Neutron stars mergers in a stochastic chemical evolution model: impact of time delay distributions
MMNRAS , 1–14 (2020) Preprint 22 September 2020 Compiled using MNRAS L A TEX style file v3.0
Neutron stars mergers in a stochastic chemical evolutionmodel: impact of time delay distributions
L. Cavallo, G. Cescutti , and F. Matteucci , , Dipartimento di Fisica, Sezione di Astronomia, Universit ˜A˘a di Trieste, Via G. B. Tiepolo 11, 34143 Trieste, Italy INAF, Osservatorio Astronomico di Trieste, via G. B. Tiepolo 11, 34143 Trieste, Italy IFPU, Istitute for the Fundamental Physics of the Universe, Via Beirut, 2, 34151, Grignano, Trieste, Italy INFN, Sezione di Trieste, via A. Valerio 2, 34127 Trieste, Italy
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
We study the evolution of the [Eu/Fe] ratio in the Galactic halo by means of a stochas-tic chemical evolution model considering merging neutron stars as polluters of eu-ropium. We improved our previous stochastic chemical evolution model by adding atime delay distribution for the coalescence of the neutron stars, instead of constantdelays. The stochastic chemical evolution model can reproduce the trend and the ob-served spread in the [Eu/Fe] data with neutron star mergers as unique producers ifwe assume: i) a delay time distribution ∝ t − . , ii) a M Eu = . × − M (cid:12) per event,iii) progenitors of neutron stars in the range − M (cid:12) and iv) a constant fraction ofmassive stars in the initial mass function (0.02) that produce neutron star mergers.Our best model is obtained by relaxing point iv) and assuming a fraction that varieswith metallicity. We confirm that the mixed scenario with both merging neutron starsand supernovae as europium producers can provide a good agreement with the datarelaxing the constraints on the distribution time delays for the coalescence of neutronstars. Adopting our best model, we also reproduce the dispersion of [Eu/Fe] at a givenmetallicity, which depends on the fraction of massive stars that produce neutron starmergers. Future high-resolution spectroscopic surveys, such as 4MOST and WEAVE,will produce the necessary statistics to constrain at best this parameter. Key words:
Galaxy: evolution – Galaxy: halo – stars: abundances – stars: neutron– ˆa ˘A¸S nuclear reactions, nucleosynthesis, abundances – binaries: close
The majority of all nuclei that are heavier than the iron-peak element (A ≥
70) are produced by neutron-capturereactions. The neutron capture processes are divided intotwo different classes: rapid or r-process (neutron capturetimescale shorter than β decay) and slow or s-process (in thiscase the neutron-capture timescale is longer than β decay).Most neutron-capture elements are produced by both r ands-process, but for some of these heavy nuclei, the productionis dominated by only one process. A series of works found aspread of r-process elements in the metal-poor environmentof the Galactic halo (McWilliam 1998; Koch & Edvardsson2002; Honda et al. 2004; Fulbright 2000). This spread canreach 2 dex at [Fe/H] ∼ − dex. On the other hand, [ α /Fe]ratios (where α stands for α -elements) show a smaller scatterthan r-process elements. The α -element spread, if real andnot due to observational uncertainties, can be due to cosmicselection effects favoring contributions from supernovae in acertain mass range (see Ishimaru et al. 2003; Karlsson, T. & Gustafsson, B. 2005). In literature Eu is often indicatedas a good r-process tracer for two basic reasons: i) morethan 90 % of Eu in the solar system has been produced byr-process (Cameron 1982; Howard et al. 1986; Bisterzo et al.2015). ii) Europium is one of the few r-process elements thatshows clean atomic lines in the visible part of the electro-magnetic spectrum, and this makes Eu abundances easier tomeasure than other r-process elements (Woolf et al. 1995).Two main astrophysical sites have been proposed for Euproduction: i) core-collapse SNe (Type II SNe during explo-sive nucleosynthesis (Cowan et al. 1991; Woosley et al. 1994;Wanajo et al. 2001). However, there are still many uncertain-ties in the physical mechanism involved in Eu production inType II SNe (Arcones et al. 2007). ii) neutron star mergers(NSM) can provide a strong Eu production (Freiburghauset al. 1999; Wanajo et al. 2014; Panov et al. 2008; Symbal-isty & Schramm 1982; Oechslin et al. 2007; Bauswein et al.2013; Hotokezaka et al. 2013; Perego et al. 2014). Each eventcan produce a total amount of Eu from − to − M (cid:12) (Ko-robkin et al. 2012). © a r X i v : . [ a s t r o - ph . GA ] S e p L. Cavallo et al.
