To power the X-ray plateaus of gamma-ray bursts through larger amplitude electromagnetic waves

The origin of gamma-ray burst (GRB) X-ray plateau, especially the internal plateau is still unclear, but could be related to GRB's central engine of magnetar. It is generally believed that the spin-down power of the magnetar is injected into forward external shock, however we propose here that most of the power will be dissipated behind the GRB jet through larger amplitude electromagnetic wave (LAEMW). The relevant physical conditions and observational implications are analyzed and discussed, and various kinds of X-ray light curves could be reproduced. Although it is still a matter of debate about the chromatic multi-band afterglow in the standard external afterglow fireball model, we can explain naturally this feature under this scenario. Furthermore, we predict that the X-ray emission of spin-down wind could possibly precede the prompt emission of GRB jet if the energy of LAEMW is dissipated first but shock-induced radiation in the jet is produced later. It is emphasized that both the GRB jet and the spin-down wind should have significant observational consequences in the magnetar scenario, and would be focused equally in GRB physics.

Generally, the plateau followed by a power-law de-dushuang@pku.edu.cn, r.x.xu@pku.edu.cn cay with an index of q <∼ −3 is called 'internal plateau' (e.g., GRB 070110, Troja et al. 2007; GRB 090515, Rowlinson et al. 2010). For clarity, in this paper, we call those plateaus with q > −3 as ordinary plateaus (e.g., GRB 060729, Evans et al. 2007). A number of models are proposed to explain both types of X-ray plateaus. Some interpretations do not depend on specific central engine but invoke structured GRB jet (Eichler & Granot 2006;Toma et al. 2006;Yamazaki 2009), jet with appropriate distribution of bulk Lorentz factor (Rees & Mészáros 1998;Uhm & Beloborodov 2007), jet with evolving microphysical parameters (Panaitescu et al. 2006;Ioka et al. 2006), jet with delayed deceleration (Granot & Kumar 2006;Shen & Matzner 2012;Duffell & MacFadyen 2015), the exchange of jet and circumburst medium (Kobayashi & Zhang 2007;Shao & Dai 2007) and jet viewed slightly off-axis (Beniamini et al. 2019). Those models may account for the ordinary plateau but are usually incapable to confront the internal plateau due to the existence of steep decay. The leading model to explain both the ordinary plateau and internal plateau suggests a long-lasting central engine. Without considering the interaction be-tween the assumed long-lasting magnetized wind and GRB jet, Lyutikov & Jaramillo (2017) proposed that the ordinary X-ray plateau can be produced through the interaction of the wind with the shocked circumburst medium 1 . To account for the internal plateau, we reiterate that energy dissipation of the continuous outflow should trace the energy injection. Let's have a briefing.
Under the long-lasting BH-disc central engine scenario (Kumar et al. 2008), the different segments of outflow, as well as the light curve, are related to the accretion of different parts of the GRB progenitor star. But, this model can only be applied to collapser. Besides, the accretion should be inhomogeneous (as shown in rapidly changing light curves of GRB prompt emission), the"late prompt" emission in the long-lasting outflow is inevitable (but this emission is assumed to be smoothly by Ghisellini et al. (2007) and ignored by Beniamini & Mochkovitch (2017)), such that a smooth plateau (e.g., GRB 070110, Troja et al. 2007) can hardly appear.
Under the magnetar scenario, on one hand, the ordinary plateau would be powered by the spin-down wind of stable magnetar with a certain braking index. On the other hand, the gravitational collapse of unstable magnetar due to spin down could account for the steep decay of internal plateau phenomenologically . However, it is generally believed that the energy of spin-down wind will be directly injected into the forward external shock (e.g., Dai & Lu 1998b;Zhang & Mészáros 2001), so that the decay follows the internal plateau should be a normal decay (decay index ∼ −1.2, Zhang et al. 2006) instead of steep decay after the magnetar collapsing. Some authors suggest that there may be some unknown internal dissipations in the magnetized winds (Fan, & Xu 2006;Yu et al. 2010).
