FRB as Pulsar Lightning

TL;DR

This paper proposes that Fast Radio Bursts (FRBs) are analogous to lightning in planetary atmospheres, occurring in neutron star magnetospheres with vacuum gaps, and introduces the concept of 'electrars' powered by electrostatic energy release.

Contribution

It introduces a novel model linking FRBs to lightning phenomena and suggests they are powered by electrostatic energy, expanding the understanding of FRB energetics and origins.

Findings

FRBs share phenomenological similarities with lightning.

FRBs may occur in neutron star magnetospheres with vacuum gaps.

Energetic constraints imply shorter spin-down lifetimes for more energetic FRBs.

Abstract

There are striking phenomenological similarities between Fast Radio Bursts and lightning in the Earth's and planetary atmospheres. Both have very low duty factors, $\lesssim 10^{-8}$--$10^{-5}$ for FRB and (very roughly) $\sim 10^{-4}$ for the main return strokes in an active thundercloud. Lightning occurs in an electrified insulating atmosphere when a conducting path is created by and permits current flow. FRB may occur in neutron star magnetospheres whose plasma is believed to be divided by vacuum gaps. Vacuum is a perfect insulator unless electric fields are sufficient for electron-positron pair production by curvature radiation, a high-energy analogue of electrostatic breakdown in an insulating gas. FRB may be "electrars" powered by the release of stored electrostatic energy, counterparts to Soft Gamma Repeaters powered by the release of stored magnetostatic energy (magnetars). This frees pulsar FRB models from the constraint that their power not exceed the instantaneous spin-down power. Energetic constraints imply that the sources of more energetic FRB have shorter spin-down lifetimes, perhaps shorter than the three years over which FRB 121102 has been observed to repeat.

Equations

$$n_e=\frac{\vec{B}\cdot {\vec{\omega }}_m}{2\pi ec}\approx 1.1\times 10^{17}{\vec{B}}_{15}\cdot {\vec{\omega }}_{m,4}{\text{cm}}^{-3},$$

$$\Gamma \equiv \frac{B^2/8\pi }{n_e{m}_e{c}^2}=\frac{Be}{4\omega m_{e}c}\approx 4.4\times 10^{17}\frac{B_{15}}{{\omega }_4}⋙1.$$

$$\vec{E}+\frac{{\vec{\omega }}_m\times \vec{r}}{c}\times \vec{B}=0,$$

$${\mathcal{E}}_{el}\sim \frac{{\omega }_m^2{B}^2{R}^5}{2c^2}\sim 5\times 10^{46}{\omega }_m^2{B}_{15}^2\text{ergs},$$

$$\Delta {\mathcal{E}}_{el}\sim ({\omega }_{m,i}^2-{\omega }_{m,f}^2)\frac{B^2{R}^5}{2c^2}\sim 5\times 10^{46}\Delta {\omega }_4^2{B}_{15}^2\text{ergs},$$

$${\mathcal{E}}_{cap}\sim \frac{{\omega }^4{B}^2{R}^8}{2h{c}^4}\sim 5\times 10^{45}\frac{R}{h}{B}_{15}^2{\omega }_4^4\text{ergs},$$

$$E\sim \frac{{\omega }^2{R}^{3}B}{h{c}^2}\sim 10^{14}\frac{R}{h}{\omega }_4^2{B}_{15}.$$

$$E_{SHES}\approx 0.05\frac{m_e^2{c}^3}{e\hslash }\approx 2\times 10^{12}.$$

$${\mathcal{E}}_{cap}\sim \frac{1}{2}h{R}^2{E}_{SHES}^2\sim 2\times 10^{42}\frac{h}{R}\text{ergs}.$$

