Constraining the GRB-magnetar model by means of the Galactic pulsar population
Nanda Rea, Miguel Gullon, Jose' A. Pons, Rosalba Perna, Maria G. Dainotti, Juan A. Miralles, Diego F. Torres
DDraft version October 8, 2018
Preprint typeset using L A TEX style emulateapj v. 04/17/13
CONSTRAINING THE GRB-MAGNETAR MODEL BY MEANS OF THE GALACTIC PULSAR POPULATION
N. Rea , M. Gull´on , J. A. Pons , R. Perna , M. G. Dainotti , J. A. Miralles , D. F. Torres Draft version October 8, 2018
ABSTRACTA large fraction of Gamma Ray Bursts (GRBs) displays an X-ray plateau phase within < sfrom the prompt emission, proposed to be powered by the spin-down energy of a rapidly spinningnewly born magnetar. In this work we use the properties of the Galactic neutron star population toconstrain the GRB-magnetar scenario. We re-analyze the X-ray plateaus of all Swift
GRBs with knownredshift, between January 2005 and August 2014. From the derived initial magnetic field distributionfor the possible magnetars left behind by the GRBs, we study the evolution and properties of asimulated GRB-magnetar population using numerical simulations of magnetic field evolution, coupledwith Monte Carlo simulations of Pulsar Population Synthesis in our Galaxy. We find that if theGRB X-ray plateaus are powered by the rotational energy of a newly formed magnetar, the currentobservational properties of the Galactic magnetar population are not compatible with being formedwithin the GRB scenario (regardless of the GRB type or rate at z=0). Direct consequences would bethat we should allow the existence of magnetars and ”super-magnetars” having different progenitors,and that Type Ib/c SNe related to Long GRBs form systematically neutron stars with higher initialmagnetic fields. We put an upper limit of ≤
16 ”super-magnetars” formed by a GRB in our Galaxyin the past Myr (at 99% c.l.). This limit is somewhat smaller than what roughly expected from LongGRB rates, although the very large uncertainties do not allow us to draw strong conclusion in thisrespect.
Subject headings: (stars:) gamma-ray burst: general — stars: magnetars — (stars:) pulsars: general INTRODUCTION
Gamma Ray Bursts (GRBs) are one of the most ex-treme and powerful transient phenomena in the Universe.They are generally divided in two groups, which havebeen proposed to have two distinctly different origins:Long GRBs (LGRBs), connected to the Type Ib/c Core-Collapse Supernovae, and Short GRBs (SGRBs), origi-nating from the merger of two neutron stars or a neutronstar and a black hole.Independently of the progenitor scenario, the prompt γ -ray emission is followed by intense longer-wavelengthemission (afterglow). According to the standard ”Fire-ball” theory, this radiation arises from the formation of ablast wave, due to a relativistic outflow pushing throughthe interstellar medium (Meszaros & Rees 1997; Sari,Piran & Narayan 1998). In the past decade, thanksto Swift , the sample of Long and Short GRBs with agood multi-band monitoring of the afterglow became suf-ficiently large to enable a statistical study of the after-glow characteristics and energetics (Nousek et al. 2006;O’Brien et al. 2006; Zhang et al. 2006; Willingale etal. 2007; Evans et al. 2009; Dainotti et al. 2011b,Margutti et al. 2013, Dainotti et al. 2015b). It was Anton Pannekoek Institute for Astronomy, University ofAmsterdam, Postbus 94249, NL-1090 GE Amsterdam, theNetherlands. Instituto de Ciencias de l’Espacio (ICE, CSIC–IEEC), Cam-pus UAB, Carrer Can Magrans s/n, 08193 Barcelona, Spain. Departament de Fisica Aplicada, Universitat d’Alacant, Ap.Correus 99, 03080 Alacant, Spain. Department of Physics and Astronomy, Stony Brook Univer-sity, Stony Brook, NY, 11794, USA. Physics Department, Stanford University, Via Pueblo Mall382, Stanford, CA, USA. Instituci´o Catalana de Recerca i Estudis Avanats (ICREA),E-08010 Barcelona, Spain. observed that most GRBs do