Neutron star binary orbits in their host potential: effect on early r-process enrichment
Matteo Bonetti, Albino Perego, Massimo Dotti, Gabriele Cescutti
MMNRAS , 1–18 (2019) Preprint 11 September 2019 Compiled using MNRAS L A TEX style file v3.0
Neutron star binary orbits in their host potential: effect onearly r-process enrichment
Matteo Bonetti, , , (cid:63) Albino Perego, , † Massimo Dotti and Gabriele Cescutti DiSAT, Universit`a degli Studi dell’Insubria, Via Valleggio 11, 22100 Como, Italy Dipartimento di Fisica “G. Occhialini”, Universit`a degli Studi di Milano-Bicocca, Piazza della Scienza 3, 20126 Milano, Italy INFN, Sezione di Milano-Bicocca, Piazza della Scienza 3, 20126 Milano, Italy Dipartimento di Fisica, Universit`a degli Studi di Trento, via Sommarive 14, 38123 Trento, Italy INAF, Osservatorio Astronomico di Trieste, Via G.B. Tiepolo 11, I-34143 Trieste, Italy
Accepted 2019 September 9. Received 2019 September 9; in original form 2019 May 28
ABSTRACT
Coalescing neutron star binary (NSB) systems are primary candidates for r -processenrichment of galaxies. The recent detection of r -process elements in ultra-faint dwarf(UFD) galaxies and the abundances measured in classical dwarfs challenges the NSBmerger scenario both in terms of coalescence time scales and merger locations. In thispaper, we focus on the dynamics of NSBs in the gravitational potentials of differenttypes of host galaxies and on its impact on the subsequent galactic enrichment. We findthat, for a ∼ t − delay time distribution, even when receiving a low kick ( ∼
10 km s − )from the second supernova explosion, in shallow dwarf galaxy potentials NSBs tend tomerge with a large off-set from the host galaxy. This results in a significant geometricaldilution of the amount of produced r − process elements that fall back and pollute thehost galaxy gas reservoir. The combination of dilution and small number statisticsproduces a large scatter in the expected r -process enrichment within a single UFDor classical dwarf galaxy. Comparison between our results and observed europiumabundances reveals a tension that even a systematic choice of optimistic parameters inour models cannot release. Such a discrepancy could point to the need of additional r -process production sites that suffer less severe dilution or to a population of extremelyfast merging binaries. Key words: galaxies: dwarf; stars: neutron; galaxies: abundances; methods: numer-ical
The merger of compact binaries comprising at least one neu-tron star (NS) has long been thought to be the site forthe production of a significant fraction of the heavy ele-ments above the iron group via the so called r -process nu-cleosynthesis (e.g. Lattimer & Schramm 1974; Eichler et al.1989; Freiburghaus et al. 1999). The recent detection of akilonova transient (AT2017gfo) associated with the grav-itational wave (GW) signal produced by two NSs in thelate phases of their inspiral (GW170817, e.g. Abbott et al.2017b,a; Drout et al. 2017; Tanaka et al. 2017; Pian et al.2017; Tanvir et al. 2017; Kasen et al. 2017; Nicholl et al.2017; Chornock et al. 2017) has finally provided strong ob-servational support to these ideas and confirmed that NSBmergers are one of the major (if not the main) production (cid:63) E-mail: [email protected] † E-mail: [email protected] site for r -process nucleosynthesis elements (see e.g. Thiele-mann et al. 2017; Rosswog et al. 2018; Hotokezaka et al.2018; Cˆot´e et al. 2018).The unprecedented quality of the kilonova detection hasprovided a glimpse of the potential variety associated withthis new class of transients. The optical and infrared electro-magnetic data are well explained by the radioactive decayof ∼ .
05 M (cid:12) of material (e.g. Cowperthwaite et al. 2017;Rosswog et al. 2018; Tanaka et al. 2017; Tanvir et al. 2017;Smartt et al. 2017; Kasen et al. 2017; Perego et al. 2017).The presence of different peaks in the light curves and thespectral evolution can be modelled by different componentsin the outflows with different compositions and, possibly, dif-ferent physical origins. Modelling of the matter outflow andof its properties is presently accomplished by numerical sim-ulations of the merger and of its aftermath. Matter expelledwithin the first milliseconds after the NS collision is dubbeddynamical ejecta (e.g. Korobkin et al. 2012; Bauswein et al. © a r X i v : . [ a s t r o - ph . H E ] S e p M. Bonetti et al. r -process elements in the atmosphere ofmetal-poor stars in our galaxy and in nearby classical dwarfgalaxies hint at the occurrence of r -process nucleosynthesisalso in metal-poor environment, corresponding to the veryearly stages of the galaxy evolution (McWilliam 1998; Sne-den et al. 2003; Shetrone et al. 2003; Honda et al. 2006;Fran¸cois et al. 2007; Sneden et al. 2008; Roederer et al. 2014;Ural et al. 2015; Jablonka et al. 2015; Hill et al. 2018). More-over, the abundance of europium (an element synthesisedmainly by r -process nucleosynthesis) presents a large scat-ter at very low metallicities ( [ Fe / H ] < − ), suggesting that r -process elements must be synthesized in rare and isolatedevents that inject a significant amount of heavy elementsinto a relatively small amount of gas (e.g. Sneden et al. 2008,and references therein). This rare-event/high-yield scenariois also corroborated by the comparison of iron and plutoniumabundances in deep-sea sediments (Hotokezaka et al. 2015).This hypothesis was also tested in stochastic and inhomo-geneous chemical evolution models (Cescutti et al. 2015a;Wehmeyer et al. 2015), which succeed to explain the chem-ical spread of r-process abundances in halo stars, but onlyassuming very short delay for the NSB merger.Among the smallest dwarfs, called ultra-faint dwarfgalaxies (UFD), Reticulum II (Ji et al. 2016a; Roederer et al.2016; Ji et al. 2016b) and very likely also Tucana III (Drlica-Wagner et al. 2015; Simon et al. 2017; Hansen et al. 2017;Marshall et al. 2018) show a large excess of r -process ele-ments, while for all the other UFDs robust upper limits ontheir r -process element abundances have been set (Frebelet al. 2010b; Simon et al. 2010; Koch et al. 2013; Fran¸coiset al. 2016). The presence of r -process elements in UFDsposes serious challenges to NSB mergers as origin of the r -process nucleosynthesis elements. First, these galaxies havelow escape velocities (Walker et al. 2015). The kick impartedto any newly born NS by its CCSN explosion could poten-tially eject the NS from the galaxy, even when the stellarbinary system survives the second supernova explosion. Sec-ond, their old stellar population is thought to be the resultsof a fast star formation episode, that ends within the firstGyr of the galaxy evolution after the first CCSN explosionsexpel a significant fraction of baryons (Brown et al. 2014b;Weisz et al. 2015).A firm understanding of the r -process enrichment indwarf galaxies is also essential to understand metal poorstars in the Milky Way halo. Indeed both observations(Frebel et al. 2010a; Ivezi´c et al. 2012) and theoretical mod-els (Helmi 2008; Griffen et al. 2016) point to the fact thatthe dwarf satellite galaxies that we observe nowadays aroundour Galaxy are the remnants of a large population of dwarfsthat long ago merged with it to form the galactic halo stel- lar population. Moreover, recent results coming from GAIA(Gaia Collaboration et al. 2016) point to the fact that alarge fraction of the Milky Way halo was actually formedthrough a merger with a single and relative massive satellite(Haywood et al. 2018; Helmi et al. 2018).Assuming that fast mergers of compact binary systemsrequire a high natal kick, Bramante & Linden (2016) dis-favoured NSB mergers as the source of the r -process ma-terial observed in Reticulum II. In contrast, further studiesconcerning the observed distributions of the orbital param-eters of double NS systems (Beniamini & Piran 2016) andthe formation channels of such systems (Tauris et al. 2017)suggest that a large fraction of NSB systems could have re-ceived a rather small kick and have ejected a small amountof mass as a consequence of the second CCSN explosion.These conclusions imply that a large fraction (up to 60%) ofdouble NS systems could be retained even by UFD galaxiesand to merge within the first Gyr of galaxy evolution (Be-niamini et al. 2016a). Also Safarzadeh et al. (2019) reachedopposite conclusions with respect to Bramante & Linden(2016) by considering the possibility that high-kick NSBsare either on highly eccentric orbits or form with very shortseparations due to an additional mass-transfer between thefirst-born neutron star and a naked helium star, progenitorof the second neutron star.Due to the low stellar content of UFD galaxies and thesubsequent small number of NSB systems expected in thesegalaxies, objects like Reticulum II and Tucana III should bea minority. Moreover, in this scenario these UDFs shouldhave hosted a single NSB merger that was able to signifi-cantly enrich them in r -process material. These first analyseswas later refined by Beniamini et al. (2018), who consideredthat some of the r -process material synthesised during themergers could still escape from the galaxy, thanks to thelarge kinetic energy that characterizes NSB merger ejecta( ∼ erg). By performing a more detailed analysis of theenrichment in iron from CCSNe and in r -process elementsfrom NSB mergers in UFD galaxies, they confirmed the com-patibility between the abundances observed and an r -processenrichment due to the ejection of material from rare eventstaking place inside dwarf galaxies with low escape velocities.In many of the above mentioned studies, it was assumedthat a NSB bound to its host coalesces always well inside thegalaxy. The potential relevance of the merger location, andthus of the imparted kick, in explaining the abundances ob-served in UFDs was first underlined by Safarzadeh & Scan-napieco (2017). In particular, if the merger time is not ex-tremely short (Bonetti et al. 2018; Safarzadeh et al. 2019),the binary will start orbiting the galaxy and there is a highchance that the merger will still happen far from the regionsof the galaxy where the next generation of stars will form.In this work, we systematically explore the potential effectof the dilution of the ejecta on enrichment of the r -processmaterial due to the location of NSB mergers relative to thehost galaxy (see also Safarzadeh & Cˆot´e 2017, for a simi-lar analysis focused on the Milky-Way). Under a commonset of minimal assumptions about the properties of the hostgalaxy, the star formation rate inside it, the NSB birth andcoalescence as well as the ejecta properties, we investigate awide sample of galaxy masses and we compute the fractionof the ejecta material retained by the galaxy.The paper is structured as follows: in sections 2-5 we MNRAS000
05 M (cid:12) of material (e.g. Cowperthwaite et al. 2017;Rosswog et al. 2018; Tanaka et al. 2017; Tanvir et al. 2017;Smartt et al. 2017; Kasen et al. 2017; Perego et al. 2017).The presence of different peaks in the light curves and thespectral evolution can be modelled by different componentsin the outflows with different compositions and, possibly, dif-ferent physical origins. Modelling of the matter outflow andof its properties is presently accomplished by numerical sim-ulations of the merger and of its aftermath. Matter expelledwithin the first milliseconds after the NS collision is dubbeddynamical ejecta (e.g. Korobkin et al. 2012; Bauswein et al. © a r X i v : . [ a s t r o - ph . H E ] S e p M. Bonetti et al. r -process elements in the atmosphere ofmetal-poor stars in our galaxy and in nearby classical dwarfgalaxies hint at the occurrence of r -process nucleosynthesisalso in metal-poor environment, corresponding to the veryearly stages of the galaxy evolution (McWilliam 1998; Sne-den et al. 2003; Shetrone et al. 2003; Honda et al. 2006;Fran¸cois et al. 2007; Sneden et al. 2008; Roederer et al. 2014;Ural et al. 2015; Jablonka et al. 2015; Hill et al. 2018). More-over, the abundance of europium (an element synthesisedmainly by r -process nucleosynthesis) presents a large scat-ter at very low metallicities ( [ Fe / H ] < − ), suggesting that r -process elements must be synthesized in rare and isolatedevents that inject a significant amount of heavy elementsinto a relatively small amount of gas (e.g. Sneden et al. 2008,and references therein). This rare-event/high-yield scenariois also corroborated by the comparison of iron and plutoniumabundances in deep-sea sediments (Hotokezaka et al. 2015).This hypothesis was also tested in stochastic and inhomo-geneous chemical evolution models (Cescutti et al. 2015a;Wehmeyer et al. 2015), which succeed to explain the chem-ical spread of r-process abundances in halo stars, but