Nuclear-dominated accretion and subluminous supernovae from the merger of a white dwarf with a neutron star or black hole
aa r X i v : . [ a s t r o - ph . H E ] J un Mon. Not. R. Astron. Soc. , 1– ?? (2011) Printed 13 November 2018 (MN L A TEX style file v2.2)
Nuclear-dominated accretion and subluminous supernovae from themerger of a white dwarf with a neutron star or black hole
B. D. Metzger , ⋆ Department of Astrophysical Sciences, Peyton Hall, Princeton University, Princeton, NJ 08544, USA NASA Einstein Fellow
Received / Accepted
ABSTRACT
We construct one dimensional steady-state models of accretion disks produced by thetidal disruption of a white dwarf (WD) by a neutron star (NS) or stellar mass black hole (BH).At radii r ∼ < . − cm the midplane density and temperature are su ffi ciently high to burnthe initial white dwarf material into increasingly heavier elements (e.g. Mg, Si, S, Ca, Fe,and Ni) at sequentially smaller radii. When the energy released by nuclear reactions is com-parable to that released gravitationally, we term the disk a nuclear-dominated accretion flow(NuDAF). At small radii ∼ < cm iron photo-disintegrates into helium and then free nuclei,and in the very innermost disk cooling by neutrinos may be e ffi cient. At the high accretionrates of relevance ∼ − − . M ⊙ s − , most of the disk is radiatively ine ffi cient and prone tooutflows powered by viscous dissipation and nuclear burning. Outflow properties are calcu-lated by requiring that material in the midplane be marginally bound (Bernoulli constant ∼ < ∼ > − ∼ − − − M ⊙ of radioactive Ni and, potentially, a trace amount of hydrogen.Depending on the pressure dependence of wind cooling, we find that the disk may be ther-mally unstable to nuclear burning, the likelihood of which increases for higher mass WDs.We use our results to evaluate possible electromagnetic counterparts of WD-NS / BH merg-ers, including optical transients powered by the radioactive decay of Ni and radio transientspowered by the interaction of the ejecta with the interstellar medium. We address whetherrecently discovered subluminous Type I supernovae result from WD-NS / BH mergers. Ulti-mately assessing the fate of these events requires global simulations of the disk evolution,which capture the complex interplay between nuclear burning, convection, and outflows.
Key words: nuclear reactions, nucleosynthesis, abundances - accretion disks - super-novae:general - stars: white dwarf
Sensitive wide-field optical surveys are revolutionizing our under-standing of time-dependent astrophysical phenomena. New typesof stellar explosions, that were once too faint or rare to be detected,are now routinely discovered by e ff orts such as the Palomar Tran-sient Factory (Law et al. 2009; Rau et al. 2009) and PanSTARRs(Kaiser et al. 2002). As our census of the transient universe ex-pands, it becomes increasingly important to evaluate the expectedelectromagnetic counterparts of known astrophysical events.Several Type I supernovae (SNe) have recently been discov-ered that are dimmer and / or more rapidly evolving than normal ⋆ E-mail: [email protected]
SNe Ia or Ib / c. Included among these is a class of peculiar IaSNe characterized by lower luminosities and lower ejecta velocitiesthan normal Ia’s (‘SN 2002cx-like events’; Li et al. 2003; Jha et al.2006), and which occur predominantly in star-forming galaxies(Foley et al. 2009). SN 2008ha is an extreme example, with a peakbrightness and rise time of only M V ≃ −
14 and ∼
10 days, re-spectively (Valenti et al. 2009; Foley et al. 2009, 2010). Anotherrecently identified class of subluminous transients are SNe Ib withCa-rich [but S, Si, and Fe-poor] spectra (e.g. SN 2005e; Perets et al.2010). These events appear to result from ejecta that have un-dergone helium burning (Perets et al. 2010, 2011; Waldman et al.2010) and, unlike 2002cx-like events, are associated with an olderstellar population located in the outskirts of their host galaxies. c (cid:13) B. D. Metzger
Even more rapidly-evolving SNe such as 2002bj (Poznanski et al.2010) and 2010X (Kasliwal et al. 2010) may be related to this class.Lacking hydrogen in their spectra, Type I SNe are generallythought to originate from compact progenitors such as white dwarfs(WDs) or massive stars without extended envelopes. Because com-pact stars lose most of their initial thermal energy to adiabaticexpansion during the explosion, their emission must be poweredby continued energy input from the radioactive decay of isotopessuch as Ni. The rapid evolution and low luminosities of eventslike 2002cx and 2005E thus require both a significantly lower Niyield, and total ejecta mass, than characterize normal SNe.It is economical to associate new classes of SNe with antici-pated variations of well-studied models, such as the core collapseof a massive star or the thermonuclear explosion of a WD. Eventslike 2002cx may, for instance, result from the pure deflagration ofa Chadrasekhar or sub-Chandrasekhar mass WD (e.g. Branch et al.2004; Phillips et al. 2007), as opposed to the ‘delayed detonation’models that best describe normal Ia SNe (Nomoto et al. 1984).They may alternatively result from weak core collapse explosions(Valenti et al. 2009; Moriya et al. 2010). For Ca-rich Ib SNe like2005E, a promising explanation is the detonation of a helium shellon the surface of a C / O WD (Woosley et al. 1986; Iben & Tutukov1991; Livne & Arnett 1995; Bildsten et al. 2007; Shen et al. 2010;Woosley & Kasen 2010). Though promising, none of these expla-nations is yet definitive (e.g. Woosley & Kasen 2010). The possi-bility thus remains that at least some of these events represent anentirely new type of stellar explosion.In this paper we examine the observable signatures of themerger of a WD with a binary neutron star (NS) or black hole (BH)companion. Four Galactic WD-NS binaries are currently knownthat will merge due to the emission of gravitational radiation withina Hubble time (see § ff ects of nucleosynthesis on the thermodynamics and com-position of the disk and outflows. Although multidimensional nu-merical simulations are ultimately necessary to evaluate the evolu-tion and fate of these systems, the large range in spatial and tem-poral scales involved make such a study numerically challenging.Here, as a first step, we instead construct a simplified model of thedisk and its outflows that we believe captures some of the essentialphysics. We use our results for the mass, velocity, and compositionof the ejecta to quantify the associated electromagnetic counter-parts of WD-NS / BH mergers for a wide range of systems.This paper is organized as follows. In § § § § .
1) and outflow properties ( § . § . § . § / WD-BH mergers, includ-ing subluminous Type I SNe ( § .
2) and radio transients ( § . § A standard scenario for the formation of tight WD-NS / BHbinaries invokes common envelope evolution of an initiallywide binary, consisting of a neutron star or black hole anda intermediate mass ∼ < − M ⊙ main sequence companion(e.g. van den Heuvel & Bonsdema 1984). For a limited range oforbital periods, unstable Roche lobe overflow (RLOF) begins onlyafter the secondary has left the main sequence. Depending on theevolutionary state of the stellar core when this occurs, the end re-sult is a NS or BH in orbit with a WD with either (from lowestto highest WD mass) a pure He, He-C-O (‘hybrid’), C-O, or O-Necomposition. Close WD-NS / BH binaries can alternatively form di-rectly by collisions in dense stellar regions, such as the centers ofgalaxies or globular clusters (Sigurdsson & Rees 1997).After the WD-NS / BH binary is brought close together,the system continues to lose angular momentum on a longertimescale to gravitational wave emission. Angular momentumlosses may be enhanced by the eccentricity induced by Kozai os-cillations if a tertiary companion is present (Thompson 2010).Of the ∼ >
20 WD-NS binaries identified in our Galaxy (Lorimer2005), only four are su ffi ciently compact that they will mergein ∼ < yr (PSR J0751 + M WD = . , . , .
99, and 0.2 M ⊙ , respectively (e.g. Bailes et al.2003). From the first three of these systems, Kim et al. (2004) esti-mate that the WD-NS merger rate in the Milky Way is 10 − − − yr − , although correcting for pulsar beaming increases this rateby a factor of several. Population synthesis models predict some-what higher rates ∼ − − − yr − , but with larger uncer-tainty (Portegies Zwart & Yungelson 1999; Tauris & Sennels 2000;Davies et al. 2002). No confirmed WD-BH systems are currentlyknown.Once the orbital period decreases to ∼ < / BH companion. Approximating theWD as a
Γ = / R WD ≃ M WD . M ⊙ ! − / − M WD M ch ! / / (cid:18) µ e (cid:19) − / km , (1)where M ch = . M ⊙ and µ e is the mean molecular weight per elec-tron (Nauenberg 1972). The orbital separation at Roche contact isapproximately (Eggleton 1983) R RLOF = R WD . q / + ln(1 + q / )0 . q / , (2)where q = M WD / M and M is the mass of the primary BH or NS.If orbital angular momentum is conserved, then mass trans-fer is unstable for q ∼ > . − .
