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# Magnetar as a Central Engine for Long-Duration $$\gamma$$-ray Bursts and Superluminous Supernovae

Neutron stars typically have magnetic field strengths of $$\sim 10^{12} \rm \ G$$. However, if a neutron star’s rotation period is comparable to the convective overturn time, magnetic fields can be amplified by helical motion in a mean field dynamo. These highly magnetized neutron stars, magnetars, are born with short periods of $$\sim 1 \ \rm ms$$, which allow them to support an efficient $$\alpha-\Omega$$ dynamo, resulting in large magnetic dipole fields of $$10^{14}-10^{15} \rm \ G$$ (Duncan 1992). A magnetar of mass $$1.4 M_{\odot}$$, $$R = 10 \ \rm km$$, and $$P = 1 \ \rm ms$$ has a rotational energy of $E_{\rm rot} = I \omega^2 /2 = \dfrac{1}{5}MR^2 \left(\dfrac{2\pi}{P}\right)^2 \simeq 4 \times 10^{51} \ \rm erg$ The strong magnetic stresses and torques damped the rotation and release a large fraction of $$E_{\rm rot}$$ on the time scale of $\label{eqn:timescale} \tau \simeq 0.6 B_{15}^{-2} (P/1 \ {\rm ms})^2 \ \rm hr$ (Duncan 1992). This amount of energy is similar to that released in $$\gamma$$-ray bursts and superluminous supernovae (SLSNe), thus making magnetar an attractive candidate for the central engine that power these explosions. In the past two decades, a connection has been made between supernovae and at least some subclass of long-duration $$\gamma$$-ray bursts (LGRBs), supporting the case that they are powered by the same mechanism.

Around the same time that magnetars were proposed, a realization had emerged that GRBs could have cosmological origins. This would require a total energy of $$\sim 10^{51} \ \rm erg$$ to explain the observed flux assuming isotropic emission (e.g. Paczynski, 1991). Interestingly, the first proposal of highly magnetized neutron stars as central engines of these bursts, Usov (1992), was published only a day after Duncan et al. (1992). The proposed scenario is such that a magnetar forms from a white dwarf via accretion induced collapse (AIC). The WD magnetic field of $$\sim 10^9 \ \rm G$$ is amplified to $$10^{15} \rm \ G$$ by magnetic flux conservation. The rotational period of $$\sim 1 \ \rm ms$$ is a result of angular momentum conservation. 1 The newly formed neutron star then loses its rotational energy quickly due to electromagnetic torque, generating electric fields that accelerate particles to ultra-relativistic energies, which eventually give out $$\gamma$$-ray. The timescale of energy release due to magnetic dipole luminosity and gravitational wave emission of $$\sim 20 \ \rm s$$ for a typical magnetar is consistent with the timescale of long-duration GRBs. Near the magnetar’s surface out to the light cylinder, the optical depth to this radiation due to Compton scattering, absorption, and pair-cration is large. The radiation has to propagate out to a photosphere radius of $$\sim 10^8 \ \rm cm$$ before it is released. The typical radiated $$\gamma$$-ray energies of $$0.1-1 \ \rm MeV$$ is also consistent with those observed from GRBs (Usov 1992).

Recent works have shown that while magnetars formed via AIC may be responsible for some GRBs, magnetars formed in the core collapse of massive stars like those proposed by Duncan et al. (1992) are probably more prevalent. The modern view of the mechanism for core-collapse GRB as laid out by Metzger et al. (2011) is as following. Shortly after the core bounce, a non-relativistic wind heated by neutrino blows through the cavity carved out by the supernova (SN) shock into a bipolar jet. The relativistic jet from the newly formed magnetar follows, and emerges as a GRB prompt emission. After $$30-100 \ \rm s$$, the maximum Lorantz factor increases to $$\sigma_{0} \gg 1$$ rendering magnetic dissipation and jet acceleration ineffective. This ended the prompt GRB within the observed timescale of $$\sim 20 \ \rm s$$. After the prompt emission ended, the spin-down of the central magnetar continues to power the GRB into its X-ray plateau phase with a correlation between the plateau luminosity and duration (LT correlation). The observed correlation is given by $$\log L_{\rm X} = a + b \log T_a$$ where $$L_{\rm X}$$ is the plateau luminosity and $$T_a$$ is the rest frame plateau end time. The magnetar model predicts this correlation with $$b = 1$$ and $$a = \log (10^{52} I^{-1}_{45} P_{0,-3}^{-2})$$ which match observations (Rowlinson 2014). Metzger et al. (2011) also showed that the magnetar model is able to produce the evolution of $$\sigma_{0}$$ that matches observations with no need of fine-tuning, unlike models in which GRBs are powered by rapidly accreting BHs 2.

