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On the Nature of Pulsar Radiation
NZ. Kazbeg~, G.Z" Machabeli, and G.I. Melikidze
Abastumani Astrophysical Observatory
INTRODUCTION
A key question in the interpretation of the emission of pulsars is that of
the excitation and propagation of waves in a magnetospheric plasma. The
magnetosphere of a pulsar has an extremely complex structure and there
are many difficulties in the development of its self-consistent model, the
bases of which were considered in Goldreich and Julian (1969~; Sturrock
(1971~; and Ruderman and Sutherland (l975~. At present there exist some
sufficiently well-grounded models not exactly agreeing with each other
(e.g. Ruderman and Sutherland 1975; Cheng and Ruderman 198~, Arons
and Sharlemann 1979; and Arons 1981~. However, the creation of a dense,
relativistic, electron-positron plasma in the polar regions of rotating neutron
star magnetospheres is the point of similarity among these models. The
pulsar radiation should be generated in such a plasma.
A spinning magnetized neutron star generates the electric field which
extracts electrons from the star surface and accelerates them forming low
density (nb = 710-2 BoP~i, where P is the pulsar period and Be the
magnetic field at the star surface) and the energetic (the Lorentz-factor
of particles is orb = 3.106 . - 107 for Apical pulsars) primal beam. In a
weakly curved magnetic field, electrons generate Quanta which produce in
turn electron-positron pairs. Further energetic radiation will be produced,
and this will go on to produce more pairs and so on until the plasma
becomes dense and screens the electric field (Goldreich and Julian 1969;
Sturrock 1971~. As a result, sufficiently dense (no ~ l0i6 _ 10~7 cm~3)
electron-positron plasma with an averaged Lorentz-factor up = 3 - 10 is
formed. The investigation of He kinetics of this avalanche process shows
225
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226
AMERICAN AND SOVIET PERSPECTIVES
that the plasma flowing along the open magnetic force lines possesses the
aysmmetrical distribution function with a tail stretched out in the direction
of the positive momenta (from the pulsar to the observer, e.g., Arons 1981~.
Due to the strong magnetic field, the transversal components (with respect
to the magnetic field Bo) of the particles momenta p ~ decay, and the
distribution function tends to be one-dimensional.
Most probably the magnetic field near the pulsar has a complicated
structure differing greatly from a dipole. Though at a sufficiently large
distances from the stellar surface up to the light cylinder, the magnetic field
can be considered as dipolar: B = Bo(Ro/R)3 if Ro << R < c/Q, where
Ro = 106 cm Is a neutron star radius, Q is the pulsar angular velocity.
The dependence of the plasma density from the distance Is the same n =
n<~(RO/R)3, where index "O" denotes the values taken at the star surface.
In our opinion the maser emission mechanisms (Ginzburg and Zhelez-
niakov 1975) are the only valid and well-grounded among the others (e.g.
antenna mechanisms; for more details see Lominadze et al. 19861.