onlyassuming very short delay for the NSB merger.Among the smallest dwarfs, called ultra-faint dwarfgalaxies (UFD), Reticulum II (Ji et al. 2016a; Roederer et al.2016; Ji et al. 2016b) and very likely also Tucana III (Drlica-Wagner et al. 2015; Simon et al. 2017; Hansen et al. 2017;Marshall et al. 2018) show a large excess of r -process ele-ments, while for all the other UFDs robust upper limits ontheir r -process element abundances have been set (Frebelet al. 2010b; Simon et al. 2010; Koch et al. 2013; Fran¸coiset al. 2016). The presence of r -process elements in UFDsposes serious challenges to NSB mergers as origin of the r -process nucleosynthesis elements. First, these galaxies havelow escape velocities (Walker et al. 2015). The kick impartedto any newly born NS by its CCSN explosion could poten-tially eject the NS from the galaxy, even when the stellarbinary system survives the second supernova explosion. Sec-ond, their old stellar population is thought to be the resultsof a fast star formation episode, that ends within the firstGyr of the galaxy evolution after the first CCSN explosionsexpel a significant fraction of baryons (Brown et al. 2014b;Weisz et al. 2015).A firm understanding of the r -process enrichment indwarf galaxies is also essential to understand metal poorstars in the Milky Way halo. Indeed both observations(Frebel et al. 2010a; Ivezi´c et al. 2012) and theoretical mod-els (Helmi 2008; Griffen et al. 2016) point to the fact thatthe dwarf satellite galaxies that we observe nowadays aroundour Galaxy are the remnants of a large population of dwarfsthat long ago merged with it to form the galactic halo stel- lar population. Moreover, recent results coming from GAIA(Gaia Collaboration et al. 2016) point to the fact that alarge fraction of the Milky Way halo was actually formedthrough a merger with a single and relative massive satellite(Haywood et al. 2018; Helmi et al. 2018).Assuming that fast mergers of compact binary systemsrequire a high natal kick, Bramante & Linden (2016) dis-favoured NSB mergers as the source of the r -process ma-terial observed in Reticulum II. In contrast, further studiesconcerning the observed distributions of the orbital param-eters of double NS systems (Beniamini & Piran 2016) andthe formation channels of such systems (Tauris et al. 2017)suggest that a large fraction of NSB systems could have re-ceived a rather small kick and have ejected a small amountof mass as a consequence of the second CCSN explosion.These conclusions imply that a large fraction (up to 60%) ofdouble NS systems could be retained even by UFD galaxiesand to merge within the first Gyr of galaxy evolution (Be-niamini et al. 2016a). Also Safarzadeh et al. (2019) reachedopposite conclusions with respect to Bramante & Linden(2016) by considering the possibility that high-kick NSBsare either on highly eccentric orbits or form with very shortseparations due to an additional mass-transfer between thefirst-born neutron star and a naked helium star, progenitorof the second neutron star.Due to the low stellar content of UFD galaxies and thesubsequent small number of NSB systems expected in thesegalaxies, objects like Reticulum II and Tucana III should bea minority. Moreover, in this scenario these UDFs shouldhave hosted a single NSB merger that was able to signifi-cantly enrich them in r -process material. These first analyseswas later refined by Beniamini et al. (2018), who consideredthat some of the r -process material synthesised during themergers could still escape from the galaxy, thanks to thelarge kinetic energy that characterizes NSB merger ejecta( ∼ erg). By performing a more detailed analysis of theenrichment in iron from CCSNe and in r -process elementsfrom NSB mergers in UFD galaxies, they confirmed the com-patibility between the abundances observed and an r -processenrichment due to the ejection of material from rare eventstaking place inside dwarf galaxies with low escape velocities.In many of the above mentioned studies, it was assumedthat a NSB bound to its host coalesces always well inside thegalaxy. The potential relevance of the merger location, andthus of the imparted kick, in explaining the abundances ob-served in UFDs was first underlined by Safarzadeh & Scan-napieco (2017). In particular, if the merger time is not ex-tremely short (Bonetti et al. 2018; Safarzadeh et al. 2019),the binary will start orbiting the galaxy and there is a highchance that the merger will still happen far from the regionsof the galaxy where the next generation of stars will form.In this work, we systematically explore the potential effectof the dilution of the ejecta on enrichment of the r -processmaterial due to the location of NSB mergers relative to thehost galaxy (see also Safarzadeh & Cˆot´e 2017, for a simi-lar analysis focused on the Milky-Way). Under a commonset of minimal assumptions about the properties of the hostgalaxy, the star formation rate inside it, the NSB birth andcoalescence as well as the ejecta properties, we investigate awide sample of galaxy masses and we compute the fractionof the ejecta material retained by the galaxy.The paper is structured as follows: in sections 2-5 we MNRAS000 , 1–18 (2019)
SB orbits and r-process enrichment present the model we have adopted for our calculations andwe detail all its components. The results we have obtainedare presented in section 6, while in section 7 we discuss ourresults and compare them with observations. We finally con-clude in section 8. Modelling the evolution of galaxies and of their chemicalenrichment is an extremely complex task, since it requiresto follow many different processes that span a huge range ofscales (both in space and in time). In this section, we firstpresent a summary of the (simplified) model adopted in thiswork to study the enrichment in r -process material due toNSB mergers.We consider disc galaxies with a different content ofbaryonic matter M b , ranging from M (cid:12) up to M (cid:12) , aswell as a model with × M (cid:12) . The former interval is ex-pected to correspond to the initial gas content of ultra-faintand classical dwarf galaxies while the latter value is the oneof a MW-like galaxy. During the cosmic history, gas is con-verted into stars with a certain star formation rate ( f SFR )starting from the galaxy formation ( t = ) up to t ≈ T SF .For a MW-like galaxy, T SF ∼
14 Gyr and the final stellarcontent is thought to be comparable to the initial gas mass.For smaller galaxies SN feedback and/or environmental pro-cesses (e.g. tidal perturbations and ram pressure stripping)are expected to quench the conversion of gas into stars onshorter timescales and to remove a fraction of the gas fromthe galaxy (Revaz & Jablonka 2018). In the case of UFD, weassume T SF (cid:46) , while for classical dwarfs T SF ∼ − .The SFR has a non-trivial dependence on the cosmic timeand on the individual history of each galaxy. Since we arenot interested in the detailed time evolution of the metalcontent nor in the reproduction of a specific galaxy model,we adopt an exponential dependence on time, possibly de-pendent on the initial gas mass. The stellar content at anytime t , M ∗ ( t ) is then computed as the integral of f SFR overtime.For each galaxy, we generate a pool of N initial condi-tions for NSBs forming from stellar binaries as a consequenceof a double CCSN explosion. For each NSB, the relevant ini-tial conditions after the second SN are the initial positionin the galaxy, the NS masses ( m , m ), the semi-major axisand eccentricity ( a , e ), and the center of mass (CoM) ve-locity of the NSB with respect to galaxy frame ( V CM ). Forall the dwarf galaxies in the our sample, the expected num-ber of merger is (cid:46) , therefore we choose N (cid:29) N merg toproperly span the parameter space. For the MW-like galaxy,the number of mergers could potentially exceed a few mil-lions. We expect such a high number of configurations to belarge enough to significantly cover the whole NSB parameterspace. Thus we consider a pool of N ∼ × NSB configu-rations. We evolve each initial condition by integrating thetrajectory of the CoM of the NSB for the GW driven coales-cence time T gw ( a , e ) , unequivocally determined by the initialbinary parameters. Thus, we can associate to each NSB inour pools the time at which the merger happens and thecorresponding location.To compute the amount of r -process material produced by NSB mergers that has enriched a specific galaxy at a time t we proceed as follows: • starting from the galaxy stellar mass at t , M ∗ ( t ) , weestimate the amount of CCSN explosions, N CCSN , and fromthat the expected number of stellar binaries surviving a dou-ble CCSN explosion and producing a NSB system within t , N merg ; • we sample the actual number of NSB mergers N r froma Poisson distribution with average equal to N merg , and werandomly choose N r cases from the galaxy NSB pool. Westress here that a (possibly large) fraction of these binariescould merger on a time T gw (cid:46) t . In the case of the MW-like galaxy, if N r > N r , max = we select N r , max NSB fromthe pool of N ∼ × elements. In this case, at the endof the analysis, the true values of the total and retainedmasses are obtained by rescaling the computed quantities bya factor N r / N r , max . This approach is motivated by the needof reducing the necessary computations and the amount ofdata. However, it is justified by the large number of expectedmergers with respect to their intrinsic variability; • for each sampled binary merging within t , we model theproperties of the material ejected during the coalescence,and its expansion due to the interaction with the diffusegaseous halo of the host in order to determine the fractionand composition of the ejecta that gets injected in the galaxydisc. We consider that the ejected mass can be in form ofwind ejecta (isotropically distributed) as well as dynamicalejecta, characterized by both a polar and an equatorial com-ponent.Finally, in order to estimate both the average and thescatter in the distribution of the total ejected and retainedmaterial, for each galaxy model we perform 200 differentrealizations of the expected NSB populations. For each real-ization we assume full mixing of the ejecta. Despite being asimplification, this hypothesis is not a limitation because wewill compare with estimates of the total galactic abundances.More detailed studies assuming inhomogeneous mixing andstochastic stellar enrichment are planned for the future.In the following we fully detail the procedure adoptedin our model. In section 3, we start with the modeling ofthe host galaxy and with the determination of the numberof stars, SNae and merging NSBs for each galaxy model. Wealso discuss the modelling of the host galaxy potential, whichdetermines the velocity of the NSB progenitor binary sys-tem, its CoM orbital evolution as well as the escape velocity.We then discuss the initialization of the intrinsic parametersof each NSB (section 4.1): the masses of the two NSs ( m and m ), the semi-major axis and eccentricity of the the bi-nary ( a and e ), and the kick that the NSB gets due to thetwo SNae explosions ( V CM ). The initialization of the initialposition and velocity of the NSB CoM is described in sec-tion 4.2. The ejecta properties immediately after the NSBcoalescence are described in section 4.3, while their evolutionwithin the host halo is described in section 4.4. We consider five different disc galaxies spanning a wide rangeof possible masses, M b : four dwarf galaxy models with to-tal baryonic mass of M (cid:12) , M (cid:12) , M (cid:12) , and M (cid:12) MNRAS , 1–18 (2019)
M. Bonetti et al. respectively, and a MW-like host modelled with five dynam-ical baryonic components (galactic bulge + thin and thickstellar disks + HI and H disks, see next and Barros et al.2016, for full details) surrounded by a dark halo.The stellar population of UFD galaxies seem to be dom-inated by very old ( ∼
12 Gyr ) stars (Brown et al. 2014a,and references therein), pointing to a rapid SF, (cid:46) .Old stars ( (cid:46)
10 Gyr ) dominate also dwarf galaxies (Grebel1997), indicating that the bulk of the SF in them happenswithin 2-3 Gyr. To approximately catch the dependence ofthe duration of the star formation on the initial baryonicmass M b , we assume a power-law dependence normalized tothe MW-like case: T SF ( M b ) = T SF ( M MW ) (cid:18) M b M MW (cid:19) α (1)with α = . and T SF ( M MW ) =
14 Gyr . For the star formationrate, we adopt the following exponential dependence: f SFR ( t , M b ) = (cid:40) A M b exp (cid:16) − t τ ( M b ) (cid:17) if t < T SF . (2)where A is a constant fixed by the requirement that the MW-like model reproduces the presently observed star formationrate in the MW, i.e. f SFR (
