By the construction of $\tau_{\epsilon}$ and the strong solution of $X$,

$\displaystyle\tau_{\epsilon}$ $\displaystyle=\inf\\{0\leq s\leq
T-t:X_{t+s}^{x+\epsilon}\leq b(t+s)\\}$ $\displaystyle=\inf\bigg{\\{}0\leq
s\leq T-t:W_{s}\leq\frac{\log{\frac{b(t+s)}{x+\epsilon}}-\beta
s}{\sigma}\bigg{\\}}=\inf\\{0\leq s\leq T-t:-W_{s}\geq g(s)\\},$

where $\beta=r-\frac{\sigma^{2}}{2}$ and
$g(s)=\frac{-\log{\frac{b(t+s)}{x+\epsilon}}+\beta s}{\sigma}$. As
$\epsilon\to 0$ (recalling that $x=b(t)$),

$\displaystyle g(s)\to\frac{-\log{\frac{b(t+s)}{b(t)}}+\beta
s}{\sigma}\leq\frac{\beta s}{\sigma},$

where the inequality is from the map $t\mapsto b(t)$ being increasing and
since $s\mapsto\frac{\beta s}{\sigma}$ is the lower function of the Brownian
motion, so is $g(s)$, implying that $P(\tau_{\epsilon}=0)=1$, which is
precisely the claim. ∎

#### 9\. The proof of equation (3.81):

By the definition of $U$, the strong Markov property of $X$ and smoothing
lemma, we see that

$\displaystyle
E_{t,x}\left(e^{-rs}U\left(t+s,X_{t+s}\right)-\int_{0}^{s}e^{-ru}H\left(t+u,X_{t+u}\right){I}\\{X_{t+u}<c(t+u)\\}du\right.$
$\displaystyle\qquad-\left.\frac{1}{2}\int_{0}^{s}e^{-ru}{I}\\{c(t+u)\geq
L\\}\bigl{(}G_{x}\left(t+u,L+\right)-G_{x}\left(t+u,L-\right)\bigr{)}dl_{u}^{L}(X)\right)$
$\displaystyle=E\left(e^{-rs}E\left(e^{-r(T-t-s)}G(T,X_{T})-\int_{t+s}^{T}e^{-r(u-t-s)}H\left(u,X_{u}\right){I}\\{X_{u}<c(u)\\}du\right.\right.$
$\displaystyle\qquad\qquad\qquad\,\,\,\,\left.\left.-\frac{1}{2}\int_{t+s}^{T}e^{-r(u-t-s)}I\\{c(u)\geq
L\\}\bigl{(}G_{x}\left(u,L+\right)-G_{x}\left(u,L-\right)\bigr{)}dl_{u}^{L}(X)\right.\bigg{|}\mathcal{F}_{t+s}\right)$
$\displaystyle\qquad\,\,\,\,\,\,\,\,-\left.\int_{t}^{t+s}e^{-r(u-t)}H\left(u,X_{u}\right){I}\\{X_{u}<c(u)\\}du\right.$
$\displaystyle\qquad\,\,\,\,\,\,\,\,\,-\left.\frac{1}{2}\int_{t}^{t+s}e^{-r(u-t)}{I}\\{c(u)\geq
L\\}\bigl{(}G_{x}\left(u,L+\right)-G_{x}\left(u,L-\right)\bigr{)}dl_{u}^{L}(X)|\mathcal{F}_{t}\right)$
$\displaystyle=E_{t,x}\left(e^{-r(T-t)}G(T,X_{T})-\int_{t+s}^{T}e^{-r(u-t)}H\left(u,X_{u}\right){I}\\{X_{u}<c(u)\\}du\right.$
$\displaystyle\qquad\,\,\,\,\,\,\,\,\,\,\left.-\frac{1}{2}\int_{t+s}^{T}e^{-r(u-t)}I\\{c(u)\geq
L\\}\bigl{(}G_{x}\left(u,L+\right)-G_{x}\left(u,L-\right)\bigr{)}dl_{u}^{L}(X)\right.$
$\displaystyle\qquad\,\,\,\,\,\,\,\,\,\,-\left.\int_{t}^{t+s}e^{-r(u-t)}H\left(u,X_{u}\right){I}\\{X_{u}<c(u)\\}du\right.$
$\displaystyle\qquad\,\,\,\,\,\,\,\,\,\,-\left.\frac{1}{2}\int_{t}^{t+s}e^{-r(u-t)}{I}\\{c(u)>L\\}\bigl{(}G_{x}\left(u,L+\right)-G_{x}\left(u,L-\right)\bigr{)}dl_{u}^{L}(X)\right)$
$\displaystyle=E_{t,x}\left(e^{-r(T-t)}G(T,X_{T})-\int_{0}^{T-t}e^{-ru}H\left(t+u,X_{t+u}\right){I}\\{X_{t+u}<c(t+u)\\}du\right.$
$\displaystyle\qquad\qquad\,\,\,\,\,\,\,\,\,\,\left.+\frac{K-L}{2}\int_{0}^{T-t}e^{-ru}I\\{c(t+u)\geq
L\\}Z_{x}(t+u,L-)dl_{u}^{L}(X)\right)=U(t,x),$ (A.88)