$$\begin{array}{rl}{\nu }_p& =\frac{1}{2\pi }\sqrt{\frac{4\pi n_e{e}^2}{m_e}⟨\frac{1}{{\gamma }^3}⟩}=\frac{1}{2\pi }\sqrt{\frac{2e\vec{B}\cdot {\vec{\omega }}_m}{m_{e}c}⟨\frac{1}{{\gamma }^3}⟩}\\ & \approx 3.0\times 10^{12}\sqrt{{\vec{B}}_{15}\cdot {\vec{\omega }}_{m,4}⟨\frac{1}{{\gamma }^3}⟩}\text{Hz},\end{array}$$

$$\tau =\frac{3c^{3}I}{2B^2{\omega }^2{R}^6}\approx \frac{4\times 10^2\text{s}}{B_{15}^2{\omega }_4^2},$$

$$\tau =\frac{3}{2}\frac{cI{e}^2}{(2\pi {\nu }_p{)}^4{m}_e^2{\gamma }^6}\approx 8\times 10^{15}{\left(\frac{{\nu }_p}{\text{1GHz}}\right)}^{-4}{\gamma }^{-6}\text{s}.$$

$$\tau \approx 2\times 10^9\frac{\Delta ϵ}{{\mathcal{E}}_{FRB40}}\text{s}$$

Full text

††pubyear: 2017††pagerange: Fast Radio Bursts as Pulsar Lightning–References

1 Introduction

The mechanism and astronomical sites of FRB emission remain mysterious, even after the discovery of the repeating FRB 121102 and its identification with a dwarf galaxy at redshift $z=0.19$ (Chatterjee, et al., 2017). The requirement that energies of as much as $10^{40}$ ergs, assuming isotropic emission (Thornton, et al. (2013); see, however, Katz (2017)) be radiated in $\lesssim 1$ ms points to sources in regions of high power and high energy density (Katz, 2016a). The natural candidates are neutron stars, and possible energy sources are magnetostatic, as in Soft Gamma Repeaters (SGR), or rotational, as in radio pulsars.

SGR have low duty factors and short rise times analogous to those of FRB, but their candidacy suffers from the fact that the radiation of SGR is almost entirely thermal soft gamma rays. Although at least one SGR has been observed as a radio source (Frail, Kulkarni & Bloom, 1999), its emission lasted for weeks, resembled that of an expanding cloud rather than a fast transient, and offers no explanation of the FRB phenomenon. The environment of SGR has an intense flux of thermal radiation (up to $10^{34}$ erg/cm2/s at the neutron star, corresponding to a black body temperature of $3\times 10^{9\,\circ}$K or 300 keV) which is inhospitable to coherent (or any nonthermal) radio radiation because of rapid Comptonization of energetic electrons. In addition, one FRB was fortuitously the subject of radio observation (Tendulkar, Kaspi & Patel, 2016) during its giant outburst, but no FRB was observed. As a Galactic object, had it been the source of a FRB an extraordinarily strong signal, even in the far sidelobes of a radio telescope, would have been expected (for possible loopholes see Katz, 2016b).

Giant pulses from radio pulsars may be more plausible explanations of FRB. The fact that nanoshots from two Galactic radio pulsars (Soglasnov, et al., 2004; Hankins & Eilek, 2007) had brightness temperatures exceeding even those of FRB points in this direction. No detailed mechanism has been proposed, but after nearly 50 years in which theorists have failed to present a convincing detailed mechanism of even “ordinary” coherent radio pulsar emission, this should not be a fatal objection. Even the basic electrodynamics of pulsar magnetospheres remains controversial (Michel, 1990, 2003; Melrose & Yuen, 2016).

Pulsar magnetospheres contain “gaps” in which ${\vec{E}}\cdot{\vec{B}}\neq 0$ (Melrose & Yuen, 2016). In these gaps intense electric fields accelerate electrons and positrons. Their curvature radiation consists of gamma-rays of sufficiently high energy to produce further electron-positron pairs by a single-photon process in the strong neutron star magnetic field (Ruderman & Sutherland, 1975). The purpose of this paper is to point out an analogy between the breakdown of a resistive atmosphere in a lightning stroke and the breakdown of the resistive vacuum in a gap in a pulsar magnetosphere.