not show a smooth decayin X-ray flux after the prompt emission, as expected for apure fireball model, but present rather ubiquitous X-rayplateaus at times < s, eventually pointing to a con-tinuos energy injection in the first hours/day after theGRBs. These X-ray plateaus are generally interpretedas due to: a newly-born rapidly-spinning magnetar (seei.e. Metzger et al. 2011), an accreting black hole (see i.e.Kumar et al. 2008) or a top-heavy jet evolution (Du-ell & MacFadyen 2015). The similarity of these plateauphases between the two classes of GRBs was ascribedto a common injection scenario. The fluence of theseplateaus in both LGRBs and SGRBs is comparable withthat of the prompt emission (never lower than an order ofmagnitude), and their luminosities and durations are ob-served to be anti-correlated (Dainotti et al. 2008, 2010,2011a,2013a; Rowlinson et al. 2013, 2014).The latter correlation, combined with the fact that anewly born magnetar could be formed either via the col-lapse of a massive star (hence via a LGRB), or duringthe merger of two neutron stars (hence via a SGRB),motivated the interpretation of these X-ray plateaus asresulting from the delayed injection of rotational energy(with ˙ E rot ∼ − erg s − ) from a fast spinningmagnetar (Usov 1992; Zhang & M´esz´aros 2001; Metzgeret al. 2011).Further support to the GRB-magnetar scenario wasprovided by the successful fitting of a large sample ofLong and Short GRB afterglows (Zhang & M´esz´aros2001; Troja et al. 2007; Lyons et al. 2010; Dall’Osso etal. 2011; Bernardini et al. 2012; Rowlinson et al. 2010,2013; L¨u & Zhang 2014) by modeling the plateau du-ration and luminosity in terms of the spin-down energyrelease timescale ( τ sd ) and luminosity ( L sd ): a r X i v : . [ a s t r o - ph . H E ] O c t N. Rea et al. !" (cid:239) $ (cid:239) % (cid:239) & (cid:239) ' (cid:239) ( (cid:239) ) !"! * + , - ., / + / (8 (8 . : ; < = *+,-.,/0>/:,-2430? ) (cid:239) :,60H+,5D.5G (cid:239) :,60H+,5D.50C2-D0EFED4:-0@AB5 Long GRB Long GRB with SN X-ray Flash X-ray Flash with SN Short GRB . . M a gn e ti c f i e l d B p , ( G a u ss ) Spin Period P (ms)100% efficiency Long GRBsLong GRBs with SNX (cid:239) ray FlashesX (cid:239) ray Flashes with SNShort GRBs ε /(1-‐cos θ )=1 Long GRB Long GRB with SN X-ray Flash X-ray Flash with SN Short GRB . . M a gn e ti c f i e l d B p , ( G a u ss ) Spin Period P (ms)10% efficiency Long GRBsLong GRBs with SNX (cid:239) ray FlashesX (cid:239) ray Flashes with SNShort GRBs ε /(1-‐cos θ )=0.1 Long GRB Long GRB with SN X-ray Flash X-ray Flash with SN Short GRB
Fig. 1.—
Top panel : the rest-frame plateau durations versus the luminosity (1–10000 keV) at the end of the plateaus for all the GRBs inthe sample (black = Long GRBs, Blue = Short GRBs, and Red = X-ray Flashes).
Bottom panels : derived dipolar fields and initial spinperiod assuming the GRB-magnetar model for two different values of efficiency ( (cid:15) ) versus opening angle ( θ ) relation (see text for details).The shaded grey area excludes the rotational periods that would exceed the mass shedding limit under any reasonable neutron star EoSassumption. T (cid:39) τ sd = 2 .
05 ( I B − p, P ms R − ) L (cid:39) L sd = ( B p, P − ms R ) ,where T is the plateau duration in 10 s, L is theplateau luminosity in 10 erg s − , I is the moment ofinertia in units of 10 g cm , B p, is the magnetic fieldstrength at the poles in units of 10 G, R is the radius ofthe neutron star in 10 cm, and P ms is the initial periodof the pulsar in milliseconds (Zhang & M´esz´aros 2001;Metzger et al. 2011) . In this scenario, the spin down lu-minosity and duration are expected to be anti-correlatedas: log( L sd ) = a − log( τ sd ), where a = log(10 I − P − − ).Fitting the intrinsic plateaus it has been obtained that a = 52 . ± . L ∝ T ( − . ± . (Dainotti et al.2013a; Rowlinson et al. 2014).In our Galaxy we have discovered in