central engine (likely a black hole) injects relativistic shells of plasma that collide and produce the gamma-rays observed in the prompt emission. Shell collisions within the burst provide ideal conditions for particle acceleration via the Fermi mechanism (Bell 1978 ;Blandford & Ostriker 1978 ), hence GRBs have long been speculated as promising sources of UHECRs (Milgrom & Usov 1995 ;Vietri 1995 ;Waxman 1995 ).

Recently, a high-energy photon thought to originate from the interaction of an UHECR emitted by GRB980425 has reinvigorated interest into the possibility that some high-energy protons are indeed due to GRBs (Mirabal 2022a , b ). GRB980425 is a representative of the low-luminosity GRB (LLGRB) population (Liang et al. 2007 ;Dereli et al. 2017 ), which have been theoretically explored as potential sources of UHECRs (see e.g. Murase et al. 2006 ;Wang et al. 2007a ;Liu, Wang & Dai 2011 ;Zhang et al. 2018 ;Boncioli, Biehl & Winter 2019 ;Rudolph et al. 2022 ); although see Samuelsson et al. ( 2019Samuelsson et al. ( , 2020 ) ). Hence it is tempting to see if the observed rate of such events can account for the production of some or all UHECRs.

In this paper, we use a sample of LLGRBs from Dereli et al. ( 2017 ) to determine their occurrence rate in the local Universe. From that rate, we infer the production rate of UHECRs from LLGRBs and compare our result with observations. This paper is organized as follows. In Section 2 , we describe the data set used in our statistical study, and in Section 3 we outline our method for computing the rates of LLGRBs and UHECRs. In Section 4 , we discuss the nature of LLGRBs, and provide constraints on the number of UHECRs that can be accelerated in the jet by comparing our derived UHECR flux to the detected flux. In Section 5 , we investigate potential suppression mechanisms that could bias the results, and discuss the implications our results have on the contents of a GRB jet. For our analysis, we assume a flat lambda-cold dark MNRAS 530, 555-559 ( 2024) matter ( -CDM) cosmology with H 0 = 67.66 km s -1 Mpc -1 and M = 0.30966 (Planck Collaboration VI 2020 ).

So far, only a tentative association of an UHECR has been found for GRB980425 (Mirabal 2022a , b ). This event is one of the closest GRBs ever recorded on Earth (Galama et al. 1998 ), but also one of the most underluminous to date (Amati 2006 ). Dereli et al. ( 2017 ) has shown that this burst was representative of a whole population of faint events that are not visible at large ( z 1) redshifts, but are o v erabundant in the local Universe. We considered that this population of nearby events would be the principle source of UHECRs produced by GRBs. Thus, we use a sample of bursts similar to GRB980425 from Dereli et al. ( 2017 ) that were selected by their X-ray properties, which act as a proxy for the total fireball energy measurement. We rejected from that sample events with no known redshift, as we study the occurrence rate of these events corrected for the variation in detection distances. The final sample comprises of 41 LLGRBs with known redshift. They are listed in Table 1 , with some of their properties.

The sample consists of events detected between 1998 and 2016. Over such a large time span, several instruments were used to perform the detection of each burst, including BeppoSAX (GRBM, Boella et al. 1997 ), INTEGRAL (IBAS, Mereghetti et al. 2003 ), and the Neil Gehrels Swift Observatory (BAT, Gehrels et al. 2004 ). We list in Table 2 the ef fecti ve time span for each instrument together with the field of view of its gamma-ray detector. To date, this is the most recent complete sample published for LLGRBs. Since we take into consideration a time-averaged rate of LLGRBs within the detection volume in our method, the addition of more recent GRBs would be balanced by an increase of the operation period, making the final rate value nearly constant.

