SPIN EVOLUTION OF MILLISECOND MAGNETARS WITH HYPERACCRETING FALLBACK DISKS: IMPLICATIONS FOR EARLY AFTERGLOWS OF GAMMA-RAY BURSTS

The successful launch of the Swift satellite (Gehrels et al. 2004) has opened a new era of the study of cosmological gamma-ray bursts (GRBs). In this era, there have been many important discoveries led by Swift (see Zhang 2007 and Gehrels et al. 2009 for recent reviews), one of which is the identification of a canonical X-ray afterglow light curve, described by broken power laws F_ν(t)∝t^−α (Nousek et al. 2006; Zhang et al. 2006): an initial steep decay phase with α ∼ 3 (or a steeper slope) extending to ∼10²–10³ s is followed by a shallow decay phase with α ∼ 0.5 or a flatter slope. This shallow decay phase usually lasts ∼10³–10⁴ s. A subsequent normal decay phase has α ∼ 1.2, being in agreement with the standard afterglow model. No spectral evolution across the shallow-to-normal decay break is observed. A post-jet-break decay phase with α ∼ 2, predicted by the jet model, is sporadically observed following the normal decay phase. Besides these four phases, one or multiple X-ray flares also appear in nearly one-half of GRB early afterglows. The observed X-ray flares typically have very steep rising and decaying slopes (Burrows et al. 2005; Falcone et al. 2007).

These observations suggest that the GRB central engine may be in a long-lasting activity for two reasons. On one hand, the rapid rising and decaying timescales and their distributions of X-ray flares require that the central engine restarts at a later time (Lazzati & Perna 2007). This conclusion can also be drawn from the fact that the peak time of the X-ray flare observed by Swift is nearly equal to the ejection time of the outflow from the central engine, by assuming that the decaying phase of an X-ray flare is due to the high-latitude emission from a relativistic outflow (Liang et al. 2006).

On the other hand, the shallow decay phase of early afterglows is currently understood as being due to energy injection into a relativistic blast wave, assuming an injection luminosity L(t)∝t^−q (Zhang et al. 2006; Nousek et al. 2006). One natural scenario invokes a strongly magnetic millisecond pulsar, which spins down through magnetic dipole radiation (Dai & Lu 1998a, 1998b; Zhang & Mészáros 2001). An early version of this scenario is that a newborn pulsar loses its rotational energy in the form of Poynting flux. Numerical calculations based on this version by Fan & Xu (2006), Yu & Huang (2007), and Dall'Osso et al. (2011) show that it provides a satisfactory fitting to the observed shallow decay phase. Furthermore, the observed lightcurve plateaus of the early X-ray afterglows from GRBs 050319 (Huang et al. 2007), 050801 (De Pasquale et al. 2007), 060729 (Grupe et al. 2007), 070110 (Troja et al. 2007), 080913 (Greiner et al. 2009), and 090515 (Rowlinson et al. 2010) indicate that q ≃ 0, a pulsar-type energy injection. A recent, more physical version of the energy-injection scenario of a pulsar assumes that the pulsar may continuously eject an ultrarelativistic electron–positron pair wind, the interaction of which with a circumburst medium leads to a relativistic wind bubble (Dai 2004). The relativistic reverse shock emission from this bubble can fit observed shallow decays and even plateaus in some GRB afterglows (Yu & Dai 2007; Mao et al. 2010).

An alternative scenario for energy injection requires ejecta with a wide-Γ (Lorentz factor) distribution from a newborn black hole (Rees & Mészáros 1998; Sari & Mészáros 2000), in which scenario the low-Γ ejecta catch up with a blast wave when the high-Γ ejecta are decelerated. Two features of this scenario are that all the materials with a wide-Γ distribution are impulsively released from the central engine during the prompt emission phase and that the resulting reverse shock during the shallow decay phase is non-relativistic. On the contrary, the central engine activity is long-lasting in the energy-injection scenario of a pulsar and the reverse shock is ultrarelativistic. This difference has some astrophysical implications for testing the two scenarios (Dai 2004; Yu et al. 2007; Corsi & Mészáros 2009).