~ consider the curvature of the magnetic field lines exactly the pylin-
dnc coordinates x, r, S° will be used below. The x-axis is directed transversely
to the plane where the curved field lines lie, r- is the radial and A- the
azimuthal coordinates. The latter describes the curvature of the field line
(torsion is neglected and BRB/0r = 0, RB - is the curvature radiUS of the
field line). In such geometry one has the following integrals of motion: A,
pa - W,B~ r/c, pa r. Here pi = vi y/c and vi are the particle momentum and
velocity, wale = en B/m c. The particle distribution function should depend
on the integrals of motion. Using the method of integration along the par-
ticle trajectory one obtains the components of the dielectrical permeability
tensor:
zz 1 '2 ~ w2 J ~ :`w k~v~-2k~u2/c)( 1+ ~1 of
+2w' 2 Q7o of A;
Err = 1 - 2 ~ P2 | dry Tt,, _ ~ it, _ ~ it, Nt 1 1
c~ = 1 + ~ Pa
a
l tow-Have-Aria) He+ T Q_ J Jay,
J dip 2QO it;
_ 1_ ~ PO / PA T(W-k~v~-kru=) i Q+-Q_ ) fa i;
~J
~ i'
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Representative terms from entire chapter:
z machabeli
HIGH-ENERGY ASTROPHYSICS
6'Z = - 2 ~ P2 J PY {(A-kyvy-2kyu2/c)(Q+ _ Q_)f~};
1 p2 dry vy 1 1 1 1
6=Y = 2 ~ w2 J ~ C ([kSc(Q+ + Q_)-ikrc((Q+ _ Q )] If
+2w UP ~ aft );
1 Spa ~ rln t, /1
6
228
AMERICAN AND SOVIET PERSPECTIVES
( 4/~)B 4)
my ( 16~)
(2)
(3)
note that formulae (2) and (3) are obtained for co < Up, and expression
(3) describes only the low-frequency branch of the it-waves. The phase
velocity of the high-frequengy branch exceeds the speed of light and cannot
interact with particles. The pulsar emission mechanisms provide infor-
mation on wave excitation and propagation in the magnetosphere plasma
along with their emergence in vacuum. The electromagnetic waves leave
the magnetosphere without transformation.
THE PULSE RADIATION MECHANISMS
Sagdeev and Shafranov (1960) were the first to point out the existence
of the cyclotron instability in plasma with the anisotropic temperature.
Let us invest/gate the possibility of t-wave generation on me Cyclotron
resonance:
-k~c-knurl BE =0
(4)
Substituting formula (2) and the expressions k = k~(1 + kl2/2k~p2) and v<,,
= c(1 - 1/2~2 - u2/2c2) in formula (4) one obtains:
k2 + 1 ~ 1 (k~
2k2 272e5 2 ~ kit
~ 2
UC! ~ _ ~ = ~ B (5)
c J k~pc~re~
As it was shown by Machabeli and Usov (1979) the l-waves are being excited
in the pulsar magnetosphere only at the anomalous Doppler-effect (upper
sign of the equation (5)). The lower sign of equation (5) corresponds to
wave damping. In the plasma rest frame these conditions are fulfilled for
the velocities satisfying the condition (Lominadze et al. 1986):
v cL'ok~v~+ ~ COB ~ ~WB + k2 V2 _ w2
C=
~)B + k2 V2
(~6)
It is obvious that for the whole range of frequencies we = kc < ~B7p
damping (lower sign in formula (6)) can occur only on particles having
negative velocities, and for the distribution function given on Figure 1,
at we << Wisp it is absent. In the case when the resonance condition
we - live,-Mu - CUB/7 = 0 is nevertheless fulfilled, waves are strongly
HIGH-ENERGY ASTROPHYSICS
f
l
/
l
l
/
/ I I _
-
_
up ~fib
FIGURE 1 The particle distribution function in the pulsar magnetosphere.
229
damped and cannot reach an observer. Hence, for energetic particles only
the resonance at the anomalous Doppler-effect (upper sign of equation
(5~) can be satisfied. Assuming that the electron and positron distribution
functions are about the same, and considering the perturbations propagating
nearly along the magnetic field in the k ~ v ~ /wB ~ 1 approximation one
obtains the following dispersion relation for t-waves:
k2c2
2 = e==
Cat
(7)
For the development of the cyclotron instability it is necessary to satyr the
following conditions (Machabeli and Usov 1979~:
1. The distance R between the star and the instability development
region should be less than the pulsar light cylinder radius:
( ) ~ ( ) ~
(8)
2. The characteristic time He of the instability development should be
less than me time of We plasma escape from the light cylinder ret
Using the definition of the rib (Goldreich and Julian 1969) and that rip
nip the condition (8) can be rewritten in me following way:
~p<5 10 orb (~012G) (Is) (9)
230
AMERICAN AND SOVIET PERSPECTIVES
In the case of the "typical" pulsar (Be ~ 10~2G, P ~ Is) for the beam
particles we have the following limitation on the average Lorentz-factor of
the particles of the bunk of plasma up < 5~, as for the particles of the tail
the limitation is Up < 10.