The variation of the field of view is a bias of this sample, which we deconvolve by taking into account the fact that the GRBs are isotropically distributed. The fact that we do not consider more recent LLGRBs, such as those detected by Fermi (GBM, Meegan et al. 2009 ), also contribute to the reduction of this bias, as we would be introducing variations in the size of the fields of view. Most of the bursts were detected by the Neil Gehrels Swift Observatory (38 of them), with BeppoSAX performing two detections and INTEGRAL only one.

In this work, we only consider UHECRs composed of baryons. For the purposes of our study, we require a point of comparison between our predicted UHECR flux and the observed UHECR flux. We have used the results of the Pierre Auger Observatory (Abraham et al. 2004 ), where the canonical UHECR flux is obtained by numerically integrating the cosmic ray energy spectrum over the ranges for UHECRs abo v e 10 19 eV. The most recent estimate for this value is approximately 1 particle km -2 yr -1 (e.g. Pierre Auger Collaboration 2017 ; Aab et al. 2020 ).

We follow the method of Guetta & Valle ( 2007 ) and Howell et al. ( 2014 ) to compute the expected rates of LLGRBs, like GRB980425, in the local Universe. We calculate the rates as:

.

(1) In this equation, represents the sky coverage, as listed in Table 2 . T is the time elapsed between the mission launch and the most recent GRB in our sample, as listed in Table 2 . We do not correct for any duty cycle effect, as this is a negligible effect. Finally, V max represents the maximum detection volume of each burst:

where d V /d z refers to the number density of GRBs at a given redshift that remains constant within the Hubble flow. The normalized Hubble flow h ( z) is then determined by:

Hence, the expected rate of each GRB can be found using the redshifts listed in Table 1 . Note that Table 1 presents the computed rate for each burst. We also calculate the total expected rate of GRBs R tot as the sum of the individual GRB rates R GRB ,

to be 8.340 × 10 -7 Mpc -3 yr -1 . Therefore, the rate of GRBs within the GZK horizon (which we assume to be d GZK = 50 Mpc; e.g. Abbasi et al. 2008 ) that are directed face on is

By definition, we can only observe face on events. Therefore, we need to provide a beaming correction, f b , to account for the population missed that are not directed face on. Then, under the assumption that GRBs emit N UHECR particles, we can determine the particle luminosity as

We can finally compute the particle flux on Earth from sub-GZK GRBs as:

where S GZK is the surface area of the GZK sphere.

There is currently no clear consensus on the nature of LLGRBs, and sev eral e xplanations hav e been invoked in order to explain their origin and properties. The detection of rare Type Ic supernovae with these bursts suggests a common origin with that of typical long GRBs (for instance, compare GRB980425 and GRB130427A: Galama et al. 1998 ;Xu et al. 2013 ), hence the difference may lie with either the properties of the jet or the nature of the central engine.

One popular hypothesis is that LLGRBs are powered by a shock breakout, either from the progenitor star itself (e.g. Campana et al. 2006 ;Wang et al. 2007b ;Nakar & Sari 2012 ) or from an extended low-mass stellar envelope (Nakar 2015 ). However, some LLGRBs do not support this argument (e.g. GRB120422A, Zhang et al. 2012 ). Instead, others have proposed that perhaps the cause lies with the central engine allowing for only low bulk Lorentz factors of the flow that results in the observed low emissivity of the jet (Liang et al. 2010 ;Irwin & Che v alier 2016 ), or indeed it may simply be due to a viewing angle effect (Zhang et al. 2004 ). However, in the study by Daigne & Mochkovitch ( 2007 ), it was shown that the viewing angle effect cannot explain the dim afterglows of these bursts, and the probability of detecting off-axis GRB980425-like GRBs should be rather low. Thus, the detection of other LLGRBs as well as their less-ener getic after glows supports the notion that these bursts are powered by either a mildly relativistic/energetic outflow, or by shock breakout.