the past fewdecades about 20 magnetars (Duncan & Thompson 2002;see Mereghetti 2008, Rea & Esposito 2011 for recentreviews, and the McGill Magnetar Catalog ). Theyare characterized by relatively bright X-ray luminosities These equations apply to the electromagnetic dominated spindown regime, since the gravitational wave dominated regime wouldbe extremely rapid and produce a negligible electromagnetic signal.It is also assumed that the loss of rotational energy is given by themagneto-dipole formula, whose validity in this scenario is highlyquestionable. ∼ pulsar/magnetar/ ( L X ∼ − erg s − ), rotational periods in the 0.3–12 s range, strong X/ γ -ray flares and outburst activity,dipolar magnetic fields in the 6 × − G, and es-timated ages between ∼ − kyr.In this paper we investigate the possible GRB origin ofthe magnetars in our Galaxy, as well as derive the limitson the GRB-magnetar scenario imposed by the proper-ties of the Galactic magnetars. In Sec. 2 we re-analyzethe Swift data of GRBs with good redshift measurements.Fitting them with the GRB-magnetar model (see alsoRowlinson et al. 2014), we derive initial magnetic fieldsand spin period distributions for the sample. In Sec. 3we use state-of-the-art magnetar evolution models (Vi-gan`o et al. 2013) coupled with Pulsar Population Syn-thesis simulations (Gull´on et al. 2014, 2015) to constrainthe current properties of possible magnetars formed viaa GRB in our Galaxy in the past Myr, by comparingsynthetic populations with the observed Galactic popu-lation. In Sec. 4 we discuss our results as well as the is-sue of how many and which GRBs are expected to leavebehind a long-lived stable magnetar, and the large un-certainties in the local GRB rates. We summarize ourresults and draw conclusions in Sec. 5. FITTING MAGNETAR-DRIVEN PLATEAUS TO THESWIFT LIGHT CURVES
We re-analyzed the sample of all GRB X-ray after-glows, detected by
Swift from January 2005 up to August
Simulated Observed
Observed Observed Simulated Simulated S xobs > 10 -13 erg/cm s S xobs > 10 -12 erg/cm s S xobs > 10 -12 erg/cm s S xobs > 10 -13 erg/cm s Fig. 2.—
Results of the Population Synthesis Simulations of our X-ray pulsar population plus 100 ”stable” magnetars formed via a GRBin the past Myr in our Galaxy.
Top panel : logN–logS of the simulated sample (red) compared with the observed X-ray fluxes (black).
Bottom panels : Spin period distribution of the observed X-ray pulsar population compared with the ”observable” simulated sample ofsynthetic GRB-magnetars, and relative P − ˙ P diagram. In the latter, the black and grey symbols represent the observed objects, while thesynthetic sample is displayed in red and blue. Grey and blue dots represent pulsars with X-ray fluxes of > − erg s − cm − ; black andred dots are objects with fluxes 10 − < f X < − erg s − cm − . and in the Circulars Notice archive (GCN). Thetotal number of GRBs with known redshift (in the 0.033-9.4 range) observed by Swift until 2014 August 14th is283 (63 of which are SGRBs, and 25 are X-ray Flashes;XRFs), but not all these GRBs show a well-defined X-ray plateau emission. We found that among those, 176GRBs afterglows (14 of which are SGRBs, and all the25 XRFs) are well fitted by an X-ray plateau model, asdescribed in Dainotti et al. (2013a). The fitting proce-dure fails either when it gives unreasonable values of theerrors or when the determination of confidence intervalin 1 σ does not fulfill the Avni (1976) prescriptions .We plot in Fig. 1 (left panel) the rest-frame plateauluminosities in the Swift bolometric band ( E min =1, E max =10000 keV) at the time T a (end time of theplateau). The luminosities are computed from: http://heasarc.nasa.gov/xanadu/xspec/manual/ L X ( E min , E max , T a ) = 4 πD L ( z ) F X ( E min , E max , T a ) × K , where D L ( z ) is the GRB luminosity distance (wehave assumed a ΛCDM flat cosmological model withΩ M = 0 .
