Preamble

Pinch instability: an alternative mechanism for pulsar coherent radio radiation Shuang Du , Renxin Xu

ABSTRACT

The provenance of the pulsed radio emissions emanating from pulsars has long been a subject of enduring and intricate perplexity. A common viewpoint posits that the production of this coherent radio radiation necessitates robust electric currents. Noting that pinch instability is likely to occur concomitantly with the emergence of intense electric currents, this manuscript delves into an exploration of the conditions under which pinch instability develops within the magnetospheres of pulsars. We find that, under typical pulsar parameters, pinch instability could be able to develop effectively, and the charged bunches necessary for pulsed coherent radio radiation would subsequently be generated as a result of this type of instability.

Subject headings: pulsars, radio pulsars, radio astronomy

1. Introduction

Given the effect of unipolar induction and the excellent electrical conductivity of s- tars, pulsar magnetospheres should be replete with charged particles (Goldreich & Julian 1969). As indicated by the existence of pulsar wind nebulae (Rees & Gunn 1974) and the requirement to produce pulsed electromagnetic radiation (Sturrock 1971), pulsars should be constantly injecting a large amount of charged particles into their surroundings. There- fore, to maintain the plasma environment, pulsars and their magnetospheres must be able to continuously generate charged particles. Such a dynamic magnetosphere may allow the development of some plasma instabilities, for example, the widely discussed two-stream in- stability. 1 School of Mathematics and Computer Science, Tongling University, Tongling, 244000, Anhui, China;

For pulsar physics, the two-stream instability is invoked to explain pulsar pulsed radio radiation (Ruderman & Sutherland 1975). However, it remains uncertain whether this type of instability can be effectively developed within pulsar magnetospheres (Melrose 2017). The newly-developing forceful scenario presents that the fluctuating process involving intermit- tent sparks of electron-positron production can directly drive coherent radio radiation of pulsars (Philippov et al. 2020; Mestel et al. 1985; Beloborodov 2008; Timokhin & Arons 2013). While this charge-fluctuation scenario has not been fully fleshed out , the usually im- plicit assumption that coherent curvature radiation emanating from charged bunches formed through mechanisms such as the two-stream instability (Usov 1987) no longer appears to be the resigned choice. Regardless of whether it is the mentioned charge-fluctuation scenario or the bunch scenario, the fundamental requirements stay consistent, namely, electron-positron pair cascades should be initiated in a strong electric field. Hence, whether the demand stems from observational or theoretical perspectives, strong electric currents flowing along pulsar magnetic fields ought to emerge within pulsar magnetospheres. Given that such a current will induce a toroidal magnetic component, it is natural to ponder: can pinch instabilities effectively develop within a pulsar magnetosphere? Furthermore, can this kind of instability segment these electric currents into the bunches necessary for coherent radio radiation?

In the next section, by employing the inner gap model (Ruderman & Sutherland 1975), we show that the current bunches necessary for coherent curvature radiation in radio band could be produced through pinch instabilities within pulsar magnetospheres, coincidently, under typical pulsar parameters.

2.1. The basic physical picture

Within the framework of the inner gap model, the electromotive force provided by u- nipolar induction must surpass the voltage required by sparks. Consequently, the thickness of the gap will exceed the mean free path for electron-positron pair conversions of the cur- vature photons emitted by initial charged particles. Therefore, secondary electron-positron pairs would be spatially separated by the electric field of the gap. As shown in Figure 1 [FIGURE:1], supposing that electron-positron pair cascades in a pulsar magnetosphere result in an elec- tric current with density flowing along the pulsar magnetic field with strength . The current induces a toroidal magnetic component, . According to the principle of magnet-

ic flux conservation, there is a rough critical condition for the shrink of the current that 2, where is the radius of the current. Since the current is gradually increas- ing during the cascading process, at least before a certain point in time, the two components of the magnetic field satisfy 2 at the boundary of the current. Therefore, the pinch instability is restrained initially. However, the pulsar magnetic field may decay as , where is the magnetic field strength on the pulsar surface and is the radius of the pulsar, the condition, 2, could be satisfied in the pulsar magnetosphere finally, and then the pinch instability is able to develop.

