Fast Radio Bursts from Interacting Binary Neutron Star Systems

Repeating FRBs seem to have lower luminosities than apparently non-repeating ones, with a typical isotropic value of a few ${10}^{41}\ \mathrm{erg}\,{{\rm{s}}}^{-1}$ (Luo et al. 2020a). Given that the typical duration of repeating FRBs is a few milliseconds, the isotropic energy of each burst can be estimated as ${E}_{\mathrm{iso}}\sim {10}^{39}$ erg. The average isotropic-equivalent FRB production power from the source may be estimated as

$$\begin{eqnarray}&&{\bar{L}}_{\mathrm{FRB},\mathrm{iso}}\sim \dot{N}{E}_{\mathrm{iso}}=({10}^{42}\ \mathrm{erg}\ {\mathrm{yr}}^{-1}){\dot{N}}_{3}{E}_{\mathrm{iso},39},\end{eqnarray} \tag{ 1 }$$

where $\dot{N}={10}^{3}\ {\mathrm{yr}}^{-1}{\dot{N}}_{3}$ is the bursting rate (beaming toward Earth) per year from a particular source. For an FRB source lasting for a duration $\tau =({10}^{2}\,\mathrm{yr})\ {\tau }_{2}$, the total isotropic-equivalent energy output in FRBs is

$$\begin{eqnarray}&&{E}_{\mathrm{FRB},\mathrm{iso}}={\bar{L}}_{\mathrm{FRB},\mathrm{iso}}\tau =({10}^{44}\ \mathrm{erg}){\dot{N}}_{3}{E}_{\mathrm{iso},39}{\tau }_{2}.\end{eqnarray} \tag{ 2 }$$

When beaming is considered, this energy budget is reduced. Let us assume that each FRB has a beaming angle of $\delta {\rm{\Omega }}\ll 4\pi$ (e.g., of the order of $\sim \pi {\gamma }^{-2}$ in our scenario, where γ is the characteristic Lorentz factor of electrons in the bunch), and that the bulk of FRBs are concentrated in a solid angle of ${\rm{\Delta }}{\rm{\Omega }}\lt 4\pi$ (which is expected for the interacting model discussed here). The true energy of each burst is smaller by a factor $\delta {\rm{\Omega }}/4\pi$, and the total number of bursts is increased by a factor ${\rm{\Delta }}{\rm{\Omega }}/\delta {\rm{\Omega }}$. As a result, the true FRB energy budget is

$$\begin{eqnarray}&&{E}_{\mathrm{FRB}}={f}_{b}{E}_{\mathrm{FRB},\mathrm{iso}}=({10}^{43}\ \mathrm{erg}){f}_{b,-1}{\dot{N}}_{3}{E}_{\mathrm{iso},39}{\tau }_{2},\end{eqnarray} \tag{ 3 }$$

where ${f}_{b}\equiv {\rm{\Delta }}{\rm{\Omega }}/4\pi$. This energy should be the minimum energy budget in the system.

To estimate the total energy budget in the BNS system, we take the double-pulsar system PSR J0737-3039A/B (Kramer & Stairs 2008) as the nominal system. This system is the only BNS system where both members have measured spin parameters. For reference, we list the relevant parameters of the two pulsars in Table 1. One can see that relatively speaking PSR A has a shorter period (P), lower polar cap magnetic field (B_p), but a higher spindown power ($\dot{E}$) than PSR B. We do not list the current orbital parameters of the system, since we envisage a much later stage of the evolution as the magnetospheres of the two pulsars interact. We do not assume longer periods of the two pulsars than observed in PSR J0737-3039A/B, since the observed BNS merger systems by LIGO/Virgo have shorter lifetimes than Galactic BNS systems in order to merge within the Hubble time. In the following, we normalize the parameters of the two pulsars as the measured values from the PSR J0737-3039A/B system (Kramer & Stairs 2008), i.e., ${P}_{{\rm{A}}}=(0.0227\ {\rm{s}})\ {p}_{{\rm{A}}}$, ${B}_{{\rm{p}},{\rm{A}}}=(1.3\times {10}^{10}\ {\rm{G}})\ {b}_{{\rm{p}},{\rm{A}}}$, ${R}_{\mathrm{LC},{\rm{A}}}=1.1\times {10}^{8}\ \mathrm{cm}\ {r}_{\mathrm{LC},{\rm{A}}}$, ${\dot{E}}_{{\rm{A}}}=5.7\times {10}^{33}\ \mathrm{erg}\,{{\rm{s}}}^{-1}\ {\dot{e}}_{{\rm{A}}};$ ${P}_{{\rm{B}}}\,=(2.77\ {\rm{s}}){p}_{{\rm{B}}}$, ${B}_{{\rm{p}},{\rm{B}}}=(3.2\times {10}^{12}\ {\rm{G}})\ {b}_{{\rm{p}},{\rm{B}}}$, ${R}_{\mathrm{LC},{\rm{B}}}\,=\,1.3\times {10}^{10}\,\mathrm{cm}{r}_{\mathrm{LC},{\rm{B}}}$, ${\dot{E}}_{{\rm{A}}}=1.6\times {10}^{30}\ \mathrm{erg}\,{{\rm{s}}}^{-1}\ {\dot{e}}_{{\rm{B}}}$.

