FRB 200428: An Impact between an Asteroid and a Magnetar

Similar to the accretion of matter onto a compact object in binary systems, an instant accretion column will form in our scenario (see Figure 1). As the accreting matter flows along magnetic field lines to the polar cap, it will become highly supersonic and essentially in freefall. Before landing on the magnetar's surface, since the infalling material needs to be decelerated to subsonic, some sort of strong shock must occur in the accretion stream. After the shock, the hot gas settles to the NS surface in the magnetically confined column. Since our understanding of the column structure/shock characteristics is far from complete, here, we adopt a simple model to estimate basic parameters for the instant accretion column.

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The isotropic fluence of the X-ray burst is ∼7 × 10⁻⁷ erg cm⁻², lasting for δt ∼ 150 ms. We attribute the X-ray burst energy to the release of gravitational potential energy of the asteroid, i.e.,

$$\begin{eqnarray}&&{E}_{X}=\eta \displaystyle \frac{{{GM}}_{\mathrm{NS}}{m}_{{\rm{a}}}}{{R}_{\mathrm{NS}}},\end{eqnarray} \tag{ 5 }$$

where η is the efficiency of energy transforming. Taking E_X = 8 × 10³⁹ erg and typical η ∼ 0.1, the asteroid mass needed is m_a = 4 × 10²⁰ g. This mass is roughly in the mass range of normal asteroids. The average X-ray luminosity is ${L}_{{\rm{X}}}={E}_{X}/\delta t=5\times {10}^{40}$ erg s⁻¹, and the corresponding accretion rate is $\dot{M}={m}_{{\rm{a}}}/\delta t=2.7\times {10}^{21}$ g s⁻¹. The X-ray luminosity of the burst exceeds the Eddington luminosity (${L}_{\mathrm{Edd}}\approx 2\times {10}^{38}{M}_{\mathrm{NS},1.4{M}_{\odot }}$ erg s⁻¹), indicating the column is radiatively dominated. Due to the increasing magnetic pressure approaching the magnetar surface, at the magnetar's surface, we assume that the cross-section of the accretion column is a thin rectangle of width d and length l. The polar angle (with respect to the magnetic axis) of this rectangle could be estimated by the landing point of accreted material, i.e., ${\theta }_{\mathrm{land}}=\arcsin (\sqrt{{R}_{\mathrm{NS}}/{R}_{{\rm{A}}}})$. Then l could be estimated as $l\approx 2\pi {R}_{\mathrm{NS}}\sin {\theta }_{\mathrm{land}}\times \delta t/P=8\times {10}^{4}\,\mathrm{cm}$. The freefall velocity at the surface is ${v}_{\mathrm{ff}}=\sqrt{2{{GM}}_{\mathrm{NS}}/{R}_{\mathrm{NS}}}$. After the shock, the plasma falls down to a velocity of ${v}_{\mathrm{fd}}={v}_{\mathrm{ff}}/7$ (Lyubarskii & Syunyaev 1982; Becker 1998) and the mass density changes as ${\rho }_{\mathrm{ps}}=\dot{M}/({{ldv}}_{\mathrm{fd}})$.

Following the calculations of Mushtukov et al. (2015), the height of the accretion column H can be estimated from

$$\begin{eqnarray}&&{L}_{X}\approx 7.6\left(\displaystyle \frac{l/d}{10}\right)\displaystyle \frac{{\kappa }_{{\rm{T}}}}{{\kappa }_{\perp }}f\left(\displaystyle \frac{H}{{R}_{\mathrm{NS}}}\right){L}_{\mathrm{Edd}},\end{eqnarray} \tag{ 6 }$$

where $f\left(\tfrac{H}{{R}_{\mathrm{NS}}}\right)=\mathrm{ln}\left(1+\tfrac{H}{{R}_{\mathrm{NS}}}\right)-\tfrac{H}{{R}_{\mathrm{NS}}+H}$, κ_T is the Thomson electron scattering opacity of the solar mix plasma (the mean number of nucleons per electron is 1.17), and κ_⊥ is the opacity across the field lines. For a strong magnetic field and a typical photon energy E_γ (∼84 keV for this burst) less than cyclotron energy E_cycl, ${\kappa }_{{\rm{T}}}/{\kappa }_{\perp }\sim {({E}_{\gamma }/{E}_{\mathrm{cycl}})}^{-2}\sim 1000$ (Canuto et al. 1971). We assume thermodynamical equilibrium deep inside the column and the plasma temperature to be T; the radiation field energy density in the deep center could be written as

$$\begin{eqnarray}&&{P}_{\mathrm{rad}}\approx {\rho }_{\mathrm{ps}}\displaystyle \frac{{{GM}}_{\mathrm{NS}}}{{R}_{\mathrm{NS}}+H}\displaystyle \frac{H}{{R}_{\mathrm{NS}}}\approx \displaystyle \frac{{{aT}}^{4}}{3},\end{eqnarray} \tag{ 7 }$$