Given the natural correspondence between the collapse of magnetar and the steep decay of internal plateau, we prefer to the magnetar scenario. In Section 2, we introduce the applicability of GRB NS/magnetar model, and propose that the GRB jet and the spin-down wind might evolve separately. In Section 3, we discuss some implications of this proposal. Section 4 is discussion and summary. Throughout this paper, parameters without primes are in the lab frame, and parameters with primes are in the rest frame co-moving with the spin-down wind.
2. WHY IS NS/MAGNETAR NEEDED? 1 We note here that the interaction between the magnetized wind and GRB jet must exist, since the Lorentz factor of magnetized wind is assumed to be ∼ 10 6 which is much larger than that of GRB jet.
GRBs can be classified into two categories based on duration T 90 (Kouveliotou et al. 1993): short GRBs with T 90 < 2 s and long GRBs with T 90 > 2 s. Observations confirm that short GRBs at least originate from double NS mergers (GW 170817/GRB170817A association, Abbott et al. 2017), and long GRBs originate from massive star collapses (e.g., GRB/type-Ic supernova associations, Galama et al. 1998;Hjorth et al. 2003;Stanek et al. 2003).
For an NS-NS merger, the merger remnant may be an NS or a BH, which depends on the total mass of the binary and the equation of state of NS. A precise measurement shows the upper limit on the rest mass of NS should satisfy M ToV > 1.97 M ⊙ (Antoniadis et al. 2013). GW 170817/GRB 170817A indicates M TOV may be ∼ 2.2 M ⊙ (Margalit & Metzger 2017, alternatively, see Cromartie et al. 2019) or even more larger (Yu et al. 2018). In this sense, the maximum mass that a rotating NS can support will be ∼ 2.6 M ⊙ or even more. Therefore, if the total mass distribution of extragalactic double NS systems is similar to that of in the Milk Way, i.e., M tot ∈ (2.5 M ⊙ , 2.9 M ⊙ ) (Lazarus et al. 2016;Stovall et al. 2018; seeÖzel & Freire 2016 for review), the remnant of NS-NS merger can be a supramassive/hypermassive NS (even a stable NS). For massive star collapse, since an NS can be born in a supernova explosion, the centre of the long GRB associated with the supernova can also be an NS (stable, supramassive, hypermassive). Therefore, NS/magnetar scenario is at least reasonable for some of GRBs.
To avoid the difficulty of mganetar scenario mentioned in the introduction and make mganetar scenario remain available to internal plateaus, we suggest an improved scenario that most of the spin-down power of magnetar is usually not injected directly into the forward external shock but continuously dissipated behind the GRB jet independently (mainly through large amplitude electromagnetic waves, LAEMWs, see below). Notably, once LAEMWs can keep the energy dissipation of the spindown wind tracing the spin-down power, which is larger than the X-ray luminosity of the GRB jet due to the external shock, the collapse of the magnetar will be naturally corresponding to the steep decay of the internal plateau. In the case of this scenario, the GRB jet and the spin-down wind evolve separately, so that both of the GRB jet and the spin-down wind should have significant observational consequences.
It is worth noting that this proposal does not depend on the type of plateaus, to be applicable for both the ordinary and the internal plateaus. Interestingly, if one believes the X-ray transient CDF-S XT2 (Xue et al. 2019) is powered by a magnetar, our proposal can also be consistent with the implication that the energy release of the magnetar spin-down wind has nothing to do with the GRB jet. In the next, we discuss some implications of our proposal. We will focus on the internal plateaus, since the discussion can be naturally generalized to the ordinary plateaus.

Large amplitude electromagnetic waves (LAEMWs)
Inspired by the observed excess in infrared radiation from Carb Nebula, Usov proposed that synchrotron radiation from electrons accelerated in the field of LAEMWs (Usov 1975, also see Usov 1994;Melatos, & Melrose 1996) can explain this observation. In this section, we apply this concept to GRB X-ray internal plateaus. Let us introduce briefly LAEMWs at first.
