An X-Ray Burst Associated with FRB 200428 from a Magnetar-asteroid Impact

Ze-Nan Liu ,Yuan-Pei Yang ,Wei-Yang Wang ,Zhao Joseph Zhang ,Qiao-Chu Li ,Ken Chen ,Jia Ren,Bin-Bin Zhang,Cheng-Kui Li,Shuang-Nan Zhang4,,and Zi-Gao Dai

1 School of Astronomy and Space Science,Nanjing University,Nanjing 210023,China

2 Key Laboratory of Modern Astronomy and Astrophysics (Nanjing University),Ministry of Education,Nanjing 210023,China

3 South-Western Institute for Astronomy Research,Yunnan University,Kunming 650500,China

4 National Astronomical Observatories,Chinese Academy of Sciences,Beijing 100101,China

5 University of Chinese Academy of Sciences,Chinese Academy of Sciences,Beijing 100049,China

6 Key Laboratory of Particle Astrophysics,Institute of High Energy Physics,Chinese Academy of Sciences,Beijing 100049,China

7 Department of Astronomy,School of Physical Sciences,University of Science and Technology of China,Hefei 230026,China;daizg@ustc.edu.cn

Abstract Fast radio bursts(FRBs)are extragalactic radio transients with millisecond duration and brightness temperature.An FRB-associated X-ray burst (XRB) was recently found to arise from the Galactic magnetar SGR J1935+2154.Following the model of Dai,in which an FRB may originate from a magnetar encountering an asteroid,we focus on explaining the spectrum of the XRB associated with FRB 200428 from SGR J1935+2154.Collisions between asteroidal fragments and the magnetar surface produce a fireball,which further expands relativistically.Due to the velocity difference among some shells in the fireball,internal shocks would form far away from the magnetar,and further emit X-ray emission.We propose that the FRB-associated XRB can be produced by synchrotron emission from the internal shocks,and then constrain the physical parameters by the observed XRB spectrum.

Key words: stars: magnetars– minor planets– asteroids: general– radiation mechanisms: general

1.Introduction

Fast radio bursts (FRBs) are a kind of extragalactic radio transient with millisecond duration(Lorimer et al.2007;Keane et al.2012;Thornton et al.2013).So far,many physical models have been proposed to explain FRBs (see details Platts et al.2019;Xiao et al.2020;Zhang 2020;Lyubarsky 2021),including close-in models and far-away models based on the distance of the radiation region from the neutron star.For example,the close-in models show that the radio emission comes from the magnetosphere of a neutron star (Cordes &Wasserman 2016;Kumar et al.2017;Zhang 2017;Ghisellini&Locatelli 2018;Katz 2018;Yang&Zhang 2018;Lu&Kumar 2019;Wang et al.2019;Wadiasingh &Timokhin 2019;Dai 2020;Geng et al.2020;Lu et al.2020;Wadiasingh et al.2020;Wang et al.2020;Yang et al.2020;Zhang 2020;Yang&Zhang 2021),which is supported by the observed light curve variability timescales and the spectro-temporal correlations(Beniamini&Kumar 2020).The far-away models suggest that an outflow interacts with the ambient medium to release energy,and radio emission is generated by synchrotron maser emission(Lyubarsky 2014;Beloborodov 2017;Ghisellini 2017;Waxman 2017;Gruzinov &Waxman 2019;Metzger et al.2019;Beloborodov 2020;Margalit et al.2020;Wu et al.2020;Xiao&Dai 2020).

