A Magnetar-asteroid Impact Model for FRB 200428 Associated with an X-Ray Burst from SGR 1935+2154

Fast radio bursts (FRBs) are mysterious millisecond-duration transients of gigahertz radio emission (Lorimer et al. 2007; Thornton et al. 2013) because their physical origin and mechanism remain unknown (for observational and theoretical reviews, see Katz 2018; Cordes & Chatterjee 2019; Petroff et al. 2019; Platts et al. 2019). This year, the first light of understanding FRBs seems to have arisen due to two discoveries. First, a ∼16 day period repeating source FRB 180916.J0158+65 was discovered (CHIME/FRB Collaboration et al. 2020a). Activity lasting a longer period of ∼160 days for the first repeating FRB 121102 was then reported (Rajwade et al. 2020). These observations suggest that FRBs could arise from periodic objects such as precessing magnetars (Levin et al. 2020; Yang & Zou 2020; Zanazzi & Lai 2020) or magnetized neutron stars in binaries (Dai & Zhong 2020; Ioka & Zhang 2020; Lyutikov et al. 2020). For the former models, however, starquake-like events occurring at a stellar fixed region are required to produce an FRB 180916.J0158+65-like periodic phenomenon.

Second, an extremely bright FRB 200428 with two pulses of intrinsic durations ∼0.60 ms and ∼0.34 ms from the direction of the Galactic magnetar SGR 1935+2154 was reported (Bochenek et al. 2020; CHIME/FRB Collaboration et al. 2020b). The two pulses are separated by ∼28.9 ms. This burst was detected to have an average fluence of ∼700 kJy ms and 1.5 ± 0.3 MJy ms by the CHIME and STARE2 telescopes, respectively, which imply the isotropic-equivalent energy release of ${E}_{\mathrm{CHIME}}={3}_{-1.6}^{+3.0}\,\times {10}^{34}$ erg and E_STARE2 = (2.2 ± 0.4) × 10³⁵ erg in two different frequency bands for the source's distance D ∼ 10 kpc. Very fortunately, an X-ray burst (XRB) with two corresponding pulses associated with FRB 200428 was simultaneously detected by high-energy satellites such as Insight-HXMT (Li et al. 2020), AGILE (Tavani et al. 2020), INTEGRAL (Mereghetti et al. 2020), and Konus–Wind (Ridnaia et al. 2020). The isotropic-equivalent emission energy release of the XRB in the soft X-ray to soft gamma-ray energy band is E_X ∼ (0.8–1.2) × 10⁴⁰(D/10 kpc)² erg (Li et al. 2020; Mereghetti et al. 2020; Ridnaia et al. 2020; Tavani et al. 2020). The spectrum seems to be fitted by either a cutoff power-law model (Li et al. 2020; Mereghetti et al. 2020; Ridnaia et al. 2020; Tavani et al. 2020) or a double-temperature blackbody model (kT₁ ∼ 11 keV and kT₂ ∼ 30 keV, Mereghetti et al. 2020; Ridnaia et al. 2020).

The physical parameters of the magnetar SGR 1935+2154 include the rotation period P ≃ 3.24 s, spin-down rate $\dot{P}\,\simeq 1.43\times {10}^{-11}\,{\rm{s}}\,{{\rm{s}}}^{-1}$, surface dipole magnetic field strength B_s ≃ 2.2 × 10¹⁴ G, and spin-down age t ∼ 3.6 kyr (Israel et al. 2016). The source is hosted in the Galactic supernova remnant (SNR) G57.2+0.8 (Gaensler 2014). However, some estimates of the distance D and age of the SNR remain highly debated, e.g., D is in a range of ∼6.6 to ∼12.5 kpc (Pavlović et al. 2013; Surnis et al. 2016; Kothes et al. 2018; Zhong et al. 2020; Zhou et al. 2020). Inferred recently from the observed dispersion measure and Faraday rotation measure, D turns out to be 9.0 ± 2.5 kpc (Zhong et al. 2020). Although this range implies that the isotropic-equivalent energy release of FRB 200428 is close to the low energy end of cosmological FRBs (Bochenek et al. 2020; CHIME/FRB Collaboration et al. 2020b), the association of the FRB with SGR 1935+2154 clearly indicates a magnetar origin at least for some FRBs. Based on the frame of a magnetar, some models for the association of an FRB/XRB were discussed (Lu et al. 2020; Lyutikov & Popov 2020; Margalit et al. 2020) in which both FRBs and XRBs are triggered by starquake-like explosions and powered magnetically. We call these models interior-driven ones.