10 Gyr , M MW ) = .
65 M (cid:12) yr − (Lic-quia & Newman 2015). For the timescale appearing insidethe exponential factor, we choose τ = T SF / . This choice isbroadly compatible with (simple) models of the MW (e.g.Snaith et al. 2014) and of classical dwarf galaxies (Northet al. 2012). Finally, the gas mass converted into stars as afunction of time and of initial baryonic mass can be easilycomputed as M ∗ ( t , M b ) = A M b T SF (cid:16) − e − t / T SF (cid:17) / t < T SF , A M b T SF (cid:16) − e − (cid:17) / . (3) We assume that the potential of the host is well describedby only two components: a baryonic disc (where we assumethat the stars and gas follow the same profile) and a darkmatter spherical halo.The baryonic density profile of the galaxy is modelledas an exponential disk: ρ d ( R , z ) = M b π R d z d exp (cid:18) − RR d (cid:19) sech (cid:18) zz d (cid:19) (4)where R and z are the cylindrical radial and vertical coor-dinates, while the length scales R d and z d are (Mo et al.1998) R d = . (cid:18) M b M (cid:12) (cid:19) / kpc , (5) z d = . R d . (6)An analytic form for the potential (and consequently for theacceleration) cannot be obtained, we therefore employ a nu-merical sampling of the disk density profile with ≈ × tracers, and splitting every tracer in 8 sub-tracers by chang-ing the sign of 1, 2 or all the 3 coordinates, in order to preserve the symmetry of the potential. The disc acceler-ation a disc ( r ) is then evaluated through direct summationover all the sampled particles, where we use a gravitationalsoftening of (cid:15) soft = . (cid:16) M b / M (cid:12) (cid:17) / kpc to avoid spu-rious strong scattering due to the finite number of tracersused (see e.g. Monaghan & Lattanzio 1985). Such a kindof exponential disk profiles fits well the stellar brightnessprofile of many observed dwarf galaxies, including dwarf el-lipticals and dwarf spheroidals, (see e.g. Faber & Lin 1983;Binggeli et al. 1984; Kormendy 1985; Graham 2002; Graham& Guzm´an 2003). Furthermore, it implies that the stellardynamics is dominated by rotation. While this seems to bethe case for the vast majority of dwarf galaxies (Kerr et al.1954; Kerr & de Vaucouleurs 1955; Swaters et al. 2009), fordwarf spheroidal galaxies with less rotational support andembedded in virialized galaxy clusters it is reasonable to as-sume that they were more rotationally supported in the past(during their star formation epoch) and lost their coherencedue to galaxy harassment (Moore et al. 1996).The dark matter density is assumed to follow a NFWprofile: ρ DM ( r ) = ρ rr h (cid:18) + rr h (cid:19) , (7)where the normalization ρ is given by ρ = M DM π r h (cid:18) log ( + C ) − C + C (cid:19) , (8)with M DM = M b , C = . and the scale radius is givenby: R h = . (cid:18) M b M (cid:12) (cid:19) / kpc . (9)The potential generated by such distribution is analytic Φ DM ( r ) = − πρ Gr h r ln (cid:18) + rr h (cid:19) , (10)which allow us to compute the acceleration directly from a DM ( r ) = −∇ Φ DM ( r ) . (11)The total acceleration is then computed as a = a DM + a disc . Following Barros et al. (2016), we model the MW by con-sidering a dark halo, a galactic bulge plus two stellar disks(thin and thick) and two gaseous disks (HI and H ). Thedark halo is modelled as a logarithmic potential, while thebulge is assumed spherically symmetric and described by anHernquist profile (Hernquist 1990). The description of the Or equivalently S´ersic profiles with a very small S´ersic index Dwarf galaxies are among the structures with the smallest bary-onic to dark matter ratios. Detailed studies (e.g. McGaugh et al.2010; Chan 2019) suggest a weak dependence on the baryonicmass, ∼ M . b . For simplicity in our study we assume one sin-gle value broadly compatible with the expected ratio in dwarfgalaxies. MNRAS000
65 M (cid:12) yr − (Lic-quia & Newman 2015). For the timescale appearing insidethe exponential factor, we choose τ = T SF / . This choice isbroadly compatible with (simple) models of the MW (e.g.Snaith et al. 2014) and of classical dwarf galaxies (Northet al. 2012). Finally, the gas mass converted into stars as afunction of time and of initial baryonic mass can be easilycomputed as M ∗ ( t , M b ) = A M b T SF (cid:16) − e − t / T SF (cid:17) / t < T SF , A M b T SF (cid:16) − e − (cid:17) / . (3) We assume that the potential of the host is well describedby only two components: a baryonic disc (where we assumethat the stars and gas follow the same profile) and a darkmatter spherical halo.The baryonic density profile of the galaxy is modelledas an exponential disk: ρ d ( R , z ) = M b π R d z d exp (cid:18) − RR d (cid:19) sech (cid:18) zz d (cid:19) (4)where R and z are the cylindrical radial and vertical coor-dinates, while the length scales R d and z d are (Mo et al.1998) R d = . (cid:18) M b M (cid:12) (cid:19) / kpc , (5) z d = . R d . (6)An analytic form for the potential (and consequently for theacceleration) cannot be obtained, we therefore employ a nu-merical sampling of the disk density profile with ≈ × tracers, and splitting every tracer in 8 sub-tracers by chang-ing the sign of 1, 2 or all the 3 coordinates, in order to preserve the symmetry of the potential. The disc acceler-ation a disc ( r ) is then evaluated through direct summationover all the sampled particles, where we use a gravitationalsoftening of (cid:15) soft = . (cid:16) M b / M (cid:12) (cid:17) / kpc to avoid spu-rious strong scattering due to the finite number of tracersused (see e.g. Monaghan & Lattanzio 1985). Such a kindof exponential disk profiles fits well the stellar brightnessprofile of many observed dwarf galaxies, including dwarf el-lipticals and dwarf spheroidals, (see e.g. Faber & Lin 1983;Binggeli et al. 1984; Kormendy 1985; Graham 2002; Graham& Guzm´an 2003). Furthermore, it implies that the stellardynamics is dominated by rotation. While this seems to bethe case for the vast majority of dwarf galaxies (Kerr et al.1954; Kerr & de Vaucouleurs 1955; Swaters et al. 2009), fordwarf spheroidal galaxies with less rotational support andembedded in virialized galaxy clusters it is reasonable to as-sume that they were more rotationally supported in the past(during their star formation epoch) and lost their coherencedue to galaxy harassment (Moore et al. 1996).The dark matter density is assumed to follow a NFWprofile: ρ DM ( r ) = ρ rr h (cid:18) + rr h (cid:19) , (7)where the normalization ρ is given by ρ = M DM π r h (cid:18) log ( + C ) − C + C (cid:19) , (8)with M DM = M b , C = . and the scale radius is givenby: R h = . (cid:18) M b M (cid:12) (cid:19) / kpc . (9)The potential generated by such distribution is analytic Φ DM ( r ) = − πρ Gr h r ln (cid:18) + rr h (cid:19) , (10)which allow us to compute the acceleration directly from a DM ( r ) = −∇ Φ DM ( r ) . (11)The total acceleration is then computed as a = a DM + a disc . Following Barros et al. (2016), we model the MW by con-sidering a dark halo, a galactic bulge plus two stellar disks(thin and thick) and two gaseous disks (HI and H ). Thedark halo is modelled as a logarithmic potential, while thebulge is assumed spherically symmetric and described by anHernquist profile (Hernquist 1990). The description of the Or equivalently S´ersic profiles with a very small S´ersic index Dwarf galaxies are among the structures with the smallest bary-onic to dark matter ratios. Detailed studies (e.g. McGaugh et al.2010; Chan 2019) suggest a weak dependence on the baryonicmass, ∼ M . b . For simplicity in our study we assume one sin-gle value broadly compatible with the expected ratio in dwarfgalaxies. MNRAS000 , 1–18 (2019) SB orbits and r-process enrichment baryonic disks requires instead additional modelling. In fact,despite from an observational point of view these disks canbe accurately fitted assuming exponential disks, as pointedout in section 3.1, they do not admit any analytical form forthe gravitational potential nor for the acceleration. This doesnot represent a real issue for the dwarf galaxy case becausethe global number of NSB that we have to simulate is rathersmall. On the contrary, for a MW-like galaxy this numbercan exceed several millions, with a severe impact on theperformance of our calculation. This motivated our choice todescribe the baryonic disks with analytical potential-densitypairs. In particular, in Barros et al. (2016), each of the fourdisk is modelled with a superposition of three Miyamoto-Nagai (MN) disks (Miyamoto & Nagai 1975), where one ofthe three mass parameter is usually negative in order tomimic the sharp decrease that characterise the density pro-file of exponential disks. We address the interested reader tosection 2 of Barros et al. (2016) for the detailed descriptionof the model, and to table 3-4 in the same paper for the val-ues of the parameter that best describe the MW and thatwe implemented in our model. Given the large uncertainties that affect the determination ofthis value, we adopt a fairly simple but physically motivatedapproach. We assume a standard stellar initial mass function(IMF), identical for all models (Kroupa 2001). For a givengalaxy model and a given time, we compute the number ofstars that have exploded as CCSNae ( N CCSN ) as the numberof stars with mass greater than (cid:12) . In doing that, we areimplicitly assuming that the evolution timescale of massivestars is much shorter than any time we are going to explore.We then assume that the number of NSB systems that formrepresents a fraction of the total number of SNe that haveexploded. In particular, we parametrise N NSB as N NSB = N CCSN / x (12)where for x we choose four values logarithmically distributedbetween an optimistic and a very pessimistic estimate, i.e. x = [ , , , ] . This broad interval covers presentuncertainties in the determination of the ratio between thenumber of exploding CCSNe and the number of formingNSB systems, as obtained in detailed population synthesismodels (Giacobbo & Mapelli 2018).In Table 1 we summarize the properties of the differ-ent dwarf galaxy models, alongside their names used in thefollowing. Both the masses and the semi-major axes of each binary aresampled from observationally constrained distributions (e.g.Tauris et al. 2017, and references therein). The masses of thetwo NSs m and m are randomly sampled from normal dis-tribution with µ = . (cid:12) , µ = .
