GRB 101225A as Orphan Dipole Radiation of a Newborn Magnetar with Precession Rotation in an Off-Axis Gamma-Ray Burst

A rapidly spinning, strongly magnetized neutron star, the so-called “millisecond magnetar”, could be served as the energy source of gamma-ray bursts (GRBs) (Usov 1992; Dai & Lu 1998; Zhang & Mészáros 2001; Metzger et al. 2011). The discovery of association among GRB, gravitational wave (GW) event, and kilonova of GRB170817A/GW 170817/AT2017gfo from the merger of neutron stars not only opens the multi-messenger era of GRB study but also renews our understanding of the GRB physics (Abbott et al. 2017a, 2017b). It was shown that GRB 170817A should be from off-axis observations to a structured jet (Margutti et al. 2017; Takahashi & Ioka 2021) and its central engine might be a magnetar (Metzger et al. 2018). A newborn magnetar loses its rotational energy via the electromagnetic (EM) dipole radiation (DR) and/or GW emission. Its EM lightcurve is featured as a steady, long-lasting emission segment with a transition to a decay phase as $t^{-\beta}$ at a characteristic timescale ($\tau_{\rm SD}$), where $\beta={-1}$ if GW emission dominates the spindown losses and $\beta={-2}$ if EM losses are dominant instead (e.g. Dai & Lu 1998; Zhang & Mészáros 2001; Lü et al. 2018). A rapid drop of the X-ray flux following the plateau is observed in some cases (Troja et al. 2007; Liang et al. 2007; Lü et al. 2014), which is regarded as evidence of a supra massive magnetar collapsing to a black hole (Troja et al. 2007). In such a jet-wind configuration, its prompt gamma-ray emission may not be able to trigger instruments or could be observed as a low-luminosity GRB, but orphan DRs or jet afterglows may be detectable for an off-axis observer. Multiple messenger observations of GRB 170817A favor such a paradigm. Xue et al. (2019) discovered such a candidate from the deep X-ray survey by the Chandra X-ray telescope. By assuming a quasi-universal Gaussian jet, Xie et al. (2020) illustrated that the observed GRBs from their jets and the X-ray emission from their DR winds powered by newborn magnetars can be reproduced via the Monte Carlo simulations.

The DR would be a probe for the evolution of a newborn magnetar. At birth of a magnetar, the spin axis $\boldsymbol{\Omega}$ is misaligned to the symmetry axis of the magnetic field B with a tiny inclination angle $\alpha$. The internal viscous dissipation will drive the magnetic symmetry axis orthogonal to the spin axis, which could cause a fast growth of $\alpha$, causing a strong GW signal (Cutler 2002; Stella et al. 2005; Dall’Osso & Stella 2007). In the early stages, the effect of the internal viscous dissipation would largely overcome the effect of the electromagnetic (EM) torques. As the magnetar spins down, the EM torques take over and dominate the $\alpha$ evolution and the magnetar precession (Goldreich 1970; Jones 1976; Dall’Osso et al. 2009; Philippov et al. 2014; Dall’Osso et al. 2015; Şaşmaz Muş et al. 2019). This is caused by its rapidly rotational and highly elliptical structure (Goldreich 1970; Spitkovsky 2006; Philippov et al. 2014; Zanazzi & Lai 2015; Şaşmaz Muş et al. 2019; Suvorov & Kokkotas 2020). In the case of GW emission dominates the spin down, $\alpha$ cannot be equal to 0. Observationally, the precession may lead to the oscillations of the observed X-ray flux from its DR (e.g. Suvorov & Kokkotas 2020).

