HOW TO SWITCH A GAMMA-RAY BURST ON AND OFF THROUGH A MAGNETAR

One of the most elusive features of gamma-ray bursts (GRBs) is the sporadic emission prior to the main prompt event observed in at least ∼15% of cases. Previous studies of precursor activity on BATSE (Koshut et al. 1995; Lazzati 2005; Burlon et al. 2009) and Swift/Burst Alert Telescope (BAT; Barthelmy et al. 2005) samples (Burlon et al. 2008, hereafter B08) showed that precursors have spectral and temporal properties similar to the main prompt emission, and smaller, but comparable, energetics. They are separated from the main event by a quiescent time that may be extremely long (up to ∼100 s, rest frame), especially if measured in terms of the typical variability timescale of the prompt emission (∼1 ms). In some cases, more than one precursor has been observed in the same burst, separated by several tens of seconds.

Current models fail to explain these characteristics. They usually involve the collapse of the progenitor star, either in two steps, first to a neutron star (the precursor) and then to a black hole (the main emission; Wang & Mészáros 2007; Lipunova et al. 2009), or via a shock breakout through the stellar envelope preceding the main GRB (Ramirez-Ruiz et al. 2002; Wang & Mészáros 2007), or when the fireball becomes transparent (Ruffini et al. 2001; Mészáros & Rees 2000; Li 2007; Daigne & Mochkovitch 2002). These models predict, however, one precursor episode, and in the last two cases, the precursor spectrum should be thermal (i.e., a blackbody), contrary to what is observed. Alternatively, the precursor emission may be related directly to the central engine activity, although this requires ad hoc conditions (Ramirez-Ruiz et al. 2001).

In this paper we propose a new scenario for which precursors are explained by assuming that the central GRB engine is a newly born magnetar (Usov 1992; Duncan & Thompson 1992; Uzdensky & MacFadyen 2007; Metzger et al. 2011), i.e., a neutron star endowed with a large magnetic field (B ∼ 10¹⁵–10¹⁶ G). In this model the precursor and the prompt emission arise from the accretion of matter onto the surface of the magnetar. The accretion process can be halted by the centrifugal drag exerted by the rotating magnetosphere onto the infalling matter, allowing for multiple precursors and very long quiescent times. After the end of the prompt emission, the GRB afterglow can also be influenced by the magnetar, being re-energized by its spin-down power (Dall'Osso et al. 2011).

We made use of the complete sub-sample of bright long GRBs observed by BAT presented in Salvaterra et al. (2012, hereafter BAT6) to constrain the observational properties of GRBs with precursors in an unbiased way, and to have a direct and independent estimate of the magnetic field and spin period of the magnetar needed to calculate the characteristic luminosities of the quiescent phase for different GRBs. Among different complete samples available in the literature, as the sample of GRBs with optical afterglow promptly observed by GROND (Greiner et al. 2011) or The Optically Unbiased Gamma-Ray Burst Host Survey (Hjorth et al. 2012), the BAT6 sample is the best candidate for the search for precursors. In fact, one of its main selection criteria is the brightness of the prompt emission as observed by Swift/BAT, providing suitable conditions for the identification and analysis of precursor activity prior to the main event.

In Section 2 we describe the sample selection criteria and the main observational properties of GRBs with precursors in the BAT6 sample. In Section 3 we discuss the model and its main implications. In Section 4 we compare the rate of magnetars with the rate of supernovae Ibc and low-luminosity and normal GRBs. In Section 5 we draw our main conclusions. We adopt standard values of the cosmological parameters: H_○ = 70 km s⁻¹ Mpc⁻¹, Ω_M = 0.27, and Ω_Λ = 0.73. Errors are given at 1σ confidence level unless otherwise stated.

We searched for precursors in all the long GRBs of the BAT6 sample (consisting of 58 GRBs) adopting the same criteria of B08, where the sample of all the GRBs observed by Swift and with redshift measurement was considered, i.e., a precursor is identified if:

• 1.  
  the peak count-rate of the precursor in the BAT 15–150 keV energy band light curve is lower than or equal to the main event;
• 2.  
  the Swift/BAT flux falls below the background level before the main event starts.