The value of up and consequently the satisfaction of the condition
(8) (or its equivalent (9~) depends on the configuration of the external
magnetic field of the pulsar. As it was shown earlier (Lominadze et al.
1983) in the dipole magnetic field the cyclotron instability can develop only
in the magnetospheres of the rapidly rotating pulsars (like PSR 0531~21
and PSR 0833~5~. Besides the energy flow earned away by the particles
from the pulsar should be of the same order as the energy released when
the rotation of the pulsar slows down. The difficulty for the pulsars with
a dipole magnetic field to satisfy the condition (9) is due to a very large
value of up > 102 . 103 (Cheng and Ruderman 1980~. On the other hand
it seems more probable that the magnetic field of a neutron star near the
stellar surface differs greatly from the dipole one, and the curvature of the
field lines RB is on the order of Ro (Ruderman and Sutherland 1975~.
Observational data on pulsars (Davies e! al. 1g84) and Faceting neutron
stars in Me binary systems, such as Her-X1 (Pines 1980) testier to the above
assumption. In the magnetic field with RB ~ RO one obtains Up < 10.
Consequently in the magnetic field of this configuration the region of the
cyclotron instability development may be inside the light cylinder both for
the beam of the primary particles and for the high-energy tail of the plasma
particles.
The resonance condition (5~(the upper sign) is satisfied only if
2 2
Ores
(10)
The latter is fulfilled at k ~ ~ 0. Thus the waves are being excited in very
narrow angles. The grown rate of Cyclotron instability is
`~2 1 u2
~o^JT 2 C
and
~ = ~"s ~ .
at that the resonant frequency is defined as
CaB
To ~
Unfree
(11)
(12)
(13)
(7T and tree are the thermal spread and the average Lorentz-factor of the
resonant particles respectively). As for the damping of t-waves, it occurs
HIGH-ENERGY ASTROPHYSICS
231
on the particles of the bulk of plasma and the corresponding decrement is
given by:
A,__ P 1
24)oC0B )~'
(14)
the frequency of damped waves is we ~ 2yp~B. However, as it was already
mentioned above, the damping for the distribution function given on Figure
1 and at we at< Pub iS absent
Note that the expressions (11) and (12) are valid only if the resonance
width is more than the growth rate (the so-called condition of kinetic
approximation). This leads to the following limitation on
<< wig AT
V
I red
(15)
From the expressions (11), (12), and (IS) it follows that for the particles of
the tail, the excitation of l-waves is possible at the distances R ~ 108 cm
from the center of the pulsar. As for the particles of the primary beam the
distance R is: R ~ 109 cm.
The interaction of the excited l-waves with the beam particles brings
about the quasilinear diffusion of particles in the momentum space. As a
result the beam particles acquire non-zero pitch-angles ¢(tg; = p ~ lips).
Besides the particles in the inhomogeneous magnetic field B(R) drift with
the velocity us. As it is shown by Machabeli and Usov (1979) in the central
parts of the magnetosphere the value of I,6 of RB2 and so it may exceed us.
In this case the resonance condition we - k~v~ - k~u~ = 0 due to the fact
that v~/c = 1 ~ll27re,2 - Nb2/2 - U=2/2C2 will take the following form:
(k )2
(16)
This type of Cherenkov resonance can be easily fulfilled. Let us consider
the possibility of the t-wave excitation by the beam particles at a Cherenkov
resonance (Kazbegi et al. 1987~. Solving the imaginary part of the dispersion
relation we obtain the growth rate of t-waves:
~ _ 1rW', {klPlOC)2 2 fib (6f~p)
t"' - 2 `~72 ~2wg J pro 1 + p2' 0 <0p
232
AMERICAN AND SOVIET PERSPECTIVES
The condition of the kinetic approximation yields tO
- < 1627T ~ 10 - 4 - 10 - 5
WO fib
(19)
The comparison of the forlllulae (16) through (19) gives an estimate rho
~ 54/3, which shows the possibility of the instability development on the
distance R ~ 5108 -. 109 cm.