When the current passes through a point where is 2, the current begins to shrink. A disturbance of the current radius, ), perpendicular to the pulsar magnetic field begins to grow, where is the time. If the pinch instability can develop effectively, the value of ) should achieve during the cross time , where is the length of the current and is the speed of light. One of the normal methods to discuss the development of a instability is the normal mode. For simplicity, we are only interesting in the solution of ) with the formalism: where is the angular frequency and is the wave number . Considering a bulk of charged particles moving along the magnetic field approximately uniformly distributed in the direc- tion perpendicular to the axis, the dispersion relation corresponding to equation (1) is given by (Galtier 2016) where is the permeability of vacuum, is the mass density of the current, is the pulsar magnetic field before the perturbation, is the toroidal magnetic component induced by the current before the perturbation, are the modified Bessel functions with the subscripts being their orders.

2.2. The parameter estimation

If the perturbation can be effectively developed, so that the current can be segmented into bunches, the relation, 1, ought to be satisfied. To ensure radiation from these bunches be coherent in radio band, the bunch length should be, at least, Consequently, the wave number of the perturbation is Since

be soon satisfied after the current passes through the point where , where is the charge density of the bunch being the Goldreich-Julian density, equation (3) can be rewrote as

η = 2 B s R 3 ∗ u 0 ρ GJ cr 0 z 3

Taking the typical pulsar parameters, where the rotation period = 1s, the magnetic field strength on the surface G, and the radius cm, the numerical dependence between is depicted in Figure 2 [FIGURE:2].

As shown in Figure 2, if we adopt , the same as the usually optimistic expectation of the cascade pair multiplicity (Timokhin & Harding 2015; de Jager et al. 1996; de Jager 2007; Bucciantini et al. 2011), the radius of the current should be cm under The width of the current can be contributed by two ways: (i) the beam width of the curvature radiation from charge particles; (ii) the broadening associated with the conversion of photons emitted via synchrotron radiation of secondary charged particles.

The beam width of curvature radiation can be estimated as Γ = 10 where cm is the mean free path of the electron-positron pair conversion of the initial curvature photons, and Γ = Γ is the Lorentz factor of the initial charged particles.

For secondary charged particles, the broadening of the current caused by the gyration radius of these particles is so minuscule that it is disregarded in the subsequent estimation. The primary contribution to the broadening is the mean free path of the conversion of photons emitted via synchrotron radiation, which is given by (Erber 1966; Shukre & Radhakrishnan 1982)

= 10 − 6 B − 1 12 exp ( 60 B 12 ξ

where is the fine-structure constant, is the reduced Planck constant, is the rest mass of electrons, is the energy of the synchrophotons, 2 If the radio radiation height is too low, we must expect that the value of increase significantly for single bunches.

-axis (parallel to . The number density corresponding to the Goldreich-Julian density adopted in the calculation is . The height of radio emission is set from cm to cm (Mitra 2017). The wave number is = 60m . To insure a non-relativistic perturbation, the radius of the current should be smaller than the length of the current ( 10cm).

G is the effect magnetic field strength for the conversion. According to equation (5), to keep cm, there should be . Since the peak energy of synchrotron radiation is 0 , where is the peak frequency with being the elementary charge, the effect Lorentz factor for synchrotron radiation of secondary positron- electron pairs should be, at least,