exhibiting the martingale property.


# The novel mechanism of pair creation in pulsar magnetospheres

Z.N. Osmanov,1,2 G.Z. Machabeli3 & N. Chkheidze3
1School of Physics, Free University of Tbilisi, 0183, Tbilisi, Georgia
2E. Kharadze Georgian National Astrophysical Observatory, Abastumani 0301,
Georgia
3 ITP, Ilia State University, Tbilisi, Georgia
E-mail<EMAIL_ADDRESS>
###### Abstract

In this paper we study the possibility of efficient pair production in a
pulsar’s magnetosphere. It has been shown that by means of the relativistic
centrifugal force the electrostatic field exponentially amplifies. As a result
the field approaches the Schwinger limit leading to pair creation process in
the light cylinder area where the effects of rotation are very efficient.
Analysing the parameters of the normal period ($\sim 1$ sec) pulsars we have
found that the process is so efficient that the number density of electron-
positron pairs exceeds the Goldreich-Julian density by five orders of
magnitude.

###### keywords:

pulsars: general - instabilities - acceleration of particles

††pagerange: The novel mechanism of pair creation in pulsar
magnetospheres–LABEL:lastpage††pubyear: 2016

##  1 Introduction

Gold (1969) has suggested that pulsars are rotating neutron stars
centrifugally energising the pulsar’s magnetospheric particles to energies
enough for producing high energy electromagnetic radiation. Generally
speaking, it is evident that the neutron stars rotate with increasing period
of rotation, $P$, characterized by the slow down rate, $\dot{P}\equiv
dP/dt>0$. The corresponding slow down power $\dot{W}=I\Omega|\dot{\Omega}|$,
then becomes

$\dot{W}\simeq 4.7\times 10^{31}\times
P^{-3}\times\frac{\dot{P}}{10^{-15}}\times\frac{M_{ns}}{1.5M_{\odot}}\;ergs\;s^{-1},$
(1)

where $\Omega$ is the angular velocity of rotation of the pulsar, the slow
down rate is normalized by the parameters of the normal period pulsars (with
period of the order of one second), $I=2M_{ns}R_{\star}^{2}/5$ is the moment
of inertia of the neutron star, $M_{ns}$ and $R_{\star}\simeq 10^{6}$cm denote
its mass and radius respectively and $M_{\odot}\simeq 2\times 10^{33}$g is the
solar mass. The value of $\dot{W}$ indicates that a pulsar is a reservoir of
enormous energy, which potentially might lead to very interesting processes.

The Gold’s idea provoked a new theory of pulsar electrodynamics proposed by
Goldreich & Julian (1969), where the authors have described the spacial
distribution of charges inside the light cylinder (LC) surface (a hypothetical
area where the linear velocity of rotation coincides with the speed of light)
and studied efficiency of acceleration of particles escaping the inner
magnetospheric area on open magnetic field lines. As it has been pointed out
by Sturrock (1971), the theory of Goldreich & Julian (1969) could not explain
an emission pattern of radio pulsars. To study this particular problem
Sturrock (1971) has examined a general case: when the dipolar momentum of a
neutron star is inclined with respect to the rotation axis. The main idea is
based on the fact that by means of interaction of very high energy (VHE)
curvature gamma rays and strong magnetic field of the order of $10^{12}$
Gauss, the high efficiency pair production process initiates at the electron
and proton polar zones. This model has been extended by Jones (1977), where
apart from the pair production by means of the curvature photons, the author
also studied the mechanism of hadronic photo-absorption. Several modifications
of pair cascading have been proposed in a series of papers (Arons, 1981;
Daugherty & Harding, 1982; Usov & Melrose, 1995) considering the role of the
strong magnetic field.