The requirement of collimated relativistic flows leading to bi-polar jets in GRBs limit the central engines to only magnetars having periods of $$1 \ \rm ms$$ and magnetic fields of $$\sim 10^{15} \ \rm G$$. However, the less extreme population of magnetars can still power quite fantastic cosmic fireworks. Duncan et al. (1992) noted that the spin down timescale given in \ref{eqn:timescale} is shorter than the SN shock breakout time, making SNe that create magnetars brighter than usual. These subclasses of brighter Type II SNe are indeed observed (e.g. Richardson et al., 2002). Recently, a number of rare superluminous SNe (SLSNe) are discovered, emitting the total radiation energy of $$\sim 10^{51} \ \rm erg$$ (e.g. SN2005ap, Quimby et al., 2007; SN2008es, Miller et al., 2008). The radioactive decay of $$\rm ^{56}Ni$$ alone cannot output this amount of energy. Kasen et al. (2010) showed that magnetars with $$B \sim 10^{14} \ \rm G$$ and initial period $$P_i \sim 2-20 \ \rm ms$$ can release rotational energy via magnetic dipole radiation on the spin-down timescale comparable to the effective diffusion time of the ejecta. They showed that these magnetars can enhance the peak luminosity to what given by $L_{\rm peak} \sim E_{\rm p} t_{\rm p}/t_{\rm d}^2 \sim 5 \times 10^{43} B_{14}^{-2} \kappa_{\rm es}^{-1} M_{5}^{-3/2}E_{51}^{1/2} \ \rm erg s^{-1}$ where $$\kappa_{\rm es}$$ is the electron scattering opacity, $$M_{5}$$ is the ejecta mass in the $$5 \ M_{\odot}$$ unit. This exceeds $$10^{43} \rm \ erg \ s^{-1}$$, making these events brighter than normal core-collapse Type II-P SNe. Figure 4 and 5 in Kasen et al. (2010) show the dependence of the peak luminosity and the time to peak on $$B$$ and $$P_i$$ for ejecta mass of $$5$$ and $$20 \ M_{\odot}$$ respectively.

If magnetars with different magnetic field strengths and birth periods are responsible for both LGRBs and SLSNe, at least some LGRBs should be accompanied by SNe. These associations between GRBs and SNe are indeed observed, with the first pair of events being SN1998bw and GRB 980425 (Kulkarni 1998). A number of other events have since been observed (see e.g. Woosley et al., 2006 for a review). Metzger et al. (2015) showed that magnetars in the transition region in the $$B-P_i$$ plane with $$P_i \sim 2 \ \rm ms$$ and the spin-down timescale of $$\sim 10^4 \ \rm s$$ can explain GRB 111209 and SN2011kl pair and the SLSN ASASSN-15lh, the most luminous SN ever observed. This association between GRBs and SNe is another evidence that these two classes of events share magnetar as their central engine. As modern transients surveys in all wavelengths continue to discover various types of peculiar events, we will have more data to constrain theoretical models, leading to a better understanding of this messy ending chapter of the stellar evolution.

1. Duncan et al. (1992) also briefly mentioned this scenario as a possible explanation for cosmological GRBs.

2. A lot can be said about the rivalry between these two competing models for the central engine of GRBs. Metzger et al. (2011) and references therein provide some introduction to both.

### References

1. Robert C. Duncan, Christopher Thompson. Formation of very strongly magnetized neutron stars - Implications for gamma-ray bursts. ApJ 392, L9 (1992). Link

2. B. Paczynski. Cosmological gamma-ray bursts. 41, 257-267 (1991).

3. V. V. Usov. Millisecond pulsars with extremely strong magnetic fields as a cosmological source of $$\upgamma$$-ray bursts. Nature 357, 472–474 (1992). Link

4. B. D. Metzger, D. Giannios, T. A. Thompson, N. Bucciantini, E. Quataert. The protomagnetar model for gamma-ray bursts. Monthly Notices of the Royal Astronomical Society 413, 2031–2056 (2011). Link

5. A. Rowlinson, B. P. Gompertz, M. Dainotti, P. T. O, R. A. M. J. Wijers, A. J. van der Horst. Constraining properties of GRB magnetar central engines using the observed plateau luminosity and duration correlation. Monthly Notices of the Royal Astronomical Society 443, 1779–1787 (2014). Link

6. Dean Richardson, David Branch, Darrin Casebeer, Jennifer Millard, R. C. Thomas, E. Baron. A Comparative Study of the Absolute Magnitude Distributions of Supernovae. The Astronomical Journal 123, 745 (2002). Link

7. Robert M. Quimby, Greg Aldering, J. Craig Wheeler, Peter Höflich, Carl W. Akerlof, Eli S. Rykoff. SN 2005ap: A Most Brilliant Explosion. ApJ 668, L99–L102 (2007). Link

8. A. A. Miller, R. Chornock, D. A. Perley, M. Ganeshalingam, W. Li, N. R. Butler, J. S. Bloom, N. Smith, M. Modjaz, D. Poznanski, A. V. Filippenko, C. V. Griffith, J. H. Shiode, J. M. Silverman. THE EXCEPTIONALLY LUMINOUS TYPE II-LINEAR SUPERNOVA 2008es. ApJ 690, 1303–1312 (2008). Link

9. Daniel Kasen, Lars Bildsten. SUPERNOVA LIGHT CURVES POWERED BY YOUNG MAGNETARS. ApJ 717, 245–249 (2010). Link

10. S. R. Kulkarni, D. A. Frail, M. H. Wieringa, R. D. Ekers, E. M. Sadler, R. M. Wark, J. L. Higdon, E. S. Phinney, J. S. Bloom. The gamma-ray burst of 980425 and its association with the extraordinary radio emission from a most unusual supernova. ArXiv Astrophysics e-prints (1998).

11. S.E. Woosley, J.S. Bloom. The SupernovaGamma-Ray Burst Connection. Annual Review of Astronomy and Astrophysics 44, 507–556 (2006). Link

12. Brian D. Metzger, Ben Margalit, Daniel Kasen, Eliot Quataert. The diversity of transients from magnetar birth in core collapse supernovae. Mon. Not. R. Astron. Soc. 454, 3311–3316 (2015). Link