Impact of the Rotation and Compactness of Progenitors on the Mass of Black Holes
Michela Mapelli, Mario Spera, Enrico Montanari, Marco Limongi, Alessandro Chieffi, Nicola Giacobbo, Alessandro Bressan, Yann Bouffanais
DDraft version March 20, 2020
Typeset using L A TEX twocolumn style in AASTeX63
Impact of the Rotation and Compactness of Progenitors on the Mass of Black Holes
Michela Mapelli,
1, 2, 3
Mario Spera,
1, 4, 5, 2
Enrico Montanari, Marco Limongi,
6, 7
Alessandro Chieffi,
8, 9
Nicola Giacobbo,
1, 2, 3
Alessandro Bressan,
10, 3 and Yann Bouffanais
1, 2 Physics and Astronomy Department Galileo Galilei, University of Padova, Vicolo dell’Osservatorio 3, I–35122, Padova, Italy INFN-Padova, Via Marzolo 8, I–35131 Padova, Italy INAF-Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, I–35122, Padova, Italy Center for Interdisciplinary Exploration and Research in Astrophysics (CIERA), Evanston, IL 60208, USA Department of Physics & Astronomy, Northwestern University, Evanston, IL 60208, USA Istituto Nazionale di Astrofisica - Osservatorio Astronomico di Roma, Via Frascati 33, I-00040, Monteporzio Catone, Italy Kavli Institute for the Physics and Mathematics of the Universe, Todai Institutes for Advanced Study, the University of Tokyo, Kashiwa,Japan 277-8583 (Kavli IPMU, WPI) Istituto Nazionale di Astrofisica - Istituto di Astrofisica e Planetologia Spaziali, Via Fosso del Cavaliere 100, I-00133, Roma, Italy Monash Centre for Astrophysics (MoCA), School of Mathematical Sciences, Monash University, Victoria 3800, Australia SISSA, via Bonomea 265, I–34136 Trieste, Italy (Received; Revised; Accepted)
Submitted to ApJABSTRACTWe investigate the impact of stellar rotation on the formation of black holes (BHs), by means of ourpopulation-synthesis code sevn . Rotation affects the mass function of BHs in several ways. In massivemetal-poor stars, fast rotation reduces the minimum zero-age main sequence (ZAMS) mass for a starto undergo pair instability and pulsational pair instability. Moreover, stellar winds are enhanced byrotation, peeling-off the entire hydrogen envelope. As a consequence of these two effects, the maximumBH mass we expect from the collapse a rotating metal-poor star is only ∼
45 M (cid:12) , while the maximummass of a BH born from a non-rotating star is ∼
60 M (cid:12) . Furthermore, stellar rotation reduces theminimum ZAMS mass for a star to collapse into a BH from ∼ −
25 M (cid:12) to ∼ −
18 M (cid:12) . Finally, wehave investigated the impact of different core-collapse supernova (CCSN) prescriptions on our results.While the threshold value of compactness for direct collapse and the fallback efficiency strongly affectthe minimum ZAMS mass for a star to collapse into a BH, the fraction of hydrogen envelope that canbe accreted onto the final BH is the most important ingredient to determine the maximum BH mass.Our results confirm that the interplay between stellar rotation, CCSNe and pair instability plays amajor role in shaping the BH mass spectrum.
Keywords: black hole physics – gravitational waves – methods: numerical – stars: mass loss INTRODUCTIONThe mass function of stellar black holes (BHs) is stillan open question in astrophysics. Gravitational wavedata are going to revolutionise our knowledge aboutBHs in the coming years: the first two observing runs ofthe LIGO–Virgo collaboration (LVC) led to the detec-tion of ten binary BHs (Abbott 2018a,b), few additional
Corresponding author: Michela [email protected] events were claimed by Venumadhav et al. (2019) andZackay et al. (2019), based on a different pipeline, andseveral new public triggers were announced during thethird observing run of the LVC, which is still ongoing.This growing population of BHs complements the sam-ple from dynamical mass measurements in nearby X-raybinaries ( ¨Ozel et al. 2010; Farr et al. 2011) and will pro-vide us with an unique opportunity to test BH formationmodels.According to our current understanding, compact ob-ject masses are strictly related to the mass evolution a r X i v : . [ a s t r o - ph . H E ] M a r Mapelli et al. and to the final fate of their progenitor stars. Massivestars ( (cid:38)
30 M (cid:12) ) can lose a significant fraction of theirinitial mass by stellar winds, depending mostly (but notonly) on their metallicity (Kudritzki et al. 1987; Vinket al. 2001) and luminosity (Gr¨afener & Hamann 2008;Vink et al. 2011). We expect that the final mass and theinner properties of a star at the onset of collapse havea strong impact on the final outcome of a core-collapsesupernova (CCSN). If the final mass of the star is suf-ficiently large (Fryer 1999; Fryer & Kalogera 2001) andthe central compactness sufficiently high (O’Connor &Ott 2011; Ugliano et al. 2012), a star might even avoidthe final explosion and collapse to a BH quietly. Basedon this reasoning, the maximum mass of BHs is pre-dicted to depend on progenitor’s metallicity, with metal-poor stars leaving more massive remnants than metal-rich ones (Heger et al. 2003; Mapelli et al. 2009, 2010,2013; Belczynski et al. 2010; Fryer et al. 2012; Speraet al. 2015; Spera & Mapelli 2017).This basic framework is complicated by uncertaintieson CCSN models (e.g. Janka 2012, 2017; Foglizzo et al.2015; Sukhbold et al. 2016; Pejcha & Thompson 2015;Burrows et al. 2018; Ebinger et al. 2019a,b), by the ex-istence of other explosion mechanisms, such as electron-capture supernovae (Nomoto 1984, 1987; Jones et al.2013), pulsational pair instability supernovae (PPISNe)and pair instability supernovae (PISNe) (Fowler & Hoyle1964; Barkat et al. 1967; Woosley et al. 2007; Woosley2017, 2019), and by the complex