On the rate of core collapse supernovae in the Milky Way
OOn the rate of core collapse supernovae in theMilky Way
Karolina Rozwadowska a , Francesco Vissani a , Enrico Cappellaro b a INFN, Laboratori Nazionali del Gran Sasso, Assergi (AQ), Italyand Gran Sasso Science Institute, L’Aquila, Italy b INAF, Osservatorio Astronomico di Padova, Padova, Italy
Abstract
Several large neutrino telescopes, operating at various sites around the world,have as their main objective the first detection of neutrinos emitted by a gravi-tational collapse in the Milky Way. The success of these observation programsdepends on the rate of supernova core collapse in the Milky Way, R . In this work,standard statistical techniques are used to combine several independent results.Their consistency is discussed and the most critical input data are identified. Theinference on R is further tested and refined by including direct information on theoccurrence rate of gravitational collapse events in the Milky Way and in the LocalGroup, obtained from neutrino telescopes and electromagnetic surveys. A conser-vative treatment of the errors yields a combined rate R = . ± .
46 (100 yr) − ;the corresponding time between core collapse supernova events turns out to be T = + − yr. The importance to update the analysis of the stellar birthrate methodis emphasized. The rate of core collapse supernovae (CCSN) in the Milky Way, R , is a quantity ofkey importance for the current and next generation neutrino telescopes, which includesmulti-kiloton detectors such as Super- [1] and Hyper-Kamiokande [2], IceCube [3],JUNO [4] and DUNE [5]; in fact, a quantitative evaluation of the supernova rate isrelevant also for multi-wavelength galactic astronomy, cosmic ray physics and astro-physics at large. Various theoretical expectations are available in the literature, andthey are in the range of one-to-few events per century. See e.g., R = . ± .
74 (100yr) − [6] for a classical result, where we updated the value of the Hubble constant [7]but the error does not include systematic uncertainties, and R = . + . − . (100 yr) − [8],for a new determination based on a SN observability model of the Milky Way, basedon the small sample of 5 historical galactic SNe (of which only two CCSN) that resultsin a very conservative and conversely not very informative rate.1 a r X i v : . [ a s t r o - ph . H E ] S e p n this work we would like to attempt a more precise evaluation by combiningvarious results that have been presented over the last twenty years in the scientificliterature. We obtain a determination of R with a formal uncertainty of 30%, which isquite precise, for which the evaluation of R based on the stellar birth rate has a possiblyundeservedly high weight. We also discuss the inclusion of the neutron star birth rate,which indicates a higher value in moderate tension with other determinations of theCCSN formation rate. We include in the determination of R the observational resultsfrom the Milky Way and in the rest of the Local group of galaxies. Our first estimate of the rate of CCSN is based on the combination of the four re-sults obtained by: i ) counts of massive stars, ii ) census of supernova explosions in thecosmos, iii ) counts of neutron stars and iv ) measurement of the galactic chemical en-richment of Al. The combined value is calculated, tested with an independent and very recent determination (based on SN remnant age determination) and discussed inSect. 2.6. Although all these results are based on methods that are well recognised asvalid and informative, it is worth noting that none of them relies directly on the ob-servations of CCSN in the Milky Way. The contribution of more direct - though lessinformative - methods to the global fit of the rate will be examined in the Section 3.
This method is based on the study of slightly more than 400 stars in the Solar neigh-bourhood to determine the birthrate of stars with mass higher than 10 solar masses [9];this mass limit is adopted to make sure that the stellar catalogue is complete and themethod assumes that the minimum mass, leading to the CCSN, is less than this value.From this sample the author derived an estimate of the galactic supernova rate that isquoted as “probably not less than ∼ ∼ R = . ± . − . Taken on face value, this result is the mostprecise among the available ones, and for this reason extensive discussion is deserved.We note that, among the possible hidden systematics, a most severe bias is relatedto the uneven star distribution in the Galaxy. In particular [10] argue that the supernovarate in a region of 1 kpc from the Sun is greater than the galactic mean value by afactor of 5-6. If this is true, it could be risky to extrapolate the local value to theentire Milky Way. In order to proceed conservatively, without discarding arbitrarilythis result, we adopt as representative the central value indicated by the analysis, butwe conservatively assume the error to 50% of the central value. Therefore, defining asusual, G ( R , ¯ R , δ R ) = exp (cid:34) − ( R − ¯ R ) δ R (cid:35) √ π δ R (1)2e will assume L stellarbirthrate ( R ) = G ( R , ¯ R sb , δ R sb ) with ¯ R sb = . δ R sb = .
