion: now a strong constrainton what post-CE remnant mass
should be at the moment when the dynamical phase of
Gm m Gm m Gm m the ejectionends. Itdoesnothaveto be the sameasthe
α λ c 2 − 1 2 = 1 e (3)
CE massoftheremnantthatweobservenow,e.g.,indouble
(cid:18) 2a 2a (cid:19) R
f i 1
white-dwarf(WD)systemsorinsub-dwarfBstars: some
1University of Alberta, Dept. of Physics, 11322-89 Ave, Ed- remaining post-CE hydrogen-rich material can be easily
monton, AB,T6G2E7,Canada
2 Ivanova
removed through strong winds similar to those on hor-
izontal branch, or asymptotic giant branch, or in Wolf-
Rayetstarsetc. Betweenthe dynamicalphase andlong-
term evolution, the core will readjust itself on a thermal
time-scale,andthishasnotbeenaddressed. Here,wead-
dress the problem of what the post-ejection mass could
be, different regimes in which a post-CE remnant can
shed its remaining hydrogen-rich mass and the conse-
quences for a companion due to post-CE mass transfer.
2. THEPOST-EJECTIONREMNANT
2.1. The divergence point
InSPHsimulationsofphysicalcollisionsbetweenaRG
and a neutron star (NS) it has been found that not all
hydrogenmaterialis ejected along with the envelope – a
tinylayerofhydrogen,fromtheH-burningshell,remains
(Lombardi et al. 2006). This event is not directly com-
parable to a typical CE event in a binary as at the time
of initial approach, at periastron, the H-burning shell of
the donor could have been in the immediate Roche lobe
of the intruder. This magnifies the mass-loss from the
H-burningshellandassuchcandecreasethemassofthe
post-CE remnant compared to a typical CE, where this
shell might never be in the Roche lobe of the spiraling- Figure 1. Comparisonofthelocalthermaltime-scaleτth(m)and
in companion. This example makes clear that even in the local dynamical time-scale τdyn(m) with the mass-loss time-
a dynamical CE some hydrogen-richmaterial always re- scale τML. Shown in the case of 18.5M⊙ (ZAMS mass 20M⊙,
mains. consideredwhenR=750R⊙).
Ontheotherhand,studiesoftheevolutionofstripped
cores of low-mass RGs, have shown that there is a min- coremassesmX. Asduringtheadvancedevolutionstages
imum “envelope” mass δm that has to be left on stars can shrink, we ensured that the chosen giants had
e,min
the core in order for the star to reexpand; if less mass expanded to their current radius for the first time. On
is left on the core the star will contract and become a these giants, we imposed very fast (“adiabatic”) mass
WD (Deinzer & von Sengbusch 1970). This expansion loss, 1M⊙/year. Such timescale for the mass-loss τML –
or contractionof the remaining shell occurs on the ther- about several initial binary orbits – is comparable with
mal timescale of remaining layer, τth. a fast CE event. The lower bound on τML & 1yr should
It is plausible therefore to suppose that there is a be clear as a CE event has to happen over at least one
unique “divergence”pointm inside the hydrogenburn- binary period at the initial Roche lobe overflow.
d
ingshell,suchthatifapost-CEstarhasanymassabove We do not imply that a CE ejection features a con-
this point, the star will continue to expand on τ . If its stant fast mass loss, and also do not study the reaction
th
final mass is less than m , the star will shrink, also on of the outer (convective) envelope. We are interested in
d
its τ . We recognise that τ might mean different val- the reaction of the inner layers,which are most likely to
th th
uesinthe caseofdegeneratecore(applicableonlyto the remain after the envelope ejection has occurred.
remaining shell) or non-degenerate core (where it likely Foreachmasscoordinate,letuscompareτML withthe
to depend on the core conditions). We expect that the local thermal timescale τTH(m) = Ebind(m)/L(m) and
material above the divergence point, if left, will expand, the local dynamical timescale τdyn(m): The mass-loss
in order to obtain thermal equilibrium, but is not re- andthestarevolutionwillbeadiabaticifτML ≪τTH(m).
quiredto escapetoinfinity withoutanadditionalenergy The evolution can be described by hydrostatic approxi-
source (such as the orbital energy). During this thermal mation if τML ≫ τdyn(m), as a star will always acquire
readjustment (TR), it may also fill its Roche lobe. itshydrostaticequilibriumwithinadynamicaltime,and
its state at the hydrostatic equilibrium is defined by its
2.2. Calculations
thermal structure.
