# Relativistic effects of rotation in $\gamma$-ray pulsars - Invited Review

Osmanov Z.N  School of Physics, Free University of Tbilisi, 0183, Tbilisi,
Georgia  E. Kharadze Georgian National Astrophysical Observatory, Abastumani
0301, Georgia

###### Abstract

In this paper we consider relativistic effects of rotation in the
magnetospheres of $\gamma$-ray pulsars. The paper reviews the progress
achieved in this field during the last three decades. For this purpose we
examine direct centrifugal acceleration of particles and corresponding
limiting factors: constraints due to the curvature radiation and the inverse
Compton scattering of electrons against soft photons. Based on the obtained
results generation of parametrically excited Langmuir waves and the
corresponding Landau-Langmuir-centrifugal drive is studied.

###### keywords:

pulsars; gamma-rays; multi-wavelength

##  1 Introduction

One of the fundamental problems of $\gamma$-ray pulsar astrophysics is the
generation of VHE emission. In this context it is worth noting that last
decade has been very fruitful in detecting VHE $\gamma$-rays of pulsars [1, 2,
3, 4, 5]. On the other hand, it is clear that generation of VHE emission
requires existence of relativistic particles and the process of their
acceleration is still under debate.

Generally speaking, it is observationally evident that the pulsars are
rotating neutron stars 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 the slow down rate is normalized by the typical value for normal period
pulsars (with period of the order of one second) and we have taken into
account that the moment of inertia of a sphere is expressed as
$I=2M_{ns}R_{\star}^{2}/5$, $M_{ns}$ denotes the neutron star’s mass,
$R_{\star}\simeq 10^{6}$ cm is its radius and $M_{\odot}\simeq 2\times
10^{33}$ g is the solar mass. It is clear from the aforementioned expression
that slowing down of pulsar’s rotation provides enormous power, which might be
even higher for millisecond pulsars. Therefore there is no question about a
source of energy, but the problem concerns a particular mechanism which might
be responsible for acceleration of particles up to relativistic energies.

In the framework of the so-called polar-cap models by means of the strong
electric field parallel to magnetic field lines (parallel electric field) the
particles are uprooted from the surface of the neutron star [6], generation of
electric fields in turn, is a direct result of neutron star’s rotation. In the
vicinity of the neutron stars the electrons might undergo further acceleration
provided either by magnetic field line curvature [7], or space-charge-limited
flow [8], or effects of inertial frame [9]. On the other hand, generation of
parallel electric field might be strongly suppressed either by means of the
field curvature [10] or the IC scattering of electrons against photons [11].
Therefore, to explain the origin of VHE particles outer gap models became
actual. In the framework of this approach the electrons gain energy in the
outer magnetosphere in the so called vacuum gap [12, 13], but as it turned
out, in these scenarios the gap size is not enough to explain the observed VHE
emission from pulsars. Some attempts to enlarge the gap size [14, 15, 9]
although resulted in a more efficient acceleration process, but still could
not solve the problem. By considering the mechanism of energization along the
magnetic field lines in the force-free pulsar magnetospheres has been studied
in detail by Contopulous et al. (see [16]) where the authors have pointed out
that acceleration is significantly suppressed by the force-free topology of
the magnetic field. The acceleration process of a two component electron-
positron plasma in a monopole configuration near the LC (a hypothetical zone,
where the linear velocity of rotation exactly equals the speed of light) area
has been examined in [17] and it has been shown that the maximum Lorentz
factors of particles is of the order of $\sigma\equiv B^{2}/(8\pi
nm_{e}c^{2})$, where $B$ is the magnetic field, $n$ is the particle number
density, $m_{e}$ is the electron’s mass and $c$ is the speed of light.

Figure 1: Sketch of the magnetospheric Pulsar model from [26]. The
centrifugally accelerated co-rotating particles in the nearby zone of the LC
area emit in a relatively narrow cone along the tangential direction.

A rather different mechanism of acceleration which might take place in the LC
zone has been proposed by Gold [18, 19], who suggested that extremely strong
magnetic field forces particles to slide along the field lines, which are
trajectories of particles in the rotating frame of reference. This is an
effective picture: the electrons gyrate in magnetic field and a trajectory
averaged by time for long time-scales is a certain trajectory, which in a very
strong magnetic field approximation is a line, along which the particles
slide. In particular, the magnetic field in the pulsar’s magnetosphere is
assumed to be dipolar $B\simeq B_{st}\times(r_{s}/r)^{3}$, where $B_{st}\simeq
3.2\times 10^{19}\sqrt{P\dot{P}}$ Gauss is the magnetic induction on the
neutron star’s surface [35] and $r_{s}\simeq 10$ km is the pulsar’s radius.
Then, for electrons with relativistic factors of the order of $10^{5}$, the
gyro-radius in the LC area (a hypothetical zone where the linear velocity of
rigid rotation exactly equals the speed of light) for normal period pulsars is
of the order of $0.0016$ cm, which is by many orders of magnitude than the
corresponding length-scale of the LC radius: $10^{9}$ cm. Therefore, the
electrons are frozen into the magnetic field and follow the co-rotating field
lines. The corresponding condition of the frozen-in state ${\bf
E}+\frac{1}{c}{\bf v}\times{\bf B}=0$ explains the drift of particles in the
direction of rotation as follows: by means of the radial magnetic field and
the generated electric field perpendicular to the equatorial plane the ${\bf
E}\times{\bf B}/c^{2}$ drift occurs, which actually is the co-rotation
velocity, ${\bf v}$, of the particle.

On the other hand, it is clear that the total velocity cannot be greater than
the speed of light and therefore, because of the co-rotation of the pulsar’s
magnetosphere [20], by reaching the LC surface (if the rigid rotation is
preserved) the longitudinal velocity will tend to zero and electrons will have
only the azimuthal component, leading to the emission along the tangential
direction Fig. 1. Here $\Omega$ denotes the angular velocity of neutron star’s
rotation, $r_{{}_{L}}$ is the radius of the LC and $\alpha$ is the angle
between the dipole magnetic moment and the axis of rotation.

In this review paper we consider the centrifugally generated relativistic
electrons as for normal period pulsars as well as for millisecond ones and
present the progress achieved in this field during almost the last three
decades. The paper is organized in the following way: Sec. 2 gives a detailed
description of different phenomena which take place due to the centrifugal
effects. We apply the model of centrifugal acceleration to the well known Crab
and Vela pulsars and normal period pulsars and in Sec. 3 we outline the
obtained results.