We applied these selection criteria to the Swift/BAT light curves binned with a signal-to-noise ratio S/N = 5, and we identified 10 GRBs out of 58 of the BAT6 complete sample (17%; see Table 1) fulfilling these criteria. Six of them are also in the B08 sample. GRB 061222A and GRB 070306 have two clear and well-separated precursors, while GRB 060904A and GRB 080602 have a long-lasting precursor emission with several peaks, although not well separated. Since we are dealing with a rather small sample, when possible we supported our results with those found in previous works on larger samples.

Table 1. Summary of the Prompt Emission Properties of GRBs with Precursors in the BAT6 Sample as Observed by BAT (15–150 keV)

Name	z	F p	f p	$T_{90}^{\rm p}$	$t_{\rm pk}^{\rm p}$	Γp	F me	f me	$T_{90}^{\rm me}$	$t_{\rm pk}^{\rm me}$	Γme	fmin
050318	1.44	2.5	12.6 ± 1.5	6.65	0.82	2.11 ± 0.24	10.7	22.9 ± 1.6	...a	29.13	1.90 ± 0.08	15.9
050401	2.90	74.1	62.0 ± 18.0	9.14	0.49	1.51 ± 0.09	37.8	99.9 ± 7.8	8.00	24.00	1.38 ± 0.08	6.6
060210	3.91	4.6	7.7 ± 2.8	9.65	−224.00	1.55 ± 0.26	63.9	23.4 ± 2.2	167.16	0.44	1.46 ± 0.06	1.6
060904A-1	...	12.9	5.9 ± 1.1	35.90	5.08	1.46 ± 0.08	57.4	43.6 ± 2.0	37.90	55.80	1.52 ± 0.03	...
060904A-2	...	7.9	7.9 ± 1.1	18.50	30.60	1.64 ± 0.08	...	...	...	...	...	...
061007	1.26	58.7	56.2 ± 5.2	12.79	6.82	1.12 ± 0.05	380.7	154.9 ± 3.6	54.40	45.19	1.04 ± 0.01	5.8
061121	1.31	6.6	12.5 ± 1.3	7.83	2.04	1.63 ± 0.08	133.1	199.5 ± 3.7	29.45	74.48	1.31 ± 0.03b	5.9
061222A-1	2.09	2.8	6.1 ± 1.4	14.3	1.99	1.41 ± 0.14	67.1	79.4 ± 1.8	47.20	87.00	1.22 ± 0.02	1.3
061222A-2	2.09	6.8	18.6 ± 1.7	20.00	26.30	1.76 ± 0.10	...	...	...	...	...	...
070306-1	1.50	3.1	6.5 ± 2.0	7.70	−115.60	1.29 ± 0.28	38.7	29.5 ± 1.4	67.60	98.27	1.64 ± 0.03	3.3
070306-2	1.50	10.2	6.9 ± 2.0	26.00	2.03	1.54 ± 0.20	...	...	...	...	...	...
080602	∼1.4c	15.2	19.1 ± 3.6	17.66	0.73	1.34 ± 0.14	13.1	28.2 ± 2.0	9.96	60.10	1.06 ± 0.07	...
091208B	1.06	7.9	37.1 ± 6.0	3.95	0.48	1.97 ± 0.20	22.9	104.7 ± 7.2	5.78	8.79	1.75 ± 0.12	40.2

Notes. Name and redshift (z); fluence (F); 1s peak flux (f); duration (T₉₀); peak time (t_pk); photon index (Γ) of the spectrum assuming a power-law model of the precursor(s) (superscript "p") and main event (superscript "me"); minimum accretion flux necessary to penetrate the centrifugal barrier (f_min; see Appendix B). Fluences are given in units of 10⁻⁷ erg cm⁻², fluxes in 10⁻⁸ erg cm⁻² s⁻¹, times in s. ^aThe event data of part of the main emission are not available. ^bSpectrum fitted with a cutoff power-law model. Cutoff energy is E_cpl = 499 ± 66 keV. ^cPhotometric redshift on the basis of the most probable host galaxy association in the XRT error circle (Rossi et al. 2012).

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The Swift/BAT data were retrieved from the public archive⁷ and processed with the standard Swift analysis software included in NASA's HEASARC software (HEASOFT, ver. 6.12) and the relevant calibration files. The analysis has been performed in the 15–150 keV energy band, unless otherwise stated. We extracted background-subtracted light curves binned with S/N = 5 for the identification of the precursors and main event, and binned at 4 ms for the identification of the interesting time intervals with battblocks. The precursor and main event spectral analysis was performed with XSPEC (ver. 12.6.1). The spectra were fitted with either a single (PL) or a cutoff power-law (CPL) model. A model with a larger number of free parameters was not required to improve the best fit of the spectra, also due to the limited energy band of the BAT instrument. The results of our analysis are detailed in Table 1.