γ ± > 17 . 8 m 2 e c 3

G, there is

750. By considering the radiation reaction of curvature

radiation , the acceleration of electrons and positrons in the gap is approximatively where is the electric field intensity of the gap, and is the radius of curvature of the magnetic field lines on the pulsar surface. For a typical gap with potential V, electrons and positrons cannot be able to achieve the saturation Lorentz factor, Γ , determined by . Therefore, the Lorentz factor of the primary electrons and positrons can be estimated as , and the corresponding energy of the typical curvature photons can be up to MeV, where cm. The effect Lorentz factor for syn- chrotron radiation of the electron-positron pairs converted from the initial curvature photons ) = 10 . Hence, the condition 750 can be achieved by the most of secondary particles. According to the exponential term in equation (5), the distribution of the charged particles converted from synchrophotons emitted by secondary positron-electron pairs with the momentum perpendicular to the pulsar magnetic field is unconsoli- dated, so that the contribution of these particles to the current is negligible. Therefore, it is possible to achieve a dense current with the radius being cm. Consequently, provided that the spatial distribution of the initial particles for sparks and the sparks themselves are inhomogeneous, it is reasonable to expect the generation of a series of snatchy currents in the magnetosphere.

2.3. The stretch of bunches

In the preceding subsection, we have shown that, under some typical values of pulsar parameters that = 1s, , and cm, the 3 The impeding of resonant inverse Compton scattering Sturner (1995); Zhang & Harding (2000) can be secondary as long as the acceleration of the primary particles passes through the epoch with Γ Timokhin & Harding (2015).

bunches necessary for coherent radio radiation (at cm) can be formed due to pinch instability. However, velocity dispersion, ∆ , of the charged particles in a bunch will stretch the bunch, so that the superposition of phase positions of radio waves emitted by the bunch may turn into incoherent. According to the relation between velocity, , and Lorentz factor, , that , we simply have ∆ , where ∆ is the dispersion of Lorentz factors of the particles in the bunch. Considering a maximum radiation altitude cm, the stretch, 10cm, gives ∆ 10 under

800. Even

the distribution function of particles in the bunch is unknown, the nearly monochromatic requirement seems to be unpractical.

Two elements may relax above difficulty. (i) Although the Lorentz factors of the charged particles within a bunch can span a broad range, the bunch can maintain a dense core, provided that the Lorentz factors of the majority of particles are confined within a relatively narrow range. (ii) As the strength of the pulsar’s magnetic field diminishes, pinch instability has the potential to recur, as long as the dense core of the bunch retains sufficient density.

As shown in Figure 2, if the maximum radiation altitude is cm, when a bunch with reaches this point, the shrink of the bunch still can grow effectively. Therefore, even in the presence of velocity dispersion and a notable decline in the number density of particles within the bunch over time (from ), pinch instability can still occur due to the effect decay of the pulsar magnetic field.

Comparing cm and cm, even if particles are evenly distributed within a range of ∆ 100, it remains feasible to form appropriate bunches. This significantly eases the aforementioned difficulty. From an observational perspective, the brightness temperature of the radio radiation is expected to undergo a significant reduction at high latitudes.

3. Summary and discussions

In this manuscript, we discussed the effect of pinch instability on pulsar pulsed radio radiation. We found that under the typical values of pulsar related parameters, pinch insta- bility can result in the formation of current bunches which may be the launcher of coherent radio radiation.

The above discussion is presented in the framework of the inner gap model (Ruderman & Sutherland 1975). Under the slot gap model (Arons & Scharlemann 1979), the pinch instability is still able to develop provided that the current is sufficiently intense. could be the difference between the mechanism discussed in this manuscript and the work of Philippov et al.(Philippov et al. 2020). Because, if coherent radio radiation is triggered by charge fluctuations, under the inner gap model, the radio radiation ought to occur near

pulsars surfaces, which may conflict with the reported high-altitude radio emissions (Mitra 2017; Sun et al. 2025).

Clarifying above issue holds fundamental significance for pulsar physics due to the treat- ment of the binding energies of the electrons and ions in pulsars surfaces. If both the charge- fluctuation scenario and the reported high-altitude radio radiation are reliable, the inner gap model should be only alive in bare strange stars (Xu et al. 1999). Even so, we still require an additional channel to produce potential high-altitude radio radiation emanating from bare strange stars, such as the mechanism proposed in this manuscript.

Acknowledgements This work is supported by a research start-up fund of Tongling University, and by National SKA Program of China (2020SKA0120100).

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