where a is the radiation constant. An empiric value of l/d ∼ 10–100 is usually adopted in previous studies (Basko & Sunyaev 1976; Mushtukov et al. 2015). We find that our scenario could well explain the spectral feature of the X-rays if we chose d = l/10 = 8 × 10³ cm. Taking L_X = 5 × 10⁴⁰ erg s⁻¹, we obtain H/R_NS = 0.3, T ≈ 2 × 10⁹ K, or k T ≈ 170 keV (k is the Boltzmann constant). On the other hand, the effective temperature (T_⊥) corresponding to the escaping flux (perpendicular to the field lines) is determined by the emergent flux

$$\begin{eqnarray}&&{F}_{\perp ,\mathrm{esp}}\approx \displaystyle \frac{4{{cP}}_{\mathrm{rad}}}{{\rho }_{\mathrm{ps}}{\kappa }_{\perp }d}={\sigma }_{\mathrm{SB}}{T}_{\perp }^{4},\end{eqnarray} \tag{ 8 }$$

where ${\sigma }_{\mathrm{SB}}$ is the Stefan–Boltzmann constant. This relation gives kT_⊥ ≈ 40 keV, roughly consistent with the cutoff energy ∼80 keV of the X-ray burst (Li et al. 2020).

More detailed processes should be considered for the radiation from the spot. High temperatures of the column result in the creation of electron–positron pairs, so that soft X-ray photons will be bulk Comptonized by ${e}^{+}{e}^{-}$ pair plasmas before escaping. The pair number density could be given by (Mushtukov et al. 2019)

$$\begin{eqnarray}&&{n}_{\pm }\simeq 1.8\times {10}^{30}{e}^{-{m}_{e}{c}^{2}/{kT}}{\left(\displaystyle \frac{{kT}}{{m}_{e}{c}^{2}}\right)}^{3/2}\,{\mathrm{cm}}^{-3},\end{eqnarray} \tag{ 9 }$$

from which we get n_± ≃ 3 × 10²⁸ cm⁻³ using k T = 170 keV. Considering the total energy of accreting material is going into pair creation, the Lorentz factor of the pairs γ_± could be obtained from

$$\begin{eqnarray}&&({{\rm{\Gamma }}}_{\mathrm{ff}}-{{\rm{\Gamma }}}_{\mathrm{fd}}){\rho }_{\mathrm{ps}}{c}^{2}=2{\gamma }_{\pm }{n}_{-}{m}_{e}{c}^{2},\end{eqnarray} \tag{ 10 }$$

where ${{\rm{\Gamma }}}_{\mathrm{ff}/\mathrm{fd}}={(1-{({v}_{\mathrm{ff}/\mathrm{fd}}/c)}^{2})}^{-1/2}$, which gives γ_± ≃ 8. The scattering between the seed thermal/bremsstrahlung photons of ∼1 keV and ${e}^{+}{e}^{-}$ pairs will result in a hard photon energy of 60 keV (Becker & Wolff 2007), which is also consistent with the cutoff energy of this X-ray burst. To obtain an accurate photon spectrum, we need to solve the Kompaneet's equation, which is beyond the scope of this work. Scattering, recoil and escaping from a finite medium should be included. However, according to studies of unsaturated Compton scattering in X-ray binaries, the emerging spectrum is expected to be a cutoff power-law shape (Ferrigno et al. 2016), which is consistent with the spectral feature of the X-ray burst here.

The X-ray burst showed two hard peaks with a separation of ∼30 ms. This could be understood by supposing that the elongated asteroid breaks into two main fragments around R_b (Dai 2020). We can infer that the distance between these two fragments should be roughly 3.5 × 10⁶ cm, if we attribute the time separation of X-ray peaks to the difference of the arrival time of two fragments (using Equation (3) in Geng & Huang 2015). This distance is comparable with the radius of the asteroid before elongation, making the explanation more plausible.

FRB 200428 consists of two subbursts with durations of ∼0.6 ms and ∼0.34 ms, respectively, separated by ∼30 ms (The CHIME/FRB Collaboration et al. 2020b). For the first subburst, the primary beam detection from the Survey for Transient Astronomical Radio Emission 2 (STARE2) gives that its isotropic-equivalent luminosity is L_FRB = 4.2 × 10³⁸ erg s⁻¹ at band ∼1.4 GHz (Bochenek et al. 2020). The second subburst is a bit fainter. Below, we will take the first subburst as the typical case in our interpretation.

