Exploring an Alternative Channel of Evolution Towards SNa Ia Explosion
aa r X i v : . [ a s t r o - ph . S R ] J a n Mon. Not. R. Astron. Soc. , 000–000 (2014) Printed 15 October 2018 (MN L A TEX style file v2.2)
Exploring an Alternative Channel of Evolution TowardsSNa Ia Explosion
E. Chiosi , C. Chiosi , , P. Trevisan , L. Piovan , & M. Orio , Astronomical Observatory of Padova, INAF, Vicolo dell’Osservatorio 5, 35122 Padova, Italy Department of Physics and Astronomy, University of Padova, Vicolo dell’Osservatorio 2, 35122 Padova, Italy Department of Astronomy, University of Wisconsin Madison, 475 N. Charter Str., Madison WI 53706, USAE-mail: [email protected] (EC); [email protected] (CC); [email protected] (PT);
E-mail: [email protected] (LP); [email protected] (MO)Submitted: September 2014; Accepted: ****
ABSTRACT
In this paper we explore the possibility that isolated CO-WDs with mass smallerthan the Chandrasekhar limit may undergo nuclear runaway and SNa explosion, trig-gered by the energy produced by under-barrier pycno-nuclear reactions between carbonand light elements. Such reactions would be due to left over impurities of the lightelements, which would remain inactive until the WDs transit from the liquid to thesolid state. We devise a simple formulation for the coulombian potential and the localdensity in a ionic lattice affected by impurities and introduce it in the known ratesof pycno-nuclear reactions for multi-component plasmas. Our semi-analytical resultsindicate that the energy generated by these pycno-nuclear reactions exceeds the WDluminosity and provides enough energy to elementary cells of matter to balance theenergy cost for C-ignition at much younger ages than the age of the Universe, evenfor WDs with masses as low as ≃ . M ⊙ . A thermonuclear runaway may thus betriggered in isolated WDs. The explosion would occur from few hundred thousand toa few million years after the WD formation in the mass interval 0 . − . M ⊙ . Key words:
Stars – structure, evolution; White Dwarfs – pycno-nuclear reactions –Supernovae Ia
Carbon-Oxygen White Dwarfs (CO-WDs) originate fromlow and intermediate mass progenitors in the mass inter-val 0 . − . M ⊙ to 6 M ⊙ (Weidemann 1967, 1977, 1990;Chiosi et al. 1992; Weidemann 2000), which after the cen-tral H- and He-burning phases develop a highly electron de-generate CO core (electrons are fully degenerate and nearlyrelativistic). After the thermally pulsing AGB phase, thesestars eject the whole envelope, baring the CO core. A WD isthus formed by a CO core surrounded by thin (if any) layersof lighter elements.Given any reasonable initial mass function, about 97%of the stars of any generation with lifetime shorter than theage of the Universe (13 . ± . . − . M ⊙ . The peak value is 0 . M ⊙ and the range ofmasses extends from about 0.4 M ⊙ toward the low mass endto 1 . − . M ⊙ on the opposite side. According to theoretical models, there also is a clear relationship between the pro-genitor and WD masses, M i vs M WD , (Weidemann 1967,1977, 1990; Chiosi et al. 1992; Weidemann 2000; Marigo2001; Catal´an et al. 2008) for which there is a plausible ex-planation based on current theory of stellar structure andevolution (see below for more details).The most important facts of the WD theory are: (i) Ex-cept for a thin surface layer, the equation of state (EOS) canbe approximated as fully degenerate electrons ( P ≡ P e ) withkinematic conditions changing from non relativistic ( P e = K / ρ / ) to fully relativistic ( P e = K / ρ / ) with increas-ing central and mean density. (ii) The structure of a WD ofgiven chemical composition (mean molecular weight of ionsand/or electrons, µ i and µ e respectively) is fully determinedby its central density. (iii) Assuming Newtonian hydrostaticequilibrium, the WD mass has a maximum value, called theChandrasekhar mass M Ch , for which the central density isinfinite so that the EOS is fully relativistic. In this case, M Ch = 5 . /µ e with µ e the molecular weight of electrons.For a typical CO-WD, µ e ≃ M Ch ≃ . M ⊙ . However,the radius of the Chandrasekhar mass is zero. Clearly such c (cid:13) E. Chiosi, C. Chiosi, P. Trevisan, L. Piovan, & M. Orio non physical situation means that no WD can be born withthe Chandrasekhar mass, which therefore is a mere idealvalue. (iv) Along the sequence toward the Chandrasekharvalue at increasing central density, two important physicalprocesses can intervene, overall instability driven by GeneralRelativity (GR) effects, and pycno-nuclear burning. WDsmore massive than about 1 . M ⊙ , i.e. denser than about3 × g cm − , become dynamically unstable because of GR(Shapiro & Teukolsky 1983); C-O WDs denser than about6 × g cm − , may start pycno-nuclear burning during theliquid-solid regimes (Shapiro & Teukolsky 1983). The onsetof these phenomena may lead to a thermo-nuclear runaway.Therefore, all stable WDs we observe must have formed withmasses lower than 1 . − . M ⊙ .The current theory of type Ia SNe assumes that theChandrasekhar mass can be reached by accretion or merg-ing and although the details are not fully known, carbonis ignited via the pycno-nuclear channel. This is followedby C-detonation or C-deflagration (the latter is more likely)and by a thermal runaway, because the gravo-thermal spe-cific heat of the star is positive. The liberated nuclear en-ergy exceeds the gravitational binding energy and the staris thorn apart (see the recent review by Nomoto et al. 2013,and references therein). Binary models for type Ia SNe areclassified as double-degenerate , i.e. the merger of two gravi-tationally bounded WDs, and single-degenerate , i.e. the evo-lution to the explosive phases is due to the accreting mate-rial from a companion star (see e.g., Napiwotzki et al. 2003;Trimble & Aschwanden 2004; Orio 2013).Owing to the important role of the pycno-nuclear reac-tions in destabilizing a WD close to the Chandrasekhar massand triggering type Ia SNa explosions, it is worth examiningin some detail the condition under which pycno-nuclear re-actions can occur. The pycno-nuclear regime starts in verydense and cool environments, i.e. in the liquid/solid state.While both the central and mean densities are determinedby the WD mass and remain nearly constant if the mass doesnot change, the temperature decreases because the WD isradiating energy from the surface. Because there are no nu-clear sources and the electrons are highly degenerate, whilethe WD radiates, the ions must cool down and transit fromgaseous to liquid, and eventually solid (crystallised) con-ditions, i.e. carbon and oxygen ions form a lattice whichis pervaded by a gas of electrons. As the temperature de-creases, the ions eventually reach the fundamental energystate under a coulombian potential that approximately hasthe form of a harmonic oscillator. In quantum physics, theenergy of the fundamental state of a harmonic oscillator is1 / ~ ω ( ω is the plasma frequency). This means that evenat extremely low temperatures there is a finite probabilityfor the C and O ions to penetrate the repulsive coulom-bian barrier. In this scheme, the rates of pycno-nuclear re-actions between C and/or O ions were first calculated bySalpeter & van Horn (1969) and then refined over the years.In their pioneering study, Salpeter & van Horn (1969)first noticed that impurities may enhance the pycno-nuclearreactions among the nuclei of the lattice. The enhancementis due to local over-densities in the sites of the impurities.Furthermore, even the contaminant nuclei themselves mayreact with the lattice nuclei, thus contributing to the totalenergy generation. If this happens, WDs that are less denseand/or less massive than the limits we discussed above be- come unstable to the ignition of pycno-nuclear reactions trig-gered by impurities. Since at the distance scales correspond-ing to the densities of old WDs, electric forces dominate thescene, impurities due to light elements, such as hydrogen orhelium, may induce higher local over-densities so that lightelements can more easily cross the coulombian barriers.In order to explore this idea, first we evaluate the changeinduced by the light element contamination on the typi-cal inter-ion distance R o . Second, we present the reactionrates in the pycno-nuclear channel for reactions like H+C,He+C, etc. Finally, we explore the possibility that even iso-lated WDs with mass significantly smaller than the Chan-drasekhar limit and relatively low density (i.e. in the ranges0 . − . M ⊙ and 10 − g cm − , respectively) may un-dergo nuclear runaway, because of the energy produced byunder-barrier nuclear reactions by contaminant elements,when the WDs reach the liquid/solid state.The paper is organized as follows. In Section 2 we firstshortly review the history of a WD progenitor and the cool-ing and crystallization processes of ions in the WD, and in-troduce the relationships between the progenitor mass andthe WD mass and between the WD mass and its centraldensity. In Section 3 we describe the fundamentals of nuclearburnings in WDs, summarize two current sources for the re-action rates in both the thermal and pycno-nuclear regimesfor single and multi-component fluids, and present a prelim-inary comparison of the reaction rates. In Section 4 we eval-uate the changes in the rates due to impurities, first in thetransmission probability of the coulombian potential barrierpenetration and then in the local density. Since impuritiesby light elements are more efficient than those by heavy ele-ments, in Section 5 we estimate the abundances of hydrogenand helium left over by previous evolutionary phases. Af-ter an initial phase at the beginning of the cooling sequencein which part of the energy may be of nuclear origin, for along time the only source of energy is the thermal energyof the ions and all nuclear sources are turned off. Thereforethe light elements remain inactive for a long time until theWD reaches the conditions for the activation of the pycno-nuclear regime. In Section 6 we estimate and compare theenergy production by reactions like H + C and He + Cshowing that they can produce the typical luminosity of anold WD in the pycno-nuclear stage. In Section 7, we followthe evolution of WDs of different mass along their coolingsequences towards the pycno-nuclear regime and explore theeffect of different abundances of light elements (H in particu-lar) on the energy release by nuclear reactions. We estimatethe abundance of contaminant elements and the epoch atwhich the nuclear energy generation first equals and then ex-ceeds the WD luminosity. In addition to this we formulate anew condition for C-burning ignition in a liquid, semi-solidmedium and propose a two-steps mechanism (named ”thefuse C-ignition” ) for igniting carbon in elementary volumesdefined by the mean free path of thermal conduction. Forplausible values of H and He abundances, WDs with massesas low as ≃ . − . M ⊙ (and above) reach the criticalcondition for rekindling the nuclear energy production andinitiate a thermal runaway. Finally, in Section 8 we drawsome conclusions about the possible implications of theseresults for the progenitors of type Ia SNe. c (cid:13) , 000–000 xploring an Alternative Channel of Evolution Towards SNa Ia Explosion The structure and evolution of low and intermediatemass stars, the progenitors of CO-WDs, can be de-scribed with the aid of three milestone masses (for all de-tails see Iben & Renzini 1983; Kippenhahn & Weigert 1990;Chiosi et al. 1992). We define low mass stars those that de-velop an electron degenerate helium core, shortly after leav-ing the main sequence toward the red giant branch (RGB).When the mass of the He-core has grown to a critical value(0.45-0.50 M ⊙ , the precise value depends on the composi-tion, star mass, and input physics), a He-burning runaway(called He flash) starts in the core and continues until elec-tron degeneracy is removed. Then nuclear burning proceedsquietly. The maximum initial mass of the star for this to oc-cur is M HeF . Stars more massive than M HeF are classifiedas intermediate-mass or massive depending on the physicsof carbon ignition in the core. After core He-exhaustion, in-termediate mass stars develop a highly degenerate CO core,and undergo helium shell flashes or thermal pulses as asymp-totic giant branch (AGB) stars. The AGB phase is termi-nated either by envelope ejection and formation of a CO-WD(with initial mass M i in the range M HeF M i M w ) orby carbon ignition and deflagration in a highly degeneratecore, once it has grown to the Chandrasekhar limit of 1.4 M ⊙ .The limit mass M w is regulated by the efficiency of massloss by stellar wind during the RGB and AGB phases. Theminimum mass of the CO core, below which carbon igni-tion in non degenerate condition fails and the above schemeholds, is 1.06 M ⊙ . The initial mass to reach a core of 1.06 M ⊙ is called M up . The exact value of M HeF , M up , and M w depends on many details of stellar physics. M w is mainlycontrolled by mass loss during the AGB phase and is about6 M ⊙ . The values of M HeF and M up are ≃ . − . M ⊙ and ≃ − M ⊙ , respectively, in absence of convectiveovershooting. These ranges become M HeF ≃ . − . M ⊙ and M up ≃ M ⊙ when convective overshooting is included(Chiosi et al. 1992). Since the ignition mass for a fully de-generate fully relativistic CO core is 1.46 M ⊙ , the possibilitythat C-ignition may occur in single CO-WDs is definitelyruled out. . The interiors of CO-WDs are made of ions ofC, O, traces of other elements, and free electrons. Ions arefully ionized and electrons form a uniform background. Inother words, there is a multi-component mixture (custom-arily named multi-component plasma, MCP) of ion species i = 1 , , ... with mass numbers A i , atomic number Z i , andnumber densities n i . The total number density is n = P i n i .For one component plasma (OCP) the suffix i is omitted. Atwo components medium is defined a binary ionic mediumor BIM.The number density is related to the mass density ρ ofthe matter by n i = X i ρA i m u (1)where X i is the mass fraction or abundance of ions i , and m u the atomic mass unit ( m u = 1 . × − g ). If the densityis not very high, the total mass fraction contained in thenuclei is X N = P i X i ≈
1, whereas for a density higherthan ∼ × g cm − above which neutrons may be free, X N <
1. Introducing the fractional number x i = n i /n , with P i x i = 1, we use two groups of useful relationships: h Z i = X i x i Z i , h A i = X i x i A i (2)where h Z i h A i are the mean atomic and mass number ofions and n e = n h Z i , ρ = m u n h A i X N , x i = X i /A i P j X j /A j . (3)The physical state is best described by the Coulombcoupling parameter Γ i for ions i :Γ i = ( Z i e ) a i k B T = Z i e / a e k B T (4) a e = (cid:20) πn e (cid:21) / , a i = Z / i a e , a i = (cid:20) πn i (cid:21) / (5)where T is the temperature, k B the Boltzmann constant, a e the electron-sphere radius, and a i the ion -sphere radius(a radius of a sphere around a given ion, where the electronand ion charge balance each other). The coupling parameterΓ i is the ratio between the Coulomb energy, E C = ( Z i e ) a i ,to the thermal energy E th = k B T of the ions. If Γ i << i > i ∼ i ≃
175 via a weak second-order phasetransition.In most cases, BIMs or MCPs are supposed to representthe composition of WDs, therefore it may be useful to intro-duce the mean ion Coulombian parameter h Γ i = P j x j Γ j .Strongly coupling occurs if h Γ i >
1, causing a transitionfrom plasma to liquid. The temperature T L at which thisoccurs is given by k B T L = X j (cid:20) Z j e a j (cid:21) x j ≡ k B T h Γ i . (6)With decreasing temperature, the ion motions can no longerbe considered as classical, but must be quantized. The nucleiform a Debye plasma with temperature T P associated withthe ion plasma frequency ω P T P = ~ ω P k B , ω P = vuutX j πZ j e n j A j m u . (7)This is the critical stage at which the specific heat of thematerial is determined by nuclei oscillations of frequency ω P rather than by free thermal motions.Increasing h Γ i further, by either lowering the temperatureor increasing the density or both, the matter crystallizes c (cid:13) , 000–000 E. Chiosi, C. Chiosi, P. Trevisan, L. Piovan, & M. Orio into a rigid Coulomb lattice. The solidification (or melting)temperature T M is given by T M = 1 k B X j (cid:20) Z j e a j (cid:21) x j = T L h Γ i M (8)where h Γ i M = 175 (Dewitt et al. 2001). At such high den-sities, even the small zero point oscillation allow the neigh-bouring nuclear wave functions to overlap, inducing nuclearreactions that depend on density and not on temperature.This is the pycno-nuclear regime. WD cooling . After a short lived initial phase, duringwhich the energy supply is sustained by some nuclear burn-ing in the two progressively extinguishing surface shells andthe large energy losses by neutrinos emission, the evolution-ary rate of the WD is driven only by the internal energy ofthe ions and electrons. Owing to the very different specificheat at constant volume of ions and electrons, the ions arethe dominating source. While the ions cool down, the WDundergoes several phase transitions. When the temperatureis ≃ K , the WD is gaseous and the ions behave like a per-fect Maxwell-Boltzmann gas. As the temperature decreases,first the ions become liquid and eventually they form a solidlattice. Because the electrons are fully degenerate, they can-not cool down.At this stage, in the energy conservation equation L r dM r = ǫ n + ǫ g + ǫ ν (with obvious meaning of all the sym-bols) ǫ n and ǫ ν can be neglected and the gravitational term ǫ g = − C v ˙ T + Tρ ( ∂P∂T ) v ˙ ρ is approximated as ǫ g ≃ − C v ˙ T .Therefore the luminosity of a WD is given by L = − Z Mo C v ˙ T dM r (9)where C v and ˙ T are function of the position and time.As the interior of a WD cools down, the ion specificheat C ionv per gram gradually changes from C ionv ≃ k B Am u to C ionv ≃ k B Am u (with the usual meaning of all the symbols) whereby thefirst relation is for the hot gaseous phase and the secondone is for the phase in which the temperature has decreasedbut it is still far from the Debye value (whereby quantumeffects become important). The increase by a factor of 2 isdue to increasing correlations of the ion positions driven bythe growing importance of the Coulomb interaction energiesas compared to the thermal energies of the ions. In otherwords, the spatial scale of the coulombian interactions iscomparable to the inter-ion spacing determined by the den-sity. When the temperature is close to the Debye value, C ionv decreases dramatically and becomes proportional to T .As far as electrons are concerned, the specific heat ofthe electrons can be neglected with respect to that of ionsand their contribution to the cooling rate of the WD can beignored (Kippenhahn & Weigert 1990). To conclude, as theions are the main contributors to the luminosity of a WD,the above relationships are used to calculate the cooling se-quence of a WD (see Section 3.5 below). Liquefaction and crystallization.