For a rotating NS with inclined dipole magnetic field in vacuum, the magnetic dipole radiation generated by this magnetic dipole can be solved analytically under a given accuracy. But in reality, the NS is contained in a magnetosphere filled with plasma due to unipolar induction effect (Goldreich & Lynden-Bell 1969). Therefore, this low-frequency magnetic dipole radiation will be screened at near field.
Nevertheless, the rotational energy of the NS must be extracted by the approximativly isotropic outflow (spindown wind) consisting of plasma and magnetic field frozen inside. Certainly, since there is an inclination angle between the rotation axis and the magnetic axis of the NS, the frozen magnetic field in the spin-down wind along the rotation axis (as well as the GRB jet) can be alternating-direction (i.e., striped magnetic field, see, e.g., Coronoti 1990;Usov 1999;Spruit et al. 2001). As the spin-down wind moves away from the NS and the wind density decreases, there is a moment that displacement currents can not be screened any more and an induced electric field occurs. Eventually, the striped magnetic field is transformed into low-frequency electromagnetic waves, i.e., LAEMWs, at far field.

How does the spin-down wind dissipate?
In this subsection, we discuss conditions under which the spin-down wind dissipated through LAEMWs. To result in an almost flat X-ray plateau and a steep decay, energy of the spin-down wind should be released rapidly and smoothly. Therefore, (i) the spin-down wind should be highly magnetized initially, i.e., where B is the magnetic field strength, ρ is the mass density, v is the speed of the spin-down wind, S is the cross area, c is the speed of light, L sd is the spin-down power, and Γ w is the bulk Lorentz factor of the spin-down wind; (ii) the spin-down power L sd should be balanced by the magnetic-energy dissipation rate L dis approximatively, i.e., where n ± is the number density of accelerated electrons (and positrons), P e is the synchrotron radiation power of a single electron and l is the distance that the wind travels per second 2 ; (iii) when the magnetic energy in the volume Sv is dissipated totally, the electrons should also be cooling down, i.e., the radiative lifetime of these electrons, τ , satisfies where E e is the energy of a single electron. Condition (i) makes the energy of the spin-down wind almost be released totally once the magnetic energy is completely dissipated. Conditions (ii) and (iii) guarantee that the magnetic energy can be dissipated fastly and the energy in the accelerated electrons does not accumulate, so that the corresponding light curve keeps flat. For magnetic-field-dominated relativistic wind, according to conditions (ii) and (iii), we have and where γ e is the Lorentz factor of electrons.
On the other hand, the plateau usually appears at t a ∼ 100 s after the burst, so the spin-down wind at least dissipates at r d = ct a ∼ 10 12 cm away from the source. Since magnetic field in the spin-down wind is dominated by the toroidal component at a large distance from the source but the dipole magnetic field in the light cylinder, 2 Strictly, l should be the radiative damping length of LAEMWs. However, to result in a plateau instead of a rising lightcurve during the hydrodynamic evolution time scale of the spin-down wind, ∼ r d /Γ 2 w c, with r d the radius of the dissipation region and Γ w,0 the Lorentz factor of the spin-down wind, we need l to be small enough, i.e., l/c ≤ r d /Γ 2 w,0 c, such that the energy of the LAEMWs would be dissipated near r d . Note that r d ∼ 10 12 cm (see later in this sub-section) and the order of magnitude of Γ w,0 should be ∼ 10 (see section 3.4), there is l/c ∼ 1. This argument also is consistent with the estimations of Asseo et al. (1978) and Usov (1994). Therefore, we simply consider l as the distance that the wind travels per second. the magnetic field strength at r d is (see also Usov 1994) where B dip , r lc = cP/2π and P are the surface magnetic field strength, the light cylinder radius and spin period of the magnetar, respectively. For simplicity, the radius of the magnetar R * = 10 6 cm is adopted. In equation (7), we also assume that the magnetic energy release is not important before the spin-down wind reaching r d . The reason is that instabilities which cause magnetic field dissipation need time to propagate. The typical propagating time scale is the Alfvén crossing time over the length scale that magnetic field in the outflow changes, this gives that the distance r from the source where magnetic energy release becomes important is ∼ 10 12 cm (Spruit et al. 2001). It is clear that r ∼ 10 12 cm matches r d well.