The comparisons of the energy-injection scenario of a pulsar with the observations (Lyons et al. 2010; Yu et al. 2010) show that the central engine of some GRBs may be a millisecond magnetar, a type of millisecond pulsar whose surface magnetic field strength exceeds the critical one. The conventional magnetic-dipole-radiation model predicts q ≃ 0 within the spin-down timescale. This naturally explains the shallow decay phase and plateaus (α ∼ 0). As analyzed by Liang et al. (2007), however, we note that in some early afterglows α is obviously smaller than zero. This requires that q < 0. In this paper we propose a new model to explain this significant brightening phenomena by considering a hyperaccreting fallback disk around a newborn millisecond magnetar. Zhang & Dai (2008, 2009) first investigated the properties of a hyperaccreting disk around a neutron star, and Zhang & Dai (2010) further studied the effects of a magnetar-strength magnetic field on the disk. The present paper focuses on the effects of a hyperaccreting fallback disk on spin evolution of a newborn millisecond magnetar.

This paper is organized as follows. Section 2 describes both the generation of a GRB within the framework of a millisecond magnetar and the magnetar–disk interactions. Section 3 studies the magnetar's spin evolution during fallback accretion analytically and numerically. We show that for typical values of the surface magnetic field strength, initial rotation period, and accretion rate, sufficient angular momentum of the accreted matter is transferred to the magnetar and spins it up. It is this spin-up that leads to a dramatic increase of the magnetic-dipole-radiation luminosity with time and thus significant brightening of an early afterglow. Section 4 fits the early afterglows of 12 GRBs in the relativistic pulsar wind bubble model originally proposed by Dai (2004) and reconsidered by Yu & Dai (2007). The final section presents a discussion and conclusions.

2.1. Generation of a GRB

2.2. Magnetar–Disk Interactions

The materials ejected during the supernova explosion must have a velocity distribution, some of which fail to achieve escape velocity and eventually fall back onto the central millisecond magnetar. The minimum free-fall time of fallback matter is denoted by t_fb. This time might be extended because the falling matter needs to overcome the resistance of a low-density neutrino-heated bubble (MacFadyen et al. 2001). If this effect is neglected, t_fb corresponds to the minimum radius around which matter starts to fall back, r_fb = (2GMt²_fb)^1/3 ≃ 1.0 × 10¹⁰(M/1.4 M_☉)^1/3(t_fb/50 s)^2/3 cm, where M is the magnetar mass. However, the matter staying around r_fb cannot be immediately accreted. This is because a relativistic wind (with luminosity of L_w ∼ 10⁴⁹–10⁵¹ erg s⁻¹) from the central magnetar may exert an outward ram pressure, which stops fallback accretion. Once L_w decreases dramatically just at t_KH, fallback accretion may be able to proceed. Therefore, we obtain the time when fallback accretion is expected to start, t₀ = max (t_fb, t_KH) ∼ 10² s. This time is similar to the one from numerical simulations of MacFadyen et al. (2001). For simplicity, we set t₀ = 100 s in the following calculations.

Following MacFadyen et al. (2001) and Zhang et al. (2008), we parameterize the fallback accretion rate

$$\begin{equation} \dot{M}=\big(\dot{M}_{\rm early}^{-1}+\dot{M}_{\rm late}^{-1}\big)^{-1}, \end{equation} \tag{ 1 }$$

where

$$\begin{equation} \dot{M}_{\rm early}=10^{-3}\eta t^{1/2}\,M_\odot \,{\rm s}^{-1}, \end{equation} \tag{ 2 }$$

and

$$\begin{equation} \dot{M}_{\rm late}=10^{-3}\eta t_1^{13/6}t^{-5/3}\,M_\odot \,{\rm s}^{-1}. \end{equation} \tag{ 3 }$$

Here η ∼ 0.01–10 is a factor that accounts for different explosion energies (smaller η corresponds to a more energetic explosion) and t₁ ∼ 200–10³ s is the time at which the mass accretion rate starts to drop (longer t₁ for smaller η). For t ≫ t₁, Equation (1) shows that the late-time fallback accretion behavior follows $\dot{M}\propto t^{-5/3}$, as suggested by Chevalier (1989). Assuming that M₀ is the initial baryonic mass of the magnetar, from Equation (1), we obtain the stellar total baryonic mass at time t,

$$\begin{equation} M_b(t)=M_0+\int _0^t\dot{M}dt. \end{equation} \tag{ 4 }$$

Because a non-negligible fraction of this mass becomes binding energy and is radiated away in the form of neutrinos (as discussed by Lattimer & Prakash 2001), the time-dependent gravitational mass of the accreting magnetar with radius R_s is

$$\begin{equation} M=M_b(t)\left[1+\frac{3}{5}\frac{{\it GM}_b(t)}{R_sc^2}\right]^{-1}. \end{equation} \tag{ 5 }$$