The liquefactionand crystallization theory (van Horn 1968) predicts that in WDs ions start to liquefy and freeze in an ordered cristallinelattice from the center to the outer layers when the temper-ature falls below T L and T M . The phase transition froman isotropic Coulomb liquid to a crystalline solid implies adiscontinuity in the distribution of the plasma ions. Becausesymmetry cannot be achieved instantaneously, the transitionis a first-order phase change. Therefore, for h Γ i = h Γ i m =172 −
175 (Kitamura 2000) latent heat is released (van Horn1968).For a ionic mixture with more than one species of ions,chemical separation may occur, either at solidification orin the fluid phase. The mixture behaviour, in this case,derives from the peculiar shape of the state diagram. Be-cause a phase separation, with the companion stratificationof elements, is a source of gravitational energy (caused bysinking of the heavier ions), able to deeply modify the WDcooling time, it is of fundamental importance to obtain de-tailed phase diagrams for the BIM or MCP of interest. More-over, for an accreting WD, chemical separation may producechemical stratification, thus affecting the electron capture,opacity and fusion rates.The most difficult problem with a strongly coupledMCP at low temperatures is understanding its actual state.MonteCarlo simulations of the freezing of classical OCPby DeWitt et al. (1992) indicate that it freezes into im-perfect body-centered cubic (BCC) or faced-centered cubic(FCC) micro-crystals. Unfortunately there are no reliablesimulations of freezing for MCPs. Cold MCPs are muchmore complex than OCPs; they can be regular lattices withimpurities or an amorphous uniformly mixed structure ora lattice of one phase with admixtures of other ions; oreven a mosaic of phase separated regions. Fortunately, theseextreme conditions are seldom reached. In a typical CO-WD (with X C = X O = 0 . . log ρ . . log T .
5. At ρ > × g cm − the carbon nucleicannot survive in dense matter because of beta captures. At ρ > × gcm − , the oxygen nuclei are also destroyedby beta captures. In this plane, the loci of T L , T P and T M are straight lines whose terminal points [log ρ , log T ] are: T L [8.4, 9.5], T P [10.5, 8.8], and T M [10.5, 7.9]. There is somemarginal effect that depends on the fractional abundances x j (see Yakovlev et al. 2006, for more details). Because in first approximation, the EOS of WDs doesnot depend on the temperature but only on the density,the polytropic description can be used, i.e. the mechan-ical and thermal structure of the WDs can be treatedseparately (Chandrasekhar 1939). However, whenever thethermal history of a WD is required to estimate correctlythe nuclear energy release or the luminosity as a func-tion of time, or other details of the cooling sequence, com-plete models of WDs are required. Both types of mod-els have been calculated by many authors and have beenmade available in the literature. To mention a few, re-cent state-of-the-art models of WDs have been calculatedby Althaus & Benvenuto (1997, 1998), Althaus et al. (2009,2012, 2013), Miller Bertolami et al. (2013), Panei et al.(2007), Renedo et al. (2010), and Salaris et al. (2010, 2013).Such models will be used in our analysis. c (cid:13) , 000–000 xploring an Alternative Channel of Evolution Towards SNa Ia Explosion Table 1.
Masses M G (in units of M ⊙ ) and radii R (in units of R ⊙ ) of CO WDs at varying the central density ρ c (in g cm − ).The mass is the gravitational mass; it tends to the Chandrasekharlimit as the density goes to infinity.log ρ c M G R log ρ c M G R × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − Table 2.
Properties of the progenitor star at the start of the TP-AGB phase prior to the formation of the WD. M i , M co , M WD are in M ⊙ , ρ c is in g cm − , the age is in years. M WD is derivedfrom eqn. (10). ρ c is interpolated from Table 1.Y=0.26 Z=0.017 M i M co M WD log ρ c Age ( yr )6.0 0.956 1.14 7.95 7.881 × × × × × × × In concluding this section, we need to present andshortly discuss a few relationships between important pa-rameters of WDs such as (i) the initial mass M i of the pro-genitor star, (ii) the CO core mass, M CO , at the beginningof the TP-AGB phase, (iii) the central density and the ageof the progenitor at the end of the AGB phase, (iv) themass M WD of the descendent WD (which is also named thegravitational mass M G ≡ M WD ), and finally (v) the centraldensity, radius, and age.A recent empirical estimate of the relationship between M WD and M i has been given by Catal´an et al. (2008) towhom we refer whenever necessary M WD = (0 . ± . M i + (0 . ± . i < . ⊙ M WD = (0 . ± . M i + (0 . ± . i > . ⊙ . (10)The relationships between the central density ρ c , themass M G ≡ M WD , and the total radius are derived fromAlthaus & Benvenuto (1997, 1998) and are reported in Ta-ble 1. The cooling sequences for the same WD massesare also from Althaus & Benvenuto (1997, 1998). The mass M CO of the CO core at the beginning of the TP-AGB phaseand the age of the progenitor are from the Padova Libraryof stellar models (Bertelli et al. 2008, 2009) and are listedin Table 2. Owing to the very short duration of the TP-AGB phase, the stellar age at the beginning of this phasenearly coincide with the age of the progenitor star when theCO core is exposed and the WD phase begins. We neglecthere the effects of the initial chemical composition on M CO , M WD and ages, and focus on the case [Y=0.26, Z=0.017]typical of a young population. If the chemical compositionis taken into account, because of the different total lifetimesof stars of the same mass but different chemical compositionand other details of stellar structure, at the low mass end ofthe WD mass distribution not all combinations of M i and M WD correspond to realistic cases. The lower mass limit isdetermined by the age of the Universe, i.e. 13 . ± . M CO core of TP-AGB stars, the core increases little duringthis phase in low mass stars (up to about 3 M ⊙ ) while itincreases significantly in stars with mass in the range 3 M ⊙ to 6 M ⊙ . Furthermore, the upper mass limit M up for AGBphase to occur (and to WDs to be generated) depends onthe initial chemical composition; the detailed stellar modelsby Bertelli et al. (2008, 2009) show that it can vary from5 M ⊙ to 6 M ⊙ . Therefore the M WD vs M i relationship ofTable 2 has to be considered as only indicative of the overalltrend. In the first stages of the WD life, nuclear burning can occurin two shells, close to the surface (see Renedo et al. 2010;C´orsico & Althaus 2014, and references therein). Hydrogenmay still be burning via the CNO-cycle, because a smallamount of it (mass abundances in the range 10 − X H − ) is left on the surface, and the temperature is still suffi-ciently high to sustain nuclear burning. Similarly, somewhatdeeper inside there may be also residual He-burning in ashell surrounding the inert CO-core.In general, as the temperature decreases, all thermalnuclear burnings are turned off for billions of years untilthe WD has cooled down to very low temperature, so thatthe pycno-nuclear regime is reached. This is possible onlywhen the internal energy of ions and the WD luminosity arevery low, and the baryonic matter (mostly C and O ions)is crystallized, usually when the WDs are quite old (severalGyrs). An exception to the above picture are the resultsof Miller Bertolami et al. (2013) who have recently shownthat for low-mass WDs resulting from low-metallicity pro-genitors, residual H-burning constitutes the main contrib-utor to the stellar luminosities for luminosities as low aslog L/L ⊙ ≃ − . c (cid:13) , 000–000 E. Chiosi, C. Chiosi, P. Trevisan, L. Piovan, & M. Orio increase the nuclear burning rates (see Section 4). This sug-gestion is the starting point of this study.Other formulations of nuclear reactions in thepycno-nuclear regime are by Schramm & Koonin(1990), Ogata et al. (1991), Ichimaru et al. (1992),Kitamura & Ichimaru (1995), Brown & Sawyer (1997),Ichimaru & Kitamura (1999a), Kitamura (2000),Gasques et al. (2005), Yakovlev et al. (2006), Beard(2010), and references therein. All these authors consid-ered and derived the pycno-nuclear reaction rates for alattice composed of a single element (the one-componentplasma OCP of Gasques et al. 2005), two elements (thebinary ionic medium BIM of Kitamura 2000), or a multi-component plasma (MCP) of (Yakovlev et al. 2006). Theyalso discussed electron screening and phase transitions fromgaseous to liquid to solid phases. However, they did notdiscuss the presence of impurities (see Section 3.5).
We are interested in nuclear fusion reactions( A i , Z i ) + ( A j , Z j ) → ( A c , Z c )where A c = A i + A j , Z c = Z i + Z j refer to the compoundnucleus c . To study these reactions we must extend the for-malism presented in Sect. 2.2 and introduce the ion-spherequantities a ij = a i + a j , µ ij = m u A i A j A c , Γ ij = Z i Z j e a ij k B T (11) T Lij = Z i Z j e a ij k B T Pij = ~ k B ω ij ω Pij = s πZ i Z j e n ij µ ij T Mij = Z i Z j e a ij Γ Mij (12)where µ ij is the reduced mass, a ij characterizes the equi-librium distance between neighbouring nuclei (the corre-sponding number density is n ij = 3 / πa ij ), Γ ij describestheir coulomb coupling, T Lij is the temperature of the onsetof strong coupling (or liquefaction temperature), T Pij is theDebye temperature for the oscillations of ions i and j , and T Mij is the melting or solidification temperature . Finally, weneed the generalized Bohr radius r Bij = ~ µ ij Z i Z j e (13)which becomes the ion Bohr radius for equal ions i = j ,and the parameter λ ij , corresponding to the parameter λ introduced by Salpeter & van Horn (1969) λ ij = r Bij (cid:16) n ij (cid:17) / = 2 r Bij ( Z / i + Z / j ) (cid:18) ρX N h Z i h A i m u (cid:19) / (14) In the case of an OCP all the symbols reduce to those alreadydefined in Sect. 2.2
In WDs the central density ranges from 10 g cm − to10 g cm − depending on the initial mass of the star, andthe density decreases from the center to the surface, so wemust consider a wide range of densities. The temperaturealso spans a wide range, from 10 K at the beginning of thecooling sequence to typical values of about 10 K after afew Gyrs. In the plane T − ρ , the WD undergoes two phasetransitions: from gas to liquid and from liquid to solid (crys-tallization). In parallel to this, there are five burning regimes(see Salpeter & van Horn 1969; Yakovlev et al. 2006; Beard2010): (i) the classical thermo-nuclear; (ii) the thermo-nuclear with strong electron screening; (iii) the thermo-pycno-nuclear; (iv) the thermally enhanced pycno-nuclear;(v) and eventually the zero temperature pycno-nuclear. Fora complete description of the regimes see Beard (2010). Thefive regimes have the following characteristics:i) The classical thermo-nuclear one takes place whenΓ ij <<
1. The mean inter-nuclear distance is much greaterthan the typical scale at which the particles feel electrostaticinteraction: the nuclei are fully stripped and there is a smallscreening effect from the background electron gas. The bulkmatter behaves like an ideal gas.ii) The second regime is bounded by the temperatures T Lij and T Pij with T Pij T T Lij . For temperatures lower than T Lij the ions are in the liquid phase, and for temperaturesclose to T Pij , the ions cannot be considered a free gas but partof a lattice with vibrations ω Pij . The thermonuclear burningassociated to very strong electron screening operates in thistemperature range.iii) The third regime corresponds to the temperaturerange T Mij T T Pij and Γ ij >
1. Nuclei are bound to thelattice sites, so that the reactions occur between highly ther-mally excited bound nuclei, which oscillate with frequencieshigher than the plasma frequency and have energy greaterthan the zero point energy of the plasma. The nuclei are alsoembedded in a highly degenerate electron gas, so that thereaction rates are enhanced by the charge screening electronbackground. For Γ
Mij > T Mij , the nuclei form a solid lattice.iv) In the thermally enhanced pycno-nuclear regime,Γ ij >> ij > T ≃ T = 0 pycno-nuclear rates. There are some important differences in the hypotheses un-derlying thermo-nuclear and pycno-nuclear reactions thatare worth highlighting. In the thermo-nuclear reactions each c (cid:13) , 000–000 xploring an Alternative Channel of Evolution Towards SNa Ia Explosion particle can interact with all the others, therefore the ratecontains the product of the two densities of the two inter-acting species. The reaction rate is R ij = N i N j < σv > (15)The probability of interaction is proportional to the productof two factors: the Maxwell-Boltzmann statistical factor andthe probability of tunneling. Their product gives origin tothe so-called Gamow window. The first factor is proportionalto e − E/ ( K B T ) (16)whereas the second one is proportional to e − √ E G /E (17)where E G is the Gamow energy. If we define E pkij = √ E G k B T we can write the thermo-nuclear reaction rateas R thij = 4 n i n j δ ij E pkij µ ij ! / S ( E pk ) k B T e − τ ij (18)with τ ij = E pkij k B T .The rate for the pycno-nuclear reactions is quite differ-ent. Each nucleus in the lattice can interact only with thenearest neighbours. Therefore, the rate is R pyci = n i < ν i p i > (19)where p i is the reaction probability between a pair of ionsand ν i is the number of nearest neighbours. According toSalpeter & van Horn (1969), the reaction rate probabilitycan be written as p i = D pyc λ − C pl i S ( E pki ) ~ r Bi exp − C exp λ / i ! (20)where λ i is the ratio of the Bohr radius to the latticespacing. For more details on the meaning of the varioussymbols see Salpeter & van Horn (1969); Beard (2010) andShapiro & Teukolsky (1983). In the literature on pycno-nuclear reactions, there are basi-cally three different formulations or models: (i) the classicalone, based on the simple harmonic oscillator at zero tem-perature proposed long ago by Salpeter & van Horn (1969),that we will shortly revisit to explore the effect of impuri-ties (Sect. 4), (ii) the model by Kitamura (2000, and refer-ences) and finally (iii) the models by Gasques et al. (2005)and Yakovlev et al. (2006). The last two models include alsothe effect of temperature, and they are mutually consistentalthough they make use of different analytical expressions.