physics of massive starevolution.In particular, population-synthesis models used to in-vestigate the mass function of (single and binary) BHs(e.g. Bethe & Brown 1998; Portegies Zwart & Yungel-son 1998; Belczynski et al. 2002, 2008, 2010; Mapelliet al. 2013; Mennekens & Vanbeveren 2014; Spera et al.2015; Spera & Mapelli 2017; Eldridge & Stanway 2016;Stevenson et al. 2017; Mapelli et al. 2017; Mapelli & Gi-acobbo 2018; Giacobbo et al. 2018; Giacobbo & Mapelli2018; Kruckow et al. 2018; Spera et al. 2019; Eldridgeet al. 2019; Mapelli et al. 2019; Stevenson et al. 2019)usually do not include stellar rotation among their in-gredients. This might be a serious issue, because stel-lar rotation can dramatically affect the evolution of theprogenitor star (Limongi & Chieffi 2018; Dvorkin et al.2018; Groh et al. 2019). Rotation has (at least) two com-peting effects on stellar evolution. It enhances chem-ical mixing (Meynet & Maeder 2005; Ekstr¨om et al.2012; Chieffi & Limongi 2013; Marchant et al. 2016; deMink & Mandel 2016; Mandel & de Mink 2016), lead-ing to the development of larger stellar cores, and atthe same time enhances mass loss, quenching the finalstellar mass (see e.g. Limongi 2017 for a review). Stars with He core 135 (cid:38) M He / M (cid:12) (cid:38)
64 are expected to un-dergo a PISN leaving no compact remnant. Stars with64 (cid:38) M He / M (cid:12) (cid:38)
32 experience enhanced mass loss be-cause of pulsational pair instability. Since stellar rota-tion leads to the formation of more massive He cores,especially at low metallicity where winds are quenched,the minimum zero-age main sequence (ZAMS) mass fora rotating star to undergo PISN and PPISN can be sig-nificantly smaller than the minimum ZAMS mass for anon-rotating star.Moreover, most population synthesis codes model theoutcome of a CCSN explosion based on the carbon-oxygen mass of the progenitor star, following the pre-scriptions in Fryer et al. (2012), but hydrodynamicalsimulations of CCSNe suggest that this approach mightbe incomplete. For example, O’Connor & Ott (2011)propose that the outcome of a CCSN, for a given equa-tion of state, can be estimated, to first order, by thecompactness of the stellar core at bounce, defined as ξ M = M/ M (cid:12) R ( M ) / , (1)where R ( M ) is the radius that encloses a baryonic massequal to M at core bounce and M is a given mass (usu-ally M = 2 . (cid:12) ).Here we present a new version of the population-synthesis code sevn (Spera et al. 2015; Spera & Mapelli2017; Spera et al. 2019) in which we include stellar ro-tation by means of the franec stellar evolution tracks(Limongi et al. 2000; Chieffi & Limongi 2004; Limongi& Chieffi 2006; Chieffi & Limongi 2013; Limongi & Chi-effi 2018). We discuss the impact of stellar rotationon compact-object mass. We also add a new simpleprescription to include compactness and we comparethe outcomes of CCSNe described by compactness withFryer et al. (2012) prescriptions. METHODS2.1.
SEVN sevn ’s main difference with respect to most popula-tion synthesis codes is the approach to stellar evolution(Spera et al. 2015; Spera & Mapelli 2017; Spera et al.2019). While the vast majority of population synthesiscodes implements stellar evolution through the polyno-mial fitting formulas initially derived by Hurley et al.(2000), sevn describes stellar evolution through look-up tables, obtained from stellar evolution tracks . Thelook-up tables contain information on star mass and core combine (Kruckow et al. 2018) is the only other binary popu-lation synthesis code (besides sevn ) that adopts look-up tablesand has been used to study binary compact objects. mpact of rotation and compactness on black holes parsec stellar evolution tracks (Bressanet al. 2012; Tang et al. 2014; Chen et al. 2015; Marigoet al. 2017). In this work, we describe the implementa-tion of new tables derived from franec (see the nextsection). The interpolation algorithm adopted in sevn is already described in Spera & Mapelli (2017) and Speraet al. (2019). The main advantage of using look-up ta-bles with respect to polynomial fitting formulas is thatstellar evolution in sevn can be updated very easily bychanging the current set of look-up tables with a newone, while polynomial fitting formulas are bound to thestellar evolution model they were extracted from.Binary evolution is implemented in sevn followingthe prescriptions by Hurley et al. (2002). We includea treatment of tides, decay by gravitational-wave emis-sion, mass transfer and common envelope as already dis-cussed in Spera et al. (2019). The main novelty with re-spect to Hurley et al. (2002) consists in the descriptionof common envelope and stellar mergers. Thanks to theinterpolation algorithm, the mass and the stellar type ofthe outcome of a common envelope or a stellar mergerare derived from the look-up tables directly, without theneed for a collision matrix or other fitting formulas.Here below, we describe the new tables derived from franec and the updates to the description of CCSNoutcomes in sevn .2.2. franec stellar evolution tracks The stellar models adopted in this paper have beencomputed by means of the latest release of the franec code. Here, we summarize their main features, whilewe refer to Limongi & Chieffi (2018) for a full descrip-tion of the models and the code . The initial masses are13, 15, 20, 25, 30, 40, 60, 80 and 120 M (cid:12) , the initialmetallicities are [Fe/H]= 0 , − , − , −