We tested this idea of “divergence” point on giants For most stars, τ is much shorter than any local
ML
of several initial masses (1,2,10,20,30 M⊙). The stars thermal timescale (see Fig. 1). We note however that
were evolved using the stellar code and input physics in the inner layers that are close to the cores of our
described in Ivanova & Taam (2004). This code is ca- most massive stars (20 and 30 M⊙), the complete mass-
pable ofperforming both hydrostaticandhydrodynamic loss sequence can take up to 10% of the local thermal
stellar evolution calculations. For a Roche lobe over- timescale ofa few hundredyears. Thus, evensucha fast
flow evolution in binaries, it finds mass loss rates im- mass-loss produces only approximately adiabatic evolu-
plicitly. Massive stars, where wind loss are important, tion: somethermalevolutionproceedsandisexpectedto
wereevolvedwithwindlossratesaccordingtoVink et al. beresponsible,inparticular,forsomeexpansionofinner
(2001), or, where Vink rates are not applicable, accord- non-degenerate layers during the CE phase. As a san-
ing to Kudritzki & Reimers (1978). ity check, we calculated additional mass-loss sequences
For each initial mass, we chose 2-4 evolutionary states for massive stars,with faster and slowermass-loss rates,
withinthegiantstagewithdifferenthydrogen-exhausted andfoundonlyminordifferencesintheregionofinterest
Mass and fate of the post-CE remnant 3
between the runs with 0.1,1 and 10 M⊙yr−1. the energy generation rate is maximum (Tauris & Dewi
On the other hand, local dynamical time-scales are 2001) — does not coincide with the divergence point.
longest at the surface and significantly shorter for in- We found that in the considered models m is close
div
nermost layers (Fig. 1). τ (m) is comparable by the to the “maximal compression point” m , which is the
dyn cp
orderofmagnitude to τ in the outer layerofthe mas- masszonewiththemaximumvalueof‘compression’P/ρ
ML
sive giants. τ (m) is however by 3 or more orders of in the hydrogen shell.
dyn
magnitude smaller than τ in the Helium rich layers,
ML
andcloser to Hydrogenexhausted core,it is ∼10−5τ , 2.4. Non-degenerate cores
ML
even in our most massive considered stars. Therefore,
For massive giants, as previously, a post-CE remnant
although the evolution of the outer layers is indeed de-
also has divergence point that corresponds to the min-
pendent on the inclusion of hydrodynamical terms, the
imum post-CE expansion of the remnant, although the
inner layers always have enough time to regain hydro-
overall response is different from the case of the giants
static equilibrium, and are therefore insensitive under
with degenerate cores:
the adopted mass-loss rate. In summary, we find that
for studies of the thermal reaction of inner layers that
• Forallpossibleremnantmasseswithm.m ,the
will form the remnant after the fast envelope ejection, cp
core slightly adiabatically expands during the fast
the hydrostatic version of the code is sufficient.
adiabaticmassloss. Once westopthe massloss,it
As a result of mass loss evolution, we obtained se-
can veryslightly (a few per cent) expand andthen
quencesof(post-CE)remnantswithdifferentfinal(post-
shrink dramatically, becoming smaller than it was
CE) masses, each of which then was evolved for several
before the CE.
τ , to check if this post-CE star is expanding or con-
th
tracting. We note that in our code the value of post-CE • For larger remnant masses, as previously, the con-
mass could be resolved no better than pre-CE resolu- vectiveenvelopeis re-formed,andthe starremains
tion in hydrogen shell, this is specifically important for asanextendedgiantforawhile. Theenvelopecan
low-mass giants (e.g., we have about 50 mesh points per become larger than the pre-CE giant2.
0.02M⊙ H-shell in 2 M⊙ RG with a mX =0.52M⊙).
To summarize, we separate the CE event into two • For the intermediate range of remnant masses,
stages: one resulting in the envelope ejection, and the above the m , but below the boundary where the
cp
subsequent thermal readjustment of the remnant. The convective zone redevelops, the core experiences a
latterpartisadistinctphaseunlessthespiral-in(includ- pulse on ∼τ , being able to expand by up to few
th
ing the ejection of the envelope) takes place on a time- hundred times more than this mass had as a ra-
scalecomparabletothe shortestthermaltime scale,sev- dius coordinatebeforetheCE.After thepulse, the
eral hundred years; and has not been heretofore treated post-CE star shrinks significantly, also becoming
in the literature. smaller than prior the CE.