GRB 101225A is a long GRB at redshift $z=0.847$ (Levan et al. 2014). It is interesting due to its extremely unusual gamma-ray and X-ray lightcurves. Its gamma-ray lightcurve shows a plateau up to 1700 s post the $Swift$ Burst Alert Telescope (BAT) trigger (Cummings & Sakamoto 2010). The gamma-ray plateau is followed by X-ray flares whose peak fluxes decay as $F_{X}\propto t^{-1}$ (Lü et al. 2018). The X-ray emission rapid drops with a decay slope of $-6$ at $t>2.5\times 10^{4}$ s. This GRB was claimed as an ‘ultra-long’ GRB of a unique population from different progenitors (Levan et al. 2014), such as a tidal disruption event of a minor body around an isolated Galactic neutron star (Campana et al. 2011), a merger of a helium star with a neutron star (Thöne et al. 2011). Lü et al. (2018) proposed that GRB 101225A is driven by the DR of a newborn magnetar when its rotation energy is lost via the GW emission before its collapses to a black hole (see also Zou et al. 2019). The temporal feature of its optical afterglows is completely different from the X-ray emission. It presents a plateau up to $\sim 2\times 10^{5}$ s, then decays as $t^{-1.3}$ (Campana et al. 2011).

We revisit the nature of GRB 101225A motivated by the discovery of off-axis GRB 170817A and CDF-S XT2. We propose that this GRB is a typical GRB powered by a newborn magnetar viewing off-axis to the GRB jet, being similar to GRB 170817A. In Section 2, we model the orphan optical afterglow lightcurve and estimate the viewing angle with respect to the jet axis in the structured jet scenario. In Section 3, we investigate X-ray flares as signature of the magnetar precession in GRB 101225A. Our discussions and conclusions are presented in Section 4. Throughout, a concordance cosmology with parameters $H_{0}=70\text{ km s}^{-1}\text{ Mpc}^{-1}$, $\Omega_{M}=0.3$, and $\Omega_{\Lambda}=0.7$ is adopted.

Fig.1 shows the multiple lightcurves of GRB 101225A. One can observe that the X-ray and optical lighcurves are completely different. The highly variable X-ray flares with a sharp drop at $2.5\times 10^{4}$ s, which indicates that the X-ray emission is from an internal energy dissipation process. The optical flux keeps almost constant, then decay as $t^{-1.3}$ at $t>2\times 10^{5}$ s. We propose that the optical emission results from off-axis observation of a structure jet. We model the optical data with the standard external shock model by parameterizing the jet structure as a Gaussian jet (Ren et al. 2020), i.e.

where $E_{\rm k,iso,c}$ is the kinetic isotropic energy ($E_{\rm k,iso}$) observed within the characteristic half opening angle $\theta_{c}$. The medium profile surrounding the jet is taken as an inter-stellar medium (ISM). The physical origin of the last X-ray data, $F_{\rm X}=2.34\times 10^{-13}$ erg cm^-2 s^-1 at $t=5.6\times 10^{4}$ s, is uncertain. We take it as an upper limit of the X-ray afterglow. We employ the Markov Chain Monte Carlo technique to evaluate the likelihoods of the model parameters. Our best fit to the data is shown in Fig.1. It is found that the optical data can be well represented with the following parameters, $\log E_{\rm k,iso,c}/{\rm erg}=53.81$, the initial fireball bulk Lorentz factor $\log\Gamma_{\rm 0,j}=2.03$, the electron energy partition $\log\epsilon_{e}=-1.10$, the magnetic energy partition $\log\epsilon_{B}=-3.89$, the medium density $\log n/{\rm cm^{3}}=0.12$, $p=2.42$, $\theta_{\rm c}=1.67^{\circ}$, $\theta_{\rm j}=3.00^{\circ}$, and $\theta_{v}=3.70^{\circ}$. Note that $\theta_{\rm c}$ represents the half-opening angle of energetic jet core, and $\theta_{\rm j}$ is the outer boundary of the Gaussian structured jet. It is found that $E_{\rm k,iso,c}$ of the GRB is comparable to the bright GRBs (e.g. Zou et al. 2019), and the viewing angle is slightly larger than the jet opening angle.