##  2 Results

In this section we consider effects of rotation of the pulsar’s magnetosphere
in the LC area and study several problems: Centrifugal acceleration of
particles; Rotationally driven Langmuir waves; Landau Langmuir centrifugal
drive and reconstruction of pulsars’ magnetospheres by means of the
centrifugal effects.

###  2.1 Centrifugal acceleration

As we have already mentioned in the introduction, the longitudinal velocity
component should decrease in the nearby zone of the LC. Deceleration of the
radial velocity and thus the centrifugal force reversal has been investigated
in the Schwarzschild black-hole metrics [21]. The corresponding problem but in
a special relativistic case has been studied in [22], where authors have
considered a gedanken experiment: a bead freely sliding on a rotating wire and
studied particle’s dynamics. It has been demonstrated that effect of
”relativistic mass” leads to the similar phenomenon: if the rigid rotation is
preserved, the radial velocity gradually decelerates and vanishes on the LC
zone. For studying the problem of acceleration one can perform calculations in
the co-rotating reference frame. then, if one introduces an interval for a
straight field line inclined by the angle $\alpha$ with respect to the
rotating axis [22, 23]

$ds^{2}=-c^{2}dt^{2}\left(1-\frac{\Omega_{ef}^{2}r^{2}}{c^{2}}\right)+dr^{2},$
(2)

where $\Omega_{ef}\equiv\Omega\sin\alpha$ is the effective angular velocity of
rotation for a field line inclined by the angle $\alpha$ with respect to the
rotation axis.

The particles moving along the co-rotating trajectories are characterised by
the following Lagrangian

$L=\frac{1}{2}G_{\mu\nu}\frac{dx^{\mu}}{d\tau}\frac{dx^{\nu}}{d\tau},$ (3)

and the corresponding equation of motion

Figure 2: Here we plot $u$ and $\upsilon$ versus dimensionless radial
distance, $x$. The set of parameters is $x_{0}=0.001$ and $u_{0}=0$.

$\frac{\partial L}{\partial x^{\mu}}=\frac{d}{d\tau}\left(\frac{\partial
L}{\partial\dot{x}^{\mu}}\right)$ (4)

where $G_{00}=-(1-\Omega_{ef}^{2}r^{2}/c^{2})$, $G_{11}=1$,
$\mu,\nu=\\{0,1\\}$, $x^{0}=ct$, $x^{1}=r$.

Considering the radial component of the equation of motion one can show that
the corresponding equation in the dimensionless form is given by

$\frac{d^{2}x}{d\tau^{2}}=\frac{x}{1-x^{2}}\left[1-x^{2}-2\left(\frac{dx}{d\tau}\right)^{2}\right],$
(5)

where $x\equiv r/r_{{}_{L}}$, $r$ is the radial coordinate,
$r_{{}_{L}}=c/\Omega$ denotes the LC radius, $\tau\equiv t\Omega$ is the
dimensionless time, $dx/d\tau=\upsilon_{r}/c\equiv u$ and $\upsilon_{r}$
denotes the radial velocity. In Fig. 2 we show the behaviour of radial
velocity, $u(x)$ and total velocity, $\upsilon(x)\equiv\sqrt{x+u(x)^{2}}$ in
the dimensionless forms for the following initial values $x_{0}=0.001$ and
$u_{0}=0$ (note that the linear velocity of rotation in the mentioned
dimensionless form coincides with the radial coordinate). As it is clear from
the plots, initially the radial velocity increases, but in due course of time,
by reaching the LC ($x=1$) it starts decelerating and the total velocity of
the particle tends to the speed of light ($\upsilon\rightarrow 1$). By
considering a plasma flow in the co-rotating pulsar’s magnetosphere it has
been shown that in the limit of strong magnetic field the system of fluid
equations reduce to a single particle approach described by Eq. (5).

Eq. (3) for $\mu=0$ leads to energy - constant of motion

$E=\gamma mc^{2}\left(1-\frac{\Omega r}{c^{2}}\upsilon_{\varphi}\right),$ (6)

where $m$ is the particle’s mass, $\upsilon_{\varphi}$ is the azimuthal
velocity component and
$\gamma=\left(1-\upsilon_{r}^{2}/c^{2}-\upsilon_{\varphi}^{2}/c^{2}\right)^{-1/2}$
is the Lorentz factor of the particle. If one imposes the condition of
constancy the radial behaviour of the relativistic factor writes as [25, 26]

$\gamma(r)=\gamma_{0}\frac{1-r_{0}^{2}/r_{{}_{L}}^{2}}{1-r^{2}/r_{{}_{L}}^{2}},$
(7)

where $\gamma_{0}$ and $r_{0}$ denote initial values of the Lorentz factor and
radial coordinate respectively. Eq. (7) confirms that particle’s energy
asymptotically increases close to the LC.

From the observations it is clear that particles originating in the pulsar
magnetospheres, leave these regions and dynamically become force-free. This
means that the trajectory of the particle in the laboratory frame of reference
is a straight line. Since the particles are in the frozen-in condition and
follow the magnetic field lines, the shape of the latter in the co-rotating
frame of reference should be given by the Archimedes’ spiral, lagging behind
the rotation [27]. On the other hand, some of the $\gamma$-ray pulsars exhibit
jet-like structures [28], therefore one should introduce 3D configuration of
magnetic field lines co-rotating with the neutron star. It is worth noting
that the field lines will become more and more twisted in a narrow region of
the LC. This particular problem has been considered in detail in [29]. It was
shown that in the nearby zone of the LC the curvature drift of plasma
particles driving perpendicular to the equatorial plane leads to generation of
a toroidal component. This in turn, changes the shape of the field lines into
a configuration of the Archimedes’ spiral. For this purpose, in the co-
rotating frame of reference we examine the field line configuration

$\varphi=\varphi(r),~{}~{}z=f(r).$ (8)

Imposing the aforementioned prescribed trajectory on the the space-time metric

$ds^{2}=-dt^{2}+r^{2}d\phi^{2}+dr^{2}+dz^{2},$ (9)

will reduce it to the form

$ds^{2}=G_{00}dt^{2}+2G_{01}dtdr+G_{11}dr^{2},$ (10)

with

$G_{\alpha\beta}=\left(\begin{array}[]{ccc}-1+\omega^{2}r^{2},&\omega\varphi^{\prime}\rho^{2}\\\
\omega\varphi^{\prime}r^{2},&1+\varphi^{\prime 2}r^{2}+\acute{f}^{2}\\\
\end{array}\right),$ (11)

where $\varphi^{\prime}=d\varphi/dr$, $f^{\prime}=df/dr$, $c=1$. Then, in the
same manner as before, one can derive an expression for the energy for the
non-diagonal metrics