In this Letter, we propose a different model for the association of FRB 200428 with an XRB from SGR 1935+2154, in which a magnetar encounters an asteroid. We show that such an impact can interpret all of the observed features self-consistently. The impact and radiation physics were discussed in detail when a moderately magnetized pulsar encounters an asteroid (Dai et al. 2016), in which case an asteroid can freely fall onto the stellar surface and lead to a bright cosmological FRB. For a magnetar, however, an asteroid during its free infall must be impeded around the Alfvén radius by an ultra-strong magnetic field and then accreted onto the poles along magnetic field lines, colliding with the stellar surface instantaneously and generating an XRB (see Figure 1). Although it is undetected at cosmological distances, such an XRB in the Galaxy is bright enough to be observed by X-ray satellites (for a discussion, see Dai et al. 2016). It should be pointed out that this gravitationally powered model does not exclude magnetar-based interior-driven models (for a brief summary of four kinds of energy sources, see Dai et al. 2017), some of which, together with our mechanism, might be able to take place for an FRB/XRB. This Letter is organized as follows. We describe our model in Section 2 and constrain the model parameters in Section 3. We present our conclusions in Section 4.

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The Hubble Space Telescope observations of the magnetar SGR 1935+2154 with an ultra-strong magnetic dipole field show that this magnetar is moving at a high proper velocity V_p = (600 ± 400)(D/10 kpc)km s⁻¹ (Levan et al. 2018). This leads to our assumption that the magnetar could catch up with decelerated supernova ejecta and encounter a rocky asteroid of mass m_a ∼ 10²⁰ g. Such an asteroid, which is further assumed to include an iron–nickel component of mass m ∼ 10¹⁷ g, could be one of the asteroids hosted by the magnetar's progenitor before supernova explosion (corresponding to the first scenario of Tremaine & Zytkow 1986) or could be formed during the collapse of a part of supernova ejecta in SNR G57.2+0.8 or could happen to wander nearby the magnetar from the outside. The rate of asteroid-neutron star collisions in the Milky Way has been estimated in different scenarios (Tremaine & Zytkow 1986; Wasserman & Salpeter 1994; Siraj & Loeb 2019). In particular, the rate of such events may be as high as ∼0.1–1 per day under reasonable conditions (Wasserman & Salpeter 1994) and could even reach ∼10 per day at flux ∼1 Jy in the radio band (Siraj & Loeb 2019).

The stellar mass, radius, and surface dipole field strength are taken to be M, R_*, and B_s, respectively. The asteroid is first disrupted tidally into a great number of fragments in the stellar gravitational field at radius R_d ∼ (M/m_a)^1/3r_a ∼ 6.1 × 10¹⁰(M/1.4M_⊙)^1/3 cm, where r_a ∼ 2.0 × 10⁶(m_a/10²⁰ g)^1/3 cm is the rocky asteroid's original radius, and then two major fragments of mass ∼m are further distorted tidally by the magnetar at breakup radius, R_b = 1.3 × 10⁹(m/10¹⁷ g)^2/9(M/1.4M_⊙)^1/3 cm, where the fragmental tensile strength and original mass density have been taken for iron–nickel matter (Colgate & Petscheck 1981). The time difference of arrival of leading and lagging parts of a fragment at any radius is estimated by

$$\begin{eqnarray}\begin{array}{rcl}{\rm{\Delta }}t & \simeq & \displaystyle \frac{12{r}_{0}}{5}{\left(\displaystyle \frac{{R}_{{\rm{b}}}}{{GM}}\right)}^{1/2}\\ & = & \ 0.57{\left(\displaystyle \frac{m}{{10}^{17}{\rm{g}}}\right)}^{4/9}{\left(\displaystyle \frac{M}{1.4{M}_{\odot }}\right)}^{-1/3}\,\mathrm{ms},\end{array}\end{eqnarray} \tag{ 1 }$$

where r₀ is the fragmental original radius (Dai et al. 2016). This timescale is independent of freefall radius (R) and thus can be considered as the duration of an FRB (Geng & Huang 2015). A requirement of the first-pulse intrinsic duration Δt ∼ 0.6 ms of FRB 200428 (Bochenek et al. 2020; CHIME/FRB Collaboration et al. 2020b) leads to the fragmental mass

$$\begin{eqnarray}&&m\simeq 1.1\times {10}^{17}{\left(\displaystyle \frac{{\rm{\Delta }}t}{0.6\mathrm{ms}}\right)}^{9/4}{\left(\displaystyle \frac{M}{1.4{M}_{\odot }}\right)}^{3/4}\,{\rm{g}}.\end{eqnarray} \tag{ 2 }$$

In the following, we discuss the geometry of an FRB-emitting region and observed features of an FRB/XRB.