Finally, an admonishment is in order: several esti-
matesexistfortheCEduration,neitheroneofthemcan Toillustratethesethreetypesoftheresponse,inFig.2
boast conclusive observational evidence or indeed self- weshowthetypicalcaseof9.75M⊙ (ZAMSmass10M⊙)
consistency. E.g., a ‘slow’ CE could last for 100 years star, taken when it had radius of 300 R⊙. For compar-
and longer (Podsiadlowski 2001). We can not justify ison, we show 18.5M⊙ (ZAMS mass 20M⊙) star with
which CE evolution timescale is more appropriate, and radius of 750 R⊙ (see Fig. 3). Even though this giant
this is not the purpose of this paper. We concentrate has profile of hydrogen qualitatively different from the
on the ‘fast’ event, however, we see no reason why our considered above 10M⊙ star, it shows similar behavior.
results should not be applicable in the ‘slow’ case: the The main difference with a 10 M⊙ star is that there is a
core reaction will be similar, albeit lagging by the time more contrasting response between inside m and out-
cp
the ejection takes. We also note that a 10-times slower side it. In a 30M⊙ giant this difference even stronger as
loss rate did not produce a significant difference in our m is located just below the bottom of hydrogen burn-
cp
calculations. ing convective zone.
2.3. Degenerate cores
2.5. The adiabatic response
Indeed, as in previous studies, we found that every
We recognize that the response we discuss above is
low-mass giant with a degenerate core has a unique di-
non-adiabatic, but is the equilibrium response (the one
vergence point m such that if post-CE mass is less
div that a star experiences in order to obtain its thermal
thanm ,itcontractsonτ . Allpost-CEremnantwith
div th equilibrium). Another important response to consider is
masses above m expand, create new outer convective
div the reaction of a star when the dynamical event ends,
zone and keep expanding even after τ .
th true adiabatic response. It is established that adiabatic
After locating m , we analyzed pre-CE giants struc-
div response of the surface layers depend on whether they
ture to find what characteristics these points had in ini-
are radiative or convective (Hjellming & Webbink 1987;
tial giants, before the stripping began. For this, all gi-
Soberman et al. 1997). In particular, radiative layers on
ants, including massive, were used. We noticed that
dynamicaltimescaletendtoshrink,andconvectivelayer
among all the giants, m could have initially a wide
div
range of hydrogen content, X = 0.08−0.58, and so a
2 Here, we can not fully separate the post-CE TR expansion
criterion involving specific constant value of hydrogen
from a normal stellar evolution along a giant branch, however we
abundance could not be satisfactory. Similarly, another findthatapost-CEstarobtainsaftertheCEalargeradiusfaster
criterion discussed in the literature – the location where thanitwouldhaveotherwise.
4 Ivanova
Figure 2. Sizeof a post-CE remnant for a9.75M⊙ star (ZAMS Figure 4. Theinitialbindingenergy(solidredline)andavailable
mass 10M⊙, considered when R = 300 R⊙). Rpre−CE is the ra- orbitalenergiesintheinitialandfinalconfigurations(dottedgreen
diuscoordinateofeachconsideredremnantmassbetweenthemass anddashed bluelines)for an18.5M⊙ star (ZAMSmass 20M⊙)
loss, Rpost−CE is the radius of a post-CE star when the fast adi- withR=750R⊙. Whencalculatingthepost-CE∆Eorb,thesizes
abatic mass loss stopped and Rmax is the maximum radius that ofthepost-CEremnants foreverygivenmasswereusedtodefine
this post-CE star had obtained within τth (time to reach maxi- af. Themasscoordinateisnormalizedtomcp (inthisstarmcp=
mum ranged from 100 to 2,100 years). Shown are the ratios of 6.47M⊙). The∆Eorbarecalculatedassumingthatthecompanion
Rpost−CE andRpre−CE (blueline,opencirclescorrespondtoeach massis1.82M⊙.
calculated model), Rmax and Rpre−CE (green line, solid circles),
RmaxandRpost−CE(redline,opensquares). X =0isthemassof eredmodelthelocationofthedivergencepointandfound
thehydrogenexhaustedcore,εn=maxiswherethenuclearburn- that all of them are located within initially (pre-CE) ra-
inghasmaximumenergygenerationrate,mcp isthe“compression diative layers. We conclude that immediate response for
point” (where P/ρ has the maximum value within the layer be-
mass removalto the divergencepoint is alwaysa shrink-
tween X = 0 and mbcz) and mbcz is the mass coordinate of the
bottom oftheconvective zoneinthepre-CEstar. age and therefore does not affect our conclusions based
on the thermal response.