We find that precursors show very similar temporal behavior to their main event, albeit with a shorter duration, from simple Fast Rise and Exponential Decay shapes to complex multi-peaked structures. The peak flux of the precursor is lower than the main event (by definition), with a median value for their ratio ∼23%, but spanning a wide range: from 6% for GRB 061121 to 68% for GRB 080602. Similarly, the median value of the ratio between the precursor and main event fluence is 20%, with the record of GRB 080602 where the precursor fluence slightly exceeds the main event total fluence. Precursors and main events thus have comparable brightness and energetics (see Table 1).

Precursors and main events have also consistent spectral indices (a Kolmogorov–Smirnov (K-S) test gives a probability P = 0.97 that they are drawn from the same population; see Table 1), similar to what was found by B08 with a larger sample of GRBs observed by BAT. The spectra in the BAT energy range are non-thermal, being well fit in all cases by a PL or CPL model. We therefore confirm the B08 findings, that a pure thermal spectrum can be excluded in the majority of cases (see also Lazzati 2005 with a large sample of BATSE GRBs with precursors).

2.1. Are Precursors Correlated to the Main Event?

2.2. The Multiwavelength Properties of GRBs with and without Precursors in BAT6

All the observed properties of GRBs with precursors that we reported in our analysis can be explained if the central GRB engine is a newly born magnetar. This proposal is based on the assumption that the GRB prompt emission originates from a newly born magnetar accreting material from an accretion disk, and the observed power is proportional to the mass accretion rate. Close to the surface of the magnetar, the behavior of the infalling material is dominated by the large magnetic field of the neutron star, so that matter is channeled along the field lines onto the magnetic polar caps. We considered a pure dipolar magnetic field. Higher order multipoles have a lower influence on the infalling matter motion due to their steeper radial dependence. The magnetic field begins to dominate the motion of matter at the magnetospheric radius r_m, defined by the pressure balance between the magnetic dipole of the magnetar and the infalling material. Accretion onto the surface of the magnetar proceeds as long as the material in the disk rotates faster than the magnetosphere (see Figure 1, right upper panel). In the opposite case, accretion can be substantially reduced due to centrifugal forces exerted by the super-Keplerian magnetosphere: the source is said to enter the "propeller" phase (Illarionov & Sunyaev 1975; Campana et al. 1998; see Figure 1, right lower panel).

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In this scenario, during the collapse of the progenitor, a proto-magnetar is formed and the confining pressure and inertia of the infalling stellar envelope act to collimate the outflow into a jet (Uzdensky & MacFadyen 2007). This matter forms a disk and accretes onto the newly born magnetar, giving rise to the GRB power. If the conditions are profitable, the system can enter the propeller phase: accretion is inhibited and the GRB becomes quiescent. During this phase, matter continues to pile up at r_m (at a few neutron star radii) until its pressure is high enough to overcome the centrifugal barrier again. Accretion onto the surface of the neutron star then restarts, giving rise to another high-energy event. All the emission episodes are produced by the same mechanism and, thus, have the same observational properties.

Every emission episode should lie above the characteristic luminosity corresponding to the onset of the propeller phase L_min, providing a strong observational test for the consistency of this model. This is indeed confirmed for the GRBs with precursors in the BAT6 sample, with typical values for the magnetic field and the spin period of a newly born magnetar (Duncan & Thompson 1992; see Figure 1, left panel, where GRB 061121 is portrayed as an example, and Table 2). It is possible to have multiple precursors if the centrifugal barrier is penetrated more than once, i.e., if centrifugal forces are weaker than for single precursor emission. This is the case for, e.g., GRB 070306 (see Appendix B).