34 M (cid:12) and σ = .
14 M (cid:12) .The semi-major axis a is evaluated from a log-uniform dis-tribution where upper and lower limit are set selecting twice the maximum and half the minimum of “observed” NS bi-nary orbital periods, i.e. P ∈ [ . , ] days. Tidal circularization acting after the first SN explosiondoes not allow to constrain the kick velocity experienced bythe first-born NS. As a consequence, we sample the first kickexperienced by the binary CoM ( V CM , ) assuming isotropyand a uniform magnitude distribution between 10 and 20km/s, as constrained by the observed velocities of High-MassX-Ray Binaries (see e.g. Coleiro & Chaty 2013). The kick velocity associated to the second SN ( V kick , )is isotropically generated in the m rest frame. The kick mag-nitude ( V kick , ) is distributed as follows: following Beniamini& Piran (2016), we assume that the distribution is bimodal,with 60-70 % of the cases following a log-normal ”low-kickdistribution” p l ( V kick , ) = √ π V kick , σ ln V , l exp (cid:32) ln ( V kick , / ¯ V l ) σ V , l (cid:33) , (13)with ¯ V l = km/s and σ ln V , l = . , and the remaining 40-30 % of the cases are sampled from a second log-normal distribu-tion p h ( V kick , ) with the corresponding parameters ¯ V h = km/s and σ ln V , h = . . The mass loss associated to the second SN is also sam-pled from a log-normal distribution: p l , h ( ∆ M ) √ π ∆ M σ ln ∆ M , l , h exp (cid:32) ln ( ∆ M / ¯ ∆ M l , h ) σ ∆ M , l , h (cid:33) , (14)with ¯ ∆ M l = . (cid:12) , ¯ ∆ M h = (cid:12) , and σ ln ∆ M , l = σ ln ∆ M , h = . . Conservation of linear momentum implies that the ve-locity acquired by the CoM is: V CM = V CM , + m m + m V kick , + (cid:18) ∆ Mm + m (cid:19) (cid:18) m m + ( m + ∆ M ) V kep (cid:19) , (15)where the second and third terms on the right hand sideof the equation correspond to the contributions to the NSBCoM velocity from the kick experienced by the second rem-nant and from the (assumed instantaneous) mass change ofthe binary system (Postnov & Yungelson 2014).The last parameter we estimate is the eccentricity e of the NSB immediately after the second SN. e is obtainedassuming the conservation of energy and angular momentumafter the second SN: a a = − χ (cid:169)(cid:173)(cid:171) V , , x + V , , z + ( V kep + V kick , , y ) V (cid:170)(cid:174)(cid:172) , (16) − e = χ (cid:16) a a (cid:17) (cid:169)(cid:173)(cid:171) V , , z + ( V kep + V kick , , y ) V (cid:170)(cid:174)(cid:172) , (17) Assuming a total mass of ∼ . (cid:12) these limits correspond toa minimum and maximum semi-major axis of about ∼ . and ∼ solar radii. For this reason, we do not need to consider the effect of themass lost during the first SN. The value of the chosen parameters are selected in order tobroadly reproduce the merger time and the eccentricity distribu-tion of observed NSB.MNRAS , 1–18 (2019)
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Model M b M DM T SF M ∗ ( T SF ) R d z d N CCSN ( T SF ) N NSB ( T SF )[ M (cid:12) ] [ M (cid:12) ] [Gyr] [ M (cid:12) ] [ kpc ] [ kpc ] min maxDwarf 1 . × . × . × . × . × . × . × Table 1.
Summary of the most relevant properties of the different dwarf galaxy models employed in our study (full details about MW-likeparameters can be found in Barros et al. (2016), tables 3-4). M b and M DM are the baryonic and dark matter masses, T SF is the durationof the star formation, M ∗ ( T SF ) is the mass converted in star at T SF , R d and z d the scale radius and height of the disk, N CCSN ( T SF ) theaverage number of CCSN exploded by T SF ) , while the two values of N NSB represent the minimum and the maximum average number ofNSB formed. The least (most) optimistic case is obtained considering one NSB every 50 (1350) CCSNe. where χ = (( m + ∆ M ) + m )/( m + m ) ≥ is the fractionalchange in mass, and V kep = (cid:112) G (( m + ∆ M ) + m )/ a . .The time to coalescence due to GW emission T gw is thenevaluated from ( a , e , m , m ) by performing the integration(Peters 1964) T gw ( a , e , m , m ) = c β G( e ) (18)where β = G m m ( m + m ) c , c = a ( − e ) e / (cid:18) + e (cid:19) − / , G( e ) = ∫ e d e (cid:48) e (cid:48) / (cid:16) + / e (cid:48) (cid:17) / ( − e (cid:48) ) / . (19)In Fig. 1 we present the probability distribution of themerger times, t m computed as equation 18, for a large set( ) of NSBs whose properties have been generated usingthe above mentioned distributions for the internal parame-ters and kick velocities. In particular, we consider the case p l = . . In 42% of the cases the NSB merges within theHubble time. This percentage decreases to 27% and 14%if we limit t m to 1 Gyr and 0.1 Gyr, respectively. We thusnotice that our choice of a log-uniform distribution in thebinary semi-major axis down to a (cid:46) R (cid:12) and the inclusion ofhighly eccentric cases (mainly for the high kick population)introduce a significant population of fast merging binaries(see e.g. Belczynski et al. 2002; O’Shaughnessy et al. 2008;Beniamini & Piran 2016). However, this percentage is pos-sibly smaller (by a factor (cid:46) ) than the one implied by theobserved distribution of Galactic NSBs (Beniamini & Piran2019). We sample the position of the CoM of the NSB at the timeof the second SN explosion from the stellar disk profile (i.e.the exponential disk for dwarf galaxies or the superpositionof MN for MW case), and we assume an initial CoM velocity V CM , i = V CM + V c , (20) Note that the assumption here on the direction of the secondaryvelocity immediately before the second SN is only made for clarityreasons. In the actual sampling V kep is assumed isotropic. log t m [yr] . . . . . . . p ( t m ) Figure 1.
Probability density function of coalescence time forbinaries generated in our model. Vertical dashed line indicatesthe current age of the universe. Despite the log-uniform distri-bution of the semi-major axis p ( t m ) is not flat because we donot assume zero eccentricity, determining a relatively fast coa-lescence also for system with larger initial semi-major axis. Notealso that the lower limit adopted for the semi-major axis distri-bution determines the existence of fraction of rapidly-coalescingNSB systems. where V CM is the velocity acquired due to the SN kicks (seeequation 15), while V c = (cid:112) R d Φ / d r is the circular velocityin the galactic plane (i.e. at z = ) at the radius at whichthe NSB is initialized. We then integrate the motion of theCoM of the NSB in the chosen galactic system by numer-ically solving the equations of motion with an 8th orderadaptive stepsize Dormand-Prince-Runge-Kutta algorithm(DOPRI853). We stop the integration at T gw and we denotethe NSB position at that time, x f , as the merger location.As a final step, we sample the direction of the NSB angularmomentum from an isotropic distribution in order to inferthe orientation of the binary orbital plane with respect tothe galactic reference frame. MNRAS000
Probability density function of coalescence time forbinaries generated in our model. Vertical dashed line indicatesthe current age of the universe. Despite the log-uniform distri-bution of the semi-major axis p ( t m ) is not flat because we donot assume zero eccentricity, determining a relatively fast coa-lescence also for system with larger initial semi-major axis. Notealso that the lower limit adopted for the semi-major axis distri-bution determines the existence of fraction of rapidly-coalescingNSB systems. where V CM is the velocity acquired due to the SN kicks (seeequation 15), while V c = (cid:112) R d Φ / d r is the circular velocityin the galactic plane (i.e. at z = ) at the radius at whichthe NSB is initialized. We then integrate the motion of theCoM of the NSB in the chosen galactic system by numer-ically solving the equations of motion with an 8th orderadaptive stepsize Dormand-Prince-Runge-Kutta algorithm(DOPRI853). We stop the integration at T gw and we denotethe NSB position at that time, x f , as the merger location.As a final step, we sample the direction of the NSB angularmomentum from an isotropic distribution in order to inferthe orientation of the binary orbital plane with respect tothe galactic reference frame. MNRAS000 , 1–18 (2019)
SB orbits and r-process enrichment The NSB is assumed to eject a certain amount of mass dur-ing and immediately after the merger. We consider two kindsof ejecta: dynamical and disc wind ejecta.
Dynamical ejecta is expelled on a timescale of a few mil-liseconds due to tidal torques and hydrodynamics shocks.To model the properties of the dynamical ejecta, we use theparametric fit reported in Radice et al. (2018b) for the totalejecta mass and average speed. To compute the NS compact-ness parameters associated with the two NS masses m , andrequired by these fits, we choose a nuclear equation of statecompatible with all present nuclear and astrophysical con-straints (SLy, Haensel & Potekhin 2004). Numerical simula-tions of NSB mergers show that this mass ejection happenspreferentially along the equatorial plane. We thus adopt a sin θ dependence of the mass spatial distribution on the po-lar angle measured with respect to the NSB rotational axis(Perego et al. 2017; Radice et al. 2018b). Moreover, neutrinoirradiation, more intense inside the polar funnels, increasesthe electron fraction above 0.25 for θ (cid:46) π / and θ (cid:38) π / .As a consequence, we assume that polar ejecta produces r -process nucleosynthesis yields between the first and second r -process peaks, for which we consider an atomic mass num-ber interval ≤ A ≤ . On the other hand, the moreneutron rich equatorial ejecta produces elements betweenthe second and third r -process peaks for which we assumean r -process nucleosynthesis interval ≤ A ≤ . The ejection of matter in the form of disc winds happens ontimescales longer than the ones of the dynamical ejecta (upto a few hundreds milliseconds). It is due to neutrino absorp-tion, magnetic and viscous processes inside the remnant. Toestimate the total mass contained inside this ejecta, we as-sume that a fixed fraction of the disc ξ wind becomes unboundand is launched with an average velocity v disk . In this workwe assume ξ wind = . and v disk = . (see e.g. Just et al.2015; Fahlman & Fern´andez 2018). The disc mass is deter-mined through the fitting formula reported in Radice et al.(2018b). As in the case of the dynamical ejecta, the calcula-tion of the dimensionless tidal coefficient of the merging NSBrequired by this fitting formula is computing assuming thesame SLy nuclear equation of state used for the dynamicalejecta. This ejecta is expected to be more isotropic, both interms of mass and electron fraction distribution. However,neutrinos could also affect its nucleosynthesis (Perego et al.2014; Martin et al. 2015; Lippuner et al. 2017). In particular,if the merger results in a long-lived massive NS, the ejectaelectron fraction could be systematically shifted above 0.25such that the production of heavy r -process elements is pre-vented and the nucleosynthesis produces only nuclei betweenthe first and second r -process peaks, i.e. for ≤ A ≤ .Otherwise, if a black hole forms promptly or on a timescalesmaller than the disc viscous timescale, the production ofall r -process elements is foreseen. In this case we assume a mass number interval ≤ A ≤ for the r -process nu-cleosynthesis. To distinguish between the first and the sec-ond case we use an empirical threshold value suggested bythe CoRe NSB merger database (Dietrich et al. 2018): if ( m + m ) < . M NS , max (where M NS , max ≈ .