Since fallback matter has sufficient angular momentum, a geometrically thin hyperaccreting disk forms around a magnetar, similar to the black hole disk of Chen & Beloborodov (2007). One main difference between these two types of disk is that the black hole disk extends to the innermost stable orbit but the magnetar disk is truncated at some radius by the magnetic field. This inner termination radius is one of the most pivotal physical quantities for the magnetar–disk interactions, as it affects the flow of energy and angular momentum in the accretion process. According to the popular viewpoint (Davidson & Ostriker 1973; Illarionov & Sunyaev 1975), since it is fixed around the magnetospheric radius, the inner termination radius increases smoothly across the corotation radius as the mass accretion rate decreases. Thus, once the inner termination radius is beyond the corotation radius, matter may be ejected from the system by the super-Keplerian magnetosphere, that is, the disk is in the propeller regime. This viewpoint was recently adopted to explore the spin evolution of a newborn millisecond magnetar and propose a propeller-powered supernova as a new mechanism for supernovae (Piro & Ott 2011).

We next define three useful radii within the accretion disk. The first radius is the corotation radius at which the Keplerian angular velocity (Ω_K) is equal to the rotation angular velocity of the central magnetar (Ω_s),

$$\begin{equation} r_c=\left(\frac{{\it GM}}{\Omega _s^2}\right)^{1/3}. \end{equation} \tag{ 6 }$$

The second radius is the magnetospheric radius defined by

$$\begin{equation} r_m=\left(\frac{\mu ^4}{{\it GM}\dot{M}^2}\right)^{1/7}, \end{equation} \tag{ 7 }$$

where μ = B₀R³_s is the magnetic dipole moment of the magnetar and B₀ is the surface magnetic field. The third radius is the distance from the stellar center to the light cylinder,

$$\begin{equation} R_{\rm L}=\frac{c}{\Omega _s}. \end{equation} \tag{ 8 }$$

Within this radius, the vertical magnetic field component of the disk is assumed to have the dipolar form, B_z = μ/r³. In addition, the fastness parameter is defined as the ratio of the stellar rotation frequency to the Keplerian angular velocity at the magnetospheric radius,

$$\begin{equation} \omega =\frac{\Omega _s}{\Omega _{\rm K}(r_m)}=\left(\frac{r_m}{r_c}\right)^{3/2}. \end{equation} \tag{ 9 }$$

The accretion disk is assumed to be truncated at r = r_m. If r_m < r_c, the matter at r_m is accreted onto the magnetar along some magnetic field lines and forced to corotate with the magnetar, so that angular momentum of the accreted matter is always transferred to the magnetar. This provides a positive torque for the magnetar, $\tau _0=\dot{M}\sqrt{{\it GMr}_m}$. If r_m ⩾ r_c (viz., in the propeller phase), however, the accreted matter at r_m initially rotates at the Keplerian angular frequency and immediately at the stellar angular velocity by the "magnetic slingshot" mechanism. This propeller effect thus leads to a negative torque exerted on the magnetar, $\tau _0=\dot{M}\sqrt{{\it GMr}_m}(1-\omega)$.

For r > r_m, the differential motion between the Keplerian disk and the magnetar generates an azimuthal field component, B_ϕ. Wang (1995) derived some different expressions for B_ϕ that are dependent on the field dissipation mechanisms. Here we adopt a simple but physically plausible expression for B_ϕ as a function of radial distance (Livio & Pringle 1992; Wang 1995; Rappaport et al. 2004; Kluźniak & Rappaport 2007),

$$\begin{equation} B_\phi =B_z\left(1-\frac{\Omega _{\rm K}}{\Omega _s}\right)=\left(\frac{\mu }{r^3}\right)\left[1-\left(\frac{r_c}{r}\right)^{3/2}\right]. \end{equation} \tag{ 10 }$$