Table 2. Best-fitting Values of the Plateau Phase for the GRBs with Precursors (Superscript "p") in the BAT6 Sample

Name	z	Bp	Pp	k'p	$t_\circ ^{\rm p}$	$E_\circ ^{\rm p}$	$L_{\rm min}^{\rm p}$	Lp(rm)	$M_{\rm acc}^{\rm p}$
		(1015 G)	(ms)		(s)	(1050 erg)	(1050 erg s−1)	(1050 erg s−1)	(M☉)
050318a	1.44	4.00	3.06	...	...	...	4.7	1.1	0.004
050401	2.90	5.67 ± 0.27	2.61 ± 0.04	0.8	31.6	5.57 ± 0.18	13.7	3.6	0.101
060210	3.91	2.34 ± 0.07	1.83 ± 0.02	0.9	15.8	1.00b	5.4	1.8	0.088
061007a	1.26	4.00	3.06	...	...	...	4.7	1.1	0.252
061121	1.31	6.03 ± 0.12	4.40 ± 0.03	0.9	63.1	0.70 ± 0.04	4.6	0.9	0.068
061222A	2.09	2.79 ± 0.04	2.25 ± 0.01	0.9	31.7	5.06 ± 0.37	4.7	1.4	0.056
070306	1.50	2.33 ± 0.10	3.60 ± 0.05	0.9	15.8	0.32b	1.1	0.2	>0.022
091208B	1.06	18.6 ± 1.1	9.70 ± 0.21	0.5	19.9	0.36 ± 0.07	6.9	0.8	0.005

Notes. Name and redshift (z); magnetic field (B) and spin period (P) of the magnetar; parameter that accounts for our ignorance about the microphysical parameters and the density profile of the ambient medium (k'); shallow decay onset rest-frame time (t_○); initial energy of the forward shock (E_○); minimum bolometric accretion luminosity necessary to penetrate the centrifugal barrier (L_min) and bolometric accretion luminosity just after the onset of the centrifugal barrier (L(r_m)), calculated with the best-fitting parameters B and P; mass accreted during the prompt emission (M_acc). ^aThese GRBs do not allow us to perform the fit of the late-time X-ray emission with the model in Equation (A3). We therefore assumed as B and P the median values of their distributions, and calculated the luminosities accordingly. ^bNot a free parameter.

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During the propeller phase the luminosity does not drop to zero. A smaller luminosity L(r_m) is expected resulting from the gravitational energy release of the infalling matter up to r_m, escaping through the pre-excavated funnel. This provides an upper limit to the observed quiescent time luminosity since only a fraction of it may leak out from the jet base. The Swift/BAT was triggered by the precursor of GRB 060124 (Romano et al. 2006), a burst that displayed a very long quiescent time (∼90 s in the cosmological rest frame), allowing the Swift satellite to slew to the source and detect the quiescent emission with the X-Ray Telescope (XRT; Burrows et al. 2005a). This quiescent luminosity is almost constant and ∼100 times fainter than the main event, well below the predictions for L(r_m) (see Figure 2, left panel).

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An expectation for the propeller model is that the longer the quiescent time, the stronger and longer the subsequent emission episode, due to the larger amount of matter stored during the propeller phase. For the special cases of GRB 070306 and GRB 061222A with two precursors,¹⁰ these scalings can be tested within the same burst. In particular we found that in GRB 061222A both the total energy emitted in the BAT energy band and the duration (T₉₀¹¹) are proportional to the previous quiescent time for the second precursor and the main event, which is what we expect from an accumulation mechanism at work, such as the propeller one. Precursors and the main event in GRB 070306 are separated by comparable time intervals, while the energetics of the main event is a factor of three larger than in the previous precursor. This might indicate that, at variance with the previous case, the mass accretion rate is not constant during the evolution of this burst. A correlation between the quiescent time and the subsequent emission episode duration was reported in the BATSE sample (Ramirez-Ruiz & Merloni 2001).

Being time-symmetric, the propeller model can still work after the main event, producing "postcursors." The giant flares observed in the X-ray light curve of the GRB afterglows (e.g., GRB 050502B, Burrows et al. 2005b) could be explained in this way.