We have modeled the optical afterglows of GRB 101225A with the external-forward shock model by parameterizing the jet energy structure as a Gaussian function of $E_{\rm k,iso}(\theta)/10^{52}{\rm erg}=64\exp[-(\theta/1.67^{\rm o})^{2}/2]$ with an initial Lorentz factor of 120 and a half opening angle of $3^{\rm o}$. The derived viewing angle is $\theta_{v}=3.7^{\rm o}$. The geometrically-corrected jet energy is $\sim 3\times 10^{50}$ ergs, being comparable to that of GRB 170817A. This result indicates that GRB 101225A should be an off-axis event. The global gamma-ray/X-ray lightcurve is featured as an internal plateau with transition to a decaying phase as $t^{-1.1}$ at $t=2\times 10^{3}$ seconds, being consistent with the DR of a newborn magnetar by losing its spin energy via GW emission. Analysis to the X-ray flares observed in the time interval of $[4900,\ 7500]$ s post the BAT trigger yields QPO signatures of $P=488$ seconds and $P=250\sim 300$ seconds with a confidence level of 90%. We find that the lightcurve of GRB 101225A and its QPO signature can be represented with the DR model of a magnetar by considering its precession motion and the Doppler boosting effect of its DR ejecta, assuming that the magnetar spindown is dominated by the GW emission and the DR ejecta is co-axial with the GRB jet axis. Adopting an efficiency of 0.05 for the DR wind, the bulk Lorentz factor of the DR wind is as constrained as $\gamma=8$, which is much lower than that of the GRB jet. The derived parameters of the magnetar are $B_{\rm p}=7.9\times 10^{13}$ Gauss, $P_{\rm s,0}=0.98$ ms, $\epsilon=6.3\times 10^{-6}$, and $w=0.4$.

GRB 170817A is believed to be an extremely off-axis GRB associated with GW 170817 (Takahashi & Ioka 2021). We compare its X-ray afterglow lightcurve with GRB 101225A in Fig.1 and fit the lightcurve with a two-component jet model. Our fitting curve is also shown in Fig.1. The model parameters are $\epsilon_{e}=8.3\times 10^{-3}$, $\epsilon_{B}=3.5\times 10^{-3}$, $p=2.1$, $n=3.9\times 10^{-5}$ cm^-3, $E_{\rm k,iso}=3.8\times 10^{52}$ erg, $\Gamma_{0}=120$, $\theta_{j}=2.9^{\rm o}$ for the narrow component and $\epsilon_{e}=5\times 10^{-2}$, $\epsilon_{B}=5\times 10^{-3}$, $p=2.24$, $n=3.9\times 10^{-5}$ cm^-3, $E_{\rm k,iso}=1.6\times 10^{50}$ erg, $\Gamma_{0}=6.4$, $\theta_{j}=28.7^{\rm o}$ for the wide component. Its geometrically-corrected jet energy is $7\times 10^{49}$ erg of GRB 170817A, which is comparable to that of GRB 101225A ($\sim 3\times 10^{50}$ erg). The derived viewing angle of GRB 170817A is $\theta_{v}=15.8^{\rm o}$, which is much larger than that of GRB 101225A ($3.7^{\rm o}$). The observed longer delay of the afterglow onset of GRB 170817A than GRB 101225A should be due to its larger viewing angle. Note that the initial Lorentz factors of the narrow and wide components are similar to the GRB jet and the DR ejecta of GRB 101225A derived in this analysis. It is possible that the nature of the two component jets is a combination of a co-axial GRB jet and DR ejecta, as shown in Xie et al. (2020).

GRB 101225A like X-ray transients (XT) is of great interest. Similar events were reported in literature. Using the data from the Deep Chandra Field-South (CDF-S) survey, Xue et al. (2019) reported an X-ray transient CDF-S XT2, which is associated with a galaxy at redshift $z=0.738$. It lies on the outskirts of its star-forming host galaxy with a moderate offset from the galaxy center, favoring that the source is from the merger of two neutron stars (Xue et al. 2019; Lü et al. 2021). Its X-ray flux keeps almost a constant, then decays with a temporal slope of -2.43 as shown in Fig.1 in comparison with GRB 101225A. The decay slope post the plateau is much steeper than that observed in GRB 101225A ($-1.1$), and even steeper than the prediction in the case that the EM emission dominates the spindown of the magnetar ($-2$). More recently, Lin et al. (2021) reported a similar event (XRT 210423) with the Chandra X-Ray Telescope. It starts with a fast rise within a few tens of seconds, then keeps a flux of $4\times 10^{-13}$ erg s^-1 cm^-2 to $4.1\times 10^{3}$ seconds, followed by a steep decay of $t^{-3.6}$. The X-ray lightcurves of CDF-S XT2 and XRT 210423 are analog to GRB 101225A, although they are much dimmer than GRB 101225A. GRB 101225A would be among the brightest ones of this kind of fast X-ray transients. It was proposed that such a population of X-ray transients may be observed with Swift/XRT (Xie et al. 2020). They would be promising EM counterparts of GW events and the associated kilonovaes if the magnetars are born in merger of two neutron stars.