Kitamura (2000, see also references therein) considerablyimproved the simple description based on the harmonic os-cillator. He derived analytic expressions for the reactionrates, taking into account the dielectric functions of rela-tivistic and non-relativistic electrons, the screening poten-tials based on the Monte Carlo simulations, and the interac-tion free energies in dense electron screened BIMs. He foundthat under-barrier reaction rates can be expressed as theproduct of three terms R = R G A (0) ij A ( e ) ij (21)where R G is the so-called Gamow channel, representing thebasic binary interaction between any two particles, and isexpressed as the Gamow rate. It is dominant in tenuousplasmas in which the effect of surrounding particles is negli-gible, so the nuclei interact via the bare Coulomb potentialand the rate is expected to be strongly dependent on tem-perature. The other two terms take the effects of the sur-rounding particles into account. The so-called few-particlesinteractions are expressed by A (0) ij and occur independentlyof the aggregation state of nuclei. The shielding effect stemsfrom local variations of particle density with respect to thebackground (also referred to as polarization). The net effectis to reduce the Coulomb potential barrier and to stronglyenhance the rate. Once the temperature is below a certainvalue, the rate is expected to increase as the temperaturedecreases. Finally the third term A ( e ) ij is due to the many-particles processes that may occur when the electrons canbe considered as a uniform background. This is expected toproduce a small effect at the typical temperatures and den-sities of liquid-solid WDs, because the so-called screeningtemperature T S , at which this effect is important, is muchlower. Therefore, the third term is expected to be small andto become dominant only at very low temperatures, unlikelyto be reached in WDs.The most important pycno-nuclear reactions are thefew-particles ones rather than the many-particles interac-tions. Quoting Ichimaru & Kitamura (1999b, page 2694) ”Since the enhancement factors generally increase steeplywith lowering of the temperature, a maximum pycno-nuclearrate may be attained in a liquid-metallic substance near theconditions of freezing transitions; reaction rates in a solid,depending on the amplitudes of atomic vibrations, increasewith the temperature” .Kitamura (2000)’s rates are particularly useful to un-derstand the effects of impurities because they includeboth BIMs and OCPs, and provide a better descriptionof the physics during the solid transition, which implies asmoother discontinuity in the reaction rates. We will applythe Kitamura (2000) rates over the whole range of temper-atures and densities, for any kind of reacting pairs.In a BIM of two elements “ i ” and “ j ”, with charges Z i and Z j , mass number A i and A j , mass fraction ρ m andtemperature T , the reaction rates are c (cid:13) , 000–000 E. Chiosi, C. Chiosi, P. Trevisan, L. Piovan, & M. Orio R ij (reactions cm − s − ) = 11 + δ ij X i X j ( A i + A j ) Z i Z j ( A i A j ) · (cid:2) ρ m (cid:0) g cm − (cid:1)(cid:3) · [ S ij ( E eff ) (MeV barns)] R (22)where δ ij is the Kr¨onecker delta function ( δ ij = 1 if i = j and δ ij = 0 otherwise), S ij is the nuclear cross-section factorfor the analyzed reaction and R depends the aggregationstate of the matter.For the liquid phase, if T s is the critical screening tempera-ture and T > T s , we have R fluid0 =2 . × τ ij vuut − " tanh (cid:18) T s T (cid:19) / · exp − α ij π s D s r ∗ ij + ξ ij ! (23a)if instead T < T s R fluid0 =1 . × (cid:18) D s r ∗ ij (cid:19) / ( " , (cid:18) D s r ∗ ij (cid:19) / − · (cid:18) TT s (cid:19) ) exp − α ij π s D s r ∗ ij + ξ ij ! (23b)where T s is the critical screening temperature at which theGamow peak energy equals the electrostatic screening en-ergy. The two expressions coincide for T = T s .The nuclear reaction rates for the solid state are ob-tained by substituting in eqn.(22), the following expressionfor R : R solid0 = 4 . · ( R ijs ) . · exp (" − .
506 + 0 . (cid:18) a ij D s (cid:19) +0 . (cid:18) a ij D s (cid:19) + F ( Y sij ) ( R ijs ) / ) . (24)The definition of all terms and meaning of all symbols usedin eqns. (23) and (24) above can be found in Kitamura(2000) to whom the reader should refer. We note that thefirst term in the exponential factor of eqns. (23) correspondsto the Gamow thermo-nuclear channel, while the secondterm corresponds to the under-barrier reaction channel. Inthis paper, we examine the reactions listed in Table 4. Theexperimental value of the cross-section factor and Q -valuefrom Fowler et al. (1975) are also in Table 4. Gasques et al. (2005) elaborated a model for OCPs to cal-culate the reaction rates in several regimes, from thermo-nuclear to pycno-nuclear. Subsequently Yakovlev et al.(2006) and Beard (2010) extended the model to MCPs.These three papers are based on the Sao Paulo potential,that takes into account the effect of Fermi statistic on the nucleons of the interacting nuclei. The purposes of thesestudies are: (a) to evaluate the rate for the pycno-nuclearpart R pycij and (b) to apply a phenomenological expressionfor the temperature and density dependent part ∆ R ij ( ρ, T ).All the auxiliary quantities contained in the expressions be-low have already been introduced in Sect. 2.2 or are givenin Table 3.The pycno-nuclear component is R pycij = 10 C pyc ρX N x i x j A i A j h A i Z i Z j (1 + δ ij ) A c S ( E pkij ) × ˜ λ − C pl ij exp ( − C exp (˜ λ ij ) / ) cm − s − (25)whereas the phenomenological expression for the tempera-ture and density dependent part of the rate that combinesall the burning regimes is R ij ( ρ, T ) = R pycij ( ρ ) + ∆ R ij ( ρ, T ) , (26)∆ R ij ( ρ, T ) = n i n j δ ij S ( E pkij ) ~ r Bij
P F, (27) F = exp ( − ˜ τ ij + C sc ˜Γ ij φe − Λ ˜ T ( p ) ij /T − Λ ˜ T ( p ) ij T ) ,P = 8 π / √ / ( E aij k B ˜ T ) γ (28)where E aij = 2 µ ij Z i Z j e / ~ (29) φ = p Γ ij / [( C scij /ζ ij ) + Γ ij ] / (30)˜ τ ij = 3( π / ( E aij k B T ) / ˜Γ ij = Z i Z j e a ij k B ˜ T , ˜ T = q T + C T ( T ( p ) ij ) , (31) γ = ( T γ + ( ˜ T ( p ) ij ) γ ) / ( T + ( ˜ T ( p ) ij ) )) (32) E pkij = ~ ˜ ω ( p ) ij + ( Z i Z j e a ij + k b T τ ij exp ( − Λ ˜ T ( p ) ij T ) (33)where γ = 2 / γ = (2 / C pl + 0 . The rates predicted by Kitamura (2000, dashed lines) andGasques et al. (2005, solid and dotted lines ) for the C+ C reaction are shown as a function of density in Fig. 1. Forthe Gasques et al. (2005) rate we adopt the numerical co-efficients for MCPs that are listed in Table 3 and show thecases of maximum (solid lines) and minimum (dotted lines)rates. For each case we show separately the sole pycno-nuclear component (red lines) and the pycno-nuclear plusthermal channel (blue lines). The rates by Kitamura (2000)fall always in between those by Gasques et al. (2005) for allregimes. In general, the three cases have a very similar trend,the only difference being the density at which the transi-tion from thermal+pycno to pycno-nuclear regimes occursand the absolute value of the rates. However, we note that c (cid:13) , 000–000 xploring an Alternative Channel of Evolution Towards SNa Ia Explosion Table 3.
Coefficients in the interpolation expressions for a reaction rate for the optimal model of nuclear burning and for the modelswhich maximize and minimize the rate. The parameters CT, α λij , α ωij are different for the multi-component plasma (MCP) and onecomponent plasma (OCP) (the values for OCP are given in brackets). For a MCP, the models assume a uniformly mixed state (seeYakovlev et al. (2006) for details).Model Cexp Cpyc Cpl CT α λij α ωij ΛOptimal 2.638 3.90 1.25 0.724 (0.724) 1.00 (1) 1.00 (1) 0.5Maximum rate 2.450 50 1.25 0.840 (0.904) 1.05 (1) 0.95 (1) 0.35Minimum rate 2.650 0.5 1.25 0.768 (0.711) 0.95 (1) 1.05 (1) 0.65
Figure 1.
Comparison of the Gasques et al. (2005) and Kitamura(2000) rates for the C + C reaction. The coefficients of theGasques et al. (2005) rates are those listed in Table 3. The ther-mal branch and pycno-nuclear channel are calculated with tem-peratures of 10 K and 10 K, respectively. The mass abundanceof carbon is equal to X C = 1. Table 4.
Parameters for the reaction rates of most common re-actions in CO-WDs.Reaction Products Q -value [MeV] S (0) [MeV barn] C+ C Mg . × C+ O Si . × O+ O S . × there is a large difference between the maximum and mini-mum efficiency for the Gasques et al. (2005) rates, amount-ing to about ten orders of magnitude, and that the Kitamura(2000) rates run closer to the Gasques et al. (2005) case withmaximum efficiency. There are large differences in the ratesat all densities, causing a large difference in the final results. When matter becomes solid, the C and O ions are in a fixedconfiguration in a crystal whose structure is known fromsolid state physics. We cannot exclude the possibility thatsome impurities of elements of any type are present in thelattice.The dominant element with the lowest atomic numberis carbon ( Z C =6), followed by oxygen ( Z O = 8), so impu-rities can be grouped according to their atomic number: i)heavy elements like magnesium, etc...with Z > Z C ; ii) lightelements like hydrogen and helium with Z < Z C .We are particularly interested here in impurities of light ele-ments, like hydrogen and helium. In the next section, we willsee that the pycno-nuclear rates for these elements may beparticularly high, so the amount of light elements necessaryfor relevant effects is extremely low. For instance, in the caseof hydrogen, abundances in the range 10 − X H − are sufficient to produce nuclear energy in amounts compa-rable to the luminosity of typical WDs at suitable ages.This should not be confused with the surface contentof hydrogen that can be as high as X H ≃ − withouttriggering thermo-nuclear burning (see Fujimoto 1982b,a;Renedo et al. 2010, for all details). Higher concentrationsare not allowed otherwise the associated thermo-nuclear en-ergy generation via the CNO-cycle would exceed the lumi-nosity of a WD, probably inducing a thermonuclear run-away.In this section we describe, with a simple formulation,how the coulombian potential and local density in a ioniclattice are modified by impurities. We adopt the expressionof Salpeter & van Horn (1969) and Shapiro & Teukolsky(1983) for the rate of pycno-nuclear reactions. The originaltheory was developed for reactions among the same nuclei(e.g. carbon). Since we consider at least two elements (e.g.reactions among hydrogen and carbon), the columbian po-tential must be suitably modified. We are going to presenta template, to which more sophisticated formulations mustbe compared. Using the same formalism of Shapiro & Teukolsky (1983),we consider an OCP made of a certain type of ions, e.g.either pure carbon or pure oxygen. In this framework, firstwe consider the one-dimensional potential of an array of ionsand extend it to 3D. Suppose that the lattice is composedof ions of charge Z . In one cell we substitute the charge Z with a charge Z (for instance hydrogen or helium). If x isthe displacement from the equilibrium position (see Fig.2) c (cid:13) , 000–000 E. Chiosi, C. Chiosi, P. Trevisan, L. Piovan, & M. Orio
Figure 2.
The potential governing the motion of one incidention of charge Z relative to an adjacent ion of charge Z . Zeropoint energy fluctuations in the harmonic potential well near theincident ion can lead to Coulomb barrier penetration and nuclearreactions. the coulombian potential becomes V ( x ) = Z Z e R − x + Z Z e R + x − Z e R (34)where R is the inter-ionic distance. In eqn. (34) we assumethat the energy of the lattice is zero and estimate the energyvariation. Removing a charge Z produces the negative term(interaction of the neighbouring particles with the hole) andthe addition of a charge Z at position x introduces thepositive terms. We suppose that x << R and use the Taylorexpansions to derive V ( x ) = 2 Z Z e R − Z e R + 2 Z Z e x R . (35)The constant term shifts all the energy levels. The variableterm + 2 Z Z e x R (36)is the dominant one and behaves like a harmonic oscillatorwith ω = ( 4 Z Z e R µ ) / . (37)Using the equation E = V ( r ) = V +1 / Kr = V +1 / ~ ω ,the turning point r is r = ( ~ R / Z Z µ ) / e ) / (38)The solutions of the Schr¨odinger equation for a harmonicoscillator are known. In three dimensions they become | ψ SHO | = τ π / e − τ r (39)where τ = 1 /r . Assuming that the exponential value is one, | ψ inc | ≃ | ψ SHO | ≃ r π / . (40)The transmission coefficient for an incident ion with energy E in the WKB approximation is T = exp [ − Z ba ( 2 µ ~ [ E − V − / Kx ]) dx ] (41)where E = V + 1 / ~ ω . Following the discussion byShapiro & Teukolsky (1983), we obtain the reaction rate perion pair W = v | ψ inc | T S ( E ) E K (42)where in the limit r /R << T = R r exp ( − R r ) . (43)This transmission coefficient is the leading term in the reac-tion rate and it strongly depends on the charge Z , in factthe exponent is proportional to ( Z Z ) / , which means thatthe transmission coefficient is much higher for Z < Z .We refer to our revision of the Shapiro & Teukolsky(1983) rate as the ”modified harmonic oscillator” (MHO).We will later adopt the formulations of Kitamura (2000) andYakovlev et al. (2006), but the MHO approximation is anefficient way to explore reactions between light and heavierelements when the WD reaches the pycno-nuclear regime.We note that the same formalism can be applied to the caseof MCPs because what matters are the pairs of interactingions, e.g. H+ C or H+ O. The lower coulombian barrierin the case of C and H definitely favours this ion pair, somost of discussion will be limited to these two elements. In order to evaluate the change in the local density andconsequently in the rates of energy production by pycno-nuclear reactions caused by impurities, OCPs and MCPsmust be treated separately. We can consider three cases: (i) OCPs, neglecting electrons in the electro-static force . Let us consider an array of nuclei of charge Z (e.g. carbon) as shown in the top panel of Fig. 5. If in onesite we substitute the nucleus of charge Z with a nucleus ofcharge Z , e.g. hydrogen (the impurity), the carbon nucleineighbouring the hydrogen nucleus are shifted toward it be-cause of the new equilibrium of the forces as indicated by thearrows (the situation is clearly symmetric). The equilibriumof the forces is expressed by Z Z ( R − x ) = Z ( R + x ) . (44)After developing the squares we obtain( Z Z − Z ) x + 2 R ( Z Z + Z ) x + ( Z Z − Z ) R = 0 . (45)This is a second order equation with two solutions x , = R − ( Z Z + Z )( Z Z − Z ) ± s(cid:20) ( Z Z + Z )( Z Z − Z ) (cid:21) − . (46)We evaluate the displacement for Z = 6 (carbon) and Z = 1 (hydrogen) and keep only the solution < R ′ = R − x = 0 . R , i.e.the new local inter-ion distance R ′ is smaller than in the c (cid:13) , 000–000 xploring an Alternative Channel of Evolution Towards SNa Ia Explosion unperturbed case. For the case Z = 6 (carbon) and Z = 2(helium) obtaining R − x = 0 . R . As expected, the effectis decreasing with increasing Z . Hereinafter R ′ is renamed R . (2) OCPs with electrons in the electrostaticforce . In the above expression we neglected the contributionby electrons to the balance of electrostatic forces among ions.Taking electrons into account, we assume that in the spacebetween the ions Z and Z electrons distribute uniformlyand evaluate the field due to electrons at any distance x inbetween the ions Z and Z . In this case eqn. (44) becomes − Z Z ( R − x ) + Z ( R + x ) + −
12 ( Z − Z ) Z xR + 12 ( Z − Z ) Z ( R − x ) R = 0 (47)Performing easy algebraic manipulations we obtain the fifthdegree equation − x + R x + 4 R x − R x + 3 R = 0 (48)to be solved numerically. The solution has four complexroots and only one real with physical meaning, i.e. x =0 . R . Therefore, in the case of carbon and hydrogenthe new distance between two neighbouring ions is R ′ =0 . R . (3) MCPs with electrons in the electrostaticforce . It is worth of interest to consider the case of MCPs,limited however to a mixture of C + O (the BCC configu-ration, Ichimaru 1982) and including the effects of electronsin the force balance. We consider an array of ions in the se-quence O ( Z ) - C ( Z ) - H ( Z ) - C( Z ) - O( Z ) as shownin the bottom panel of Fig. 5, and evaluate the displacementof the C nuclei bracketing the hydrogen nucleus by impos-ing the balance of electrostatic forces. In this case eqn. (4.2)becomes − Z Z ( R − x ) + Z Z ( R + x ) + −
12 ( Z − Z ) Z xR + 12 ( Z − Z ) Z ( R − x ) R = 0 (49)the associated fifth degree equation is − x +5 R x +20 R x +4 R x − R x +19 R = 0 (50)where R is the distance between the C and O ions in theBCC lattice. Also in this case there is only one real solution, x = 0 . R . The distance between the C and H ions istherefore R ′ = 0 . R . Implementing the correction into the reactionrates . The effect of this correction on the interionic distancein the pycno-nuclear rates depends on the physical model forthe rates. If we insert the new value of R into eqns. (40)and (41), the pycno-nuclear reaction of Salpeter & van Horn(1969) are enhanced by a large factor.Using the Gasques et al. (2005), and Yakovlev et al.(2006) rates, in which the pycno-nuclear term is separate Figure 3.