The accretion process ends when the mass inflow rate decreases enough for the magnetospheric radius to reach the light cylinder (i.e., the radius at which the field lines co-rotate with the neutron star at the speed of light; see Figure 2, right panel). Beyond this radius the field becomes radiative and expels much of the infalling matter. After the end of the prompt emission, the GRB afterglow can also be influenced by the magnetar, being re-energized by its spin-down power (Dall'Osso et al. 2011). We applied the solution of Dall'Osso et al. (2011) to our sample (details are given in Appendix A) to have a direct and independent estimate of the magnetic field and spin period of the magnetar (B ∼ (2–20) × 10¹⁵ G, P ∼ 2–10 ms) and to calculate the characteristic luminosities of the propeller phase for different GRBs. In all cases we find that the precursor and main GRB event occur in the accretion phase, whereas the luminosity between them falls in the propeller phase. The estimate of the spin period from late-time emission may be biased by spin-up and spin-down occurring during the accretion and propeller phases. However, it has been demonstrated that the spin period does not change appreciably assuming a reliable mass-accretion rate during supernova fallback onto a magnetar (Piro & Ott 2011).

If the initial mass inflow rate is not high enough to shut off the spin-down power, a magnetar powered GRB can occur. In this case we might expect a smoother and longer event, such as GRB 060218 (Campana et al. 2006; Soderberg et al. 2006a). At the other extreme, if during the accretion phase the magnetar accretes enough matter, then it collapses to a black hole (Piro & Ott 2011). A clear signature of this transition to a black hole is the absence of the spin-down power emitted by the magnetar able to re-energize the late-time X-ray emission, and thus no plateau in the X-ray afterglow should emerge. An example of this phenomenon in the BAT6 sample is provided by GRB 061007 (Schady et al. 2007), which shows precursor activity and a single power-law decaying X-ray afterglow. When the prompt and precursor energies are converted to accreted mass, GRB 061007 is found to be the object that has accreted the largest amount of mass among the GRBs with precursors in the BAT6 sample (see Appendix B).

The local rate of core-collapse supernovae (SNe) is ∼0.01 yr⁻¹ (Fryer & New 2011). Core-collapse SNe can be mainly divided into Type II (usually associated with supergiant progenitors) and Type Ibc (usually associated with Wolf–Rayet progenitors). SNe Ibc are ∼30% of core-collapse SNe (Li et al. 2011), leading to a local rate for the SNe Ibc ≈3 × 10⁻³ yr⁻¹. Magnetars constitute about 10% of the Galactic neutron star birth rate (Gaensler et al. 2005; Gill & Heyl 2007; Muno et al. 2008; Kouveliotou et al. 1994), and it is expected that a similar fraction of SNe Ibc give rise to a magnetar (Gaensler et al. 2005; Soderberg et al. 2006a), i.e., ∼10% of SNe Ibc host a magnetar.

Long GRBs in the local universe have been unambiguously associated with SNe Ibc. The SNe Ibc associated with GRBs have broad emission lines and are extremely energetic (called hypernovae; Iwamoto et al. 1998). At the same time, not all broad-line SNe Ibc are related to GRBs (e.g., SN 2002ap, Mazzali et al. 2002), likely indicating that there is the need for an additional ingredient. SNe Ibc were initially associated with low-luminosity GRBs (with peak luminosity L < 10⁴⁹ erg s⁻¹, such as GRB 980425/SN 1998bw, Pian et al. 1999, or GRB 060218/SN 2006aj, Campana et al. 2006; Soderberg et al. 2006a), but later on they were discovered also in normal (i.e., the ones considered in our analysis) GRBs (such as GRB 030329/SN 2003dh, Stanek et al. 2003; Hjorth et al. 2003).

The rate of GRBs has been constrained by independent and consistent estimates. The rate of SNe Ibc with an associated low-luminosity GRB is between ∼1% and 9%, while those associated with a normal GRB are about 0.3%–3%, depending on the assumption on the beaming factor (Soderberg et al. 2006a; Guetta & Della Valle 2007; Ghirlanda et al. 2013). Estimates from the radio surveys give that the incidence of low-luminosity events is <3% in SNe Ibc (Berger et al. 2003), while the limit on the fraction of SNe Ibc actually associated with a GRB, including off-axis events, is <10% (Soderberg et al. 2006b).

It is clear from the above estimates that, overall, magnetars related to SNe Ibc are consistent with the total number of observed GRBs, accounting for both low-luminosity and normal GRBs.

The main observational properties of the GRBs with precursors can be qualitatively explained if the central GRB engine is a newly born magnetar and the precursor and the prompt emission arise from the accretion of matter onto its surface. In this scenario the accretion process can be halted by the centrifugal drag exerted by the rotating magnetosphere onto the infalling matter, allowing for multiple precursors and very long quiescent times.