55 (e.g. Verbunt & Rappaport 1988;see Paschalidis et al. 2009, their Fig. 11). If strictly applicable, thiscriterion would limit tidal disruption to NS binaries with relativelymassive ∼ > . M ⊙ C-O / O-Ne WDs, and might preclude disruptionin WD-BH binaries altogether. Stable systems increase their orbitalseparation after mass transfer begins. The result is a long-lived ac-creting system, which may be observed as an ultracompact X-raybinary (e.g. Verbunt & van den Heuvel 1995).Conservative mass transfer is, however, unlikely. Tidal cou-pling during inspiral can transfer orbital angular momentum intospin. Furthermore, the accretion rate onto the NS / BH just af-ter RLOF is highly super-Eddington, in which case a com- c (cid:13) , 1– ?? mon envelope may engulf the system and outflows are likely(e.g. Ohsuga et al. 2005). Depending on the amount of angular mo-mentum lost to winds or tides, unstable mass transfer may occur insystems with even lower mass ratios q ∼ < . / O-Ne WDs with low mass ∼ M ⊙ BHs (Fryer et al. 1999). Fu-ture numerical simulations are required to address the challengingissue of stability in degenerate binary mergers (e.g. Guerrero et al.2004; D’Souza et al. 2006).If mass transfer is unstable, then the WD is tidally disrupted injust a few orbits. The WD material then circularizes and producesa disk around the central NS / BH with a total mass M WD and char-acteristic radius R d ∼ R RLOF (1 + q ) − . This disk accretes onto theNS / BH on the viscous timescale t visc ≃ α − R GM ! / HR d ! − ∼
140 s (cid:18) α . (cid:19) − M . M ⊙ ! − / (cid:18) R d cm (cid:19) / H / R d . ! − (3)and at a characteristic rate˙ M ( R d ) ∼ M WD t visc ∼ − M ⊙ s − (cid:18) α . (cid:19) M . M ⊙ ! / M WD . M ⊙ ! (cid:18) R d cm (cid:19) − / H / R d . ! (4)where M is the NS / BH mass, α parametrizes the disk viscosity( § H is the scale-height of the disk, normalized to a char-acteristic ( § α ∼ . − . ffi cient angular momentum transport(e.g. Laughlin & Bodenheimer 1994). In § . M ( R d ) ∼ − − − M ⊙ s − , depending primarily on M WD and α . More massive WDs accrete at a higher rate, because both˙ M ∝ R − / ∝ R − / (1 + q ) / and possibly α (if the disk is grav-itationally unstable) increase with q . At such high accretion rates( ∼ − times larger than the Eddington rate) the disk cannotcool through photon emission and is termed a radiatively-ine ffi cientaccretion flow (RIAF). In this section we present a one dimensional steady-state model ofthe accretion disks produced by WD-NS / BH mergers. Although thesteady state approximation is clearly invalid just after the merger,it represents a reasonable description on timescales ∼ t visc , duringwhich most of the mass accretes at the rate estimated in equation(4). We address the stability of our solutions in § . 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. Neutrino Cooled H e S i/ S / A r / C a C / O [ H e ] F e / N i O / N e / M g [ C / S i ] NS/BH
Inner Disk / Boundary Layer p / n ? d M(R ) in d6−7
R ~ 10 cm R ~ 10 cm
Figure 1.
Schematic diagram of accretion and outflows following themerger of a WD-NS / BH binary. At the outer edge of the disk (radius R d ∼ . − . cm) the accretion rate ˙ M ( R d ) is ∼ − − . M ⊙ s − , de-pending on the properties of the merging binary and the viscosity α (eq. [4]).The outer composition is He, He-C-O, C-O, or O-Ne, depending on theproperties of the disrupted WD. As matter accretes to small radii, it ex-periences higher temperatures and burns to increasingly heavier elements.Outflows are driven from the disk (directly or indirectly) by energy releasedfrom viscous dissipation and nuclear burning (Fig. 2-5). Elements synthe-sized near the midplane are transported to the surface, such that the localcomposition of the wind matches that of the midplane. The outflow com-position characterizing each range of radii are shown next to the outgoingarrows for the case of a C-O WD (the He WD case is shown in brackets).The velocity of the wind is a factor 2 η / times the local escape speed, suchthat the inner disk produces the highest velocity outflow. On larger scalesthe ejecta forms a singular homologous outflow with a much lower velocitydispersion (see text). At radii r < R in ∼ − cm the temperature is suf-ficiently high that nuclei are photodissintegrated into He and, ultimately,free neutrons and protons. Near the surface of the NS or BH, cooling byneutrinos may be important if the accretion rate is su ffi ciently high. thick, averaging over the vertical structure serves only as a crudeapproximation. In standard RIAF models, viscous heating is bal-anced by advective cooling, due either to accretion or outflows (seeNarayan et al. 1998 for a review). As we describe below, a keyaspect of accretion following WD-NS / BH mergers is the relativeimportance of heating from nuclear reactions compared to viscousheating. We term this novel accretion regime a Nuclear DominatedAccretion Flow (NuDAF).We begin by defining the (positive) steady state accretion rate˙ M ( r ) ≡ − π r Σ v r , (5)where Σ = H ρ is the surface density, ρ is the midplane density, H is the vertical scaleheight, and v r < M w , whichwe characterize by the local quantity p ( r ) ≡ ∂ ln ˙M w ∂ ln r (6)We do not demand that p ( r ) be constant with radius, but rather de-termine its functional form self-consistently from the solution, asdescribed below.We assume that the wind exerts no torque on the disk, suchthat it carries away only its own specific angular momentum,i.e. j w = j d = r Ω , where Ω is the angular velocity. This is validprovided that the disk is not threaded by a large scale field of su ffi -cient strength to establish an Alfven radius significantly above thedisk surface. We furthermore assume that no torque is applied by c (cid:13) , 1– ?? B. D. Metzger the inner boundary, such that (at radii much greater than the innerdisk edge) the radial velocity obeys v r ≃ ν Ω ∂ Ω ∂ r ≈ − ν r = − α (cid:18) Hr (cid:19) v k , (7)where we adopt a Shakura & Sunyaev (1973) kinematic viscosity ν = α P /ρ Ω with α ≪ H ≈ a / Ω < r is the vertical scaleheight; a ≡ ( P /ρ ) / defines the midplane sound speed; and inthe last expression we assume that Ω equals the Keplerian rate Ω k = ( GM / r ) / = v k / r to leading order in ( H / r ) . In adopt-ing an anamalous viscosity we have implicitely assumed that an-gular momentum by MHD turbulence or gravitational instability outwards dominates any inward transport from convection; in § . M = πν Σ .Mass continuity ∂ ˙ M d /∂ r = ∂ ˙ M w /∂ r implies that ∂ ln ρ∂ ln r + ∂ ln a ∂ ln r = p − , (8)while radial momentum conservation v r ∂ v r ∂ r + ρ ∂ P ∂ r + GMr − r Ω = ∂ ln ρ∂ ln r + ∂ ln a ∂ ln r − r ( Ω − Ω ) a = , (10)where we have neglected the first term ∝ v r ∝ ( H / r ) .The entropy equation reads˙ q adv ≡ v r T ∂ s ∂ r = v r ∂ǫ∂ r + P ∂ρ − ∂ r ! = v r a r γ − ∂ ln a ∂ ln r − ∂ ln ρ∂ ln r ! = ˙ q (11)where T is the midplane temperature, γ ∈ [4 / , /
3] is the adi-abatic index, and s and ǫ are the specific entropy and internalenergy, respectively. We have neglected the chemical potentialterm ∝ µ i ∂ Y i ∂ r because all species (including electrons) are non-degenerate at the radii where nuclear burning commences. Equa-tion (11) may be interpreted as the balance between “cooling”from advection through the disk ˙ q adv and others sources of heat-ing / cooling ˙ q = ˙ q visc + ˙ q nuc + ˙ q w , including those resulting fromnuclear reactions ˙ q nuc ( ρ, T ) (see eq. [14] below), winds ˙ q w (see be-low), and viscous dissipation˙ q visc = ν r ∂ Ω ∂ ln r ! ≃ ν Ω = ν Ω " + (cid:18) Hr (cid:19) ∂ ln ρ∂ ln r + ∂ ln a ∂ ln r ! , (12)where we have used equation (10) in the last expression and neglectterms ∝ ∂ ( H / r ) /∂ r . We again note that radiative cooling may beneglected because the timescale for photon di ff usion from the diskmidplane is much longer than the viscous time time (eq. [3]). In § . ff erentfrom the model developed in this section.An important characteristic of RIAFs without outflows ( p =
0) is that the Bernoulli parameterBe d ≡ v r + r Ω + ǫ + P ρ − GMr (13)is generically positive (Narayan & Yi 1994). This implies that, inprinciple, material in the midplane has su ffi cient energy to es-cape to infinity. This fact has been used previously to argue thatRIAFs are susceptible to powerful outflows (Narayan & Yi 1995;Blandford & Begelman 1999) that carry away a substantial portion of the accreting mass, viz. p ∈ [0 , preferentially heated . Only in this manner can the windachieve a positive Bernoulli parameter Be w = v / >