75 (2)We refer to this method as stellar birthrate . The rest of this paper shows that this resulthas a relevant weight in the combined fit. Note that, accepting the nominal error wouldgive an even higher weight to this result in the combined fit; a uniform distributionbetween 1 and 2 would produce an even stronger e ff ect. A statistical sample of extragalactic CCSN can be used to infer the expected rate on theMilky Way assuming that our Galaxy has average properties for its morphologic typeand luminosity. Ref. [11] reports R = . ± .
48 (100 yr) − , that updating the Hubbleconstant [7] becomes R = . ± .
41 (100 yr) − .This would mean a narrow Gaussian distribution G , but the attribution of morpho-logic type for the Milky Way is uncertain and requires to include a systematic errorof a multiplicative factor of ∼ Thus, we consider amultiplicative coe ffi cient x with distribution (cid:96) ( x ) = e − (log x / log2) / / ( √ π log(2) x ) (3)that has median x = (cid:82) / (cid:96) ( x ) dx = . R = x × R is obtained changing variable in the thetwo-dimensional distribution G ( R ) (cid:96) ( x ) dxdR and integrating away x : L cosmiccensus ( R ) = (cid:90) ∞ G (cid:18) Rx , ¯ R cens , δ R cens (cid:19) (cid:96) ( x ) dxx with ¯ R cens = . δ R cens = .
41 (4)This likelihood corresponds to R ∈ [0 . , .
96] (100 yr) − at 68.3% CL, which is muchwider than the range from the previous method. We call this method cosmic census ofCCSN. This is the distribution (cid:96) ( x ,µ,α ) = exp[ − (log( x / ¯ x )) / (2log( α ) )] / ( √ π log( α ) x ) that can be derived bythe Gaussian G ( y ,µ,σ ) by changing variable and rewriting the average value as µ = log( ¯ x ); since the valuesbetween α ¯ x and ¯ x /α have to correspond to the 1 σ region, we set the variance σ = log( α ). In our case wehave ¯ x = α = ( a ) : Stellar Birthrate ( Reed 2005 )( b ) : Extragalactic SN rates ( Li et al. 2011 )( c ) : Al - ( Diehl et al. 2006 )( d ) : Neutron star birthrate ( Keane & Kramer 2008 )( e ) : Combination of ( a - d )( f ) : Milky Way optical ( g ) : Milky Way neutrinos ( h ) : Andromeda ( i ) : Rest of Local Group ( j ) : Combination of ( f - i ) , this paper ( k ) : Best estimate, this paper R (
100 yr ) - Figure 1:
CCSN rate R in the Milky Way: from the existing literature (blue), computed fromdirect information from the Local Group (gray) and full result (black) of Eq. (17). Al abundance
This method models the gamma-ray emission from radioactive Al in the Milky Way.Assuming that this emission traces the ongoing nucleosynthesis pollution by CCSN inthe Milky Way, this method was used to infer the value R = . ± . − [12].The uncertainty can be used directly as a Gaussian error: L Al-26 ( R ) = G ( R , ¯ R , δ R ) with¯ R = . δ R = .
1. The shorthand for this method is simply
Al-26.
Again, note thatthis range is much wider than the range from the stellar birthrate method. For a recentstudy of the role that could be played by Al from young stars see [13].
The birthrate of Galactic neutron stars was estimated by [14] by summing 4 contri-butions, supposed to be independent: ordinary radio pulsars, rotating radio transients,X-ray dim isolated neutron stars, magnetars. Their rates, in units of objects per cen-tury are respectively: 1 . ± . , . ± . , . ± . , . ± .