3. CONSEQUENCESFORTHEENERGYBUDGET
As the core expands during semi-adiabatic mass-loss,
asurvivingbinarymustbe wider whencoreexpansionis
taken into account. Thus more binaries will merge (this
result is opposite to the claim in Deloye & Taam 2010).
As an example, the giant in Fig. 3 will easily survive a
CE event with a NS of 1.4M⊙ (with αCE =1) if its core
didnotexpand. Settingcoremass.7.44M⊙ willsatisfy
the energy budget to create a compactbinary. However,
if one takes into account the core expansion, the binary
will merge: for all core masses above m , the remnant
cp
will expand significantly and overfill its Roche lobe. A
minimumcompanionmass,forwhichbothenergybudget
and post-CE core size are taken into account, will be
∼ 1.82M⊙ and the giant core in this case should have
been removed to at least m (see Fig. 4). It can be
cp
seen from this Figure that for a fixed companion mass,
there is a unique solution where available orbital energy
canexceedbindingenergyiftheremnantexpandedafter
massloss,whereasanon-expandedremnantgivesawide
range of the possible core masses, from 5.95 to 7.5 M⊙.
Let us introduce “the energy expense”, the difference
between the required energy E (m) and the available
Figure 3. Sizeofapost-CEremnantfora18.5M⊙ (ZAMSmass orbital energy δE , normalizebdinpder E (m).
20M⊙,consideredwhenR=750R⊙). Notations asinFigure2. orb bind
E (m)−(E −E )
bind orb,i orb,f
remain the same or expand. We checked for all consid- δ = (4)
ε
E (m)
bind
Mass and fate of the post-CE remnant 5
Figure 5. Theenergyexpenseinthethelayersthatcanbecome Figure 6. Theenergyexpenseinseveralgiantswithinitialmass
apost-CEremnantfora4.75M⊙ star(ZAMSmass10M⊙)with 20M⊙. The mass coordinate is normalized to mcp. Energy ex-
R = 400R⊙. The two curves are for different companion masses penses are calculated using different assumed companion masses,
(asmarked). Themasscoordinateisnormalizedtomcp. tobringthem tothesamevaluefortheenergyminima.
4.1. Consequences for the final post-CE mass
Itisthe(normalized)excessenergyavailabletotheenve-
lopeafterallthematterabovethegivenmasscoordinate Let us consider what happens if the companion was
has been removed. Of course, a positive δε signals that massive enough so that during CE not all mass to mcp
the removal process is not possible from the energy con- had to be removed. In this case a CE would end with
siderations. Fig. 5 shows distribution of δε as a function a binary separationsuch that Roche Lobe overflowfor a
of the mass coordinate. For these calculations, Eorb,f post-CE remnant during its TR will follow. To consider
was assumed to be at the Roche lobe limited orbit for this, we took a post-CE model showed in Fig. 3, con-
the post-CEremnant(semi-adiabatic expansionis taken sidering its remnant with mpost−CE = 6.84M⊙ (larger
into account), so that available orbital energy is at its thanits mcp =6.47M⊙). Thispost-CEremnantisthen
maximum. placed in a contact binary with an arbitrary companion
Notethatmcp theenergeticallyoptimalpositiontore- of 3 M⊙: this mass self-consistently satisfies the energy
move the envelope to: it is the equilibrium point of the required to shed the mass above 6.84M⊙. For the mass
generalizedforce∂m(Ebind+Eorb). Theshapeandthelo- transfer(MT)we canchooseafully conservativeorfully
cationoftheminimumofδε onlydependsonthedonor’s non-conservative mode 3.
