Low-luminosity gamma-ray bursts as the sources of ultrahigh-energy cosmic ray nuclei

Recent results from the Pierre Auger Collaboration have shown that the composition of ultrahigh-energy cosmic rays (UHECRs) becomes gradually heavier with increasing energy. Although gamma-ray bursts (GRBs) have been promising sources of UHECRs, it is still unclear whether they can account for the Auger results because of their unknown nuclear composition of ejected UHECRs. In this work, we revisit the possibility that low-luminosity GRBs (LL GRBs) act as the sources of UHECR nuclei and give new predictions based on the intrajet nuclear composition models considering progenitor dependencies. We find that the nuclear component in the jet can be divided into two groups according to the mass fraction of silicon nuclei, Si-free and Si-rich. Motivated by the connection between LL GRBs and transrelativistic supernovae, we also consider the hypernova ejecta composition. Then, we discuss the survivability of UHECR nuclei in the jet base and internal shocks of the jets, and show that it is easier for nuclei to survive for typical LL GRBs. Finally, we numerically propagate UHECR nuclei ejected from LL GRBs with different composition models and compare the resulting spectra and composition to Auger data. Our results show that both the Si-rich progenitor and hypernova ejecta models match the Auger data well, while the Si-free progenitor models have more difficulty in fitting the spectrum. We argue that our model is consistent with the newly reported cross-correlation between the UHECRs and starburst galaxies, since both LL GRBs and hypernovae are expected to be tracers of the star-formation activity. LL GRBs have also been suggested as the dominant origin of IceCube neutrinos in the PeV range, and the LL GRB origin of UHECRs can be critically tested by near-future multimessenger observations.

Figure 1

The distribution of the nuclear mass fraction (upper panel) and the specific angular momentum $J_{\text{equator}}$ (lower panel) for presupernova model Si-F 1 (HE16F) at the onset of core collapse. The red, green, and blue lines are the estimated specific angular momentum at ISCO as a function of black hole mass, see the definition in the main text. We also show the distribution of enclosed mass as a function of radius $r$ in a brown line. The Si-F 1 (HE16F) model is evolved from a bare helium core with an initial mass $16M_{\odot }$ as the outcome of the stellar merger. The final mass at core collapse is $14.80M_{\odot }$ considering a fraction of $0.03\dot{M}$ of the standard WR mass-loss rate, and the Fe core has an angular momentum ${\mathcal{J}}_{\mathrm{Fe}\mathrm{core}}=114\times {10}^{47}\mathrm{erg}\mathrm{s}$ at core collapse after considering the effect of magnetic torques [71].

Figure 2

Same as Fig. 1, but for the presupernova model Si-F 2 (16TI). The Si-F 2 (16TI) model are evolved from a rapidly rotating star with an initial mass $16M_{\odot }$ and 1% solar metallicity. The final mass is $13.95M_{\odot }$ considering a fraction of $0.3\dot{M}$ of the standard WR mass-loss rate, and the Fe core has an angular momentum at core collapse ${\mathcal{J}}_{\mathrm{Fe}\mathrm{core}}=86.7\times {10}^{47}\mathrm{erg}\mathrm{s}$ considering the effect of magnetic torques [71].

Figure 3

Same as Fig. 1, but for presupernova model Si-R 1 (12TJ). The Si-R 1 (12TJ) model are evolved from a rapidly rotating star with initial mass $12M_{\odot }$ and 1% solar metallicity. The final mass is $11.54M_{\odot }$ considering a fraction of $0.1\dot{M}$ of the standard WR mass-loss rate, and the Fe core has an angular momentum at core collapse ${\mathcal{J}}_{\mathrm{Fe}\mathrm{core}}=150\times {10}^{47}\mathrm{erg}\mathrm{s}$ considering the effect of magnetic torques [71].

Figure 4

Same as Fig. 1, but for presupernova model Si-R 2 (16TJ). The Si-R 2 (16TJ) model are evolved from a rapidly rotating star with initial mass $16M_{\odot }$ and 1% solar metallicity. The final mass is $15.21M_{\odot }$ considering a fraction of $0.1\dot{M}$ of the standard WR mass-loss rate, and the Fe core has an angular momentum at core collapse ${\mathcal{J}}_{\mathrm{Fe}\mathrm{core}}=178\times {10}^{47}\mathrm{erg}\mathrm{s}$ considering the effect of magnetic torques [71].

Figure 5

Same as Fig. 1, but for presupernova model Si-R 3 (35OC). The Si-R 3 (35OC) model are evolved from a rapidly rotating star with initial mass $35M_{\odot }$ and 10% solar metallicity. The final mass is $28.07M_{\odot }$ considering a fraction of $0.1\dot{M}$ of the standard WR mass-loss rate, and the Fe core has an angular momentum at core collapse ${\mathcal{J}}_{\mathrm{Fe}\mathrm{core}}=230\times {10}^{47}\mathrm{erg}\mathrm{s}$ considering the effect of magnetic torques [71].