Table 1.  Parameters of the Double-pulsar System PSR J0737-3039A/B (Kramer & Stairs 2008)

Parameters	PSR J0737-3039A	PSR J0737-3039B
P	22.7  ms	2.77  s
$\dot{P}$	$1.7\times {10}^{-18}$	$0.88\times {10}^{-15}$
Bp	$1.3\times {10}^{10}$ G	$3.2\times {10}^{12}$ G
${R}_{\mathrm{LC}}$	$1.1\times {10}^{8}\,\mathrm{cm}$	$1.3\times {10}^{10}$ cm
$\dot{E}$	$5.7\times {10}^{33}\ \mathrm{erg}\,{{\rm{s}}}^{-1}$	$1.6\times {10}^{30}\ \mathrm{erg}\,{{\rm{s}}}^{-1}$
M	1.337(5)${M}_{\odot }$	1.250(5) ${M}_{\odot }$

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The ultimate energy budget in the system includes the rotation energies of the two pulsars:

$$\begin{eqnarray}&&{E}_{\mathrm{rot},{\rm{A}}}=\displaystyle \frac{1}{2}I{{\rm{\Omega }}}_{{\rm{A}}}^{2}=(3.8\times {10}^{49}\ \mathrm{erg})\ {I}_{45}{p}_{{\rm{A}}}^{-2},\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&{E}_{\mathrm{rot},{\rm{B}}}=\displaystyle \frac{1}{2}I{{\rm{\Omega }}}_{{\rm{B}}}^{2}=(2.6\times {10}^{45}\ \mathrm{erg})\ {I}_{45}{p}_{{\rm{B}}}^{-2},\end{eqnarray} \tag{ 5 }$$

as well as the orbital gravitational energy releasable until coalescence:

$$\begin{eqnarray}&&{E}_{\mathrm{orb}}=\displaystyle \frac{{{GM}}_{1}{M}_{2}}{2R}=(2.6\times {10}^{53}\ \mathrm{erg})\ {M}_{1.4}^{2}{R}_{6}^{-1},\end{eqnarray} \tag{ 6 }$$