3, obtained by averaging thethree determinations of Table 1 from [14] and using model NE2001. We sum the errorslinearly and find a conservative result, that we use in Gaussian statistics with:¯ R NS = . − , δ R NS = . − , (5)This estimate is significantly higher than the estimate derived with other methods. Thequestion of consistency of this result with the other ones was discussed in the samepaper [14]. Using our estimations, if we compute | ¯ R NS − ¯ R Al-26 | / (cid:113) δ R NS + δ R Al-26 = . .5 Model for SNR ages Recently [16, 17] used a theoretical model to reconstruct the properties of 58 super-nova remnants (SNR). The fraction of CCSN in this sample is f = . ± . Theestimation of the age of the SNRs allows to calculate their production rate r ; this, to-gether with an estimation of the incompleteness factor of the sample η , yields the totalrate of supernovae formation (CCSN + Ia) according to the formula R = r × η . How-ever [17] argues that the original theoretical model does not properly describe the caseof supernovas embedded in a medium with the density profile of a stellar wind. Afterproper analysis of the wind dominated SNR, [17] derived an e ff ective correction factor δ which is estimated to be in the range δ ∈ [0 . , . R = r × (1 + δ ) × η . If we want to limit to the rate of CCSN,we can write R = r × ( f + δ f ) × η , where f is the fraction of CCSN subject to the cor-rection. From Table 4 of [17], one finds that f ∼ . f ; thus we use the conservative range f ∈ [0 . , . R = . − . The 68.3% range ofthe coe ffi cient is in the interval [0 . , . η [17]. This is a very interesting result, and possibly along similar lines there willbe future progress. For the time being, however, we prefer to use this result as a testrather than including it with the other determinations, for a number of reasons: 1) Thevalue of r that we obtain is (0 . ± .
05) (100 yr) − , which is smaller than the value r = /
230 yr ∼ . / century cited in [17]. 2) The correction δ is large, and adoptingthe average value of the interval might introduce a bias in the analysis. Note that thefraction of CCSN decreases after including δ , becoming ( f + δ f ) / (1 + δ ) = . + . − . .3) The works [16, 17] are primarily aimed at modelling the 58 SNR, and in our un-derstanding the determination of total rate of SNR is mostly a consistency check. Inconclusion, we regard the numbers obtained above as preliminary and suggest that anindependent study of SNR, specially dedicated to determining the CCSN rate, shouldbe attempted. The four results listed above employ di ff erent methods with also di ff erent error es-timates. In fact, the methods do not report consistent systematic errors, especiallythe stellar birthrate uses several assumptions, not indicating the exact error. Anyway,these published results are consistent and can be combined to obtain a more preciseinference. Multiplying the four likelihoods, we find R combined = . ± .
55 (100 yr) − (6) This is based on the sub-sets of classified SNRs: 21 −
30 CC and 6 − This was checked in several ways: by considering the number of SNR N ( T ) whose age is less than atime T , and evaluating r = N ( T ) / T (the e ff ect of the uncertainties on the age was also estimated); fitting thecumulant distribution with a linear function N ( T ) = κ + rT , with κ = κ free to fluctuateas recommended in [17]. The value that we find does not change significantly and the range is consistentwith Poisson fluctuations. Observed CCSN in the Local Group. number of observation location observational assumptions,CCSN time [yr] approach method see A and [8]0 40 Milky Way neutrino telescopes direct0 100 Andromeda optical surveys f ∈ [0 . , . f ∈ [0 . , . ε = / which is very close to the value we derived from the SNR model as described inSect. 2.5. Unless stated otherwise, we will always quote as a rule the average values for R , that do not di ff er by much from the mode for all cases. We note that the stel-lar birthrate input remains the most important individual result in this combined fit,despite our conservative treatment of the errors. This is quite evident from the visualcomparison allowed by Fig. 1, lower part. Therefore, it is important to add independentinformation; this is the goal of the next section. The historical records of astronomical observations of the Milky Way date back to 4thcentury BC, with more reliable records starting possibly from 8th century [18]. In thelast 40 years the neutrino telescopes joined the CCSN observational campaign. Weuse the CCSN observed by means of visual and neutrino astronomy (2 and 0 CCSN,respectively) to directly constrain the galactic rate of CCSN.