Note that the typical energy of synchrotron emission is ε = 1.7 γ ′ e 10 2 2 B d 10 7 Gs keV.
In our scenario, equations (7) and (8) show that B d ∼ 10 7 Gs and γ ′ e ∼ 10 2 are basic demands when r d ∼ 10 12 cm, and meanwhile equation (6) can be easily satisfied. According to the fast dissipation condition, i.e., equation (5), if most of electrons in the spin-down wind are accelerated, there is n ± = 7.2 × 10 9 γ ′ e 10 2 −2 . ( Now, the question is which mechanism accelerates these electrons. We have argued that the spin-down wind is continuously dissipated behind the GRB jet, so that without internal collisions (Zhang & Yan 2011) the ordered striped magnetic field in the spin-down wind may be hard to be dissipated into X-ray emission violently (see Pétri 2016 for review) 3 . As introduced in subsection 3.1, LAEMWs may be another way. The critical condition for the emergence of LAEMWs is that the 3 To convert magnetic energy of spin-down wind into kinetic energy to produce a GRB, some other mechanisms are proposed (e.g., Drenkhahn, & Spruit 2002;Lyubarsky 2010). But, the prediction of these mechanisms is not supported by the observation that GRBs are produced by relativistic jets rather than approximative isotropic spin-down winds (Mooley et al. 2018;Izzo et al. 2019). Therefore, we do not consider these mechanisms. number density, n ± , decreases to the Goldreich-Julian density (Usov 1994) n cr = 1.0 × 10 9 r d 10 12 cm Since there are n ± ∝ r −2 and n cr ∝ r −1 , comparing equations (9) and (10), one can see the dissipation of magnetic energy through LAEMWs will be important when the spin-down wind reaches ∼ r d . It is worth mentioning that before the first data point is observed, Chandra Telescope had pointed at CDF-S XT2. That's to say, the X-ray emission of CDF-S XT2 is appeared suddenly without a gradual brightening as shown in its light curve. This feature is consistent with our scenario that the spin-down wind dissipates immediately when reaches r d .
3.3. Gamma-ray photons and X-ray photons, which are first?
In the above, we have shown that the spin-down wind is dissipated at least at r d ∼ 10 12 cm, since X-ray plateau usually appears at ∼ 100 s after the burst. However, the time delay, ∆t ≈ 1.7 s, between GW 170817 and GRB 170817A indicates that the prompt emission occurs at (Rees, & Meszaros 1994;Murguia-Berthier et al. 2017) R γ = 2c∆tΓ 2 jet = 1.0 × 10 15 Γ jet 10 2 2 cm, when assuming the Lorentz factor of the relativistic jet launched from the binary NS merger satisfies Γ jet ≫ 1 initially. Accordingly, if the dissipation of the spindown wind follows the generation of gamma-ray photons, there is r d ∼ R γ . Therefore, the parameters derived through equations (6), (7) and (8) However, the condition of emerging LAEMWs is still satisfied. Because, at this time, the critical number den-sity is (15) so one still has n cr ∼ n ± .
It is worth noting that if the ∼ 1.7 s time delay between GW 170817 and GRB 170817A is caused by delayed jet launch or acceleration of jet from non-relativistic case to extreme relativity (Mochkovitch et al. 1993), there is no need to demand R γ ∼ 10 15 cm . But one can't rule out the possibility that the initial X-ray photons due to the dissipation of spin-down wind can be detected earlier than the GRB prompt emission (e.g., r d ∼ 10 12 cm, R γ ∼ 10 15 cm), as long as the initial interval, ∆L, between the spin-down wind and the jet satisfying Therefore, this feature makes our proposal have a chance to be tested.