3, and the initialequatorial rotation velocities are 0, 150 and 300 km s − .We adopt the solar composition from Asplund et al.(2009), corresponding to a total heavy element massfraction of Z (cid:12) ∼ . The main properties of these models, together to their final yields,are available at the webpage http://orfeo.iaps.inaf.it. More spe-cific details about the models may be provided upon request. sponding to [Fe/H]= − , − , − ∼ × − , × − , × − , respectively. The initial velocities werechosen to roughly span the range of observed values(Dufton et al. 2006; Hunter et al. 2008; Ram´ırez-Agudeloet al. 2017).The nuclear network, fully coupled to the equationsfor the stellar structure as well as to the various kindsof mixing, includes 335 isotopes in total, from H to Bi,linked by more than 3000 nuclear reactions. This net-work is well suited to properly follow all the stable andexplosive nuclear burning stages of massive stars.Mass loss is taken into account following differentprescriptions for the various evolutionary stages, e.g.,Vink et al. (2000, 2001) for the blue supergiant phase(T eff > eff < Core-collapse supernovae (CCSNe) sevn includes five different models to describe the out-come of CCSNe: the rapid and delayed models presentedin Fryer et al. (2012), the prescriptions adopted in star-track (Belczynski et al. 2008), the compactness cri-terion (O’Connor & Ott 2011) and the two-parametercriterion by Ertl et al. (2016). The first three modelsdepend only on the carbon-oxygen mass after carbonburning and on the pre-supernova mass of the star, thefourth model depends also on the compactness ξ . , de-fined in equation 1 (assuming M = 2 . (cid:12) ), while thefifth model depends on the enclosed mass at a dimen-sionless entropy per nucleon s = 4 ( M ) and the massgradient at the same location ( µ ).2.3.1. Compactness model
In the previous version of sevn , the criterion basedon compactness and the two-parameter criterion wereimplemented in a non-self-consistent way, because thetable of compactness ξ . and that of M and µ were Mapelli et al. m CO [M fl ] . . . . . . . ξ . v = 0kms − v = 150kms − v = 300kms − Figure 1.
Compactness ξ . as a function of the carbon-oxygen core mass ( m CO ) at the onset of collapse for the franec evolutionary tracks with rotation v = 0 ,
150 and300 km s − (blue, black and red circles, respectively). Thedark red line overlaid to the data is the fit described in equa-tion 2. calculated through the mesa code (Paxton et al. 2011,2013, 2015), while stellar evolution was derived from parsec . Here, we update the treatment of compactnessin a self-consistent way. In fact, compactness can becalculated directly from franec models, because theyare evolved up to the onset of core collapse .Limongi & Chieffi (2018) have shown that there isa strong correlation between compactness and carbon-oxygen mass at the onset of collapse (see their Figure 21)and this correlation is not significantly affected by stel-lar rotation. Thus, in our new version of sevn , we inter-polate compactness among stellar models by using thefollowing fitting formula: ξ . = a + b (cid:18) m CO (cid:12) (cid:19) c , (2)where a = 0 . b = − . c = − .
0. Figure 1 showsthe fit reported in equation 2 overlaid to the data of franec .O’Connor & Ott (2011) suggest that progenitors with ξ . > .
45 most likely form BHs without explosion,while Horiuchi et al. (2014) suggest a lower thresholdvalue ( ξ . (cid:38) . ξ . = 0 . ξ . ≤ . O’Connor & Ott (2011) adopt compactness at bounce, butUgliano et al. (2012) show that compactness at the onset of col-lapse is consistent with compactness at bounce and is much easierto estimate. Hereafter, we refer to compactness at the onset ofcollapse. neutron star (NS) by CCSN explosion, while progenitorswith ξ . > . ξ . does not show amonotonic trend with the CO core (Sukhbold et al. 2018and references therein), but rather has a complicatedtrend, with several localized branches and multivaluedsolutions. This result is still a matter of debate. Weare studying this problem in detail and will discuss ourresults in a forthcoming paper. For this reason, andfor the purposes of the present paper, here we adopt aconservative approach based on the results presented inLimongi & Chieffi (2018).The compactness criterion allows us to discriminatebetween the formation of a NS (if the progenitor ex-plodes) and that of a BH (if the progenitor collapsesdirectly). When the progenitor explodes leaving a NS,the mass of the NS is assigned randomly, following aGaussian distribution with mean (cid:104) m NS (cid:105) = 1 .
33 M (cid:12) anddispersion σ NS = 0 .
09 M (cid:12) , based on the distributionof observed NSs in binary NS systems ( ¨Ozel & Freire2016).When the progenitor undergoes a direct collapse, themass of the BH is derived as m BH = m He + f H ( m fin − m He ) , (3)where m fin and m He are the total mass and the He coremass of the star at the onset of collapse, respectively (theHe core, by definition, includes also heavier elementsinside the He core radius), while f H is a free parameterwhich can assume values from 0 to 1. The presenceof f H accounts for the uncertainty about the collapseof the H envelope (if the progenitor star retains a Henvelope to the very end). Some studies (e.g. Nadezhin1980; Lovegrove & Woosley 2013; Sukhbold et al. 2016;Fern´andez et al. 2018) stress that is quite unlikely thatthe H envelope collapses entirely, even during a directcollapse, because it is loosely bound. In the following,we consider the two extreme cases in which f H = 0 (theH envelope is completely lost) and f H = 0 . f H = 0 and f H = 0 .
9, we are able to bracket themain uncertainties on direct collapse.In the compactness model, we assume that the effi-ciency of fallback is negligible, following recent hydro-dynamical simulations (e.g. Ertl et al. 2016).2.3.2.