Chinese astronomy flourished particularly after the astronomer Yi Xing, namely, after8th century (Tang dynasty; see [18]): therefore we assume an essentially completeastronomical coverage for T =
13 centuries. In this period, 2 CCSN have been seen inthe Milky Way: SN 1054 and SN 1181 [19], which are the very same CCSN used inthe analysis of [8]. The occurrence of CCSN events is a Poisson process, which impliesthat the most plausible number of visible CCSN accessible to optical astronomy isprecisely two.We know that, most CCSN in the Galaxy are hidden in the visible light due todust obscuration. The fraction of visible CCSN according to the calculations of [8] is Chinese historical records exist since the 4th century BC, and 1 (perhaps 2) CCSN have been seen inthis period: SN 393 (and SN 185). We do not use these data as the uncertainty is considerable. A clear demonstration of the existence of such objects is given by Cas A, which is a supernova remnant = /
16; in fact, they estimate that the best-fit in the last millennium is 3.2 CCSN / (100yr). The error is not estimated, however, the comparison of the value of ε with thatobtained by another method of calculation [8] suggests that it is small and does notgive an e ff ect as important as the one of statistical fluctuations; therefore, we neglect it.For these reasons we will adopt the following likelihood over R for the historicalgalactic CCSN P Milky Wayastronomy = µ e − µ µ = ε R T (7)
Thanks to the neutrino detectors Artemovsk, Baksan, Kamiokande-II, LVD, Super-Kamiokande etc. we are sure that we have not seen neutrinos from CCSN events inthe Milky Way in the last ∼
40 years. The information concerning neutrinos appliesdirectly to CCSN being therefore particularly important; see [20, 21] for further dis-cussion.The time of observation of the Milky Way is T = . T = + /
12, of the beginning of data taking in Baksan (0.2 kt ofscintillator [23]) T = + /
12, of the beginning of data taking in Kamiokande-II(2.14 kt of water [24]) T = T = + /
12. Then wecalculate T = ε A ( T − T ) + ε B ( T − T ) + ε C ( T − T ) (8)where we estimate the e ffi ciency to detect a CCSN as ε A = . ε B = . / (2018 − . = .
86 in Baksan [23] and, after thebeginning of Kamiokande-II operation, we set ε C = .
99; the third term is by far thelargest contribution.We adopt again a Poisson likelihood for R , that describes the lack of neutrino ob-servation of the CCSN within the Milky Way, P Milky Wayneutrino = e − µ with µ = R T (9)
We can perform an external check on the consistency of the above estimate by exam-ining the record of CCSN events in the Local Group, outside the Galaxy.The Local Group of galaxies includes more than 50 galaxies. The Galaxy and M31(Andromeda) are the largest; all other are small or very small and do not add much to associated with a X-ray point-like source that is about 300 years old, but was not seen at the time. a priori larger.The galaxies of the Local Group have been monitored for optical transients since1885 at least when a type Ia supernova exploded in M31. Actually, the first SN searchwas started by Fritz Zwicky only in 1935. More recently, the core collapse SN1987Ain the LMC was discovered. In this manner, we have learned that in about one centuryonly 1 CC SN was found in the Local Group.The corresponding likelihoods are P = µ n exp( − µ ) with µ = f R × T (10)where for T we take conservatively 100 yr.The rates in Andromeda and the rest of the Local Group were expressed as f R ,where we remember that R is the rate in the Milky Way: therefore, f describes therelative rate of CCSN compared to that in the Milky Way and the possibility that someof the events were not observed, due to, for example, the absorption of the emittedlight. In order to proceed conservatively, we do not fix the value of f as was done in[25], but we average the Poisson likelihoods over f , replacing P with (cid:104)P(cid:105) = f − f (cid:90) f f P ( f RT ) d f (11)that have simple analytical expressions. Based on the observed samples of the su-pernova remnants in the Local Group we assume f ∈ [0 . , .
75] for M31, and f ∈ [0 . , .
05] for the rest of the Local Group, for the reasons discussed in details in A.
The four likelihoods discussed in this section are shown in Fig. 2. By combining them, L Local CCSN = P MilkyWayastronomy × P
MilkyWayneutrino × (cid:104)P
M31 (cid:105) × (cid:104)P rest of Localgroup (cid:105) (12)we find the distribution shown in black in Fig. 2. Let us discuss this distribution. Itsmode, median and average are slightly di ff erent among them: R Local CCSN = . , . , .
62 (100 yr) − (13)The 68.3% interval, built using two equal tails at the left and right of the distribution,turns out to be R Local CCSN = (0 . , .