$E=-\gamma\left(G_{00}+G_{01}v\right)=const,$ (12)

where $v\equiv dr/dt$ is the radial velocity and

$d\tau/dt\equiv
u^{t}\equiv\gamma=\left(-G_{00}-2G_{01}v-G_{11}v^{2}\right)^{-1/2}$ (13)

is an expression for the relativistic factor in the general case - $3D$ curved
trajectories. One can explicitly solve Eqs. (12,13) for the radial velocity

$v={\frac{\sqrt{G_{00}+E^{2}}}{(G_{01}^{2}+E^{2}G_{11})}}{\left[-G_{01}\sqrt{G_{00}+E^{2}}\pm
E\sqrt{G_{01}^{2}-G_{00}G_{11}},\right]},$ (14)

where different signs correspond to different initial conditions.

Figure 3: The dependence of the Lorentz factor on the radial coordinate for
different initial relativistic factors $\gamma_{0}$.The calculations are done
for an Archimedean spiral with the critical Lorentz factor $\gamma_{cr}=7$.
All particles are launched from $r_{0}=0$. As it is evident from the plot, the
curve $\gamma_{cr}$ attracts all other curves. The figure is taken from [26].

From the Fig. 2 it is clear that a particle sliding along a rectilinear co-
rotating field lines never cross the LC zone. The reason follows from a nature
of rigid rotation of straight field lines. On the other hand, as we have
already mentioned the particles originating in the pulsars’ magnetospheres
might leave this region becoming dynamically free. This in turn, leads to a
specific shape of the field lines in the local frame of reference -
Archimede’s spiral [28].

As a first example we consider a curved field lines having the shape of the
Archimedean spiral located in the equatorial plane

$\varphi=ar,\;z=0.$ (15)

For a spiral which lags behind the rotation ($a<0$) one can find a critical
velocity when there is no ”reaction force” acting from the curved trajectory.
In this case the angular velocity of the particle in the laboratory frame,
$\Omega_{l}=\Omega+a\upsilon$, equals zero and therefore the corresponding
critical radial velocity is defined as $\upsilon_{cr}=-\Omega/a$.

In Fig. 3 we plot the behaviour of the Lorentz factor versus the radial
coordinate for different initial relativistic factors $\gamma_{0}$.The
calculations are performed for an Archimedean spiral with the critical Lorentz
factor $\gamma_{cr}=7$. All particles are launched from $r_{0}=0$. It is clear
from the figure that if the initial value of $\gamma$ equals $7$ the particle
does not experience the reaction force and therefore it stays constant. Unlike
this case, if the initial values are different from the critical value of the
relativistic factor, it asymptotically tends to $\gamma_{cr}$. From the plots
is evident that particles go beyond the LC zone. The reason is following:
since in due course of time the Lorentz factor tends to its critical value,
the effective angular velocity, $\Omega_{l}$, tends to zero and consequently
the corresponding LC radius ”inflates” and the particle always stays inside
$c/\Omega_{l}$ but outside the distance, $c/\Omega$.

A second example we would like to consider is the field lines located in a
common plane with the rotation axis:

$\varphi=const,\;z=f(r).$ (16)

After imposing this condition, from Eq. (14) one can derive a longitudinal
velocity (along a field line) as follows

$v_{{}_{\parallel}}=\left[\left(1-\omega^{2}r^{2}\right)\left(1-\frac{1-\omega^{2}r^{2}}{E^{2}}\right)\right]^{\frac{1}{2}}.$
(17)

where

$E=\gamma_{0}\left(1-\omega^{2}r_{0}^{2}\right),$ (18)

and $\gamma_{0}$ and $r_{0}$ are initial values quantities. Centrifugal
mechanism of acceleration can explain the Crab pulsar’s jet-like wind’s
velocity. From observations is evident that jets are collimated flows.
Therefore, one can model them as the particles moving along the field lines
with the following asymptotic structure: $r\rightarrow R$ and
$z\rightarrow\infty$ ($R<r_{{}_{L}}$). The aforementioned velocity in the unit
of the speed of light has its maximum value, which can be straightforwardly
obtained from Eq. (17) (see for details [28]) $v_{{}_{\parallel}}^{max}=0.5$.
On the other hand, the velocity averaged across the jet should be less than
the possible maximum value, which is known from observations:
$\upsilon_{Jet}\simeq 0.4$ [28].

As a last example it is natural to consider a field configuration given by

$\varphi=ar,\;z=f(r)$ (19)

which, on the one hand is characterised by a $3D$ geometry and one the other
hand has a property of the Archimedean spiral.

After straightforward mathematical manipulations From Eq. (14) one can derive
an asymptotic behaviour of the radial velocity

$v\rightarrow-\frac{\omega}{a}+\frac{1}{r^{2}\omega
a}\left(E^{2}-1+\frac{E\sqrt{a^{2}-\omega^{2}\left(1+\acute{f}^{2}\right)}}{a}\right).$
(20)

It is evident from this expression that the effective angular velocity
$\Omega_{l}\propto 1/r^{2}$ is a continuously decreasing function of the
distance and asymptotically tends to zero, implying that the particle becomes
dynamically force-free.

Figure 4: The figure is from [28]. On the top panel we show the particle
trajectories in (a) the rotational frame of reference and (b) the laboratory
frame of reference respectively. On graphs (c) and (d) the behaviour of the
curvature ($\rho\equiv 1/R_{c}$, where $R_{c}$ is the curvature radius of
magnetic field lines) and jet velocity is shown respectively. The set of
parameters is: $a=-28$, $f^{\prime}=11$, $v_{0}=0.01$ and $r_{0}=0$.

On Fig. 4 we show the trajectories of particles in (a) the rotational frame
(RF) and (b) the laboratory frame (LF) respectively. we also show (c) the
behaviour of the curvature (normalized by the initial value) of the trajectory
in the LF and (d) the jet velocity, thus the velocity along the $z$ axis. The
set of parameters is: $a=-28$, $f^{\prime}=11$, $v_{0}=0.01$ and $r_{0}=0$. It
is clear that despite the fact that the particle’s trajectory in the RF is
given by a regularly shaped 3D-Archimedean spiral, in the LF the trajectory is
more complex (see (b)). Initially the trajectory is more curved (see (c)) but
in due course of time the curvature vanishes and as a result the trajectory in
the LF becomes more rectilinear (see (b)). This means that dynamics tends to
the force-free regime. Indeed, from Fig. 4 (d) one can see that the $z$
component of the velocity initially increases but asymptotically saturates
indicating the transition to the zero force regime.