The magnetic torque exerted on the magnetar by the disk is given by

$$\begin{eqnarray} \tau _{\rm M} & = & -\int _{r_m}^{R_{\rm L}}r^2B_\phi B_zdr\nonumber \\ & = & -\int _{r_m}^{R_{\rm L}}\frac{\mu ^2}{r^4}\left[1-\left(\frac{r_c}{r}\right)^{3/2}\right]dr\nonumber \\ & = & -\frac{\mu ^2}{9}\left(\frac{3}{r_m^3}-\frac{3}{R_{\rm L}^3}-2\sqrt{\frac{r_c^3}{r_m^9}}+2\sqrt{\frac{r_c^3}{R_{\rm L}^9}}\right)\nonumber \\ & = & -\frac{\mu ^2}{9r_m^3} \left(3-3\epsilon ^2-\frac{2}{\omega }+\frac{2\epsilon }{\omega }\right), \end{eqnarray} \tag{ 11 }$$

where = (r_m/R_L)^3/2. Therefore, the net torque exerted on the magnetar by the accretion disk reads

$$\begin{equation} \tau _{\rm acc}=\tau _0+\tau _{\rm M}=n(\epsilon ,\omega)(\dot{M}\sqrt{{\it GMr}_m})=n(\epsilon ,\omega)\frac{\mu ^2}{r_m^3}, \end{equation} \tag{ 12 }$$

where n(, ω) is the dimensionless torque parameter,

$$\begin{equation} n(\epsilon ,\omega)=\left\lbrace \begin{array}{@{}ll@{}}(2-2\epsilon +6\omega +3\epsilon ^2\omega)/(9\omega), & {\rm for}\,\,\omega <1, \\ (2-2\epsilon +6\omega +3\epsilon ^2\omega -9\omega ^2)/(9\omega), & {\rm for}\,\,\omega \ge 1.\end{array} \right. \end{equation} \tag{ 13 }$$

Please note that neither τ₀ nor n(, ω) connects smoothly from ω < 1 to ω > 1 in our model. This is because for ω < 1 and ω > 1, the system is in two different phases, the accretion phase and the propeller phase. Recently, a discontinuity of the dimensionless torque parameter across ω = 1 was also noted by Tauris (2012). When r_m > R_L (viz., > 1), the magnetar–disk interactions are so weak that τ_acc = 0. In this case, the magnetar behaves as a normal pulsar, which spins down through magnetic dipole radiation.

Spin evolution of the magnetar is given by the following differential equation,

$$\begin{equation} \frac{d(I\Omega _s)}{dt}=\tau _{\rm acc}+\tau _{\rm dip}, \end{equation} \tag{ 14 }$$

where I = 0.35MR²_s is the stellar moment of inertia and τ_dip is the torque due to magnetic dipole radiation,

$$\begin{equation} \tau _{\rm dip}=-\frac{\mu ^2\Omega _s^3\sin ^2\chi }{6c^3}=-\frac{\mu ^2\sin ^2\chi }{6R_{\rm L}^3}, \end{equation} \tag{ 15 }$$

with χ being the inclination angle of the magnetic axis to the rotation axis. For moderately stiff-to-stiff equations of state for nuclear matter, the radius of a massive neutron star is nearly independent on the mass (Lattimer & Prakash 2001), so R_s is taken to be a constant in this paper.

3.1. Asymptotic Analytical Solutions

Before carrying out numerical calculations on Equation (14), we derive analytical solutions in three limiting cases. Generally, we have r_m ≪ R_L or ≪ 1, so n(, ω) is simplified as

$$\begin{equation} n(\epsilon ,\omega)\simeq \frac{2}{9\omega }+\frac{2}{3} > \frac{8}{9},\,\,\,{\rm for}\; \omega <1, \end{equation} \tag{ 16 }$$

and

$$\begin{equation} n(\epsilon ,\omega)\simeq \frac{3-(3\omega -1)^2}{9\omega } \le -\frac{1}{9},\,\,\,{\rm for}\; \omega \ge 1. \end{equation} \tag{ 17 }$$

Hence, we obtain

$$\begin{equation} \left|\frac{\tau _{\rm dip}}{\tau _{\rm acc}}\right|= \frac{\epsilon ^2{\rm sin}^2\chi }{|6n(\epsilon ,\omega)|}\ll 1, \end{equation} \tag{ 18 }$$

and τ_dip can be neglected compared to τ_acc in Equation (14). Then first, for ω ≪ 1 (viz., slow rotators), we find n(, ω) ≃ 2/(9ω) from Equation (13). After further assuming a constant moment of inertia and neglecting the $\dot{I}\Omega _s$ term in Equation (14), we have

$$\begin{equation} I\frac{d\Omega _s}{dt}\simeq \tau _{\rm acc}\propto \dot{M}^{9/7}\Omega _s^{-1}. \end{equation} \tag{ 19 }$$

Since $\dot{M}\propto t^{1/2}$ at early times (t₀ < t < t₁), Equation (19) becomes

$$\begin{equation} \Omega _s\propto t^{23/28}, \end{equation} \tag{ 20 }$$

showing that the magnetar spins up at early times.