In our analysis, we model the X-ray lightcurve of GRB 101225A in the scenario that the spin down of the magnetar is due to its rotation energy lost via the GW emission. This would set a lower limit of the energy release of the GW emission ($E_{\rm GW}$) as $E_{\rm GW}\geq E_{\rm EM}$, where $E_{\rm EM}$ is the kinetic energy of the DR wind. $E_{\rm GW}$ can be calculated by integrating $L_{\rm EM}$ over the time interval before the collapse of the magnetar. Based on our above analysis, we have $E_{\rm EM}\sim 4\times 10^{50}$ erg. Adopting $\epsilon=6.3\times 10^{-6}$ and $\Omega_{0}=6408$ $\rm rad$ $\rm s^{-1}$, we set the lower limit of $I$ with the condition of $E_{\rm GW}\geq E_{\rm EM}$. We get $I>6.35\times 10^{46}$ g $\rm cm^{2}$. The total rotational energy of the magnetar then is $E_{\rm rot}>2.0\times 10^{53}$ erg.

We estimate the detection probability of its GW emission with LIGO. The strain of GW emission at frequency $f$ for a rotating neutron star at distance $D_{L}$ is given by

The optimal matched filter signal-to-noise ratio can be expressed as (Corsi & Mészáros 2009)

where $h_{c}=fh(t)\sqrt{|dt/df|}$ is the characteristic amplitude, $h_{\rm rms}=\sqrt{fS_{h}(f)}$, and $dt=\tau_{\rm sd}d\Omega^{-4}/\Omega^{-4}_{0}$. In the limit, the characteristic amplitude of GW from a rotating NS can be estimated as (e.g. Dall’Osso & Stella 2007; Corsi & Mészáros 2009; Dall’Osso et al. 2009; Dall’Osso et al. 2015; Lasky & Glampedakis 2016; Dall’Osso et al. 2018)

We calculate $h_{c}(f)$ from $f=120$ Hz to $1000$ Hz (aLIGO detector sensitive band) by adopting $I=6.35\times 10^{46}$ g cm². Our results are shown in Fig.4. One can find that the GW emission cannot be detectable even if it is at a distance of 40 Mpc with current aLIGO and could be detected by the future ET.

One caveat should be addressed is that the moment of inertia of the magnetar and rational energy of the magnetar in GRB 101225A inferred from the conditions of $E_{\rm GW}\geq E_{\rm EM}$ and pure-GW dominated spindown violates the maximum moment of inertia and spin energy consistent with any NS equation of state (EoS). The inferred $E_{\rm rot}$ is $\sim 2\times 10^{53}$ erg, which is even larger than the maximum NS spin energy given by the dynamical bar mode instability, i.e. $E_{\rm rot}\lesssim 1\times 10^{53}$ erg (Chandrasekhar 1969; Dall’Osso et al. 2018). In addition, by adopting $I=6.35\times 10^{46}$ g cm², the inferred spin-down timescale for the magnetar would be $3.33\times 10^{6}$ s, which is inconsistent with the observed one, i.e. $t_{\rm sd}=2.5\times 10^{4}$ seconds. Note that the parameters derived in this analysis are from a pure GW-driven spindown scenario. It is possible that the spindown is not purely GW-dominated and the moment of inertia may be overestimated.

We very appreciate helpful comments and suggestions from the referee. We thank Cong Yu, Hou-Jun Lü, Qin-He Yang, and Kuan Liu for useful discussion. We acknowledge the use of the public data from the Swift data archive and the UK Swift Science Data Center. This work is supported by the National Natural Science Foundation of China (Grant No.12133003 and U1731239), Guangxi Science Foundation (grant No. 2017AD22006), and Innovation Project of Guangxi Graduate Education (YCBZ2020025).