2.1. Geometry of an FRB-emitting Region

Dai et al. (2016) analyzed the fragmental size and mass density as functions of R during the freefall. Physically, the fragment is initially elongated as an incompressible flow from R_b and subsequently further transversely compressed to a cylinder (Colgate & Petscheck 1981). Here we present two evolutional results. First, the radius of the cylindrical fragment at R is written as

$$\begin{eqnarray}&&r=1.9\times {10}^{4}{\left(\displaystyle \frac{{\rm{\Delta }}t}{0.6\mathrm{ms}}\right)}^{1/2}{\left(\displaystyle \frac{R}{{10}^{7}\mathrm{cm}}\right)}^{1/2}\,\mathrm{cm}.\end{eqnarray} \tag{ 3 }$$

Second, the freefall fragment is significantly affected by the stellar magnetic field, whose interaction radius (R_m) is approximately equal to the Alfvén radius (Ghosh & Lamb 1979). At the latter radius, the kinetic energy density of the fragment is equal to the magnetic energy density. Assuming that the magnetic dipole moment $\mu ={B}_{{\rm{s}}}{R}_{* }^{3}$ and the freefall velocity v_ff = (2GM/R)^1/2, we thus derive

$$\begin{eqnarray}\begin{array}{rcl}{R}_{{\rm{m}}} & \simeq & 1.2\times {10}^{7}{\left(\displaystyle \frac{{\rm{\Delta }}t}{0.6\mathrm{ms}}\right)}^{-1/18}{\left(\displaystyle \frac{M}{1.4{M}_{\odot }}\right)}^{-15/54}\\ & & \times \ {\left(\displaystyle \frac{\mu }{2.2\times {10}^{32}{\rm{G}}{\mathrm{cm}}^{3}}\right)}^{4/9}\,\mathrm{cm},\end{array}\end{eqnarray} \tag{ 4 }$$

where following Litwin & Rosner (2001) we have assumed that at R ∼ R_m the plasma in the fragment becomes thoroughly "threaded" by the magnetic field so that a strong electric field E₂ is induced and meanwhile the fragment is significantly slowed because around this radius the external magnetic field is commonly believed to penetrate the plasma (Lamb et al. 1973; Burnard et al. 1983; Hameury et al. 1986). Inserting Equations (2) and (4) into Equation (3), we further obtain the cylindrical radius at R_m,

$$\begin{eqnarray}\begin{array}{rcl}r({R}_{{\rm{m}}}) & \simeq & 2.1\times {10}^{4}{\left(\displaystyle \frac{{\rm{\Delta }}t}{0.6\mathrm{ms}}\right)}^{17/36}{\left(\displaystyle \frac{M}{1.4{M}_{\odot }}\right)}^{-15/108}\\ & & \times \ {\left(\displaystyle \frac{\mu }{2.2\times {10}^{32}{\rm{G}}{\mathrm{cm}}^{3}}\right)}^{2/9}\,\mathrm{cm}.\end{array}\end{eqnarray} \tag{ 5 }$$

This radius together with R_m determines an FRB-emitting region (i.e., yellow shaded region in Figure 2), whose relevant disk looks like an openmouthed clam and its inclination angle from the symmetric plane is

$$\begin{eqnarray}\begin{array}{rcl}{\theta }_{{\rm{i}}}\simeq \displaystyle \frac{r({R}_{{\rm{m}}})}{{R}_{{\rm{m}}}} & \simeq & 1.7\times {10}^{-3}{\left(\displaystyle \frac{{\rm{\Delta }}t}{0.6\mathrm{ms}}\right)}^{19/36}{\left(\displaystyle \frac{M}{1.4{M}_{\odot }}\right)}^{15/108}\\ & & \times \ {\left(\displaystyle \frac{\mu }{2.2\times {10}^{32}{\rm{G}}{\mathrm{cm}}^{3}}\right)}^{-2/9}.\end{array}\end{eqnarray} \tag{ 6 }$$

It will be seen in Section 3 that this angle is much smaller than the inverse of the typical bulk Lorentz factor of FRB-emitting electrons (i.e., γ ∼ 120).