41) (100 yr) − (14)8 R (
100 yr ) - L i k e li hood MW optical ( ) MW neutrinos ( ) Andromeda ( ) rest of Local Group ( ) combined Figure 2:
Likelihoods of the CCSN rate R in the Milky Way from the historical evidence ofCCSN in the Galaxy and Local Group. For comparison, we show also the likelihood discussedin Sect. 2.6, indicated with the label ‘combined’. while if this interval is built integrating the likelihood, when it is above a certain value(which resembles a bit the Gaussian construction of confidence levels) this is R Local CCSN = (0 . , .
04) (100 yr) − (15)The fact that these two interval do not coincide exactly, just as the mode, the median andthe average do, shows that the combined likelihood is slightly non Gaussian. However,Fig. 2 does not indicate any critical inconsistency of these input data among them, orwith those discussed in Sect. 2.6. This is further illustrated visually in the upper part ofFig. 1. For this reason, we proceed and combine the entire dataset. When we proceed and use the full information - namely, the one from the combinedrate, together with the one from neutron star birthrate and the direct information on therate of CCSN in the Local Group, the resulting likelihood becomes L full ∝ L combined × L Local CCSN (16)This implies the following range for the CCSN rate in the Milky Way R full = . ± .
46 (100 yr) − (17)The likelihood resulting from combination of several Poissonian and Gaussian distri-butions results in a quasi-Gaussian distribution, with median 1.61 (100 yr) − and mode9.57 (100 yr) − . The χ / ( N −
1) test of our sample suggests that the input valuesare consistent. Taking all four results from Sect. 2, contributing to the combined rate and combination of historical CCSN in Milky Way and Local Group from Eq. (13), χ / ( N − = .
01 with N =
5. The highest contribution comes from the neutron starbirthrate result of [14].The value R full is quite similar to the result from Eq. (6), based only on results ofthe previous scientific literature. Eq. (17) is the best value that we can o ff er in view ofpresent information, and for convenience we will call it the best estimate . Comparisonof the cited values with our result is shown in Fig. 1. Let us comment on this result.The increase of the expected rate, due to the inclusion of the neutron star birthrate inthe analysis, is almost exactly compensated by the decrease due to the inclusion ofthe direct information from CCSN events in the Local Group. In view of the Poissoncharacter of the CCSN likelihood, and due to the very meagre dataset (i.e., 2 CCSNin Milky Way over 13 centuries and 1 CCSN in Large Magellanic Cloud) the lastdistribution will need to be updated as soon as new events are observed.The stellar birthrate results of [9], which we treat in a conservative way by increas-ing the error, is still the input with the smallest error. As a matter of fact, we have notfound any valid reason to discard this result, neither on a physical basis nor in com-parison with the other results: the combined result is almost indistinguishable to the1 σ range 1 < R <
2, that is R = . ± . − . We believe that, while these con-siderations do not motivate particular doubts towards the combined result, they o ff ergood reasons to undertake an updated analysis and more accurate assessment of the re-sults on CCSN from the stellar birth rate method. This might be particularly desirableconsidering the results of the GAIA [26] mission. In summary, we have shown that various determinations of the rate of CCSN in theMilky Way, when combined, produce a likelihood that is distributed in a quasi-Gaussianway. This likelihood indicates a rate of R = . ± .
46 CCSN per century, see Eq. (17).Note that this result is consistent but more precise than the values of [6] or of [8] citedin the introduction, and therefore quite informative, being in particular at odds with themost optimistic (higher) CCSN rates. This conclusions shows that it is urgent to assessand clarify current expectations based on neutron star birthrate [14], which is the oneamong the ones we have considered that indicates the largest value of the rate. Themost crucial individual input however remains the stellar birthrate method [9].It is also interesting to discuss the value for the time of CCSN events in the MilkyWay, T = / R . The corresponding likelihood has a considerable skewness, and con-sequently the expected range is quite asymmetric. We report the time of CCSN as a The χ can be evaluated directly from the average value ¯ x and the likelihoods L i by the formula χ = − (cid:80) Ni = log[ L i ( ¯ x ) / L i ( ¯ x i )]. When the likelihoods are almost Gaussian as in our case the results are in goodagreement. T full = + − yr (18)while the most plausible value (mode) is at 52 yr. By way of comparison, in previousauthoritative works the value of T remains firmly fixed around 50 yr: for example,the monograph of Ginzburg and Syrovatskii ii [27] uses T =
50 yr in Eq. (11.9), andthe review work of Tammann, Loe ffl er and Schroeder [28], that summarizes most ofthe information available in the last century, quotes T = / . =
47 yr already in itsabstract.A neutrino telescope that runs for a time t has the Poissonian chance P (0) = e − Rt that no galactic supernova will occur during its running time. When the uncertaintieson the rate R are described by a Gaussian distribution restricted to the physical region R >
0, the chance of observing at least one event, P ( > = − P (0), is given by P ( > = − e − Rt · (cid:20) + erf (cid:18) R − σ t √ σ (cid:19)(cid:21) · e σ t + erf (cid:18) R √ σ (cid:19) (19)where ‘erf’ indicates the Gauss error function. With the best fit values obtained abovefor the rate R and its variance σ , Eq. (17), we find that the chances of seeing at least oneevent are P ( > = . t = , ,
20 and 50 years respectively. This consideration illustrates quantitatively theimportance of disposing of neutrino telescopes that are stable and capable to operatefor a rather long time.