Previous models, such as Argast et al. (2004), computedthe evolution of Eu for the halo of our Galaxy with anin-homogeneous chemical evolution model. They concludedthat NSMs cannot be the major production site of Eu dueto their low merging rate. In this scenario NSMs failed to re-produce the observation of stars at low metallicity ([Fe/H] < − . ). Later Cescutti et al. (2006) found that, in a modelwith instantaneous mixing, SNe II can be entirely responsi-ble for the production of Eu. Moreover, he suggested thatEu originates from stars in a mass range 12-30 M (cid:12) .Matteucci et al. (2014) showed that, in a chemical modelwith instantaneous mixing approximation (I.M.A), neutronstars (NS) can be the only production site of Eu under someconditions: the time scale of coalescence cannot be longerthan 1 Myr; the yield of Eu per single event is around × − M (cid:12) ; the mass range of neutron stars progenitorsis 9-50 M (cid:12) . With similar assumptions on NSM parame-ters, Cescutti et al. (2015) proved that with a stochasticchemical evolution model these events can also explain thelarge spread of [Eu/Fe] vs [Fe/H] observed in the halo ofour Galaxy. It was also found out that the scenario whichbest reproduces the observational data is the one where bothneutron star mergers and a fraction of Type II supernovaeproduce Eu. A main assumption of the previous models isthe short coalescence time of NS systems, but some obser-vational bounds cannot be satisfied by a constant and shortcoalescence time, such as to explain the recently observedevent GW170817 which occurred in an early-type galaxywith no star formation, as well as to reproduce the cosmicrate of short Gamma-Ray Bursts (short-GRBs). RecentlyCˆot´e et al. (2019) proved that, if we assume NSM as theonly r-process site, there are some tensions between modelsand observational data when we drop the condition of shortand constant coalescence time. In particular, they found thatNSMs with a coalescence time that follows the same delaytime distribution (DTD) of SNe Ia cannot reproduce the de-creasing trend of [Eu/Fe] at [Fe/H] > − dex in the Galacticdisk. However, Sch¨onrich & Weinberg (2019) showed that,also with a DTD for NSM (with a characteristic merger time-scale t NS = 150 Myr), they were able to explain the observedabundance patterns assuming a 2-phase ISM (hot and cold).On the other hand, Simonetti et al. (2019) adopted a DTDfor NSM built from theoretical considerations and concludedthat either SNeII or a fraction of NSM variable in time canpotentially explain the [Eu/Fe] in the Galaxy as well as thecosmic rate of short-GRBs.Moreover, the effect of a DTD for NSM on the chemical evo-lution of r-process elements was also explored by Shen et al.(2015). In particular, they investigated the chemical evolu-tion of the heavy r-process elements in our Galaxy usinga high-resolution cosmological simulation (Eris) for the for-mation of a Milky-Way like galaxy. They used a power-lawslope with two different exponents: ∝ t − x ( x =
1; 2 ). Later, inthe framework of the hierarchical galaxy formation, Komiya& Shigeyama (2016) explored the effects of propagation ofNSM ejecta across proto-galaxies on the r-process chemicalevolution. Considering these effects, they found that NSMwith a DTD are able to reproduce the emergence of r-processelements at very low metallicity ([Fe/H] ∼ − < − To test the predictions of our model we use the abundancesof the halo stars contained in Roederer et al. (2014). Thesample contains abundances of 115 metal-poor stars. Wechose to test our models with a data-set provided by a sin-gle author even if the dimension of the sample is quite smallcompared to the total amount of data that are availablein literature ( (cid:39) ), e.g. JINAbase (Abohalima & Frebel2018). We have opted for this choice in order to remove po-tential off-sets between different data.
The abundances measured in halo stars show a clear largescatter in the ratio of [r/Fe], where r stands for an r-processelement, versus metallicity. Cescutti (2008) suggested thatthe wider spread observed in neutron-capture elements,compared to [ α /Fe] ratios, is a consequence of the differencein mass ranges between the production sites. This alsoimplies that, in the early Universe, the production of Eumust have been rare and prolific compared to the one of α -elements. Gamma-ray bursts display a bi-modal duration distribu-tion with a separation between the short and long-durationbursts at about 2 s. The progenitors of Long GRBs have beenidentified as massive stars. On the other hand, Short-GRBsare thought to be correlated with compact object mergers(Berger 2014; Eichler et al. 1989; Tanvir et al. 2013). Thishypothesis has been recently reinforced by the observationof a short GRB, that followed the NSM event GW170817 de-tected by LIGO/Virgo Collaboration (Abbott et al. 2017b).In particular, NGC 4993, the host galaxy of GW170817, isan early-type galaxy (Abbott et al. 2017a; Coulter et al.2017). If we assume a coalescence time constant and short,at least <
10 Myr as suggested in Matteucci et al. (2014) andCescutti et al. (2015), it will be impossible to detect a NSMin an early-type galaxy, where the star formation is over and
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MNRAS000 , 1–14 (2020) eutron stars mergers in a stochastic chemical evolution model all the NS-NS systems should have already merged. This re-quires the adoption of a DTD including long timescales. Asa caveat, we should also point out that it is not impossi-ble that the merger took place in a dwarf galaxy still starforming that we are unable to distinguish. The chemical evolution model adopted here is the same asin Cescutti et al. (2015), which is based on the stochasticmodel developed by Cescutti (2008). We review its maincharacteristics to improve the reader comprehension of thework.The Galactic halo is simulated by means of 200 stochasticrealisations. Each realisation consists of a non-interactingregion with the same typical volume. The dimensions of thetypical volume were chosen in order to neglect the interac-tions between different regions. In fact, for typical ISM den-sities, a supernova remnant becomes indistinguishable fromthe ISM before reaching ∼
50 pc (Thornton et al. 1998). Onthe other hand, we do not want a too large volume becausein that case, we would lose the stochasticity. For these rea-sons has been chosen a typical volume with a radius of ∼ M (cid:12) star, which is themaximum stellar mass considered.In each region, following the homogeneous model by Chiap-pini et al. (2008), we assume the following function for theinfall of gas with primordial composition: dGas in ( t ) dt = I nf all e −( t − t ) / σ (1)where t is set to 100 Myr, σ is 50 Myr and I nf all is equalto . × M (cid:12) M y r − . We define the Star Formation Rate(SFR) as: SFR ( t ) = ν (cid:18) σ gas ( t ) σ h (cid:19) . (2)where σ gas ( t ) is the surface density of the gas inside a volumeat each time-step, σ h =
80 M (cid:12) pc − and ν is set to (cid:12) M y r − . We also take into account an outflow that followsthe law: dGas out ( t ) dt = Wind ∗ SFR ( t ) (3)where Wind is set to 8.In all the subhaloes of the model, we assume the sameSFR and infall laws. The following lines will introduce thestochastic part contained in our model.Let us assume that we know the mass that is transformedat each time-step into stars ( M newstars ), then we generateone star with a mass sorted out with a random function,weighted on the initial mass function (IMF) of Scalo (1986)in the mass range from 0.1 to 100 M (cid:12) . After that, the massof the second star is extracted, and so on until the totalmass of newborn stars reaches M newstars . In this way, the totalamount of mass transformed into new stars is the same ineach region at each time-step, but the total number andmass distribution of stars is different. For all the stars we also know mass and lifetime. In particular, we assume thestellar lifetime of Maeder & Meynet (1989).When a star dies, it enriches the ISM with its newlyproduced elements and with the unprocessed elementspresent in the star since its birth. Our model considers adetail pollution from SNe core-collapse (M > (cid:12) ), AGBstars, NSMs, and SNe Ia, we follow the prescriptions for thesingle degenerate scenario of Matteucci & Greggio (1986).In Fig. 1, we present graphically how the chemical en-richment proceeds in our stochastic model . For clarity,only ten realisations are shown on the [Eu/Fe] vs [Fe/H]plane. The model has a constant delay time for NSM of1 Myr and it is one of model studied in Cescutti et al.(2015), namely NS00. In this Figure, we can appreciatehow the time at which the first NSM explodes and pollutestars with europium, varies among the different realisations.Although the delay between the formation of NS binaryand NSM is the same and very short, the formation of a NSbinary is stochastic. So, we can have realisations where thefirst stars present europium after only ∼ Myr, but alsorealisations where this happens later at around ∼ Myr.A short formation delay implies less chemical enrichmentof the volume. Therefore, the model results typically lie athigh [Eu/Fe] and low [Fe/H], the contrary for longer delays(lower [Eu/Fe] and higher [Fe/H]).In Fig. 1, the reader can also appreciate the differentpaths followed by each single realisation in the [Eu/Fe]vs [Fe/H] plane. These paths show some patterns, whichcan be understood in terms of the enrichment that takesplace in that region. For example, when a realisation moveshorizontally towards lower metallicities, there is no eventsenriching the ISM of iron or europium and the gas is dilutedby the infalling gas with primordial composition. Then,when an event produces Fe, the realisation moves to highermetallicities and lower [Eu/Fe] ratios. If a NSMs explodes,the realisation makes a jump towards higher [Eu/Fe] values.In general, the height of these ”jumps” varies for differentrealisations, due to the variable amount of Eu that a singleNSM can produce (see Equation 4).