05 M (cid:12) is themaximum cold NS mass predicted by the SLy nuclear equa-tion of state) then the central remnant does not collapses toa BH before the wind ejecta is expelled and the productionof all r -process elements in the disk ejecta is prevented. The ejecta expanding in the intergalactic (IGM) or interstel-lar (ISM) medium will slow down, forming a NSB mergerremnant similar to the remnant produced by SN explosions(see e.g. Montes et al. 2016). This will eventually mix withthe surrounding medium, enriching it with its nucleosyn-thesis yields. We assume the IGM/ISM to follow the samedensity profile as the model galaxy down to a floor num-ber density n , and to be formed by hydrogen atoms witha temperature of T ∼ K . The relation between the massand the number floor densities is given by ρ = n m p µ a ,where m p is the proton mass and µ a the mean atomic weight, µ a = . . Values of n around galaxies are largely unknownand strongly dependent on the environment and cosmologi-cal epoch. We consider a fiducial value n = − cm − , largerbut still comparable to the present average density of theUniverse ( ∼ × cm − ). To explore the possible impactof larger densities, expected for example in the early Uni-verse, we investigate also n = − cm − . We notice that theresulting interval − cm − (cid:46) n (cid:46) − cm − is compatiblewith the values inferred by the GRB afterglow emission ofGW170817 (Margutti et al. 2017; Ghirlanda et al. 2019).To model the evolution of the remnant, we assume thatthe two kinds of ejecta (characterized by different initial ki-netic energies) will produce two remnants, that we treat in-dependently. Moreover, for simplicity, we consider the rem-nant expansion to happen inside a uniform medium of den-sity ρ med = max ( ρ d ( x f ) , n m p µ a ) , (21)where x f = ( R f , z f ) represents the merger location in thegalactic reference frame with ρ d ( x f ) denoting the baryonicdensity (cf. section 3.1) at such point. If R rem and v rem denotethe radius and the speed, respectively, of the NSB remnantforward front, we model their time evolution as in the caseof SN remnants and we follow an approach similar to theone described in Haid et al. (2016) and in Beniamini et al.(2018). More specifically, we consider the following expan-sion phases: • free expansion: during this phase, the ejecta expandswith constant velocity v ej equal to either v dyn or v disk , de-pending on the nature of the ejecta considered: (cid:40) R rem ( t ) = v ej t ≤ t ≤ t , v rem ( t ) = v ej ≤ t ≤ t . (22)This phase lasts up to t = v ej (cid:16) m ej πρ (cid:17) / , i.e. the point whenthe ejecta has swept-up a mass comparable to its total mass. MNRAS , 1–18 (2019)
M. Bonetti et al.
Figure 2.
Cartoon representation of the three possible situa-tion that determine the amount of r − process enrichment (see sec-tion 5.1). • Sedov-Taylor expansion: this phase lasts up to the pointwhere radiative cooling becomes relevant. This transitionusually happens at t ∼ t TR = × yr n − . (Haid et al.2016). The phase t TR ≤ t < C TR t TR , with C TR ≈ . , is anintermediate phase in which the expansion cannot be ex-pressed as a self-similar solution. In this first study we ne-glect this additional complication and assume that the self-similar Sedov-Taylor expansion continues up to t = C TR t TR R rem ( t ) = R rem ( t ) + ξ (cid:16) E rem ρ ( t − t ) (cid:17) / t < t ≤ t , v rem ( t ) = ξ (cid:16) E SN ρ (cid:17) / ( t − t ) − / t < t ≤ t , (23)where ξ = /( π ) and E rem ≈ m ej v / is the initial kineticenergy of the ejecta. • snowplow expansion: in this phase, the ejecta has pro-duced a thin shell, containing most of the ejecta mass, thatexpands driven by the hot interior (pressure-driven snow-plow phase). This phase is characterized again by a self-similar solution, and continues up to the point where theejecta velocity equals the sound speed of the IGM/ISM, v lim ≈
10 km / s . If we denote this time as, t = (cid:169)(cid:173)(cid:171) v lim R rem ( t ) t / (cid:170)(cid:174)(cid:172) / , (24)then R rem ( t ) = R rem ( t ) (cid:16) tt (cid:17) / t < t ≤ t , v rem ( t ) = R rem ( t ) t (cid:16) tt (cid:17) − / t < t ≤ t . (25)The maximum remnant expansion is assumed to occur at t ,since after that the ejecta dissolves into the IGM/ISM. Let us consider a NSB that merges at a certain location x f = ( x f , y f , z f ) = ( R f , z f ) . If the resulting remnant is large enough (or alternatively the merger point is not too far from the hostgalaxy), a certain fraction of the ejecta can be retained bythe galaxy. Moreover, if the NSB inspiral and the remnantevolution are fast enough (i.e. T GW + t (cid:46) T SF ), the retainedmass will pollute the gas that will form a new generation ofstars. To compute the fraction of retained ejecta, the firststep is to define a suitable contour for the galaxy. We defineit as the iso-density contour in the baryonic density profile ρ lim = ρ d ( ξ lim R d , ) and we consider the surface S (enclosingthe volume V , i.e. the galaxy) defined by all points ( R p , z p ) that satisfy the condition ρ d ( R p , z p ) = ρ lim . We set ξ lim = and ξ lim = in order to explore the impact of the chosenthreshold on the retained mass. If ρ med denotes the densityinside which the remnant expands (see previous subsection),these are the possible scenarios (see Fig. 2 for a cartoonsketch): • if ρ d ( x f ) > ρ lim , the coalescence happens within thehost. When the ejecta reaches the maximum expansion ra-dius, R rem ( t ) , we evaluate the intersection of the correspond-ing spherical shell Σ with the volume V and we estimate theamount of the retained mass as the fraction of the spheri-cal surface that lies inside the galaxy times the total ejectedmass, i.e. m ret = max (cid:18)∫ Σ ∩V (cid:18) d m ej d Σ (cid:19) d Σ , . m ej , tot (cid:19) , (26)where d m ej / d Σ is the ejecta angular distribution on the rem-nant sphere. The maximum with . m ej , tot ensures that atleast half of the ejected mass remains inside the galaxy. Thisis due to the fact that the increasing density gradient movingfrom x f towards the galactic disc plane will certainly slowdown the ejecta moving in that direction. • if ρ d ( x f ) < ρ lim and Σ does not expand across the galaxyvolume V , then none of the ejecta can enrich the galaxy.Practically, the situation of non-intersection verifies whenone of conditions below is satisfied (cid:113) x f + y f − R rem ( t ) > ξ lim R d , | z f − R rem ( t )| > | z ρ lim | , (27)where ξ lim R d is the largest disk contained inside V , whilethe interval [− z ρ lim , z ρ lim ] represents the maximum z extentof the galaxy at a radial distance R f = (cid:113) x f + y f . • if ρ d ( x f ) < ρ lim and the remnant sphere extends insideor beyond the galaxy, then a certain amount of the ejectedmass can pollute the galaxy. In order to estimate this mass,we identify the fraction of solid angle that intersects thegalaxy, Ω f , using a Monte Carlo approach. The retainedmass is then computed as m ret = ∫ Ω f (cid:18) d m ej d Ω (cid:19) d Ω , (28)where d m ej / d Ω is the ejecta angular distribution. Here we comment on the dwarf galaxy case only (i.e. ρ d isgiven by the exponential disk), with the understanding that theprocedure followed for the MW is conceptually the same. To this end, we generate ∼ × propagation directions startingfrom x f and spanning the whole solid angle.MNRAS000
10 km / s . If we denote this time as, t = (cid:169)(cid:173)(cid:171) v lim R rem ( t ) t / (cid:170)(cid:174)(cid:172) / , (24)then R rem ( t ) = R rem ( t ) (cid:16) tt (cid:17) / t < t ≤ t , v rem ( t ) = R rem ( t ) t (cid:16) tt (cid:17) − / t < t ≤ t . (25)The maximum remnant expansion is assumed to occur at t ,since after that the ejecta dissolves into the IGM/ISM. Let us consider a NSB that merges at a certain location x f = ( x f , y f , z f ) = ( R f , z f ) . If the resulting remnant is large enough (or alternatively the merger point is not too far from the hostgalaxy), a certain fraction of the ejecta can be retained bythe galaxy. Moreover, if the NSB inspiral and the remnantevolution are fast enough (i.e. T GW + t (cid:46) T SF ), the retainedmass will pollute the gas that will form a new generation ofstars. To compute the fraction of retained ejecta, the firststep is to define a suitable contour for the galaxy. We defineit as the iso-density contour in the baryonic density profile ρ lim = ρ d ( ξ lim R d , ) and we consider the surface S (enclosingthe volume V , i.e. the galaxy) defined by all points ( R p , z p ) that satisfy the condition ρ d ( R p , z p ) = ρ lim . We set ξ lim = and ξ lim = in order to explore the impact of the chosenthreshold on the retained mass. If ρ med denotes the densityinside which the remnant expands (see previous subsection),these are the possible scenarios (see Fig. 2 for a cartoonsketch): • if ρ d ( x f ) > ρ lim , the coalescence happens within thehost. When the ejecta reaches the maximum expansion ra-dius, R rem ( t ) , we evaluate the intersection of the correspond-ing spherical shell Σ with the volume V and we estimate theamount of the retained mass as the fraction of the spheri-cal surface that lies inside the galaxy times the total ejectedmass, i.e. m ret = max (cid:18)∫ Σ ∩V (cid:18) d m ej d Σ (cid:19) d Σ , . m ej , tot (cid:19) , (26)where d m ej / d Σ is the ejecta angular distribution on the rem-nant sphere. The maximum with . m ej , tot ensures that atleast half of the ejected mass remains inside the galaxy. Thisis due to the fact that the increasing density gradient movingfrom x f towards the galactic disc plane will certainly slowdown the ejecta moving in that direction. • if ρ d ( x f ) < ρ lim and Σ does not expand across the galaxyvolume V , then none of the ejecta can enrich the galaxy.Practically, the situation of non-intersection verifies whenone of conditions below is satisfied (cid:113) x f + y f − R rem ( t ) > ξ lim R d , | z f − R rem ( t )| > | z ρ lim | , (27)where ξ lim R d is the largest disk contained inside V , whilethe interval [− z ρ lim , z ρ lim ] represents the maximum z extentof the galaxy at a radial distance R f = (cid:113) x f + y f . • if ρ d ( x f ) < ρ lim and the remnant sphere extends insideor beyond the galaxy, then a certain amount of the ejectedmass can pollute the galaxy. In order to estimate this mass,we identify the fraction of solid angle that intersects thegalaxy, Ω f , using a Monte Carlo approach. The retainedmass is then computed as m ret = ∫ Ω f (cid:18) d m ej d Ω (cid:19) d Ω , (28)where d m ej / d Ω is the ejecta angular distribution. Here we comment on the dwarf galaxy case only (i.e. ρ d isgiven by the exponential disk), with the understanding that theprocedure followed for the MW is conceptually the same. To this end, we generate ∼ × propagation directions startingfrom x f and spanning the whole solid angle.MNRAS000 , 1–18 (2019) SB orbits and r-process enrichment r -process nucleosynthesis abundances As stated in section 4.3, NSB merger ejecta is characterizedby a heterogeneous composition of heavy r -process elementspotentially produced in different components. In order toasses the contribution of each element to the mass retainedby the host galaxy, we consider that the relative abundancesof heavy elements closely follows the Solar System (SS)abundances obtained from Lodders (2003) and decomposedin its s − and r − process contributions following Sneden et al.