Rapid model
In this work, we compare the new compactness crite-rion implemented in sevn with the rapid CCSN modelby Fryer et al. (2012), which assumes that the explosion mpact of rotation and compactness on black holes <
250 ms after bounce. In the rapid model, themass of the compact object is m rem = m proto + m fb ,where m proto = 1 M (cid:12) is the mass of the proto-compactobject and m fb = f fb ( m fin − m proto ) is the mass ac-creted by fallback. In the previous expression, f fb is thefractional fallback parameter, defined as in Fryer et al.(2012).In the rapid CCSN formalism, the maximum NS massis 2 M (cid:12) , while the minimum BH mass is 5 M (cid:12) . Thisresult strongly depends on the assumptions about fall-back. In contrast, our compactness-based model cannotpredict a maximum NS mass, because the mass of theNS is derived from an observational distribution ( ¨Ozel& Freire 2016).We stress that none of the prescriptions currentlyadopted in the literature to infer the mass of compactobjects (including the rapid model and the compactness-based models adopted in this work) is sufficient to cap-ture the complexity of CCSN physics (see e.g. Burrowset al. 2018, 2019; Vartanyan et al. 2019). The aim ofour study is to compare different CCSN prescriptionsand to quantify the uncertainties on BH mass spectrumthat arise from a different choice of these simplified pre-scriptions. 2.4. PPISNe and PISNe sevn includes a treatment for PISNe and PPISNe asdescribed in Spera & Mapelli (2017), based on the re-sults of Woosley (2017). In particular, if the He coremass is 135 ≥ m He / M (cid:12) ≥
64, the star undergoes aPISN and leaves no compact object. If the He core massis 64 > m He / M (cid:12) ≥
32, the star undergoes pulsationalpair instability and the final mass of the compact objectis calculated as m rem = α P m no PPI , where m no PPI is themass of the compact object we would have obtained ifwe had not included pulsational pair instability in ouranalysis (just CCSN) and α P is a fitting parameter de-scribed in Appendix A. RESULTS3.1.
Impact of rotation on BH masses
Figures 2 and 3 show the mass of compact objects asa function of the ZAMS mass of their progenitor starsfor different CCSN models (rapid, compactness with ξ . = 0 . f H = 0, and compactness with ξ . = 0 . f H = 0 . franec stellar evolution tables for three initial equato-rial velocities of the progenitor stars: v = 0, 150 and300 km s − . For comparison, we show also the resultsof parsec stellar evolution tables with v = 0 km s − .From these Figures it is apparent the strong impact ofrotation on the minimum ZAMS mass for BH formation, regardless of progenitor’s metallicity. The minimumprogenitor mass to collapse to a BH is m ZAMS ∼ − (cid:12) for rotating stars and m ZAMS ∼ −
25 M (cid:12) for non-rotating stars (with a mild dependence on the CCSNmodel, see Table 1). This happens because stars with10 (cid:46) m ZAMS / M (cid:12) (cid:46)
30 are not particularly affected bystellar winds, regardless of their metallicity. Thus, an-gular momentum is not efficiently removed by mass lossand rotation has enough time to induce chemical mixing,leading to the growth of the stellar core. This shifts thethreshold between explosion and direct collapse towardslower ZAMS masses.Furthermore, Figures 2 and 3 show that stellarrotation has a strong impact on the (pulsational)pair-instability window for metal-poor stars ( Z =0 . , . Z =0 . , . m PPISN ∼
50 M (cid:12) and ∼
70 M (cid:12) for rotating andnon-rotating models, respectively, see Table 1). Again,this happens because chemical mixing leads to signifi-cantly larger He cores in rotating metal-poor stars. Wenote that there are no significant differences between v = 150 km s − and v = 300 km s − .We now go through different metallicities, to discusshow the effect of stellar rotation changes with Z . Inmetal-poor stars ( Z ≤ . Z = 0 . / m ZAMS (cid:46)
30 M (cid:12) and m ZAMS (cid:38)
30 M (cid:12) . If m ZAMS (cid:46)
30 M (cid:12) , stellar windsare not particularly efficient, even in rotating models.Thus, rotating stars develop larger cores and end theirlife with higher compactness than non-rotating stars.The main consequence of this is that the minimum pro-genitor mass to collapse to a BH is smaller for rotat-ing stars than for non-rotating stars. In contrast, if m ZAMS (cid:38)
30 M (cid:12) , stellar winds are efficient at Z = 0 . Z = 0 .
003 the mini-mum ZAMS mass to enter the PPISN regime is slightlylower for non-rotating models ( m PPISN ∼ −
68 M (cid:12)
Mapelli et al.
10 20 30 40 50 60 70 80 90 100 110 120 m ZAMS [M fl ] m r e m [ M fl ] Z = 0 . FRANEC , − FRANEC , − FRANEC , −
10 20 30 40 50 60 70 80 90 100 110 120 m ZAMS [M fl ] m r e m [ M fl ] Z = 0 . FRANEC , − FRANEC , − FRANEC , − PARSEC , −
10 20 30 40 50 60 70 80 90 100 110 120 m ZAMS [M fl ] m r e m [ M fl ] Z = 0 . FRANEC , − FRANEC , − FRANEC , − PARSEC , −
10 20 30 40 50 60 70 80 90 100 110 120 m ZAMS [M fl ] m r e m [ M fl ] Z = 0 . FRANEC , − FRANEC , − FRANEC , − PARSEC , − Figure 2.