Regardless of how LLGRBs are theoretically modelled, comparisons with the local Type Ib/c supernova rate indicate that they are not strongly beamed (as discussed in Liang et al. 2007 ). This is further compounded with the lack of detection of a jet break in the afterglow of these bursts (e.g. GRB060218 and GRB201015A: Soderberg et al. 2006 ;Patel et al. 2023 ); suggesting a quasi-spherical nature of the jet. With such large jet opening angles, the beaming correction for these bursts should be less than that of typical long GRBs.

On the other hand, with no certainty on the true beaming of these event, it is more conservative to use a confidence interval for f b rather than a given value.

This leads us to the question: what is a good value for f b in our calculations? As previously discussed, some of these bursts appear to display properties suggesting quasi-spherical emission, thereby a correct lower boundary is to simply assume f b = 1, indicating no beaming. In such a case, we can compute a theoretical particle flux from equation ( 7) as F particle = N × 8.8 × 10 -21 particles km -2 yr -1 mol -1 . On the other hand, we know for sure that we do observe some events, and thus that the beaming angle has to be larger than the one sustained by the Earth surface, hence f b ≤ S GZK / S Earth . Naturally, such a beaming factor is non-physical, as it vastly o v erpredicts the number of LLGRBs with respect to the local supernova rate; none the less, it is useful as an extreme upper bound as we show here. Then, the particle flux becomes F particle = N × 5.2 × 10 14 particles km -2 yr -1 mol -1 . We can thus formulate an inequality to express where the true flux lies:

Finally, it is possible to constrain the value for N within the particle flux inequality by using the observed Auger UHECR particle flux. If we express this constraint as the amount of mass accelerated within the jet then, under the assumption that 1 M contains ∼10 33 mol of baryons, in order to reach the observed 1 particle km -2 yr -1 , we require 10 -48 ≤ M ≤ 10 -13 M .

(8)

One may be tempted to compare our rate of LLGRBs with others published in the literature. Ho we ver, we emphasize that we are using a dedicated sample that has been selected from their afterglow properties, rather than from the prompt properties as in, e.g. Virgili, Liang & Zhang ( 2009 ) and Sun, Zhang & Li ( 2015 ). Indeed, it has been discussed numerous times that gamma-ray prompt properties are plagued by selection effects (such as the T 90 value which is different for each detector; or the end time of the prompt phase), and that X-ray observations are better suited for measuring physical events (e.g. the end of the prompt phase, Stratta et al. 2013 ). Another example is the classification of GRB211211A, which is a 'long short GRB': its prompt properties classify it as a long collapsar event, while its global properties indicate that it is a short event linked to a binary merger of compact objects (Rastinejad et al. 2022 ). For these reasons, by construction, our rates may differ from other published ones, as they are supposed to be less biased by instrumental effects.

MNRAS 530, 555-559 ( 2024)

None the less, our observed rate of 8.340 × 10 -7 Mpc -3 yr -1 is still highly compatible with the rate of Type Ib/c supernovae, assuming a realistic beaming factor. Estimates place the local ( d ࣠ 100 Mpc) rate of these supernovae at ∼ 9 + 3 -5 × 10 -6 Mpc -3 yr -1 (Cappellaro, Evans & Turatto 1999 ;Dahlen et al. 2004 ;Guetta & Valle 2007 ), placing our LLGRB rate as ∼0 . 9 per cent .

The first obvious suppression mechanism of our calculated UHECR output is the GZK horizon. The predominant energy loss mechanism for UHECRs travelling through the extragalactic environment is interactions with interstellar radiation fields, in particular the CMB.

It is easy to reject potential sources of UHECRs simply due to their existence out of the GZK horizon, which can vary from 50 Mpc to hundreds of Mpc. Out of the GRB population used in this study, only one burst is located within the horizon: GRB980425. It is not to be o v erlooked that the GRB rates and hence the UHECR flux calculated are primarily driven by that one event, as can be seen by its rate in Table 1 . In considering that burst alone using the rate data in Table 1 , we derive a rate of 0.336 GRB980425-like bursts per year, yielding an UHECR flux of N × 6.8 × 10 -21 ≤ F particle ≤ N × 4 × 10 14 particles km -2 yr -1 mol -1 . Assuming all the other events would be suppressed due to the horizon, this corresponds to a suppression of only ∼33 per cent.

