A Spallation Model for the Titanium-rich Supernova Remnant Cassiopeia A
aa r X i v : . [ a s t r o - ph . H E ] S e p A Spallation Model for the Titanium-rich Supernova Remnant Cassiopeia A
Rachid Ouyed, Denis Leahy, Amir Ouyed, and Prashanth Jaikumar Department of Physics and Astronomy, University of Calgary,2500 University Drive NW, Calgary, Alberta T2N 1N4 Canada Department of Physics and Astronomy, California State University Long Beach,1250 Bellflower Blvd., Long Beach, CA 90840 USA (Dated: August 14, 2018)Titanium-rich subluminous supernovae are rare and challenge current SN nucleosynthesis models.We present a model in which ejecta from a standard Supernova is impacted by a second explosionof the neutron star (a Quark-nova), resulting in spallation reactions that lead to Ni destructionand Ti creation under the right conditions. Basic calculations of the spallation products showsthat a delay between the two explosions of ∼ Ti inCas A and explains its low luminosity as a result of the destruction of Ni. Our results could haveimportant implications for lightcurves of subluminous as well as superluminous supernovae.
PACS numbers: 23.23.+x, 56.65.Dy
I. INTRODUCTION
Cas A is a young galactic supernova remnant formedin the aftermath of a core-collapse explosion of a mas-sive star. Tentative classification as TypeIIb (based onthe detection of weak Helium lines in addition to theHydrogen line) has been strengthened by infra-red stud-ies of the scattered light echo [1]. Cas A is intensivelystudied as a prototype that can unveil the pristine com-position of nucleosynthetic yields, thereby constrainingaspects of the supernova mechanism and stellar evolu-tion models. Multi-wavelength studies show that emis-sion from the remnant is dominated by a bright ring, butalso find jets and knots, as well as an X-ray bright CCO(central compact object) believed to be a rapidly cool-ing neutron star. Cas A is an unusual supernova (SN)in some respects: nuclear decay lines of Ca (1157 keV)and Sc (67.9 and 78.4 keV) detected by COMPTELand BEPPO SAX indicate a very large synthesized Timass of (0.8-2.5) × − M ⊙ [2] ( M ⊙ is the solar mass),which would imply an ejected Ni mass of at least 0.05 M ⊙ [3, 4], making Cas A extremely bright given its prox-imity; but no definitive historical record of such a brightSN at that time ( ∼
300 yrs ago) exists. The possibledetection by Flamsteed in 1680 suggests 6th magnitude.Possible solutions center on extinction due to a surround-ing dust cloud, possibly generated by a pre-SN wind fromthe massive star and an asymmetric SN explosion [5]. Al-though observations of Cas A do give indications of thesefeatures, they do not explain why no other Ti-rich SNehave been found despite searches in massive star regionsin the inner galaxy [6, 7]. It appears that Ti-producingSNe are quite rare. NuSTAR (nuclear spectroscopic tele-scope array) aims to map the Ti in more Cas A-likeremnants to solve this puzzle.In this letter, we present an alternative that can rec-oncile the sub-luminous nature of Cas A with its excess Ti production. Assuming that the progenitor of theCas A remnant was not atypical (i.e., did not producelarge amount of Ti in situ), we suggest that Ti is formed instead as a spallation product when neutron-rich mate-rial ejected from a ”second” explosion , viz., that of theneutron star (a quark-nova), impacts and destroys the Ni layer ejected by the preceding explosion (the SN).The basic picture is that a SN can produce a massive neu-tron star, which then converts explosively to a quark star(QS hereafter) (in an event called a Quark Nova or QN;[8, 9]). Such an explosion can happen if the Neutron Star(NS), in its spin-down evolution, reaches the quark de-confinement density and subsequently undergoes a phasetransition to the more stable strange quark matter phase[10–12], resulting in a conversion front that propagatestoward the surface in the detonative regime. The outerlayers of the parent NS are ejected from an expandingthermal fireball [13, 14] which allows for ejecta with ki-netic energy easily exceeding 10 erg. If the QN occursless than a few weeks after the SN, the QN energy re-lease reheats the preceding SN ejecta, and results in anevent we call a dual-shock Quark Nova (dsQN hereafter).In previous papers, we introduced the dsQN as a modelfor superluminous SNe [15] and discussed their photo-metric and spectroscopic [16] signatures. Here, for thefirst time, we explore the nuclear processing of the inner-most SN ejecta by the QN relativistic ejecta (neutronsand heavy nuclei). We show that such a model for CasA can explain why the SN is sub-luminous yet Ti-rich;in addition, the rarity of such events follows naturallyas a constraint from the time delay between the two ex-plosions. Testable predictions based on our model are,delayed Hydrogen signatures (weeks after the second ex-plosion) and a modified SN light curve. II. THE SPALLATION MODEL