For the Eu production sites, we take into account both NSMand core-collapse SNe. We define three parameters to includethe Eu production from NSM (Matteucci et al. 2014):(i) the fraction of massive stars that generate a binarysystem of neutron stars that will eventually merge, α NS .(ii) the amount of Eu produced by a single merging event, M EuNS .(iii) the delay time between the formation of the binarysystem and the merging event. From now on we willcall it coalescence time, t c .In our work, we assume that a fixed fraction of massive stars,generated during the simulation, is the progenitor of NSMs.The progenitors are chosen randomly among all the gen-erated massive stars in the mass range 8-50 M (cid:12) . We takethe progenitor mass range as suggested in Matteucci et al. MNRAS , 1–14 (2020)
L. Cavallo et al. [ Fe / H ] [ E u / F e ] NS00 t [ Myr ] Figure 1.
Results of [Eu/Fe] vs [Fe/H] for ten realisations of NS00 model. As reported in Table 2, this model has a constant delay timefor NSMs of 1 Myr. With the colour map we show the time at which the realisation pass through a certain point in the [Eu/Fe] vs [Fe/H]plane. We also report the initial point and the time at which the first NSM has exploded. (2014). We assume a similar α NS to the one contained inMatteucci et al. (2014) ( ∼ ∼
80 Myr − ).For the nucleosynthesis of Eu, we use empirical values thathave been chosen in order to reproduce the surface abun-dances of Eu in low-metallicity stars as well as the solarabundances of Eu (see Cescutti et al. 2006). These valuesare consistent with the limits calculated by Korobkin et al.(2012), who suggested that a single NSM can produce from − to − M (cid:12) of Eu.During the work, we have also considered a non-constantEu production for a single NSM. In general the variation isunknown, so we assume a range from 1 % of the average Eu( M Eu ) to 200 % of it. Since the total mass of Eu producedshould be preserved, the n th star ejects a mass of Eu that follows this equation: M EuNS ( n ) = M Eu ( . + . · Rand ( n )) (4)where Rand ( n ) is a uniform random distribution in the rage [ , ] (same as in Cescutti & Chiappini 2014).For the production of Eu from SNe II we adopt yields simi-lar to those of Matteucci et al. (2014) (Mod2SNNS Model).Since recent results showed that the conditions during asupernova Type II explosion may not be able to producemuch Eu (Arcones et al. 2007; Wanajo et al. 2011), wealso tested an alternative channel: the magneto-rotationallydriven (MRD) SNe. MRD SNe are a particular class of core-collapse supernovae. Here, we assume that the 10 % of theCC-SNe explode as MRD. This r-process site is active onlyat low metallicity (Z < − ), so it affects the model resultsonly at low metallicity. These assumptions are identical tothe ones contained in Cescutti et al. (2015). This particu-lar fate is rare, only few SNe explode as MRD-SNe, and as MNRAS000
80 Myr − ).For the nucleosynthesis of Eu, we use empirical values thathave been chosen in order to reproduce the surface abun-dances of Eu in low-metallicity stars as well as the solarabundances of Eu (see Cescutti et al. 2006). These valuesare consistent with the limits calculated by Korobkin et al.(2012), who suggested that a single NSM can produce from − to − M (cid:12) of Eu.During the work, we have also considered a non-constantEu production for a single NSM. In general the variation isunknown, so we assume a range from 1 % of the average Eu( M Eu ) to 200 % of it. Since the total mass of Eu producedshould be preserved, the n th star ejects a mass of Eu that follows this equation: M EuNS ( n ) = M Eu ( . + . · Rand ( n )) (4)where Rand ( n ) is a uniform random distribution in the rage [ , ] (same as in Cescutti & Chiappini 2014).For the production of Eu from SNe II we adopt yields simi-lar to those of Matteucci et al. (2014) (Mod2SNNS Model).Since recent results showed that the conditions during asupernova Type II explosion may not be able to producemuch Eu (Arcones et al. 2007; Wanajo et al. 2011), wealso tested an alternative channel: the magneto-rotationallydriven (MRD) SNe. MRD SNe are a particular class of core-collapse supernovae. Here, we assume that the 10 % of theCC-SNe explode as MRD. This r-process site is active onlyat low metallicity (Z < − ), so it affects the model resultsonly at low metallicity. These assumptions are identical tothe ones contained in Cescutti et al. (2015). This particu-lar fate is rare, only few SNe explode as MRD-SNe, and as MNRAS000 , 1–14 (2020) eutron stars mergers in a stochastic chemical evolution model DTD percent of NSMs exploded before Myr
Myr
Myr ∝ t − % % % ∝ t − . % % % Table 1.