Second, in the case of ω ∼ 1, we obtain $\Omega _s\sim \Omega _{\rm K}(r_m)\propto r_m^{-3/2}\propto \dot{M}^{3/7}$. Since $\dot{M}\propto t^{-5/3}$ at late times, we find

$$\begin{equation} \Omega _s\propto t^{-5/7}, \end{equation} \tag{ 21 }$$

which is roughly consistent with an approximative solution to Equation (14), Ω_s∝t^−3/7, if we also assume a constant moment of inertia and neglect the $\dot{I}\Omega _s$ and τ_dip terms as in the first case. This shows that the magnetar spins down.

Third, if ω ≫ 1 (viz., fast rotators), we find n(, ω) ≃ −ω. From Equation (14) together with $\dot{M}\propto t^{-5/3}$, we have

$$\begin{equation} \frac{d\Omega _s}{dt}=-K\Omega _st^{-5/7}, \end{equation} \tag{ 22 }$$

where K is the coefficient. An integration to Equation (22) leads to

$$\begin{equation} \Omega _s=\Omega _s(t_1)\exp \big[-(7K/2)\big(t^{2/7}-t_1^{2/7}\big)\big]. \end{equation} \tag{ 23 }$$

This also implies that the magnetar spins down.

3.2. Numerical Results

Numerical integrations to Equation (14) lead to spin evolution of the magnetar for different values of the model parameters (viz., surface magnetic field strength B₀, initial rotation period P₀, η, and t₁). This further provides the magnetic-dipole-radiation luminosity as a function of time,³

$$\begin{eqnarray} L_{\rm dip}&=&\frac{\mu ^2\Omega _s^4\sin ^2\chi }{6c^3}\nonumber\\ &=&9.6\times 10^{48}\,{\rm erg}\,{\rm s}^{-1}\sin ^2\chi \left(\frac{\mu }{10^{33}\,{\rm G}\,{\rm cm}^3}\right)^2\left(\frac{P}{1\,{\rm ms}}\right)^{-4}. \nonumber\\ \end{eqnarray} \tag{ 24 }$$

In our calculations, we take sin ²χ = 0.5. For this value and smaller values of sin ²χ (viz., weakly oblique rotators), the magnetar–disk interaction and dimensionless torque parameter are nearly identical to those for an aligned rotator (Wang 1997). The initial baryonic mass of a magnetar is assumed to be 1.4 M_☉ and the maximum gravitational mass is 2.5 M_☉, beyond which the magnetar may become a black hole. We consider this value of the maximum mass of a neutron star for two reasons: (1) very stiff nuclear equations of state lead to the maximum mass of ∼2.5 M_☉ (Lattimer & Prakash 2001); and more importantly, (2) detections of the mass of the black widow pulsar, PSR B1957+20, give M_PSR = (2.40 ± 0.12) M_☉ (van Kerkwijk et al. 2011). In addition, a postmerger millisecond pulsar with mass ⩾2.5 M_☉ has been suggested by Dai et al. (2006) to explain X-ray flares from a short-duration GRB. Figure 1 shows evolution of the stellar mass with time. We see that for typical values of η and t₁ the stellar mass does not exceed the maximum mass during the fallback accretion.

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We take the benchmark model parameters: P₀ = 3 ms, B₀ = 10¹⁵ G, η = 0.5, and t₁ = 400 s. Figure 2 plots the stellar rotation period as a function of time for these benchmark values. We can see (from the red line in this figure) that the magnetar first spins up and then spins down. The evolutional behaviors of the magnetar's spin at early and late times are consistent with the asymptotic analytical solutions given by Equations (20) and (21), respectively. The results for some other values of the model parameters are also shown in Figure 2. We also see that at late times the magnetar always spins down, being independent of whichever values the model parameters are taken to be. This is due to the fact that the fastness parameter ω is close to (and somewhat larger than) unity at late times. At early times, however, the longer initial rotation period (or weaker surface magnetic field strength or larger η or longer t₁), the more significant initial spin-up. Figure 3 shows the magnetic-dipole-radiation luminosity (L_dip) as a function of time. We find a dramatic increase of L_dip for typical values of the model parameters.