28 and H = 70kms − Mpc − ), F X is the mea-sured X-ray energy flux, K = (1+ z ) − β a is the so called K -correction, where β a is the spectral index assuming asimple power law spectrum (Evans et al. 2009; Dainottiet al. 2010). Note that, in the current paper we usethe variables L and T corrected for selection bias andredshift evolution (namely, de-evolved). This approachis slightly different from the one presented in Rowlinsonet al. (2014): in this work we derive the slope of thecorrelation directly by using the de-evolved luminosityand time observables. In Rowlinson et al. (2014) theslope is fixed to the intrinsic one and the normalizationis derived from the simulated data. This slightly differ-ent approach do not change the results, since the intrinsicslope used in Rowlinson et al. (2014) has been computedtaking into account the same evolution. Caveats on theuse of the observed slope instead of the intrinsic one havebeen discussed in Dainotti et al. (2013b). Since the restframe time and luminosity we use are already correctedfor selection bias and redshift evolution, the derived spinperiod and the magnetic field are unbiased too (this is N. Rea et al.a crucial point often omitted in the literature). In ouranalysis we have taken into account the undetected pop-ulation of GRBs through the correction of the observedvariables with the Efron & Petrosian (1992) method.In Fig. 1 (middle and left panels) we then report onthe inferred initial magnetic fields (at the neutron starpole) and spin period distributions derived from model-ing the plateau luminosities and durations with a GRB-magnetar model (Zhang & Meszaros 2001), namely: B p, (cid:39) . I R − [ L sd , ∗ (cid:15)/ (1 − cos θ )] − τ − , P , − (cid:39) . I [ L sd , ∗ (cid:15)/ (1 − cos θ )] − τ − , ,where (cid:15) is the conversion efficiency of extracting rota-tional energy from the newly born pulsar, and θ is thebeaming angle. We have studied the dependence of thederived initial B and P distributions on these two un-known quantities. Lowering the efficiency factor resultsin a net shift of the resulting B-fields and periods to-wards lower values (with several GRBs requiring unphys-ically low values of the birth rotational period; see e.g.Fig. 1 right panel). On the other hand, assuming an ex-tremely beamed emission has the opposite effect, shift-ing all inferred values toward longer spin periods butunreasonably high magnetic fields (see also below). Bystudying the set of parameters that better reproduce theluminosity-time correlation, Rowlinson et al. (2014) pro-pose a range for (cid:15)/ (1 − cos θ ) (cid:39) −
4; this range leadsto very high initial magnetic fields. For our purposes, weadopt the less problematic case (cid:15)/ (1 − cos θ ) = 1, but ourconclusions will be unchanged if we assume larger values.From the Swift
X-ray plateau modeling we can derivea B distribution for all GRBs, which we have distin-guished in different classes. As it can be noted fromFig. 1 (i.e. middle panel), there is no evidence for a dis-tinct B and P distributions as a function of the GRBclass.The B field distribution of the resulting magnetars forall GRBs is well fitted by a log-normal distribution cen-tered at log B = 15 . σ (cid:39) .
55. In-ferred rotational periods at birth range between ∼ NEUTRON STAR POPULATION SYNTHESISSIMULATIONS AND RESULTS
In Gull´on et al. (2015) we have performed a Popula-tion Synthesis analysis considering both the radio-pulsarand the thermal X-ray emitting neutron star populations(comprising the magnetars), taking advantage of 2D nu-merical simulations of the magneto-thermal neutron starevolution (Vigan`o et al. 2013). We refer to Gull´on et al.(2014, 2015) for details on the Monte Carlo simulationsused to synthesize the Galactic neutron star populations.This analysis allowed us to derive the best set of pa-rameters ( B and P distributions) consistent with boththe current pulsar and magnetar P − ˙ P distributions.The most important result of this work was the discov-ery that a single log-normal B-field distribution functioncould not explain at the same time the radio pulsars and the magnetars. Either a truncated log-normal B distri-bution, or a binormal distribution with two distinct pop-ulations, were needed. More importantly, in both casesthe current lack of detected isolated X-ray pulsars withperiods >
12 s strongly constrains the number of Galac-tic neutron stars born with B > G .We begin the simulations with the assumption of twodifferent populations: normal radio-pulsars, and mag-netars associated to GRBs. For the radio-pulsar pop-ulation, we use the best fit parameters correspondingto model D in Gull´on et al. 2015, which successfullyfits the radio-pulsar population properties. The initialmagnetic field distribution for the synthetic magnetarsis assumed to be the one consistent with the GRB-magnetar association obtained in Sec. 2 (a log-normaldistribution centered at log B = 15 . σ = 0 . P was forced to be correlated with B by the for-mula: log P = − . .