Since the magnetospheres of pulsars are full of photon field, the similar
problems have been studied for a situation when a co-rotating system is
imbedded in the isotropic photon field leading to a certain friction force
influencing dynamics of particle acceleration. Likewise the previous cases it
is shown that when the trajectories are presented by the Archimedean spiral
with arms lagging behind the rotation, the particles reach force-free regime
whatever the value of the friction force is. An interesting feature has been
found for the Archimedean spiral with arms oriented in opposite direction. In
this case the particles do not tend to the force free regime, but instead
reach a certain equilibrium location. The similar results have been obtained
for 3D configurations of the field lines (for details a reader is referred to
[30, 31]).

####  2.1.1 Co-rotation constraints

It is natural to suppose that for almost straight field lines rigid rotation
cannot be maintained up to the LC surface, therefore in order not to violate
the relativity principle the pulsar’s magnetosphere must reconstruct in a
proper way. In particular, the rigid rotation is maintained by means of the
strong magnetic field. Therefore, the co-rotation condition is satisfied if
the magnetic field energy density, $B^{2}/8\pi$, exceeds the plasma energy
density, $\gamma n_{{}_{GJ}}Mm_{e}c^{2}$, where $B$ denotes the magnetic
field, $n_{{}_{GJ}}={\bf\Omega B}/(2\pi ec)$ is the Goldreich-Julian number
density of electrons in the pulsar’s magnetosphere [32], $e$ is the electron’s
charge and $M$ is the multiplicity factor of particles. In this context one
has to note that in the pulsar models it is assumed that primary particles,
accelerated by the parallel electric field radiate due to the curvature
radiation. The interaction of the emitted photons with the strong magnetic
field will inevitably lead to the very efficient pair creation process
($\gamma+{\bf B}\rightarrow e^{+}+e^{-}+{\bf B}$), newly generated secondaries
repeat the same cascading mechanism until the plasma particles screen the
electric field [33, 34].

By taking the magnetic field strength and pulsar’s radius $r_{s}\simeq 10$ km,
into account combining with the aforementioned condition and Eq. (7) one can
show that for $r_{0}<<r$ the maximum attainable relativistic factor of
electrons moving on the open field lines in the LC zone writes as [26, 36]

$\gamma_{cor}\simeq 1.5\times
10^{7}\times\left(\frac{\gamma_{0}}{10^{4}}\times\frac{sin^{3}\alpha}{\\\
cos\alpha}\right)^{1/2}\times\left(\frac{P}{P_{{}_{Cr}}}\right)^{5/4}\times\left(\frac{\dot{P}}{\dot{P}_{{}_{Cr}}}\right)^{1/4},$
(21)

where the initial Lorentz factor is normalized by its typical value in the
Crab-like pulsars [36], $P$ and $\dot{P}$ are normalized by the Crab pulsar’s
parameters, $P_{{}_{Cr}}\simeq 0.0332$ s and $\dot{P}_{{}_{Cr}}\simeq
4.21\times 10^{-13}$ ss-1. The condition $r\simeq r_{{}_{L}}/\sin\alpha$ has
been taken into account and we considered the open field lines with curvature
radii exceeding the LC length scale, which means that in the framework of this
approximation the field lines are assumed to be straight. As it is clear from
the aforementioned expression in the Crab-like pulsar’s magnetospheres the
particles might achieve extremely high energies. One also can
straightforwardly show that for normal pulsars ($P\simeq 1$ s, $\dot{P}\simeq
10^{-15}$ ss-1) the co-rotation condition might guarantee the maximum
attainable relativistic factor of the order of $2.6\times 10^{5}$.

####  2.1.2 Emission constraints

The approach presented in the previous section is simplified for several
reasons: in realistic astrophysical scenarios dynamics of particles is
strongly affected either by the IC scattering of relativistic particles
against soft photons or curvature radiation. Unlike them, the synchrotron
mechanism can not impose constraints on the motion of electrons. In
particular, by taking the single particle synchrotron power into account,
$P_{syn}\simeq 2e^{2}\omega_{B}^{2}\gamma^{2}/3c$, where $\omega_{B}=eB/mc$ is
the cyclotron frequency, the synchrotron cooling time-scale,
$t_{syn}\simeq\gamma mc^{2}/P_{syn}$ is of the order of $6\times 10^{-9}$ sec
for $\gamma=10^{5}$. This value is very small compared to the kinematic time-
scale of pulsars (rotation period), therefore immediately after becoming
relativistic the electrons lose their perpendicular momentum, go to the ground
Landau level and continue sliding along the field lines, which in turn
guarantees the centrifugal mechanism of acceleration.

In the previous subsection we have shown that efficiency of rotation energy
pumping significantly increases in the LC area. If one assumes a straight co-
rotating field line inclined by the angle $\alpha$ with respect to the
rotation axis, then the acceleration time-scale on the LC

$t_{acc}=\frac{\gamma}{d\gamma/dt},$ (22)

reduces to

$t_{acc}\simeq\frac{r_{{}_{L}}}{2c\sin\alpha}\times\left(1-\frac{r_{0}^{2}\sin^{2}\alpha}{r_{{}_{L}}^{2}}\right)^{1/2}\times\left(\frac{\gamma_{0}}{\gamma}\right)^{1/2}.$
(23)

From Eq. (21) it is evident that in the nearby zone of the LC the relativistic
factor increases. This in turn, leads to the decrease of the acceleration
time-scale (see Eq. (23)), indicating high efficiency of the acceleration
process. On the other hand, the pulsar magnetosphere is full of soft - thermal
photons originating from the neutron star’s surface and the IC scattering
seems to be important. In this context one has to note that very often the
role of IC mechanism with thermal photons is supposed to be negligible because
it has been accepted that the photons and relativistic electrons travel in the
same - radial direction [37]. Contrary to this, following the idea, originally
proposed by Gold (see [18, 19]), strong magnetic field (see the previous
subsection) guarantees the frozen-in condition of electrons almost up to the
LC zone resulting in the co-rotation of particles, having only the azimuthal
velocity component, therefore, the angle of interaction is $\sim 90^{0}$ and
consequently the IC scattering of electrons against thermal photons should be
significant.