0, while theremaining disk material is accordingly cooled and remains bound(Be d ∼ < v w is the asymptotic velocity of the wind.Following previous work (e.g. Kohri et al. 2005), we quantifythe e ffi ciency of wind heating by a parameter η w ≡ v / v k , whichequals the ratio of Be w to the local gravitational binding energy.This prescription is certainly not unique; for instance, v w could in-stead scale with the disk sound speed a . Although for our purposeshere this distinction is unimportant since v k and a have similar val-ues, the pressure-dependence of wind cooling is di ff erent betweenthese cases, which has an important e ff ect on the thermal stabilityof the disk ( § . d η w , this uniquelyspecifies the wind mass outflow rate p ( r ) (eq. [6]). Although weadopt a ‘two zone’ model (disk + outflow), we do not specify thesource of wind heating explicitly. In most of our calculations weadopt values for η w ∼ few, because a terminal speed of the orderof the escape speed is a common feature of thermally-driven winds(e.g. Lamers & Cassinelli 1999).If the wind is heated, then the disk necessarily cools at the rate˙ q w = − ( v / − Be d )2 π Σ r ∂ M d ∂ ln r = − p ( η w − Be ′ d ) ν Ω , (14)where Be ′ d ≡ Be d / v k is the ‘normalized’ disk Bernoulli function.By combining equations (11) − (14) we find that (cid:18) Hr (cid:19) " γ − γ − ∂ ln a ∂ ln r + ∂ ln ρ∂ ln r = −
23 ˙ q nuc ν Ω k + p ( η w − Be ′ d ) −
32 (15)Finally, nuclear reactions change the composition of the ac-creting material according to v r ∂ X A ∂ r = ˙ X A | P (16)where X A is the mass fraction of isotope with mass number A , and˙ X A | P is the nuclear reaction rate. Equation (16) shows that materialburns for approximately the local accretion timescale ∼ r / v r at anyradius. We assume that burning occurs at constant pressure becausein general the burning timescale is long compared to the dynamicaltimescale over which vertical pressure balance is established.We neglect the e ff ects of convective / turbulent mixing in thedisk, which may act to smooth radial abundance gradients; mix-ing could, in principle, be modeled by including an additional term ∝ ν mix ∇ X A to the right hand side of equation (16), where ν mix isthe di ff usion coe ffi cient. Neglecting mixing is justified as a first ap-proximation because compositional changes typically occur overa radial distance ∼ > H . Studies of the di ff usion of passive contami-nants in numerical simulations of the MRI furthermore demonstratethat ν mix < ν (e.g. Carballido et al. 2005), such that it appears un-likely that burned material will di ff use upstream.The nuclear reaction rates ˙ X A ( ρ, T ) and nuclear heating rate˙ q nuc ( ρ, T ) in equations (11) and (16) are calculated using a 19 iso-tope reaction network that includes α -capture, heavy-nuclei, and One possible source of coronal heating is the dissipation of MHD waves.Waves may be excited by turbulence due to the MRI or vertical convection,the latter of which is likely due to the strong temperature dependence of thenuclear heating rate. See c (cid:13) , 1– ?? ( α ,p)(p, α ) reactions (Timmes 1999). The temperature T ( r ) and adi-abatic index γ ( r ) are calculated using a standard equation of state,which includes ideal gas, radiation, and degeneracy pressure.We calculate solutions by integrating equations (8), (15), (16)from the outer boundary r = R d inwards, in order to obtain a ( r ), ρ ( r ) and X A ( r ). The wind mass loss rate p ( r ) (eq. [6]) is determinedby demanding that the Bernoulli function is regulated by outflowsto a fixed value Be ′ d = − . − M d ( R d )(eq. [4]) and by assuming that the inflowing material reaches a self-similar evolution with a ∝ v k ∝ r − / , such that ∂ lna ∂ lnr | R d = − / Weterminate our solutions at the radius R in ∼ < cm at which heavynuclei are photo-disintegrated into free nuclei. Interior to this point,neutrinos from the reactions e − + p → n + ν e and e + + n → p + ¯ ν e become an important source of cooling (e.g. Narayan et al. 2001;Chen & Beloborodov 2007). Depending on ˙ M and α , the disk maybecome radiatively e ffi cient and geometrically thin, such that strongoutflows may be suppressed close to the surface of the centralNS / BH. Figure 1 shows a schematic diagram of the structure ofthe disk and outflows.The composition at the outer boundary r = R d is that of thetidally disrupted WD. For C-O WDs we adopt an initial compo-sition X = . X = . X = . X = . X = . X = . X = .
4; Han et al. 2000), and pure He WDs ( X = § M ⊙ BHcompanions. Table 1 enumerates the models calculated in the nextsection.
Figures 2-5 and Tables 1 and 2 summarize the results of our calcu-lations. In this section we describe the disk structure ( § . § . § . § . ff ects of convection on our resultsand in § . Figure 2 shows our baseline model for accretion following themerger of a 0 . M ⊙ C-O WD with a 1.4 M ⊙ NS. In this exam-ple we adopt characteristic values for the initial accretion rate˙ M ( R d ) = × − M ⊙ s − , viscosity α = .
1, disk Bernoulli pa-rameter Be ′ d = − . η w =
2. The toppanel shows the radial profiles of quantities characterizing the diskthermodynamics. Note the following: (1) the disk is geometricallythick, with aspect ratio H / r ≈ . P gas exceeds radiation pressure P rad , althoughthey are comparable at small radii ∼ < cm; (3) ˙ M decreases by afactor ∼
10 between R d ≈ cm and the inner radius R in ≈ cm, such that ∼ >
80% of the accreting mass escapes in outflows. This Regardless of the precise outer boundary condition adopted, the self-similar solution rapidly obtains (Narayan & Yi 1994). Self-similarity ismaintained until nuclear burning becomes important, which introduces anadditional scale into the problem. (a) Radial Profile of Disk Properties(b) Heating / Cooling Terms in the Entropy Equation(c) Composition Profile
Figure 2.
Accretion following the merger of a 0.6 M ⊙ WD with a 1.4 M ⊙ NS. We adopt fiducial values η w = ffi ciency, Be ′ d = − . α = . M ( R d ) ≃ × − M ⊙ s − at the outer edge of the disk R d = cm (eq. [4]). Top:
Disk tem-perature T ( Blue ), surface density Σ ( Red ), aspect ratio H / r ( Turquoise ),ratio of radiation to gas pressure P rad / P gas ( Green ), negative of the electronchemical potential µ e divided by kT ( Pink ), accretion rate ˙ M normalized to˙ M ( R d ) ( Brown ), and wind mass loss index p ( Orange ; eq. [6]), as a func-tion of radius.
Middle:
Sources of heating and cooling (eq. [11]), in ratio tothe viscous heating rate ˙ q visc : nuclear reactions ˙ q nuc ( Solid Blue ); advectivecooling within the disk ˙ q adv ≡ Tv r ( ∂ s /∂ r ) ( Solid Dotted ); and wind cooling˙ q w ( Dashed Green ; eq. [14]).
Bottom:
Mass fraction X A of various isotopesin the disk midplane.c (cid:13) , 1– ?? B. D. Metzger (a) Radial Profile of Disk Properties(b) Heating / Cooling Terms in the Entropy Equation(c) Composition Profile
Figure 3.
Similar to Figure 2, but calculated for a pure He WD. Values of η w =
2, Be ′ d = − .
1, and α = . M ( R d ) ≃ − M ⊙ s − due to thelarger outer radius R d = × cm. behavior is reflected in the wind mass loss index p (eq. [6]), whichfluctuates with radius, depending on the local heating / cooling fromnuclear reactions.The middle panel in Figure 2 shows sources of heating andcooling that contribute to the entropy equation (eq. [11]), in ratio tothe viscous heating rate ˙ q visc (eq. [12]). At large radii ( r ∼ > × (a) Radial Profile of Disk Properties(b) Heating / Cooling Terms in the Entropy Equation(c) Composition Profile
Figure 4.
Same as Figure 3, but calculated for a ‘hybrid’ He-C-O WD. cm), both viscous dissipation and nuclear burning contribute to theheating, while cooling results from advection to smaller radii ˙ q adv and mass loss due to winds ˙ q w (eq. [14]). The nuclear heating rateachieves local maxima at radii where the temperature reaches thethreshold necessary to burn the next element (e.g. carbon burns at r ≈ × cm and T ≈ . × K). At these locations the energygenerated by nuclear reactions is significant compared to that re-leased by viscous dissipation; this general behavior has importantimplications for the thermal stability of the disk ( § . c (cid:13) , 1– ?? (a) Radial Profile of Disk Properties(b) Heating / Cooling Terms in the Entropy Equation(c) Composition Profile
Figure 5.
Similar as Figures 2-4, but calculated for the merger of a O-NeWD with a 3 M ⊙ BH. We adopt values of η w =
2, Be ′ d = − .
2, and α = . M ( R d ) ≃ × − M ⊙ s − , R d = × cm. radii ( r ∼ < × cm), by contrast, ˙ q nuc < cooling the disk. This occurs once temperatures are su ffi cientlyhigh ( T ∼ > × K) to photo-disintegrate nuclei endothermically .The bottom panel of Figure 2 shows the mass fractions of in-dividual elements X A as a function of radius. The composition atthe outer edge of the disk is that of the initial WD: half C and half O. Moving to smaller radii, carbon first burns to Ne and He, the latter of which rapidly capture onto O, eventually forming Mg.At smaller radii and higher temperatures, O burns to Si. Mov-ing yet further in, photo-dissociation releases α − particles, whichimmediately capture onto heavier nuclei (‘Si burning’), producing S, Ar, Ca, Ni, and Fe. Finally, at the smallest radii, Fephoto-disintegrates into He, which itself then disintegrates intofree protons and neutrons.In Figure 3 we show similar results to Figure 2, but for thecase of a 0.3 M ⊙ pure He WD. We adopt the same values for η w = ′ d = − .