energy profile up to the Roche radius, and does not de- As expected, the post-CE remnant rapidly expanded
pendonthecompanionmass. Thelatteronlydetermines and started the MT; initially at a very high rate (∼
thTehmuasgintitiusdeeasoyf tthoedeenteerrgmyineexctehses (mseineimFiugm. 5)m.ass of a 1 − 5 × 10−2M˙⊙/yr), in accordance to τth of the re-
maining hydrogen rich layer. After removing most of
companion which allows the survival of the binary: as it
the layer above m , it slowed down to τ of the core
cp th
can be seen from the Fig. 5, it is such that will result in
removal of mass to m , precisely. (∼ 10−4M˙⊙/yr). The MT continued till MT rates be-
cp come comparable to the TR time-scale so that the core
We verified that the coincidence of the minimum of
can shrink faster than it expands due to mass loss. In
the energy expense with shedding the envelope to about
this particular example the core reached almost exactly
m for minimum likely companionmass holds for many
ofcpthe studied giants, though does not hold for giants the divergence point, shedding ∼0.4M⊙ during the MT
sothatthefinalmasswasm . WealsoperformedaMT
that are early on the giant branch which only recently cp
calculation in the fully non-conservative regime. In this
developconvectiveenvelopes. E.g. inanearly10M⊙ we
case the final mass of the core after the TR phase is the
observeonemoreenergyminimum,atahighercoremass
same as in the conservative calculations.
∼2.45M⊙;thelocalenergyminimumatmcpnonetheless We also considered the case when the companion is
holds. In more massive early giants, the energy expense
minimumisbetweenm andm (Fig.6),itslocationis a NS. We note that with a 20M⊙ donor then, even for
cp X α = 1, the energy requirements for envelope ejection
closer to the location of ε =max than to m ; the star CE
n cp would not be satisfied if the core expands as much as
still has the same reaction on expansion or contraction
with respect to m as other stars.
cp 3 Partial conservation, limited to Eddington rates can be con-
sideredas well, but withthe MT rates that wefind and describe,
4. POST-CEMASSTRANSFER thiscasedoesnotdiffermuchwiththefullynon-conservativecase.
6 Ivanova
we find after our fast mass loss (i.e. the binary would tion when the radiation diffusion dominates in this case
merge). The final mass of the remnant of the massive (10−3M⊙yr−1)ishigherthantheonefoundtoworkdur-
giantisthesame,mcp,asforthe3M⊙companion. Ifthe ing a CE event (2×10−4M˙⊙yr−1).
masstransferis fully conservative,the NSis presumably If the latter estimate is more proper than in the stud-
spun-up, as it would accumulate 0.34M⊙. Again, in the ies listed above, then it is possible that a CE hyper-
caseofa fully non- conservativeregime,we findthat the accretion, as having too low mass accretion rate, does
final mass of the post-CE remnant is mcp. not lead to a significant accumulation of the material.
Nextweconsideredasystemwitha10M⊙ giant(same Post-CEcoreexpansion,however,ineithercasecanlead
as shown in Fig. 2), considering as the post-CE core toahyper-accretionregime,asduringthisthermalpulse
2.54 M⊙ (mcp = 2.04). In our MT simulations with a it provides a much higher mass accretion rate. This can
NS companion, less material above mcp has been trans- lead to a NS spun up. In our calculations, MT rates
ferred, only 0.24 M⊙, the final remnant mass is 2.3 M⊙. exceeded 10−3M⊙yr−1 long enough to accrete in hyper-
It might be connected to the fact that 10 M⊙ star does criticalregimeonaNS0.29M⊙inthecaseof20M⊙giant
not have such a sharp profile as a 20M⊙ in the post-CE and 0.09M⊙ in the case of a 10M⊙.
thermal pulse zone, and its post-CE expansion is more We note that the observed double NSs are generally
flatteruntil aboutthis mass(see Fig.2). With asmaller mildlyrecycled,havingperiods0.024−2.7seconds(Stairs
companion mass, more of the post-CE remnant mass is 2004). The mass distribution of those with mass mea-
stripped off. surement errors . 0.02M⊙ are such that the difference
betweenthemassesoftheNSsis.0.1M⊙ (e.g.,seedata
4.2. Consequences for the companion in Stairs 2004; Kiziltan et al. 2010). Their location on
The MT rates that we encounter in the post-CE TR the P −P˙ diagramfor galacticfield NSs is alsointerme-
phase are highly super-Eddington and we face the obvi- diate between the millisecond pulsars and non-recycled
ous question whether the MT is approximately conser- ones (Arzoumanian et al. 1999), and the post-accretion
vative or almost non-conservative, as this is crucial for period is likely to be not millisecond (Lorimer et al.
a companion. The question what happens if the mass- 2005). It couldbe a signthat the veryrapidmasstrans-
accretionrateonaNSexceedsEddingtonlimithasbeen fer followed the CE andpreceding the second NS forma-
discussed extensively in the literature, in particular, the tionis not capableto fully spin up andefficiently reduce
regimeinwhichitexceedsM˙ by many orders of mag- the magnetic field on a NS that was formed first. Same
Edd
nitude. mechanismcanleadtoaformationofmildlyrecycledbi-
Begelman(1979)showedthatiftheaccretionrateisex- nary pulsars with low-mass white dwarf companions (Li
tremelyhigh,fewtimes 10−4M⊙ yr−1,thenwithinsome 2002; Deloye 2008).