Very recently,Bochenek et al.(2020) and CHIME/FRB Collaboration et al.(2020b) reported a short radio burst,FRB 200428,with two pulses from the general direction of the Galactic magnetar SGR J1935+2154.The reported FRB 200428 has two pulses with intrinsic durations of 0.60 ms and 0.34 ms,respectively(CHIME/FRB Collaboration et al.2020b;Bochenek et al.2020),and the two pulses are separated by ～29 ms.Moreover,the intriguing X-ray burst(XRB)associated with FRB 200428,which has two corresponding pulses,was coincidentally observed by high-energy satellites such as Insight-HXMT (Li et al.2021),INTEGRAL (Mereghetti et al.2020),Konus-Wind (Ridnaia et al.2021) and AGILE (Tavani et al.2021).Remarkably,since Insight-HXMT has the wider energy band observation,the extended low energy emission was observed with Insight-HXMT in the 1–250 keV energy band,and the XRB spectrum is dominated by the non-thermal component (Li et al.2021).Therefore,in the following discussion,we constrain the model parameters using Insight-HXMT data (see Section 3).Noticeably,the spectrum of the XRB is consistent with other magnetar bursts,but the cutoff energy in this event is significantly higher than the typical burst energy(Ridnaia et al.2021;Younes et al.2021).It is suggested that the associated event might be special compared with normal XRBs.Some works considered a trapped fireball and a relativistic outflow to explain the hard XRB (Ioka 2020;Yamasaki et al.2022).Subsequently,there were over 217 bursts observed during the first 1120 s from SGR J1935+2154(Younes et al.2020),but their characteristics were different from those of the XRB associated with FRB 200428,which exhibited a higher peak energy and a lower flux density at peak energy (Ridnaia et al.2021;Younes et al.2021).

Remarkably,some recent works involved the interaction between neutron stars and asteroids in explaining FRBs (Dai 2020;Geng et al.2020;Decoene et al.2021).Following Dai(2020),in this work,we propose that the XRB associated with FRB 200428 is produced by synchrotron emission from internal shocks formed by collisions between fast and slow shells produced by a relativistically expanding fireball,which can constrain the model parameters using the observed XRB spectrum.The magnetar-asteroid impact satisfies conservation of energy.The gravitational energy of the asteroid and the magnetic energy of the magnetar’s magnetosphere are converted into the kinetic energy of ejected shells.The ejected shell velocity would have some differences after impacting the magnetar’s surface due to the components of multipolar fields at the magnetar’s surface.In our model,a freely-falling asteroid with a mass ofma～1020g is impeded by an ultra-strong magnetic field at Alfvén radiusRA,and then some major fragments with a mass ofm～1019g move along the magnetic field lines and generate pulses from FRB 200428 and its associated XRB(Dai 2020).This work is organized as follows.We describe the physical process of an expanding fireball generated by the magnetar-asteroid impact in Section 2.1.We analyze the internal shock model to account for the synchrotron emission spectrum in Section 2.2.The best fitting results are shown in Section 3.The results are discussed and summarized in Section 4.The conventionQx=Q/10xin cgs units is used throughout this paper.

2.An Expanding Fireball and X-Ray Emission from Internal Shocks

The impact of an asteroid/comet with a neutron star was proposed early as an origin of gamma-ray bursts (GRBs,Colgate&Petschek 1981;Howard et al.1981;van Buren 1981;Mitrofanov &Sagdeev 1990;Shull &Stern 1995) and soft gamma repeaters(SGRs,Livio&Taam 1987;Boer et al.1989;Katz et al.1994).In our model,when an asteroid collides with the polar surface of a magnetar,a conical hot plasma(fireball)is ejected,which is different from the fan-shaped outflow discussed by Colgate &Petschek (1981).

2.1.An Expanding Fireball

We consider an asteroid with a mass ofma～1020g to freely fall in the gravitational field of a magnetar,which is disrupted tidally into some fragments at ～1010cm from the magnetar’s center.Some major fragments are distorted tidally at breakup radius (Colgate &Petschek 1981)

where ρ0,r0,sandMare the iron-nickel asteroid’s original mass density,radius,tensile strength and stellar mass,respectively.When an asteroid falls toward the magnetar’s surface,electrons are torn off the asteroidal surface by an induced parallel electric field.These accelerated electrons move along magnetic field lines,and emit an FRB by coherent curvature radiation (Dai et al.2016).Subsequently,some fragments are accreted onto the footpoints of closed magnetic field lines (see Figure 1(a)).A fireball with electrons and baryons is generated during the collision.The fireball expands relativistically due to an extremely high initial temperature,and blackbody emission is generated from the expanding fireball at the photosphere radius(see Figure 1(b)).Synchrotron emission from collisions between different shells in the relativistic fireball produces an XRB (see Figure 1(c)).