In the standard theory, the temperature of neutron star’s surface strongly
depends on their age, $\tau=P/(2\dot{P})$ [38]

$T(\tau)\simeq 5.9\times
10^{5}\times\left(\frac{10^{6}\;yr}{\tau}\right)^{0.1}K\;\;\;\;for\;\tau\leq
10^{5.2}\;yr,$ $T(\tau)\simeq 2.8\times
10^{5}\times\left(\frac{10^{6}\;yr}{\tau}\right)^{0.5}K\;\;\;\;for\;\tau>10^{5.2}\;yr,$
(24)

therefore, for the Crab-like pulsars the age is of the order of $10^{3}$ yr
and the surface temperature is $T\simeq 1.2\times 10^{6}$ K and for the normal
pulsars ($P=1$ s, $\dot{P}=10^{-15}$ ss-1) the age is three orders of
magnitude bigger and consequently $T\simeq 2.4\times 10^{5}$ K. For the peak
frequencies $\nu\simeq 2.8\;kT/h$ and the corresponding photon energies one
has $\nu\simeq 6.7\times 10^{16}$ K and $\epsilon_{max}\simeq 0.3$ KeV for the
Crab-type pulsars and $\nu\simeq 4.1\times 10^{15}$ K and
$\epsilon_{max}\simeq 0.02$ keV for the normal pulsars respectively (here $h$
denotes the Planck’s constant and $k$ is the Boltzmann’s constant). In
general, the IC scattering might occur in two extreme regimes, in the so-
called Thomson regime for very low energy particles and in the Klein-Nishina
regime, for very energetic electrons. In particular, the former takes place if
the following condition is satisfied $\gamma\epsilon_{ph}/(mc^{2})<<1$ [39]
and in the latter case $\gamma\epsilon_{ph}/(mc^{2})>>1$, where
$\epsilon_{ph}$ is the photon’s energy. Then, one can straightforwardly check
that for both class of pulsars, where we have already shown that $\gamma\simeq
10^{7}$ (Crab-like pulsars) and $\gamma\simeq 10^{5}$ (normal pulsars) the IC
process occurs in the KN regime when the photons gain almost the total energy
of electrons. Then, after taking into account the spherically symmetric photon
field, the KN Compton power is given by [39]

$P_{{}_{KN}}\simeq\frac{\sigma_{{}_{T}}\left(mckT\right)^{2}}{16\hbar^{3}}\left(\ln\frac{4\gamma
kT}{mc^{2}}-1.981\right)\left(\frac{r_{s}}{r}\right)^{2}.$ (25)

Logarithmic dependence on the Lorentz factor implies that the IC cooling time-
scale in the LC area

$t_{{}_{IC}}=\frac{\gamma mc^{2}}{P_{{}_{KN}}}\propto\gamma$ (26)

is a continuously increasing function of $\gamma$, which indicates that the IC
process does not impose any significant constraints on the maximum attainable
energies. Therefore, the maximum energy is limited by the co-rotation
constraint and consequently after scattering photons will have energies

$\epsilon_{{}_{ph}}\simeq
7.6\times\left(\frac{\gamma_{0}}{10^{4}}\times\frac{sin^{3}\alpha}{\\\
cos\alpha}\right)^{1/2}\times\left(\frac{P}{P_{{}_{Cr}}}\right)^{5/4}\times\left(\frac{\dot{P}}{\dot{P}_{{}_{Cr}}}\right)^{1/4},\;TeV$
(27)

in Crab-type pulsars and $130$MeV in normal pulsars.

Despite the fact that the IC mechanism does not affect the maximum achievable
energy of electrons, the produced VHE luminosity might be quite high. For
calculating this particular quantity, one should emphasise that the
acceleration is extremely efficient in a narrow thickness, where the
relativistic factor asymptotically increases and the corresponding scaling
factor - rate of change, $d\gamma/dr$, increases as well. Therefore,
acceleration is more efficient for highly nonuniform $\gamma$’s and hence the
corresponding thickness length-scale can be estimated as to be
$d\simeq\frac{\gamma}{d\gamma/dr}\simeq\gamma_{0}r_{{}_{L}}/(2\gamma)$ where
we have taken into account Eq. (21) for $r_{0}=0$. The azimuthal length-scale
involved in the emission process is of the order of $\delta l\simeq
r_{{}_{L}}\theta$, where $\theta\simeq 2\pi\xi/10$ ($\xi<10$) and consequently
for the active volume one has $\Delta V\simeq(\delta l)^{2}d$.

We have discussed in the previous subsection that the pair cascading is very
efficient. The corresponding number density of electron-positron pairs is
significantly increased and as is shown in [32] the multiplicity factor equals
$1/(1-r^{2}/r_{{}_{L}}^{2})$, which combined with Eq. (7) leads to
$M=\gamma/\gamma_{0}$. After taking into account all these quantities the
expression for the bolometric IC luminosity is given by

$L_{{}_{IC}}\simeq 2n_{{}_{GJ}}M\Delta VP_{{}_{KN}}\simeq$ $\simeq 6.1\times
10^{29}\times\xi^{2}\times\sin^{5}\alpha\times\left(\frac{T}{1.2\times
10^{6}\;K}\right)^{2}\times\left(\frac{P_{{}_{Cr}}}{P}\right)^{5/2}\times\left(\frac{\dot{P}}{\dot{P}_{{}_{Cr}}}\right)^{1/2}\;ergs\;s^{-1},$
(28)

where we have taken into account that in generation of the TeV emission the
most energetic photons are involved. The normal pulsars (with $T\simeq
2.4\times 10^{5}$ K) in the VHE domain ($\sim 100$ MeV) might provide
luminosity of the order of $\sim 1.6\times 10^{21}$ ergs s-1.

Even if the field lines are almost straight, due to the co-rotation condition
the particles follow the field lines and therefore in the laboratory frame of
reference their trajectories might be significantly curved leading to the
mechanism of curvature radiation. In the context of pulsars this mechanism has
been investigated in [40, 41] where the authors discussed in detail the
generation of radio emission by means of the curvature radiation mechanism. We
will see that this process might account for the generation of much higher
energies.