22 log B , being the observedcorrelation between these quantities intrinsic, and en-compassing all different kind of GRBs (see Fig. 1 andDainotti et al. 2013a; Rowlinson et al. 2014). We stressthat the particular choice of P is completely irrelevantfor our results, since the high average magnetic field ofthe population makes them spin-down very fast to reachhigher periods. Assuming an initial period of 1 or 10 msmakes no difference for the results discussed below.The only parameter that still needs to be fixed is therelative normalization of the number of radio pulsars andGRB-magnetars, the latter being expected to be propor-tional to the product of the GRB rate ( ρ GRB ) at z=0,and the fraction of GRBs expected to leave as a remnantcompact object a ”stable” magnetar ( f mag ; i.e. that sur-vives subsequent collapse to a black hole). Given thelarge uncertainties in these two quantities, and the dif-ferences related to the different GRB types, we first runsimulations and derive probabilities as a function of ageneral ρ GRB ∗ f mag ≡ N gen : namely the number of ”sta-ble” magnetars that were formed via a GRB in the MilkyWay in the past million years, regardless of the GRB type(this is allowed by the fact that all types have a similar B distribution; see Fig. 1). We then discuss differencesin our conclusions for different GRB types in Sec. 4. Notethat there is no GRB beaming effect involved in the de-tectability of the remnant as an X-ray pulsar, so therecould be unseen GRBs leaving behind a visible magne-tar. Initial positions in the Galaxy are drawn from theradial probability distribution of Yusifov & Kucuk (2004)within the Galactic spiral arms. The position of eachmagnetar is then evolved until its present age, by solvingthe Newtonian equations of motion under the influenceof the smooth Galactic gravitational potential (Kuijken& Gilmore 1989; Carlberg & Innanen 1987). The age ofeach star is randomly uniform in the interval [0 ,
1] Myr.Spatial kick velocities and the inclination angle (betweenrotational and magnetic axes) are also randomly selected.In order to account for the detections in the X-ray bandwe assume blackbody emission plus Resonant ComptonScattering, as typical of magnetars’ spectra (Rea et al.2008; Zane et al. 2009). The photoelectric absorptionalong the line of sight is also considered (see Gull´on etal. 2015 for further details).In Fig. 1 we report the results of a typical realiza-tion with N gen = 100 magnetars (plus the large numberof radio-pulsars fitting the radio-pulsar population) byshowing their predicted P − ˙ P and log N − log S diagramsfor the visible X-ray pulsars at present, compared withthe observed sources. We show results with two differentcut-offs in absorbed X-ray fluxes, at 10 − and 10 − ergs − cm − . In the left panel, we see that the total numberof X-ray pulsars we observe in our Galaxy (after filteringfor selection effects) is roughly consistent with the simu-lated radio pulsars plus 100 GRB-magnetars. This con-firms our initial assumption of ρ GRB ∗ f mag ≡ N gen = 100in order to explain the currently observed ∼
20 magnetars(after selection effects). However, it is clear that theirdistribution of X-ray fluxes and spin periods are quite dif-ferent from the observed population. As expected, fromthe extremely high B inferred from the GRB plateaus,the simulated GRB magnetars are too bright and tooslow compared with the observed magnetars. These dis-crepancies are even more pronounced if we change theefficiency/beaming factor ( (cid:15)/ (1 − cos θ )) within the GRB-magnetar scenario, and cannot be mitigated by changingthe magneto-thermal evolutionary model, or the crustalmicrophysics assumptions within reasonable values (Vi-gan`o et al. 2013; Pons, Vigan`o & Rea 2013).Note that if some GRBs not showing a plateau phasestill have magnetar central engines, i.e. with lower ini-tial B-fields (hence with X-ray plateaus too faint to bedetected over the afterglow), this will not change ourconclusions, because the initial GRB-magnetar B-fielddistribution will not change systematically to lower val-ues, but the log-normal distribution will only be slightlyskewed to include also these putative additional GRB-magnetars with lower B .Hence, our first result is that our observed populationof magnetars cannot be formed via a GRB (regardlessof the assumptions on the rates or the different GRBtypes) because they would have luminosity and perioddistributions largely inconsistent with the observationaldata.We can now estimate the maximum number of ”stable”GRB-magnetars in the Milky Way left behind in the pastMyr, that is still compatible with the observations.As shown in Gull´on et al. 2015, the number of de-tectable synthetic magnetars in each realization closelyfollows a Poissonian distribution with mean value λ = N gen ∗ p , where p is the detection probability (note thatwe assume the GRB rate constant over 1 Myr timescale).We have calculated this probability by performing a largenumber of runs ( ∼
500 realizations), and we obtained p = 0 .
18 and p = 0 .