1, and α = .
1, but the latter corresponds to a lower ac-cretion rate ˙ M ( R d ) ≃ − M ⊙ s − than in the C-O WD case, dueto the larger outer radius R d = × cm. The qualitative diskstructure is similar to the C-O case, but a few di ff erences shouldbe noted. First, mass loss is considerably greater from the outerportions of the disk ( p ∼ > O, Ne and Mg are largely bypassed (e.g. in comparison to the C-O case)because additional α captures are rapid following the onset of the[rate-limiting] triple- α reaction.Figure 4 shows results for a 0.4 M ⊙ ‘hybrid’ He-C-O WD,again for an accretion rate ˙ M ( R d ) = − M ⊙ s − . Although the innerstructure of the disk resembles the C-O case (Fig. 2), an importantdi ff erence is again the onset of He burning at relatively large radii(low temperature), this time via the reaction He + O → Ne + γ .As we discuss in § .
3, the sensitive temperature dependence of thisreaction may render such disks thermally unstable.Figure 5 shows results for a 1.2 M ⊙ O-Ne WD accreting ontoa 3 M ⊙ BH. The compact outer radius of the disk in this case R d ≃ × cm results in a high accretion rate ˙ M ( R d ) ≃ × − M ⊙ s − for α = .
1. Without He or C present, the first element to burn inthis case is Ne at T ≈ × K, but otherwise the compositionprofile is similar to the C-O case (Fig. 2).In addition to the calculations presented above, we have ex-plored the sensitivity of our results to variations in the binary pa-rameters, WD composition, viscosity α , and wind mass loss pa-rameter η w (see Table 2). In general, we find that increasing η w atfixed Be ′ d decreases the amount of mass loss because wind coolingis more e ff ective. By contrast, changing α (along with a compen-sating change in ˙ M ( R d ); eq. [4]) has a smaller e ff ect. Our results arealso robust to realistic variations in the composition of the disruptedWD.Notably absent from the discussion thus far is the merger of amassive C-O or O-Ne WD with a NS companion. The large massratio q ∼ R d ∼ cm) witha very high initial accretion rate ˙ M ( R d ) ≈ . M ⊙ s − for α = . ffi ciently high that nuclearburning begins already at r ≈ R d during the circularization pro-cess itself. Since virialization has not yet occurred, the WD mate-rial forming the disk may still be degenerate, potentially leading toan explosive situation. Furthermore, as we discuss in § .
3, accre-tion in these systems may be thermally unstable, even under non-degenerate conditions. Clearly, many of the simplifications adoptedin our model break down in the case of massive WD-NS mergers.Obtaining even a qualitative understanding of such events may re-quire multi-dimensional simulations that include the e ff ects of nu-clear burning, even during the merger process itself. c (cid:13) , 1– ?? B. D. Metzger
A schematic diagram of the outflows from WD-NS / BH mergerdisks is shown in Figure 1. The total mass loss rate from thedisk is approximately given by ˙ M w ≃ ˙ M ( R d ) − ˙ M ( R in ), where R in ∼ . − . cm is the radius interior to which the disk isprimarily free nucleons. We adopt this inner boundary because ifneutrino cooling is important at radii r < R in , then additional massloss will be suppressed (although in § . M ( R d ) (eq. [4]),then the total ejecta mass is M ej ≈ f ej M WD , where f ej = h ˙ M w ˙ M d ( R d ) i isthe fraction of the accreted mass lost in winds. Depending on η w ,we find that f w ∼ . − . . − .
99] for C-O[He] WDs (Table2), corresponding to M ej ∼ . − . . M ⊙ .The radial structure of the disk is imprinted on the composi-tion of the outflows (Fig. 1). Table 2 provides the mass fractions ineach element of the wind ejecta, which are calculated as X w , A = M w Z R d R in X A ∂ ˙ M w ∂ r dr (17)Here we have assumed that the local surface composition of thedisk is similar to that in the midplane due e.g. to turbulent mix-ing. To compare with our C-O disk models, we also list theabundances predicted by the well-studied ‘W7’ model for SN Iafrom the delayed-detonation of a Chandrasekhar-mass C-O WD(Nomoto et al. 1984). To compare with our He disk models, welist the range in ejecta abundances predicted from models for un-stable He shell burning on the surface of a C-O WD, as calculatedby Shen et al. (2010) for di ff erent values of the mass of the WD andHe layer.In our fiducial C / O model (Fig. 2) the majority of the ejectais unburnt C (26%) and O (39%), with the remainder in Si(10%), Mg (7%), He (6%), Ne (5%), Fe (3%), S(2%), Ni(1 . Ar (0 . Ca (0 . / O, Ne, and Mgare significantly higher. Despite these di ff erences, the abundancesof the elements Si, S, and Ca, generally responsible for producingthe most prominent spectral features in normal Ia, are similar to theW7 model within a factor ∼ < −
3. Importantly, the yield of radioac-tive Ni is substantially lower than in normal Ia SN ejecta; this hasimportant implications for the luminosities of optical transients as-sociated with WD-NS / BH mergers ( § . He (63%), with the remainder in C (15%), Si (10%), S(8%), Ar (2%), Fe (0 . Mg (0 . Ca(0 . Ne (0 . O (0 . Ni (0 . He fraction is similar to the He shell burning models ofShen et al. (2010), the predicted abundances of the radioactive iso-topes Ti, Cr, and Ni are all significantly lower, while thoseof intermediate mass elements are significantly higher. In additionto the above elements, disk outflows may in all cases also containa tiny fraction of H from the very inner disk, the implications ofwhich are briefly discussed in § . v w ∝ v k ∝ r − / than those from further out. Nevertheless, mostof the ejecta will ultimately reside in a single ‘shell’ of materialwith a much lower velocity dispersion. This is because, initiallyafter the disruption (on timescales t ≪ t visc ; eq. [3]), the disk isconcentrated at large radii ≈ R d . Slow outflows from this early stagewill thus have time to encase the system before outflows have evenbegun in earnest from the inner disk on a timescale t ∼ > t visc . Fast ejecta from the inner disk will thus collide with the slower materialejected prior. Because this interaction conserves energy (radiativelosses are negligible at early times), the ejecta will achieve a meanvelocity ¯ v ej , which may be estimated by averaging over the totalkinetic energy of winds from each disk annulus:12 M ej ¯ v = Z R d R in ∂ ˙ M w ∂ r v w dr (18)Table 2 shows that for C-O WD / NS mergers ¯ v w is typically ∼ − × km s − , while for He[O-Ne] mergers v w is somewhatlower[higher].The reason that ¯ v w can exceed the typical velocity of SNIa ejecta is the deeper gravitational potential well in the case ofNS / BH accretion. As a toy example, if mass loss were to occurwith a constant power-law index p = const < v ej ≃ h p η w − p GMR in (cid:16) R in R d (cid:17) p i / , p < h η w GMR d ln (cid:16) R d R in (cid:17)i / , p = , (19)where in the first expression we have assumed that R d ≫ R in . Alarger wind e ffi ciency factor η w thus increases ¯ v ej , both because v w ∝ η / w and because larger η w decreases the mass loss rate p ( r )required to cool the disk. Equation (19) shows that (for η w ≃
1) theminimum value of v ej ( p ≈
1) is of the order of the orbital veloc-ity at r ≈ R d . This varies from ≈ × km s − for He WD-NSmergers to ≈ km s − for O-Ne WD-NS / BH mergers.
The calculations presented in the previous sections assumed steady-state accretion. This is not valid, however, if the disk is thermallyunstable, which might be expected due to the sensitive temperaturedependencies of nuclear reactions. The criterion for thermal stabil-ity may be written (Piran 1978) ∂ ln q + ∂ ln H (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Σ < ∂ ln q − ∂ ln H (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Σ → ∂ ln q + ∂ ln P (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Σ < ∂ ln q − ∂ ln P (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Σ (20)where q + = ˙ q visc + ˙ q nuc and ˙ q − = ˙ q adv + ˙ q w are the total heatingand cooling rates, respectively (see eq. [11]), and the second con-dition follows because P = ρ a ∝ Ω Σ H . Viscous heating obeys˙ q visc ∝ P , while advective cooling obeys ˙ q adv ∝ ˙ q visc ( H / r ) ∝ P .In our model, we have assumed that the terminal speed of diskwinds v w is proportional to the local escape speed ∝ v k (eq. [14]),such that ˙ q w ∝ ˙ q visc ∝ P . However, the precise mechanism respon-sible for driving the wind is uncertain; we could equally well haveassumed that v w is proportional to the sound speed a , in which case˙ q w ∝ ( H / r ) v k ∝ P . Considering both cases, the stability criterionin equation (20) can equivalently be written as an upper limit onthe fraction f nuc ≡ ˙ q nuc / ˙ q + of the disk heating supplied by nuclearreactions, viz. f nuc < ( − f w ) n − ≡ f nuc , th1 , v w ∝ v esc2 n − ≡ f nuc , th2 , v w ∝ a , (21)where n ≡ (cid:16) ∂ ln˙ q nuc ∂ ln P (cid:17) | Σ and f w ≡ ˙ q w / ˙ q − is the fraction of the coolingresulting from winds. Equation (21) shows that accretion is stableat a given radius if either (1) nuclear burning is absent entirely; or(2) the burning rate has a weak pressure dependence n < − f w or n <
4, in the cases v w ∝ v k and v w ∝ a , respectively. c (cid:13) , 1– ?? Table 1.