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We also calculate the magnetar's rotation parameter β = T/|W|, where T = IΩ²_s/2 and |W| is given by (Lattimer & Prakash 2001)

$$\begin{equation} |W|\simeq 0.6Mc^2\frac{{\it GM}/R_sc^2}{1-0.5({\it GM}/R_sc^2)}, \end{equation} \tag{ 25 }$$

which is shown in Figure 4. We see β < 0.14 for typical values of the model parameters (except for the blue line in the right-upper panel), implying that some instabilities such as dynamical bar-mode instabilities and secular instabilities can be neglected. This is because the occurrence of these instabilities requires β > 0.27 (Chandrasekhar 1969) and β > 0.14 (Lai & Shapiro 1995), respectively.

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The interaction of an ultrarelativistic wind from the millisecond magnetar with its ambient medium is in physics similar to the well-observed Crab Nebula. In order to explain the Crab Nebula, it was proposed (Rees & Gunn 1974; Kennel & Coroniti 1984; Begelman & Li 1992; Chevalier 2000) that a realistic, continuous wind from the Crab pulsar is ultrarelativistic and dominated by the kinetic energy flux of electron–positron pairs. From the viewpoint of evolution, even if this wind is initially Poynting flux-dominated, the fluctuating component of the magnetic field in the wind can be dissipated by magnetic reconnection and used to accelerate the wind to an ultrarelativistic velocity (Coroniti 1990; Michel 1994; Kirk & Skæjaasen 2003). Recently, Aharonian et al. (2012) suggested that the acceleration should take place abruptly in the narrow cylindrical zone with radius between 20R_L and 50R_L and the wind's Lorentz factor γ_w ∼ 10⁶ to fit the spectral energy distribution of the pulsed high-energy γ-ray radiation from the Crab pulsar, even though this suggestion challenges current models on wind acceleration. In the case of a GRB afterglow, therefore, if the central engine is a millisecond magnetar, we assume that the magnetar's wind with luminosity of L_w ≃ L_dip is accelerated to a Lorentz factor of γ_w ∼ 10⁶ within a cylinder of radius much less than the typical deceleration radius (∼10¹⁶–10¹⁷ cm) of a relativistic GRB fireball in an interstellar medium (ISM). Please note that this assumption relaxes the requirement of abrupt acceleration of an ultrarelativistic wind suggested by Aharonian et al. (2012), but still keeps γ_w ∼ 10⁶. A similar value of γ_w has been adopted for some pulsar wind nebulae, e.g., G0.9+0.1 (Tanaka & Takahara 2011), and required by Suzaku observations of PSR B1259−63 (Uchiyama et al. 2009). As we find in our calculations, a large value of γ_w favors the occurrence of a lightcurve plateau or brightening of an early afterglow, although an exact value of γ_w remains highly uncertain in the literature.

The interaction of an ultrarelativistic wind with its ambient medium leads to a relativistic wind bubble (Dai 2004; Yu & Dai 2007). This can be regarded as a relativistic version of the Crab Nebula. The relativistic wind bubble should include two shocks: a reverse shock that propagates into the cold wind and a forward shock that propagates into the ambient medium. Thus, there are four regions separated in the bubble by these shocks: (1) the unshocked medium, (2) the forward-shocked medium, (3) the reverse-shocked wind gas, and (4) the unshocked cold wind, where regions 2 and 3 are separated by a contact discontinuity. Dai (2004) analyzed the wind bubble's dynamics and emission features, and found a plateau of the reverse shock emission light curve. Yu & Dai (2007) and Mao et al. (2010) carried out numerical calculations and confirmed such a plateau feature for typical values of the model parameters. This feature is due to the fact that for a magnetar without any accretion the wind luminosity L_w is nearly a constant at early times less than the typical spin-down timescale. As in Sections 2 and 3, the fallback accretion spins up the magnetar, leading to an increase of the wind luminosity with time at early times, for typical values of the model parameters. It is thus expected that the reverse emission gives rise to significant brightening of an early afterglow.