The substitution of the it-waves spectrum (3) in the Cherenkov reso-
nance condition yields to
ok j2 w2
And the growth rate of the it-waves, when ~f 0 is given by
r it, ~ 2 ~ k ~ 2 ~
(20)
(21)
The kinetic approximation condition is the same, i.e. (19~. Note that
when k ~ ~ O it- and l-waves turn into one purely electromagnetic branch
describing the perturbations with an arbitrary polarization.
According to its definition the drift velocity us of the beam, with the
motion of particles from a pulsar to the light cylinder increases (us or
(~Ro~a, where a ~ 2 depending on the dependence of RB from R). In
the region where us > ~ the Cherenkov resonance condition looks like
[_ k2 _ 1 (U=_ k~52 a
for the l-waves and
(22)
U2
_ _ _ _ , _ .
C2 k2 8wp27p ~
us
(23)
for the it-waves. Making use of form. (1) one obtains the growth rate of
the t- and it-waves that equal each other in this case:
~ ~ sib orb ~ kr
Wo Stop AT k'
(24)
The condition of the ldnetic instability for both types of waves is given by:
r {U2~2 AT
wo c J orb
(25)
HIGH-ENERGY ASTROPHYSICS
2~33
It is clear that for the considered parameters of the typical pulsar the drift
motion causes the I- and it-wave excitation at the distances R ~ 109 cm
and in the radio frequency range we ~ 108 10~° Hz
Therefore, the wave excitation in the plasma of the pulsar magne-
tosphere is possible. The first possibility originates when the cyclotron
instability develops at the anomalous Doppler-effect resonance between
the l-waves and both the particles of the beam and the high-energetic par-
ticles of the tail of the distn~ution function. The excited waves propagate
along the magnetic force lines with k' ~ O producing the "core"-~e
emission Note that this mechanism is most stable among the others and
can be applied to the young pulsars. The development of the cyclotron
instability is accompanied by the quasilinear diffusion of waves with the
energetic particles. As a result, the beam particles obtain non-zero pitch
angles (p ~ ~ 0~. At the same tune and the same distance (R ~ 109 cm for
the typical pulsars) the particle drift motion caused by the magnetic field
inhomogeneity becomes substantial. In this case the Cherenkov resonance
can be satisfied in different parts of the magnetosphere: in the central part
taking into account that ~ > Tic, and at the edges where ~ < uric. In
both cases I- and it-waves are being excited. These mechanisms are valid
mainly for pulsars with P > 0.1 s producing the "cone"-bpe radioemission.
APPLICATION TO 1ME CENTRAL OBJECT IN SN1987A REMNANT
The detection of neutrinos from supernova 1987A suggests the for-
mation of a neutron star in its intenor. While the star left over from the
supernova explosion is now obscured by the ejects material, a pulsar can
be detected within the next few years. The period of a new-born pulsar can
be even 1 -. 2 ms (Friedman et al. 1986; Ostriker 1987~. But such values of
periods require very high luminosities of a pulsar L ~ 104° 104i e/s. By
this moment one can put an upper limit on the possible luminosity L < 1038
e/s. Hence the period increases too. The recently discovered radioem~sion
from SN (Chin) et al. 1988) suggests the following parameters of a pulsar:
P = 43 ms and L = 1038 e/s (Salvati e! al. 1989~. At the same time it is
supposed that pulsars are born having sufficiently long periods P ~ 0.1s
and even more (Narayan 1987~. Let us assume that the new-born pulsar
in SN1987A has the period of the order of P ~ 0.05 . 0.1s. Our task is
now to predict the range of frequencies in which the pulsar having these
periods will emit.