Properties of Disk Models model M ( a ) M WD R ( b )d ˙ M ( R d ) α Be ′ ( c )d η ( d )w X A ( R d ) ( e ) ˙ M wind ˙ M ( R d ) ( f ) ¯ v ( g )ej unstable? ( h ) ( M ⊙ ) ( M ⊙ ) (cm) ( M ⊙ s − ) (10 km s − )NS C-O 1 1.4 0.6 10 × − X = . , X = . × − × − × − × − X = . , X = . , X = . × − ( i ) × − X = . , X = . × × − X = . , X = . × × − × × − × × − X = . , X = . × × − × − X = × − × − ( a ) Mass of the central NS or BH. ( b ) Outer edge of the disk R d = R RLOF (1 + q ) − , where q ≡ M WD / M and R RLOF is the binary separation at RLOF (eq. [2]). ( c ) Normalized Bernoulli parameter in the disk (eq .[13]). ( d ) Ratio of the specific kinetic energy of the wind from any radius to the local gravitational bindingenergy. ( e ) Initial composition of the disk from the disrupted WD. ( f ) Ratio of the total mass loss rate in the wind ˙ M w ≡ ˙ M ( R d ) − ˙ M ( R in ) to the initial accretionrate ˙ M ( R d ) at the outer edge of the disk. ( g ) Mean velocity of the ejecta from disk winds (eq. [18]). ( h ) Whether the disk is thermally unstable due to nuclearburning at any radius, according to the most stringent criterion f nuc < f nuc , th2 given in equation (20). ( i ) Characteristic values shown for illustration only. Thedisk model presented in this paper cannot be applied to the merger of massive WD-NS binaries because nuclear burning begins already at r ≈ R d (see § . Table 2.
Elemental Mass Fractions in the Wind Ejecta X w , A (eq. [17]) model He C O Ne Mg Si S Ar Ca Ti Cr Fe Fe NiW7 Ia ( a ) . × − . × − - 0.054 0.42NS C-O 1 0.063 0.26 0.39 0.049 0.065 0.10 0.023 0.0043 0.0031 2 . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − NS C-O 4 0.070 0.21 0.36 0.057 0.076 0.13 0.029 0.0055 0.0040 2 . × − . × − . × − . × − . × − . × − × − . × − . × − . × − . × − . × − . × − × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − ( b ) . − .
45 - - - - - - 6 − × − . − .
04 0 . − . . − . . − . . × − . × − . × − . × − NS He 2 0.60 0.23 0.0031 0.0036 0.016 0.13 0.013 7 . × − × − . × − . × − . × − . × − NS He 3 0.87 0.077 4 . × − . × − × − . × − . × − . × − . × − . × − ( a ) W7 Type Ia SN model (Nomoto et al. 1984), ( b ) Range of models for unstable He shell burning from Shen et al. (2010).
Figure 6 shows f nuc ( solid line ), f nuc , th1 ( solid lines ), and f nuc , th2 as a function of radius, calculated for our solution from § M ⊙ C-O WD (Fig. 2). Fig-ure 6 shows that in the case v w ∝ a , the disk is stable at all radii,i.e. f nuc < f nuc , th2 . If, however, we instead assume that v w ∝ v k , thenwe find that the disk is unstable ( f nuc > f th1 ) at the radii ∼ few × cm where carbon burning peaks. Although we do not show this caseexplicitly, we find a simlilar result for He WDs: the triple- α reac-tion is not su ffi ciently pressure-sensitive for instability if v w ∝ a (i.e. n < v w ∝ v k the disk is unstable at radii ∼ − × cm where He burning peaks. We conclude that whether C-O or HeWD disks are thermally stable depends on the pressure dependenceof wind cooling, which in turn depends on the [uncertain] mecha-nism responsible for driving the wind.In the case of O-Ne and hybrid WDs, thermal instability seemsmore assured. In the bottom panel of Figure 6 we show f nuc and f nuc , th2 (eq. [21]) for cases corresponding to the merger of a 0.4 M ⊙ He-C-O WD with a NS (Fig. 4); a 1 . M ⊙ O-Ne WD with a 3 M ⊙ BH (Fig. 5); and a 1 . M ⊙ C-O WD with a 3 M ⊙ BH. In all threecases we find that over some range of radii, f nuc exceeds eventhe more conservative threshold for instability f nuc , th2 . This impliesthat our calculated solutions are thermally unstable, indpendent ofwhether v w ∝ v k or v w ∝ a . Although addressing the implicationsof unstable burning is beyond the scope of this paper, we speculatethat the end result may be a complicated, time-dependent evolu-tion. One possibility is ‘limit-cycle’ behavior, in which the initialinflow is halted by runaway burning, before resuming again laterwith fresh fuel (a possible analog are dwarf nova outbursts of cat-aclysmic variables; e.g. Cannizzo 1993). Though it is not obvious,the steady-state solutions constructed here may remain a reasonabledescription of the average flow over many cycles. c (cid:13) , 1– ?? B. D. Metzger (a) 0.6 M ⊙ WD / NS(b) 0.4 M ⊙ He-C-O / NS, 1.0 M ⊙ WD / BH, and 1.2 M ⊙ WD / BH Figure 6.
Analysis of the thermal stability of the accretion solutions pre-sented in § .
1. Quantities shown are the fraction of the total heating due tonuclear burning f nuc ≡ ˙ q nuc / ˙ q + ( dotted line ) and the threshold fractions forthermal instability f nuc , th1 ( dashed line ) and f nuc , th2 ( solid line ) under theassumption that v w ∝ v k and v w ∝ a , respectively (eq. [21]). Top : Solu-tions corresponding to accretion following the merger of a 0.6 M ⊙ M ⊙ WD with a NS at the rate ˙ M ( R d ) = × − M ⊙ s − (Fig. 2). Note that al-though f nuc < f nuc , th2 at all radii, f nuc exceeds f nuc , th1 across some range inradius. This shows that whether the disk is in fact stable depends on the de-tailed pressure-dependence of wind cooling. Bottom : Accretion followingthe merger of a 1 M ⊙ C-O WD with a 3 M ⊙ BH (
Black ); 1 . M ⊙ O-Ne WDwith a 3 M ⊙ BH (
Blue ); ‘hybrid’ He-C-O WD with a 1.4 M ⊙ NS (
Red ). In allthree cases f nuc > f nuc , th2 over some range in radii. The solutions are thusunstable to runaway heating from nuclear burning, regardless of whether v w ∝ v k or v w ∝ a . ff ects of Convection It is well known that ADAFs are unstable to radial con-vection (Narayan & Yi 1994; Igumenshchev et al. 2000;Quataert & Gruzinov 2000). In the case of NuDAFs, the diskmay also be unstable to vertical convection because the nuclearheating peaks sharply near the midplane due to its sensitivetemperature dependence. Although our calculations have thusfar neglected its e ff ects on the transport of angular momentum,energy, and composition, in this section we describe how strongconvection could alter our conclusions.If convection transports angular momentum outwards − similar to MHD turbulence produced by the MRI − then it would act simplyto enhance the e ff ective value of “ α ”, which we have shown doesnot qualitatively a ff ect our results. If, on the other hand, convectiontransports angular momentum inwards (Ryu & Goodman 1992),then an qualitatively di ff erent type of solution could obtain: a “con-vection dominated accretion flow” (CDAF; Quataert & Gruzinov2000; Narayan et al. 2000). In CDAFs, the angular momentumtransported outwards by viscosity is balanced by the inward trans-port by convection, such that the net mass accretion rate is verysmall. Because in steady state the convective energy flux F c ∝ ρ c s r is constant with radius, CDAFs are characterized by a moreshallow density profile ρ ∝ r − / than for normal ADAF solutions( ρ ∝ r − / ). If present, the CDAF would extend from the inner edgeof the RIAF at R in ∼ − cm to much larger radii, where theoutflowing energy is released through radiation or outflows.Coincidentally, the density profile of a CDAF scales the sameway with radius as in ADIOS (wind) models with “maximal” massloss ( p =
1; ˙ M ∝ r ). Our steady-state calculations with high massloss rates (low η w ) thus may also be applied to the CDAF case.Note, however, that the physical interpretation is completely dif-ferent: in wind models, the outflowing mass escapes the systementirely, while in CDAFs it remains bound and is simply ‘recycled’by mixing to large radii, from which it accretes again later.If the outer edge of the disk remains near the initial radiusof the disrupted WD R d , then convection simply increases thetimescale for matter to accrete from that given in equation (3) bya factor ∼ ˙ M ( R d )˙ M ( R in ) ∼ R d R in ∼ − , or from minutes to hours ordays. The radial structure of the disk is similar to the maximal windcase, but the observational signature of the event as described in § § . § . Ni fraction would be smaller.A more radical possibility is that the outward convective en-ergy flux will ‘feed back’ on the outer disk, alterating its dynam-ics entirely. If pressure forces cause the outer reservoir of mass toexpand significantly from its initial radius ∼ R d , then rotationalsupport will no longer be important. As a much larger, quasi-spherical accreting envelope, the structure may thus come to re-semble a similar ‘Thorne-Zytkow object’ (Thorne & Zytkow 1975)or ‘quasi-star’ (e.g. Begelman et al. 2008). In this case, the accre-tion timescale will be even long than estimated above, because itinstead depends on the rate that the convective energy escapes theouter boundary in radiation or winds. A quantitative exploration ofsuch a model is an important and interesting excercise for futurework, but is beyond the scope of this paper.Despite the important possible e ff ects of convection discussedabove, it is not clear that CDAFs are in fact relevant to WD-NS / BHmergers. As already noted, whether the flow is best described as aCDAF or ADAF (with winds) depends on the strength (and direc-tion) of convective angular momentum transport relative to othertorques. Although radial convection in hydrodynamic simulationsof RIAFs indeed appears to balance outwards viscous transport for α ∼ < .