Following Yu & Dai (2007), we calculate the dynamics of a relativistic wind bubble expanding in an ISM and the emission fluxes of forward and reverse shocks. As in Sari et al. (1998), we assume that p is the spectral index of the shock-accelerated electrons, and the electron and magnetic energy densities behind a shock are fractions, _e and _B, of the total energy density of the shocked matter respectively. Of course, these parameters may be different for forward and reverse shocks, as the unshocked medium and the unshocked wind may have different magnetic fields and compositions. Figure 5 shows the light curves of forward and reverse shock emissions for the benchmark values of the model parameters (i.e., P₀ = 3 ms, B₀ = 10¹⁵ G, η = 0.5, and t₁ = 400 s). From this figure, we can see significant brightening of an early afterglow, being due to a dramatic increase of the magnetar's wind luminosity with time.

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We search GRBs detected by Swift⁴ and find the early-time significant brightening of 12 afterglows. Figure 6 show fits to these early afterglows for the assumed initial rotation period, surface magnetic field, and the parameters involved in the accretion rate. The required shock parameters are shown in Table 1. This table also presents the stellar gravitational mass at 10⁶ s (i.e., M(10⁶ s)), the maximum value of the rotation parameter (β_max), and the minimum magnetospheric radius (r_{m, min}) for each of 12 GRBs. We can see M(10⁶ s) < 2.5 M_☉, β_max < 0.14, and r_{m, min} > R_s. In addition, R_L = c/Ω_s = 47.7(P/1 ms) km is much greater than R_s. These ensure that our model is self-consistent. Figure 6 together with Table 1 shows that our model can well explain the significant brightening of 12 early afterglows. This explanation requires that the fallback accretion is sufficiently strong.

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Table 1. Shock Parameters and Some Other Model Parameters for Fitting Light Curves of X-Ray Afterglows of Some GRBs

GRB Name	Eiso, 51	z	e, f	B, r	pr	M(106 s)/M☉	βmax	rm, min
								(km)
051016B	1	0.94	0.001	0.5	2.05	1.55	0.083	17.8
060109	1	1.0a	0.01	0.001	2.5	1.80	0.049	23.5
060510A	1	1.0a	0.01	0.5	2.1	2.43	0.084	17.7
061121	10	1.31	0.1	0.5	2.3	2.07	0.12	15.5
070103	0.1	1.0a	0.01	0.5	2.05	1.42	0.097	16.7
070714B	1	0.92	0.01	0.0006	2.5	1.79	0.11	17.8
080229A	10	1.0a	0.1	0.4	2.2	2.30	0.13	15.0
080310	1	2.43	0.1	0.002	2.2	1.92	0.11	17.5
091029	1	2.75	0.1	0.4	2.2	1.66	0.073	18.6
110213A	10	1.46	0.1	0.005	2.2	2.32	0.13	15.4
120118B	1	1.0a	0.01	0.0001	2.5	2.29	0.053	20.7
120404A	10	2.87	0.1	0.002	2.2	2.07	0.083	17.8

Notes. The ISM number density, the magnetic field equipartition factor in the forward shock, the electron equipartition factor in the reverse shock, the spectral power-law index of forward-shocked electrons, the initial bulk Lorentz factor of the fireball, and the bulk Lorentz factor of the relativistic wind are taken to be n = 1 cm⁻³, _{B, f} = 0.1, _{e, r} = 1 − _{B, r}, p_f = 2.2, γ₀ = 300, and γ_w = 10⁶, respectively for all GRBs (where the subscripts f and r denote forward and reverse shocks, respectively). In this table, E_{iso, 51} is the post-burst initial fireball energy in units of 10⁵¹ erg, M(10⁶ s) is the mass of the accreting magnetar at 10⁶ s, β_max is the maximum value of the magnetar's rotation parameter, and r_{m, min} is the minimum value of the magnetospheric radius. The redshifts of GRBs are taken from GCN (GRB 051016B (Soderberg et al. 2005), GRB 061121 (Bloom et al. 2006), GRB 070714B (Graham et al. 2007), GRB 080310 (Prochaska et al. 2008), GRB 091029 (Chornock et al. 2009), GRB 110213A (Milne & Cenko 2011), and GRB 120404A (Cucchiara & Tanvir 2012)), while the redshifts of GRBs with no redshift measurement are artificially taken to be 1.0 (with superscript of a).