Using the complete BAT6 sample to estimate the fraction of long GRBs with precursors in an unbiased way, we end up with ∼17% of GRBs showing precursor activity. The fraction of GRBs powered by a magnetar is even higher if we interpret the X-ray plateau in the GRB afterglows as due to continuous energy injection from the magnetar spin-down power, resulting in at least ∼80%. The remaining GRBs can be powered by a newly born black hole or by a magnetar collapsed to a black hole (GRB 061007 alike). Indeed, the collapse of a massive star accommodates both the direct collapse to a black hole and the formation of a proto-magnetar in those cases where fast-rotating cores produce a magnetorotational explosion. Despite the uncertainties, recent simulations seem to indicate that proto-magnetars are more easily produced than black holes by current stellar evolutionary models (Dessart et al. 2012). Within current uncertainties, the rate of magnetar-powered SNe Ibc is comparable to the rate of (low-luminosity and normal) GRBs, indicating that a large fraction of the long GRBs are powered by a magnetar.

The propeller mechanism as an explanation for the quiescent time could also be extended to short GRB precursors (Troja et al. 2010). In this case the central magnetar would originate from a binary white dwarf merger, a white dwarf accretion induced collapse, or, perhaps, the merger of a double neutron star binary (Levan et al. 2006; Metzger et al. 2008).

The authors thank the referee for useful comments. The authors acknowledge support from ASI grant I/004/11/0 and PRIN-MIUR 2009 grants. D.B. is funded through ARC grant DP110102034. M.G.B. and S. Campana want to thank L. Stella and S. Dall'Osso for conversations, P. Romano and R. Margutti for having supplied data of GRB 060124, and A. Hawken for language editing.

The observation of a flattening in the X-ray light curve (plateau) in a large fraction of GRBs (46% in the BAT6 sample) can be explained as an injection of energy into the forward shock (the GRB afterglow; Zhang et al. 2006). This fraction is even larger (80% in the BAT6 sample) if we also include those GRBs displaying a shallow decay phase without the initial steep decay (Bernardini et al. 2012; D'Avanzo et al. 2012). A natural source for this energy is the power emitted by a spinning-down newly born magnetar (Dai & Lu 1998; Zhang & Mészáros 2001; Corsi & Mészáros 2009; Dall'Osso et al. 2011). This proposal has been successfully tested both for long (Lyons et al. 2010; Dall'Osso et al. 2011; Bernardini et al. 2012) and short (Rowlinson et al. 2013) GRBs. In particular, the plateau luminosity and its temporal duration are directly related to the spin-down luminosity and timescale and, thus, to the magnetic field (B) and the spin period (P) of the magnetar. The analysis of the plateau phase in the X-ray light curves provides a direct estimate of these parameters.

We refer to the model proposed by Dall'Osso et al. (2011), which calculated analytically the contribution to the forward shock of the power emitted by a millisecond spinning, ultramagnetized neutron star at time t as

$$\begin{equation} \frac{{dE}(t)}{dt}=L_{\rm sd}(t)-k^{\prime } \frac{E(t)}{t}=\frac{L_i}{(1+at)^2}-k^{\prime } \frac{E(t)}{t}, \end{equation} \tag{ A1 }$$

where L_i = IB²R⁶/(6Ic³P⁴)∝B²/P⁴ is the initial spin-down luminosity (I is the moment of inertia, R is the radius of the magnetar,¹² c the speed of light), a = 1/t_b2 = 2B²R⁶/(6Ic³P²)∝B²/P² is the inverse of the spin-down timescale t_b2, E is the forward shock energy, and k' is a parameter that accounts for our ignorance about the microphysical parameters and on the density profile of the ambient medium (in general 0 < k' < 1). A solution of this equation is

$$\begin{equation} E(t)=\frac{L_i}{t^{k^{\prime }}}\int ^t_{t_\circ } \frac{t^{k^{\prime }}}{(1+at)^2}+E_\circ \left(\frac{t_\circ }{t} \right)^{k^{\prime }}, \end{equation} \tag{ A2 }$$

where t_○ is any time chosen as initial condition and E_○ the initial energy. The solution of the above integral can be expressed in terms of the real valued hypergeometric function ₂F₁(a, b, c; (1 + at)⁻¹). The total bolometric luminosity is, then