05 (Naryan et al. 2000; Narayan et al. 2004), higher val-ues of α ∼ . / BH mergers, angular momentum transport due to gravita-tional instabilities may also compete with convection. Finally, it hasbeen argued that global MHD simulations of RIAFs more closely c (cid:13) , 1– ?? resemble the ADIOS picture of outflows employed in this paper(Hawley et al. 2001; Hawley & Balbus 2002), although this con-clusion may depend on the [uncertain] saturation strength of theMRI and the ultimate fate of the mass leaving the grid. We conclude this section with an analytic estimate of the impor-tance of nuclear burning in hyper-accreting disks, i.e. we addresswhen a disk is in the NuDAF regime. Focusing on the burningof one isotope A , the nuclear heating rate may be written ˙ q nuc =∆ ǫ nuc / m n t burn , where ∆ ǫ nuc = X A Q / A is the nuclear energy releasedper nucleon, Q is the Q-value of the reaction, m n is the nucleonmass, t burn = ∆ r burn / v r ( r burn ) is the time spent burning at radius r = r burn , v r = ν/ r is the accretion velocity, and ∆ r burn is theradial distance over which most of the energy is released. The ratioof nuclear to viscous heating can thus be written˙ q nuc ˙ q visc (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r = r burn = ∆ ǫ nuc / m n t burn (9 / ν Ω k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r = r burn = ∆ ǫ nuc r burn GMm n r burn ∆ r burn (22)where we have used equation (12), neglecting terms ∝ ( H / r ) .Due to the temperature sensitivity of nuclear burning rates, r = r burn generally occurs within a factor ∼ < T = T burn , which depends primarily just on the element underconsideration. Because the radius and temperature of burning arerelated by r burn ≃ µ GMm n kT burn (cid:16) Hr (cid:17) , P gas ≫ P rad (cid:18) ˙ MG / M / πα aT rH (cid:19) / , P rad ≫ P gas , (23)in gas and radiation-dominated regimes, respectively, we can write˙ q nuc ˙ q visc (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r = r burn = H ∆ r burn × . (cid:16) ∆ ǫ nuc MeV (cid:17) (cid:16) T burn K (cid:17) − , P gas P rad ≫ (cid:16) ∆ ǫ nuc MeV (cid:17) (cid:16) M . M ⊙ (cid:17) − . (cid:16) α . (cid:17) − . (cid:16) ˙ M − M ⊙ s − (cid:17) . (cid:16) T burn K (cid:17) − . , P rad P gas ≫ , . (24)where we have assumed that H / r = . µ =
2. Assuming nominal gas pressure dominance, the ratioof radiation to gas pressure is given by P rad P gas ≃ παµ / m / p k − / b a r ( GM ) ˙ M − (cid:18) Hr (cid:19) ≈ (cid:18) α . (cid:19) H / r . ! M . M ⊙ ! (cid:18) T burn K (cid:19) / ˙ M − M ⊙ s − ! − . (25)Turbulent mixing likely sets a lower limit of ∆ r burn ∼ > H onthe radial extent of burning. On the other hand, burning is unlikelyto occur over a much larger radial region ∆ r burn ≫ r burn ∼ H due to the sensitive temperature dependence of the nuclear reactionrates and the power-law behavior of T ( r ). The bracketed quantityin equation (24) is thus a reasonable estimate of ˙ q nuc / ˙ q visc . The ap-proximate equality ˙ q nuc ∼ > ˙ q visc may be considered a loose defini-tion for whether a disk is in the NuDAF regime on a radial scale r ≈ r burn .As an example, consider carbon burning ( Q / A ≃ .
38 MeVnucleon − ; T burn ≈ . × K) in a NS disk with a mass fraction X = .
5, for which ∆ ǫ nuc ≈ . q nuc ∼ < ˙ q visc for ˙ M = − M ⊙ s − and α = .
1, a result seemingly
Figure 7.
Bolometric light curve of supernova-like emission from WD-NS / BH mergers, powered by the decay of Ni in the wind-driven ejecta.Three models shown correspond to the merger of [1] a 0.6 M ⊙ C-O WDwith a 1.4 M ⊙ NS ( solid ; ¯ v w ≃ . × km s − ; M ej ≃ . M ⊙ ; M Ni = × − M ⊙ ; Fig. 2); [2] a 0.3 M ⊙ He WD with a 1.2 M ⊙ NS ( dotted ;¯ v w ≃ . × km s − ; M ej ≃ . M ⊙ ; M Ni = × − M ⊙ ; Fig. 3); and[3] a 1.2 M ⊙ O-Ne WD with a 3 M ⊙ BH ( dashed ; ¯ v w ≃ . × km s − ; M ej ≃ M ⊙ ; M Ni = × − M ⊙ ; Fig. 5) independent of ˙ M or the mass of the central object. However, thisresult holds only if gas pressure dominates ( ˙ M ∼ > − ( α/ . M ⊙ s − ; eq. [24]). At lower accretion rates, radiation pressure domi-nates, in which case ˙ q nuc / ˙ q visc decreases ∝ ˙ M . M − . . In this section we use our results from § / BH mergers.
A previously discussed EM counterpart of WD-NS / BH mergers isa long duration Gamma-Ray Burst, powered by accretion onto thecentral BH (Fryer et al. 1999) or NS (King et al. 2007). Althoughthe formation of a relativistic jet from the inner disk is certainlypossible, our calculations show that the accreted mass reachingthe NS surface or BH event horizon may be a factor ∼ − § . ffi cient mass accretes to produce a powerful jet, onlya small fraction of GRB jets are pointed towards the Earth. For o ff -axis events, the prompt and afterglow emission are much dimmerdue to relativistic debeaming. Thus, although it would be unsur-prising if WD-NS / BH mergers were accompanied by non-thermaljetted emission at some level, bright high energy emission may notbe a ubiquitous feature.
A promising source of isotropic EM emission from WD-NS / BHmergers is a supernova(SN)-like transient, powered by the radioac-tive decay of Ni from the wind-driven ejecta. We argued in § . c (cid:13) , 1– ?? B. D. Metzger large scales. Depending on the binary parameters and the wind ef-ficiency η w , the ejecta has a characteristic mass M ej ∼ . − M ⊙ ,mean velocity ¯ v ej ≈ − × km s − , and Ni mass M Ni ≈ X M ej ∼ − . − − . M ⊙ (Table 2). More massive WDs generally producemore massive ejecta, with a higher speed and larger Ni yield. Thisconclusion should be taken with caution, however; thermal insta-bility may alter the disk evolution, especially for high mass (andpossibly hybrid) WDs ( § . / BH mergers, calculated using the modelof Kulkarni (2005) and Metzger et al. (2008a) for ejecta propertiescorresponding to di ff erent examples of WD-NS / BH systems from §
4. In all cases the light curve peaks on a timescale ∼ week. Bycontrast, the peak luminosity L bol varies from ∼ ergs s − to ∼ . ergs s − (absolute magnitude M B ≃ − ≃ − Ni mass, which dependsprimarily on the mass of the disrupted WD.