Estimated mass of the compact object ( m rem ) as a function of the zero-age main sequence (ZAMS) mass of theprogenitor star ( m ZAMS ). The outcome of CCSNe is described by the rapid model (Fryer et al. 2012). From top to bottom andfrom left to right: Z = 0 . franec (Limongi &Chieffi 2018) with initial equatorial rotation speed v = 300 km s − . Black dashed line: franec (Limongi & Chieffi 2018) with v = 150 km s − . Blue dot-dashed line: franec (Limongi & Chieffi 2018) with v = 0 km s − . Green solid line: stellar evolutionis described by parsec (Bressan et al. 2012). We do not have parsec models with metallicity Z = 0 . m He = 32 M (cid:12) ( m He = 64 M (cid:12) ), corresponding to the minimummass to undergo PPISN (PISN). for v = 0 km s − , Table 1) than for rotating models( m PPISN ∼
80 M (cid:12) for v = 300 km s − , Table 1), withan opposite behavior with respect to more metal-poorstars. Stars with m ZAMS ≤
120 M (cid:12) and Z = 0 .
003 donot develop He cores >
64 M (cid:12) , thus they do not enterthe PISN regime.Finally, metal-rich stars ( Z = 0 . ∼ Z (cid:12) ) with m ZAMS ≤
30 M (cid:12) behave similarly to metal-poor stars:they are only mildly affected by mass loss; hence, rotat- ing stars grow larger He cores than non-rotating stars,causing the minimum ZAMS mass for BH formationto shift to lower values in rotating models. In con-trast, stellar winds are so efficient in metal-rich starswith m ZAMS (cid:38)
30 M (cid:12) that they do not enter eitherthe PPISN or PISN window, regardless of their rotationspeed (with the exception of the parsec model, whichundergoes PPISNe at m ZAMS (cid:38)
94 M (cid:12) ). At high Z ,stellar rotation does not affect significantly the maxi- mpact of rotation and compactness on black holes
10 20 30 40 50 60 70 80 90 100 110 120 m ZAMS [M fl ] m r e m [ M fl ] Z = 0 . ξ . = 0 . , f H = 0 ξ . = 0 . , f H = 0 . , − FRANEC , − FRANEC , −
10 20 30 40 50 60 70 80 90 100 110 120 m ZAMS [M fl ] m r e m [ M fl ] Z = 0 . ξ . = 0 . , f H = 0 ξ . = 0 . , f H = 0 . , − FRANEC , − FRANEC , − PARSEC , −
10 20 30 40 50 60 70 80 90 100 110 120 m ZAMS [M fl ] m r e m [ M fl ] Z = 0 . ξ . = 0 . , f H = 0 ξ . = 0 . , f H = 0 . , − FRANEC , − FRANEC , − PARSEC , −
10 20 30 40 50 60 70 80 90 100 110 120 m ZAMS [M fl ] m r e m [ M fl ] Z = 0 . ξ . = 0 . , f H = 0 ξ . = 0 . , f H = 0 . , − FRANEC , − FRANEC , − PARSEC , − Figure 3.
Same as Figure 2, but CCSNe are described with the compactness criterion. Thick lines: we assume ξ . = 0 . f H = 0 .
0; thin lines: we assume ξ . = 0 . f H = 0 . mum BH mass, which is ∼ −
24 M (cid:12) , regardless of theassumed CCSN model.3.2.
Impact of CCSN model on BH masses
Figure 2 shows the mass of compact objects we obtainassuming the rapid CCSN model described in Fryer et al.(2012). In contrast, Figure 3 is based on the compact-ness criterion. By considering these different models, wewant to quantify the uncertainty on BH mass derivingfrom CCSN prescriptions.The main sources of uncertainty are the amount offallback, the minimum value of the compactness (orcarbon-oxygen mass) required for direct collapse and thefate of the hydrogen envelope (if any). The rapid model by Fryer et al. (2012) assumes that fallback can be ef-ficient (mass accreted by fallback m fb ≥ . (cid:12) ) andthat stars with carbon-oxygen core mass m CO ≥
11 M (cid:12) collapse to BH directly, including their hydrogen enve-lope (if any). In contrast, in the compactness model weassume no fallback at all and we require that stars withcompactness ξ . ≥ . f H = 0 . f H = 0 .
9) we assume that the hydrogen enve-lope does not collapse (90 % of the hydrogen envelopecollapses) to BH.The main difference between the rapid model and thecompactness model, which manifests regardless of stellarrotation and metallicity, is the minimum ZAMS mass to
Mapelli et al. m rem [M fl ] − − − P D F v=0km s − Z=0 . ξ . , f H = 0 . ξ . , f H = 0 . m rem [M fl ] − − − P D F v=0km s − Z=0 . ξ . , f H = 0 . ξ . , f H = 0 . m rem [M fl ] − − − P D F v=0km s − Z=0 . ξ . , f H = 0 . ξ . , f H = 0 . m rem [M fl ] − − − P D F v=150km s − Z=0 . ξ . , f H = 0 . ξ . , f H = 0 . m rem [M fl ] − − − P D F v=150km s − Z=0 . ξ . , f H = 0 . ξ . , f H = 0 . m rem [M fl ] − − − P D F v=150km s − Z=0 . ξ . , f H = 0 . ξ . , f H = 0 . m rem [M fl ] − − − P D F v=300km s − Z=0 . ξ . , f H = 0 . ξ . , f H = 0 . m rem [M fl ] − − − P D F v=300km s − Z=0 . ξ . , f H = 0 . ξ . , f H = 0 . m rem [M fl ] − − − P D F v=300km s − Z=0 . ξ . , f H = 0 . ξ . , f H = 0 . Figure 4.
Probability distribution function (PDF) of compact object masses. We assume a Kroupa (2001) IMF for theprogenitor stars with minimum mass m min = 13 M (cid:12) and maximum mass m max = 120 M (cid:12) . Orange line: rapid model for CCSNe(Fryer et al. 2012); purple line: compactness criterion with ξ . = 0 . f H = 0; dark red line: compactness criterion with ξ . = 0 . f H = 0 .