Figure 6

Constraints on the initial radiation luminosity $L_{\mathrm{rad},0}$ and the location of jet base $r_0$ for the survival of nuclei with various chemical species: He, O, Si, and Fe during the initial loading at jet base. The blank (shaded) region beyond (below) the iron curve has an optical depth ${\tau }_{A\gamma }<1$ (${\tau }_{A\gamma }>1$) in the lower panel and effective optical depth $f_{A\gamma }>1$ ($f_{A\gamma }>1$) in the upper panel. The optical depth ${\tau }_{A\gamma }$ represents the interaction rate, while the effective optical depth $f_{A\gamma }$ indicates the energy loss efficiency of nuclei.

Figure 7

Constraints on the isotropic equivalent radiation luminosity $L_{\gamma \mathrm{iso}}$ and Lorentz factor $\mathrm{\Gamma }$ for nuclei with various chemical species with an energy $E_A={10}^{20}\mathrm{eV}$ to survive in the internal shock region. The upper panel corresponds to the energy loss time scales and the lower panel corresponds to the interaction time scales. We assume the radius where internal shock taking place is $R\simeq 1.2\times {10}^{15}\mathrm{cm}$ and the break energy of prompt emission is ${ϵ}_b=500\mathrm{eV}$ in the jet comving frame.

Figure 8

The GRB luminosity function used in this work. The parameters for LL GRBs are $L_{\min }^{\mathrm{LL}}={10}^{46}\mathrm{erg}{\mathrm{s}}^{-1}$, $L_{\max }^{\mathrm{LL}}={10}^{49}\mathrm{erg}{\mathrm{s}}^{-1}$, $L_b^{\mathrm{LL}}={10}^{47}\mathrm{erg}{\mathrm{s}}^{-1}$, ${\alpha }_1^{\mathrm{LL}}=0.0$, ${\alpha }_2^{\mathrm{HL}}=3.5$, and the parameters for HL GRBs are $L_{\min }^{\mathrm{HL}}={10}^{49}\mathrm{erg}{\mathrm{s}}^{-1}$, $L_{\max }^{\mathrm{HL}}={10}^{54}\mathrm{erg}{\mathrm{s}}^{-1}$, $L_b^{\mathrm{HL}}=10^{52.35}\mathrm{erg}{\mathrm{s}}^{-1}$, ${\alpha }_1^{\mathrm{HL}}=0.65$, ${\alpha }_2^{\mathrm{HL}}=2.3$ [39]. Schematically we indicate that the UHECRs in LL GRBs have a composition dominated by nuclei, while HL GRBs are likely to have a proton-rich or mixed composition.

Figure 9

The UHECR nuclei spectrum and distribution of $⟨X_{\max }⟩$ and $\sigma (X_{\max })$ calculated from model Si-F 1. The blue data points are taken from Auger [10], and we also show the magenta data points measured by TA for comparison [11]. The maximum acceleration energy is $Z{E}_{p,\max }^{\prime }=10^{18.2}Z{L_{\gamma \mathrm{iso},47}}^{1/2}\mathrm{eV}$ and ${\delta }_E=-0.14$.

Figure 10

Same as Fig. 9 but for model Si-F 2. The maximum acceleration energy is $Z{E}_{p,\max }^{\prime }=10^{18.2}Z{L_{\gamma \mathrm{iso},47}}^{1/2}\mathrm{eV}$ and ${\delta }_E=-0.14$.

Figure 11

Same as Fig. 9 but for model Si-R 1. The maximum acceleration energy is $Z{E}_{p,\max }^{\prime }=10^{18.2}Z{L_{\gamma \mathrm{iso},47}}^{1/2}\mathrm{eV}$ and ${\delta }_E=0.14$.

Figure 12

Same as Fig. 9 but for model Si-R 2. The maximum acceleration energy is $Z{E}_{p,\max }^{\prime }=10^{18.2}Z{L_{\gamma \mathrm{iso},47}}^{1/2}\mathrm{eV}$ and ${\delta }_E=0.14$.

Figure 13

Same as Fig. 9 but for model Si-R 3. The maximum acceleration energy is $Z{E}_{p,\max }^{\prime }=10^{18.2}Z{L_{\gamma \mathrm{iso},47}}^{1/2}\mathrm{eV}$ and ${\delta }_E=0.14$.

Figure 14

Same as Fig. 9 but for hypernova model. The maximum acceleration energy is $Z{E}_{p,\max }^{\prime }=10^{18.2}Z{L_{\gamma \mathrm{iso},47}}^{1/2}\mathrm{eV}$ and ${\delta }_E=0.14$.

Figure 15

The UHECR nuclei spectrum and distribution of $⟨X_{\max }⟩$ and $\sigma (X_{\max })$ calculated from the model Si-R 2 with a power-law injection spectrum. The maximum acceleration energy is $Z{E}_{p,\max }^{\prime }=10^{18.7}Z{L_{\gamma \mathrm{iso},47}}^{1/2}\mathrm{eV}$; spectral index is $s_{\mathrm{esc}}=0.5$, and ${\delta }_E=0.06$.

Figure 16

Same as Fig. 12 but with the HL GRB contribution added. The maximum acceleration energy is $Z{E}_{p,\max }^{\prime }=10^{18.2}Z{L_{\gamma \mathrm{iso},47}}^{1/2}\mathrm{eV}$ and ${\delta }_E=0.14$.