(2008). If A min and A max are the minimum and maximummass number produced by the r -process nucleosynthesis ina certain ejecta component, the mass fraction of each ele-ment E i (whose isotopes are such that A min ≤ A i ≤ A max )inside the retained mass of each component can be expressedas X ( E i ) = m ej , ret ( E i ) m ej , ret = ( Y ( E i )) (cid:12) (cid:104) A i (cid:105) (cid:205) k ( Y ( E k )) (cid:12) (cid:104) A k (cid:105) , (29)where Y ( E i ) is the elemental abundance of E i , (cid:104) A i (cid:105) its meanmass number, as obtained by the SS abundances, and thesum at denominator ranges over all produced elements. Once A min and A max are given, X ( E i ) is unambiguously de-termined and allows us to obtain the mass composition ofthe r -process nucleosynthesis retained material in each com-ponent.The total retained mass of each element is then obtainedas the sum of the element retained masses in all components.Finally, the abundance ratio of two elements, defined as [ E i / E j ] = log (cid:18) Y ( E i ) Y ( E j ) (cid:19) − log (cid:18) Y ( E i ) Y ( E j ) (cid:19) (cid:12) , (30)can be equivalently computed as [ E i / E j ] = log (cid:18) m ( E i )(cid:104) A ( E j )(cid:105) m ( E j )(cid:104) A ( E i )(cid:105) (cid:19) − log (cid:18) n i n j (cid:19) (cid:12) , (31)where the last term is a constant determined by the SS abun-dances.In addition to the retained mass in r -process elements,we also compute the iron mass produced by CCSNe. Accord-ing to observations, 2/3 of CCSNe explode as type II SN andproduce, on average, 0.02 M (cid:12) of iron; 1/3 are SNIbc andproduce, on average, 0.2 M (cid:12) of iron (Li et al. 2011; Droutet al. 2011). For each galaxy realization, if N CCSN is the num-ber of CCSNe occurred within a time t , we sample N CCSNII type II SNe from a Poisson distribution with average equal to N CCSN / , and compute N CCSNIbc = N CCSN − N CCSNII . SinceCCSNe are expected to explode inside the galaxy, we fur-ther assume that all the iron is retained inside the galaxyand we estimate its amount from the average quantity pro-duced per event. More detailed studies taking into accountthe kinetic energy of the ejecta as well as the elemental mix-ing (Beniamini et al. 2018; Emerick et al. 2018), showed thatthe amount of retained elements could sensitively depend onthe source. Thus, our values provide upper limits to the iron Throughout the literature, the SS abundance of an element E i is usually quoted in terms of the astronomical logarithmic scale A ( E i ) = log ( n i / n H ) + , with n H the number density of Hy-drogen atoms or according to the cosmochemical scale, which in-stead normalises the number of Silica atoms to . The relationbetween the two scale is given by A ( E i ) = . + log N cosmo ( E i ) . enrichment due to CCSNe. For example, Beniamini et al.(2018) estimated a retain factor of 0.2-0.9 for ejecta prod-ucts exploded within a dwarf galaxy. In this section we present the result obtained when all the in-gredients presented in sections 2-5 are combined. We first fo-cus on a fiducial case characterised by n = − cm − , ξ lim = , x = , p l = . in Sec. 6.1. We then discuss the impact ofparameters by varying each of them one at a time in Sec.6.2. A first qualitative interpretation of our results is given byFig. 3, where we report the x − z projection of the NSBmerger locations (black dots in the figure) as well as theexpansion sphere (circles in the projections) of the ejected r − process material. In this plot and for the other that fol-low, we consider, from left to right, three galaxy models withtotal baryonic mass of , , ∼ × M (cid:12) , indicativelyrepresentative of UFDs, classical dwarf and MW-like galax-ies. From the figure it is evident that two factors primarydetermine the level of enrichment of the progenitor galaxy(dark blue contour in the figure): the number of NSBs thatform in the galaxy and merge within T SF , and the distancefrom the galaxy at which each NSB travels prior to coales-cence. Both factors crucially depend on the baryonic massof the galaxy. In very low mass galaxies (e.g. left panel ofFig. 3) the enrichment level is quite low since the averagenumber of formed NSB is usually small, around unity or frac-tions of it. In addition, the potential well provided by thegalaxy is rather shallow, therefore the SN kicks can imprintenough velocity such that a NSB system can substantiallyrecede from the parent galaxy. Even if a NSB is gravitation-ally bound to the galaxy, it often spends most of its orbitaltime rather far from the galactic disc. On the contrary, formassive galaxies (e.g. MW-like ones, right panel of Fig. 3)the level of enrichment is much higher given the increasednumber of massive stars that can produce NSB systems, thelonger T SF , as well as the higher escape velocity that preventNSB to travel far away from the galaxy (the energetics ofthe SNae is reasonably assumed to be the same irrespectiveof the galaxy mass). Between the above situations, galaxieswith intermediate masses continuously connect the two ex-tremes, as visible for example in the central panel of Fig. 3.This can be more quantitatively inferred from Fig. 4, inwhich we report the mass distribution of the whole amountof produced r − process material (dashed magenta lines) com-pared to the r − process mass actually retained by the hostgalaxy (solid black lines). For each galaxy we also consideronly NSBs that coalesce within the time available for starformation ( T SF ), since a galaxy with no more star formationactivity cannot form stars enriched with heavy elements.In the least massive galaxy case ( M b = M (cid:12) , leftpanel), only a small fraction of the realizations ( < )presents at least one NSB merger happening within T SF = . The small probability that the corresponding ejectaintersects the galactic disc further reduces the probabilitythat a galaxy realization displays a r -process material en-richment ( ∼ ). For more massive galaxies the distributions MNRAS , 1–18 (2019) M. Bonetti et al.
Figure 3.
Projection in the x − z galactic frame of NSB coalescence positions (black dots) and expansion sphere of ejected material.Dynamical ejecta are displayed in red, while wind ejecta in green. Left panel : Dwarf 1 case.
Central panel : Dwarf 2 case.
Right panel :MW-like case. In all panels the dark blue profiles represent the galaxy density iso-contour equal to the chosen ρ lim (see section 5.1 fordetails). − − − log M ejecta [M (cid:12) ] N r e a li s a t i o n Dwarf model 1, T SF = 1 . − − − log M ejecta [M (cid:12) ] Dwarf model 3, T SF = 2 . retained r-process ejectatotal r-process ejecta log M ejecta [M (cid:12) ] MW model, T SF = 10 . Figure 4.
Comparison between distributions of the total ejected mass (dashed purple lines) compared to the actual retained mass (solidblack line) for the standard case.
Left panel : Dwarf model 1.
Central panel : Dwarf model 3.
Right panel : MW model. Note how thediscrepancy between the two masses decreases with increasing galaxy mass. are more peaked with an higher number of realisations pro-viding some enrichment, meaning that in these systems NSBmergers are more likely and generally closer to the galaxy.Moreover, from the figure it is particularly clear that thereexists a dilution factor between the r − process mass producedby NSB mergers and the actual amount captured by the hostgalaxy. This dilution factor is simply due to the weakening ofthe galactic gravitational field and increases with decreasinggalaxy mass. For instance, in the least massive galaxy casethe amount of retained material is diminished by at least afactor 10 with respect to the produced one. Overall, thereis a potential discrepancy of more than three order of mag-nitude for the lightest galaxies between the produced andthe retained r -process material. For larger galaxy masses,as of a consequence of the larger numbers of merging NSBsand of the deeper gravitational potential of the galaxy, theamount of r -process material increases and the discrepancybetween the produced and the retained material decreases. A global dilution factor of the order of ten is still visible forthe M b = M (cid:12) model (central panel). Finally, no signifi-cant discrepancy is visible for the most massive case (rightpanel), where a large number of NSB merger occur within t =
10 Gyr < T SF , most of them pollute the galaxy, and nosignificant dilution factor applies. In the following, unlessdifferently specified, we will always refer to r − process massactually retained by galaxies.A deeper insight into the specific origin of r − processmaterial is given in Fig. 5 and Fig. 6, for the CCSN toNSB ratio x =
150 and 50, respectively. Upper panels show r − process mass generated from dynamical ejecta, both fromequatorial (solid red lines) and polar components (dashedblue lines). Middle panels report instead the mass comingfrom disk wind ejecta, labelled as “wind 1” for the windcoming from more massive NSB that do form a BH beforethe disk wind emerges (solid orange lines) and “wind 2” forthe ejecta coming from NSB merger that form a long-lived MNRAS000
150 and 50, respectively. Upper panels show r − process mass generated from dynamical ejecta, both fromequatorial (solid red lines) and polar components (dashedblue lines). Middle panels report instead the mass comingfrom disk wind ejecta, labelled as “wind 1” for the windcoming from more massive NSB that do form a BH beforethe disk wind emerges (solid orange lines) and “wind 2” forthe ejecta coming from NSB merger that form a long-lived MNRAS000 , 1–18 (2019)
SB orbits and r-process enrichment Dwarf model 1, T SF = 1 . dynamical eqdynamical pol Dwarf model 3, T SF = 2 . MW model, T SF = 10 . N r e a li s a t i o n wind 1wind 2 − − − − log M ret /M ejecta , tot r-process massEu mass − − − − log M ret /M ejecta , tot − − − − log M ret /M ejecta , tot Figure 5.
Distribution of retained mass that enriches the galaxy over total ejected mass for a CCSN to NSB ratio x = . Left panels :Dwarf model 1.
Central panels : Dwarf model 3.
Right panels : MW model.
Top panels : dynamical polar ejecta (blue dashed line) anddynamical equatorial ejecta (red solid line).
Central panels : wind 1 ejecta (green dashed line) and wind 2 ejecta (orange solid line).
Bottom panels : total retained r − process (black solid line) and total retained Europium (cyan dashed line). Dwarf model 1, T SF = 1 . dynamical eqdynamical pol Dwarf model 3, T SF = 2 . MW model, T SF = 10 . N r e a li s a t i o n wind 1wind 2 − − − − log M ret /M ejecta , tot r-process massEu mass − − − − log M ret /M ejecta , tot − − − − log M ret /M ejecta , tot Figure 6.
Same as Fig. 5, but considering the optimistic case with x = . NS (dashed green lines). We recall that the latter cases donot produce a significant amount of Eu. Finally, lower pan-els show the distributions of total retained r − process mass(solid black lines) as well as retained europium mass (dashedcyan lines). For the dynamical ejecta, the sin θ mass distri-bution, as well as the larger solid angle, favours pollutionfrom the equatorial component, inside which europium isproduced. For the disk winds, lighter NSB mergers producemore massive disc whose ejecta can pollute the galaxy more significantly, but without producing significant amount ofheavy r -process elements, including europium. Summing upof the relevant contributions, the amount of retained eu-ropium is thus usually three orders of magnitudes smallerthan the total amount of r -process material. Results for themore optimistic case ( x = , Fig. 6) are qualitatively verysimilar to the x = ones, but showing an higher numberof successful events and larger amount of r -process materialand europium masses. Still for the least massive case the ra- MNRAS , 1–18 (2019) M. Bonetti et al. N r e a li s a t i o n M r − process Dwarf model 1, T SF = 1 . Dwarf model 3, T SF = 2 . MW model, T SF = 10 . − − − − log M ret /M ejecta , tot N r e a li s a t i o n M Eu x = 50 x = 150 x = 450 x = 1350 − − − − log M ret /M ejecta , tot − − − − log M ret /M ejecta , tot Figure 7.