9. At Z = 0 . v = 0 km s − ; middle row: v = 150 km s − ; lower row: v = 300 km s − . Left-hand column: Z = 0 . Z = 0 . Z = 0 . form a BH (Table 1). This difference arises mostly fromthe adopted threshold for direct collapse. In fact, directcollapse happens in the rapid model if m CO ≥
11 M (cid:12) ,which (according to equation 2) corresponds to compact-ness threshold ξ . ≥ .
45. By increasing the thresholdfor direct collapse from ξ . = 0 . ξ . = 0 .
45, thecompactness models produce approximately the sameminimum ZAMS mass for BH formation as the rapidmodel. Another feature of the rapid model which does notshow up in the compactness-based models, regardless ofstellar metallicity and rotation, is the complex behaviorof BH mass for m ZAMS (cid:46)
40 M (cid:12) . This is a consequenceof the sophisticated fitting formulas for fallback derivedfrom Fryer et al. (2012).If m ZAMS (cid:38)
40 M (cid:12) , metallicity and rotation matter,as we have seen in the previous section. If stellar metal-licity is high ( Z = 0 . m ZAMS (cid:38)
40 M (cid:12) , the mpact of rotation and compactness on black holes Z = 0 . Z ≤ . m ZAMS (cid:38)
40 M (cid:12) , the initial rotation becomes the cru-cial ingredient. If the star rotates, the minimum ZAMSmass for PPISN and PISN decreases significantly (seethe previous section) and stellar winds are efficient evenat low metallicity. The combination of these two effectsremoves the hydrogen envelope and even a fraction ofthe He core. For this reason, the rapid model and thetwo compactness models are indistinguishable for rotat-ing stars with Z ≤ .
003 and m ZAMS (cid:38)
40 M (cid:12) .If the star does not rotate, the BH mass for 40 (cid:46) m ZAMS /M (cid:12) (cid:46)
80 and Z ≤ . f H = 0 .
9) predict a BH mass m BH ∼
60 M (cid:12) ,almost twice as large as that expected from the compact-ness model with f H = 0 in this range of ZAMS masses.Finally, non rotating stars with m ZAMS (cid:38)
80 M (cid:12) ejecttheir H envelope entirely. Thus, the three CCSN modelspredict similar BH masses for extremely massive metal-poor non rotating stars.In summary, if we look at the maximum BH mass,rotating models predict m BH , max ≤
45 M (cid:12) (originat-ing from stars with m ZAMS ∼ −
100 M (cid:12) and Z ≤ . m BH , max ∼
60 M (cid:12) (orig-inating from stars with m ZAMS ∼ −
70 M (cid:12) and Z ≤ . m BH , max ∼ −
50 M (cid:12) (originating from stars with m ZAMS ∼ −
120 M (cid:12) and Z ≤ . Impact of rotation, CCSN model and metallicityon BH mass function
For each considered metallicity, for each rotationspeed and for each CCSN model separately, we have gen-erated a set of 10 single stars distributed according to aKroupa initial mass function (IMF, i.e. dN/dm ∝ m − α with α = 2 .
3, Kroupa 2001), with minimum ZAMS mass m min = 13 M (cid:12) and maximum ZAMS mass m max = 120M (cid:12) . Figure 4 shows the mass function of compact objectsfor the three considered rotation speeds, for three metal-licities ( Z = 0 . Z = 0 . franec tracks is m ZAMS = 13 M (cid:12) .Smaller masses will be included in follow-up works.In general, the mass function of single BHs can beapproximated with a power law, but the slope of thepower law depends on metallicity, on rotation speed andon the assumed CCSN prescription. If we make a linearfit of log PDF = D log m rem + G across our models,we find a preferred value of D ≈ − .
5, with a verylarge scatter. Binary evolution can change this scalingdramatically and will be included in a follow-up study.The main differences among all the considered mod-els are the number of NSs and the minimum mass ofBHs. Because of the difference in the minimum ZAMSmass to form a BH (see Sections 3.1 and 3.2), stars with v = 300 km s − and minimum mass m ZAMS = 13 M (cid:12) adopting a compactness-based CCSN criterion do notform NSs, regardless of their metallicity. For these ex-tremely fast rotating models to produce NSs, we needto assume a significantly higher ξ . threshold.The minimum BH mass spans from ∼ . (cid:12) to ∼
15 M (cid:12) , depending on the CCSN prescription (thecompactness-based model with f H = 0 . m rem (cid:38)
60 M (cid:12) form only from non-rotating models).At solar metallicity, the three CCSNe models andthe three rotation speeds produce very similar BHpopulations (almost identical in the case of the twocompactness-based models). The reason is that stellarwinds peel-off massive stars, regardless of their initialrotation velocity and of the assumed CCSN model. Incontrast, at lower metallicities the differences betweenthe three CCSN models become important.In this section we assumed that stars in the same stel-lar population have the same initial rotation speed. Thisis clearly a simplistic assumption because stars mightform with different initial speed. Data of stellar rota-tion in the Milky Way show that stellar speeds should bedistributed according to a Gaussian with average speed ∼
200 km s − and dispersion ∼
100 km s − (Duftonet al. 2006). In follow-up studies we will consider a dis-tribution of initial stellar rotation velocities.0 Mapelli et al.
10 20 30 40 50 60 70 80 90 100 110 120 m ZAMS [M fl ] m r e m [ M fl ] v=0kms − Z =0 . Z =0 . Z =0 . Z =0 .
10 20 30 40 50 60 70 80 90 100 110 120 m ZAMS [M fl ] m r e m [ M fl ] v=150kms − Z =0 . Z =0 . Z =0 . Z =0 .
10 20 30 40 50 60 70 80 90 100 110 120 m ZAMS [M fl ] m r e m [ M fl ] v=300kms − Z =0 . Z =0 . Z =0 . Z =0 . Figure 5.