It is still uncertain how coherent radio emission is produced from pulsars. Coherent mechanisms, either by charged bunches (e.g., Ruderman & Sutherland 1975; Benford & Buschauer 1977; Lyutikov et al. 2016; Yang & Zhang 2018; Kumar & Bošnjak 2020) or a maser mechanism (e.g., Blandford 1975; Luo & Melrose 1992; Lyubarsky 2014; Metzger et al. 2019) due to growth of plasma instabilities are proposed to produce radio emission. The same situation has also confronted us in the FRB. Since the time delay between the radio signal and the X-ray burst is in good agreement with the dispersion delay, both of them are suggested to originate within the magnetosphere, which motivates us to explain the radio pulse with curvature radiation from bunches.

As mentioned in the previous subsection, the landing of accretion material onto the NS surface produces abundant ${e}^{+}{e}^{-}$ pairs. Considering that the landing point of the column θ_land is beyond the gap region of the NS, only a small portion of these pairs will flow into the gap region. On the other hand, the large electric field in the gap can accelerate electrons and produce pair cascade within a very short timescale ∼1 μs (Mitra 2017). During the rising phase of the X-ray burst, the arrival of the charges will screen the gap electric field due to the continuous increasing supply of charges. The screen is destructed once the charge density from the accretion column decreases to a threshold during the decline phase of the X-ray burst. At that time, these charges from the column will be accelerated (with Lorentz factors of γ) to flow along the open field lines. After the leaving of this plasma cloud, the discharge from the gap can initiate soon within a period of ∼1 μs, which produces a following plasma cloud. When it caught up with the plasma cloud emitted from accretion material, the overlapping of the slow- and fast-moving particles will lead to the two-stream instability in the plasma, hence the coherent emission. The height of the gap is (Mitra 2017)

$$\begin{eqnarray}&&h=95{B}_{\mathrm{NS},14}^{-4/7}{P}^{1/7}{\dot{P}}_{-11}^{-2/7}{R}_{\mathrm{NS},\ 6}^{2/7}\,\mathrm{cm}.\end{eqnarray} \tag{ 11 }$$

The velocity difference between the slow- and fast-moving particles is about ${\rm{\Delta }}v=c/(2{\gamma }^{2})$, and the time for the particles to overlap is h/Δv. Hence, the instability can develop at a distance of ${r}_{\mathrm{emi}}\sim {ch}/{\rm{\Delta }}v\simeq 2{\gamma }^{2}h$ from the NS. The corresponding characteristic frequency of curvature emission is

$$\begin{eqnarray}&&{\nu }_{c}={\gamma }^{3}\displaystyle \frac{3c}{4\pi {r}_{{\rm{c}}}},\end{eqnarray} \tag{ 12 }$$

where r_c is the curvature radius and is related to r_emi by ${r}_{{\rm{c}}}\simeq \tfrac{4{r}_{\mathrm{emi}}}{3\sin {\theta }_{\mathrm{emi}}}$ according to the magnetosphere geometry, and θ_emi is the poloidal angle of the emission region. Therefore, we could obtain

$$\begin{eqnarray}&&\gamma =677\left(\displaystyle \frac{{\nu }_{{\rm{c}}}}{1.4\,\mathrm{GHz}}\right)\left(\displaystyle \frac{h}{65\,\mathrm{cm}}\right){\left(\displaystyle \frac{\sin {\theta }_{\mathrm{emi}}}{0.05}\right)}^{-1}.\end{eqnarray} \tag{ 13 }$$

This is within the range of Lorentz factor (${10}^{2}\mbox{--}{10}^{4}$) of ${e}^{+}{e}^{-}$ accelerated from the NS gap.

Assuming the length of the bunching shell is Δ, the duration δt of the FRB pulse implies Δ ≃ cδt. We further assume that the ratio of the shell's solid angle to 4π is f, then the emission volume is ${V}_{\mathrm{emi}}=4\pi {{fr}}_{\mathrm{emi}}^{2}{\rm{\Delta }}$. Electrons radiate coherently in patches with a characteristic radial size of $\lambda =c/{\nu }_{c}$, and they can be casually connected in the relativistic beam of angle of 1/γ; the corresponding volume of one small patch is ${V}_{\mathrm{coh}}=(4/{\gamma }^{2}){r}_{\mathrm{emi}}^{2}\lambda$. The number of the patches writes as ${N}_{\mathrm{pat}}\approx {V}_{\mathrm{emi}}/{V}_{\mathrm{coh}}$. Then the coherent curvature emission luminosity can be estimated as

$$\begin{eqnarray}&&{L}_{\mathrm{coh}}=({P}_{e}{N}_{\mathrm{coh}}^{2})\times {N}_{\mathrm{pat}},\end{eqnarray} \tag{ 14 }$$

where ${N}_{\mathrm{coh}}={n}_{e}\times {V}_{\mathrm{coh}}$ is the number of electrons in each patch. Using ${L}_{\mathrm{coh}}={{fL}}_{\mathrm{FRB}}$, we can constrain the electron number density to be