The energy generated by the reactions C+ C (blueline labelled C+C), H + C (green line labelled H+C), and He + C (red line labelled He+C) according to the MHO rates.The abundances by mass of C, H and He are ≃
1, 10 − , and3 × − , respectively. This corresponds to an ideal WD made ofsole carbon with traces of hydrogen or helium. The rates for thereactions H + C (magenta line labelled [H+C]’) and He + C (cyan line labelled [He+C]’) have been recalculated includingalso the effect on the local density caused by the presence ofimpurities as discussed in Section 4.2. Finally, the dotted blackline represents the mean luminosity expressed in erg s − cm − ofa typical WD. First all the rates intersect the mean luminositywhich is the signature of a potential nuclear runaway, second theintersection occurs at higher and higher densities with increasingatomic number. Third, the effect of impurities is conspicuous (e.g.compare the green and magenta lines for hydrogen impurities orthe red and cyan lines for He impurities). from the thermal one, the functions a ij must be redefined.Finally, the reaction rate Kitamura (2000) is more compli-cated to treat, because the temperature dependence is in-trinsic in the formulation and cannot be singled out easily.(i) Gasques and Yakovlev rates . There are two approxi-mations depending on the composition of the plasma. If theplasma is made of pure C with H impurities, the new a ij are a ij = λ [ 34 πn ] / (51)where λ = 0 .
56 for the reaction H + C, and λ = 0 . He + C. Other combinations of reactants(such as H + O) can be easily obtained from the generalformalism above, and the effect of impurities is expectedto be smaller. While using the rate by Salpeter & van Horn(1969), the correction is directly applied to R , using theGasques et al. (2005) and Yakovlev et al. (2006) rates, theold a ij coefficients of eqn. (11) are replaced by eqn. (51. c (cid:13) , 000–000 E. Chiosi, C. Chiosi, P. Trevisan, L. Piovan, & M. Orio
Figure 4. C + C pycno-nuclear reaction rates for thesimple MHO model (blue continuous line) compared with theSalpeter & van Horn (1969) model (red dashed line). They areroughly similar.
Figure 5. Top Panel : OCP made of carbon in which a nucleusof carbon ( Z ) is replaced by a nucleus of hydrogen ( Z ). Weevaluate the forces acting on the nuclei Z adjacent to the im-purity (charge Z ) and the corresponding displacement x of Z given by the new condition of force equilibrium. Bottom Panel :the same but for a MCP made of carbon and oxygen nuclei inBCC configuration in which a nucleus of oxygen is replaced by anucleus of hydrogen. a ij = 12 (cid:2)(cid:0) Z i n e (cid:1) / + (cid:0) Z j πn e (cid:1) / (cid:3) . (52)In such a case the effect is less evident and smaller thanbefore.In the case of a MCP (for instance C + O ), the new a ij coefficients are a ij = 0 . a CO = 0 . / + 8 / )2 (cid:2) m u π . ρ (cid:3) / (53)where a CO is derived from eqn. (52) above.(ii) Kitamura Rate . In this case, at low densities the a ij are those of the classical formulation given by eqn. (52),whereas at high density they must coincide with those ofeqn. (4.2). In order to transit smoothly from the thermal tothe pycno-nuclear regime and to take the correction of localdensities due to impurities into account, we assume that the a ij coefficients can be linearly interpolated in log ρ accordingto a ′ ij = a pycij + ( a thij − a pycij ) 10 − log ρ − a pycij is defined according to eqn. (4.2) at log ρ = 10,and a thij is defined according to eqn. (52) at log ρ = 6. In the MHO formalism derived above, the total number ofreactions per second per cm is given by P = n el W (55)where n el is the number density of the reacting elements.This is related to the abundance by number f el by n el = f el N/V where N is the total number of ions in the starand V is the volume. In the simple case of reactions amongidentical ions, converting number densities to mass densi-ties, mass abundances, and number abundances is straight-forward, whereas in the case of reactions among differentelements the procedure is more complicated. If N j , X j and A j are the number of ions of species j with mass abundance X j and mass number A j , X j = N j A j m u P N l A l m u (56)where m u is the mass unit. If a WD, made of pure carbon,is polluted by traces of hydrogen and helium ( N H << N and N He << N ) we obtain X H = f H
12 and X He = f He
412 = f He . (57)For a WD of carbon and oxygen, whose mass abundancesare related as X C / X O = α , we derive X H = f H
12 [ α + (3 / α + 1) X He = f He
412 [ α + (3 / α + 1) (58)for typical values of α , X H is about a factor of 10 lower than f H , and X He is a factor of 3 lower than f He .Since the mass and structure of a WD depend on its centraldensity, it is useful to express the reaction rates as a functionof this parameter. In Fig. 3 we show the energy generationrates for the H + C, He + C and C + C reactions.These are calculated with the MHO rates with and withoutthe impurities and associated density enhancement. We alsoshow the case of the C + C reaction calculated with theMHO. The astrophysical factors S (0) and Q -values of the c (cid:13) , 000–000 xploring an Alternative Channel of Evolution Towards SNa Ia Explosion Table 5.
Parameters for the nuclear reactions of hydrogen andhelium with carbon. S(0) QMeV barn MeV H + C 1 . × − He + C 1 . × − reactions H + C, He + C are given in Table 5, those forthe C + C are in Table 4. The abundances by numberand by mass of the elements are f H = 10 − , f He = 10 − and f C = 1 (or equivalently X H = 10 − , X He = 3 × − ,and X C = 1). For the sake of comparison we show also themean luminosity of a typical CO-WD, 10 erg s − . In orderto express the energy generation rates and WD luminosityin units of erg s − cm − , the WD luminosity is divided bythe WD volume estimated assuming a mean radius of 5000km. The results of Fig 3 are of paramount importance: theyclearly show that in the case of hydrogen impurities (with f H = 10 − ) the energy produced by the pycno-nuclear re-actions at zero temperature may exceed the typical lumi-nosity of a WD at a density ρ ≃ × g/cm , correspond-ing to a mass of about 1.05 M ⊙ , significantly smaller thanthe Chandrasekhar mass, see Table 1. This is a potentiallyexplosive situation. In the case of helium, the intersectiondensity and corresponding WD mass are higher but still ofinterest ( ρ ≃ × g cm − and about 1 . M ⊙ , respectively).Finally, the intersection with the C + C line is at about9 . − and the corresponding WD mass is nearly equalto the Chandrasekhar limit.As already discussed in Shapiro & Teukolsky (1983),the MHO expression for the reaction rate is not very differ-ent from the original one obtained by Salpeter & van Horn(1969) taking into account other effects, like electron screen-ing and anisotropy. As a test, we compare the rates forthe C + C reaction derived from the MHO above andSalpeter & van Horn (1969). The results are shown in Fig.4.The agreement is remarkable. Furthermore, these rates arealso in good agreement with those by Gasques et al. (2005)for the same reaction. The existence of traces of light elements in the core of CO-WDs is at the basis of our investigation. Light elementsare the best candidates to consider, because the very highcoulombian barriers of high Z elements would quench thepenetration probabilities to zero.Undoubtedly, hydrogen and helium on the WDsurfaces in small but significant abundances are pre-dicted theoretically (10 − X H − according toMiller Bertolami et al. 2013; Renedo et al. 2010) and are ob-served (Bergeron et al. 1990). How much hydrogen or he-lium can be present in the interior?Following Kippenhahn & Weigert (1990), the ratherhigh internal temperatures (from 10 to 10 K) of an ag-ing WD set a limit to the possible hydrogen content in theinterior. If hydrogen were present with a mass concentra- tion X H , we would expect H-burning via the pp-chain. Foraverage values T = 5 × K, ρ = 10 g cm − , the energyreleased by the pp-chain ǫ pp ≃ × X H erg g − s − towhich for a M = 1 M ⊙ the luminosity would be L/L ⊙ ≃ M/M ⊙ × ǫ pp ≃ . × X H . The mean observed luminosityof WD L/L ⊙ ≃ − allows X H × − , the upper limitfor X H in WD interiors. Stability considerations indeed ruleout that the luminosity of normal WD is generated by ther-monuclear reactions, which was first pointed out by Mestel(1952b,a). Nuclear burning could only be expected in nearlycold configurations near T = 0 by pycno-nuclear reactions.According to Kawaler (1988) sedimentation and diffu-sion of elements will bring hydrogen and helium from theinterior to surface and viceversa. Although this is a slowprocess, the initial content of hydrogen can be as high as10 − M ⊙ . It is worth noting that diffusion of hydrogen in-wards is inhibited by the electron degeneracy (see for in-stance Iben & MacDonald 1985). In brief, while the tail ofthe hydrogen distribution chemically diffuses inward as theWD cools down, actually it may reach a maximum depth,and with further cooling, begins to retreat outwards. Thisis because the increasing electron degeneracy halts the in-ward diffusion of hydrogen. Hence, surface hydrogen neverreaches very deep layers.We argue and try to demonstrate here that traces oflight elements (below the upper limits given above) left overby core nuclear burnings during the previous phases can stillbe present when the WD is cooling. This would be the analogof the situation in neutron stars, where traces of electrons,protons and even heavy nuclei are expected to exist in themedium of free neutrons (Shapiro & Teukolsky 1983). Weshow that hydrogen abundances in the range 10 − X H − may be present and cause ignition of pycno-nuclearreactions. Analogous considerations can be made for helium. The correct answer to the above question would be detailednumerical models of stellar interiors in which the abun-dances of all elements undergoing nuclear processing, con-vective mixing, diffusion and sedimentation (whenever ap-propriate) are followed in a great detail to even very low val-ues. In current stellar models, the abundances of the chemi-cal elements are calculated by solving complicated networksof differential equations governing the local creation / de-struction of the elemental abundances according to the ratesof the nuclear reactions that are involved, and the efficiencyof the diffusive/convective mixing.A widely adopted short-cut to the above complex inte-gration technique and quite heavy computational burden isto set equal to zero the abundances of elements when theyfall below some reasonably small values, unless otherwise re-quired. This is particularly true for those elements that ini-tially are very abundant, such as hydrogen and helium. Theyindeed are customarily set to zero when X H ≃ − andX He ≃ − . In the case of the central H- and He-burningsthis is also taken to mark the end of the evolutionary phase(see for instance Bertelli et al. 1994; Bressan et al. 2012,and references therein) when describing the physical ingre-dients and technical details of the Padova stellar evolution-ary code. Furthermore, requesting that all elements that are c (cid:13) , 000–000 E. Chiosi, C. Chiosi, P. Trevisan, L. Piovan, & M. Orio
Figure 6. Left Panel : energy rates per unit volume produced by the reaction C( H , γ ) N with hydrogen abundance X H = 10 − as a function of density and four different temperatures of the thermal branch. The nuclear rates are from Kitamura (2000).The dottedlines represent the energy without local density correction while the continuous lines represent the model that considers local densitycorrection. Right panel : the same as in the left panel but for reaction rates according to Yakovlev et al. (2006).The dotted lines representthe energy without local density correction while the continuous lines represent the model that considers local density correction. gradually transformed into heavier species (and therefore de-crease their abundance) are followed to extremely low valuesis hardly viable and time consuming when extended grids ofstellar models are considered. Therefore, proving or disprov-ing that traces of hydrogen and helium (at the abundancesthat will be used in the present study) can survive previousburnings and be present in WDs cannot be easily assessedwith current evolutionary codes to disposal.However, the elementary theory of thermo-nuclear re-actions suggests that elements at very low abundances cansurvive even the nuclear phase in which they are burned intoheavier species. In brief, looking at the case of hydrogen, thetypical lifetime of a proton against a proton in the pp-chainor against CNO nuclei in the CNO-cycle is expected to de-crease with increasing the temperature and to increase withdecreasing X H (see Iben 2013a,b). In other words, as theabundance of hydrogen decreases, it is more and more dif-ficult to destroy it further, i.e. the abundance of hydrogennever becomes zero. Similar considerations can be made forhelium (and any other element in general).We try here two simple experiments designed to esti-mate the abundances X H and X He left from the nuclear burn-ings along the whole evolutionary history from the zero agemain sequence to the beginning of the TP-AGB phase thatterminates with generation of CO WD (the TP-AGB phasecan be neglected because of its short lifetime compared tothat of previous phases). Hydrogen and helium expected inWDs have survived all previous nuclear burnings.To this aims we consider the evolutionary sequencesof the 3, 5 and 6 M ⊙ stars with solar-like composition[X=0.723, Y=0.260, Z=0.017] that are the progenitors ofmassive CO WDs e.g. the entries of Table 2. The sequencesare taken from the Padova library of stellar models byBertelli et al. (2009). For each star we know all relevantphysical quantities as a function of time both in the cen-tre (temperature, density, abundances of elements, size of the convective and H-exhausted core, etc.) and at the sur-face (total and partial luminosities, effective temperature,etc.).Since the central H- and He-burning (CNO and 3 α , re-spectively) in these stars are point-like sources at the centre,we may use simple energy conservation arguments and write ǫ i ( X i , T, ρ ) = L i M (59)where the index i stands for H- or He-burning, and L i are the partial luminosities (the total luminosity is L = L H + L He + L G , where L G is the contribution from gravi-tational contraction, usually negligible). This equation sim-plifies the complexity of energy production (nuclear burningand gravitational release in convective/radiative conditions,the latter depending on the stellar mass) to an ideal situ-ation in which a gram of matter produces the amount ofenergy radiated by the surface per unit mass and unit time.Using the above equation we implicitly neglect the contri-bution of the gravitational contraction/expansion to the lu-minosity (a reasonable approximation for most of the stellarlifetime). Furthermore, for the sake of simplicity, we con-sider only the nuclear burning occurring in the core andignore the nuclear burning in the shell. In the 3, 5, 6 M ⊙ stars, hydrogen burns via the CNO-cycle with a small con-tribution from the pp-chain, and helium burns via the 3 α process. Finally, in first approximation we consider nuclearburning only in radiative conditions, neglecting continuousrefueling by convection which may occur in stars of 3, 5 and6 M ⊙ .For the energy release by H- and He-burning weadopt analytical expressions from classical textbooks(Kippenhahn & Weigert 1990; Weiss et al. 2004): c (cid:13) , 000–000 xploring an Alternative Channel of Evolution Towards SNa Ia Explosion Figure 7.