At least in the case of C-O and O-Ne mergers, both the brightnessand duration of the predicted transient are broadly consistent withthe properties of recently discovered subluminous Type I SNe suchas 2005E, 2008ha, and 2010X ( § / BHmergers results in several potential di ffi culties.Although the ejecta mass, Ni mass, and presence of C, O,Si, S, Ca, Fe from C-O / O-Ne WD mergers are all broadly con-sistent with the observed properties of SN 2008ha (Valenti et al.2009; Foley et al. 2009, 2010), the velocity of the ejecta ∼ < kms − inferred for this and other ‘2002cx-like’ events are much lowerthan the predictions of our baseline models. One caveat is that ourcalculations of ¯ v ej assume that winds from all radii in the disk con-tribute to a single homologous body of ejecta, whereas in reality ¯ v ej could be smaller if the fast wind from the inner disk escapes alongthe pole without e ffi ciently coupling its kinetic energy.The ejecta mass, Ni mass, velocity, and presence of He, Ca,and O of our C-O and He WD models are similarly consistent withthe properties of SN 2005E. However, the large Ca mass ∼ > . M ⊙ inferred in the ejecta of this and related events (Perets et al. 2010)is much higher than we predict. If this disagreement can be rec-onciled, we speculate that the mysterious location of 05E-like ob-jects in the outskirts of their host galaxies could be explained byour model if WD-NS binaries are given a ‘kick’ during their super-nova, which removes them from the disk of the galaxy by the timeof merger.For SN 2010X, again many of the elements produced byWD-NS / BH mergers are seen in the spectra (and some, like Cand other intermediate mass elements, are not expected in alter-native .Ia models; Kasliwal et al. 2010). However, Ti is also ob-served, despite its low predicted quantity in our models. Further-more, none of our models produce enough Ni to explain SN2002bj (Poznanski et al. 2010), an event which otherwise sharesseveral properties with 2010X. Again, these conclusions must bemoderated due to our ignorance of the outcome of high mass ratiomergers and the uncertain e ff ects of thermal instabilities.As a final note, although this possibility has been neglectedthus far, free nucleons from the very inner disk may also con-tribute a small fraction of the ejected mass (Fig. 1). These may belost to winds heated by viscous dissipation (as discussed here) orby neutrino irradiation from the very inner disk or boundary layer (e.g. Metzger et al. 2008, Metzger et al. 2008b). Once neutrons de-cay, the net e ff ect is contamination of the ejecta with hydrogen.Since even a small quantity of H may be detectable due to its stronglines, its presence in an event otherwise classified ‘Type I’ wouldsupport the WD-NS / BH merger model, due to its unlikely presencein other WD models (its absence would, however, not rule out themodel).
A final source of transient emission from WD-NS / BH mergers isnon-thermal synchrotron emission powered by the deceleration ofthe ejecta with the surrounding interstellar medium (ISM). Shockemission peaks once the ejecta sweeps up its own mass in the ISM;this occurs on a timescale t dec ≈ M ej . M ⊙ ! / ¯ v ej km s − ! − (cid:18) n cm − (cid:19) − / yrs , (26)where n is the ISM density. Following Nakar & Piran (2011) (theirequation [14]), we estimate that the number of radio transients de-tectable with a single 1.4 GHz snapshot of the whole sky down toa limiting flux F lim is given by N . ≈ M ej . M ⊙ ! . ¯ v ej km s − ! . × (cid:18) n cm − (cid:19) (cid:18) ǫ B . (cid:19) . (cid:18) ǫ e . (cid:19) . F lim . ! − . R − yr − ! (27)where R ∼ − − − yr − is the merger rate per galaxy (see § p = .
5. Here ǫ e ( ǫ B ) arethe fraction of the energy density behind the shock imparted to rel-ativistic electrons and magnetic fields, respectively, normalized tovalues characteristic of radio supernovae (Chevalier 1998). Equa-tions (26) and (27) show that future wide-field radio surveys at ∼ GHz frequencies (e.g. Bower et al. 2010) will preferentially detectthose events with the highest ejecta mass and velocity, as charac-terize higher mass WD mergers in our model. This is due both tothe sensitive dependence of N . on M ej and ¯ v ej and the requirementthat t dec be su ffi ciently short to produce an appreciable change inthe radio brightness over the characteristic lifetime of the survey. Although the e ff ects of nuclear burning on accretion havebeen explored in previous work (e.g. Taam & Fryxell 1985;Chakrabarti et al. 1987), these e ff orts focused on systems withmuch lower accretion rates, such as X-ray binaries. Unphysicallylow values of the disk viscosity α ∼ < − are necessary in thesecases to achieve su ffi ciently high densities for appreciable nucle-osynthesis. Here we have shown that nuclear burning has an im-portant e ff ect on the dynamics of accretion following the tidal dis-ruption of a WD by a NS or stellar mass BH, even for more re-alistic values of α ∼ > . − .
1. Under conditions when the heat-ing from nuclear burning is comparable to, or greater than that, re-leased by viscous dissipation, we have introduced the concept of a‘Nuclear-Dominated Accretion Flow’ (NuDAF; see the discussionsurrounding eq. [24] for a more concrete definition of this regime).The dependence of the disk thermodynamics on nuclear burning isparticularly acute because inflow is already radiatively-ine ffi cient c (cid:13) , 1– ?? and hence marginally bound (Narayan & Yi 1994), even withoutan additional heat source.In § / BH mergers. In addition to their EM signa-tures, WD-NS / BH binaries are considered a promising source ofgravitational wave emission at frequencies ∼ < mHz. Estimates sug-gest that a spaced-based interferometer such as LISA could resolve ∼ −
100 WD-NS / BH systems within our galaxy (Nelemans et al.2001; Cooray 2004; Paschalidis et al. 2009). Unfortunately, mostof the discovered systems will require ∼ > years to merge. Al-though the odds are not favorable, if a WD-NS / BH binary in theMilky Way or a nearby galaxy were su ffi ciently compact to mergeon a shorter timescale, it should be relatively easy to detect andlocalize. EM counterpart searches could then be triggered at theexpected merger time.The results presented here may be applied to other con-texts. One immediate possibility is the tidal disruption of aWD by an intermediate mass BH (e.g. Rosswog et al. 2009;Clausen & Eracleous 2011). For nearly circular orbits prior tomerger (as we have discussed), burning is unlikely to be impor-tant if the primary mass M is too large. This is because whenradiation-pressure dominates (as is likely in this case; eq. [25]),the ratio of nuclear to viscous heating ˙ q nuc / ˙ q visc ∝ M − . (eq. [24])is small due to the deep potential well of the BH. On the otherhand, for highly elliptical orbits with small pericenter radii, tidalforces during the disruption can themselves trigger a nuclear run-away (Rosswog et al. 2009); because the initial orbit has a very lowbinding energy, the nuclear energy released will have a more sig-nificant e ff ect on what matter is unbound.A perhaps more promising application of our results is to ac-cretion following the core collapse of a massive star, as in the‘collapsar’ model for long duration GRBs (MacFadyen & Woosley1999). Notably, the outer layers of the Wolf-Rayet progeni-tors of long GRBs are predicted to have a He-C-O composi-tion (Woosley & Heger 2006), similar to that of a ‘hybrid’ WD(Fig. [4]). Due to its low temperature threshold, the reaction He + O → Ne + γ may have a particularly important influenceon the disk, if material circularizes at a su ffi ciently large radius.Note that current numerical simulations of collapsar disks gen-erally neglect the e ff ects of nuclear reactions, except perhaps atvery high temperature ∼ > × K (MacFadyen & Woosley 1999;Lindner et al. 2010; Milosavljevic et al. 2010). Our results showthat may not be valid, as nuclear energy generation can be im-portant, even for temperatures as low as ∼ K. The e ff ects ofnuclear burning may be similarly important in accretion followingthe merger of a helium star with a NS or BH (Fryer & Woosley1998). The recent GRB 101225A was interpreted as resulting fromaccretion following following NS-helium star merger (Th¨one et al.2011); we note that the small Ni mass inferred from the weak SNassociated with this event is consistent with our estimated yieldfrom He WD-NS mergers.There is much room for improvement on the model pre-sented here in future work. For one, we have assumed a time-steady, height-integrated disk model. In reality, the disk may havea complex vertical structure, due in part to the interplay betweenMRI-driven turbulence and convection, the latter driven by thestrong temperature dependence of nuclear heating. Complex time-dependent behavior is also expected, associated with both the secu-lar viscous evolution of the disk and, potentially, shorter timescalevariability associated with thermal instability (possibly resultingin limit cycle behavior; § . η w and Be ′ d , whose val- ues depend on the mechanism and heating source responsible fordriving outflows from the disk corona. The pressure dependenceof wind cooling has important implications for the thermal sta-bility of the disk ( § . ff ectsof convection on energy and angular momentum transport in thedisk ( § . ff ects of nu-clear reactions will be required to fully characterize the evolutionand fate of these systems.Finally, the discussion of optical and radio EM counterparts in § . § . § . . ff erent WD-NS / BH systems.
ACKNOWLEDGMENTS
I thank J. Goodman for many helpful conservations and for en-couraging my work on this topic, as well as for a thorough readingof the text. I thank C. Kim and D. Lorimer for helpful informa-tion on WD-NS binaries. I also thank D. Giannios, E. Quataert,R. Narayan, A. Piro, R. Foley, M. Kasliwal, and L. Bildsten forhelpful conversations and information. I thank F. Timmes for mak-ing his nuclear reaction codes available to the public. BDM is sup-ported by NASA through Einstein Postdoctoral Fellowship grantnumber PF9-00065 awarded by the Chandra X-ray Center, which isoperated by the Smithsonian Astrophysical Observatory for NASAunder contract NAS8-03060.