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When the fallback accretion rate is so small that the magnetospheric radius r_m exceeds the light-cylinder radius R_L (viz., > 1), both the fallback accretion and the propeller effect stop and the torque exerted on the magnetar by the accretion disk disappears. Meanwhile, the magnetar spins down only via the magnetic-dipole-radiation mechanism, which has been shown to be able to explain the shallow decay phase or the plateau phase of early afterglows (Dai & Lu 1998a, 1998b; Zhang & Mészáros 2001; Dai 2004; Yu & Dai 2007; Mao et al. 2010). In this case, therefore, our present model turns out to be the conventional energy-injection scenario of a pulsar.

Several physical explanations of the shallow decay phase or the plateau phase of early afterglows discovered by Swift include energy injection invoking a long-lasting central engine, energy injection from ejecta with a wide-Γ distribution, two-component jets, dust scattering, varying microphysical parameters, and so on (see Zhang 2007). The two leading scenarios are based on energy injection to a relativistic blast wave. The first scenario invokes a millisecond magnetar, while the second scenario in fact requires a stellar-mass black hole. Thus, these two scenarios have different central engines. Three astrophysical implications are discussed to test them.

First, in the first scenario, the magnetic field in the reverse-shocked region of a relativistic wind bubble consists of two components: a large-scale toroidal field and a random field. If the toroidal field dominates over the random component, one would expect high polarization of an early afterglow during the shallow decay phase, as discussed by Dai (2004). This could be used to distinguish between the relativistic-wind-bubble model and the other explanations including the second scenario.

Second, the compositions of winds in the two scenarios are also different: the wind is lepton-dominated in the first scenario and baryon-dominated in the second scenario. Yu et al. (2007) studied the dynamics of winds and calculated the corresponding high-energy photon emission by considering synchrotron radiation and inverse Compton scattering of electrons. Even though in the two scenarios there is a plateau (or a bump) in high-energy light curves during the X-ray shallow decay phase, the first scenario predicts more significant high-energy gamma-ray afterglow emission than the second scenario does. This is because a considerable fraction of the injecting energy in the first scenario is shared by a relativistic long-lasting reverse shock and the reverse-shock energy is almost carried by leptons (electrons and positrons), while the energy of a non-relativistic reverse shock is mainly carried by baryons in the second scenario.

Third, if the initial period of a newborn magnetar is as small as ∼1 ms and even smaller, then the stellar rotation parameter β may be larger than 0.14 (e.g., the blue line in the right-upper panel of Figure 4). In this case, some instabilities such as dynamical bar-mode instabilities and secular instabilities could not only be motivated so that they affect the stellar spin evolution, but also the resultant gravitational waves would be detectable with the future advanced-LIGO detector for a nearby GRB. Meanwhile, the magnetar's rotational energy could be injected to a post-burst blast wave via magnetic dipole radiation, leading to the shallow decay phase of an early afterglow (Corsi & Mészáros 2009).

To summarize, it is well known that the conventional model of a pulsar predicts a nearly constant magnetic-dipole-radiation luminosity within the spin-down timescale. This provides an explanation for the shallow decay phase or the plateau phase of early afterglows. However, we note that significant brightening occurs in some early afterglows, which apparently conflicts with the conventional model. In order to explain this significant brightening phenomena, we here investigate the effect of a hyperaccreting fallback disk on the spin evolution of a newborn millisecond magnetar. We show that for typical values of the model parameters, sufficient angular momentum of the accreted matter is transferred to the magnetar and spins it up. It is this spin-up that leads to a dramatic increase of the magnetic-dipole-radiation luminosity with time and thus significant brightening of an early afterglow. Furthermore, we carry out numerical calculations and fitted well early afterglows of 12 GRBs assuming sufficiently strong fallback accretion. Furthermore, It is worth noting that if the accretion is very weak, our present model turns out to be the previously proposed energy-injection scenario of a pulsar. Therefore, our model can provide a unified explanation for the shallow decay phase, plateaus, and significant brightening of early afterglows. In addition, possible detections of high polarization, gravitational waves, and/or high-energy gamma rays during the shallow decay phase would be used to test this model in the future.

We thank the referee for useful comments and X.-D. Li and S. Rappaport for helpful discussions. This work was supported by the National Natural Science Foundation of China (grant No. 11033002).