In general, strong electric fields also can contribute to the efficient pair
cascading process. In particular, according to electrodynamics, in vacuum the
virtual particles and antiparticles are continuously producing and
annihilating. If on the other hand, strong electric field is present, it can
separate the virtual charges significantly, leading, via the quantum
tunnelling, to pair creation. This happens when the work done on the Compton
wave-length, $\lambda_{{}_{C}}$, equals the necessary energy of materialised
pairs, $eE\lambda_{{}_{C}}\simeq 2mc^{2}$, where $m$ and $e$ are the
electron’s mass and charge respectively and $c$ is the speed of light.
Original works dedicated to this problem has been considered in a series of
papers (Sauter, 1931; Heisenberg & Euler, 1936; Schwinger, 1951). The role of
electric fields in pair cascading has been studied in the context of the
relativistic ionic Coulomb fields (Cheng & Ruderman, 1977) and an oscillating
electric fields surrounding a pulsar (Parker & Tiomno, 1972). In the latter,
the authors considered the radial electric field induced by means of the
radial currents and investigated the rate of pair production. It is worth
noting that the pulsar current sheets provide induced electric fields small
compared to the Schwinger limit $E_{{}_{S}}=\pi m^{2}c^{3}/(e\hbar)\simeq
1.4\times 10^{14}$ statvolt cm-1 (Schwinger, 1951), where $\hbar$ is the
Planck constant. Therefore, it is clear that the aforementioned mechanism is
not very efficient. On the other hand, in the pulsar magnetospheres the
electric field might be induced by means of neutron star’s rotation, ${\bf
E}={\bf\upsilon\times B}/c$, which is maximum close to the star’s surface and
for normal pulsars equals $2\times 10^{8}$ statvolt cm-1 and still is by
several orders of magnitude less than $E_{{}_{S}}$.

Since the pulsars are rapidly rotating neutron stars, it is clear that the
centrifugal effects might be very important Gold (1969). As it has been shown
by Machabeli et al. (2005) the centrifugal force in the LC area is different
for different species of particles (magnetospheric electrons and positrons).
This in turn, might lead to charge separation creating the Langmuir waves. On
the other hand, since the centrifugal force is time dependent (Machabeli et
al., 2005), it acts as a parameter and the driven electrostatic field is
parametrically amplified (Mahajan et al., 2013; Osmanov et al., 2015, 2014).
In the mentioned papers it was found that this mechanism is so efficient that
the induced electric field exponentially increases, gradually approaches the
Schwinger limit resulting in the efficient pair production. This is a
completely new mechanism of pair creation in the pulsar magnetosphere and in
this paper we study it.

The paper is organized as follows: in Sec.2 we introduce the theoretical
model, in Sec. 3 we apply the approach to pulsars and obtain results and in
Sec. 4 we sumarize them.

##  2 Theoretical model

In this section we present the theory of parametric instability of
centrifugally excited Langmuir waves and apply the amplified electrostatic
field to the pair production process. For this purpose, for the nearby zone of
the LC we consider the linearized system of equations (Mahajan et al., 2013;
Osmanov et al., 2014) composed by the Euler equation

$\frac{\partial p_{\beta}}{\partial t}+ik\upsilon_{\beta
0}p_{\beta}=\upsilon_{\beta
0}\Omega^{2}r_{\beta}p_{\beta}+\frac{e_{\beta}}{m}E,$ (2)

the continuity equation

$\frac{\partial n_{\beta}}{\partial
t}+ik\upsilon_{{\beta}0}n_{\beta},+ikn_{{\beta}0}\upsilon_{\beta}=0$ (3)

and the Poisson equation

$ikE=4\pi\sum_{\beta}n_{\beta 0}e_{\beta},$ (4)

where $\beta$ denotes the index of species (electrons and positrons),
$p_{\beta}$ is the perturbation of momentum per unit mass,
$r_{\beta}(t)\simeq\frac{V_{0\beta}}{\Omega_{e}}\sin\left(\Omega
t+\phi_{\beta}\right)$, $\upsilon_{\beta}\simeq V_{0\beta}\sin\left(\Omega
t+\phi_{\beta}\right)$ (Mahajan et al., 2013) and $e_{\beta}$ are respectively
the radial coordinate, radial velocity perturbation ($\upsilon_{\beta 0}$ is
the unperturbed value) and charge of the corresponding species, $V_{0\beta}$
is the velocity amplitude, $\phi_{\beta}$ denotes the phase, $E$ denotes the
induced electrostatic field of the driven wave, $k$ \- is its wave number and
$n_{\beta}$ and $n_{\beta 0}$ represent respectively perturbed and unperturbed
particle number densities. Since the magnetic field in the pulsar’s
magnetosphere is very high, we assume that the particles are in the frozen-in
condition ${\bf E_{0}+\frac{1}{c}v_{0\beta}\times B_{0}=0}$ and are co-
rotating with the field lines. The latter are assumed to be almost straight,
because it is believed that the emission is formed by the particles sliding
along the open field lines.