Percentage of NSM already merged at different timesfor DTD of different shapes. Those values are for DTDs with a t minc = Myr mentioned in Winteler et al. (2012), it should be more likelyto happen at low metallicity (Yoon et al. 2006).
In this section, we present the different types of coalescencetime scale for NSM that we consider in our models. TheDTD functions assumed in this work are ∝ t − and ∝ t − . and defined as follows: DT D ( t ) = if t < t cmin A x t − x if t cmin < t < Gyr if t > Gyr w ith x = { , . } and A x = / ∫ τ − x d τ ; (5)where t cmin is the minimum coalescence time (in our modelscan assume two values: and Myr), and A x is the nor-malisation constant.We also discuss possible tensions with observations. As we mentioned in the Introduction, NSM with a shortand constant coalescence delay are able to reproduce thedecreasing trend of [Eu/Fe] (also called knee) starting from[Fe/H] ∼ − α -elements. As showed by Ces-cutti et al. (2015), they can also explain the [Eu/Fe] spreadin metal-poor stars in the Milky Way halo. However, a shortand constant coalescence time is incompatible with severalobservations (see Simonetti et al. 2019; Cˆot´e et al. 2019).To begin with, if we assume that NSMs are progenitors ofshort-GRBs (Berger 2014), they cannot explain the obser-vation of short GRBs in early-type galaxies, where star for-mation has stopped several Gyr ago. Furthermore, a shortcoalescence timescale ( <
100 Myr) is inconsistent with thetheoretical estimation of merging times of the seven knownNS-NS binary systems, indeed their coalescence timescaleranging from 86 to 2730 Myr (Tauris et al. 2017). Finally, asalready mentioned, a NSM scenario with short and constanttimescales cannot explain the event GW170817 observed inan early-type galaxy. ∝ t − In Literature, a lot of authors have derived the DTD func-tion of SNe Ia from observations. Most of the studies suggestthat, SNe Ia follow a DTD with the form ∝ t − (see Totaniet al. 2008; Maoz & Badenes 2010; Graur et al. 2011; Maoz& Mannucci 2012; Rodney et al. 2014). This slope is also in agreement with predictions from population synthesis mod-els. Similar techniques can be applied to derive the DTD ofshort-GRBs (i.e. the DTD of their progenitors: the NSMs).Fong et al. (2017) found that, the DTD of short-GRBs canhave the form of t − . A power-law with a − slope is also inagreement with population synthesis studies (see Dominiket al. 2012; Chruslinska et al. 2018). Assuming a similarDTD for NSMs and SNe Ia (i.e. ∝ t − ) is also consistentwith the fact that SNe Ia and short GRBs are detected insimilar proportion in early-type galaxies. However, with thisassumption on the DTD of NSMs, it was already shown thatNSM cannot reproduce the decreasing trend of [Eu/Fe] inthe Galactic disk (see Cˆot´e et al. 2019; Simonetti et al. 2019).In our work, we tested this functional form for the DTDwith three different lower bounds in the coalescence time:1 Myr, 10 Myr, and 100 Myr. In order to include the coa-lescence timescales of NS-NS systems contained in (Tauriset al. 2017), we should have chosen an upper limit equal to ∞ . In this work, we choose an upper limit of 10 Gyr becauseif we assume a larger one it would have changed only thenormalization of the DTD, without a significant impact onthe results. ∝ t − . We also tested a DTD ∝ t − . . This kind of slope is consis-tent with the distribution function of short-GRBs derivedby D’Avanzo (2015). In particular, a steeper DTD functionof the form of t − . is not in agreement with the fact thatthe observed fractions of short-GRBs and SNe Ia are simi-lar. This disagreement could be eased if the DTD functionof SNe Ia has also a t − . form, as suggested by Heringeret al. (2016), which showed that SNe Ia follow a DTD with apower-law slope in the range from − . to − . . On the otherhand, in the environment of a chemical evolution model,SNe Ia with a DTD ∝ t − . are not able to reproduce all the[X/Fe] vs [Fe/H] trends in the Galaxy since, with a DTD ∝ t − . , the explosion time-scales of SNe Ia are too short(see Matteucci et al. 2006).In the light of what we discuss in Sect. 3.2.2, this DTD isnot in agreement with the one for SNe Ia and provides coa-lescence timescales that are variable but still short. In fact,as shown in Table 1, more than 90 % of NSMs explode before100 Myr. In the following, we summarise the results of the models wecomputed, as shown in Table 2. They are distinguished infour classes. a ) In this scenario only NSM can produce Eu.In this class, is also assumed a constant coalescence time. b ) These models test the effects of different DTDs on europiumenrichment in the a-class scenarios. c ) There both NSM andMRD SNe are r-process sites. For NSM systems we still as-sume a constant coalescence time. We also assume that, atmetallicity (Z < − ), 10 % of CC-SNe explode as a MRD. d ) There we test the effects of relaxing constancy of coalescencetime in a NS+MRD scenario. We assume the same α =0.1 forMRD. e ) With these models we test a variable α NS versus[Fe/H] in a NS-only scenario. i ) Last, we test the dependencebetween the value of α NS and the dispersion of the results at MNRAS , 1–14 (2020)
L. Cavallo et al.
Model Name DTD t minc [Myr] α N S M Eu ; N S M [ M (cid:12) ] M Eu ; M RD [ M (cid:12) ] Ψ NS00 a no 1 0.02 . × − (varying as eq. 4) no productionNS01 a ” 10 ” ” ”NS02 a ” 100 ” ” ”NSt1 b ∝ t − × − (varying as eq. 4) no productionNSt2 b ” 10 ” ” ”NSt3 b ∝ t − . . × − (varying as eq. 4) ”NSt4 b ” 10 ” ” ”NS+MRD00 c no 1 0.02 . × − (varying as eq. 4) . × − (varying as eq. 4)NS+MRD01 c ” 10 ” ” ”NS+MRD02 c ” 100 ” ” ”NS+MRDt1 d ∝ t − . × − (varying as eq. 4) . × − (varying as eq. 4)NS+MRDt2 d ” 10 ” ” ”NS+MRDt3 d ∝ t − . d ” 10 ” ” ”NSt3+ α e ∝ t − . α N S = . ) × − (varying as eq. 4) no productionNSt1+ α e ∝ t − i ∝ t − . α N S = . ) × − (varying as eq. 4) no productionTest2 i ” ” varying as eq. 6 ( α N S = . ) . × − (varying as eq. 4) ”Test3 i ” ” varying as eq. 6 ( α N S = . ) . × − (varying as eq. 4) ” Table 2.