The energy loss rate by means of the curvature radiation of the electron is
given by

$P_{c}=\frac{2}{3}\frac{e^{2}c}{R_{c}^{2}}\gamma^{4},$ (29)

where $R_{c}$ denotes the radius of the curvature. The characteristic cooling
time-scale, $t_{c}\equiv\gamma mc^{2}/P_{c}$, then becomes

$t_{c}=\frac{3mcR_{c}^{2}}{2e^{2}\gamma^{3}}.$ (30)

It is clear from this expression that $t_{c}$ is vanishing faster than the
acceleration time-scale, $t_{acc}\propto 1/\gamma^{1/2}$, (see Eq. (22)),
therefore, the maximum attainable Lorentz factor, $\gamma_{c}$, is achieved
when $t_{acc}\simeq t_{c}$, leading to the following expression of
$\gamma_{c}$

$\gamma_{c}\simeq\frac{1}{\gamma_{0}^{1/5}}\times\left(\frac{3mPc^{3}\sin\alpha}{2\pi
e^{2}}\right)^{2/5}\times\left(\frac{R_{c}}{r_{{}_{L}}}\right)^{4/5}\simeq$
$\simeq 4.9\times
10^{7}\times\left(\frac{10^{4}}{\gamma_{0}}\right)^{1/5}\times\left(\frac{R_{c}}{r_{{}_{L}}}\right)^{4/5}\times\left(\frac{P}{P_{{}_{Cr}}}\right)^{2/5}\times\sin^{2/5}\alpha,$
(31)

where we have taken into account that the curvature radius, $R_{c}$ of the
trajectory in the nearby zone of the LC is of the order of $r_{{}_{L}}$, which
is clearly seen from Eq. (7). In particular, close to the mentioned area the
Lorentz factor asymptotically increases and the linear velocity of rotation
tends to the speed of light, therefore, the radial velocity tends to zero and
consequently the curvature radius equals the LC radius.

Since the derived value of the maximum relativistic factor is higher than the
Lorentz factor limited by the co-rotation constraint (see Eq. (21)), the
latter is responsible for reaching the maximum energy of electrons, leading to
the following energy of curvature photons [36]

$\epsilon_{cur}\simeq\frac{3ch}{4\pi R_{c}}\gamma^{3}\simeq$ $\simeq
0.6\times\left(\frac{\gamma_{0}}{10^{4}}\times\frac{sin^{3}\alpha}{\\\
cos\alpha}\right)^{3/2}\times\left(\frac{P}{P_{{}_{Cr}}}\right)^{11/4}\times\left(\frac{\dot{P}}{\dot{P}_{{}_{Cr}}}\right)^{3/4}\;GeV.$
(32)

As we see, the curvature radiation provides the emission in the GeV range for
the millisecond pulsars. For the total curvature luminosity one obtains

$L_{c}\simeq 2n_{{}_{GJ}}M\Delta VP_{c}\simeq$ $\simeq 2.8\times
10^{34}\times\left(\xi\times\frac{r_{{}_{L}}}{R_{c}}\times\frac{\gamma_{0}}{10^{4}}\times\frac{\sin^{3}\alpha}{\cos\alpha}\right)^{2}\times\left(\frac{P}{P_{{}_{Cr}}}\right)^{5/2}\times\left(\frac{\dot{P}}{\dot{P}_{{}_{Cr}}}\right)^{3/2}\;ergs\;s^{-1}.$
(33)

Despite the fact that the curvature emission mechanism is not responsible for
limiting the maximum attainable energy of particles, most of the energy is
emitted by means of it.

For the normal pulsars ($P\simeq 1$ s, $\dot{P}\simeq 10^{-15}$ ss-1) one can
straightforwardly check that the co-rotation still remains the major mechanism
limiting the maximum energy and the curvature radiation provides emission in
the upper UV domain ($\epsilon_{c}\simeq 100$ eV) with the luminosity of the
order of $2.4\times 10^{22}$ ergs s-1.

###  2.2 Parametric excitation of electrostatic waves and Langmuir Landau
centrifugal drive

In the previous subsection we have shown that centrifugal mechanism of
particle acceleration might be very efficient in millisecond as well as normal
pulsar’s magnetospheres. It is worth noting that different species of
centrifugally accelerated particles experience different forces leading to
charge separation and a consequent generation of the electrostatic field. On
the other hand, as it is found in [23, 42] the relativistic centrifugal force
is time dependent and therefore, the excitation process is parametrically
amplified.

For studying dynamics of a plasma flow following the magnetic field lines, it
is assumed that the field lines are almost rectilinear, which is satisfied on
distances less than the curvature radius. For simplicity one can consider
dynamics of plasma particles in the local frame of reference of a rotating
pulsar. If one considers the $1+1$ formalism [43] in the framework of a single
particle approximation, for the chosen reference frame the zero angular
momentum observers - ZAMOs, will measure the following proper time
$d\tau\equiv\zeta dt$, where $dt$ is the universal time,
$\zeta\equiv\sqrt{1-\Omega_{ef}^{2}r^{2}/c^{2}}$ is the so-called lapse
function. The equation of motion then is given by the following equation [23]

$\frac{d{\bf p}}{d\tau}=\gamma{\bf g}+\frac{e}{m}\left({\bf
E}+\frac{1}{c^{2}}{\bf v}\times{\bf B}\right),$ (34)

where ${\bf E}$ is the electric field, ${\bf p}\rightarrow{\bf p}/mc$ is the
dimensionless momentum, $\gamma{\bf g}$ is the effective centrifugal force
[24] with ${\bf g}=-{\bf\nabla}\zeta/\zeta$ and in the relativistic factor
$\gamma=\left(1-V^{2}/c^{2}\right)^{1/2}$ the velocity is defined according to
the $1+1$ approach ${\bf V}=d{\bf r}/d\tau$. In order to rewrite the
aforementioned equation for a magnetospheric fluid one should emphasise a
complex composition of plasmas. In Sec. 2.1.1. we have briefly discussed that
initially accelerated particles radiate by means of the curvature process.
Newly produced photons under certain conditions might generate secondary
electron positron pairs. Finally magnetospheric $e^{+},e^{-}$ population may
be divided into two major components: (I) the bulk component - the basic
plasma mass with relatively mild Lorentz factors and (II) the beam component
with high values of relativistic factors.Then, if one takes into account the
identity $d/d\tau=\partial/(\zeta\partial t)+\left({\bf V}{\bf\nabla}\right)$,
Eq. (34) will reduce to