Following the method developed in (Mahajan et al., 2013; Osmanov et al., 2015)
we introduce the anzatz

$n_{\beta}=N_{\beta}e^{-\frac{iV_{0\beta}k}{\Omega}\sin\left(\Omega
t+\phi_{\beta}\right)},$ (5)

which reduces the aforementioned set of equations to non-autonomous ”Mode”
equations for electrons and positrons

$\frac{d^{2}N_{e}}{dt^{2}}+{\omega_{e}}^{2}N_{e}=-{\omega_{e}}^{2}N_{p}e^{-i\chi},$
(6)
$\frac{d^{2}N_{p}}{dt^{2}}+{\omega_{p}}^{2}N_{p}=-{\omega_{p}}^{2}N_{e}e^{i\chi},$
(7)

where $\omega_{e,p}\equiv\sqrt{4\pi e^{2}n_{e,p}/m\gamma_{e,p}^{3}}$ and
$\gamma_{e,p}$ are the relativistic plasma frequencies and the relativistic
factors for the beam components and $\chi=b\cos\left(\Omega
t+\phi_{+}\right)$, $b=\frac{2ck}{\Omega_{e}}\sin\phi_{-}$,
$2\phi_{\pm}=\phi_{p}\pm\phi_{e}$. After making the Fourier transform of Eqs.
(6,7) one straightforwardly derives the dispersion relation of the Langmuir
wave

$\omega^{2}-\omega_{e}^{2}-\omega_{p}^{2}\;J^{2}_{0}(b)=\omega^{2}\omega_{p}^{2}\sum_{\mu\neq
0}\frac{J^{2}_{\mu}(b)}{\left(\omega+\mu\Omega\right)^{2}}.$ (8)

To study the instability the frequency must be decomposed by the real and
imaginary parts $\omega=\omega_{r}+\Delta$. The former should satisfy the
resonance condition $\omega_{r}=\mu_{r}\Omega_{e}$ for the most efficient
process (Mahajan et al., 2013; Osmanov et al., 2014, 2015). Then, one obtains
the following cubic equation

$\Delta^{3}=\frac{\omega_{r}{\omega_{p}}^{2}{J_{\mu_{r}}(b)}^{2}}{2},$ (9)

having two complex solutions characterising a growth rate of the centrifugally
excited Langmuir waves

$\Gamma=\frac{\sqrt{3}}{2}\left(\frac{\omega_{e}{\omega_{p}}^{2}}{2}\right)^{\frac{1}{3}}{J_{\mu_{r}}(b)}^{\frac{2}{3}},$
(10)

where $\omega_{r}=\omega_{e}$ and $J_{\mu_{r}}$ denotes the Bessel function of
the first kind.

Therefore the time evolution of the electrostatic field is given as follows

$E(t)=E_{0}e^{\Gamma t},$ (11)

where $E_{0}$ is its initial value which can be approximated straightforwardly

$E_{0}\simeq 4\pi n_{{}_{GJ}}e\Delta r,$ (12)

where

$n_{{}_{GJ}}=\frac{\Omega B}{2\pi ec}\times\frac{1}{1-r^{2}/R_{lc}^{2}},$ (13)

is the Goldreich-Julian number density (Goldreich & Julian, 1969),
$R_{lc}=c/\Omega$ denotes the LC radius and $\Delta r$ is the characteristic
length-scale of charge separation, which takes place close to the LC area. Its
value can be derived from the fact that the process of CA occurs in the LC
zone, where the Lorentz factor asymptotically increases (Rieger & Mannheim,
2000)

$\gamma=\frac{\gamma_{0}}{(1-r^{2}/R_{lc}^{2})},$ (14)

therefore, $\Delta r$ should characterise a scale factor of non-uniformity,
which can be expressed as $\Delta r=\frac{\gamma}{\partial\gamma/\partial r}$,
finally leading to $\Delta r\simeq\gamma_{0}R_{lc}/(2\gamma)$ (Osmanov et al.,
2014). Here $\gamma_{0}$ is the initial relativistic factor.