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2.2. Features of an FRB/XRB

2.2.1. An FRB

When the fragment crosses the stellar magnetic field lines over R_m, as shown in Dai et al. (2016), an electric field (E₂ = −v_ff × B/c) is not only induced outside of the fragment but it also has such a strong component parallel to the stellar magnetic field that electrons are torn off the fragmental surface and accelerated to ultra-relativistic energies instantaneously. Subsequent movement of these electrons along magnetic field lines leads to coherent curvature radiation. This emission component can account for the following features of an FRB.

First, as they move along a magnetic field line with curvature radius ρ_c at radius R_m, ultra-relativistic electrons produce curvature radiation and their typical Lorentz factor γ at R_m can be calculated by

$$\begin{eqnarray}\begin{array}{rcl}\gamma & \equiv & \chi {\gamma }_{\max }\simeq \chi {\left(\displaystyle \frac{6\pi e| {E}_{2}| }{{\sigma }_{T}{B}^{2}}\right)}^{1/2}\\ & \simeq & \ 140\chi {\left(\displaystyle \frac{{\rm{\Delta }}t}{0.6\mathrm{ms}}\right)}^{-5/72}{\left(\displaystyle \frac{M}{1.4{M}_{\odot }}\right)}^{-7/72}\\ & & \times \ {\left(\displaystyle \frac{\mu }{2.2\times {10}^{32}{\rm{G}}{\mathrm{cm}}^{3}}\right)}^{1/18},\end{array}\end{eqnarray} \tag{ 7 }$$

where σ_T is the Thomson scattering cross section, the parameter χ is introduced by Dai & Zhong (2020), and the maximum Lorentz factor γ_max is given by Equation (12) of Dai et al. (2016). Therefore, the characteristic frequency of curvature radiation observed at an angle (θ_v) from the symmetric plane of the openmouthed-clam-shaped disk becomes

$$\begin{eqnarray}\begin{array}{rcl}{\nu }_{\mathrm{curv}} & \simeq & 2.0{\chi }^{3}\delta {\left(\displaystyle \frac{{\rm{\Delta }}t}{0.6\mathrm{ms}}\right)}^{-5/24}{\left(\displaystyle \frac{M}{1.4{M}_{\odot }}\right)}^{-7/24}\\ & & \times \ {\left(\displaystyle \frac{\mu }{2.2\times {10}^{32}{\rm{G}}{\mathrm{cm}}^{3}}\right)}^{1/6}{\left(\displaystyle \frac{{\rho }_{{\rm{c}}}}{{10}^{7}\mathrm{cm}}\right)}^{-1}\,\mathrm{GHz}\\ & \simeq & \ 2.5{\chi }^{3}\delta {\left(\displaystyle \frac{{\rm{\Delta }}t}{0.6\mathrm{ms}}\right)}^{-11/72}{\left(\displaystyle \frac{M}{1.4{M}_{\odot }}\right)}^{-1/72}\\ & & \times \ {\left(\displaystyle \frac{\mu }{2.2\times {10}^{32}{\rm{G}}{\mathrm{cm}}^{3}}\right)}^{-5/18}\,\mathrm{GHz},\end{array}\end{eqnarray} \tag{ 8 }$$

where $\delta =1/\{2{\gamma }^{2}[1-\beta \cos ({\theta }_{{\rm{v}}}-{\theta }_{{\rm{i}}})]\}$ is the factor related with the Doppler effect: δ = 1 for θ_v ≤ θ_i, δ ≃ 1/[γ(θ_v − θ_i)]² for 1/γ ≪ θ_v − θ_i ≪ 1, and otherwise δ ≃ 1/(2γ²) (see Lin et al. 2020). It is noted that the second equality of Equation (8) has used ρ_c = 0.635R_m near the equator from Appendix G of Yang & Zhang (2018).

Second, by considering the region of coherent curvature radiation (as shown in Figure 2), the total luminosity (L_tot) of a beamed FRB has been given by Equation (15) of Dai et al. (2016),

$$\begin{eqnarray}\begin{array}{rcl}{L}_{\mathrm{tot}} & \sim & 2.0\times {10}^{36}\displaystyle \frac{1}{{\chi }^{3}}{\left(\displaystyle \frac{m}{{10}^{17}{\rm{g}}}\right)}^{8/9}{\left(\displaystyle \frac{M}{1.4{M}_{\odot }}\right)}^{19/12}\\ & & \times \ {\left(\displaystyle \frac{\mu }{2.2\times {10}^{32}{\rm{G}}{\mathrm{cm}}^{3}}\right)}^{3/2}{\left(\displaystyle \frac{{R}_{{\rm{m}}}}{{10}^{7}\mathrm{cm}}\right)}^{-23/4}\\ & & \times \ {\left(\displaystyle \frac{{\rho }_{{\rm{c}}}}{{10}^{7}\mathrm{cm}}\right)}^{-1}\,\mathrm{erg}\,{{\rm{s}}}^{-1},\end{array}\end{eqnarray} \tag{ 9 }$$