$\frac{1}{\zeta_{i}}\frac{\partial{\bf p_{i}}}{\partial t}+\left({\bf
V_{i}}{\bf\nabla}\right){\bf p_{i}}=\gamma_{i}{\bf
g_{i}}+\frac{e_{i}}{m}\left({\bf E}+\frac{1}{c}{\bf V_{i}}\times{\bf
B}\right),$ (35)

for several species of particles denoted by $i=b,e,p$, where $b$ corresponds
to bulk particles (mostly electrons), $e$ \- to electrons and $p$ \- to
positrons. One should note that ZAMO’s momentum is coincident with the
momentum measured in the inertial frame of reference. Indeed, by combining the
definition of momentum, ${\bf p}=\gamma{\bf V}$, where
$\gamma=\zeta\gamma^{\prime}$ and ${\bf V^{\prime}}=d{\bf r}/d\tau$ (prime
denotes the quantities in the inertial frame of reference) one can
straightforwardly show ${\bf p}={\bf p^{\prime}}$. Then for the Eq. (35)
rewritten in the inertial frame one has (omitting primes)

$\frac{\partial{\bf p_{i}}}{\partial t}+\left({\bf
v_{i}}{\bf\nabla}\right){\bf
p_{i}}=-c^{2}\zeta_{i}\gamma_{i}{\bf\nabla}\zeta_{i}+\frac{e_{i}}{m}\left({\bf
E}+\frac{1}{c}{\bf v_{i}}\times{\bf B}\right).$ (36)

As we have already discussed, the centrifugal force, acting on different
species, leads to charge separation, which automatically creates the
electrostatic field. Therefore, for describing the physical system momentum
equation should be complemented by the continuity equation and the Poisson
equation

$\frac{\partial n_{i}}{\partial t}+{\bf\nabla}\left(n_{i}{\bf
v_{i}}\right)=0,$ (37) ${\bf\nabla}{\bf E}=4\pi\;\sum_{i}e_{i}n_{i},$ (38)

where $n_{i}$ denotes the number density of a corresponding component. It is
worth noting that strong magnetic field guarantees the frozen-in condition of
plasmas, ${\bf E_{0}}+\frac{1}{c}{\bf v_{0i}}\times{\bf B_{0}}=0$, which is
supposed to be a leading state of the system. By taking into account this
condition the momentum equation reduces to Eq. (5) which in a dimensional form
is given by

$\frac{d^{2}r}{dt^{2}}=\frac{\Omega_{ef}^{2}r}{1-\frac{\Omega_{ef}^{2}r^{2}}{c^{2}}}\left[1-\frac{\Omega_{ef}^{2}r^{2}}{c^{2}}-2\frac{\upsilon^{2}}{c^{2}}\right],$
(39)

where $\upsilon=dr/dt$ denotes the radial velocity. In [22] it has been shown
that for ultrarelativistic particles ($\gamma>>1$) Eq. (39) has the following
solution

$r(t)\simeq\frac{c}{\Omega_{ef}}\sin\left(\Omega_{ef}t+\phi\right),$ (40)
$\upsilon_{0}(t)\simeq c\cos\left(\Omega_{ef}t+\phi\right),$ (41)

where $\phi$ denotes the phase. After taking into account these solutions, one
can straightforwardly show that the analog of the centrifugal force,
$-\zeta_{i}\gamma_{i}{\bf\nabla}\zeta_{i}$, is time dependent and drives the
parametric excitation of the Langmuir waves. In this paper we intend to study
the linear generation of the electrostatic waves. Therefore, all physical
quantities are expanded around the leading state $\Psi^{0}$

$\Psi\approx\Psi^{0}+\Psi^{1},$ (42)

where $\Psi=(n,{\bf v},{\bf p},{\bf E},{\bf B})$ and $\Psi^{1}$ denotes a
small perturbation of the corresponding quantity. It is straightforward to
show that the anzatz

$\Psi^{1}(t,{\bf r})=\Psi^{1}(t)\;exp\left(i{\bf kr}\right),$ (43)

leads to the following linearized versions of Eqs. (36-38)

$\frac{\partial p_{i}^{1}}{\partial
t}+ik\upsilon_{0}p_{i}^{1}=\upsilon_{0}\Omega_{ef}^{2}rp_{i}^{1}+\frac{e_{i}}{m}E^{1}.$
(44) $\frac{\partial n_{i}^{1}}{\partial
t}+ik\upsilon_{i0}n_{i}^{1}+ikn_{i0}\upsilon_{i}^{1}=0,$ (45)
$ikE^{1}=4\pi\;\sum_{i}e_{i}n_{i}^{1}.$ (46)

By introducing the so-called plasma quantities for perturbations
$n_{pl}^{1}=n_{e}^{1}-n_{p}^{1}$,
$\upsilon^{1}_{pl}={\upsilon^{1}_{e}}-{\upsilon^{1}_{p}}$ (the plasma
component) with the unperturbed quantities of electrons and positrons
$n^{0}_{e}=n^{0}_{p}$ and $\upsilon^{0}_{e}=\upsilon^{0}_{p}$, and considering
the bulk component and the plasma component respectively, after
straightforward but tedious mathematical manipulations one arrives at the
coupled non-autonomous differential equations governing the centrifugal
generation of the electrostatic instability [44]

$\frac{d^{2}N_{1}}{dt^{2}}+\omega_{1}^{2}N_{1}=-\omega_{1}^{2}N_{2}e^{i\Xi},$
(47)
$\frac{d^{2}N_{2}}{dt^{2}}+\omega_{2}^{2}N_{2}=-\omega_{2}^{2}N_{1}e^{-i\Xi},$
(48)

where $\omega_{1,2}=\sqrt{8\pi e^{2}n_{1,2}/m\gamma_{1,2}^{3}}$ and $N_{1}$
and $N_{2}$ relate to the first order perturbations of number densities of two
species in the following way

$N_{\beta}=n_{\beta}\exp\left[\frac{ick}{\Omega}\sin\left(\Omega
t+\phi_{\beta}\right)\right],\;\beta=1,2,$ (49)

where $\phi_{\beta}$ denote the corresponding phases and $\Xi$ is expressed by
the following way

$\Xi=b\cos\left(\Omega t+\phi_{+}\right),$ (50)

and

$b=2\frac{ck}{\Omega}\sin\phi_{-},$ (51)

with $2\phi_{\pm}=\phi_{1}\pm\phi_{2}$. We have assumed that energy is almost
uniformly distributed among the species, $n_{1}\gamma_{1}\simeq
n_{2}\gamma_{2}\simeq n_{{}_{GJ}}\gamma_{b}$. After Fourier transforming Eqs.
(47,48) by applying the tools developed in [23], one can arrive to the
following dispersion relation