where $I={10}^{45}\,{\rm{g}}\,{\rm{c}}{{\rm{m}}}^{2}\,{I}_{45}$ is the moment of inertia of the neutron star, ${M}_{1}={M}_{2}=(1.4{M}_{\odot }){M}_{1.4}$, and $R={10}^{6}\ \mathrm{cm}\ {R}_{6}$ is the radius of the neutron stars. Several remarks should be made: (1) The magnetic energies of the two pulsars are ${E}_{{\rm{B}},{\rm{A}}}\,=(1/6){B}_{{\rm{A}}}^{2}{R}^{3}=(2.8\times {10}^{37}\ \mathrm{erg})\ {b}_{{\rm{A}}}^{2}$ and ${E}_{{\rm{B}},{\rm{B}}}=(1/6){B}_{{\rm{B}}}^{2}{R}^{3}\,=(1.7\times {10}^{42}\ \mathrm{erg})\ {b}_{{\rm{B}}}^{2}$, respectively. These energies (especially that of PSR B) can be directly dissipated to power FRB emission. However, after dissipation, it is likely that the fields would be replenished from the rotation energies of the neutron stars (by analogy with the magnetic cycle of the Sun). So we list the rotation energies of the two neutron stars (rather than their magnetic energies) as the ultimate energy sources. (2) Based on the face values of the spindown rates of the two pulsars, the usable spin energy during the period of τ is only $\sim \dot{E}\tau$, which is $(1.8\times {10}^{43}\ \mathrm{erg})\ {\dot{e}}_{{\rm{A}}}{\tau }_{2}$ and $(5.0\times {10}^{39}\ \mathrm{erg})\ {\dot{e}}_{{\rm{B}}}{\tau }_{2}$ for PSRs A and B, respectively. This is barely enough to meet the repeating FRB energy budget unless ${\rm{\Delta }}{\rm{\Omega }}\ll 1$ or ${\dot{e}}_{{\rm{A}}}\gg 1$. However, due to the close interactions between the magnetospheres of the two pulsars, additional braking is possible to tap the spin energies of both pulsars, which are limited by Equations (4) and (5) and are more than enough to power the observed FRBs. (3) The majority of the orbital energy (Equation (6)) is carried away by gravitational waves. However, it is likely that a small fraction of the orbital energy is dissipated due to the interaction between the two magnetospheres (e.g., Palenzuela et al. 2013a, 2013b; Carrasco & Shibata 2020). If this fraction is greater than 10⁻⁹, it would also provide another relevant energy budget to power repeating FRBs.⁵

There are several characteristic timescales in a BNS system. The first two are the rotation periods of the two pulsars, which are typically of the order of tens of milliseconds and seconds, respectively. Since the triggers of bursts depend on the complicated magnetic configurations in the system, the arrival times of the detected bursts would not follow the same rotation phase as in radio pulsars so that no strict periodicity is expected.⁶ In any case, the imprints of the two spin periods may still exist, probably in the form of some quasi-periodic features in the burst arrival times. This prediction can be tested with future repeating FRB data.

The third timescale is the orbital period, which we estimate below. Since PSR A is much more energetic than PSR B, its pulsar wind will significantly distort the magnetosphere of the latter. The pressure balance at the interaction front may be written as

$$\begin{eqnarray}&&\displaystyle \frac{{\dot{E}}_{{\rm{A}}}}{4\pi {r}_{{\rm{A}}}^{2}c}=\displaystyle \frac{{B}_{{\rm{p}},{\rm{B}}}^{2}}{8\pi }{\left(\displaystyle \frac{R}{{r}_{{\rm{B}}}}\right)}^{6},\end{eqnarray} \tag{ 7 }$$

where ${r}_{{\rm{A}}}$ and ${r}_{{\rm{B}}}$ are distances of the interaction front from PSRs A and B, respectively, and a dipolar magnetic field configuration has been assumed for B's magnetosphere. Significant interaction occurs as the separation between the two pulsars is comparable to the size of the distorted B's magnetosphere. This corresponds to ${r}_{{\rm{A}}}\sim {r}_{{\rm{B}}}$. Solving Equation (7), one gets the separation between the two pulsars

$$\begin{eqnarray}&&a\sim 2{r}_{{\rm{A}}}=2{\left(\displaystyle \frac{{B}_{{\rm{p}},{\rm{B}}}^{2}{R}^{6}c}{2{\dot{E}}_{{\rm{A}}}}\right)}^{1/4}\simeq 4.5\times {10}^{9}\ \mathrm{cm}\ {b}_{{\rm{p}},{\rm{B}}}^{1/2}{\dot{e}}_{{\rm{A}}}^{-1/4}.\end{eqnarray} \tag{ 8 }$$

Assuming a circular orbit and again ${M}_{1}={M}_{2}=1.4{M}_{\odot }$, one can derive the orbital period of the system

$$\begin{eqnarray}&&{P}_{\mathrm{orb}}=\displaystyle \frac{4{\pi }^{2}{a}^{3}}{{GM}}\simeq 100\ {\rm{s}}\ {\left(\displaystyle \frac{a}{4.5\times {10}^{9}\mathrm{cm}}\right)}^{3/2}{M}_{2.8}^{-1/2},\end{eqnarray} \tag{ 9 }$$

where $M={M}_{1}+{M}_{2}=(2.8{M}_{\odot }){M}_{2.8}$ is the total mass of the system. It would be interesting to look for a characteristic timescale of this order in the repeating FRB data.