This table summarize the parameters of the models that we test during this work. It is organised as follows: in column 1, wereport the name of the model, in column 2, the assumed DTD for coalescence time, in column 3, the minimum delay time for NSM,in column 4, the assumed fraction of massive star that could lead to NSM, in column 5, the assumed yield for NSM, in column 6, theassumed yield for MRD SNe. Ψ When we take into account the Eu production by MRD-SNe we set α M RD =0.10. moderate metallicity ( ∼ − . dex). In Figure 2; 3; 4; 5; 6 areshown the results, in the [Eu/Fe]vs[Fe/H] plane, from ourmodels. In the plots, at [Eu/Fe] = − . dex, we also reportthe long-living stars formed without Eu (formally [Eu/Fe] = −∞ ). In Fig. 2 is shown the distribution of the long-living starsin the [Eu/Fe]-[Fe/H] plane, as predicted by our stochasticmodels with the following assumptions: i) Eu is producedonly from NSM whose progenitors are in the mass rangefrom 9 to 50 M (cid:12) . ii) The amount of Eu produced from asingle event follows equation 4 with an average value ( M Eu ) of . × − M (cid:12) . iii) of massive stars are in binary sys-tems with the right characteristics to lead to NSM. iv) Theminimum value for the coalescence time is fixed at 1 Myr.The plotted models are NS00, NSt1 and NSt3 (cfr. Table 2).In the left panel of Fig. 2 is seen that the NS00 model isin agreement with the data for stars with [Fe/H] > − < < −
3. Finally, the model does not predict stars with[Eu/Fe] < − < − < ∝ t − the situation is even worse. In the modelNSt1 (middle panel) fails to reproduce the distribution ofthe observational data. Finally, when we take into accounta DTD ∝ t − . (NSt3) we obtain similar results of NS00model. In fact, as shown in Fig. 2 (right panel), the model MNRAS000
3. Finally, the model does not predict stars with[Eu/Fe] < − < − < ∝ t − the situation is even worse. In the modelNSt1 (middle panel) fails to reproduce the distribution ofthe observational data. Finally, when we take into accounta DTD ∝ t − . (NSt3) we obtain similar results of NS00model. In fact, as shown in Fig. 2 (right panel), the model MNRAS000 , 1–14 (2020) eutron stars mergers in a stochastic chemical evolution model [ Fe / H ] [ E u / F e ] NS00 [ Fe / H ]NSt1 [ Fe / H ]NSt3 log N stars Figure 2.
Left panel : results of [Eu/Fe] vs [Fe/H] for model NS00. This model has a constant delay time for NSMs of 1 Myr, Euproduction that vary as equation 4 and a mean value of . × − M (cid:12) , no Eu production from CC-SNe. Model NS00 is the same as NS00contained in Cescutti et al. (2015). Central panel : same as left panel but for model NSt1. The only difference from the previous modelis the assumption on the coalescence time; in fact in this case we assume a delay time with a DTD ∝ t − for NSMs. Right panel : same asprevious panels but for model NSt3. In this model for the coalescence time of NSMs we assume a DTD ∝ t − . . Note that all the modelscontained in this figure has minimum coalescence time set to 1 Myr. The long-living stars formed without Eu (formally [Eu/Fe] = −∞ )are shown at [Eu/Fe] = − . dex. [ Fe / H ] [ E u / F e ] NS01 [ Fe / H ]NSt2 [ Fe / H ]NSt4 log N stars Figure 3.
Left panel : results of [Eu/Fe] vs [Fe/H] for model NS01. This model has a constant delay time for NSMs of 10 Myr, Euproduction that vary as equation 4 and a mean value of . × − M (cid:12) , no Eu production from CC-SNe. Model NS01 is the same as NS01contained in Cescutti et al. (2015). Central panel : same as left panel but for model NSt2. The only difference from the previous model isthe assumption on the coalescence time; in fact in this case we assume a delay time with a DTD ∝ t − for NSMs. Right panel : same asprevious panels but for model NSt4. In this model for the coalescence time of NSMs we assume a DTD ∝ t − . . Note that all the modelscontained in this figure has minimum coalescence time set to 10 Myr. cannot explain the presence of stars with [Eu/Fe] < − < − < -0.2 inthe metallicity range − < [Fe/H] < − − M (cid:12) . ii) The amount of Eu produced from a sin- MNRAS , 1–14 (2020)
L. Cavallo et al. [ Fe / H ] [ E u / F e ] NS+MRD00 [ Fe / H ]NS+MRDt1 [ Fe / H ]NS+MRDt3 log N stars Figure 4.
Left panel : results of [Eu/Fe] vs [Fe/H] for model NS+MRD00. This model has a constant delay time for NSMs of 1 Myr, Euproduction from NSMs that varying as equation 4 and a mean value of . × − M (cid:12) , Eu production from MRD-SNe (10 % of CC-SNeonly at Z < − ) that vary as equation 4 and a mean value of . × − M (cid:12) . Central panel : same as left panel but for model NS+MRDt1.The only difference from the previous model is the assumption on the coalescence time; in fact in this case we assume a delay time witha DTD ∝ t − for NSMs with. Right panel : same as previous panels but for model NS+MRDt3. In this model for the coalescence time ofNSMs we assume a DTD ∝ t − . . Note that all the models contained in this figure has minimum coalescence time set to 1 Myr. [ Fe / H ] [ E u / F e ] NSt3+ [ Fe / H ] [ E u / F e ] NSt1+
Figure 5.