This is only an analogy. The fundamental physical processes are different and the parameter regimes differ by many orders of magnitude. Estimates are necessarily qualitative, or, at best, order-of-magnitude. Even were it possible to construct a more quantitative theory, the range of plausible pulsar parameters is several orders of magnitude in both spin frequency and magnetic field. It includes such hypothetical objects as neutron stars with both surface fields of $\sim 10^{15}$ gauss and millisecond periods. The constraint of a lifetime $>3\,$y inferred from the repeating FRB 121102 need not apply to non-repeating FRB that might spin down much faster.

IS91a; IS91b have suggested “lightning” triggered by external $\gamma$-rays as a source of RRAT emission. An important difference between their work and this paper is the suggestion here that “lightning” is powered by stored electrostatic energy, and thus may involve and radiate a power greater, perhaps by orders of magnitude, than the spindown power. This is likely necessary to explain FRB, but not to explain RRAT.

Although there is a fairly detailed semiquantitative understanding of terrestrial cloud charging and lightning, it is unlikely that we would have predicted their existence had we not known of it. This should remind us that pulsar magnetospheres may also behave in unexpected ways. Unable to predict their phenomenology, our understanding may be limited to delineating the range of possibilities permitted by physical laws.

2 Lightning

Lightning has been the subject of scientific study since the work of Ben Franklin nearly 300 years ago. The historic and modern literatures are reviewed by Uman (1969) and Rakov & Uman (2003). Cumulonimbus clouds (thunderheads) are charged by differential motion among oppositely charged water droplets, graupel (partially frozen hail) and ice. Potential drops are $\sim 10^{8}\,$V over 5–10 km and discharges carry $\sim 10$ Coulombs, releasing $\sim 10^{9}\,$J of electrostatic energy. This is $\gtrsim 10\%$ of the total electrostatic energy of the cloud, a capacitor of dimension $\sim 10\,$km in each of three dimensions (capacitance $\sim 1\,\mu$F). Not only does the air break down along the visible channel of the lightning stroke (bolt), but there must be sufficient conductivity outside the channel itself to collect a substantial fraction of the charge distributed throughout the cloud.

3 Pulsar Magnetospheres

Much less is known about pulsar magnetospheres. Unlike thunderclouds, they cannot be imaged or probed; we only observe their radiation that is directed toward us and its variation as the neutron star rotates. Both their structure and the mechanisms of production of pulsar radiation remain controversial, nearly 50 years after their discovery (Michel, 1990, 2003; Melrose & Yuen, 2016).

It is generally assumed that the magnetic fields of pulsar magnetospheres are determined by currents within the dense matter of the neutron star that change only on a very long ($>10^{6}\,$y) resistive time scale. This is in contrast to SGR, that are believed to result from the sudden release of magnetostatic energy in a reorganization of the magnetic field (Katz, 1982; Thompson & Duncan, 1992, 1995) resulting from reconnection of magnetospheric currents. These currents may be relics of the formation of the neutron star that dissipate in $\lesssim 10^{4}\,$y, the empirical lifetimes of SGR (Katz, 2016b).

3.1 States

If there are two or more possible states, with different charge distributions and electrostatic energies, of a pulsar magnetosphere, with the more energetic state metastable, then there may be sudden transitions between them with the release of their energy difference. This would be analogous to the transition of a thundercloud from a more to a less energetic charge distribution as charge flows in a lightning bolt.

Pulsar magnetospheres are highly relativistic. The characteristic plasma density of a co-rotating magnetosphere with angular velocity $\omega_{m}$ (Davis, 1947; Deutch, 1955; Hones & Bergeson, 1965; Goldreich & Julian, 1969) is

[TABLE]

where $B_{15}\equiv B/10^{15}\,\text{gauss}$ and $\omega_{m,4}\equiv\omega_{m}/10^{4}\,\text{s}^{-1}$. The ratio of its magnetic energy density to its characteristic rest mass energy density

[TABLE]

This is very large for all pulsars, even low field millisecond pulsars.