The fourth timescale is the time toward the coalescence, which can be estimated as

$$\begin{eqnarray}&&\tau \simeq 500\ \mathrm{yr}{\left(\displaystyle \frac{{P}_{\mathrm{orb}}}{100{\rm{s}}}\right)}^{8/3}{\left(\displaystyle \frac{2.8{M}_{\odot }}{M}\right)}^{2/3}\left(\displaystyle \frac{0.7{M}_{\odot }}{\mu }\right),\end{eqnarray} \tag{ 10 }$$

where $\mu ={M}_{1}{M}_{2}/M$ is the reduced mass of the binary system. This is the typical lifetime of a repeating FRB source. Noticing the sensitive dependence (index 8/3) on ${P}_{\mathrm{orb}}$, this timescale may range from decades to centuries when a range of PSR parameters $({p}_{{\rm{A}}},{b}_{{\rm{A}}},{p}_{{\rm{B}}},{b}_{{\rm{B}}})$ are considered.

Within this model, the FRBs are conjectured to be produced during sudden reconnection of magnetic field lines. The magnetic geometry of an interacting BNS is complicated. It is difficult to provide concrete predictions on when a burst could be generated. Nonetheless, one may imagine that for certain configurations, magnetic field lines with opposite polarities from the two pulsars would encounter and reconnect, leading to active bursting episodes. The quiescent states correspond to the epochs when the magnetic configurations are not favorable for reconnection, or when the depleted magnetic fields are being replenished. Dedicated numerical simulations may reveal the complicated interaction processes in such systems.

The FRB radiation mechanism is very likely bunching coherent curvature radiation (Katz 2016; Kumar et al. 2017; Yang & Zhang 2018).⁷ In particular, Yang & Zhang (2018) showed that a sudden deviation of the electric charge density from the nominal value (e.g., the Goldreich-Julian value; Goldreich & Julian 1969) would induce coherent bunching curvature radiation. Such a condition is readily satisfied in a dynamically interacting system. The emission configuration is very similar to that of the "cosmic comb" model (Zhang 2017), so that the estimate of the characteristic frequency and duration from that model can be directly applied, i.e.,

$$\begin{eqnarray}&&\nu =\displaystyle \frac{3}{4\pi }\displaystyle \frac{c}{\rho }{\gamma }_{e}^{3}\simeq (7.2\times {10}^{8}\ \mathrm{Hz})\ {\rho }_{10}^{-1}{\gamma }_{e,3}^{3},\end{eqnarray} \tag{ 11 }$$

and

$$\begin{eqnarray}&&{\rm{\Delta }}t\sim \displaystyle \frac{a}{v{\gamma }_{e}}\simeq (3.3\,\mathrm{ms})\ {a}_{10}{\beta }_{-1}^{-1}{\gamma }_{e,3}^{-1}.\end{eqnarray} \tag{ 12 }$$

Here ρ is the curvature radius, which is comparable to the separation a between the two pulsars, ${\gamma }_{e}\sim {10}^{3}{\gamma }_{e,3}$ is the typical Lorentz factor of the electrons accelerated from the reconnection regions, and $\beta =v/c$ is the dimensionless field-line-sweeping velocity of the emission region, which is normalized to ∼0.1 of the light cylinder radius of PSR B.

The predicted FRB luminosity depends on the intrinsic properties of each reconnection and how the beamed emission intersects with the line of sight. Only very energetic events or the events whose beam squarely sweeps across Earth would produce rare, extremely bright FRBs. Most FRBs should be less luminous and would follow a power-law distribution in the apparent luminosity with the concrete power-law index depending on model details. The reconnection-injected particles likely slide along field lines after synchrotron cooling. Coherent bunching curvature radiation is preferentially produced in open field line regions (Yang & Zhang 2018), which may interpret the observed subpulse down-drifting patterns in some bursts (Wang et al. 2019). There should be an associated high-energy emission for each burst with a luminosity ${L}_{\mathrm{HE}}\sim {10}^{43}\ \mathrm{erg}\,{{\rm{s}}}^{-1}\ {\eta }_{-2}$, which depends on the radio efficiency parameter η (normalized to 10⁻²).⁸ A millisecond-duration X-ray or γ-ray burst with such a luminosity at a typical FRB distance is way below the sensitivity of the current high-energy detectors.