We compare the rates of the pycno-nuclear reac-tions C( H , γ ) N (dashed red lines) and C( C , γ ) Mg(solid blue line) according to the rates by Yakovlev et al. (2006)and different values of X H as indicated. From top to bottomlog X H = −
18, -20 and -22. (i) pp-chain, ǫ pp = 2 . × f pp g pp ρX H T − . e ( − . T − . ) (60) g pp = 1 + 0 . T . + 0 . T . + 0 . T f pp = 1 + 0 . ρ . T − . (ii) CNO-cycle ǫ CNO = 8 . × g , X CNO X H ρT − / e − . /T / (61) g , = 1 + 0 . T / − . T / − . T (iii) and 3 αǫ α = 5 . × f α X He ρ T − e ( − . /T ) (62) f α = exp (2 . × − ρ / /T / )where T and T are the temperatures in units of 10 and 10 K, respectively, the functions g , and g α are the screen-ing factors, X H , X He , X CNO are the abundances by massof hydrogen, helium and CNO group, respectively; ǫ , and ρ are in cgs units, erg g − s − and g cm − , respectively; fi-nally, the above expressions refer to situations in which allintermediate steps are at equilibrium.(i) Analytical . An estimate of the hydrogen abundanceleft from the nuclear burning after the main sequence andthe helium burning phase can be given analytically in thefollowing way considering the relative weigh of pp-chain andCNO-cycle in the H-burning process. Let X H,r be the hy-drogen abundance at a certain time t . At proceeding nuclearburning over a time interval ∆ t , the fraction of consumedhydrogen is X H,c = X H,i − X H,r , where X H,i is the initialabundance of hydrogen. Therefore ∆ X H,c = − ∆ X H,r . The
Table 6.
Hydrogen abundances in the innermost regions at theend of the Main Sequence and core He-burning, and beginning ofthe TP-AGB phases according to two simple models.Mass Phase X H X H Analytical Numerical3 M ⊙ Hb 2 . × − . × − Heb 3 . × − . × − AGB 2 . × − . × − M ⊙ Hb 8 . × − . × − Heb 2 . × − . × − AGB 1 . × − . × − M ⊙ Hb 3 . × − . × − Heb 2 . × − . × − AGB 1 . × − . × − fraction X H,c is in turn governed by the equation∆ X H,c ρm p = ǫρ ∆ tQ (63)where ǫ is a function of X H,r . In particular for the case ofthe pp-chain, ǫ it is proportional to X H,r . Therefore we canwrite in differential form − dxx = A pp dt (64)where x stands for the current value of X H,r . Upon integra-tion we obtain x f = 1 x i + A pp ∆ t (65)where x i and x f are the values of X H,r at the beginning andat the end of the generic time step.In the case of the CNO-cycle ǫ is a linear function in X H,r . Therefore we have − dxx = A cno dt (66)that gives x f = x i e − A cno ∆ t (67)Taking the weighted mean value of the two channels andextending the integration over the whole time interval frothe zero age main sequence to the end of he-burning, weobtain h X i = A pp X ppH + A cno X cnoH A pp + A cno ≃ − − − (68)for a progenitor from 3 to 6 M ⊙ . The evolutionary history ofthe stars comes in via the central temperature and densityas functions of the time. The results for the central hydrogencontent along the evolutionary sequences are shown in Table6. (ii) Numerical . The above result can be refined mak-ing use of eqn. (59), considering that in principle the pp-chain and CNO-cycle can occur simultaneously, howeverwith significantly different efficiency, and taking into consid-eration the structure of the stars during their evolutionaryhistory. In principle if one or more nuclear burning regionsare present in a star, eqn. (59) implies X i Z M X j ǫ i,j dm = X i L i (69) c (cid:13) , 000–000 E. Chiosi, C. Chiosi, P. Trevisan, L. Piovan, & M. Orio
Figure 8. Left Panel : We plot the pycno-nuclear energy generation rates based on the Kitamura (2000) model for the reaction C( H , γ ) N and three different values of hydrogen (from X H = 10 − to X H = 10 − ) as shown by the blue continuous lines andcompare these with the luminosity of the WD (red dotted line) according to the models of Althaus & Benvenuto (1997, 1998) for a 1.2 M ⊙ WD. The released nuclear energy and WD luminosity are per unit volume in units of erg s − cm − . To do so, the luminosity of theWD is divided by the WD volume using the data provided by Althaus & Benvenuto (1997, 1998). The nuclear rates are calculated usingthe central values of temperature and luminosity of the WD model we have adopted. The green lines represent the energy without localdensity correction while the blue lines represent the model that considers local density correction. Right Panel : the same as in the leftpanel but for the nuclear energy rates based on the Yakovlev et al. (2006) and different values of X H from X H = 10 − to X H = 10 − . where i indicates the phase, and j the type of burning (pp,CNO, and 3 α as appropriate). If only central H-burning ispresent (main sequence), eqn. (69) reduces to ǫ H × M = L H ;when central and shell H-burnings are present it becomes ǫ H,core × M core + ǫ H,shell × M shell = L H,core + L H,shell = L H ;with more complicated nuclear stratifications and phases thegeneralization of eqn. (69) is obvious. The first case is sim-ple to treat and L H is the true luminosity generated by coreH-burning. The second case is more complicated and cannotbe solved analytically for obvious reasons. Fortunately, someapproximation are possible: first compared to the main se-quence lifetime it is short lived and can be neglected. To thisaim we have remove the contribution to the nuclear energygeneration by the the shell and suitably rescaled the lumi-nosity L H (this is possible because all details of the stellarmodels are to our disposal).In the case of central core H-burning, isolating the de-pendence on X H in eqns. (61) and (62), we can write ǫ pp + ǫ cno ≃ ǫ opp X H + ǫ ocno X H = L H,core /M (70)where the quantities with superscript o are of obvious defi-nition and are immediately known from eqns. (61) and (62),and L H,core is the contribution to the total luminosity bythe sole core H-burning. This quantity is known from thestellar models we are using. Eqn. (70) is a second order al-gebraic equation to be solved for X H as a function of thetemperature and density of the mass element (typically asmall volume at the center) along the evolutionary sequenceof a star. The results at the end of the core H-burning, coreHe-burning, and beginning of the TP-AGB phase are listedin Table 6.Similar reasoning can be followed to derive the helium abundance. We recast eqn. (70) as ǫ α = ǫ o α X He = L He,core /M (71)and solve it for X He . With this procedure we estimate thatthe abundance of helium in the core at the start of TP-AGBphase is about X He = 10 − in the three stellar models.No estimate is made of the amounts of central hydrogenand helium consumed during the TP-AGB phase because ofthe complexity of the stellar evolution during this phase.However, this is less of a problem because the TP-AGB isshort-lived compared to the previous evolutionary history sothat it can be neglected. X H and X He during the WDcooling sequence At the beginning of the WD cooling sequence the luminos-ity is sustained by CNO burning on the surface. We wantto assess whether part of the luminosity may still be due tominor nuclear burning in the interior (if it can occur at all),thus possibly further lowering the inner content of hydrogen.In order to follow the hydrogen and helium consumption bynuclear burnings during the early stages of WD cooling, weused the WD evolutionary sequences of Renedo et al. (2010)in which the surface burnings (in the H and He-shells) arefollowed in great detail so that they can be taken as a ref-erence models. Using the same procedure described above,we estimate that the central hydrogen and helium abun-dances in our test models of 3, 5 and 6 M ⊙ during thetime interval in which the surface shells are active, approx-imately ∼ × yr, decrease only by a modest amount(roughly less than a factor of 10), i.e. the new abundancesare X H = 10 − − − and X He = 10 − . c (cid:13) , 000–000 xploring an Alternative Channel of Evolution Towards SNa Ia Explosion Figure 9.
The Coulomb coupling parameter Γ ij = Z Z e a ij k B T asfunction of age for the two extreme values of the WD masses un-der consideration. The underlying nuclear reaction is H + C.The enhancement in the local density induced by the contami-nant hydrogen is included (see eqn. 48 and the solution R ′ =0 . R ). The blue lines show the 0 . M ⊙ WD with X H = 10 − ,whereas the red lines are the same but for the 1 . M ⊙ WD withX H = 10 − . All other combinations of mass and X H abundancesfall in between. The corresponding Coulomb coupling parameterwith no enhancement in the local density are obtained by scalingthe values on display by the factor 1/0.5178. To conclude, the experiment and arguments presented abovesuggest that traces of hydrogen and helium can be left overin the interiors of a WD at the end of the whole nuclear his-tory via the thermal channels of nuclear reactions. Howeverit is worth emphasizing that major drawbacks and uncer-tainties are present. In brief(i) There is a large difference between the results ob-tained with the two methods. We prefer to consider thosefrom the numerical approach to be more realistic becausethey are tightly related to detailed stellar models (tempera-tures, densities and luminosities).(ii) We have used the classical reactions rates for thepp-chain, CNO-cycle and 3 α group at the equilibrium. Thisprevents us from following the temporal history of the abun-dances of the intermediate elements and reactions. To high-light the issue, if some hydrogen survives the stage of core H-burning, we would expect that any trace of hydrogen shouldbe completely burnt during the helium core burning, as a re-sult of the reactions C + H → C and C + H → N. Even-tually, all N should be converted into Ne via α -captures.This sequence of events cannot be described correctly by ourCNO-cycle equilibrium rates. This is perhaps the point ofmajor uncertainty.(iii) Detailed evolutionary models would be the rightway of assessing whether traces of light elements could bepresent in the interiors of WDs at the beginning of theircooling sequence and, if so, whether there exists any trendand/or lower limit at varying the initial mass of the progen-itor star. Figure 10.
The nuclear energy rates R ij ( ρ, T ) = R pycij ( ρ ) +∆ R ij ( ρ, T ) of Yakovlev et al. (2006). The first term is the pycnocontribution (short dashed and solid red lines) and the secondone the thermal component (long dashed and dotted blue lines).Two values of the WD mass and two values of X H are displayed,i.e. 0.8 M ⊙ with X H = 10 − and 1.2 M ⊙ with X H = 10 − . Theenergy generation rates are in erg cm − s − . (iv) An alternative to calculating complete stellar mod-els could be the ideal case of an elementary cell of matterin which extended networks of nuclear reactions among anumber of elementary species are let occur as a function oftemperature, density and patter of initial chemical parame-ters. To simulate the centre of a real star, temperature anddensity as a function of time could be taken from detailedstellar models.Work is in progress along the lines of items (iii) and (iv).Despite the above limitations of the present approachand waiting for careful numerical investigations, we considerthe above estimates of X H and X He as free parameters withinthe ranges we derived above, and proceed to investigate theeffects they would induce on the evolution of WDs of differ-ent mass. The Kitamura (2000), Gasques et al. (2005) andYakovlev et al. (2006) formalisms (see also Beard 2010) wereoriginally tailored for reactions like C + C, C + O, O + O. However, they can be extended to reactions in-volving impurities of light elements, and to incorporate theenhancement in the local density. We have used the reactionrates of Kitamura (2000) to study the reactions H + Cand/or He + C. We will not examine here the similarreactions occurring with oxygen because of the higheratomic number Z . The relative S factors and Q values ofthe two reactions are listed in Table 5. The abundance ofcarbon in these test calculations is X C ≃
1. The resultsare more general than those of the Salpeter & van Horn(1969) rates because they take also into account the effectsof the temperature, extending from the thermal to the c (cid:13) , 000–000 E. Chiosi, C. Chiosi, P. Trevisan, L. Piovan, & M. Orio
Figure 11.
The pycno-nuclear energy rates based on theYakovlev et al. (2006) model for an abundance concentration ofhydrogen X H = 10 − (blue continuous line) compared to the lu-minosity of the WD (red dotted line) according to the models ofAlthaus & Benvenuto (1997, 1998) for a 1.2 M ⊙ WD. The dashedmagenta line represents the energy of the ions. The luminosity isin the same units as the energy generation rates (erg cm − s − ). pycno-nuclear regime. However, our results match thoseobtained with the Salpeter & van Horn (1969) rates in thelow temperatures and high densities regime. Results forthe C( H , γ ) N reaction with X H = 10 − are presentedin left panel of Fig. 6, in which the generated energy isplotted as a function of central density. The same procedurehas been applied to the Yakovlev et al. (2006) formalism.The results for the same reaction and hydrogen abundanceare shown in the right panel of Fig. 6. In both cases, fourdifferent temperatures have been considered.In both panels the thermal branches are visible only forthe 10 and 10 K temperatures. All curves merge togetherbeyond a certain value of the density that depends on thetemperature and source of the rates. The two groups of ratessignificantly differ both in the thermal and pycno-nuclearregime: the Kitamura (2000) rates are higher in the thermaland lower in the pycno-nuclear regime compared to thoseYakovlev et al. (2006). The rates are enhanced by the in-crease in the local density. This dependence is shown by theresults in Fig. 7 which displays the H + C and C + Creaction rates, limited to the case of Yakovlev et al. (2006),for three values of X H as indicated. These results can becompared to those of Fig. 3, scaled to the same energy unitsand multiplied by the Q -value. The energy generation rateexceeds the mean luminosity of a WD at relatively low den-sities. WDs of given masses follow cooling sequences along whichthe luminosity varies with time. Therefore we relax now thesimplifying assumption of a constant mean luminosity by us-ing the evolutionary sequences of WDs of different mass and compare them with the energy released by nuclear burning.The cooling sequence in the luminosity vs effective temper-ature plane is determined by the central density and meanmolecular weight of the electrons (i.e. the chemical composi-tion of the WD). Therefore the really meaningful comparisonis between the energy production of the WD governed by itstemperature and density (the central values) and its currentluminosity.