Left panel : results of [Eu/Fe] vs [Fe/H] for model NSt3+ α . Comparing this panel with the right one of Fig. 2 is clear that avariable α N S has a great impact on the stars distribution predicted by our models. In particular, an α N S that depends on the metallicityallows models to generate stars Eu-enriched at lower metallicity.
Right panel : same as left panel but for model NSt1+ α . In this case avariable α N S has the same effect and improves the compatibility between model results and observational data in the metallicity range − < [Fe/H] < − gle event follows equation 4 with an average value ( M Eu ) of . × − M (cid:12) . iii) of massive stars are in binary systemswith the right characteristics to lead to merging NS. iv) Theminimum value for the coalescence time is fixed at 10 Myr.The plotted models are NS02, NSt2 and NSt4 (see Table 2).In the left panel of Fig. 3 we can notice that the model NS01 does not predict the presence of stars with [Eu/Fe] < − > − MNRAS000
Right panel : same as left panel but for model NSt1+ α . In this case avariable α N S has the same effect and improves the compatibility between model results and observational data in the metallicity range − < [Fe/H] < − gle event follows equation 4 with an average value ( M Eu ) of . × − M (cid:12) . iii) of massive stars are in binary systemswith the right characteristics to lead to merging NS. iv) Theminimum value for the coalescence time is fixed at 10 Myr.The plotted models are NS02, NSt2 and NSt4 (see Table 2).In the left panel of Fig. 3 we can notice that the model NS01 does not predict the presence of stars with [Eu/Fe] < − > − MNRAS000 , 1–14 (2020) eutron stars mergers in a stochastic chemical evolution model we drop the constancy of the coalescence time and we alsoassume a DTD ∝ t − , models completely fail to reproducethe observational data. Finally, model NSt4, which assumesa DTD ∝ t − . , fails to reproduce europium abundancesof stars with [Fe/H] < − . × − M (cid:12) for the Eu production, it also cannot repro-duce some stars of the upper envelope of the observed stardistribution. As seen in the previous section, a scenario, where NSMs arethe only r-process site, fails to predict the presence of Eu instars with metallicity [Fe/H] < − M (cid:12) . ii) The amount of Eu produced from asingle NSM event follows equation 4 with an average value M Eu ; NSM = . × − M (cid:12) . v) of massive stars are in bi-nary systems with the right characteristics to lead to merg-ing NS. iv) At low metallicity (Z < − ), 10 % of CC-SNeexplode as MRD. v) the amount of Eu produced by a sin-gle MRD explosion follows equation 4 with an average value M Eu ; MRD = . × − M (cid:12) (same as NSMs). The plottedmodels are NS+MRD00, NS+MRDt1 ,and NS+MRDt3 (cfr.Table 2).Model NS+MRD00 (right panel of Fig. 4) is in good agree-ment with the observational data and it well predicts thepresence of stars with [Fe/H] < − < α NS =0.02, every NSMproducing a constant amount of Eu equal to . × − M (cid:12) and a single MRD SN produces, on average, × − M (cid:12) and 10 % of stars in the mass range −
80 M (cid:12) explodes asMRD SNe. α NS Another possible way to solve it is to relax the assumptionof constancy of the fraction of massive stars that can gen-erate a binary system of neutron stars which will eventually merge, α NS .Several works investigate the formation of double NS sys-tems (Bogomazov et al. 2007; Ivanova et al. 2008; Mennekens& Vanbeveren 2014; Shao & Li 2018). In particular, as shownin Giacobbo & Mapelli (2018), metallicity plays a crucial rolein the formation of binary systems of compact objects. Forthese reasons, we decide to test this scenario with our chem-ical evolution model.We assume a dependence of α NS on [Fe/H] (see Fig. 7) simi-lar to the one assumed in model 4AV2 contained in Simonettiet al. (2019). With this assumption α NS varies as α NS ( [Fe/H] ) = α NS if [Fe/H] ≤ − . α NS ( − ln ( [Fe/H] + z ) + z ) if [Fe/H] > − . α minNS if α NS < α minNS z = . dex ; z = ln ( . ) (6)In order to test this scenario we have built a new model(NSt3+ α ) with the following assumptions: i) Eu is producedonly from NSM, whose progenitors are in the mass rangefrom 9 to 50 M (cid:12) . ii) The amount of Eu produced from asingle event follows equation 4 with an average value ( M Eu ) of × − M (cid:12) . iii) The parameter α NS depends on [Fe/H]and varying as equation 6; α NS is set to 0.275. iv) The co-alescence time distribution of NSMs follows a DTD ∝ t − . .This systems has a minimum delay time of 1 Myr. The pre-dictions of NSt3+ α are plotted in left panel of Fig. 5. It isseen that this model is in good agreement with the obser-vational data, but it is not able to predict the presence ofstars with low [Eu/Fe] ( < − < − α model predicts also the presence of stars with [Eu/Fe] < − > − < − α NS variable and the DTD ∝ t − . We built NSt1+ α model with the following assump-tions: i) Eu is produced only from NSM, whose progenitorsare in the mass range from 9 to 50 M (cid:12) . ii) The amount ofEu produced from a single event follows equation 4 withan average value ( M Eu ) of × − M (cid:12) . iii) The parameter α NS depends on [Fe/H] and varies as equation 6; α NS is setto 0.275. iv) The coalescence time of NSMs follows a DTD ∝ t − and has a minimum value of 1 Myr. The results ofthis model are shown in the right panel of Fig. 5. As youcan notice, the model does not predict the presence of starswith [Eu/Fe] < − < − − α NS as a function of [Fe/H] has a great im-pact on the results of our chemical evolution models. In par-ticular, an α NS that varies with metallicity, can substituteMRD-SNe in the framework of explaining the low-Eu tail ofmetal-poor Halo stars ([Fe/H] < − MNRAS , 1–14 (2020) L. Cavallo et al. [ Fe / H ] [ E u / F e ] Test1 [ Fe / H ]Test2 [ Fe / H ]Test3 log N stars Figure 6.