As it turns out from Eq. (11) the field is exponentially increasing and in due
course of time the electric field will inevitably reach the Schwinger limit
and thus, the probability of pair production might increase very rapidly.

In general for a constant electric field the pair production rate ($R$ \-
number of produced pairs per unit of time and unit of volume) is given by
(Schwinger, 1951; Marinov & Popov, 1977)

$R\equiv\frac{dN}{dtdV}=\frac{e^{2}E^{2}}{4\pi^{3}c\hbar^{2}}\sum_{n}\frac{1}{n^{2}}exp\left({-\frac{\pi
m^{2}c^{3}}{e\hbar E}n}\right).$ (15)

Generally speaking, if the electrostatic field is varying the pair production
rate is higher, but in the considered scenario the plasma oscillation
frequency can be maximum $10^{4-5}$ Hz, which is by many orders of magnitude
less than $\nu\simeq 2mc^{2}/h\sim 10^{20}$ Hz. Therefore, the constant field
case is a physically realistic approximations and hence, the aforementioned
formula is valid.

##  3 Discusion

Figure 1: Behaviour of the instability time-scale versus $\gamma_{2}$. The set
of parameters is: $P=1$ sec, $\dot{P}=10^{-15}$ s s-1, $\gamma_{1}=10^{3}$,
$\gamma_{0}=\\{10^{3};10^{4}\\}$.

Figure 2: Two plots for the behaviour of the pair production rate versus time
are shown. The set of parameters is the same as in Fig. 1. On the top panel we
show the plot for $\gamma_{0}=10^{3}$ and on the bottom panel we show the
results for $\gamma_{0}=10^{4}$.

Figure 3: Dependence of the multiplicity factor versus the initial Lorentz
factor. The set of parameters is the same as in Fig. 2 except $\gamma_{0}$
which changes in the interval $[10^{3}-10^{4}]$.

In this section we consider the normal period pulsars and study the
possibility of pair production by means of the centrifugally driven
electrostatic fields. On the LC length-scale, where the CA is most efficient
the magnetic induction (Manchester & Taylor, 1978)

$B_{st}\simeq
10^{12}\times\left(\frac{P}{1\;s}\times\frac{\dot{P}}{10^{-15}\;ss^{-1}}\right)^{1/2}\;\;Gauss,$
(16)

guarantees frozen-in condition, leading to a one-dimensional geometry of
particles’ kinematics along the magnetic field lines.

Using the fact that CA lasts until the plasma energy density, $\gamma
mc^{2}n_{{}_{GJ}}$, exceeds that of magnetic field, $B^{2}/(8\pi)$, for the
maximum Lorentz factor one obtains

$\gamma_{m}\simeq 2.6\times 10^{5}\times
P^{5/4}\times\left(\frac{\gamma_{0}}{10^{4}}\right)^{1/2}\times\left(\frac{\dot{P}}{10^{-15}s\;s^{-1}}\right)^{1/4},$
(17)

where we have taken into account Eq. (14), the radial behaviour of the dipolar
magnetic field and the corresponding expression on the LC, $B\simeq
B_{st}R^{3}_{st}/R_{lc}^{3}$ . Here $R_{st}\simeq 10$km denotes the neutron
star’s radius.

To demonstrate efficiency of the Langmuir wave’s amplification, we consider
two different initial Lorentz factors $\gamma_{0}=\\{10^{3};10^{4}\\}$. In
Fig. 1 we plot the dependence of the instability time-scale on $\gamma_{2}$.
The set of the rest of parameters is: $P=1$ sec, $\dot{P}=10^{-15}$ s s-1,
$\gamma_{1}=10^{3}$, $R_{st}=10$ km. In the framework of this paper we assume
the equipartition approximation: when energy is uniformly distributed among
all species: $\gamma_{m}n_{{}_{GJ}}\simeq n_{1}\gamma_{1}\simeq
n_{2}\gamma_{2}$. As it is clear from the plots the instability time-scales
are small compared to the escape time-scale (which is of the order of the
rotation period of the pulsar $P\simeq 1$ sec), indicating high efficiency of
the amplification process.