Taking the WD structure into account . In ouranalysis we need to compare the total luminosity with thetotal nuclear energy generation inside ( L ≡ R ǫdM ) alongthe cooling sequence. To this aim we need the internal struc-ture of the WD. The cooling sequence provides the age fromthe beginning of the WD phase, the luminosity, effectivetemperature, and the central temperature and density. Ateach stage we integrate the structure equations of a WD ofgiven mass, chemical composition, and central density in theso-called zero-temperature approximation. This provides thestratification of mass and density as functions of the radius.Owing to the high thermal conductivity of degenerate elec-trons (see below), the temperature tends to become moreand more uniform from the center to the surface at increas-ing age of the WD (lowering of the mean temperature bycooling). In order to evaluate the reaction rate as a functionof the position it is sufficient to assume that the temperaturethroughout the star is equal to its central value. This maynot be correct toward the surface, but the reaction rates arenearly zero over there because of the low density (the nu-clear energy production, if any takes place, is mostly limitedto the central regions). Therefore we assume that the rateof energy production along the radial distance is ǫ [ T c , ρ ( r )],and proceed to calculate the mean energy production perunit volume, h ǫ i , and unit mass, h ǫρ i . The luminosity perunit volume of a WD is given by LV = 4 πR σT eff V (72)where V is the total volume of the WD. This luminositymust be compared with h ǫ i = R R ǫ [ T c , ρ ( r )] × πr dr R R πr dr (73)in erg cm − s − .In addition to this we will make use of the energy gen-erated per unit mass and time given by h ǫρ i = R R ǫ [ T c , ρ ( r )] ρ − × πr dr R R πr dr (74)in erg g − s − .These quantities can be expressed by those evaluatedat the center [ ǫ ] c and [ ǫρ ] c by means of the relations h ǫ i =[ ǫ ] c ( V c /V ) and h ǫρ i = [ ǫ ] c × ( V c /V ), where V c is the volumeof a small sphere of radius R c around the center. The radius R c is found to range from 6 . × to 2 . × cm whenthe WD increases from 0.6 to 1.2 M ⊙ mass. This radius isabout a factor 1/20 smaller than the real radius of WDsof the same mass. This means that the nuclear energy ismainly generated in a small sphere around the center, thusjustifying the approximation h ǫ i ≃ ǫ c × ( V c /V ) and h ǫ/ρ i ≃ ( ǫ c /ρ c ) × ( V c /V ). c (cid:13) , 000–000 xploring an Alternative Channel of Evolution Towards SNa Ia Explosion Results . For the sake of illustration, let us considerthe evolutionary sequence of a CO-WD of 1.2 M ⊙ (the bor-der value above which the mean density is high enoughto enter the General Relativity domain) calculated byAlthaus & Benvenuto (1997, 1998). All necessary informa-tion about central temperature, central density and luminos-ity, etc is available. The composition of the WD is X C ≃ . O ≃ .
5. We are particularly interested in determin-ing when the energy rates given by Kitamura (2000) or byYakovlev et al. (2006), modified for the effects of contami-nants on the local densities, exceed the luminosity. Fig. 8shows the variation of the WD luminosity (the red thickdotted curve) as a function of time and the energy gener-ated by the reaction C( H , γ ) N for different abundancesX H and keeping X C constant (the thin dashed blue and andthe thin dotted green curves). The dotted green lines arefor nuclear rates that neglect the local density enhancementcaused by impurities, whereas the dashed blue lines are fornuclear rates that take this into account.There are a number of important points to note:(i) When the enhancement in local density is not consid-ered, the nuclear rates show a similar trend, they initiallydecrease, but at given WD age they become flat (see thegreen curves in both panels of Fig. 8). The little kick in theKitamura (2000) rates is a numerical artefact in his relationspassing from fluid to solid state. The initial decrease of thenuclear rate is the signature of the thermal regime and thatthe main source of energy is the internal energy of the ions,whereas the levelling of the rate is when the pycno-nuclearregime takes over.(ii) When the enhancement in local density is takeninto account, this trend is only typical of the Kitamura(2000) rates. This is partially because we used a linear in-terpolation to calculate the interionic distance in transitingfrom the low-density, thermal regime to the high-densitypycno-nuclear regime (see eqn. 4.2). The nuclear rates ofYakovlev et al. (2006) are nearly flat all over the age range.The pycno-nuclear regime dominates the rate from the verybeginning. This is partially due to the mass we have cho-sen in which the density is very high ( ρ c ≃ . × g cm − for the 1 . M ⊙ WD). For lower masses the previous trend isrecovered.(iii) For the Kitamura (2000) model the pycno-nuclearenergy exceeds the luminosity of the WD for X H = 10 − after 6 Gyr (see the left panel of Fig. 8). Assuming insteadthe Yakovlev et al. (2006) rates, we reach the critical situa-tion for X H = 10 − even before 1 Gyr (see the right panelof Fig. 8).To clarify the above issues first we examine the variationof the Coulomb coupling parameter Γ ij along the cooling se-quences of the WD models. Γ ij for the H + C reaction isshown in Fig.9. It is soon evident that the reaction occurs inthe third and fourth regimes of the group of five that havebeen discussed in Section 3.3 as long as the age is youngerthan 1 Gyr for the 1 . M ⊙ and 8 Gyr for the 0 . M ⊙ , respec-tively. More precisely, initially the nuclei are bound to thelattice sites, so that the reaction occurs between highly ther-mally excited nuclei, which oscillate with frequencies higherthan the plasma frequency and have energies greater thanthe zero point energy of the plasma; later they enter theso-called thermally enhanced pycno-nuclear regime but the melting temperature is not reached yet. Only WDs olderthan the above limits enter the pure pycno-nuclear regime.Second we look at the separate contributions to the to-tal rate by the thermally enhanced and pure pycno-nuclearregimes according to the Yakovlev et al. (2006) formalism.In eqn. (26), the total rate is made by the sum of two terms R ij ( ρ, T ) = R pycij ( ρ )+∆ R ij ( ρ, T ), where the first term is thepycno contribution and the second one the thermal compo-nent. The evaluation is made for two values of the WD massand two values of X H , i.e. 0.8 M ⊙ with X H = 10 and 1.2 M ⊙ with X H = 10 − . The results are shown in Fig. 10,where the blue long dashed and dotted curves are for thethermally enhanced component, and the red short dashedand solid curves are for the pure pycno component, respec-tively.As already anticipated, the contribution from the ther-mally enhanced channel is significant for the 0 . M ⊙ WD,it gradually decreases as the WD mass (density) increases,and is fully masked by the pycno-nuclear one for the 1.2 M ⊙ star and beyond. X H abundances and ages We now include all energy sources, i.e. the generation by anyof the five possible regimes for the pycno-nuclear reactions(including also the so-called thermally enhanced terms inthe early stages of WD cooling sequences) and the thermalenergy of the ions.
The proto-type WD of . M ⊙ . FollowingYakovlev et al. (2006) we consider only the C( H , γ ) Nreaction, activated by the traces of hydrogen left over bythe previous phases and still present in the WD. This is oneof the three starting reactions that ignite the CNO-cycle,which however cannot be completed, owing to the extremelylow abundances of the intermediate elements. This reactionhas a Q -value of 1.94 MeV compared to the 25.02 MeVwhen the cycle is completed, releasing a factor of 12.5less energy. The nuclear energy release, the ion internalenergy, and the comparison luminosity are all expressedper unit volume of the WD. The volume is calculatedfrom the luminosity and effective temperature of the WDalong its cooling sequence. All quantities are providedby Althaus & Benvenuto (1997, 1998) and Renedo et al.(2010).The temporal evolution of the WD luminosity (red dot-ted line), nuclear energy release (blue solid line) and thermalenergy of the ions (magenta dashed line) are shown in Fig.11. Since the ion internal energy is derived here from simpleexpressions, neglecting the effect of degenerate electrons andthe variation of the degrees of freedom in the crystal latticeat very low temperatures, we do not extend it to very highdensities or low temperatures, characteristic of very old ages.First of all, we notice that the abundance X H of the residualhydrogen cannot exceed 10 − , otherwise the nuclear energyrelease by the sole C( H , γ ) N reaction during the initialstages of WD cooling is comparable to or even exceeds thetotal luminosity, thus bringing the WD to a risky regime atwhich a nuclear runaway may start. See also (Renedo et al.2010) for the energy production by the surface CNO duringthe same phase. We will see below that the upper limit to X H to avoid early nuclear runaway changes with the WD mass, c (cid:13) , 000–000 E. Chiosi, C. Chiosi, P. Trevisan, L. Piovan, & M. Orio
Figure 12.
Energy rates per unit volume produced by hydrogen impurities with different abundances of X H as indicated for WD ofdifferent mass. The reaction on display is the C( H , γ ) N according to Yakovlev et al. (2006). The luminosity is in the same units asthe energy generation.
Table 7.
Logarithm of the age in yr at which the energy generation h ǫ/V i by the nuclear reaction C( H , γ ) N in erg cm − s − crossesthe WD luminosity in erg cm − s − for different M WD in M ⊙ and hydrogen abundances X H . Ages in italic corresponds to very earlyintersections of the two curves when the temperature is very high and neutrino cooling is very efficient. In these cases, the nuclear energyis most likely carried away and radiated by the star. The other ages in roman correspond to cases in which the energy produced bynuclear reactions could be trapped inside the WD because the energy production by nuclear reactions overwhelms the luminosity andneutrinos have become much less efficient in removing the energy. The luminosities and radii (volumes) of the WDs are from the coolingsequences of Althaus & Benvenuto (1997, 1998). log X H M WD -15 -16 -17 -18 -19 -20 -21 -22 -230.6 ; 9.8 ; > > ; 8.4 ; 9.2 9.81.0 ; 8.2 8.8 9.61.1 8.2 8.81.2 7.0 8.2 8.9 it increases at decreasing mass. Incidentally, this constraintmay be a way of estimating and calibrating the maximumcontent of residual hydrogen in a CO-WD.As cooling proceeds, the energy generation by nuclearreactions in the thermally-enhanced regime decreases andthe light contaminants do not have any effect until the purepycno-nuclear channel begins. In the case of the 1 . M ⊙ starwith the hydrogen abundance of X H = 10 − , the nuclearenergy production exceeds the luminosity at the age of about2 . × as shown in Fig. 11. X H greater than the abovevalue would lead the WD to intersect the luminosity curveduring the thermally enhanced phase, i.e. soon after (orshortly later) the formation of the WD itself (a possibil-ity that cannot be firmly excluded and is temporarily leftaside, see below). Exploring the solution space . Since the abundancesX H cannot be assessed a priori, it is safe to consider them asfree parameters whose values fall within a plausible rangebounded by (a) the maximum value above which the en-ergy production exceeds the WD luminosity during the earlystages of the thermally enhanced regime, (b) the lower valuebelow which the energy production equals the WD luminos- ity at ages older than the age of the Universe. This grid ofpossible solutions is shown in the various panels of Fig. 12and the corresponding ages of intersection are listed in Table7 as function of the WD mass and X H . It is worth notinghere that intersections occurring during the earliest stages ofthe cooling sequences are unlikely because cooling by neu-trinos is very efficient so that a release of nuclear energyper unit volume and time exceeding the luminosity per unitvolume does not trigger any potential instability but simplythe energy excess is carried out by radiation and neutrinos.In contrast, the intersections occurring at later stages whenthe pycno-nuclear regime is already in place and neutrinocooling is over are much more interesting because they canpotentially lead to unstable situations.As expected, at given M WD the intersection occurs laterwith decreasing X H . For values lower than those indicated,the intersection occur at ages much older than the currentage of the Universe (and thus of little interest here). We notethat the intersection occurs at lower and lower abundancesand younger ages with increasing WD mass. In fact, theluminosity per unit volume (evaluated at the typical ageof 10 yr) increases by a factor of ten from the 0.8 to the c (cid:13) , 000–000 xploring an Alternative Channel of Evolution Towards SNa Ia Explosion M ⊙ WD. Also the central and the mean density of aWD increase with the mass, and so does the efficiency ofthe pycno-nuclear reactions. Consequently, the values of X H for which the intersection may occur within the age of theUniverse systematically decrease with increasing WD mass:X H ≃ − for the 0 . M ⊙ star and X H ≃ − for the1 . M ⊙ .Finally, we note that the systematic decrease of the”permitted” values of X H at increasing WD mass is com-patible with the past thermal history of the progenitor star.In fact, the mean temperature in a low mass progenitor islower than the mean temperature in a more massive one,clearly having effects on nuclear burning. For this reason itis tempting to suggest that X H decreases with increasingWD mass.Concluding this section, we remind the reader that therates of energy generation as function of age have been cal-culated at constant X H and X C . In reality, X H (and X C )should decrease with time. Therefore a real WD should movealong a path in the panel corresponding to its mass grad-ually shifting to lines of decreasing X H . Consequently, theage at which the liberated energy would equal the luminos-ity cannot be exactly determined without the aid of realmodels of WD stars including the energy generated by thereactions between contaminants and the carbon (oxygen)nuclei. The ages reported in Table 7 are merely indicative ofthe expected trend. Adapting Hamlet’s famous soliloquy to our contest, thequestion is whether or not WDs containing traces of lightelements (hydrogen in the case we have considered) duringcooling produce enough nuclear energy via the H + C re-action to balance and exceed the WD luminosity. A fractionof the energy would be kept inside the WD creating thephysical conditions for C+C-burning, and likely initiating anuclear runaway followed by explosion.
The condition h ǫ i > L WD /V does not necessarily imply thatC-burning is started by the [H + C] reaction and, by pro-ceeding toward C-deflagration (or C-detonation), cause theexplosion of the WD. When the energy-luminosity conditionoccurs in late cooling stages, i.e. when neutrino productiondoes not occur, it only implies that part of the energy canbe trapped in the star. For the nuclear runaway a mecha-nism must provide the threshold energy for carbon-carbonignition.According to Nomoto (1982b,a) and Kitamura (2000),the condition to ignite the Carbon-Carbon reaction overa wide range of temperatures (10 − K) and densities(10 − g cm − ) in dense matter is R CC × Q CC /ρ =10 − W g − ≡ erg g − s − where R CC is the reaction rate(see Sect.3 above) and Q CC is the energy released per reac-tion ( Q CC = 13 .
931 MeV). This condition is hardly reachedvia the sole [H+C]-reaction in presence of very low abun-dances of hydrogen.
We suggest that in the physical situa-tion of the WD interiors, in which Γ > is soon reached by cooling and the nuclear burning occurs in regimes from ther-mally enhanced to pure pycno-nuclear, the threshold limitfor C+C-ignition is reached in a different way . Elementary burning cells . Let us start consideringa single H + C reaction. It occurs in a medium in whichthe carbon and oxygen nuclei are already in liquid, partiallycrystallized or even fully crystallized conditions, and there-fore have reduced their mobility. Consequently, the energydeposited by this reaction will preferentially be given to theneighbouring nuclei, and then shared with a wider environ-ment by conduction and radiation with opacities κ c and κ r ,respectively, and total opacity κ = κ c × κ r κ c + κ r . In the physi-cal conditions of WDs, thermal conduction dominates overradiative transport (see the results of detailed calculationsdescribed by Iben 2013a,b, and references therein), therefore κ ≃ κ c . For the purposes of this study we adopt the analyticfits of Iben (1968, 1975) of the numerical conductive opac-ities by Hubbard & Lampe (1969) and Canuto (1970). Theassociated mean free path of thermal conduction is λ = 1 κρ (75)Both κ c and λ vary significantly with the temperature anddensity and in turn with the evolutionary stage and mass ofthe WD. Looking at the case of the 0 . M ⊙ WD, the conduc-tive opacity in the centre varies from 4 × − cm g − at thebeginning ( T c ≃ × K , ρ c ≃ . × g cm − ) to about3 × − at the end ( T c ≃ × K , ρ c ≃ . × g cm − )of the cooling sequence. The mean free path of the energytransportation is of the order of λ ≃ . × − cm and 16.7cm, respectively. Furthermore, within a WD of given mass,the conductive opacity κ c increases from the centre to thesurface and the mean free path does the opposite. In Fig.13 we show the run of the central value of the conductiveopacity along the cooling sequences for all the WD modelsto our disposal (left panel) and limited to the case of the1 M ⊙ the variation of κ c from the centre to the surface forthree selected models characterized by their central temper-ature (right panel). However, since the nuclear energy by the[H+C]-reaction is expected to occur within a central sphereof rather small dimensions, in the analysis below we may ap-proximate the mean effective conductive opacity of a WD toits central value. In any case, the mean free path of conduc-tion is much longer than the interionic distance a ij whichvaries from 8 × − to 2 × − cm from a WD of 0 . M ⊙ to a WD of 1 . M ⊙ .In this scenario, it is plausible to conceive that most ofthe energy emitted by a single [H+C]-reaction is acquiredby the nuclei contained in a small volume, which is approxi-mated to a cube of edge λ centered on the H-nucleus. In thevolume λ there are N C nuclei of carbon and N O nuclei ofoxygen. This volume is named the elementary burning cell .The ratio ( λ/a ij ) indicates of the total number of nucleiinside the cell, N C + N O ≡ × N C = ( λ/a ij ) . Dependingon the local values of total opacity and density, we estimate N C ≃ − . The Q HC ≃ .
