On the pulsar secondary electron-positron plasma production in low-energy region
An analytical description of low-energy falling-off of the secondary electron and positron distribution functions is proposed with the account of the effect of the inverse Compton scattering (ICS) on particle acceleration and plasma production. The resulting particle distribution with the described...
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nasplib_isofts_kiev_ua-123456789-1110202025-02-09T13:38:45Z On the pulsar secondary electron-positron plasma production in low-energy region Про генерацію вторинної електронно-позітронної плазми у магнітосфері пульсару в області малих енергій О генерации вторичной электронно-позитронной плазмы в магнитосфере пульсара в области малых энергий Kontorovich, V.M. Flanchik, A.B. Kinetic theory An analytical description of low-energy falling-off of the secondary electron and positron distribution functions is proposed with the account of the effect of the inverse Compton scattering (ICS) on particle acceleration and plasma production. The resulting particle distribution with the described low-energy falling-off may lead to the instability in magnetosphere plasma. Пропонується аналітичний опис низькоенергетичного завалу функцій розподілу електронів та позитронів, які породжуються у магнітосфері пульсару. Враховується вплив зворотнього комптонівського розсіювання на прискорення часток та народження пар Комптонівськими фотонами. Виникаючий розподіл часток вторинної плазми с завалом в області малих енергій може приводити до розвитку нестійкості. Предложено аналитическое описание низкоэнергетического завала функций распределения электронов и позитронов, порождаемых в магнитосфере пульсара, с учетом влияния обратного комптоновского рассеяния на ускорение частиц и рождение пар Комптоновскими фотонами. Возникающее распределение частиц вторичной плазмы с завалом в области малых энергий может приводить к развитию неустойчивости. The authors thank V.S. Beskin for useful discussion of the problem setup. 2007 Article On the pulsar secondary electron-positron plasma production in low-energy region / V.M. Kontorovich, A.B. Flanchik // Вопросы атомной науки и техники. — 2007. — № 3. — С. 312-316. — Бібліогр.: 10 назв. — англ. 1562-6016 PACS: 97.60.-Gb, 41.75.-Ht, 52.25.-Dg https://nasplib.isofts.kiev.ua/handle/123456789/111020 en Вопросы атомной науки и техники application/pdf Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Kinetic theory Kinetic theory Kontorovich, V.M. Flanchik, A.B. On the pulsar secondary electron-positron plasma production in low-energy region Вопросы атомной науки и техники |
| description |
An analytical description of low-energy falling-off of the secondary electron and positron distribution functions is proposed with the account of the effect of the inverse Compton scattering (ICS) on particle acceleration and plasma production. The resulting particle distribution with the described low-energy falling-off may lead to the instability in magnetosphere plasma. |
| format |
Article |
| author |
Kontorovich, V.M. Flanchik, A.B. |
| author_facet |
Kontorovich, V.M. Flanchik, A.B. |
| author_sort |
Kontorovich, V.M. |
| title |
On the pulsar secondary electron-positron plasma production in low-energy region |
| title_short |
On the pulsar secondary electron-positron plasma production in low-energy region |
| title_full |
On the pulsar secondary electron-positron plasma production in low-energy region |
| title_fullStr |
On the pulsar secondary electron-positron plasma production in low-energy region |
| title_full_unstemmed |
On the pulsar secondary electron-positron plasma production in low-energy region |
| title_sort |
on the pulsar secondary electron-positron plasma production in low-energy region |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2007 |
| topic_facet |
Kinetic theory |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/111020 |
| citation_txt |
On the pulsar secondary electron-positron plasma production in low-energy region / V.M. Kontorovich, A.B. Flanchik // Вопросы атомной науки и техники. — 2007. — № 3. — С. 312-316. — Бібліогр.: 10 назв. — англ. |
| series |
Вопросы атомной науки и техники |
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2025-11-26T08:09:33Z |
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2025-11-26T08:09:33Z |
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| fulltext |
ON THE PULSAR SECONDARY ELECTRON-POSITRON
PLASMA PRODUCTION IN LOW-ENERGY REGION
V.M. Kontorovich and A.B. Flanchik
Institute of Radioastronomy of NAS of Ukraine, Kharkov, Ukraine;
e-mail: vkont@ira.kharkov.ua; alex_svs_fl@vk.kh.ua
An analytical description of low-energy falling-off of the secondary electron and positron distribution functions
is proposed with the account of the effect of the inverse Compton scattering (ICS) on particle acceleration and
plasma production. The resulting particle distribution with the described low-energy falling-off may lead to the in-
stability in magnetosphere plasma.