REFERENCES
Bailes M., Ord S. M., Knight H. S., Hotan A. W., 2003, ApJL,595, L49Begelman M. C., Rossi E. M., Armitage P. J., 2008, MNRAS, 387,1649Bildsten L., Shen K. J., Weinberg N. N., Nelemans G., 2007,ApJL, 662, L95Blandford R. D., Begelman M. C., 1999, MNRAS, 303, L1Bower G. C., et al., 2010, ApJ, 725, 1792Branch D., Baron E., Thomas R. C., Kasen D., Li W., FilippenkoA. V., 2004, PASP, 116, 903Cannizzo J. K., 1993, ApJ, 419, 318Carballido A., Stone J. M., Pringle J. E., 2005, MNRAS, 358,1055Chakrabarti S. K., Jin L., Arnett W. D., 1987, ApJ, 313, 674Chen W.-X., Beloborodov A. M., 2007, ApJ, 657, 383Chevalier R. A., 1998, ApJ, 499, 810Clausen D., Eracleous M., 2011, ApJ, 726, 34Cooray A., 2004, MNRAS, 354, 25Davies M. B., Ritter H., King A., 2002, MNRAS, 335, 369Davis S. W., Stone J. M., Pessah M. E., 2010, ApJ, 713, 52 c (cid:13) , 1– ?? B. D. Metzger
D’Souza M. C. R., Motl P. M., Tohline J. E., Frank J., 2006, ApJ,643, 381Edwards R. T., Bailes M., 2001, ApJL, 547, L37Eggleton P. P., 1983, ApJ, 268, 368Foley R. J., Brown P. J., Rest A., Challis P. J., Kirshner R. P.,Wood-Vasey W. M., 2010, ApJL, 708, L61Foley R. J., Chornock R., Filippenko A. V., Ganeshalingam M.,Kirshner R. P., Li W., Cenko S. B., Challis P. J., Friedman A. S.,Modjaz M., Silverman J. M., Wood-Vasey W. M., 2009, AJ, 138,376Freire P., Wex N., 2010, ArXiv e-printsFryer C. L., Woosley S. E., 1998, ApJL, 502, L9 + Fryer C. L., Woosley S. E., Herant M., Davies M. B., 1999, ApJ,520, 650Guerrero J., Garc´ıa-Berro E., Isern J., 2004, A&A, 413, 257Guti´errez J., Canal R., Garc´ıa-Berro E., 2005, A&A, 435, 231Han Z., Tout C. A., Eggleton P. P., 2000, MNRAS, 319, 215Hawley J. F., Balbus S. A., 2002, ApJ, 573, 738Hawley J. F., Balbus S. A., Stone J. M., 2001, ApJL, 554, L49Iben Jr. I., Tutukov A. V., 1991, ApJ, 370, 615Igumenshchev I. V., Abramowicz M. A., Narayan R., 2000, ApJL,537, L27Jha S., Branch D., Chornock R., Foley R. J., Li W., Swift B. J.,Casebeer D., Filippenko A. V., 2006, AJ, 132, 189Kaiser N., et al., 2002, in J. A. Tyson & S. Wol ff ed., Society ofPhoto-Optical Instrumentation Engineers (SPIE) Conference Se-ries Vol. 4836 of Society of Photo-Optical Instrumentation En-gineers (SPIE) Conference Series, Pan-STARRS: A Large Syn-optic Survey Telescope Array. pp 154–164Kasliwal M. M., et al., 2010, ApJL, 723, L98Kaspi V. M., Lyne A. G., Manchester R. N., Crawford F., CamiloF., Bell J. F., D’Amico N., Stairs I. H., McKay N. P. F., MorrisD. J., Possenti A., 2000, ApJ, 543, 321Kim C., Kalogera V., Lorimer D. R., White T., 2004, ApJ, 616,1109King A., Olsson E., Davies M. B., 2007, MNRAS, 374, L34King A. R., Pringle J. E., Livio M., 2007, MNRAS, 376, 1740Kohri K., Narayan R., Piran T., 2005, ApJ, 629, 341Kulkarni S. R., 2005, ArXiv Astrophysics e-printsLamers H. J. G. L. M., Cassinelli J. P., 1999, Introduction to Stel-lar WindsLaughlin G., Bodenheimer P., 1994, ApJ, 436, 335Law N. M., et al., 2009, PASP, 121, 1395Li W., Filippenko A. V., Chornock R., Berger E., Berlind P.,Calkins M. L., Challis P., Fassnacht C., Jha S., Kirshner R. P.,Matheson T., Sargent W. L. W., Simcoe R. A., Smith G. H.,Squires G., 2003, PASP, 115, 453Lindner C. C., Milosavljevi´c M., Couch S. M., Kumar P., 2010,ApJ, 713, 800Livne E., Arnett D., 1995, ApJ, 452, 62Lorimer D. R., 2005, Living Reviews in Relativity, 8, 7Lundgren S. C., Zepka A. F., Cordes J. M., 1995, ApJ, 453, 419MacFadyen A. I., Woosley S. E., 1999, ApJ, 524, 262Metzger B. D., Piro A. L., Quataert E., 2008a, MNRAS, 390, 781Metzger B. D., Piro A. L., Quataert E., 2008b, MNRAS, 390, 781Metzger B. D., Thompson T. A., Quataert E., 2008, ApJ, 676,1130Milosavljevic M., Lindner C. C., Shen R., Kumar P., 2010, ArXive-printsMoriya T., Tominaga N., Tanaka M., Nomoto K., Sauer D. N.,Mazzali P. A., Maeda K., Suzuki T., 2010, ApJ, 719, 1445Nakar E., Piran T., 2011, ArXiv e-prints Narayan R., Igumenshchev I. V., Abramowicz M. A., 2000, ApJ,539, 798Narayan R., Mahadevan R., Quataert E., 1998, inM. A. Abramowicz, G. Bjornsson, & J. E. Pringle ed.,Theory of Black Hole Accretion Disks Advection-dominatedaccretion around black holess. pp 148– + Narayan R., Piran T., Kumar P., 2001, ApJ, 557, 949Narayan R., Yi I., 1994, ApJL, 428, L13Narayan R., Yi I., 1995, ApJ, 444, 231Nauenberg M., 1972, ApJ, 175, 417Nelemans G., Yungelson L. R., Portegies Zwart S. F., 2001, A&A,375, 890Nomoto K., Thielemann F.-K., Yokoi K., 1984, ApJ, 286, 644Ohsuga K., Mori M., Nakamoto T., Mineshige S., 2005, ApJ, 628,368O’Shaughnessy R., Kim C., 2010, ApJ, 715, 230Paschalidis V., MacLeod M., Baumgarte T. W., Shapiro S. L.,2009, Phys. Rev. D. , 80, 024006Perets H. B., Badenes C., Arcavi I., Simon J. D., Gal-yam A.,2011, ApJ, 730, 89Perets H. B., et al., 2010, Nature, 465, 322Phillips M. M., et al., 2007, PASP, 119, 360Piran T., 1978, ApJ, 221, 652Portegies Zwart S. F., Yungelson L. R., 1999, MNRAS, 309, 26Poznanski D., Chornock R., Nugent P. E., Bloom J. S., FilippenkoA. V., Ganeshalingam M., Leonard D. C., Li W., Thomas R. C.,2010, Science, 327, 58Quataert E., Gruzinov A., 2000, ApJ, 539, 809Rau A., et al., 2009, PASP, 121, 1334Rosswog S., Ramirez-Ruiz E., Hix W. R., 2009, ApJ, 695, 404Ryu D., Goodman J., 1992, ApJ, 388, 438Salaris M., Dominguez I., Garcia-Berro E., Hernanz M., Isern J.,Mochkovitch R., 1997, ApJ, 486, 413Shakura N. I., Sunyaev R. A., 1973, A&A, 24, 337Shen K. J., Kasen D., Weinberg N. N., Bildsten L., ScannapiecoE., 2010, ArXiv e-printsSigurdsson S., Rees M. J., 1997, MNRAS, 284, 318Taam R. E., Fryxell B. A., 1985, ApJ, 294, 303Tauris T. M., Sennels T., 2000, A&A, 355, 236Thompson T. A., 2010, ArXiv e-printsTh¨one C. C., et al., 2011, ArXiv e-printsThorne K. S., Zytkow A. N., 1975, ApJL, 199, L19Timmes F. X., 1999, ApJS, 124, 241Valenti S., Pastorello A., Cappellaro E., Benetti S., Mazzali P. A.,Manteca J., Taubenberger S., Elias-Rosa N., Ferrando R., Haru-tyunyan A., Hentunen V. P., Nissinen M., Pian E., Turatto M.,Zampieri L., Smartt S. J., 2009, Nature, 459, 674van den Heuvel E. P. J., Bonsdema P. T. J., 1984, A&A, 139, L16Verbunt F., Rappaport S., 1988, ApJ, 332, 193Verbunt F., van den Heuvel E. P. J., 1995, in W. H. G. Lewin, J. vanParadijs, & E. P. J. van den Heuvel ed., X-ray Binaries Formationand evolution of neutron stars and black holes in binaries.. pp457–494Waldman R., Sauer D., Livne E., Perets H., Glasner A., MazzaliP., Truran J. W., Gal-Yam A., 2010, ArXiv e-printsWoosley S. E., Heger A., 2006, ApJ, 637, 914Woosley S. E., Kasen D., 2010, ArXiv e-printsWoosley S. E., Taam R. E., Weaver T. A., 1986, ApJ, 301, 601Yungelson L. R., Nelemans G., van den Heuvel E. P. J., 2002,A&A, 388, 546 c (cid:13) , 1–, 1–