Figure 1.Schematic collision geometry between an elongated,compressed asteroid and a magnetar.(a)Some fragments(we just show one fragment in our picture) freely fall in the gravitational field of the magnetar until the Alfvén radius RA,and then are accreted onto the footpoints (orange stars) of closed magnetic field lines.The three purple parts illustrate the falling process of a fragment.(b) A fireball is generated by the collision,which then expands relativistically.Blackbody emission is generated from a photosphere.(c)Synchrotron emission from the collisions between different shells produces an XRB.The different fragments would contribute to the different shells.

Figure 2.The spectrum of the XRB associated with FRB 200428 from SGR J1935+2154.We adopt the data observed with Insight-HXMT(Li et al.2021).(Top)The observed photon count spectrum over-plotted with the best-fit model and residuals.(Bottom) The panel shows the de-convolved photon spectrum,and the data points represent the “observed” photon flux which is obtained by de-convolving the observed count spectrum using instrument responses.The synchrotron radiation model can fit the spectrum corresponding to the slow cooling case plus a blackbody component.The solid green,red and brown dashed lines signify the total,blackbody emission and synchrotron emission,respectively.The blue and purple solid lines represent the synchrotron selfabsorption frequency and cooling frequency,respectively.The residuals are defined as (data-model)/error.

When an asteroid is slowed down around the Alfvén radiusRAby an ultra-strong magnetic field of the magnetar,some major fragments with mass ～1019g move along magnetic field lines fromRAonto the footpoints of closed magnetic field lines(Dai 2020).For a dipole magnetic field configuration,the radial direction positions (R*,θ1) and (R*,θ2) signify that the boundary of a fragment corresponds to the surface of the magnetar,whereR* is the surface radius of the magnetar.We have

whereRmax?RA,Rmaxis the maximum distance of magnetic field lines across the equator andlais the length of the compressed asteroid.At the Alfvén radius,the kinetic energy density of the fragment is equal to the magnetic energy density,and one has(Dai 2020)

Combining Equations (2),(3) and (4),the initial size of the trapped fireball ?0,Xcan be expressed as

Under the assumption of thermal equilibrium,the temperature of the fireballT0can be estimated by Thompson&Duncan(1995)

wherea=7.56×10?15erg cm?3K?4corresponds to the radiation energy density constant.SincekT0is greater than the electron rest energy,the huge optical depth due to photonphoton annihilation reactions in the fireball (see Dai 2020)inevitably generates a dense population of electron and positron pairs and further makes the fireball highly collisional and opaque.The magnetic field on the surface of a magnetar will be higher due to the components of multipolar fields.The higher magnetic field will increase the initial temperature to contribute to blackbody emission from the hot spot that will produce a fireball.Meanwhile,the collisions between different relativistic shells in the fireball can produce non-thermal emission components,which will be discussed in Section 2.2.

After generating a fireball by the collisions,some baryons might be contaminated,similar to the SGR 1806-20 giant flare in 2004 (Granot et al.2006).According to the conservation of baryon number and energy,we can get the magnitude of bulk Lorentz factor (i.e.,the fireball can be accelerated to the maximum Lorentz factor),which is limited by the total entropy per baryon in the fireball as (Yamasaki et al.2022)Γi=1+(kT0/mec2)?10,wherekis the Boltzmann constant.