Because magnetospheres are relativistic, the release of energy in a transition between two states occurs in a time $\sim R/c\sim 3\times 10^{-5}\,$s, where $R$ is the neutron star’s radius, consistent with empirical upper bounds to FRB widths at their sources (before propagation broadening). There is no generally accepted solution for pulsar magnetospheres (much less two solutions, one of which is more energetic but metastable) so we only make order of magnitude estimates.

The fundamental equation of a co-rotating force-free magnetosphere is

[TABLE]

where the co-rotation frequency $\omega_{m}$ need not be the same as the neutron star’s rotation frequency $\omega$. Ruderman & Sutherland (1975) present an explicit solution for an aligned rotator with $\omega_{m}\neq\omega$. Within a connected corotating region ${\vec{E}}\perp{\vec{B}}$; $E_{\parallel}=0$. Distinct corotating regions with differing ${\vec{\omega}}_{m}$ are separated by gaps in which $E_{\parallel}\neq 0$.

From Eq. 3 the electrostatic energy may be estimated

[TABLE]

and the energy released in a transition between an initial $\omega_{m,i}$ and a final $\omega_{m,f}$

[TABLE]

where the fractional change in $\omega_{m}^{2}$ $\Delta\equiv(\omega_{m,i}^{2}-\omega_{m,f}^{2})/\omega^{2}$. $\Delta$ is unknown, as is the efficiency $\epsilon$ of conversion of this energy into coherent FRB emission. Both may be $\ll 1$, but if both $\omega$ and $B$ are large there may still be sufficient energy to power FRB.

3.2 Capacitor

Alternatively, treat the magnetosphere as a capacitor with capacitance $C\sim R^{2}/h$, where $h$ is the width of its gap, charged to the characteristic voltage in a polar cap model (Ruderman & Sutherland, 1975) $V\sim\omega^{2}R^{3}B/c^{2}$. The resulting energy would be

[TABLE]

The value of $h$ is unknown, though it cannot exceed $R$. For sufficiently strongly magnetized and rapidly rotating neutron stars the electrostatic energy may be more than sufficient to power FRB, the most energetic of which radiated, assuming isotropic emission, $10^{40}$ ergs, even if the efficiency $\epsilon\ll 1$.

3.3 Parameters

The required values of $B$ and $\omega$ are found in known pulsars, but not in combination. That is attributed to their ages (Eq. 11, below). A pulsar-FRB might be as young as the time required for a supernova remnant to dissipate to transparency, and therefore could have both large $B$ and large $\omega$. Neutron star formation by collapse without expulsion of a remnant or a visible supernova may also be possible (codes have had more difficulty producing a supernova than neutron star formation without one). If so, the only lower bound on the duration of FRB activity would be the observed activity of a repeating FRB, permitting simultaneously large $B$ and large $\omega$. The duration could be even shorter, and $B$ and $\omega$ even larger, for non-repeating FRB, for which no lower bound on their duration of activity can be established.

3.4 Vacuum Breakdown

In the absence of a magnetic field, very large electric fields are not sustainable. For example, Eq. 6 assumes an electric field (in cgs units, statvolts/cm)

[TABLE]

If $\omega_{4}^{2}B_{15}>0.02h/R$ this exceeds the Sauter-Heisenberg-Euler-Schwinger (Sauter, 1931; Heisenberg & Euler, 1936; Schwinger, 1951) vacuum pair breakdown field (in statvolts/cm)

[TABLE]

Pair production cascades driven by curvature radiation may break down the insulating vacuum at lower (but still very high) fields (Ruderman & Sutherland, 1975). “Schwinger sparks”, produced by fields exceeding the limit of Eq. 8, have been proposed (Stebbins & Yoo, 2015) as the source of pulsar nanoshots. The FRB model proposed here does not require that limit to be exceeded.