Acknowledgements
This work was partially supported by the research grant 2017W4HA7S “NAT-NET:Neutrino and Astroparticle Theory Network” under the program PRIN 2017 fundedby the Italian Ministero dell’Istruzione, dell’Università e della Ricerca. We thankM.L. Costantini, W. Fulgione, A. Gallo Rosso, C. Mascaretti and C. Volpe for pre-cious discussions. 11able 2:
Number of SNR and fraction of CCSN in the Local Group galaxies. The observationalmethods are in the order from most to least significant surveys.
Galaxy SNR CCSN Observational Ref.sample fraction methods
MW 294 . . . optical, radio, X-ray [29]M31 156 0.75 optical, X-ray, radio [30]M33 155 0.85 optical, radio, X-ray [31, 32]LMC 59 0.52-0.59
X-ray, optical, radio [33]SMC 21 0.73-0.84
X-ray, optical, radio [34]
A Core collapse supernovae in the Local Group
In the main text, we introduce the relative rate of CCSN in M31 in comparison with theMilky Way and the same for the rest of the Local Group (mostly M33, SMC, LMC).These relative rates, indicated by f , are based on the number of observed supernovaremnants (SNRs).The information that we use is summarized in Tab 2. The galactic sample containsalmost 300 SNRs listed in [29] catalogue. The studies by [30] of M31 report a sampleof 156 SNRs, while [31] identify 155 SNRs with multiwavelength coverage in M33.Surveys of Magellanic Clouds produced lists of 21 [34] and 59 [33] SNRs in SMCand LMC, respectively. These samples are most complete, thanks to the favourablenearby position of Magellanic Clouds, with small absorption in the line of sight. MilkyWay, M31 and M33 samples are likely to be incomplete, limited by the sensitivity ande ffi ciency of the observations, confusion with other sources and absorption in the lineof sight.The exact classification of SNRs to the Ia-type or to the CCSN-type is di ffi cult,we assume that the fraction of core collapse SNRs contribution to the entire sample issimilar for the galaxies in the Local Group and lies in the range n CC ( n CC + n Ia ) ∈ [0 . , . f ∈ [0 . , .
75] forM31 and f ∈ [0 . , .
05] for the rest of the Local Group.Thus, the whole Local Group is expected to have about one CCSN in about 30years, i.e., roughly twice than the Milky Way alone.The estimates of f agree with simple comparison of the galaxy masses and typestogether with probability of missing observation of the events. The mass of M31 issimilar to that of Milky Way, but correction for the high inclination angle of M31 andtherefore probable absorption in the galactic disk is needed. In the rest of the LocalGroup the main SN contribution is expected from: relatively small M33; MagellanicClouds with small masses compared to the Milky Way, but higher supernovae rate dueto irregular structure and high probability of observation from Earth taking into accountclose distance and good visibility. 12 eferences [1] Y. Fukuda et al. The Super-Kamiokande detector. Nucl. Instrum. Meth. A , 501:418–462,2003.[2] K. Abe et al. Hyper-Kamiokande Design Report. arXiv e-prints , page arXiv:1805.04163,May 2018.[3] R. Abbasi et al. IceCube sensitivity for low-energy neutrinos from nearby supernovae.
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