Interaction between the spin-down wind and GRB jet
Considering there is an interval between the spin-down wind and GRB jet (due to Γ w < Γ jet initially, or the delayed spin-down wind launch induced by fall-back accretion), when the magnetic energy in the spin-down wind dissipates, part of this energy may convert to the kinetic energy of the spin-down wind. So, Γ w will increase. Energy conservation gives where σ 0 and σ d are the magnetization factors before and after the spin-down wind dissipation, Γ w,0 and Γ w,d are the Lorentz factors of the spin-down wind before and after the dissipation, and η is the efficiency with which magnetic energy is converted into kinetic energy. Since the circumburst medium is swept up by the GRB jet, we assume that Γ w,0 and Γ w,d are constants. If Γ w,d is large enough that the remnant spin-down wind can catch up with the GRB jet (Top of Figure  1), a collision will happen. Through the collision, the forward shock propagating in the GRB jet always can produce, however the emergence of reverse shock depends on the residual magnetic field in the spin-down wind. The critical condition is that the pressure of the shocked jet matter equals to the magnetic pressure of the spin-down wind (Zhang 2018). For rough dimensional analysis, we assume that the ratio of the baryon number density of the spin-down wind to the baryon number density of the jet, n w /n jet , is order of the ratio of the spin-down luminosity to the jet power. Therefore, there is where σ cri is the critical magnetized factor. Following the plateau segment, the steep decay segment should be produced by the dissipation of the residual magnetic field in the remnant spin-down wind. Through equation (1), one has where σ c is the magnetized factor of the spin-down wind at the collision, t b is the break time of the plateau, t c is the time when the spin-down wind collides with the GRB jet, and q is the decay index of the steep decay. Substituting equation (17) into equation (19), there is Considering that the isotropic energy of some GRB X-ray plateaus can be as high as ∼ 10 52 erg (e.g., GRB 170714A, Du et al. 2019), η at most ∼ 0.1 since the most energy of the spin-down wind is released in X-ray emission. On the basis of −q > 3 for the steep decay, nonnegligible mass loading (see equation 18) and equations (20) and (21), to make σ c > σ cri , the value of σ 0 must be unacceptably large. Therefore, one can expect that reverse shock always can develop when the spin-down wind catches up with the GRB jet. Correspondingly, similar to the internal shock model (Rees, & Meszaros 1994;Kobayashi et al. 1997), a flare-like X-ray bump can be produced (e.g., GRB 070110, Troja et al. 2007;and GRB 170714A, Hou et al. 2018). Furthermore, Γ w,d ≫ 1 and n w ≪ n jet lead the reverse shock to be relativistic, we have t p − t c ∼ t b , with t p the peak time of the X-ray bump (Top of Figure 2). So the internal energy release during the collision is (1 − α)ηL sd t b , with α the value of Γ jet /Γ w,d at t c , and the peak luminosity of the X-ray bump is ∼ (1 − α)ηL sd . It's worth reminding that if Γ w,d < Γ jet (Middle of Figure 1), such that the spin-down wind can not catch up with the jet, no X-ray bump can appear (Middle of Figure 2).
However, if the spin-down wind and GRB jet are connected together until the spin-down wind reaching r d , the acceleration of the spin-down wind can not be described by equation (17). Actually, when the GRB jet pass through r d totally, the corresponding distribution of the Lorentz factor of the remnant spin-down wind between r d and the 'tail' of the jet should be from Γ w,d to Γ jet (Bottom of Figure 1). Therefore, under this situation, the interaction between the spin-down wind and GRB jet is more likely the general energy injection (Dai & Lu 1998b;Zhang & Mészáros 2001), so that only a faint X-ray bump can arise (Bottom of Figure  2), since the remanent energy of the spin-down wind, ∼ 10 51 erg, is usually smaller than the kinetic energy of the jet, ∼ 10 52 − 10 54 erg.