If the available electrostatic energy in a capacitor model is limited by the energy density $E_{SHES}^{2}/8\pi$, then

[TABLE]

This is less than the limit of Eq. 6 for rapidly spinning or strongly magnetized neutron stars, particularly if $h$ is small. It is still sufficient to power the observed FRB provided $h\sim 10^{4}$ cm (similar to gap widths estimated by Ruderman & Sutherland (1975)), the electrostatic energy difference between initial and final configurations is comparable to that given by Eq. 9, and emission is efficient and roughly isotropic. Collimated emission (Katz, 2017) would mitigate the energy requirement. The efficiency is, of course, unknown.

There can be no pair production unless $E_{\parallel}\neq 0$. If $E_{\parallel}\neq 0$ in one frame, the Lorentz invariance of ${\vec{E}}\cdot{\vec{B}}$ implies that $E_{\parallel}\neq 0$ in all frames. In gaps, where there is a parallel component of $\vec{E}$, breakdown can occur and $E$ has a limit like Eq. 8 (Kim & Page, 2006).

If there be a magnetic field with $B^{2}>E^{2}$ and ${\vec{E}}\perp{\vec{B}}$, as expected in a neutron star magnetosphere outside of “gaps”, the Lorentz invariance of $B^{2}-E^{2}$ and of ${\vec{E}}\cdot{\vec{B}}=0$ implies the existence of a frame in which ${\vec{E}}=0$, and there can be no pair production. Pair production cannot set an upper bound on $E$, other than $B$, in corotating regions in which ${\vec{E}}\perp{\vec{B}}$. The limit of Eq. 9 is not applicable to the electrostatic energy stored, and released, in regions where ${\vec{E}}\perp{\vec{B}}$. Energies as great as those of Eq. 5 may be available to power FRB.

4 Plasma Frequency

FRB might be produced by conversion of longitudinal electrostatic plasma waves, propagating along the magnetic field (with the same dispersion relation as in a nonmagnetic plasma), to transverse electromagnetic waves. The plasma frequency corresponding to the characteristic plasma density Eq. 1, allowing for the possibility that the electrons have relativistic speeds, reducing their acceleration by electric fields parallel to their velocities (that are parallel to $\vec{B}$) by a factor $\gamma^{-3}$, is

[TABLE]

where $\gamma$ is a mean Lorentz factor of electron or positron motion along the magnetic field, assumed to be the direction of propagation of the longitudinal plasma wave.

5 Spin-Down Time

The spin-down time of a rotating magnetic neutron star, taking the magnetic moment as perpendicular to its angular momentum, is

[TABLE]

where $I\approx 10^{45}\,$g-cm2 is its moment of inertia. The product $B\omega$ appears as in Eq. 10 for the plasma frequency, so that $\tau$ can be expressed as a function of $\nu_{p}$:

[TABLE]

If the plasma frequency is that of nonrelativistic electrons, then the spin-down time is comparable to that of a typical observed Galactic pulsar. Such pulsars are numerous, with $\sim 10^{4}$ in the Galaxy, while FRB are extremely rare, with a detected event rate $\sim 10^{-5}$/galaxy-y, arguing against the hypothesis that FRB are produced by pulsars resembling observed Galactic pulsars. If the plasma is relativistic, the sensitivity of $\tau$ to $\gamma$ is so extreme that no quantitative conclusion can be drawn from Eq. 12.

Eq. 11 can be compared to the observed lower bound on the lifetime of the repeating FRB 121102 to infer $\omega_{4}^{2}B_{15}^{2}<4\times 10^{-6}$. Comparing to Eq. 5 then leads to an upper bound on the available energy $\sim 10^{41}$ ergs, consistent with the observed energy of its FRB of $\sim 10^{38}$ ergs. Equivalently, the spin-down time of a pulsar emitting FRB of energy ${\cal E}_{FRB}$ with efficiency $\epsilon$ is

[TABLE]

where ${\cal E}_{FRB40}\equiv{\cal E}_{FRB}/10^{40}$ ergs. Unfortunately, we cannot estimate $\Delta$ or $\epsilon$, and if FRB emission is collimated (Katz, 2017) ${\cal E}_{FRB}$ may be much less than that inferred with the assumption of isotropic emission. Therefore, it is not possible to make quantitative predictions from Eq. 13.