$-\omega^{2}+\omega_{1}^{2}+\omega_{2}^{2}\;J^{2}_{0}(b)=-\omega^{2}\omega_{2}^{2}\sum_{\mu\neq
0}\frac{J^{2}_{\mu}(b)}{\left(\omega+\mu\Omega\right)^{2}}.$ (52)

From the obtained expression it is clear that the resonance frequency is given
by
$\omega_{r}=\left(\omega_{1}^{2}+\omega_{2}^{2}\;J^{2}_{0}(b)\right)^{1/2}$.
Following the method described in [23], expanding the frequency,
$\omega=\omega_{r}+\Delta$, and emphasising that the near the resonance the
basic contribution to the r.h.s. of the equation comes form the term
$\omega_{r}=-\mu_{0}\Omega$, one obtaines the following equation

$\Delta^{3}=\frac{\omega_{r}\;\omega_{2}^{2}}{2}\;J^{2}_{\mu_{0}}(b).$ (53)

Solving this cubic equation, one can show that the imaginary part of the
solution, thus the growth rate of the instability, writes as [45]

$\Gamma=\frac{\sqrt{3}}{2}\left(\frac{\omega_{r}\;\omega_{2}^{2}}{2}\right)^{1/3}\;J^{2/3}_{\mu_{0}}(b).$
(54)

Already amplified electrostatic waves will inevitably Landau damp with the
following increment [46]

$\Gamma_{LD}=\frac{n_{{}_{GJ}}\gamma_{b}\;\omega_{b}}{n_{p}\gamma_{p}^{5/2}}.$
(55)

It is worth noting that the most optimal regime is when both growth rates are
of the same order of magnitude. For example, if one considers the Crab pulsar
with the possible maximum relativistic factor provided by the direct CA,
$1.5\times 10^{7}$, (see Eq. (21)), one can see that two streams with
$\gamma_{1}\simeq 10^{4}$ and $\gamma_{2}\simeq 7\times 10^{4}$ drive the
electrostatic instability with the corresponding time-scale, $1/\Gamma$, of
the order of $0.01$sec. For the aforementioned stream parameters (considering
the stream 1 component as to be the plasma component) both growth rates are
equal and since the time-scale is less than the kinematic time-scale (period
of rotation) of the Crab pulsar, the process is efficient. Therefore the
energy pumped by the primary beam by means of the Landau process should be
enormous.

In particular, by considering the force responsible for the particle
acceleration, $F_{r}\simeq 2mc\Omega\;\zeta^{-3}$ [45], the energy accumulated
in the primary beam after Landau damping is given by [45]

$\epsilon\simeq\frac{n_{p}F_{r}\delta r}{n_{{}_{GJ}}}$ (56)

and for the aforementioned two streams is of the order of $3.7$PeV. Here
$\delta r\simeq c/\Gamma$ is the corresponding length-scale where the process
of re-acceleration takes place. After applying the same mechanism to normal
pulsars ($P\simeq 1$ s, $\dot{P}\simeq 10^{-15}$ ss-1) with the beam Lorentz
factor of the order of $2.6\times 10^{5}$, one can straightforwardly check
that two streams with $\gamma_{1}\simeq 10^{2}$ and $\gamma_{2}\simeq 5\times
10^{2}$, lead to the parametric instability of generation of Langmuir waves
with a high growth rate $\Gamma\sim\Gamma_{L}\simeq 46\;s^{-1}$. Consequently,
by Eq. (56) one can check that the re-acceleration process in normal pulsars,
although cannot guarantee such high energies as in the Crab-like pulsars, but
still energies will be in the VHE domain, $360$ GeV.

In subsection 2.1.2. we have shown that the IC scattering and synchrotron
mechanism can not affect the dynamics of particles. The Langmuir waves
propagate along the magnetic field lines, and the resulting re-acceleration
will not cause the pitch angle significantly suppressing the synchrotron
process. We have already discussed that from observations it is evident that
cosmic particles leave the pulsar’s magnetospheres, which indicates that they
are in the force-free regime. Therefore, as is studied in [28, 29, 47] the
field lines lag behind the rotation, having the shape of the Archimedean
spiral characterised by the rectilinear trajectories in the laboratory frame
of reference. This means that the curvature emission is also strongly
suppressed and does not impose any significant constraints on maximum
achievable energies of electrons.

##  3 Conclusions

Centrifugal processes occurring in the pulsar’s magnetospheres might lead to
new and very interesting phenomena. This may allow new insights into the
physics of these extreme environments. The detection of deceleration of
pulsars indicates that a significant fraction of neutron stars’ rotation
energy is extracted from the central object. A certain fraction of it can be
pumped into the magnetospheic electrons and electromagnetic radiation.

In particular, we have demonstrated that direct CA is a very efficient
mechanism providing the observed jet velocity of the Crab pulsar. this process
is responsible also for generation of relativistic particles with the Lorentz
factors of the order of $10^{5}$ (for normal pulsars) and $10^{7}$ (for
millisecond Crab-like pulsars).

It has been found that IC scattering might provide photon energies of the
order from $100$ MeV up to several TeV depending on a pulsar’s type. The total
luminosity for Crab-like pulsars might be at least $10^{29}$ ergs s-1 and for
relatively slowly rotating pulsars - $10^{21}$ ergs s-1.

Curvature mechanism, although provides electromagnetic radiation in the GeV
(Crab-like pulsars) and UV (normal pulsars) domain, the corresponding
luminosities are higher than in the IC scattering: $10^{34}$ ergs s-1 (Crab-
like pulsars) and $10^{22}$ ergs s-1 (Normal pulsars).

We have also examined the effects of CA on the excitation of Langmuir waves.
It has been found that by means of the strong centrifugal force in the LC zone
the electrostatic waves are parametrically excited. The corresponding
instability time-scale is much less than the rotation period of a neutron
star, indicating high efficiency of the driving process.

The amplification of the Langmuir waves will be terminated by means of the
Landau damping resulting in extremely high energies of particles. We have
found that the electrons in Crab-like pulsars after re-acceleration via the
mechanism of Langmuir-Landau-centrifugal-drive might reach energies in the PeV
domain. For the normal pulsars the achieved energies of electrons is of the
order of several hundreds of GeV.

The centrifugal parametric excitation of Langmuir waves has a very interesting
consequence. In particular, as we have shown, the growth rate of the
instability is so large that the electrostatic waves are efficiently
amplified. During this process the field can reach the Schwinger limit and
consequently an efficient pair creation will take place. This in turn might
significantly influence physical processes in the $\gamma$-ray pulsar’s
magnetospheres and sooner or later we are going to study this particular
problem.

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