where the factor 1/χ³ is added as χ is introduced in Equation (7). The isotropic-equivalent emission energy of the FRB observed at angle θ_v is thus estimated by

$$\begin{eqnarray}&&{E}_{\mathrm{radio}}\simeq \displaystyle \frac{{\delta }^{3}}{f}\times {L}_{\mathrm{tot}}\times {\rm{\Delta }}t,\end{eqnarray} \tag{ 10 }$$

where f ≡ ΔΩ/(4π) ≃ 1/(2γ) is the beaming factor because the FRB's solid angle ${\rm{\Delta }}{\rm{\Omega }}\simeq 2\pi \times \max ({\theta }_{{\rm{i}}},1/\gamma )=2\pi /\gamma$ (see Figure 2). Inserting Equations (2), (4), (7), and (9) into Equation (10), therefore, we obtain

$$\begin{eqnarray}\begin{array}{rcl}{E}_{\mathrm{radio}} & \sim & 1.4\times {10}^{35}\displaystyle \frac{{\delta }^{3}}{{\chi }^{2}}{\left(\displaystyle \frac{{\rm{\Delta }}t}{0.6\mathrm{ms}}\right)}^{119/36}{\left(\displaystyle \frac{M}{1.4{M}_{\odot }}\right)}^{145/36}\\ & & \times \ {\left(\displaystyle \frac{\mu }{2.2\times {10}^{32}{\rm{G}}{\mathrm{cm}}^{3}}\right)}^{-13/9}\,\mathrm{erg}.\end{array}\end{eqnarray} \tag{ 11 }$$

2.2.2. An XRB

If an infalling asteroid was diamagnetic, it would become an accretion sheet near a magnetar under the compression of an ultra-strong magnetic field (Colgate & Petscheck 1981). In this case, a fan-shaped hot plasma would be ejected during the collision of the asteroid with stellar surface, even though the collision physics is a little complicated. However, following Litwin & Rosner (2001), as widely argued in the literature (e.g., Lamb et al. 1973; Burnard et al. 1983; Hameury et al. 1986), a broken-up asteroid is thoroughly threaded by an ultra-strong magnetic field so that it is significantly slowed around R_m and then moves along magnetic field lines from R_m onto the poles, possibly giving rise to two accretion columns (Meszaros 1992). The total asteroid-magnetar gravitational energy is given by

$$\begin{eqnarray}\begin{array}{rcl}{E}_{{\rm{G}}}\simeq \displaystyle \frac{{{GMm}}_{{\rm{a}}}}{{R}_{* }} & \sim & 1.9\times {10}^{40}\left(\displaystyle \frac{{m}_{{\rm{a}}}}{{10}^{20}\,{\rm{g}}}\right)\\ & & \times \ \left(\displaystyle \frac{M}{1.4{M}_{\odot }}\right){\left(\displaystyle \frac{{R}_{* }}{{10}^{6}\mathrm{cm}}\right)}^{-1}\,\mathrm{erg}.\end{array}\end{eqnarray} \tag{ 12 }$$

This energy is released in a timescale (Dai et al. 2016)

$$\begin{eqnarray}\begin{array}{rcl}{t}_{{\rm{a}}} & \simeq & \displaystyle \frac{12{r}_{{\rm{a}}}}{5}{\left(\displaystyle \frac{{R}_{{\rm{d}}}}{{GM}}\right)}^{1/2}\\ & \sim & \ 0.1{\left(\displaystyle \frac{{m}_{{\rm{a}}}}{{10}^{20}{\rm{g}}}\right)}^{1/3}{\left(\displaystyle \frac{M}{1.4{M}_{\odot }}\right)}^{-1/3}\,{\rm{s}}.\end{array}\end{eqnarray} \tag{ 13 }$$