Left panel : results of [Eu/Fe] vs [Fe/H] for model Test1. This model is identical to NSt3+ α model. We plot it again toemphasise the consequences of the variation of α N S and M Eu . Central panel : same as left panel but for model Test2. In this model,the equation 6 is up-shifted by 0.04, the function shows two plateau with α N S =0.315 and 0.06. In order to maintain constant the totalamount of produced Eu, we reduce M Eu to . × − M (cid:12) . Right panel : same as left panel but for model Test3. This model, the α N S versus [Fe/H] relation, is up-shifted by 0.08 therefore the function, described by equation 6, shows two plateau with α N S =0.355 and 0.1.In this case M Eu is reduced to . × − M (cid:12) .[Fe/H] (dex) Test1 Test2 Test3mean [Eu/Fe] (dex) sigma (dex) f mean [Eu/Fe] (dex) sigma (dex) f mean [Eu/Fe] (dex) sigma f − − − − − − − − − Table 3.
Here are summarise the results of our analysis. The table is organised as follows: in column 1, the metallicity value to whichwe compute mean and standard deviation of the [Eu/Fe] values, in column 2, mean [Eu/Fe] at some metallicity for a specific model, incolumn 3, the standard deviation at some metallicity for a specific model, in column 4, fraction of Eu-free ( f = N Eu − f ree / N tot ). Thisstructure is repeated for the three different models contained in this section. In this section, we explore the correlation between the dis-persion of the [Eu/Fe] values and the fraction of massivestars that can produce a NSM, namely the parameter α NS .We start from the assumptions of our best model (NSt3+ α )and then we increase the value of α NS . With these prescrip-tions we create three different models (see Table 2): i) Test1is exactly the same as NSt3+ α . ii) Test2; for this model, weassume α NS = 0.315 (see equation 6). This implies a slightincrease of NSMs at the lowest metallicities, but it also im-plies an increase of a factor of 3 at [Fe/H] > − ∼ M Eu ; NSM to . × − M (cid:12) . iii) Test3; in this case we set α NS = 0.375. As a consequence of this, the total number of NSM is increased by a factor ∼
5. Also, in this case, we re-duce the mean Eu produced to . × − M (cid:12) .On the results of these models, we select different bins inmetallicity and in these we compute mean and standard de-viation of [Eu/Fe] values. The mean and standard deviationfor each model are reported in Table 3. In Fig. 6 are plottedthe results of the three models: Test1, Test2, and Test3.Looking at Fig. 6, focusing at the region at the interme-diate metallicity, it can be easily noticed that the obser-vational data cannot exclude any of the tested models. In-deed, stochastic models with a large variation of α NS (from0.02 to 0.10) predict differently the enrichment at interme-diate metallicity regime. On the other hand, the observa-tional data are affected by relatively large uncertainties (i.e. ∼ . − . dex in [Eu/Fe]); moreover, the sample selectedis certainly measured in a homogeneous way, but it is notlarge enough to apply safely a statistical approach. Adding MNRAS000
5. Also, in this case, we re-duce the mean Eu produced to . × − M (cid:12) .On the results of these models, we select different bins inmetallicity and in these we compute mean and standard de-viation of [Eu/Fe] values. The mean and standard deviationfor each model are reported in Table 3. In Fig. 6 are plottedthe results of the three models: Test1, Test2, and Test3.Looking at Fig. 6, focusing at the region at the interme-diate metallicity, it can be easily noticed that the obser-vational data cannot exclude any of the tested models. In-deed, stochastic models with a large variation of α NS (from0.02 to 0.10) predict differently the enrichment at interme-diate metallicity regime. On the other hand, the observa-tional data are affected by relatively large uncertainties (i.e. ∼ . − . dex in [Eu/Fe]); moreover, the sample selectedis certainly measured in a homogeneous way, but it is notlarge enough to apply safely a statistical approach. Adding MNRAS000 , 1–14 (2020) eutron stars mergers in a stochastic chemical evolution model [ Fe / H ] N S contant NS varying NS Figure 7.
The evolution with metallicity of α N S in two differentscenarios. [ Fe / H ] N E u f r ee / N t o t Test1Test2Test3Roederer et al.(2014)Cescutti et al.(2015)
Figure 8.