95 Mev of a single [H+C]-reaction is the source of energy for the nuclei of carbon andoxygen in the cell. The amount of energy to be shared perunit mass in the cell λ is Qρλ (76)where Q HC is expressed in erg and ρλ is the mass of the cell. c (cid:13) , 000–000 E. Chiosi, C. Chiosi, P. Trevisan, L. Piovan, & M. Orio
Table 8.
Columns (1) through (11) are: (1) M WD the WD mass in M ⊙ ; (2) log X H ; (3) log Age in yrs; (4) the central density ρ c in g cm − ; (5) the central temperature T c in K; (6) the reaction rate R H,C in n s − cm − ; (7) the energy generation rate ǫ in erg cm − s − ;(8) the energy generation rate ǫ λ ρ per gram in the elementary volume λ in erg g − s − ; (9) the energy difference ∆ E = WD luminosityper unit volume - nuclear energy generation per unit volume in erg cm − s − . The negative sign in brackets and in front of log | ∆ E | means that ∆ E is negative; (10) the time in sec required to process all [H+C]-reactions in a gram of matter; (11) the total opacity κ ≃ κ c in cm g − . The mean free path of thermal conduction is λ = 1 / ( ρκ ). The reaction and energy generation rates per unit mass and timeare R H,C /ρ and ǫ/ρ , respectively. The Q-value of the [H+C]-reaction is 3 . × − erg/reaction. The threshold value for C-burning ǫ CC = 10 erg g − s − . Finally, ǫ λ ρ is expected to be equal to ǫ CC . M WD log X H log Age log ρ c log T c log R H,C log ǫ log ǫ λ ρ log | ∆ E | log T ime log κ (cid:13) , 000–000 xploring an Alternative Channel of Evolution Towards SNa Ia Explosion Figure 13. Left Panel : The run of the central conductive opacity κ c (in cm g − ) as a function of the central temperature T c alongthe cooling sequence in WDs of different mass as indicated. Right Panel : the run of the conductive opacity from the center to thesurface in the 1.0 M ⊙ WD for three different values of the central temperature T c . The other masses have a similar trend. The coolingsequences and models of internal structure are from Althaus & Benvenuto (1997, 1998). The above ratio for typical values of density (10 g cm − )and mean free path (4 × − cm) turns to be in the range0.01 to 0.001. We multiply this quantity by the total rate R (in N cm − s − ) of the [H+C]-reactions to obtain the totalenergy to be shared. This energy is mostly used to heat thecarbon and oxygen nuclei in the elementary cell.The above limit for C-ignition of about 10 erg g − s − has to be applied to the quantity of carbon in the elementaryburning cell (the dimensions of which already take conduc-tion into account).In this model in which the WD medium is populatedby many elementary cells around the hydrogen nuclei, thecondition for C-ignition must be suitably scaled and appliedto the environment of the cells. We define the quantities ǫ λ = ǫ × cm λ (77) ǫ g,λ = ǫ g × cm λ = ǫ λ ρ (78)where ǫ is the energy per cm , and ǫ λ the energy per cm inthe volume λ , and ǫ g,λ the energy per gram in the same vol-ume. We show that the energy deposited by a single [H+C]-reaction, when trapped into the elementary cell, is sufficientto heat and finally ignite the C-nuclei inside the cell (see theentries in Table 8).Because of the above considerations, along the evo-lutionary sequence of a WD of assigned mass we calcu-late the reaction rate and associated energy productionrate of the [H+C]-reaction both per unit volume and unitmass (expressed in N cm − s − , erg cm − s − , N g − s − anderg g − s − , as appropriate) and also the quantity ǫ λ /ρ (theenergy generated in the elementary volume λ sampled bythe mean free path of conductive energy transport). We re-mind the reader that the above reaction rate depends alsoon X H (here considered as a free parameter). Criterion for initiating a nuclear runaway . Given these premises, along the cooling sequence of a WD of agiven mass we identify the time at which the energy inputby the [H+C]-reaction to the elementary cell, [ ǫ λ /ρ ] falls be-low the threshold value for C-ignition ǫ CC = 100 erg g − s − .Since we are moving along a cooling sequence from high tolow values of T c (the density remains nearly constant), dur-ing all previous stages, the energy ǫ λ /ρ is always above thethreshold value. Along the same cooling sequence we searchthe time at which the condition h ǫ i > L WD /V is verified,thus initiating the energy trapping. The two conditions mustbe simultaneously satisfied to start C-burning in a situationof energy trapping .The results of these calculations are summarized in Ta-ble 8 which for each value of X H lists a few key quantitiesat the stage at which the nuclear energy generation by the H + C reaction satisfies the conditions h ǫ i > L WD /V and ǫ λ /ρ erg g − s − . The physical quantities listed in Ta-ble 8 are: (1) the WD mass M WD in solar units; (2) thehydrogen abundance log X H ; (3) the logarithm of the agein years at which for the first time the elementary cell doesnot reach the conditions for C+C-ignition; (4)-(5) the corre-sponding central density and temperature ρ c in g cm − and T c in K ; (6) the reaction rate per unit volume and time R H,C in cm − s − ; (7) the energy generation rate per unit volumeand time ǫ in erg cm − s − ; (8) the energy generation rateper unit mass and time within the elementary cell of volume λ ǫ λ ρ in erg s − g − ; (9) the difference ∆ E = h ǫ i − L WD /V in erg cm − s − . To display ∆ E on a logarithmic scale, wetake the absolute value adding a negative (positive) signwhen the difference is negative (positive). The positive signis for the nuclear energy generation exceeding the luminos-ity; (10) logarithm of the time in s required to process all[H+C]-reactions in a gram of matter; and finally (11) thetotal opacity κ ≃ κ c in cm g − .At this stage of the analysis, we use three parameters:the abundance X H , the age ( τ ) at which the nuclear energygeneration rate per unit volume exceeds the WD luminosity c (cid:13)000
Columns (1) through (11) are: (1) M WD the WD mass in M ⊙ ; (2) log X H ; (3) log Age in yrs; (4) the central density ρ c in g cm − ; (5) the central temperature T c in K; (6) the reaction rate R H,C in n s − cm − ; (7) the energy generation rate ǫ in erg cm − s − ;(8) the energy generation rate ǫ λ ρ per gram in the elementary volume λ in erg g − s − ; (9) the energy difference ∆ E = WD luminosityper unit volume - nuclear energy generation per unit volume in erg cm − s − . The negative sign in brackets and in front of log | ∆ E | means that ∆ E is negative; (10) the time in sec required to process all [H+C]-reactions in a gram of matter; (11) the total opacity κ ≃ κ c in cm g − . The mean free path of thermal conduction is λ = 1 / ( ρκ ). The reaction and energy generation rates per unit mass and timeare R H,C /ρ and ǫ/ρ , respectively. The Q-value of the [H+C]-reaction is 3 . × − erg/reaction. The threshold value for C-burning ǫ CC = 10 erg g − s − . Finally, ǫ λ ρ is expected to be equal to ǫ CC . M WD log X H log Age log ρ c log T c log R H,C log ǫ log ǫ λ ρ log | ∆ E | log T ime log κ (cid:13) , 000–000 xploring an Alternative Channel of Evolution Towards SNa Ia Explosion Figure 13. Left Panel : The run of the central conductive opacity κ c (in cm g − ) as a function of the central temperature T c alongthe cooling sequence in WDs of different mass as indicated. Right Panel : the run of the conductive opacity from the center to thesurface in the 1.0 M ⊙ WD for three different values of the central temperature T c . The other masses have a similar trend. The coolingsequences and models of internal structure are from Althaus & Benvenuto (1997, 1998). The above ratio for typical values of density (10 g cm − )and mean free path (4 × − cm) turns to be in the range0.01 to 0.001. We multiply this quantity by the total rate R (in N cm − s − ) of the [H+C]-reactions to obtain the totalenergy to be shared. This energy is mostly used to heat thecarbon and oxygen nuclei in the elementary cell.The above limit for C-ignition of about 10 erg g − s − has to be applied to the quantity of carbon in the elementaryburning cell (the dimensions of which already take conduc-tion into account).In this model in which the WD medium is populatedby many elementary cells around the hydrogen nuclei, thecondition for C-ignition must be suitably scaled and appliedto the environment of the cells. We define the quantities ǫ λ = ǫ × cm λ (77) ǫ g,λ = ǫ g × cm λ = ǫ λ ρ (78)where ǫ is the energy per cm , and ǫ λ the energy per cm inthe volume λ , and ǫ g,λ the energy per gram in the same vol-ume. We show that the energy deposited by a single [H+C]-reaction, when trapped into the elementary cell, is sufficientto heat and finally ignite the C-nuclei inside the cell (see theentries in Table 8).Because of the above considerations, along the evo-lutionary sequence of a WD of assigned mass we calcu-late the reaction rate and associated energy productionrate of the [H+C]-reaction both per unit volume and unitmass (expressed in N cm − s − , erg cm − s − , N g − s − anderg g − s − , as appropriate) and also the quantity ǫ λ /ρ (theenergy generated in the elementary volume λ sampled bythe mean free path of conductive energy transport). We re-mind the reader that the above reaction rate depends alsoon X H (here considered as a free parameter). Criterion for initiating a nuclear runaway . Given these premises, along the cooling sequence of a WD of agiven mass we identify the time at which the energy inputby the [H+C]-reaction to the elementary cell, [ ǫ λ /ρ ] falls be-low the threshold value for C-ignition ǫ CC = 100 erg g − s − .Since we are moving along a cooling sequence from high tolow values of T c (the density remains nearly constant), dur-ing all previous stages, the energy ǫ λ /ρ is always above thethreshold value. Along the same cooling sequence we searchthe time at which the condition h ǫ i > L WD /V is verified,thus initiating the energy trapping. The two conditions mustbe simultaneously satisfied to start C-burning in a situationof energy trapping .The results of these calculations are summarized in Ta-ble 8 which for each value of X H lists a few key quantitiesat the stage at which the nuclear energy generation by the H + C reaction satisfies the conditions h ǫ i > L WD /V and ǫ λ /ρ erg g − s − . The physical quantities listed in Ta-ble 8 are: (1) the WD mass M WD in solar units; (2) thehydrogen abundance log X H ; (3) the logarithm of the agein years at which for the first time the elementary cell doesnot reach the conditions for C+C-ignition; (4)-(5) the corre-sponding central density and temperature ρ c in g cm − and T c in K ; (6) the reaction rate per unit volume and time R H,C in cm − s − ; (7) the energy generation rate per unit volumeand time ǫ in erg cm − s − ; (8) the energy generation rateper unit mass and time within the elementary cell of volume λ ǫ λ ρ in erg s − g − ; (9) the difference ∆ E = h ǫ i − L WD /V in erg cm − s − . To display ∆ E on a logarithmic scale, wetake the absolute value adding a negative (positive) signwhen the difference is negative (positive). The positive signis for the nuclear energy generation exceeding the luminos-ity; (10) logarithm of the time in s required to process all[H+C]-reactions in a gram of matter; and finally (11) thetotal opacity κ ≃ κ c in cm g − .At this stage of the analysis, we use three parameters:the abundance X H , the age ( τ ) at which the nuclear energygeneration rate per unit volume exceeds the WD luminosity c (cid:13)000 , 000–000 E. Chiosi, C. Chiosi, P. Trevisan, L. Piovan, & M. Orio per unit volume (see the intersections shown in Fig 12 andthe entries of Table 7), and the age τ at which ǫ λ /ρ fallsbelow ǫ CC in the elemental burning cell for the first time.The abundance X H can be further constrained by imposingthat the nuclear rate per unit volume during the first coolingstages remains smaller than the WD luminosity per unitvolume. This avoids very early ignition, which is unlikely forthe majority of WDs since we do observe most of them to liveand cool for a long time. This means that for each WD massthere is an upper limit of X H which decreases with increasingWD mass as already suggested by the intersections in Fig.12. An explosion may only occur with τ τ , as illustratedin Fig. 14 where the WD luminosity per unit volume, theenergy per unit volume produced by the [H+C]-reaction fortwo reasonable values of X H , the energy ǫ λ /ρ for the samevalues of X H , and finally the threshold value ǫ CC , all ofthem as function of time and on the same logarithmic scale.Looking at the panels of Fig. 14, the condition can be met byWDs with mass 0.9, 1.0, 1.1 and 1.2 M ⊙ whereas it is missedby WDs with mass smaller that 0.8 M ⊙ . In the latter stars,the energy production falls below ǫ CC at ages τ << τ . Thecondition could be met, however, by a 0.85 M ⊙ WD, andfinally the 0.8 M ⊙ WD is border line.
Global estimates of energy production . In orderto better highlight the above numerical results, we presentsome analytical estimates of (i) the total energy released bythe [H+C]-reactions, (ii) the fraction of this energy releasedinside an elementary cell, to assess whether it is sufficient toincrease the temperature of C-nuclei contained in the cell tothe threshold value for C-burning, and (iii) the total amountof energy produced by C-burning in the whole WD:(i) [H+C]-burning. In a WD there are X H × M WD × M ⊙ (the M WD mass is in solar units) grams of hydrogen orequivalently a total number of hydrogen nuclei N H = X H × M WD × M ⊙ A H × m u approximating M WD = 1 M ⊙ , M ⊙ = 1 . × g andusing A H = 1 we obtain N H = 1 . × × X H Adopting for X H a typical value of 10 − and considering theenergy generated by each [H+C]-reaction, i.e. 3 . × − erg, we obtain a total energy release of E [ H + C ] = 10 erg.For a typical WD volume of ≃ . × cm the nuclearenergy per unit volume is about 2 × , much higher thanthe WD luminosity per unit volume ≃ × .(ii) [C+C]-ignition. In order to verify whether the en-ergy deposited by the [H+C]-reactions in the elementarycell may increase the temperature of the latter to that ofC-ignition ( ≃ . − . × K), we start from Nomoto(1982b,a); Kitamura (2000) condition for C-ignition, the cellacquires the energy ǫ λ ρ × ( ρ h R ij i× λ ). This energy goes intothe total energy of the particles in the elemental cell, i.e. k B T × N V , where N V is the total number of particles inthe cell, i.e. the equation ǫ λ ρ × (cid:18) ρ h R ij i × λ (cid:19) = 32 k B T N V (79)is verified. According to its definition, ǫ λ /ρ ≡ ǫ CC ≃ erg g − s − and h R ij i = h ǫ ij i /Q HC . In addition, eachelementary cell contains only one nucleus of hydrogen, soin a cell the [H+C]-reaction occurs only once and thereforethe cell acquires only this specific amount of energy. Conse-quently, T = 23 ǫ CC × (cid:18) A C × m u k B × λ (cid:19) × (cid:18) h R ij i × λ (cid:19) × η (80)where N V has been replaced by ( ρ × λ ) / ( A C × m u ), A C =12, and all other symbols have their usual meaning. For typ-ical values of λ ≃ − − − and h R ij i ≃ − , weestimate a temperature in the range 10 − K. In theabove equation, we have also introduced the η parameterto take into account that part of the energy may escapefrom the cell. While the lower limit temperature is belowthe threshold for C-burning, the temperature at upper limitwould require η ≃ − . This means that a small adjust-ment of the parameters and/or the effective energy acquiredby the cell would yield the right temperature. The numericalcalculations show that the ignition temperature is reachedall cases we have considered.(iii) [C+C]-burning. Once C-burning is ignited in theelementary cells, how much energy is released by burning allthe carbon nuclei contained in the cells? The C+C reactionhas a Q CC -value of 13.93 Mev (2 . × − erg). The numberof C pairs in a cell is about N C / − s − ) at thetemperature of 6 × K and density of 10 g cm − . Therate is about 2 . × cm − s − and the energy generationrate is 4 . × erg g − s − . With the above rates, all carbonnuclei in a cell are destroyed in a tiny fraction of a second.Since the total number of burning cells is equal to thetotal number of hydrogen nuclei and the total number of[C+C]-reactions per cell is N C /
2, the total energy release E C + C is E C + C = 2 . × − × M ⊙ × M WD × N × X H × N C
2= 1 . × × M WD × X H × N C (81)where M WD is in solar units, and N = 6 . × theAvogadro number. For X H = 10 − , N C ≃ − ,and M WD = 1 the above equation yields E C + C ≃ − erg. This energy is comparable to or larger than thetotal gravitational energy of a WD, varying betwen | Ω | =5 . × erg for the 0 . × M ⊙ to | Ω | = 5 . × erg forthe 1 . × M ⊙ . Once started, [C+C]-burning will likely causea thermal runaway followed by the explosion of the star. The fuse for C-ignition . Since in any elemental vol-ume, thanks to the energy released by the [H+C]-reactionthe condition for local, mild C-ignition is met, we name thistwo steps process the ” fuse for C-ignition ”. Most likely, oncethe fuse is activated, the additional release of energy by C-burning makes it even stronger, propagating it to neighbour-ing regions, and activating complete C-burning that gradu-ally moves into the thermally driven regime. c (cid:13) , 000–000 xploring an Alternative Channel of Evolution Towards SNa Ia Explosion Figure 14.