When the asteroid collides with the stellar surface, a conical hot plasma (or more likely a hemispherical fireball, as shown by the pink shaded region in Figure 1) is ejected, which is different from the fan-shaped outflow discussed by Colgate & Petscheck (1981). This hot fireball is initially trapped by closed magnetic field lines and thus its temperature is estimated by (Thompson & Duncan 1995)

$$\begin{eqnarray}&&{T}_{\mathrm{fb}}\sim {\left(\displaystyle \frac{{B}^{2}}{8\pi a}\right)}^{1/4}=1.8\times {10}^{10}{\left(\displaystyle \frac{B}{{10}^{14}{\rm{G}}}\right)}^{1/2}\,{\rm{K}},\end{eqnarray} \tag{ 14 }$$

where a is the radiation energy density constant and B is the magnetic field strength. Owing to the fact that kT_fb is greater than the electron rest energy, a huge optical depth to photon–photon annihilation reaction in the fireball, ${\tau }_{\gamma \gamma }\sim {\sigma }_{T}{E}_{{\rm{G}}}/[(2\pi {r}_{{\rm{i}}}^{2})(2.7{{kT}}_{\mathrm{fb}})]$ ∼ $3\times {10}^{10}{({E}_{{\rm{G}}}/{10}^{40}\mathrm{erg})({r}_{{\rm{i}}}/{10}^{5}\mathrm{cm})}^{-2}{({T}_{\mathrm{fb}}/{10}^{10}{\rm{K}})}^{-1}$ (where r_i is the fireball's initial radius and 2.7kT_fb is the mean thermal photon energy), inevitably leads to a dense population of e^± pairs, further making the fireball become highly collisional and opaque. This thus drives a relativistic outflow that traps radiation near the star and releases energy at radius much larger than R_* (Thompson & Duncan 1995; Kaspi & Beloborodov 2017). This process resembles what happens in cosmological gamma-ray bursts (Kumar & Zhang 2015). On the one hand, thermal X-rays are emitted from a photosphere of the outflow, and the superposition of thermal emission from different photospheres (corresponding to different fragments colliding with the stellar surface) possibly generates a multitemperature spectrum of an XRB, whose duration is of order ∼t_a estimated by Equation (13). On the other hand, collisions between different shells in the relativistic outflow could produce an additional nonthermal emission.

FRB 200428 has two pulses separated by ∼28.9 ms. Their intrinsic durations are ∼0.6 ms and ∼0.34 ms, respectively, and their fluence ratio is ξ ∼ 480/220 = 2.2 (CHIME/FRB Collaboration et al. 2020b). The isotropic-equivalent energy release of the first pulse as an example is thus E_radio ≃ [ξ/(1 + ξ)] × (E_CHIME + E_STARE2) ∼ 1.7 × 10³⁵(Δt/0.6 ms)erg for D ∼ 10 kpc (for D also see Zhong et al. 2020), where E_CHIME and E_STARE2 are the isotropic-equivalent radio emission energies observed by the CHIME and STARE2 telescopes (Bochenek et al. 2020; CHIME/FRB Collaboration et al. 2020b), respectively. Therefore, we can constrain the model parameters.

First, from Equation (2), we find the fragmental mass

$$\begin{eqnarray}&&m\sim 1.1\times {10}^{17}{\left(\displaystyle \frac{{\rm{\Delta }}t}{0.6\mathrm{ms}}\right)}^{9/4}\,{\rm{g}}.\end{eqnarray} \tag{ 15 }$$

If B_s = 2.2 × 10¹⁴ G, M = 1.4M_⊙, and R_* = 10⁶ cm are adopted, we obtain the magnetic interaction radius

$$\begin{eqnarray}&&{R}_{{\rm{m}}}\sim 1.2\times {10}^{7}{\left(\displaystyle \frac{{\rm{\Delta }}t}{0.6\mathrm{ms}}\right)}^{-1/18}\,\mathrm{cm},\end{eqnarray} \tag{ 16 }$$

the cylindrical radius

$$\begin{eqnarray}&&r({R}_{{\rm{m}}})\sim 2.1\times {10}^{4}{\left(\displaystyle \frac{{\rm{\Delta }}t}{0.6\mathrm{ms}}\right)}^{17/36}\,\mathrm{cm},\end{eqnarray} \tag{ 17 }$$

and the inclination angle of an FRB-emitting region

$$\begin{eqnarray}&&{\theta }_{{\rm{i}}}\sim 1.7\times {10}^{-3}{\left(\displaystyle \frac{{\rm{\Delta }}t}{0.6\mathrm{ms}}\right)}^{19/36}.\end{eqnarray} \tag{ 18 }$$

Equations (16)–(18) give the parameters of the geometry of FRB 200428's emitting region in our model.