6 Discussion

This paper proposes that FRB are “electrars” powered by electrostatic energy, in analogy to SGR that are powered by magnetostatic energy (“magnetars”). In each case energy is released by a transition between two configurations, the more energetic of which is metastable. Neither of these phenomena were predicted before their discovery; Nature is cleverer than scientists at finding ways to tap stored energy.

The energetic requirements may be met in the magnetospheres of fast high-field pulsars. None of these are known, but their spin-down lifetimes would be very short, conceivably as short as minutes, and most may be hidden within opaque young supernova remnants. This is consistent with the extreme rarity of FRB.

A weakness of the hypothesis suggested here is that there is no theory that demonstrates the existence of two (or more) magnetospheric configurations with distinct energies, nor that such an energy difference can be released suddenly. Without an understanding of the magnetospheric structure of pulsars that may not be a fatal flaw, but it is a serious one. There is, however, a precedent from SGR: Although any magnetosphere has a lower energy state in which there is no magnetic field, a priori the only path to it is by very slow resistive dissipation that does not make outbursts. The fact that sudden magnetic reconnection can and does occur is empirical and not based on fundamental understanding. Perhaps electrostatic energy too may be released in a burst, as it is in a thundercloud.

Combination of Eq. 5 with Eq. 11 implies that the more energetic FRB are likely to have lifetimes much shorter than the lower bound of 3 years observed for the repeating FRB 121102. This conclusion cannot be quantified without some knowledge of $\Delta$ and $\epsilon$, but it is qualitatively consistent with the absence of repetitions of more energetic FRB. The bursts of FRB 121102 had particularly low energy; the mean fluence of 17 bursts (Spitler, et al., 2016; Scholz, et al., 2016) was 0.275 Jy-ms. Assuming isotropic emission at $z=0.19$, the mean energy radiated, integrated over the 1214–1537 MHz band of observation, was $1.0\times 10^{38}\,$ergs, less than the inferred energies of $\sim 3\times 10^{38}\text{--}10^{40}\,$ergs of other FRB.

The efficiency $\epsilon$ of coherent radio emission of pulsars is low. However, when the bunching of charges is extreme, as it must be to explain FRB, radiation is rapid. For example, KLB17 estimate energy loss times $\sim 10^{-15}\,$s. Then, in order for the emission to be at GHz frequencies (rather than $\sim 10^{15}$/s) radiative losses must be very closely balanced by the accelerating electric field on the energy loss time scale. This explains the efficient radiation as a self-consistent property of extreme charge bunching.

If more energetic FRB are produced by pulsars with particularly short spin-down times, searches for repetitions must be conducted immediately after their first occurrence. The observability of FRB may depend on the rapid dissipation of a supernova remnant, a patchy remnant, or neutron star formation without a supernova at all. These requirements may help explain the fact that the FRB rate, despite the possibility of repetitions, is less than $10^{-3}$ of the supernova rate.

The broad range (approximately -10 to +14) of spectral indices observed (Spitler, et al., 2016) in bursts from the repeating FRB 121102 might appear to indicate comparatively narrow-band emission with power peaking at frequencies sometimes above, sometimes below, and sometimes in the middle of the band of observation. However, the rapid variation of burst widths indicates that scintillation is strong. Spitler, et al. (2016) report widths of $6.6\pm 0.1\,$ms and $3.06\pm 0.04\,$ms for two bursts (their bursts 8 and 11) 413 s apart. A factor of two variation in broadening suggests ${\cal O}(1)$ variation in fluence and frequencies of spectral peaks and minima, indicating that the variation in fitted spectral indices is the result of scintillation, not an intrinsic property of the source.