Mass of the compact object as a function of the progenitor’s ZAMS mass for the main models we have considered,in comparison with some previous studies. Left: rotation velocity v = 0 km s − . Middle: v = 150 km s − . Right: v = 300km s − . The solid lines show the mean value of m rem we obtain by averaging over the three CCSN models considered in thisstudy, while the shaded areas show the maximum differences between the three CCSN models. Open triangles (LC2018): Rmodel from Limongi & Chieffi (2018). Open stars (GM2018): compact object mass predicted by mobse (Giacobbo & Mapelli2018), adopting the delayed CCSN model by Fryer et al. (2012). Open circles (SM2017): compact object mass estimatedwith sevn (Spera & Mapelli 2017), adopting the delayed CCSN and the parsec stellar tracks (Bressan et al. 2012). Openpentagons (B2016): compact object mass estimated with bse (Hurley et al. 2002), adopting the same stellar winds, PPISN andPPISN model as startrack (Belczynski et al. 2016). In all panels and for all symbols and lines, red: progenitor’s metallicity Z = 0 . Z = 0 . Z = 0 . Z = 0 . Comparison with previous work
Figures 2 and 3 show that there is not much differencebetween parsec models and franec models with v =0 km s − when implemented inside sevn and treatedwith the same model for CCSNe, PISNe and PPISNe.It is worth noting that while the typical difference inthe maximum BH mass between franec and parsec is ∼
10 % at low metallicity, the difference becomes ∼
27 %at solar metallicity ( Z = 0 . franec tracks and accounting for the uncertainties induced bythe CCSN model with a shaded area) with the massspectrum obtained in previous studies, as a function ofthe ZAMS mass. In particular, we plot the mass spec-trum from Spera & Mapelli (2017), hereafter SM2017,from Giacobbo & Mapelli (2018), hereafter GM2018,and from Limongi & Chieffi (2018), hereafter LC2018.We also consider a version of bse (Hurley et al. 2000,2002) that includes the same stellar-wind, PISN andPPISN prescriptions as startrack (Belczynski et al.2016), hereafter B2016.Our results are similar to the mass spectrum obtainedwith mobse (GM2018), although the maximum BHmass in mobse ( m BH , max ∼
65 M (cid:12) at Z = 0 . ∼ sevn . Metal-poor stars with m ZAMS ∼ −
80 M (cid:12) seem to retain a more generous portion of their hydro- gen envelope at collapse when integrated with mobse .Our results are also broadly consistent with SM2017 formetal-poor progenitors, while at Z = 0 . ∼ −
30 % larger BH masses (up to ∼ (cid:12) ), explained by the fact that SM2017 adopt parsec tracks. The models labelled as B2016 predict a maxi-mum BH mass ∼
40 M (cid:12) , significantly smaller than ourmodel with f H = 0 . f H = 0). However, B2016 assumethat the H envelope, when present, collapses with therest of the star. In their model, metal-poor stars with m ZAMS ∼ −
80 M (cid:12) lose their hydrogen envelope al-most completely for the different treatment of luminousblue variable stellar winds and of pulsational pair insta-bility.LC2018 adopt the same franec tracks we use here.Figure 5 shows their model R which assumes that starswith m ZAMS ≤
25 M (cid:12) explode as CCSNe, while starswith m ZAMS >
25 M (cid:12) collapse to BH directly, with m rem = m fin (no mass ejection). Thus, the trianglesshown in Figure 5 represent the upper limit to BHmasses we can obtain with franec if m ZAMS >
25 M (cid:12) .Finally, several previous studies investigate the impactof stellar rotation on PPISNe and PISNe (Chatzopou-los & Wheeler 2012a,b; Yoon et al. 2012; Yusof et al.2013; Takahashi et al. 2018; Uchida et al. 2019). Ourmain findings agree with their results: i) the minimumZAMS mass to undergo a PPISN and a PISN lowerssignificantly if stellar rotation is accounted for (Chat-zopoulos & Wheeler 2012a), and ii) rotating models of mpact of rotation and compactness on black holes CONCLUSIONSWe have investigated the impact of rotation and com-pactness on the mass of black holes (BHs), by imple-menting rotating stellar evolution models (Limongi &Chieffi 2018) into our population synthesis code sevn (Spera et al. 2015; Spera & Mapelli 2017; Spera et al.2019).Rotation has two major effects on BH formation.First, rotation reduces the minimum ZAMS mass fora star to collapse into a BH from ∼ −
25 M (cid:12) to ∼ −
18 M (cid:12) (according to the assumed CCSN prescrip-tions), because intermediate-mass ( m ZAMS ∼ − (cid:12) ) rotating stars develop a larger carbon-oxygen coreand a higher compactness than non-rotating stars.Secondly, rotation reduces the maximum BH massfrom metal-poor progenitors. This result comes fromtwo combined effects: i) rotation increases stellar windefficiency; thus, rotating metal-poor ( Z = 0 . − . m ZAMS ∼ −
80 M (cid:12) lose their Henvelope entirely, while non-rotating metal-poor starspreserve most of it; ii) chemical mixing induced by ro-tation increases the mass of the He core, reducing theminimum ZAMS mass for PPISNe and PISNe to hap-pen.If we assume that the entire final mass of a star (in-cluding its residual hydrogen envelope) can collapse to aBH directly, the maximum BH mass from non-rotatingstars is ∼
60 M (cid:12) , while the maximum BH mass fromfast rotating stars is ∼
45 M (cid:12) .Besides rotation, the mass of BHs is also strongly af-fected by the assumed CCSN model, especially by theamount of fallback, by the adopted threshold for directcollapse (based on ξ . or on m CO ) and by the differentfraction of hydrogen envelope that is able to collapse( f H ).In particular, the minimum ZAMS mass for a star toform a BH depends on the assumed threshold of com-pactness ξ . (larger values of the threshold leading tohigher minimum ZAMS masses) and on the efficiency offallback.The maximum BH mass that we expect from non ro-tating metal-poor ( Z = 0 . − . ∼
60 M (cid:12) ,approximately 1.5 times higher than if we assume thatonly the He core is able to collapse. This assumption is not important for metal-poor massive rotating stars andfor metal-rich (both rotating and non-rotating) stars,because stellar winds remove their hydrogen envelopeentirely, leveling these differences.Here, we consider only single stars. In future works,we will investigate how binary evolution and star clus-ter dynamics affect our conclusions. We anticipate thatclose binary evolution should lead to a further strippingof the hydrogen envelope, affecting the maximum BHmass (see e.g. Giacobbo & Mapelli 2018). On the otherhand, star cluster dynamics can lead to the formationof binary BHs that incorporate the most massive BHsformed from single star evolution and from the mergerof massive binaries (see e.g. Mapelli 2016; Di Carlo et al.2019), making the final scenario even more complex.The methodology we presented here might be appliedto estimate upper limits on BH spins. For all our modelswe find an upper limit to the final spin close to maxi-mally rotating BHs. However, our models do not includemechanisms for efficient angular momentum dissipation,such as the Tayler-Spruit dynamo (Spruit 2002; Fulleret al. 2019). Efficient angular momentum transport canlead to significantly lower BH spins ( a BH (cid:46) . Software: sevn (Spera & Mapelli 2017), parsec (Bressan et al. 2012), franec (Limongi et al. 2000), mobse (Giacobbo et al. 2018)APPENDIX2
Mapelli et al.