The possibility of GRB980425 being a source of UHECRs has also been explored by Mirabal ( 2022a , b ). In these studies, the UHECRs produced by the burst were deflected by the IGMF, and subsequently cascaded into secondary gamma rays. Their main argument is that depending on the strength of the IGMF, one would expect UHECRs arriving from this source within the next 100 yr. Thus, if LLGRBs are indeed the sources of UHECRs as our calculations have shown, one would expect that historical 980425-like GRBs may explain current observations of the UHECR flux.

A second suppression mechanism is the role of Galactic and Intergalactic Magnetic Fields (GMF and IGMF, respectively) on the trajectory and arri v al time of UHECR particles. While our understanding of UHECR propagation within the GMF has increased o v er the decades (see e.g. Medina Tanco, de Gouveia Dal Pino & Horvath 1998 ;Tin yako v & Tkachev 2005 ;Jansson & F arrar 2012 ), the IGMF is less understood, and can vary in both strength and coherence length by several orders of magnitude. At first glance, the GMF dipole can indeed introduce a potential break into the isotropy hypothesis of the common -CDM model. Ho we ver, magnetic fields cannot introduce anisotropies (Eichmann & Winchen 2020 ). As GRBs are isotropically distributed, we do not expect the GMF to affect the results. The situation is less clear for the IGMF. Numerical simulations have been performed to understand the propagation and deflection of UHECRs in the IGMF, indicating that the charge and rigidity of the cosmic rays play an important role in their behaviour (e.g. Erdmann et al. 2016 ;Hackstein et al. 2018 ;Magkos & Pavlidou 2019 ). Given the uncertainties in the IGMF, one could naively suppose that the magnetic deflections are responsible for a large dilution in the UHECR fluxes produced by the GRBs. It is that dilution which would cause the very low detection rate observed on Earth, despite a larger production rate at the acceleration sites.

On the other hand, the IGMF is unorganized enough so that we can consider that the same amount of particles deflected from us will also deflected towards us from other directions, recalling that the distribution of GRBs is isotropic (Meegan et al. 1992 ). The IGMF thus should not interfere in the suppression of the flux, but instead with the arri v al time. Even so, as noted in the case for GRB980425, the arri v al times should be delayed by hundred of years for sub-GZK events. As for the GMF, there is no observational signature of the dipole in cosmic ray arri v al directions (Aab et al. 2015 ), thus again the same previous argument should apply.

In this work, we explored the theoretical contribution of the UHECR flux by GRBs. We concentrated on the contribution of the lowluminosity bursts, which form the bulk of the nearby population of GRBs. We found that in order to reconcile our theoretical flux with the observed flux, these bursts should accelerate at most 10 -13 M of UHECRs.

Such a hypothesis can be tested by studying the composition of the UHECRs and comparing it with the composition of the surrounding medium of GRBs obtained through spectroscopy. In optical, current experiments like X-Shooter already gives access to the composition of the host galaxy a few days after the e vent. Ho we ver, one may expect a medium rich in metals expelled by the stellar progenitor before the burst. In that case, the ATHENA + instruments (Barret et al. 2023 ) could perform a study of such environments in X-rays.

As for the particle experiments, an excess of heavy nuclei (compared to simple protons) for the highest energies would then confirm the electromagnetic observations. It is therefore very important that particle observatories and large spectroscopic missions are able to work together during the next decade on those topics.

Lastly, improving neutrino detectors is also a top priority to solve this issue. Neutrinos, being by-products of hadronic interactions, and the lack of their detection in association with GRBs (in particular the prompt emission) is indicative of a lack of a significant proportion of protons into the jet. Proving the detection of an absence (and not an absence of detection) is by definition a tricky task, but in future more powerful neutrino detectors could for sure be able to do so.