The independently evolved GRB jet and spin-down wind
As discussed above, the dissipation of spin-down wind is independent of the GRB jet. Since the opening angle of the spin-down wind (approximately isotropic) is much larger than that of the GRB jet (∼ 10 • ), there is a situation that only the radiation from the spin-down wind is observed. In other words, an X-ray plateau without the corresponding GRB can be detected, such as, CDF-S XT2.
On the other hand, GRBs are transient events, however, spin-down winds form NSs will last for a long time. These spin-down winds will exert long-lasting impacts on the evolutions of pulsar wind nebulae. There is a Figure 2. Schematic diagram of the X-ray light curves powered by a supramassive magnetar. Top: there is an interval between the spin-down wind and the GRB jet, and Γ w,d > Γjet. Middle: there is still an interval between the spin-down wind and the GRB jet, but Γ w,d < Γjet. Bottom: the spin-down wind and the GRB jet are connected together.
conflict between theories and observations that the spindown wind is usually dominated by the Poynting flux with σ ≫ 1, but the analyses of Crab nebula furnish σ ≪ 1 (e.g., Begelman & Li 1994, see Pétri 2016 for review). The spin-down wind should be abruptly dissipated before reaching termination shock.
Our scenario provides a uniform explanation to these two different sources.

DISCUSSION AND SUMMARY
In the above, the acceleration of electrons during the magnetic energy release in the spin-down wind is not discussed, since this is a complicated problem in astrophysics and beyond the scope of this paper. Nonetheless, the acceleration of electrons through LAEMWs is not sensitive to σ 0 , as long as σ 0 ≫ 1, but depends on the ratio of spin-down power to the flux of electrons (Usov 1994). For a suitable Lorentz factor (e.g., γ ′ e ∼ 10 2 ), the number density of electrons should satisfy equation (9).
In this paper, we propose an improved proposal to explain the GRB internal plateaus under magnetar scenario that the magnetized spin-down wind can almost be completely dissipated through LAEMWs behind the GRB jet. Under this scenario, the interaction between the remanent spin-down wind and the jet may result in a flare-like or a faint X-ray bump following the steep decay. Especially, the acceleration mechanism of electrons in the spin-down wind under our scenario can be different with that of the standard external shock scenario, such that the multi-band afterglow light curve of these two may show different features. For example, during the plateau segment (including ordinary plateau, if the magnetar is stable), the multi-band afterglow of the former is chromatic, however the latter is more likely achromatic. This difference may be useful for distinguishing the origins of the plateaus.
It is worth noting that although we claim that the spin-down wind can be dissipated by LAEMWs, the proposal that the spin-down power is not injected into the forward external shock but continuously dissipated behind the GRB jet is compatible with other mechanisms as long as they can dissipate the spin-down wind instantaneously at a certain distance from GRB central engines. For example, if the Lorentz factor has an appropriate distribution in the spin-down wind (e.g., with a slower head and faster tail due to timeevolution mass loading), such that the spin-down wind can shrink enough during travelling, there may be a certain time that the striped magnetic field can be dis-sipated through the magnetic reconnection induced by the self-compression of the wind. We just believe that LAEMWs are a much simpler and more natural mechanism to power the X-ray internal plateau in GRB physics.
Determining the origin of the plateaus has more than just astronomical implications. It is also important for understanding the equation of state of supranuclear matter, a problem relevant to non-perturbative quantum chromo-dynamics (QCD, see, e.g., Xu 2018). Although we have entered the era of multi-messenger astronomy, the current gravitational-wave detectors can not provide effective information of the remnant of GW170817-like event. However, GRB X-ray plateaus may provide opportunities to achieve this goal, as well as the cognition of the non-perturbative QCD, as the observational behavior of plateau is strongly related to the life of a supramassive/hypermassive NS and thus to the stiffness of equation of state (i.e., M TOV ).