28 for fluxes > − and > − ,respectively.Since the most constraining observational fact is thelack of X-ray pulsars with periods greater than 12 s, wecan calculate the probability of not detecting any pul-sar with P >
12 s, which is e − λ . In Fig. 3 we plot theno-detection probability of magnetars with spin period >
12 s as a function of N gen , the number of ”stable”magnetars formed in the Milky Way via a GRB in thepast Myr. The figure compares the results for two dif-ferent flux thresholds, 10 − erg s − cm − (dashed red)and 10 − (solid blue) erg s − cm − . These two fluxesroughly encompass the range in which we believe ourX-ray sample of detected X-ray pulsar is nearly com-plete. Thus, assuming our sample is complete above
99% confidence level Number “stable” GRB-magnetars ( ρ GRB f mag ) [Gal -1 Myr -1 ] S xobs > 10 -13 erg/cm s S xobs > 10 -12 erg/cm s N o - de t e c t i on p r obab ili t y Fig. 3.—
Probability of the no-detection of a synthetic magnetarwith a spin period >
12 s as a function of the injected number of”stable” magnetars for two cuts fluxes: 10 − (dashed red) and10 − (solid blue) erg s − cm − . The grey line indicates the 99%confidence level. fluxes > − erg s − cm − , only in one per cent ofthe simulations we do not find any visible pulsar with P >
12 s, which means that we can establish the upperlimit ρ GRB ∗ f mag <
16 with a 99% confidence level.The above conclusions are in principle valid for anyGRB type leaving behind a ”stable” magnetar. However,as we will discuss further in the following section, theupper limit we derived is eventually meaningful only forLGRBs, given that SGRBs are hardly expected to leaveany ”stable” magnetar, and expected to collapse into ablack hole soon after the X-ray plateau phase in most ofthe cases. DISCUSSION
We have performed neutron star population synthe-sis simulations, to set constraints on the GRB-magnetarscenario by means of the Galactic population of highlymagnetized neutron stars. By assuming that the X-rayplateau phases of GRB afterglows are powered by the ro-tational energy of a newly born, rapidly spinning magne-tar central engine, we derived from the
Swift
GRB samplethe resulting initial B-field and spin period distributionof such newly born magnetars. Using these distributions,we simulated the properties of a synthetic population ofmagnetars formed in our Galaxy in the past Myr via aGRB.We found that, if we assume ∼
100 GRBs leaving be-hind a ”stable” magnetar in the past Myr, the numberof ”observable” objects (considering the predicted prop-erties of such simulated magnetars and all the observa-tional biases) roughly agrees with the number of mag-netars we currently observe in the Milky Way ( ∼ B -field distribution should not allow fields in ex-cess of 10 G, otherwise the limiting spin period ob-served in isolated pulsars ( ∼
12 s) cannot be reconciled(Gull´on et al. 2015; Popov 2015). The magnetars thatGRBs need to form to supply spin-down energy to theX-ray plateaus, are ”super-magnetars”, having initial B-fields significantly larger than those extrapolated for ourGalactic magnetars.Given that the number of observable magnetars is re-produced, but their general properties are not, we cansafely conclude that assuming the GRB-magnetar sce-nario in its present formulation, in particular that X-rayplateaus are powered by spin-down energy, our Galacticmagnetars (regardless the assumed GRB type or rate atz=0) should be mostly formed by a distinct formationpath than a GRB. Most likely a type of Core-CollapseSNe different from the Type Ib/c connected to LongGRBs. In this contest, this would also mean that GRB-like SNe should systematically produced stronger mag-netic fields in the proto-magnetar than other CC-SNe . General estimates of the fraction of expected stablemagnetars ( f mag ), and GRB rates at z=0 To put our simulations in contest, we discuss here cur-rent estimates of the local GRB rates, and of the proba-bility for a magnetar born associated to a GRB to surviveor collapse to a black hole after the X-ray plateau phase.Note that both these quantities are extremely uncertain.If we proceed observationally to derive f mag , within theGRB-magnetar model, we can assume that if an X-rayplateau is observed the GRB formed a magnetar. Fromthe Swift
GRB reanalysis we derived that, for LGRBs, in70% of the cases we can reasonably fit a plateau phase(137 cases over 195), for SGRBs, a plateau improves thefit in 23% of the cases, and for XRFs in 100%. We thenassume this percentage as the minimum percentage ofGRBs having a magnetar engine powering the plateaus(in the others the plateau could had been missed or toofaint).Subsequently, we assume to zeroth order that if thereis no collapse onto a black hole (i.e. due to residualaccretion onto the newly formed magnetar), at the endof the X-ray plateau there is no sharp decay in time, andthe afterglow decays as t − α , with α ≤
2. To estimate thefraction of magnetars that collapse, we have counted inhow many cases we found a subsequent t − α decay with α >
2. We find that such steep decay after the X-rayplateau is detected in 14 LGRBs, among the 137 withan X-ray plateau. We can then roughly estimate thatthe fraction of LGRBs leaving a ”stable” magnetar is f mag ∼ . ∼