volume (the “trapping radius”, e.g. King & Begelman
1999) around the star the diffusion of photons outward 5. CONCLUSIONS
cannotovercometheadvectionofphotonsinward. While
Weanalyzedasetofgiantmodelswithrespecttotheir
a black hole can swallow all the material in this case, if
likely post-CE response. Although our set was not ex-
theaccretorisaNS,radiationpressureneartheNS’ssur-
haustive, it did exhibit a clear trend that allowed us to
faceresistsinflowinexcessofthe Eddingtonlimit, likely
conclude that: i) every giant has a well-defined post-
leading to creation of a Thorne-Zy´tkov object. Blondin
CEremnantafterithasbeenthermallyreadjusted,most
(1986) has also found that when MT rates exceed the
likely given by the divergence point (see discussion be-
Eddington rate by 103×L /c2 or more, the accretion
Edd low); ii) the divergence points, at the current resolution,
proceeds in a hypercritical regime.
arebestapproximatedbythepointinthehydrogenburn-
Hypercritical accretion was then argued to be respon-
ing shell that had maximal compression (local sonic ve-
sible for such efficient material accumulation during a
locity)m prior to CE.This definition allowsus to find
cp
CE event, that a NS is likely to convert to a black hole
quickly a post-CE core mass for any giant without per-
(Chevalier 1989). Brown (1995) used this argument to
forming mass loss calculations.
understand double NS formation. He showed that in-
We remark that this divergence point does not nec-
deed in a CE event the Bondi-Hoyle-Lyttletonaccretion
essarily mark the final mass of the remnant (e.g., the
rate is about 104×M˙Edd and a NS can accumulate up stellarwindinHe richstarscouldquicklyandeffectively
to 1 M⊙. He arguedthat inthis case,consideringthat a remove the remaining hydrogen-richenvelope) or imme-
number of the discovered double NS have masses closer diatepost-ejectionmass,howeveritmarksthemassafter
to the lowest possible NS mass limit, a double NS can the thermal core readjustment.
be formed only from a binary with almost similar initial The post-TR core, defined by m , most likely co-
cp
masses,evolvingthen viadouble CEevent,beforeeither incides with the post-ejection mass in low-mass giants,
of the NSs was formed. where reestablishing of a convective envelope for masses
Houck & Chevalier (1991) considered neutrino losses above the divergence point happens on a few dynami-
during accretion on a NS. They studied in detail the cal timescales, leading to another (likely unstable) MT
regimes of the mass accretion 10−4 .M˙ .104M⊙yr−1, event. As the Ebind of remaining shells during this pe-
and found that radiation diffusion becomes important riodhasnotbeenchangedmuchcomparedtothepre-CE
when the accretion rate falls below 10−3M⊙yr−1, for state, the whole sequence of events can be considered as
smaller rates the radiation pressure can not support an one CE event with a final core being m .
cp
envelope around NS surface and can not cease the in- For giants with non-degenerate cores the situation is
fall of the material. We note that the accretion rate more complicated. It is not possible to claim that the
that separate the hypercritical accretion with the accre- divergence point defines the post-CE core (dynamical
Mass and fate of the post-CE remnant 7
phase)uniquelyfromthe energeticbudgetpointofview. is found to occur during a CE, corroborating the mass
However,the divergence point does appear to define the distributions of the observed double NSs. We conclude
remainingpost-TRcore,ifthe post-CEconfigurational- that the post-CE TR phase can be responsible for a for-
lowsRochelobeoverflowforthepost-CEremnantduring mationofmildlyrecycledpulsarsinpost-CEbinariesand
the core TR. During this period, (stable) MT can pro- specifically in double NSs.
ceed, resulting in the enrichment of a companion star
with material from the hydrogen-burning shell. 6. ACKNOWLEDGMENT
In all cases, the post-CE remnant of a giant with a
NI thanks S. Justham and C. Heinke for constructive
non-degenerate core has expanded by few times by the
comments and acknowledges support from NSERC and
end of TR. Remnant masses greater than m lead to
cp Canada Research Chairs Program.
greaterexpansion;foreachparticulargiantstar,themin-
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