One should know the plasma parameters to apply the results of the
preceding chapter to PSR in SN1987N Assume the magnetic field at the
star surface Be ~ 10~2G. Then the Goldreich-Julian density is rib ~ 710
210~2 cm. The "tale" consists of the particles of the second generation
234
AMERICAN AND SOVIET PERSPECTIVES
(Arons 1981; 1hdemaru 1973~. The estimate of "tale" particles Lorentz-
factor is ~ ~ 105 - 106 (Lominadze et al. 1983~. Substituting the obtained
parameters ~ the fo~ulae (11), (13), and (15), one finds that the instability
develops in the vicinity of the light cylinder at Me distances - R ~ 2 108
cm for P = 0.05s and R ~ 5108 cm for P = 0.1s. At these distances as it
follows from eq.(l4) the following frequencies are being excited YCYC[(P =
O.O5s)= 1014 . 1015 Hz and YCYC!~ = 0.1s) ~ 21011 . 1013 Hz).
The development of the Cyclotron instability is accompanied by the
quasilinear diffusion of the particles in the momentum space both along
and across the magnetic field. The particles obtain non-zero pitch-angles of
the order of fib ~ 10-2 . 10-3 (Machabeli and Usov 1979~. In this case the
Inequality ~ < 1/-y' is fulfilled for the "tale" particles and the synchrotron
emission peaks at the frequencies:
Oman = ~b¢72 ~ 5 102~(P = 0.05s) . 10 (P = 0.1s)
contn~uting to X-ray radiation of SN1987A that is thought lo be mainly
from the decay of radioactive Co56 (Dotani et al. 1987; Sunyaev 1987~.
Thus if the pulsar in SN1987A is formed having the considered param-
eters it should produce the energetic radiation having much in common
with Crab, Vela-pulsars and PSR 0540-69, and PSR 1509-58. The range
of emitted frequencies generated at cyclotron instability decrease with the
pulsar slow-down and if P > 0.1 0.2s the frequency range will lower
to radio frequencies and soft X-rays or UV (for radiation produced by
synchrotron mechanism), as it is in PSR 1509-58. And conversely, if the
period of the new-born pulsar is in the millisecond range Christian et al.
1989) the corresponding frequencies are ~cycZ ~ 10~7 Hz and Z~yn ~ 1024
1025 Hz (Kazbegi e! al. 1988~.
At the same time the luminosity of the pulsar in SN1987A should not
resemble that of Crab, Vela, or PSR 054~69, being of the order of Lope ~
103° . 1032 e/s and Lo ~ 1~5 2 1~6 e/s.
Despite the optically thick shell the pulsar emission should emerge
from the interior of SN1987N In our opinion (Kazbegi and Machabeli
1990) the detected radioemission from the shell with the wavelength ~ ~
1.3 mm (Chin) 1988) is a result of hydrogen atom recombination emission
at the transition from the 32 to 31 level. The photoexcitation exceeds the
pulsar rotation period. In the time 109 -. 5109s after the explosion the
shell should become transparent enough to detect pulsed radioemission.
Thus we predict the range of frequencies in which the new-born pulsar
in SN1987A with P = 0.05 . O.ls should originate. Applications to the
other periods can be easily done.
Eventual some conclusions can be drawn. The offered theory of
pulsar radiation is based on the investigation of all plasma instabilities able
HIGH^ENERGY ASTROPHYSICS
235
to develop in the electron-positron plasma of pulsar magnetospheres. As
it was shown here and previously (Kazbegi e' al. 1989a) the only plasma
instabilities are: the cyclotron developing at the anomalous Doppler-effect
resonance; and Cherenkov with the consideration of the particle transversal
momenta caused by the quasilinear diffusion in one case and by the drift
motion in the other one. These mechanisms can produce the well-known
pulse profiles (see e.g. Rankin 1983~: "core" (caused by the cyclotron
mechanism) and "cone" (by the other two). On the basis of this model
virtually all observational features listed in the papers by Rankin (1986)
and Taylor and Stinebring (1988) can be explained: mode changing, pulse
pulling and the presence of the orthogonal modes (Kazbegi et al. 1989b);
the existence and behavior of the circular polarization (Kazbegi et al.
1990a); and subpulse drift (Kazbegi et al. 1990b). The analysis of the
emission mechanisms' dependence on the pulsar rotation period P and its
application to the expected pulsar in the interior of the SN1987A shows
that it should be a source of a high-frequengy radiation.
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