Second, as for the radio properties, FRB 200428 was detected by the STARE2 telescope (Bochenek et al. 2020), implying that ν_curv ∼ 1.4 GHz, that is,

$$\begin{eqnarray}&&\gamma \sim 140\chi {\left(\displaystyle \frac{{\rm{\Delta }}t}{0.6\mathrm{ms}}\right)}^{-5/72},\end{eqnarray} \tag{ 19 }$$

and

$$\begin{eqnarray}&&{\chi }^{3}\delta \sim 0.6{\left(\displaystyle \frac{{\rm{\Delta }}t}{0.6\mathrm{ms}}\right)}^{11/72}.\end{eqnarray} \tag{ 20 }$$

The isotropic-equivalent energy release becomes

$$\begin{eqnarray}&&{E}_{\mathrm{radio}}\sim 1.4\times {10}^{35}\displaystyle \frac{{\delta }^{3}}{{\chi }^{2}}{\left(\displaystyle \frac{{\rm{\Delta }}t}{0.6\mathrm{ms}}\right)}^{119/36}\,\mathrm{erg}.\end{eqnarray} \tag{ 21 }$$

A requirement of E_radio ∼ 1.7 × 10³⁵(Δt/0.6 ms)erg leads to

$$\begin{eqnarray}&&\displaystyle \frac{{\delta }^{3}}{{\chi }^{2}}\sim 1.2{\left(\displaystyle \frac{{\rm{\Delta }}t}{0.6\mathrm{ms}}\right)}^{-83/36}.\end{eqnarray} \tag{ 22 }$$

The solution of Equations (20) and (22) is

$$\begin{eqnarray}&&\chi \sim 0.85{\left(\displaystyle \frac{{\rm{\Delta }}t}{0.6\mathrm{ms}}\right)}^{199/792},\end{eqnarray} \tag{ 23 }$$

and

$$\begin{eqnarray}&&\delta \sim 1.0{\left(\displaystyle \frac{{\rm{\Delta }}t}{0.6\mathrm{ms}}\right)}^{-119/198}.\end{eqnarray} \tag{ 24 }$$

It can be seen that χ ≲ 1 and δ ∼ 1, showing that our model is self-consistent. This also implies that our line of sight is just within the solid angle of FRB 200428. In addition, combining Equations (19) and (23), we find the electrons' typical Lorentz factor γ ∼ 120 for Δt ∼ 0.6 ms.

Third, as for the XRB properties, Equation (12) shows

$$\begin{eqnarray}&&{E}_{{\rm{X}}}\sim 1.9\times {10}^{40}\zeta \left(\displaystyle \frac{{m}_{{\rm{a}}}}{{10}^{20}\,{\rm{g}}}\right)\mathrm{erg},\end{eqnarray} \tag{ 25 }$$

where ζ is the X-ray radiation efficiency and its upper limit is ∼1/2 because at least half of the gravitational energy release E_G is transferred inward to the thermal energy of the stellar matter and eventually emitted by neutrinos. Equation (25) is consistent with the total energy of the observed XRB from SGR 1935+2154 as long as ζ(m_a/10²⁰ g) ∼ 0.5, indicating that our model can also well explain the XRB. For a more massive asteroid (i.e., m_a > 0.5 × 10²⁰ζ⁻¹ g), this conclusion is more viable. On the other hand, as the asteroid collides with the stellar surface, a resultant hot fireball has such a high temperature T_fb estimated by Equation (14) that a dense population of e^±-pairs are inevitably created. Thermal X-rays from a photosphere in the relativistically expanding fireball are emitted, and the superposition of radiation from different photospheres could generate a multitemperature spectrum of an XRB. In addition, collisions between different shells in the fireball may lead to nonthermal emission. It would be expected that these processes can explain the observed spectrum of the XRB (Li et al. 2020; Mereghetti et al. 2020; Ridnaia et al. 2020; Tavani et al. 2020), as discussed in Section 2. A full discussion of the spectrum is well beyond the scope of this Letter and will be left for future work.

Fourth, the above constraints are given for the first pulse of FRB 200428. For the second pulse of this burst, Δt ∼ 0.34 ms and E_radio ≃ [1/(1 + ξ)] × (E_CHIME + E_STARE2) ∼ 0.8 × 10³⁵(Δt/0.34 ms)erg for the distance D ∼ 10 kpc. These observed data have been used to provide similar constraints on model parameters.

Fifth, the observed light curve of FRB 200428 requires that the time interval (∼28.9 ms) between two submillisecond pulses should be smaller than the total asteroidal accretion timescale t_a (i.e., Equation (13)), implying that m_a > 3 × 10¹⁸ g. This constraint is naturally satisfied for the mass limit from XRB observations.