Beam and Target : In analogy with spallation reactionsin the laboratory, we frame our model in the context ofa ”beam” and a ”target” . The QN provides the ”beam”:a relativistic outflow of neutron-rich material from theNS surface, caused by an explosive phase transition inits core to a more compact quark phase. Recent nu-merical studies of the phase conversion front [17] sug-gest supersonic laminar motion of the conversion front,which can become unstable [18], wrinkling the conversionfront to serve as a platform for a DDT (deflagaration-to-detonation). The outcome, depending on the conversionefficiency of the shock to kinetic energy, is ejection ofabout M QN ∼ − M ⊙ [8, 9] of the NS’s outermost layersat nearly relativistic speeds with average Lorentz factorsof Γ QN ∼
10. The total number of ejected nucleons in thisbeam (mostly neutrons) is then N ∼ . × M QN , − where M QN , − is the QN ejecta mass in units of 10 − M ⊙ .Adopting these fiducial values, the neutron energy is E ∼
10 GeV. This beam of relativistic neutrons (speedsclose to c ), will overtake and strike the innermost layersof expanding SN ejecta (the ”target”) which are mov-ing much slower at a speed ” v ≪ c ”. Setting our clockby the SN explosion at t =0, this collision will happen atime t delay , the delay between the SN and the QN explo-sions. The collision between the QN and SN ejecta causesspallation and subsequently other nuclear reactions. Thecrux of our argument is that these spallation reactionscan be a mechanism to explain some of the unique fea-tures of Cas A discussed previously, if t delay is chosenappropriately.We assume an onion-like profile of the expandingshocked SN ejecta (i.e., no mixing) with the innermostejecta, viz., Ni nuclei (mass number A =56) constitutingthe target at a distance from the CCO of R in ( t )= v t delay .The target number density in the Ni layer is approxi-mately constant at n A = M A / (4 πR ∆ R ), where ∆ R isthe thickness of the Ni layer. The neutron mean free pathfor spallation in the Ni layer is λ = 1 / ( n A σ sp ), where thespallation cross-section for neutrons on a target nucleusis empirically described with σ sp ≃ A . f ( A ) mb with f ( A ) ∼ ≤
10% for energies
E > N coll . ≈ ∆ Rλ ≃ . M A, . ( A ) . ( v / s t delay , ) , (1)where M A is in units of 0 . M ⊙ . The SN ejecta must besufficiently dense that N coll . ≥
1, which for fiducial valuesof M A and v in Eq.(1), limits t delay < . Spallation Reactions : We include only the two mostrelevant reactions: spallation by neutrons and by protons(e.g. n + A → products + n × ( ζ nn + 1) + p × ζ np ) where ζ is the multiplicity. The n + A neutron multiplicity asa function of the beam energy and target material showsroughly linear dependence on the target mass number (inthe range 12 < A < ζ nn ( E ) ≃ A (0 . . E ), where the neu-tron energy E is in GeV[20]. This formula gives betterthan 10% accuracy for A >
40. The average total multi- plicity is¯ ζ ( E, A ) = ζ nn + ζ np ≃ . A (1 + 0 .
38 ln E ) Y np , (2)where Y np = (1 + ζ np /ζ nn ) is in the range 1 . < Y np < .
67 (e.g. [20]). For proton induced reactions ( p + A ),most of the results are similar to those of neutron inducedreactions [20]. We treat neutrons and protons identically.Spallation effectively ceases when ¯ ζ drops below 1, cor-responding to projectile energy of E ∼
73 MeV.
Spallation Statistics : Let us divide the Ni layer into ∼ N coll imaginary sub-layers of radial thickness ∼ λ , with k =0 denoting the innermost one. A given Ni nucleus inthis layer will be hit multiple times ( N hits ) by neutrons,resulting in a product nucleus A = A − N − X j =0 ζ ( E , A j ) . (3) E is the typical energy of neutrons impinging on the k =0 layer. To produce a realistic distribution of productnuclei from this sub-layer, we draw N and ζ fromPoisson distributions peaking at ¯ N and ¯ ζ , where¯ N ∼ σ sp N (1 − e − )4 πR (4)and ¯ ζ =¯ ζ ( E , A j ). For subsequent sub-layers, it followsthat¯ N k hits = (1 − e − ) ¯ N k − ¯ ζ k and E k = (1 − η ) E k − ¯ ζ k , (5)where η accounts roughly for the incident energy removedby radiation, nuclear excitation and other sub-products(pions, He, Deuterium etc ...). Since the pions and othermultiplicities are small, we set η = 0 [20]. A Poissoniandescription is appropriate since we have a small spalla-tion cross-section but a rapidly increasing ¯ N hits due tothe cascading effect. The total number of layers thatexperience spallation is given by min ( N coll , k max ) where E k max − ≃
73 MeV, since for a specified thickness (mass)of target material, one can run out of material beforespallation becomes insignificant.