Ratio of Eu-free stars over the total number of starsfor bins of 0.5 dex in [Fe/H]. In the figure are plotted the resultsfor the models Test1, Test2, and Test3. The blue triangles are theobservational proxy for this ratio, so the ratio between the num-ber of stars in which Eu only presents an upper limit (possiblyEu-free) over the number of stars for which at least Ba has beenmeasured (total number of stars). The horizontal error bars showthe dimension of each bin in [Fe/H]. Red triangles are the ob-servational proxy for this ratio derived from the data-set used inthis work, Roederer et al. (2014). The blue triangles are the sameobservational proxy calculated in Cescutti et al. (2015); adoptinga different data-set (cfr. Cescutti et al. 2015, , for details on thisdata-set). more authors will increase the number of data, but we riskto increase significantly the scatter among different authors.Future surveys such as 4MOST (de Jong et al. 2014) andWEAVE (Dalton et al. 2012) will surely produce larger data-set homogeneously measured and they could allow us to de-termine the value of α NS , and consequently M Eu ; NSM , moreprecisely. All the tested models have a common feature: the consideredr-process events are rare and they are only a small fraction( α NS ) of the total number of the main polluters of the ISMat low metallicity, the SNe II. It is easy to infer that, atextremely low metallicities ([Fe/H] ≤ − ), a lot of low massstars can be formed in regions where the ISM is not yetpolluted by r-process events. We also expect that loweringthe fraction α NS should lead to an increase of Eu-free stars(i.e. [Eu/Fe] = −∞ ). Moreover, a longer time delay for ther-process events will also produce a higher fraction of Eu-free stars, since for a longer time ISM will be not enrichedby r-process events.All the plots of our models (Fig. 2; 3; 4; 5; 6) show the long-living stars formed without Eu (i.e. Eu-free stars) at [Eu/Fe] = − . dex. From these plots, it is possible already to finda behavior that is in agreement with our aforementionedexpectations. For example, looking at Fig. 2, is possible tonotice that a model with a DTD function with the form ∝ t − (NSt1) predicts a higher number of Eu-free star comparedto both the short and constant delay presented NS00 andthe steeper DTD ( ∝ t − . ) of NSt3 models.However, to better study the behavior of the Eu-free starswith respect to the fraction α NS , we report in Fig. 8, theratio of Eu-free stars over the total number of stars forthe models Test1, Test2, and Test3. In this plot, it appearsclearly that increasing α NS , so moving from Test1 to Test3,the model predicts a lower fraction of Eu-free stars. Obser-vationally, it is not obvious how to put constraints to themodeling since no Eu-free star has been yet claimed. Onthe other hand, several stars have only upper limits for eu-ropium. In Fig. 8, we decide to use as a proxy of Eu-freestars, stars for which only upper limits for europium havebeen detected and barium is measured, as already assumedin Cescutti et al. (2015). In the plot, we use two data-setto compute this observational proxy. So together with theresults obtained with the stars measured in Roederer et al.(2014), we show also the results obtained in Cescutti et al.(2015) adopting a different data-set. Details of this collectioncan be found in Cescutti et al. (2015). We decide to add theseresults, because it appears clear that the number of upperlimits detected by Roederer et al. (2014) for europium arequite high and possibly due to a certain fraction of spectramissing the necessary quality to measure europium, ratherthan the real absence of this element. So, we trust more theresults from the larger sample used in Cescutti et al. (2015)toward higher metallicity, whereas for low metallicity, theyappear in reasonable agreement. In comparison, our mod-els predictions appear to follow the trend, but it is alwaysbelow the observational proxy. This can be explained by alarge fraction of false Eu-free stars, due to the difficulty ofmeasuring the weak europium lines when the abundance isreally low. Overall, our best model, (i.e. Test 1) appears in MNRAS , 1–14 (2020) L. Cavallo et al. agreement with this observational proxy, but it is hard tofind a firm conclusion from this prospective.
In this paper, we have adopted the stochastic chemicalevolution model of the Galactic halo presented by Cescutti(2008), to study the impact of relaxing the constancy of thedelay times for the coalescence of NSM, on the chemicalevolution of Eu in the metal-poor environment of theGalactic halo. To perform that, we have implemented twodifferent delay time distributions (DTDs) ∝ t − and ∝ t − . ,as suggested in Cˆot´e et al. (2019). For the Eu yields, wehave followed the prescriptions of Matteucci et al. (2014)and Cescutti et al. (2015). We have also tried to find a wayto solve the tensions between the observational data andthe results of models that assume a variable coalescencetime. In order to do that, we have explored a scenario inwhich both NSM and MRD SNe (magneto-rotationallydrive SNe) are able to produce Eu. For the same reason, wehave also implemented a fraction of massive stars that canproduce NSM systems that vary with metallicity, followingthe idea presented in Simonetti et al. (2019). Finally, wehave also studied the [Eu/Fe] vs. [Fe/H] in the Galactichalo and its correlation with the value of α NS .Our main conclusions can be summarised as follows:a) The NS-only scenario is in disagreement withobservational data, even at moderate metallicity, when weassume a DTD ∝ t − for the coalescence timescales. On theother hand, assuming a DTD ∝ t − . produces results similarto the ones with constant delay time. These conclusions aresimilar to the ones found by Cˆot´e et al. (2019), but nowwe obtain these results in the framework of a stochasticchemical evolution model.b) The mixed scenario with NS and MRD SNe is ableto explain the observed spread as shown, but only fora constant delay, in Cescutti et al. (2015). The mainassumptions, in this case, are that MRD SNe are 10 % of CC-SNe, explode only at low metallicity (Z < − ) andthe production of Eu is the same for both NSM and MRDSNe. We prove here that the models in this case agree withobservations independently by the assumed DTD.c) Our best NS-only scenario is in good agreementwith observational data under the following assumptions:i) Eu is produced only from NSM, whose progenitors arein the mass range from 9 to 50 M (cid:12) . ii) The amount of Euproduced from a single event follows equation 4 with anaverage value ( M Eu ) of × − M (cid:12) . iii) The parameter α NS depends on [Fe/H] and varies as equation 6; the required α NS is 0.275. iv) The coalescence time distribution of NSMsshould follow a DTD ∝ t − . with a minimum value of 1Myr. In this scenario, a larger fraction of NSM explodesin the early phases of the Galactic evolution, compared tonowadays (see also Simonetti et al. 2019).d) Adopting to our best model, we also show the predicteddispersion of [Eu/Fe] at a given metallicity depending on the 6; the comparison with the present literature datacannot allow us to put a stronger constraint 6. However,future high-resolution spectroscopical surveys, such as4MOST (de Jong et al. 2014) and WEAVE, (Dalton et al.2012) will produce the necessary statistic to constrain atbest this parameter.e) Our best model is in agreement with the chosenobservational proxy for Eu-free stars. However, the fractionof false Eu free stars cannot be evaluated and no firmconclusions can be raised.The models struggle to reproduce the low-metallicitytail of stars with [Eu/Fe] < − . dex at [Fe/H] < − . dex.We underline that the model at this stage does not considerseveral complexities that can play an important role to solvethis issue, for example: stochasticity in SFR and infall-law,cross-contamination of sub-haloes, pre-enrichment of theinfalling gas, and multi-phase ISM.In a future work, we will try to solve this problem by takinginto account the hierarchical formation of Galactic haloby accretion of satellite galaxies. In fact, the enrichmentof r-process elements in these objects could have beenless effective due to dynamical effects connected to theformation of binary neutron stars (see Bonetti et al. 2019). ACKNOWLEDGEMENTS
This work has been partially supported by the EuropeanUnion Cooperation in Science and Technology (COST) Ac-tion CA16117 (ChETEC). We thank Paolo Simonetti forassistance with the implementation of the time delay distri-butions.
MNRAS000
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