Because the fireball pressure is comparable to the magnetic field pressure,the fireball can expand outside along the magnetic field lines.IfR>R*,the expanding fireball size ?Xevolves as?X=?0,X(R*)32in the magnetosphere of a magnetar (Thompson &Duncan 1995).The Lorentz factor evolves as Γ ∝?X,and the temperature of the fireball evolves asT∝during adiabatic expansion (Thompson &Duncan 2001).The bulk Lorentz factor and comoving temperature of the expanding fireball evolve as

We find γshellis comparable to Γi,indicating that our derivation is self-consistent.As shown in Equation (9),the mass range of each shell will be affected by the Lorentz factor of the ejected shell.If the shell mass is too large,it is hardly accelerated to relativistic velocity.If the shell mass is too small,the Lorentz factor of the ejected shell will be larger than the maximum Lorentz factor.It is not consistent with Equation(6).Thus,the shell mass should be limited by the maximum Lorentz factor.Asteroids would fall randomly into different positions on the magnetar surface.Even though the relative collision location of different fragments is random,the fireball will expand along the open magnetic field lines,resulting in a consistent velocity direction for different fireballs.However,some differences in the magnetic field would exist due to the components of multipolar fields from the magnetar surface.The ejected shell velocity would have some differences.Therefore,internal shocks can be generated by the collisions between fireballs.

When the fireball moves to the photosphere radius,the Thomson optical depth can be given by,whereZis the atomic number,σTis the Thomson cross-section anddenotes the comoving baryon number density.For a high-load fireball,electrons and baryons dominate the number density.According to conservation of the electron number density,the radial evolution of the electron number density whenR>Rccan be given by Yamasaki et al.(2022)

whereAis the mass number.The Thomson optical depth becomes thin where the fireball expands to the size of?X?3.5×105cm,and we also obtain the temperature at the radius of the photospherekT?7 keV at 5×107cm.These results are consistent with the fitting parameters(see Section 3).

2.2.X-Ray Emission from Internal Shocks

When a fast shell catches up with a slow shell in front of it,the electrons in the shells are accelerated and produce synchrotron radiation.In general,fast and slow shells would be combined after the collision ends,and the whole process satisfies the conservation of energy and momentum.We assume that the mass,velocity,and Lorentz factor of the fast and slow shell can be expressed asM2,β2,γ2andM1,β1,γ1,respectively.The separation distance between the fast and the slow shell can be expressed bycδt,where δt～0.3 s corresponds to the time of light variable structure(Mereghetti et al.2020;Li et al.2021;Ridnaia et al.2021;Tavani et al.2021).The shell collision radius is (Paczynski &Xu 1994;Rees &Meszaros 1994)

From the time of light variables,we can get the collision radius:Rint?5×1010cm.Below,we adopt physical parameters:γ1=2,γ2=10 and δt?0.3 s.According to the conservation of energy and momentum (Paczynski &Xu 1994),the velocity,Lorentz factor and the internal random motion Lorentz factor can be given by

and then we can get the β,γ and γint

wherempis the proton mass.The kinetic energy of the shell will be converted into internal energy,which can be written asEshell(γint?1)/γint,where (γint?1)/γintis the conversion efficiency.One can obtain the magnetic energy?BEshell(γint?1)/γint,and the magnetic energy densityB2/8π.Thus,the comoving magnetic field can be given by Zhang &Mészáros (2002)

whereEshellis the shell energy.The characteristic frequency of synchrotron radiation at an electron Lorentz factor γecorresponds to photon energy

The observed peak flux at distanceDfrom the source,and the total electron numbers can be written asNe=Mshell/mp.SGR J1935+2154 is associated with the supernova remnant (SNR) G57.2+0.8 (Gaensler 2014).The distanceDof the SNR G57.2+0.8 was estimated to be in a range of ～6.6 to ～12.5 kpc(Pavlovi? et al.2013;Surnis et al.2016;Kothes et al.2018;Zhong et al.2020;Zhou et al.2020).Electrons can cool down to γcfor γm>γc(fast cooling case) and γc>γm(slow cooling case) (Sari et al.1998).We compare γmand γcto decide the fast or slow cooling case.and γc=6πmec/σTB2δtγint?10.3.Below,we adopt typical physical parameters: γ1=2,γ2=10,?e=0.1,?B=0.1,Eshell=1039erg,mshell=1019g andp=2.5.The slow cooling case is suitable for our model due to γc>γm.Thus,we consider the slow cooling case to fit the spectrum of XRB.We also find the fast cooling case corresponds to the condition of higher shell energy and γ1<<γ2.The total energy of the observed XRB from SGR J1935+2154 ～1040erg is far less than the energy released by the GRB.Thus,the GRB is always in a fast cooling regime.Moreover,if the shell in front is in a state of non-relativistic motion,it will also cause a fast cooling case.The flux at the observer in the slow cooling case can be given by Gao et al.(2013),