$$\begin{equation} L(t)=E(t)/t. \end{equation} \tag{ A3 }$$

We selected those GRBs in the BAT6 sample with redshift and with a well-sampled plateau in the X-ray light curve (16 GRBs), having or not a precursor in the prompt emission, and we assumed their 0.3–30 keV common rest-frame luminosity (Margutti et al. 2013) as a proxy of the total bolometric luminosity. In order to account for the possible collimation of the outflow θ_j and of the radiative efficiency _r, we considered the corrected luminosity L_{X, j} = (f_b/_r) L_{X, iso}, with _r = 0.1 and f_b = (1 − cos θ_j) = 0.01, which corresponds to θ_j ≃ 8°. We fitted these data with Equation (A3), using as free parameters B, P, and E_○. We fixed t_○ as the (rest-frame) starting time of the plateau phase, and k' from the decay index of the post-plateau light curve (the solution in Equation (A3) has an asymptotic behavior $\propto t^{-k^{\prime }-1}$; for details see Dall'Osso et al. 2011). Tables 2 and 3 summarize the best-fit parameters for the GRBs, grouped as GRBs with precursors and without precursors, respectively, while Figure 3 shows the results of the fit for the sample of GRBs with precursors only.

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Table 3. Best-fitting Values of the Plateau Phase for the GRBs without Precursors (Superscript "no-p") in the BAT6 Sample

Name	z	Bno -p	Pno -p	k'no -p	$t_\circ ^{\rm no{\scriptsize {-}}p}$	$E_\circ ^{\rm no{\scriptsize {-}}p}$
		(1015 G)	(ms)		(s)	(1050 erg)
060306	1.55	9.40 ± 0.90	7.30 ± 0.20	0.9	31.6	0.63 ± 0.09
060814	1.92	2.87 ± 0.10	3.79 ± 0.04	0.9	316.2	2.22 ± 0.27
061021	0.35	16.97 ± 0.48	32.21 ± 2.90	0.5	316.2	0.02 ± 0.01
080430	0.77	5.06 ± 0.21	13.42 ± 0.21	0.9	3.2	0.04 ± 0.01
080607	3.04	10.59 ± 0.57	3.61 ± 0.10	0.9	100.0	5.71 ± 0.22
081007	0.53	12.72 ± 0.77	23.58 ± 0.50	0.9	100.0	0.06 ± 0.01
081203A	2.10	19.16 ± 0.69	5.21 ± 0.10	0.9	31.6	5.75 ± 0.30
081221	2.26	7.68 ± 0.25	2.20 ± 0.01	0.9	19.9	30.13 ± 1.73
091020	1.71	12.54 ± 0.36	6.64 ± 0.07	0.7	31.7	2.18 ± 0.05
100621A	0.54	2.95 ± 0.11	6.92 ± 0.01	0.7	100.0	0.74 ± 0.03

Note. Name and redshift (z); magnetic field (B) and spin period (P) of the magnetar; parameter that accounts for our ignorance about the microphysical parameters and on the density profile of the ambient medium (k'); shallow decay onset rest-frame time (t_○); initial energy of the forward shock (E_○).

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A.1. The Properties of the Magnetar in GRBs with and without Precursors

Since the condition for the onset of the propeller phase depends on B and P, we searched for a pattern that allows us to discriminate between GRBs with (superscript "p") and without (superscript "no-p") precursors. For this reason we compared the distributions of B and P for GRBs with and without precursors in the BAT6 sample (see Tables 2 and 3, respectively). We find that both the magnetic field and the spin period distributions are centered around lower values for GRBs with precursors than for GRBs without precursors (〈log [B^p/10¹⁵ G]〉 = 0.60 while 〈log [B^no -p/10¹⁵ G]〉 = 1.00; 〈log [P^p/ms]〉 = 0.48 while 〈log [P^no -p/ms]〉 = 0.75; see Figure 4). The spin period distribution is also less scattered around its central value than in the other case ($\sigma ^{\rm p}_{\rm P}=0.26$ while $\sigma ^{\rm no{\scriptsize {-}}p}_{\rm P}=0.38$). A K-S test gives a probability P = 0.12 that the spin periods of GRBs with and without precursors are drawn from the same population, while P = 0.30 for the magnetic field distributions.