Table 1.
Most relevant masses.Stellar Ev. CCSN v (km s − ) Z m
ZAMS , min (M (cid:12) ) m PPISN (M (cid:12) ) m PISN (M (cid:12) ) m BH , max (M (cid:12) ) franec rapid 300 0.00003 18 52 97 42 franec rapid 300 0.0003 17 53 110 42 franec rapid 300 0.003 17 80 – 34 franec rapid 300 0.0135 18 – – 16 franec rapid 150 0.00003 17 50 103 43 franec rapid 150 0.0003 17 55 107 42 franec rapid 150 0.003 17 70 – 35 franec rapid 150 0.0135 18 – – 23 franec rapid 0 0.00003 24 67 – 59 franec rapid 0 0.0003 25 69 – 60 franec rapid 0 0.003 23 68 – 39 franec rapid 0 0.0135 25 – – 24 parsec rapid 0 0.0003 22 63 – 54 parsec rapid 0 0.003 22 66 – 43 parsec rapid 0 0.0135 23 94 – 33 franec ξ . = 0 . , f H = 0 300 0.00003 14 52 97 42 franec ξ . = 0 . , f H = 0 300 0.0003 ≤
13 53 110 42 franec ξ . = 0 . , f H = 0 300 0.003 ≤
13 80 – 34 franec ξ . = 0 . , f H = 0 300 0.0135 ≤
13 – – 16 franec ξ . = 0 . , f H = 0 150 0.00003 13 50 103 43 franec ξ . = 0 . , f H = 0 150 0.0003 14 55 107 42 franec ξ . = 0 . , f H = 0 150 0.003 14 70 – 35 franec ξ . = 0 . , f H = 0 150 0.0135 14 – – 23 franec ξ . = 0 . , f H = 0 0 0.00003 20 67 – 47 franec ξ . = 0 . , f H = 0 0 0.0003 21 69 – 45 franec ξ . = 0 . , f H = 0 0 0.003 21 68 – 39 franec ξ . = 0 . , f H = 0 0 0.0135 21 – – 24 parsec ξ . = 0 . , f H = 0 0 0.0003 19 63 – 45 parsec ξ . = 0 . , f H = 0 0 0.003 18 66 – 41 parsec ξ . = 0 . , f H = 0 0 0.0135 19 94 – 33 franec ξ . = 0 . , f H = 0 . franec ξ . = 0 . , f H = 0 . ≤
13 53 110 42 franec ξ . = 0 . , f H = 0 . ≤
13 80 – 34 franec ξ . = 0 . , f H = 0 . ≤
13 – – 16 franec ξ . = 0 . , f H = 0 . franec ξ . = 0 . , f H = 0 . franec ξ . = 0 . , f H = 0 . franec ξ . = 0 . , f H = 0 . franec ξ . = 0 . , f H = 0 . franec ξ . = 0 . , f H = 0 . franec ξ . = 0 . , f H = 0 . franec ξ . = 0 . , f H = 0 . parsec ξ . = 0 . , f H = 0 . parsec ξ . = 0 . , f H = 0 . parsec ξ . = 0 . , f H = 0 . Note —Column (1): Stellar evolution tables (from franec or parsec ). Column (2): model for CCSN outcome (see Section 2.3).Column (3): initial rotation speed of progenitor stars. Column (4): progenitor’s metallicity. Column (5): minimum ZAMS massto collapse to a BH (instead of producing a NS); Column (6): minimum ZAMS mass to undergo PPISN ( m PPISN ). Column (7):minimum ZAMS mass to undergo PISN ( m PISN ). Column (8): maximum BH mass ( m BH , max ). mpact of rotation and compactness on black holes A. FITTING FORMULA FOR PPISNE AND PISNEWhen PISNe and PPISNe are effective, we derive the mass of the compact object as m rem = α P m no PPI , where m no PPI is the mass of the compact remnant we would obtain without PPISN/PISN. First, we define the followingquantities F ≡ m He m fin , K ≡ . F + 0 . , S ≡ . F − . . (A1)We then express α P as a function of F , S , K and m He : α P = m He ≤ (cid:12) , ∀F , ∀S . K − m He + 0 . − K ) if 32 < m He / M (cid:12) ≤ , F < . , ∀SK if 37 < m He / M (cid:12) ≤ , F < . , ∀SK (16 . − . m He ) if 60 < m He / M (cid:12) < , F < . , ∀SS ( m He −
32) + 1 if m He ≤ (cid:12) , F ≥ . , ∀S S + 1 if 37 < m He / M (cid:12) ≤ , F ≥ . , S + 1 < . − . F + 0 . m He −
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