Distribution of retained r − process mass (upper panels) and retained europium mass (lower panels) over total ejected massthat pollute a selected galaxy model (as labelled) when considering a different ratio x between the number of CCSN and the actuallyformed NSB. We explore a range from x = (optimistic) to x = (extremely pessimistic). N r e a li s a t i o n M r − process Dwarf model 1, T SF = 1 . Dwarf model 3, T SF = 2 . MW model, T SF = 10 . − − − − log M ret /M ejecta , tot N r e a li s a t i o n M Eu n = 10 − n = 10 − − − − − log M ret /M ejecta , tot − − − − log M ret /M ejecta , tot Figure 8.
Same as Fig. 7, but considering a higher mean IGM density of n = − cm − , instead of n = − cm − used for the standardcase. Note how an higher n prevents any r − process enrichment in the least massive galaxy model, where generally NSB can cover largedistances before merging. tio of retained vs produced europium mass remains usuallyvery low, i.e. around − − − . Once again, larger galaxymasses reduce fluctuations in the distributions, leading tonarrow histograms in the MW-like cases. We now turn to explore the dependence of our results onsome of the key parameters of the model.In Fig. 7 we present the distribution of r -process mate-rial for different x , i.e. the ratio between CCSNe and formedNSBs. Also for the least massive galaxy the increased num-ber of potential NSB mergers leads to an increment of theretained mass, but still revelling a high scatter, meaning MNRAS000
Same as Fig. 7, but considering a higher mean IGM density of n = − cm − , instead of n = − cm − used for the standardcase. Note how an higher n prevents any r − process enrichment in the least massive galaxy model, where generally NSB can cover largedistances before merging. tio of retained vs produced europium mass remains usuallyvery low, i.e. around − − − . Once again, larger galaxymasses reduce fluctuations in the distributions, leading tonarrow histograms in the MW-like cases. We now turn to explore the dependence of our results onsome of the key parameters of the model.In Fig. 7 we present the distribution of r -process mate-rial for different x , i.e. the ratio between CCSNe and formedNSBs. Also for the least massive galaxy the increased num-ber of potential NSB mergers leads to an increment of theretained mass, but still revelling a high scatter, meaning MNRAS000 , 1–18 (2019)
SB orbits and r-process enrichment − − − − log M Eu /M ejecta , tot N r e a li s a t i o n Dwarf model 1, T SF = 1 . Eu from wind 1Eu from wind 1+2 − − − − log M Eu /M ejecta , tot Dwarf model 3, T SF = 2 . − − − − log M Eu /M ejecta , tot MW model, T SF = 10 . Figure 9.
Distribution of retained europium mass over total ejected r − process mass when its production is allowed also from long-livedNS (wind 2 case). that the enrichment process is dominated by small numberstatistic. A clear trend is instead observable in the othertwo cases (central and right panels of Fig. 7) showing anincrease of nearly two orders of magnitude in the peak ofretained mass when going from x = (very pessimistic)to x = (optimistic).Another crucial parameter that can affect the amountof r − process enrichment, especially at low galactic masses,is the IGM density. If NSBs merge outside from the galaxy,a higher IGM density determines a smaller expansion of theejecta bubble, preventing the ejected material to fall back onthe parent galaxy. Such trend is shown in Fig. 8, in which wecompare our standard case with an IGM number density of n = − cm − (solid blue lines) to a situation with n onehundred times higher. No sensible effect is seen in the MWcase, while for Dwarf model 3 we witness a slightly decreasein the number of successful enrichment realisations and, atthe same time, a decrease in the efficiency of the r -processenrichment. Very different is instead the case of Dwarf model1 where assuming n = − cm − no r − process enrichmentis verified (with 200 galaxy realisations).In addition, for the europium mass only, we verify thedependence on our assumptions about the the r − processnucleosynthesis in the various ejecta components. In par-ticular in Fig. 9 we compare our standard case, in whichno europium is produced in the disk wind emerging froma remnant characterized by a long-lived massive NS, to asituation in which all disc winds produce europium, i.e. weassume the same nucleosynthesis with ≤ A ≤ for allwinds. Again for the smallest galaxy case allowing this ex-tra production of europium can increase up to one order ofmagnitude the quantity of Eu, but we stress that the resultis heavily affected by small number fluctuations, with essen-tially only two realisations (out of 200) showing a significantincrease in the retained europium mass. For more massivegalaxies the impact is instead milder, determining at mostan increase of a factor of a few.Finally, we have tested the impact of other input param-eters of the models, including the extension of the galaxy incomputing the retained mass and the probability of receiv-ing of low kick, i.e. ξ and p l . In both cases, the amount ofretained material scales as expected in the case of low mass galaxies: an increase from ξ = to ξ = translates in anincrease of ∼ ( / ) in the retained mass, due to a more ex-tended disc surface. For the latter parameter, a decreasein the probability from 0.7 to 0.6 produces essentially no no-ticeable differences in the amount of retained mass, exceptfor the least massive galaxy case, where a slightly decrease ofthe captured mass arises. Any dependence on both ξ and p l becomes less and less relevant for large galaxy masses and,in particular, for a MW-like galaxy. Fig. 10 shows the amount of retained europium mass as afunction of the iron mass produced by CCSNe explosions.We stress that the nucleosynthesis contribution of SNIa isneglected in our work. This assumption is valid as long asstar formation happens significantly before SNIa start toexplode (as in the case of UFD galaxies). Otherwise, theamount of iron computed provides a lower limit. In the fig-ure we report the comparison between our results (small dotsand empty circles) and observational data points referring toUFDs (red triangles, square for Ret II and diamond for TucIII) and classical dwarfs (blue square), respectively. Eachsmall dot (empty circle) represents a successful realisationin which europium is effectively captured by the host galaxyassuming a CCSN/NSB ratio of x = ( x = ). The lattervalues have been chosen by comparing the amount of eu-ropium retained in MW-like galaxy models with estimate ofeuropium mass in our Galaxy. Assuming a mass of (cid:12) for r -process material with A ≥ (see e.g. Hotokezaka et al.2018, and references therein) and an europium mass frac-tion of m Eu / m r − proc , A ≥ = . (Lodders 2003), we found (cid:46) x (cid:46) , depending on the detailed parameter choice.Observational data points are taken from figure 1 and table Nevertheless, we expect our standard choice ( ξ = , corre-sponding to three times the radial scale of the exponential pro-file) to cover a significant fraction of the galaxy in which stars areproduced.MNRAS , 1–18 (2019) M. Bonetti et al. − M Fe [M (cid:12) ] − − − − − − − − − M E u [ M (cid:12) ] Dwarf model 1, x = 150Dwarf model 1, x = 50Dwarf model 2, x = 150Dwarf model 2, x = 50Dwarf model 3, x = 50Dwarf model 3, x = 150Dwarf model 4, x = 50Dwarf model 4, x = 150UFDs (upper limits)Ret IITuc IIIClassical dwarfs − M Fe [M (cid:12) ] − − − − − − − − − M E u [ M (cid:12) ] Dwarf model 1, x = 150Dwarf model 1, x = 50Dwarf model 2, x = 150Dwarf model 2, x = 50Dwarf model 3, x = 50Dwarf model 3, x = 150Dwarf model 4, x = 50Dwarf model 4, x = 150UFDs (upper limits)Ret IITuc IIIClassical dwarfs Figure 10.
Retained europium mass as a function of iron mass for all our considered dwarf galaxy models. Dots (red for Dwarf model1, blue for Dwarf model 2 to 4) refer to the optimistic case with x = , while open circles to the fiducial case with x = . On thesame figure we also report observational estimates of europium and iron masses for a sample of UFDs (red triangles as upper limits, redsquare for Reticulum II and red diamond for Tucana III) and classical dwarf (blue squares); we adopted these data from Beniamini et al.(2016b), apart for Tucana III, for which we calculated europium and iron mass using the same approximations. Left panel : europiumproduction is allowed in dynamical and wind 1 ejecta.
Right panel : europium can be produced in all components (i.e. also wind 2).Despite the large uncertainties, our approach broadly suggests a systematic deficiency in europium at a fixed iron mass. − − N r e a li s a t i o n Dwarf model 1, T SF = 1 . x = 150 x = 50 − − Dwarf model 2, T SF = 1 . − − [Eu/Fe] N r e a li s a t i o n Dwarf model 3, T SF = 2 . − − [Eu/Fe] Dwarf model 4, T SF = 4 . Figure 11.