3.The Fitting Results

We restrict our model parameters by the Markov Chain Monte Carlo (MCMC) method with the code PyMultiNest(Buchner et al.2014).The log-likelihood of these parameters can be given by a fitting statistic of Poisson data with Gaussian background,i.e.,

and

Hereirepresents different data from different channels,Sithe observed counts,Bithe background data andmithe predicted counts based on our model with current parameters,which has been convolved with the instrumental response matrix.tsandtbare the exposure times for the source and background spectra,respectively.By minimizing thePGfor a wide range of the parameters,we obtain the distribution of the best-fitting parameters,the central values of which maximize the log-likelihood function.

In our model,when the asteroid collides with the magnetar surface,a hot fireball with electrons and baryons is generated with a high temperatureT0estimated by Equation(6).The initial temperature of the fireball determines the relativistic expansion in the later stage due to adiabatic expansion.As the fireball expands,the size and temperature of the fireball begin to evolve(see Section 2.1).Thermal X-rays from a photosphere in the relativistically expanding fireball are emitted.In Section 2.1,we calculated the blackbody temperature and the size of expanding fireball.The Thomson optical depth becomes thin when the fireball expands to the size of ?X?3.5×105cm.We obtain the temperature at the radius of the photospherekT?7 keV at 5×107cm,and find that the size and temperature of the expanding fireball are consistent with the fitting results.

During the fireball expansion,the Lorentz factor of the fireball is also evolving (see Section 2.1).First,the Lorentz factor of the fireball increases withR.When the fireball reaches its maximum acceleration radius,the fireball will keep moving at a constant Lorentz factor.According to the conservation of energy,we calculate the Lorentz factor of the ejected shell γshell.In this section,we fit the Lorentz factors of the fast and slow shells and find that the Lorentz factor of the shell is comparable to γshell.Since the components of multipolar fields are from the magnetar surface,differences would exist in the magnetic field.Therefore,the Lorentz factor of the ejected shell would have some differences when forming the fast and slow shells.

Moreover,collisions between fast and slow shells in the fireball will cause synchrotron emission.The confined shell collision radiusRintis larger than the photosphere radius,demonstrating that our model is self-consistent.We fit the spectrum of the XRB well to constrain reasonably other parameters (e.g.,p,Ne,B) in the internal shock model.In addition,the asteroid-magnetar gravitational energy contributes ～1040erg to the XRB,which is consistent with the total energy of the observed XRB from SGR J1935+2154.Figure 3 affirms that the confined shell energy and mass are comparable to the XRB energy and the fragmental mass,respectively.

Figure 3.Corner plot of our model parameters.Histograms and contours are the log-likelihood of the parameter constraints by the McSpecFit package (Zhang et al.2016),which shows one and two-dimensional posterior probability distributions of parameters.Red crosses mark the best-fit values and error bars are 1σ uncertainties.

4.Conclusion and Discussion

In this paper,following Dai(2020)we have proposed that the XRB associated with FRB 200428 could be generated by synchrotron emission from the internal shocks between fast and slow shells formed by a relativistically expanding fireball and constrain the model parameters.The model parameters and conclusions are summarized as follows.