Generally speaking, the proposed mechanism should be somehow terminated. If
the co-rotation is maintained this will happen when the produced total energy
density of electron positron pairs equals the energy density of the
electrostatic field, leading to the following integral expression

$\sum_{n}\frac{1}{n^{2}}\int_{0}^{\tau}\exp\left(-\frac{\pi
m^{2}c^{3}n}{e\hbar E_{0}}e^{-\Gamma
t}\right)dt=\frac{\pi^{2}\hbar^{2}}{4mce^{2}},$ (18)

which actually is an equation for determining the value of $\tau$ when the
pair production is terminated. After integration one can straightforwardly
derive the corresponding algebraic equation

$\sum_{n}\frac{1}{n^{2}}\left[E_{1}\left(-\frac{\pi m^{2}c^{3}n}{e\hbar
E_{0}}e^{-\Gamma\tau}\right)-E_{1}\left(-\frac{\pi m^{2}c^{3}n}{e\hbar
E_{0}}\right)\right]=$ $=\frac{\pi^{2}\hbar^{2}}{4mce^{2}}\Gamma,$ (19)

where $E_{1}$ denotes the exponential integral.

For the same set of parameters as in Fig. 1 one can solve Eq. (19) for $\tau$.
In particular, for $\gamma_{0}=10^{3}$ and $\gamma_{0}=10^{4}$ the
corresponding termination time-scales equal $\sim 7.6\times 10^{-2}$ s and
$\sim 5.9\times 10^{-2}$ s respectively.

By taking these values into account one can estimate the pair production rate
from Eq. (15). On Fig. 2 we plot the dependence of $R$ versus time. The set of
parameters is the same as in Fig. 1. As it is clear from the plots the pair
creation rate is a continuously increasing function of time, which is a
natural result of the fact that by means of the parametric instability, the
electrostatic field exponentially amplifies (see Eq. (11)). One has to note
that the absolute values of $R$ are unrealistically high, indicating that some
additional mechanism of termination of the process must exist.

The proposed mechanism strongly depends on the effects of relativistic
centrifugal force, which takes place only if the energy density of particles
is less than that of the magnetic field. But in due course of time the number
density of magnetospheric electron-positron pairs, $n_{p}$, increases very
rapidly and as soon as the following condition

$2\gamma_{rot}mc^{2}n_{p}<\frac{B^{2}}{8\pi}$ (20)

violates, the co-rotation stops and consequently the process is terminated.
Here $\gamma_{rot}=(1-r^{2}/R_{lc}^{2})^{-1/2}$ is the Lorentz factor
corresponding to rotation. After taking into account Eq. (14) one can
straightforwardly derive the maximum number density of pairs, $n_{p}$ on the
LC zone

$n_{p}\simeq\frac{B^{7/4}}{16\pi
mc^{2}}\times\left(\frac{4mc\gamma_{0}\Omega}{e}\right)^{1/4}\simeq$ $\simeq
4.17\times 10^{5}\times
P^{-37/8}\times\left(\frac{\dot{P}}{10^{-15}}\right)^{7/8}\times\left(\frac{\gamma_{0}}{10^{4}}\right)^{1/4}.$
(21)

On Fig. 3 we plot the behaviour of the multiplicity factor $n_{p}/n_{GJ}$
versus the initial relativistic factor. The set of parameters is the same as
in Fig. 2 except $\gamma_{0}$ which varies in the interval $[10^{3}-10^{4}]$.
As one can see from the plot the multiplicity factor is a continuously
increasing function of the initial Lorentz factor and for $10^{4}$ reaches the
value of the order of $1.3\times 10^{5}$ indicating high efficiency of the
process.

##  4 Summary

We have examined centrifugally accelerated pulsar’s magnetospheric particles
and studied the parametric amplification of the electrostatic field.

Considering the typical pulsar parameters it has been shown that the electric
field exponentially increases and gradually reaches the Schwinger limit, when
efficient pair creation might occur.

Analysing constraints imposed on the process we have found that the mechanism
terminates when the energy density of produced particles exceeds that of the
magnetic field. As a result, it has been shown that this process leads to the
multiplicity factor compared to the GJ number density of the order of
$10^{5}$.

Given that the novel mechanism of pair production substantially changes the
number density of electron-positron plasmas in pulsar’s magnetospheres it
might significantly influence the physical processes there. In particular, it
is evident that the processes of particle acceleration strongly depends on the
plasma density. Consequently, the emission spectral pattern will be influenced
as well by the efficient pair creation. This in turn, might give rise to
coherent radio emission, which seems to be quite promising in the light of
modern enigma - fast radio bursts. Since all these problems are beyond the
intended scope of the paper we are going to consider them very soon.

## Data Availability

Data are available in the article and can be accessed via a DOI link.

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