III. RESULTS
Spallation products in the inner SN ejecta : Fig. 1shows the probability distribution of the spallation prod-ucts from Ni layer for t delay of 4, 5 and 6 days. Ni isdepleted, while Ti and light elements (H through Ne)are produced. The t delay = 5 day Ni target case is partic-ularly interesting for Cas A: note the Ti and C productionpeaks. Figure 2 shows normalized mass yields of spal-lated fragments η A = M A /M A , where M A is the initialamount of Ni. We observe that for 3 days < t delay < Ni-poor, Ti-rich, C-rich debris. H A = t delay = L TiHeHHLI (cid:144) Be (cid:144) B Combined H A L H A L H A = t delay = L TiCHe Combined H A L H A L H A = t delay = L TiO Combined H A L H A L FIG. 1: Spallation products in successive layers (back to front) from Ni for t delay = 4 , , Neutron-capture versus neutron-decay : What happensto the neutrons themselves? For E k max <
73 MeV fol-lowing the spallation regime, the neutrons mainly de-posit heat through inelastic collisions. Neutron-capturehappens once the neutron energy is further reduced to ≤
30 MeV. Compared to the free neutron lifetime of ∼
720 seconds, the rather long n-capture timescale of τ cap . =1 / ( n A σ cap . v n ) ∼ . v n , M A , ∆ R where v n is the neutron thermal speed and σ cap . the capture cross-section [21]) implies that t delay must be shorter than about 16 hours for significant neu-tron capture. In such cases, nucleosynthetic yields canbe slightly altered from those expected in a SN. Hydrogen Formation : There are three sources of Hin our model: (i) a direct byproduct of spallation (i.e.when A is reduced to 1) for short time delays, whichcontributes at most 1% of the total mass in the targetlayer; (ii) from spallation protons forming hydrogen viarecombination once their energy E ∼
73 MeV (protonsare about 1/3 of the total nucleons formed); (iii) fromproton recombinations following β -decay of the neutronsthat evade capture: for t delay > . M H ∼ . M ⊙ M QN , − . IV. CONCLUSIONS AND PREDICTIONS
Ni-poor, Ti-rich SNe : We suggest that the Ti-rich Ni-poor yield of Cas A can be explained if the SN was fol-lowed by a QN with t delay ∼ Ti is formed from the destructionof Ni and thus Ti-rich dsQNe will be necessarily sub-luminous. Also, the rarity of Ti-rich SNe in the massivestar populations could be because the mass-cut is abovethe Ti zone in these SNe, suppressing Ti ejection.
Onlythose experiencing a QN explosion following the SN, with
FIG. 2: Mass Yields (upper panel) and total spallation neu-trons&protons (lower panel) versus time delay for Ni target.In the lower panel, spallation layers are numbered 0-9, withspallation effectively ceasing after 10 layers for t delay = 2 days. the appropriate t delay , would show Ti produced by spal-lation (as we suggest for Cas A) . The higher mass-cutwould also favor more massive NSs, ideal candidates forthe QN transition to occur [22].