Unlike young supernova remnants, BNSs are old systems not surrounded by a matter shell in the immediate environment. As a result, one does not expect a significant contribution to DM from the vicinity of the bursting source. This is consistent with the observations of the FRBs residing in BNS-like environments (Bannister et al. 2019; Ravi et al. 2019; Marcote et al. 2020). Since the variation of other DM components is very small (Yang & Zhang 2017), one does not expect DM evolution in this scenario. This is consistent with the data of repeating FRBs so far (Spitler et al. 2016; Law et al. 2017; Luo et al. 2020a). Observations of significant DM variations would disfavor this model.⁹

The RM, on the other hand, is usually dominated by the immediate environment of the source where magnetic field strength is high. In a dynamically interacting system, one expects a complicated magnetic structure surrounding the system, so that RM, which depends on the integral of the parallel component of the magnetic field, can vary significantly within a short period of time. The evolution is also expected not to be monotonic. This is consistent with the observations of FRB 180301 (Luo et al. 2020a). A BNS system is not expected to produce extremely large RMs. Within this model, the BNS system powering FRB 121102 is located near a supermassive black hole, which gives rise to the abnormally large RM for that source. The secular RM variation could be due to the orbital motion of the system around the black hole (Zhang 2018).

Coherent curvature radiation is intrinsically linearly polarized. Pulsar radio emission shows high linear polarization degrees and a signature sweeping pattern of the polarization angle in the form of "S" or inverse "S" patterns. This has been well interpreted within the rotating vector model (Radhakrishnan & Cooke 1969) where coherent emission originates from the open field line region of an isolated rotating neutron star. In an interacting BNS system, the magnetosphere structure is much more complicated. One would expect the deviation from the simple rotating vector model and diverse polarization angle evolution patterns. These are consistent with the observations of the repeating bursts detected from FRB 180301 (Luo et al. 2020a). Under certain conditions (e.g., similar to the cosmic comb configuration as discussed in Zhang 2018), the emission region may be on nearly straight field lines. As the emission beam sweeps the line of sight, the polarization angle would not show significant evolution within single bursts. The absolute values of the polarization angles should vary among different bursts. Such a feature is not inconsistent with the observations of FRB 121102 (Michilli et al. 2018).

The event rate density of BNS mergers is estimated as ${{ \mathcal R }}_{\mathrm{BNS}}\sim {1.5}_{-1.2}^{+3.2}\times {10}^{3}\ {\mathrm{Gpc}}^{-3}\ {\mathrm{yr}}^{-1}$ from the GW170817 detection (Abbott et al. 2017). That of FRBs above ${10}^{42}\ \mathrm{erg}\,{{\rm{s}}}^{-1}$ is ${{ \mathcal R }}_{\mathrm{FRB}}(\gt {10}^{42}\ \mathrm{erg}\,{{\rm{s}}}^{-1})={3.5}_{-2.4}^{+5.7}\times {10}^{4}\ {\mathrm{Gpc}}^{-3}\ {\mathrm{yr}}^{-1}$ (Luo et al. 2020b), which is ∼20 times higher. Repeating FRBs typically have luminosities below ${10}^{42}\ \mathrm{erg}\,{{\rm{s}}}^{-1}$ (Luo et al. 2020a). Including these faint bursts, the FRB event rate density may be boosted by another $\sim (2\mbox{--}3)$ orders of magnitude. If each BNS merger system produces 10⁵ bursts during its lifetime (our nominal value), one would overproduce FRBs by about (1–2) orders of magnitude. This suggests that either the average total number of bursts produced in BNS systems is lower (i.e., FRB 121102 is abnormally active; e.g., Palaniswamy et al. 2018; Caleb et al. 2019) or some interacting systems cannot produce FRBs because of their unfavorable pulsar parameters.