Finally, during an active period of 29 XRBs from SGR 1935+2154 observed by Fermi/GBM prior to FRB 200428, the FAST radio telescope observed the magnetar but did not detect any FRB (Lin et al. 2020). This nondetection result can be understood in our model: δ ≃ 1/(2γ²) ∼ 3.5 × 10⁻⁵ for θ_v ≫ θ_i, in which case the isotropic-equivalent radio emission energy observed at θ_v is ∼0.7 × 10²² erg even if a fragment has a similar mass. This energy is too low for the FAST telescope to be able to detect any FRB. Furthermore, for a less massive fragment, any FRB-like signal from SGR 1935+2154 at large θ_v cannot be observed because of a lower intrinsical isotropic-equivalent energy release E_radio.

In this Letter, we have proposed a new model for the association of FRB 200428 with an XRB from SGR 1935+2154, in which a magnetar encounters an asteroid with mass of ∼10²⁰ g. We have shown that such an impact can self-consistently interpret the emission properties of FRB 200428 and its associated XRB. This model is different from that of Dai et al. (2016), because we here considered the magnetic interaction radius R_m, around which the asteroid during its free infall must be impeded by an ultra-strong magnetic field and then accreted onto the poles along magnetic field lines, colliding with the stellar surface and generating an XRB. Although it is undetected at cosmological distances, such an XRB in the Galaxy is bright enough to be observed by current X-ray satellites, as discussed in Dai et al. (2016). We constrained the model parameters. Our conclusions are summarized as follows.

• 1.  
  FRB 200428-emitting region looks like an openmouthed clam, whose inclination angle and magnetic interaction radius are θ_i ∼ 1.7 × 10⁻³ and R_m ∼ 1.2 × 10⁷ cm, respectively. The FRB emits along magnetic field lines around R_m.
• 2.  
  The typical Lorentz factor γ ∼ 120 of emitting electrons is found to understand a low isotropic-equivalent energy of FRB 200428 as compared to cosmological FRBs. Our line of sight is just within the solid angle of this burst. If the viewing angle is much larger than θ_i (i.e., an off-plane case), the isotropic-equivalent energy release becomes extremely low. This is why the FAST telescope has not detected any FRB-like signal during the active phase of 29 XRBs observed by Fermi/GBM.
• 3.  
  As the asteroid collides with the stellar surface, a resultant hot fireball has a temperature ∼1.8 × 10¹⁰ K. This leads to a dense population of e^± pairs. The superposition of thermal emission from different photospheres and nonthermal emission from collisions between different shells in the fireball is expected to account for the observed XRB's spectrum.³ In addition, from Equation (13), the typical duration (t_a) of an XRB is of order ∼0.1(m_a/10²⁰ g)^1/3 s, which is basically consistent with the X-ray observations.

What should be pointed out is that in this Letter we have only discussed the asteroid-neutron star direct collision case in which the impact area πb² (where b is the so-called impact parameter) is smaller than the capture cross section given by Equation (18) of Dai et al. (2016), that is, πb² < σ_a = 2.4 × 10¹⁹R_*,6(M/1.4M_⊙)V_p,7⁻² cm², where R_*,6 = R_*/10⁶ cm and V_p,7 = V_p/10⁷ cm s⁻¹, implying that $b\lt {b}_{\mathrm{cr}}={({\sigma }_{{\rm{a}}}/\pi )}^{1/2}$ = $2.8\times {10}^{9}{R}_{* ,6}^{1/2}{(M/1.4{M}_{\odot })}^{1/2}{V}_{{\rm{p}},7}^{-1}\,\mathrm{cm}$. In this case, an asteroid is not only captured by a neutron star but also eventually impacts the stellar surface. If b > b_cr, on the other hand, an asteroid cannot be captured and instead it flies out nearby the neutron star, in which case the asteroid still interacts with the stellar magnetosphere, perhaps leading to a faint burst-like electromagnetic signal due to the effect of a weak outer magnetic field.

I would like to thank the anonymous referee for helpful comments that have allowed me to improve the presentation of this Letter. I also thank Lin Lin, Xiangyu Wang, Xuefeng Wu, Yunwei Yu, Bing Zhang, and Shuangnan Zhang for their useful discussions. This work was supported by the National Key Research and Development Program of China (grant No. 2017YFA0402600) and the National Natural Science Foundation of China (grant No. 11833003).