25% of all CC-supernovae are Type Ib/c,but only 3-10% of those are related to LGRBs (Berger et Unfortunately assessing whether this is or not the case iscurrently beyond the capabilities of current simulations of magneticfield formation in proto-neutron stars (and certainly far from theaim of this work). al. 2003; Soderberg et al. 2006, 2010; Li et al. 2011; Lienet al. 2014). In the local Universe this type of SNe havea rate of ρ SN − Ib / c = 1 . × Gpc − yr − (Cappellaroet al. 1999, Soderberg et al. 2010). With one galaxyin 100 Mpc (or equivalently, with the Milky Way vol-ume of about 10 − Gpc ; Panter et al. 2007) this resultsin ∼ Swift
GRBs up to 2010 was performedby Wanderman & Piran (2010). With a low- L cutoffof L > L ≡ erg s − , they inferred a local rate of ρ LGRB(
L>L ) = (1 . ± . f − b (Gpc − yr − ). Correct-ing for a beaming factor of about f − b = 70 (see Guettaet al. 2005; Fong et al. 2012), we expect ∼ L> events in 1 Myr. We caveat here that there might be ametallicity dependence in extrapolating this GRB rateat z=0 (in particular Milky Way-like galaxies seem notto be the preferred hosts for LGRBs; see e.g. Robert-son & Ellis 2012; Salvaterra et al 2012; Trenti et al.2013, 2015). However, while some evidence points to-wards a preference of LGRBs for low-metallicity hosts(e.g. Modjaz et al. 2008; Graham & Fruchter 2013),some outliers have also been discovered (Savaglio et al.2012; Levesque 2014). The uncertain dependence of theGRB rate on metallicity and star formation, as well ason redshift, only contributes to increase the uncertaintiesof the local LGRB rate determination (Jimenez & Piran2013; Dainotti et al. 2015). For the above reasons wedo not enter in the metallicity/redshift/star formationrate dependence discussion, especially because it is notso relevant for the work presented here.As Wanderman & Piran (2010) discuss in their Sec.6.2, there are several low-luminosity LGRBs L< that arenot taken into account in their estimated rate. Giventheir faint nature, LGRBs with L < L could have arate much larger than for brighter LGRBs, but at thistime it remains even more uncertain. Current estimatesstate that they should be roughly 10 times more nu-merous than the LGRBs L> (Soderberg et al. 2006a,2010), and have very low beaming factors. Guetta &Della Valle (2007) attempted to estimate their localrates on the basis of the few known events, and inferred ∼ +620 − (Gpc − yr − ), which would result in about 38low-luminosity LGRB events in the past Myr (again withlarge errors). This is consistent with a similar estimatefound by Liang et al. (2007), assessing the rate of thelow-luminosity LGRBs as ∼ .
7% of all Type Ib/c SN.Summarizing, the different approaches estimate thatthe total (very uncertain), beaming corrected, LGRBsrate at z=0 should range within ∼ ρ SGRB = (4 ± f − b (Gpc − yr − ), where f − b is the GRBbeaming factor. Assuming f − b = 30 (see Fong et al.2012), we then expect ∼
12 SGRB events in our Galaxyin the past Myr. The estimate of f mag for SGRBs iseven more difficult than for LGRBs. Observationally,this is very much limited by the smaller sample to bemeaningful. On the other hand, theoretically, while ithas been demonstrated via general relativistic, magneto-hydrodynamical simulations that the formation of a sta-ble neutron star from the merger of two small neutronstars ( ∼ . (cid:12) ) is possible (Giacomazzo & Perna 2013;Dall’Osso et al. 2015), the formation rate depends on therate at which the small-mass neutron stars are formed atbirth, as well as on the neutron star equation of state(which determines the maximum mass of the resultingmagnetar), and on the magnitude of the subsequent rateof accretion.All in all, since the statistics of neutron star massesin binaries are still too small to draw quantitative es-timates, and the local, galactic SGRB rates are smallerthan those of LGRBs anyways, we have adopted the con-servative assumption that the possible contribution fromSGRBs to the observed galactic magnetar population isnegligible (note also that Galactic magnetars are mostlylocated in the Galactic plane and in massive star clus-ters, unlike what would be expected for the remnants ofa compact merger). Hence, even if our results are not de-pendent on the GRB type, but require only such GRB toleave a ”stable” magnetar behind, eventually our conclu-sions and constraints are meaningful only for the LGRBpopulation. Constraints on Long GRBs
With our simulations we have also estimated the prob-ability of non-detecting a GRB-formed magnetar in ourcurrent population as a function of the number of ”sta-ble” magnetars that a GRB, mainly LGRBs, have left inthe galaxy in the last Myr, namely ρ GRB ∗ f mag . We findthat, in order to reconcile at a 99% confidence level thenon-detection of a GRB-magnetar compact remnant inour Galaxy (meaning non-detecting any magnetar with P >
12 s), the quantity ρ GRB ∗ f mag should not exceed ≤
16 Gal − Myr − . This number depends mainly on thecompleteness of the X-ray sample of observed neutronstars, hence it can be revised further, and become morestringent, with the advent of new deep X-ray surveyssuch as eROSITA (Merloni et al. 2012).Extrapolating current LGRB rate estimates, we de-rived rough values of ρ LGRB ∼ −