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The luminosity of the shallow decay phase is related to the spin-down luminosity, being L_i∝B²/P⁴. A lower value of the spin period and of the magnetic field for the GRBs with precursors would result in a higher luminosity during the shallow decay phase since the luminosity depends strongly on P. Similarly, the narrower distribution of P would imply a narrower distribution of L_i, which is indeed what we found in Section 2.2 (see also Pisani et al. 2013). The magnetospheric radius depends on the magnetic field and on the mass accretion rate (given the mass and radius of the magnetar), $r_m\propto \dot{M}^{-2/7}B^{4/7}$, while the corotation radius depends only on the spin period, being r_c∝P^2/3. Thus, among the GRBs powered by a magnetar, GRBs with precursors are characterized by specific values of the magnetic field and spin period that favor the trigger of the propeller regime, responsible for the observed quiescent times.

We used the best-fit values of B and P in the case of GRBs with precursors (see Table 2) to estimate in the case of spherical accretion the bolometric accretion power corresponding to the onset of the propeller phase:

$$\begin{equation} L_{\rm min}=4\times 10^{50} B_{15}^2\, P_{-3}^{-7/3}\, {\rm erg\,s^{-1}}, \end{equation} \tag{ B1 }$$

and the bolometric power during the quiescent time:

$$\begin{equation} L(r_{\rm m})=2\times 10^{50} B_{15}^2\, P_{-3}^{-3}\, {\rm erg\,s^{-1}}, \end{equation} \tag{ B2 }$$

where B = 10¹⁵B₁₅ G and P = 10⁻³P₋₃ s (for details see Campana et al. 1998, their Equations (4) and (6); see also Bozzo et al. 2008 and Piro & Ott 2011). Equation (B2) sets an upper limit to the quiescent time luminosity, since only a fraction of it will actually escape from the jet base. The values are displayed in Table 2. In the case of GRB 050318 and GRB 061007, since it was not possible to estimate B and P from the late X-ray emission (GRB 050318 has an incomplete light curve, while GRB 061007 has a simple power-law decay), we assumed for B and P the median values obtained for GRBs with precursors B = 4.00 × 10¹⁵ G and P = 3.09 ms.

For a direct comparison with the prompt emission as observed by BAT (see Section 2), we computed the corresponding observed flux from L_min: $f_{\rm min} = ({K L_{\rm min}}/{(f_b/\epsilon _r) 4 \pi D_L^2(z)})$, where D_L(z) is the luminosity distance at redshift z and K is a K-correction to account for the limited energy band observed by BAT.¹³ The results are displayed in Table 1: the main event peak flux is always well above f_min, as well as most of the precursor(s) peak flux(es). In the worst cases, the precursor peak flux is comparable to f_min. Indeed, the smallest L_min is found for GRB 070306, which has two well-separated precursors. In this case, weaker centrifugal forces allow for multiple onsets of accretion episodes.

We estimated also the total amount of mass accreted during the prompt emission (precursor(s) plus main emission) from the total bolometric prompt emission energy, corrected for the same collimation factor: E_{γ, j} = (GM_NS/R_NS) M_acc (see Table 2; for E_{γ, iso} we refer to Table 1 in Nava et al. 2012). The largest amount of mass accreted corresponds to GRB 061007, which indeed has one among the largest E_{γ, iso} in the BAT6 sample (Nava et al. 2012).

The values we derived depend on our assumption for the ratio (f_b/_r). However, a simple scaling exists that allows us to derive all the parameters for different collimation angles and efficiencies, up to the isotropic case. In fact, the isotropic spin-down luminosity is L_{i, j} = (f_b/_r) L_{i, iso}, which implies $B_j^2/P_j^4=(f_b/\epsilon _r) B_{\rm iso}^2/P_{\rm iso}^4$. At the same time, the spin-down timescale is not influenced by collimation, t_{b2, iso} = t_{b2, j} and therefore $P_j^2/B_j^2=P_{\rm iso}^2/B_{\rm iso}^2$. Globally, we find that $P_{\rm iso}=\sqrt{(f_b/\epsilon _r)}\,P_j$ and $B_{\rm iso}=\sqrt{(f_b/\epsilon _r)}\,B_j$. Consequently, the dependence of the minimum luminosity on (f_b/_r) is $L_{\rm min}\propto B_j^2/P_j^{7/3}=(f_b/\epsilon _r)^{1/6} B_{\rm iso}^2/P_{\rm iso}^{7/3}$.