The WD luminosity per unit volume (thick red solid line), the energy per unit volume produced by the [H+C]-reactionfor two reasonable values of X H (the thin, short-dashed magenta lines) that change with the WD mass, the energy ǫ λ /ρ acquired by atypical burning cell for the same values of X H (the thin, long-dashed green lines), and finally the threshold value ǫ CC (the horizontal,dotted black line), all of which are plotted as a function of the logarithm of the age in years. The various energies are plotted on thesame logarithmic scale. From left to right, WDs with different masses are displayed, i.e. 0.6, 0.8, 0.9, 1.0, 1.1, and 1.2 M ⊙ respectively. Figure 15.
The run of the energy ǫ λ /ρ in erg g − s − produced inside an elementary cell by the [H+C]-reaction as a function of radialdistance from the centre. The distance from the centre is described by the density decreasing from the center toward the surface. Theenergy production is presented for two values of the central temperature T c (assumed also to be the temperature throughout the WD): T crit the value at the stage when ǫ λ /ρ becomes for the first time ǫ CC = 10 erg g − s − , and a lower value T c = 10 K for comparison.The dotted lines show the run of the conductive opacity for two values of T c as indicated. All the quantities are plotted on the samelogarithmic scale. Two intersections are possible for the curve labelled T crit with the horizontal line corresponding to log ǫ CC = 2 in thecase of the 1.0 and 1.1 M ⊙ WDs. The first one is at the center and the second one occurs at log ρ ≃ M ⊙ stars, respectively. Off-center C-ignition is possible. Only the intersection at the center occurs for the 1.2 M ⊙ . No off-center C-ignitionis expected.c (cid:13)000
The run of the energy ǫ λ /ρ in erg g − s − produced inside an elementary cell by the [H+C]-reaction as a function of radialdistance from the centre. The distance from the centre is described by the density decreasing from the center toward the surface. Theenergy production is presented for two values of the central temperature T c (assumed also to be the temperature throughout the WD): T crit the value at the stage when ǫ λ /ρ becomes for the first time ǫ CC = 10 erg g − s − , and a lower value T c = 10 K for comparison.The dotted lines show the run of the conductive opacity for two values of T c as indicated. All the quantities are plotted on the samelogarithmic scale. Two intersections are possible for the curve labelled T crit with the horizontal line corresponding to log ǫ CC = 2 in thecase of the 1.0 and 1.1 M ⊙ WDs. The first one is at the center and the second one occurs at log ρ ≃ M ⊙ stars, respectively. Off-center C-ignition is possible. Only the intersection at the center occurs for the 1.2 M ⊙ . No off-center C-ignitionis expected.c (cid:13)000 , 000–000 E. Chiosi, C. Chiosi, P. Trevisan, L. Piovan, & M. Orio
Off-center C-ignition . The analysis we have made sofar relies on the model of elementary cells, based on thecentral values of conductive opacity, temperature, and den-sity. We have already shown that the conductive opacity andmean free path of thermal transport vary along the radius ofa WD (see the right panel of Fig. 13). Therefore it is worthinvestigating if the energy condition for C-ignition in the el-ementary cells can be met also in more external regions (off-center ignition). To this aim we show in Fig. 15 the energyproduction by the [H+C]-reaction as a function of radial dis-tance measured here by the value of local density ρ ( r ). Theenergy is evaluated adopting the temperature of the modelslisted in Table 8, shortly indicated here by T crit , and also thevalue of 10 for comparison, and for X H the value indicatedby the results of Table 8 for each WD mass. This energyproduction is indicated in Fig. 15 by the heavy solid red linefor T crit and the dashed blue line for T = 10 K. Togetherwith this we plot the conductive opacity as a function of ρ ( r )and the two temperatures in question. Finally, we show thethreshold energy for C-ignition (horizontal line). All quan-tities are shown on the same logarithmic scale. The threepanels correspond to three values of the WD mass as indi-cated. The intersection of the threshold value ǫ CC and the T crit curve is possible for the 1.0 and 1.1 M ⊙ WDs whereasit is missing for the 1.2 M ⊙ . The intersection occurs at thecentre (situations that we have already considered) and atsome intermediate region ( ρ ≃ g cm − ). Off-center igni-tion is therefore possible. To summarize, the main results of our analysis are:(i) The above reaction always occurs in physical con-ditions with Γ ij >
1, transiting from the third (thermally-enhanced) to the fifth regime (pure pycno) of burning (seeSection 3.3). This is well evident looking at the variationof the rate of energy generation per unit time and volumeby the [H+C]-reaction as a function of the age (along thecooling sequence). The energy generation rate is accordingto Yakovlev et al. (2006), eqn. (26), i.e. made of two terms R pycij ( ρ ) + ∆ ij ( ρ, T ) whose meaning is straightforward. Athigh temperatures (early stages of the cooling sequence) thesecond term dominates and the rate steadily decreases, how-ever past a certain temperature, the first term drives therate and, since the density in a WD is essentially fixed bythe mass and remains nearly constant as long as the massdoes not change, the rate levels off. We may picture thisby saying that the density driven nuclear regime acts as awedge to stabilize the otherwise ever decreasing temperaturedependent rates.(ii) In low mass WDs, the third regime is initially moreefficient than the fifth regime and the latter overwhelms theformer only past a certain age, whereas in high mass WDsthe fifth regime always prevails. This trend is due to theincrease of central and mean density with the WD mass.(iii) Depending on X H and M WD , the sum of nuclearenergy and ion internal energy can equal the luminosity atany age along the cooling sequence.(iv) Massive WDs are more sensitive to the instabilitythreshold that can be reached even with very small hydrogenabundances. Therefore the reaction H+ C is easily ignitedand a nuclear runaway can be started as soon as massive WDs are born. For a typical value of X H ≃ − , the abovesituation should occur for M WD > .
0. Although WDs ofthis mass are rare, the possibility cannot be discarded andit may be possible to identify such progenitors.(v) WDs of lower mass, about 0.9 M ⊙ , are more stableand encounter the threshold condition only after a certainamount of time has elapsed since their formation (in thepresent analysis about 1 Gyr). WDs of 0.8 M ⊙ are at theborderline, in the sense that the two conditions could be metfor − < X H −
13. WDs of about 0.85 M ⊙ could be thetransition stars. Finally WDs of even lower mass cool downto lw temperature without reaching the threshold energy forC-ignition.(vi) We suggest that a future investigation should ad-dress the possible correlation between the WD mass (and itsprogenitor) and X H (or X He ), which translates into an ad-ditional dependence of the explosion time on the WD mass.(vii) Finally, the onset of nuclear runaway should be aconsequence of the positive gravo-thermal specific heat ofa mixture of nuclei and electrons whose equation of stateis basically driven by the highly degenerate electrons, seethe discussion by Kippenhahn & Weigert (1990) and Mestel(1952b,a).To conclude, there is an ample range of possibilitiesfor WDs contaminated by the presence of traces of lightelements in their interiors to reach the critical stage at whichnuclear burning is activated, followed by nuclear runawayand consequent disruption of the star.This semi-analytical, orders-of-magnitude analysis doesnot yield more information. Only detailed numerical modelscan give the correct answer to the Hamlet question. In this paper we investigated the possible effects of impu-rities on the pycno-nuclear reaction rates in CO-WDs. Wereviewed the present-day pycno-nuclear reaction rates, high-lighting the peculiarities of each model and the use that hasbeen done until now. We introduced two important modifi-cations in the existing expressions:i) We extended the Salpeter & van Horn (1969) andShapiro & Teukolsky (1983) reaction rate, to calculate thevariation of the coulombian potential induced by the pres-ence of a lighter ion of hydrogen or helium in an arbitrarynode of the C-O ion lattice. Thus we extended the origi-nal formulation for pycno-nuclear reaction rates conceivedfor OCPs to BIMs, including the reactions H + C and/or He + C.ii) We evaluated the displacement of nearby nuclei pro-duced by the presence of impurities of different charge andestimated the change in local density, and applied this tothe MHO, Kitamura (2000), and Yakovlev et al. (2006) ex-pressions.Using the above revision of the theoretical rates, wemade a preliminary analysis using the MHO formalism tostudy the pycno-nuclear reactions between light nuclei (hy-drogen or helium) present in extremely low abundances andheavier nuclei like carbon and oxygen, the basic constituentsof WDs.Encouraged by the results obtained with the MHOrates, we used the more sophisticated descriptions of nuclear c (cid:13) , 000–000 xploring an Alternative Channel of Evolution Towards SNa Ia Explosion reactions in high density environments by Kitamura (2000)and Yakovlev et al. (2006) to explore the effects of impuri-ties of light elements in triggering the above reactions.The main results of the present study suggest that:- The presence of hydrogen even in extremely low concen-trations (from 10 − to 10 − ) can raise the pycno-nuclearreaction rates in density intervals from 10 to 10 g cm − .The same is true for helium at somewhat higher thresholddensities.- In the case of hydrogen, the above density interval cor-responds to WD masses from ≃ .
85 to 1 . M ⊙ , well belowthe known limit of the Chandrasekhar mass.- In WDs in this mass range, the energy released bypycno-nuclear reactions like H + C may trigger the ig-nition of CC-burning in a two steps process that we havenamed the fuse of C-ignition . The age at which thisis expected to occur depends on the WD mass and abun-dance of residual hydrogen. The fuse-induced C-ignition islikely followed by thermal runaway according to the classicalmechanisms.- Even WDs with masses as low as 0.85 M ⊙ may experi-ence nuclear runaway.Our results could in principle radically change not onlythe current understanding of the structure and evolution ofWDs but also imply that single WDs may be progenitors oftype Ia SNe. We may have discovered an alternative chan-nel to SNa Ia explosions. In this way we may be able (i) toexplain the star formation rate dependence of the SNa Iarate (e.g., Mannucci et al. 2006); (ii) to provide some cluesto interpreting the observational data on the ejected massdistribution of type Ia SNe showing a significant rate of non-Chandrasekhar-mass progenitors of mass as low as 0 . M ⊙ (Scalzo et al. 2014); and (iii) to account for the SNe ex-ploding inside Planetary Nebulae (shortly named SNIPs) inalternative to the core-degenerate scenario in which a WDmerges with the hot core of an AGB star on a time interval × yr since the WD formation (see Tsebrenko & Soker2014a,b, for more details ). With our models, a single CO-WD may reach the explosion stage soon after the forma-tion if sufficiently massive ( > . M ⊙ ) and sufficiently richin residual hydrogen (X H ≃ − − − ). The expectedtime delay after formation can be as low as about a few tenof thousand years.Therefore, before proceeding further it is mandatory toremind the reader of the limitations of the present approach.- Prior to any other consideration, it is worth recallingthat the success of the proposed model for the evolution ofCO WDs relies on the existence of traces of light elementslike hydrogen and helium (the former in particular) thatsurvived previous nuclear burnings. Complete stellar mod-els from the main sequence to the WD stages in which theabundances of elements are followed throughout the variousnuclear burnings to the values used in this study (i.e. X H in the range 10 − to 10 − ) are not presently available. Tocope with this, we have presented some plausible argumentsto sustain this possibility that eventually has been adoptedas a working hypothesis. Null abundances for the light el-ements in WD interiors cannot be firmly excluded. Stellarmodels checking this major issue are mandatory.- Even if our computations rely on state-of-the-art WD models, the correct approach would be to calculate and fol-low in time stellar models responding to the new sourceof energy in the course of evolution. The present models,although acceptable for very low nuclear rates, fail to rep-resent the real physical structure of a WD in presence oflarge energy production. The WD structure may be deeplyaltered and follow a different evolutionary history that is noteasy to foresee at present.- Our calculations do not take into account yet the ad-ditional energy release due to elements stratification, solidstate transition (latent heat) and gravitational contraction.- Only complete, self-consistent models would allow us tocorrectly determine the amount of energy generated by nu-clear reactions, and to deal with the energy transport prob-lem, rigorously comparing production versus transport ofenergy.Future work should be the computation of a self-consistent model able to respond to changing physical con-ditions, indicating the exact age at which the under-barrierreactions become important and the structural response ofa WD to the novel energy input. After the WD cools to thetemperature for the activation of the under-barrier channelin the liquid/solid phase, we foresee three possible scenar-ios: (a) reheating with consequent increase of the coolinglifetime; (b) rejuvenation to another type of object; (c) ex-plosion as type Ia SNa (or a different type?).We urge reconsidering the whole subject of nuclear reac-tions in these extreme conditions, in particular in presenceof impurities that could deeply change our current under-standing of the energy sources in WDs. If the results of our exploratory project are confirmedby further investigation, important implications for the cur-rently accepted scenario for type Ia SNe will follow. The bi-nary origin of type Ia SNa explosion would be no longerstrictly necessary. Isolated WDs with masses well below theChandrasekhar limit may reach the threshold for pycno-nuclear burning and consequent SNa explosion due to thesurvival of traces of light elements. These impurities mayremain inactive for long periods of time and be activatedonly when the WDs reach the liquid-solid regime. Owing tothe large range of WD masses that could be affected by thepresence of impurities and undergo thermal runaway andconsequent SNa explosion, the nature of standard candle sofar attributed to type Ia SNe may not be true. Because of thefar reaching implications, the whole subject deserves carefulfuture investigation.
ACKNOWLEDGEMENTS
We would like to deeply thank Dr. Maurizio Salaris for hisfriendship, for the many illuminating discussions, and forproviding us his cooling sequences and WD models. We liketo thank Drs. S. Ichimaru and H. Kitamura for patientlyreplying to the numberless emails sent by PT and the manyvery useful explanations and comments. We acknowledge thecritical discussion with Dr. K. Shen and finally, the veryhelpful comments of the unknown referee. c (cid:13) , 000–000 E. Chiosi, C. Chiosi, P. Trevisan, L. Piovan, & M. Orio
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