$$\begin{eqnarray}\begin{array}{rcl}{n}_{e} & \simeq & 4.6\times {10}^{9}{\left(\displaystyle \frac{{L}_{\mathrm{FRB}}}{4.2\times {10}^{38}\mathrm{erg}{{\rm{s}}}^{-1}}\right)}^{1/2}{\left(\displaystyle \frac{\gamma }{677}\right)}^{1/2}\\ & & \times {\left(\displaystyle \frac{{r}_{\mathrm{emi}}}{6\times {10}^{7}\mathrm{cm}}\right)}^{-3/2}{\left(\displaystyle \frac{\delta t}{0.6\mathrm{ms}}\right)}^{-1/2}{\left(\displaystyle \frac{\sin {\theta }_{\mathrm{emi}}}{0.05}\right)}^{-1/2}\,{\mathrm{cm}}^{-3}.\end{array}\end{eqnarray} \tag{ 15 }$$

The corresponding plasma density near the NS surface should be $\sim {n}_{e}{({r}_{\mathrm{NS}}/{r}_{\mathrm{emi}})}^{-2}\simeq 1.7\times {10}^{13}$ cm⁻³. The Goldreich & Julian (1969) charge number density at the cap region is ${n}_{\mathrm{GJ}}({r}_{{NS}},{\theta }_{\mathrm{open}})=3.8\times {10}^{10}\,{\mathrm{cm}}^{-3}$, where $\sin {\theta }_{\mathrm{open}}=\sqrt{{R}_{\mathrm{NS}}/{R}_{\mathrm{LC}}}$. Therefore, a reasonable pair multiplicity factor of ∼400 is needed during the spark.

In our scenario, an FRB is emitted along open field lines at ${r}_{\mathrm{emi}}\sim 60\,{R}_{\mathrm{NS}}$, which indicates that the edge of the lightning cone of this magnetar may sweep the line of sight. Although the radiation position for the FRB and periodic radio pulsations may be different, the detection of periodic radio pulsations may still be possible since we have little constraints on the f factor of the FRB. Interestingly, a ∼7σ detection of periodic radio pulsations from SGR 1935+2154 was reported very recently (Burgay et al. 2020). If such a detection is confirmed in the future, our scenario would be highly favored.

Usually, the direct collision between asteroids and the magnetar may be very rare since the recurrence time of such strong direct impacts (m_a ≥ 10¹⁸ g) in a typical NS planetary system can be as large as ∼10⁶–10⁷ yr according to previous studies (Tremaine & Zytkow 1986; Litwin & Rosner 2001). However, in some special cases, it may still be possible that the NS is passing through an asteroid belt as suggested by Dai et al. (2016) so that several FRBs could be produced within a short period due to multiple collisions. A much weaker (30 mJy) radio pulse has been observed by the Five-hundred-meter Aperture Spherical radio telescope two days after the two FRB pulses (Zhang et al. 2020). It is doubtful if this weak burst can be regarded as an FRB or not because its equivalent isotropic energy is almost 10⁸ times smaller than the two pulses. If it were really from SGR 1935+2154, it may be caused by pollution of NS magnetospheres by low-level accretion from a circumpulsar debris disk (Cordes & Shannon 2008).

The landing of asteroid material onto the magnetar will increase the moment of inertia of the magnetar, leading to an abrupt change of rotational frequency of the magnetar (Huang & Geng 2014). Assuming the moment of inertia of the magnetar crust I_c is only 1% of the total moment of inertia, i.e., ${I}_{c}=0.01\times \tfrac{2}{5}{M}_{\mathrm{NS}}{R}_{\mathrm{NS}}^{2}\sim {10}^{43}$ g cm². The asteroid of mass 4 × 10²⁰ g will lead to an instantaneous change in frequency Δν ∼ −1.0 × 10⁻¹² Hz, which is too small to be detected from X-ray/radio observation. However, we have ignored the orbital angular momentum of the asteroid itself in the above estimate. The transfer of the asteroid's orbital angular momentum to the magnetar crust could result in a Δν of magnitude ∼10⁻⁸ Hz. Future observations of magnetars with radio pulses could help to identify this hypothesis.

The impact of the accretion column against the magnetar surface may also lead the crust to be more unstable, i.e., trigger the crustquakes (Thompson & Duncan 1995) or magnetic field line reconnection (Lyutikov 2003). This process heats the surface in one or more regions, from which seed thermal photons might be resonant cyclotron scattered by the electrons. As a result, an X-ray burst forest is expected to follow the FRB-associated X-ray burst. Since these following X-ray bursts come from the normal activities of an isolated magnetar, they are unlikely to be associated with further FRBs.