170 Gal − Myr − ,and f mag ∼ .
63 (from fitting the
Swift data), that re-sult in ρ LGRB ∗ f mag ∼ − <
16 at 99% confidence level). CONCLUSIONS
Our results show that the initial B-field distributionneeded to explain the GRB X-ray plateaus in terms ofa fast spinning magnetar does not reconcile the prop-erties of these GRB-magnetars with our Galactic mag-netar population, even using the most favorable choicesof efficiency/beaming factors. We should then allow theexistence of magnetars and ”super-magnetars”, with twodifferent progenitors and formation path, and differentmagnetic field formation efficiency.Even though the large uncertainties in the GRB ratesat z=0, in the metallicity and star formation rate de-pendences, and in the fraction of neutron stars collaps-ing to a black hole, do not allow anyhow to rule outthe GRB-magnetar model on the basis of the observedGalactic population, several fine-tunings are needed tomaintain the model in its present form, and keeping theinterpretation that X-ray plateaus are necessarily due tospin-down energy (i.e. we should allow some progenitorsor environments to create systematically more magneticstellar remnants than others).If those stable GRB-formed ”super-magnetars” indeedexist, their current non-detection in our Galaxy can beused to put limits on ρ LGRB ∗ f mag , that will get possiblymore and more constraining by means of future deep X-ray surveys.NR thanks A. Rowlinson, B. Metzger, R. Margutti,B. Zhang, S. Dall’Osso, A. Soderberg, R. Wijers, A.MacFayden, M. Modjaz, S. Campana, P. D’ Avanzo,M. G. Bernardini, and L. Rezzolla for useful discussionsand/or comments on the manuscript, and the referee forthe careful reading. NR is supported by an NWO VidiGrant, and kindly acknowledges Harvard ITC, NYU,Stony Brook University and the MIAPP institute inGarching, for the hospitality during the preparation ofthis work. NR and DFT are also supported by grantsAYA2012-39303 and SGR2014-1073. MG is supportedby the fellowship BES-2011-049123. MG, JAP and JAMacknowledge support by grants AYA2013- 42184-P andPrometeu/2014/69. RP acknowledges support from NSFgrant No. AST 1009396. M.G.D. is supported by FP7-PEOPLE-2013-IOF under the grant agreement number626267, and thanks the ITHES group and the Astrophys-ical Big Bang Laboratory for fruitful discussions. Thiswork is partially supported by the European COST Ac-tion MP1304 (NewCOMPSTAR). REFERENCESAvni, Y. 1976, ApJ, 210, 642Berger E., et al., 2003, Nature, 426, 154Bernardini M. G., Margutti R., Mao J., Zaninoni E., ChincariniG., 2012, A&A, 539, A3Cappellaro, E., Evans, R., & Turatto, M. 1999, A&A, 351, 459Carlberg, R. G., & Innanen, K. A. 1987, AJ, 94, 666Dainotti, M. G., Cardone, V. F., Capozziello, S., 2008, MNRAS,391, 79Dainotti, M. G., Willingale, R., Capozziello, S., Cardone, V. F.,Ostrowski M., 2010, ApJ, 722, L215Dainotti, M. G., S., Cardone, V. F., Capozziello, S. Willingale,R., & Ostrowski M., 2011, ApJ, 730, 135 Dainotti, M. G., Ostrowski, M. & Willingale, R., 2011, MNRAS,418,2202Dainotti, M. G., Petrosian, V., Singal, J., Ostrowski, M., 2013a,ApJ, 774, 157Dainotti, M. G., Cardone, V. F., Piedipalumbo, E., Capozziello,S., 2013b, MNRAS, 2337Dainotti, M. G., Del Vecchio, R., Shigehiro, N., & Capozziello, S.2015, ApJ, 800, 31Dainotti, M. G., Petrosian, V., Willingale, R., Obrien, P,Ostrowski, M. & Nagataki, S, 2015b, MNRAS, 451, 3898Dall’Osso, S., et al., 2011, A&A, 526, A121Dall’Osso, S., et al., 2015, ApJ, 798, 25