Delayed Hydrogen signatures : Decay of spallation neu-trons to protons will not immediately form hydrogensince the recombination and thermal continuum radia-tion remains trapped and ionizes the H until it escapeswhen ∆ R A = 1 / ( n e , A σ Th . ) where σ Th . is the Thompsoncross-section and n e , A the electron density in the targetlayer. For A = 56, the recombination radiation escapeswhen the innermost layers of the SN ejecta reaches a ra-dius R in , esc . ∼ . × cm × M / , . . This implies thathydrogen should be observed ∼
41 days × M / , . /v in , days following the explosion. Other signatures : The late time SN lightcurve shouldcarry signatures of Ni and Co destruction by the QNejecta. In the case of Ni an amount of it would havedecayed in t delay and converted to Co, so that, for eg.,only 57% of the original Ni will experience spallationfrom the QN for t delay = 5 days. The remaining 43% ofthe original Ni would have decayed contributing mostlyto PdV work in the SN ejecta. This means that the effi-ciency of the QN spallation determines how much of theremaining Ni is destroyed and directly influences theSN luminosity. In general, we find that for t delay < t delay ∼ t delay can result in earlier destruc-tion of Ni and thus lead to a subluminous SN withno 77 day Cobalt tail. It is interesting to consider thecase of a Si layer (with ∼
10% Ca; [23]) over a Ni layerwhere the neutrons exit the Ni layer with E n >>
73 MeV(i.e. k max >> N coll ). For fiducial values, this occurs for t delay ∼ N coll ∼
2; see Fig. 2, lower panel).This implies that the Si/Ca is destroyed by spallation,which could explain the lack of lines such as Si, Ca andIron in some superluminous SNe [24]. This delay is alsoideal for explaining reenergization of superluminous SNe[15]. A t delay ∼ Model assumptions : Our model has some fine tuningthat is unavoidable due to uncertainties regarding thenature of the hadron-quark phase transition. [25] givemass-radius constraints for the compact object in Cas A,which effectively rule out low-mass QSs based on non-interacting quark equations of state. However, large andheavy QSs may exist, so long as the quark superconduct-ing gap and strong coupling corrections are taken intoaccount (e.g. [28–30]). The issue of the mass-radius re-lation for quark stars is still a matter of debate. Spectralfitting of Cas A agrees very well with theoretical coolingmodels for NSs, when superfluidity and pair-breaking ef-fects are taken into account [31]. It is unlikely that aQS would exhibit exactly the same cooling behaviour asa NS, which is a problem for our model, but there isno comprehensive cooling simulation studies of QS andit might be purely coincidental that at this particularyoung age, a NS and a QS have the same surface tem-perature. Studies of cooling of QSs which include similarattention to physics details (e.g. color superconductiv-ity) are needed to determine whether they could be atall consistent with Cas A.
Acknowledgments:
This research is supported by anoperating grant from the National Science and Engineer-ing Research Council of Canada (NSERC). [1] O. Krause et al., Science , 1195 (2008)[2] A.F. Iyudin et al., Astron. Astrophys. , L1 (1994)[3] G. Magkotsios et al., Astrophys. J. , 66 (210)[4] S.E. Woosley and T.A. Weaver, Astrophys. J. , 181(1995)[5] K. Maeda and K. Nomoto, Astrophys. J. , 1163(2003)[6] R. Diehl, N. Prantzos, and P. von Ballmoos, P., NuclearPhysics A , 70 (2006)[7] L.-S. The et al., Astron. Astrophys. , 1037 (2006)[8] R. Ouyed, J. Dey, J., and M. Dey, M., Astron. Astrophys. , L39 (2002)[9] P. Ker¨anen, R., Ouyed, and P. Jaikumar, P., Astrophys.J. , 485 (2005)[10] A.R. Bodmer, Phys. Rev. D , 1601 (1971)[11] N. Itoh, Prog. Theor. Phys. , 291 (1970)[12] E. Witten, Phys. Rev. D , 272 (198)[13] R. Ouyed, R. Rapp, and C. Vogt, Astrophys. J. ,1001 (2005)[14] C. Vogt, R. Rapp, and R. Ouyed, Nuc. Phys. A , 543(2004)[15] D. Leahy, and R. Ouyed, Mon. Not. R. astr. Soc. ,1193 (2008)[16] R. Ouyed et al., submitted to Astron. Astrophys. [arXiv:1010.5530] (2010)[17] B. Niebergal, R. Ouyed, and P. Jaikumar, Phys. Rev. C , 062801 (2010) [arXiv:1008.4806 [nucl-th]][18] J.E. Horvath, arXiv:1005.4302 [astro-ph.HE] (2010)[19] J. R. Letaw, R. Silberberg, and C.H. Tsao, Astrophys. J. , 271 (1983)[20] J. Cugnon, C., Volant , and S. Vuillier, Nuclear PhysicsA , 729 (1997)[21] M. Heil, F. K¨appeler, and E. Uberseder, Mem. Ita. As-tron. Soc. , 922 (2006)[22] J.E. Staff, R. Ouyed, and P. Jaikumar, Astrophys. J.Lett. , L145 (2006)[23] K. Nomoto et al., Nuclear Physics A , 424 (2006)[24] R.M. Quimby et al., Nature , 487 (2011)[25] W.C.G. Ho and C.O. Heinke, Nature , 71 (2009)[26] R. Ouyed et al., Astrophys. J. , 558 (2006)[27] P. Jaikumar et al., Astron. Astrophys. , 227 (2007)[28] M. Alford et al., Nature (2007)[29] A. Kurkela, et al., Phys. Rev. D , 105021 (2010)[30] S. Weissenborn et al., arXiv:1102.2869 (2011)[31] P.S. Shternin et al., Mon. Not. R. astr. Soc.412