Element abundance of europium over iron as obtainedin our models for dwarf galaxies of increasing baryonic mass. Solidorange (dashed cyan) lines refer to a CCSN to NSB rate x = ( x = ). − M (cid:12) ), and shows a small dispersion. The europiummass instead shows a much larger scatter, especially forDwarf model 1 and 2, where variations span three ordersof magnitude. On the contrary, at increasing galaxy massthe scatter decrease. Again this result is a combination ofthe small number of events and the geometric dilution fac-tor. From Fig. 10 an increasing trend with galaxy mass forthe europium mass is clearly visible, but when comparedto the observational data points, our procedure systemati-cally underestimate m Eu for all galaxy models with baryonicmass in the range − M (cid:12) . To test potential system-atic uncertainties in our model, in the right panel of Fig. 10,we report the same quantities of the left panel but we con-sider that europium production is active also in the windemerging from long-lived remnant. This cause a shift of atmost 0.5 dex for m Eu , easing the tension, but without defini-tively solving it. Therefore, despite the large uncertaintiesthat affect the followed procedure, our results even in themost optimistic case (right panel of Fig. 10, x = case),might imply a possible tension with the scenario in whicheuropium enrichment is only generated by NSB mergers.In Fig. 11 we provide histograms of [ Eu / Fe ] for all ourfour dwarf models. Once again, results for models 1,2, and3 are systematically smaller than the values required to ex-plain abundances in Reticulum II and in classical dwarfs, asreported in table 1 of Beniamini et al. (2016b) by at leastone dex, even in the most optimistic case x = .It is still possible to argue that the effective amount of MNRAS000
Element abundance of europium over iron as obtainedin our models for dwarf galaxies of increasing baryonic mass. Solidorange (dashed cyan) lines refer to a CCSN to NSB rate x = ( x = ). − M (cid:12) ), and shows a small dispersion. The europiummass instead shows a much larger scatter, especially forDwarf model 1 and 2, where variations span three ordersof magnitude. On the contrary, at increasing galaxy massthe scatter decrease. Again this result is a combination ofthe small number of events and the geometric dilution fac-tor. From Fig. 10 an increasing trend with galaxy mass forthe europium mass is clearly visible, but when comparedto the observational data points, our procedure systemati-cally underestimate m Eu for all galaxy models with baryonicmass in the range − M (cid:12) . To test potential system-atic uncertainties in our model, in the right panel of Fig. 10,we report the same quantities of the left panel but we con-sider that europium production is active also in the windemerging from long-lived remnant. This cause a shift of atmost 0.5 dex for m Eu , easing the tension, but without defini-tively solving it. Therefore, despite the large uncertaintiesthat affect the followed procedure, our results even in themost optimistic case (right panel of Fig. 10, x = case),might imply a possible tension with the scenario in whicheuropium enrichment is only generated by NSB mergers.In Fig. 11 we provide histograms of [ Eu / Fe ] for all ourfour dwarf models. Once again, results for models 1,2, and3 are systematically smaller than the values required to ex-plain abundances in Reticulum II and in classical dwarfs, asreported in table 1 of Beniamini et al. (2016b) by at leastone dex, even in the most optimistic case x = .It is still possible to argue that the effective amount of MNRAS000 , 1–18 (2019)
SB orbits and r-process enrichment r -process material produced by NSB mergers is not so wellconstrained and a larger production could solve the problem.However, there is at least another prediction of the modelwhich we found more difficult to reconcile with observations.As previously pointed out, in our model dwarf galaxies showa significant spread of Eu enrichment compared to a relativefixed enrichment of iron. This spread is dependent on thegalaxy mass and increasing for less massive objects. Intu-itively, this is connected to the shallower potential well thatthe NSB encounter after the second SN kick in a less mas-sive host galaxy. In the model, we do not follow any possiblestochastic enrichment inside the galaxy due to finite dimen-sion of the pollution by our NSMs; the spread obtained isbetween the average Eu in single galaxies. Therefore, the ob-servational expectation is that classical dwarf galaxies candiffer in their average Eu/Fe ratio from 1 dex for the moremassive one to more than 3 dex for the lightest ones. How-ever, according to the data up to now collected, this is notthe case. Observed dwarf spheroidal galaxies with similar fi-nal stellar masses show a relative small scatter in iron (Mc-Connachie 2012), which is compatible with our model. Onthe other hand, the enrichment in Eu for classical dwarf is,in most of the cases, proportional to that of iron, differ-ently from the outcome of our numerical modelling. For theclassical dwarf, the only hint of a substantial variation ofthe Eu enrichment is the [Eu/Fe] ratio measured in stars ofSagittarius dwarf galaxy (McWilliam et al. 2013). The sit-uation is different for UFD galaxies. At the moment mostof them seem to have an extremely low enrichment of neu-tron capture elements with - at present - the only excep-tion Reticulum II (Ji et al. 2016a) and possibly Tucana III(Hansen et al. 2017). According to our model, NSB mergerscan enrich of the order of 1-2% galaxies in the mass regimeof Reticulum II. Therefore, we could have simply randomlydetected this object, although the chance are relatively lowconsidering the dozen of UFD galaxies with measured stel-lar abundances. However, in the case the measurements ofeuropium in four additional stars of Tucana III will be con-firmed (Marshall et al. 2018), then the random probabilitywill be certainly too low and a clear tension with the pre-diction would be confirmed also from this prospective. Weshould underline that the debate is still on whether UFDgalaxies used to be isolated galaxies or are just fractions oflarger tidal disrupted objects. This would relax the constrainfrom this side.If the stars that we observe in UF and classical dwarfsformed at high redshift z , the tension with our models couldpotentially increase. In fact, we expect the average particledensity in the Universe to increase as a function of z as ( + z ) . This implies that for dwarf galaxies forming stars withinthe first 3.3 Gyr after the Big-Bang (i.e. z ∼ ) the IGM couldbe 10 larger than the present Universe average and likelylarger than our standard case ( n = − cm − ). This increasecould limit the extension of the merger remnant, furtherreducing the dilution factor and, ultimately, the amount ofmatter enriching the host galaxy.Our results depends on the chosen binary semi-majoraxis distribution. Neglecting eccentricity effects, a uniformdistribution of log ( a ) translates in a t − delay time dis-tribution for NSB mergers. Beniamini & Piran (2019) re-cently showed that the observed distribution of GalacticNSBs might suggest the presence of a significant popula- tion (at least 40%) of fast merging ( (cid:46) ) binaries, largerthan the fraction implied by our distribution ( ∼ yr and yr inthe M (cid:12) and M (cid:12) dwarf galaxy model, respectively. NSB mergers are nowadays clearly recognized as one of themajor source of r -process nucleosynthesis in the Universeand a key player in galactic chemical evolution. The eventrate and the mass per event needed to explain UFD enrich-ment appears to be consistent with those in the Milky Wayand with the NSB merger properties obtained by the anal-ysis of GW170817 (see e.g. Hotokezaka et al. 2018). In thispaper, we have explored the impact of the orbital motion ofbinary systems of NSs around galaxies prior to merger onthe r -process enrichment. We have found that for low masssystems, i.e. disc galaxies with a baryonic mass M b rang-ing from up to M (cid:12) , the motion of the binary dueto the kicks imparted by the two SN explosions determinesa merger location potentially detached from the disc plane,even for gravitationally bound systems.The immediate consequence is a dilution of the amountof r -process material retained by the galaxy within its starforming age T SF (and thus potentially available for the nextgeneration of stars). This effect is more severe for low massdisc galaxies. Assuming a log-flat distribution for the semi-major axis, realistic distributions of NSB parameters, a pro-duction rate of one double NS system every 150 CCSNe, anda rather dilute IGM (with a density of n = − cm − ), inthe least massive case we have explored ( M b = M (cid:12) ) agalaxy has a ∼ probability of producing a merging NSBwithin ( T SF (cid:46) ), and a significantly lower probability ofretaining r -process elements from this single event ( ∼ ).Since the merger happens at distances comparable or largerthan the galactic disc size and outside from the disc plane, afraction ranging between 0.1 and 0.001 of the ejected mass isactually retained (corresponding to − − − M (cid:12) ) in spiteof the ejection of a few − − − M (cid:12) of r -process material,most of which stops and mixes with the IGM. This dilu-tion effects is also present in more massive galaxies, but itbecomes less and less relevant as M b increases, and it haspractically no relevance for a MW-like galaxy.We have also estimated the amount of Eu retained bythe galaxy. We considered that mass ejection in NSB mergershappens through different channels and each channel has apotentially different nucleosynthesis, based on the influenceof neutrino irradiation. We have found that the amount of re-tained Eu is usually ∼ − the amount of retained r -processmaterial. This is due to the presence of a significant fractionof binaries that produce a long-lived massive NS. For theseremnants, discs are usually more massive and the persistentneutrino irradiation suppresses the synthesis of elements be-tween the second and the third r -process peaks. The relativefraction of Eu is largely unaffected by the galaxy mass, how-ever the paucity of enrichment events in the low mass caseproduces larger fluctuations. MNRAS , 1–18 (2019) M. Bonetti et al.
We have tested the robustness of our results with re-spect to the parameters entering the model. The most rele-vant parameters are the fraction of double NS systems withrespect to the number of CCSNe ( x ) and the IGM density( n ). For the former, even considering the most optimisticcase (i.e. one double NS system every 50 CCSNe), the resultsstay within a factor of a few our standard case. For the lat-ter, a significantly larger density (which is what we expect inthe early Universe) determines less extended NSB remnantsand prevents r -process enrichment in the least massive case.The precise modelling of the composition of NSB ejectais still affected by uncertainties, as well as our knowledgeof the fraction of systems that forms a long-lived remnant.Thus, we have repeated our analysis assuming that all discwinds produce Eu in the same ratio. We verified that ourstandard results are qualitatively robust, since the amountof retained Eu increases only by a factor of few. Calculationsof the retained r -process material allows us to compare withobserved abundances of iron and europium from UFD andclassical dwarfs. Due to the dilution effect on the retained r -process material, all our realizations show a systematic de-ficiency in the europium abundance compared with observedabundances. Additionally, for a fixed galaxy model and es-pecially for low mass galaxies, the large intrinsic variabilityintroduced by the dilution process and small number statis-tics is not visible in the observations. Since the dilution ispractically negligible in the MW-like galaxy model, in thatcase we have estimated the CCSN to NSB rate within ourassumptions and we found (cid:46) x (cid:46) , with a preferenceon the low value side.Both the discrepancies on the absolute values and ontheir spread could potentially imply a tension in the expla-nation of the europium production in dwarf galaxies as pro-duced by NSB mergers only. This conclusion could dependon observational uncertainties in the estimate of the elemen-tal abundances or it could imply a significant change in oneor more of the canonical assumed parameters. However, toreconcile our models with observations a systematically op-timistic choice of parameters is required, together with asignificant revision of the observed elemental abundances.On the one hand, a possible solution to the discrepancyis to assume the existence of a fast merger population (e.g.Beniamini & Piran 2019), such that a significant fraction ofNSBs merge within yr, or even a few times yr. Underthese conditions the merger does not happen too far from thegalaxy and the dilution factor becomes (cid:38) . . If, on the otherhand, this tension is confirmed, the existence of additionalsites for the production of r -process elements is necessary. Asimilar conclusion was also obtained by other authors andmotivated by the difficulty to inject r-process elements earlyenough to explain the Eu abundances in metal-poor stars(Matteucci et al. 2014; Cescutti et al. 2015b; Wehmeyer et al.2015; Haynes & Kobayashi 2019). Moreover, also Cˆot´e et al.(2019) and Simonetti et al. (2019) further suggested an extraproduction site of europium as a possible way of reproduc-ing the decreasing trend of [Eu/Fe] in the Galactic disk, adifferent feature that is the result of ∼ Gyr of chemicalevolution.BH-NS mergers are also possible sources of r -process el-ements (e.g. Shibata & Taniguchi 2011). The lack of a MNSin the remnant favours the production of the heaviest r -process elements (e.g. Roberts et al. 2017; Lippuner et al. 2017). This could enhance the amount of retained europiumby a factor of a few. However, even if BH-NS merger ratesare highly unconstrained, we expect them to be significantlylower than NSB merger rates (Abbott et al. 2018). A possi-ble solution would thus require qualitatively different binaryparameters, for example much lower kick velocities or muchsmaller initial separations, resulting from a possibly differ-ent binary evolution. This could however be in tension withpopulation synthesis results, considering that a significantamount of ejecta from BH-NS merger requires a not toolarge mass ratio ( (cid:46) ), for moderately high BH spins.An alternative solution is represented by special classesof CCSNe, able to produce r -process elements with anamount comparable to the one of NSB mergers and witha similar rate. Possible examples include magnetically-driven CCSNe (Fujimoto et al. 2008; Winteler et al. 2012;Nishimura et al. 2015; M¨osta et al. 2018) and disk ejectafrom collapsar models (e.g. Malkus et al. 2012; Siegel et al.2019). In these cases the ejection of r -process material hap-pens still on a sufficiently short time scale and inside thegalaxy, such that no significant dilution factor affects the en-richment (Beniamini et al. 2018). Magnetically-driven CC-SNe have also the advantage that they could also explainthe chemical enrichment of Galactic halo (Cescutti & Chi-appini 2014). Further multi-physics studies combining galac-tic (chemical) evolution, binary population synthesis, NSBmergers, and taking into account nuclear uncertainties arerequired to address all the open issues in the field. In thisrespect, a crucial role is represented by the forthcoming de-termination of more precise compact binary merger rates bythe Advanced LIGO and Virgo detectors (Aasi et al. 2015;Acernese et al. 2015; Abbott et al. 2018). ACKNOWLEDGEMENTS
MB, AP and MD acknowledge CINECA, under the TEON-GRAV initiative, for the availability of high performancecomputing resources and support. G. Cescutti acknowl-edges financial support from the European Union Horizon2020 research and innovation programme under the MarieSk(cid:32)lodowska-Curie grant agreement No. 664931.
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