1.In our model,the blackbody emission of the XRB spectrum could be contributed by an expanding fireball at the photosphere radius.Moreover,we can constrain the initial size of the trapped fireball,?0,X?1.3×104cm.Then,the fireball will expand along magnetic field lines to the size of ?X?3.5×105cm,and we can get the blackbody temperaturekT?7 keV,which is consistent with observations of the XRB(Mereghetti et al.2020;Li et al.2021;Ridnaia et al.2021;Tavani et al.2021).

2.Subsequently,collisions between fast and slow shells in the relativistic fireball could generate internal shocks and synchrotron emission.We consider the internal shock model to constrain the physical parameters.

Interestingly,the onset of the XRB is tens of milliseconds ahead of the FRBs(Mereghetti et al.2020;Li et al.2021;Ridnaia et al.2021;Tavani et al.2021).Such observation is understandable well based on our physical processes.After an asteroid is disrupted tidally into some fragments,those disrupted small fragments would fall toward the magnetar’s surface earlier to contribute to the onset of the XRB,and then one of two major fragments crosses magnetic field lines at the Alfvén radius to generate a pulse of the FRB.This observation would provide strong evidence in favor of our model.In addition,Dai (2020)found that the FRB energy is proportional to,and the ratio of radio and XRB energies.Therefore,the difference in the asteroidal mass may affect this ratio.

The spectrum becomes hard at the two peaks observed by Insight-HXMT(Li et al.2021),implying that more high-energy electrons can be generated.In our model,collisions between different shells in the fireball lead to electrons being accelerated to high energy,forming more high-energy electrons.The physical process corresponds to the evolution of the spectrum from soft to hard.Moreover,in the internal shock model,the combined shells can continue colliding with other shells,leading to weaker bursts.Our model may have important implications for why there were some weaker spikes after the two peaks were observed by Insight-HXMT (Li et al.2021).

We briefly discuss the spectrum of radiation escaping the photosphere of a fireball under strong magnetic fields.The spectrum from the trapped fireball that formed during the SGR outburst can be described by Lyubarsky (2002):.The spectrum is flat at low energy (?

Furthermore,recent observations indicate that some repeating FRBs manifest periodic activity (CHIME/FRB Collaboration et al.2020;Rajwade et al.2020;Pastor-Marazuela et al.2021;Pleunis et al.2021).These observations can also be explained in the pulsar-asteroid impact model of Dai et al.(2016).Periodicity in FRB 121102 and FRB 180916.J0158+65 can be explained by the orbital period of a pulsar and a stellar-mass companion,and extragalactic asteroid belt parameters can be constrained well (Dai &Zhong 2020).In addition,some repeating FRBs show the time-frequency downward drifting patterns (Caleb et al.2020),and nearly 100% linear polarization (Day et al.2020).The frequency drifting is mainly caused by the geometric structure of a pulsar magnetosphere,and the linear polarization can be constrained (Liu et al.2020).

Acknowledgments

We are grateful to Fa-Yin Wang,He Gao,Tong Bao,Shu-Qing Zhong and Jun Yang for their helpful discussions and constructive comments.This work was supported by the National Key Research and Development Program of China (Grant Nos.2017YFA0402600,2018YFA0404204),the National SKA Program of China (Grant No.2020SKA0120300),the National Natural Science Foundation of China (NSFC,Grant Nos.11833003,U2038105) and the Program for Innovative Talents,Entrepreneur in Jiangsu.Y.P.Y.is supported by the National Natural Science Foundation of China (NSFC,Grant No.12003028)and Yunnan University(Grant No.C176220100087).C.K.L.thanks support from the National Natural Science Foundation of China (NSFC,Grant No.U1838113).B.B.Z.acknowledges support by Fundamental Research Funds for the Central Universities (14380046),the National Key Research and Development Programs of China (2018YFA0404204),the National Natural Science Foundation of China (NSFC,Grant Nos.11833003 and U2038105),the science research grants from the China Manned Space Project with NO.CMS-CSST-2021-B11,and the Program for Innovative Talents,Entrepreneur in Jiangsu.