PACS: 97.60.-Gb, 41.75.-Ht, 52.25.-Dg
1. INTRODUCTION
It is known that a pulsar activity is related to ultrare-
lativistic electron-positron plasma which is produced
near neutron star polar cap (PC) and moves along open
magnetic force lines [1,2]. The plasma production is
determinated by the neutron star rotation in super-strong
magnetic field (about 10 G). It results from the particle
acceleration in a vacuum gap above the polar cap which
is a part of open force line region where a longitudinal
(along the magnetic field) electric field exists. The
plasma production process undergoes in several steps
[2,3].
12
Electrons being torn from the pulsar surface are ac-
celerated in the longitudinal field of the gap up to
gamma-factors . There are two
mechanisms of energy losses for such electrons – a cur-
vature radiation (CR) and an inverse Compton scatter-
ing (ICS). CR arises due to motion of fast electron along
curved magnetic force line; the energy maximum of CR
belongs to frequencies ,
with being a curvature radius of magnetic force line.
ICS of primary electrons may take place by photons of
thermal emission of hot polar cap (with temperature
about K [4,5]), by the pulsar low-frequency emis-
sion near light cylinder [2] and by low-energy photons
of eigenmodes of the gap [6]. In this work we are to
consider the last factor influence.
72 10/ ≥ε=Γ mcprpr
cr ,3.0~ ωω cprcr Rc 2/3 3Γ=
cR
105
CR and ICS photons propagating in the pulsar
curved magnetic field create 1-st generation electron-
positron pairs. The electrons and positrons of 1-st gen-
eration are born at high Landau levels going down to the
normal state they emit synchrophotons which also are
capable to produce pairs (of 2-nd generation).
In previous paper [7] we obtained the low-energy
asymptotic of distribution function with falling-off
without influence of ICS. We showed that it was deter-
minated by exponential factor
)(Γf
( )2exp Γ
(Γf
2
0 /Γ− , where
is the maximum position of the function . Here we
consider the ICS acting on primary particle acceleration
and pair production in low-energy region.
0Γ
)
2. THE ICS ACTING ON THE PRIMARY
ELECTRON ACCELERATION
The primary particle equation of motion in vacuum
gap is [3]
ICS
pr
CR
pr
pr zE
mc
e
dz
d
c Γ−Γ−=
Γ
)(|| , (2.1)
where is the longitudinal electric field in the gap,
are the energy losses due to CR and ICS.
They are given by
)(|| zE
ICS
prΓCR
prΓ ,
2
22
42
3
4,
3
2
pr
TICS
pr
c
prCR
pr mc
u
Rcm
e
Γ
σ
=Γ
Γ
=Γ , (2.2)
with being the Thomson
cross-section and being the energy density of low-
energy radiation required for ICS.
222 )/()3/8( cmeT π=σ
u
The energy density is a parameter characterizing
the ICS acting on entire considered process. For the
thermal emission of the pulsar polar cap one has
. For the low-energy radiation in the
pulsar gap according to radio observation data we have
.
u
erg/cm
36 erg/cm10≤u
≤38 erg/cm10 u 31010≤
The detailed description of the primary electron ac-
celeration process using the equation (2.1) has not been
possible yet because there is not the exact solving of the
problem about the determination of the longitudinal
field with taking into account its screening by produced
plasma. To obtain simple approximate solution and ana-
lytical estimations let us use the equation for the longi-
tudinal field in framework of Arons’ model [1]:
)()1()(3)( 2
|| zzzzz
cR
BzE cc
НЗ
s −Θξ−−
Ω
≈ , (2.3)
where PCϑϑ=ξ / , , is an angle between
the magnetic axes and a line undergoing from the center
of the star to the exit point of the given force line from
polar cap.
10 ≤ξ≤ ϑ
cRНЗ /Ω
cz
PC ≈ϑ is an angle radius of the
polar cap and a value is the gap height depending on
. ξ
According to the procedure from [7], let us estimate
characteristic length at which the energy losses due to
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2007, N3 (2), p. 312-316. 312
ICS become dominant and corresponding primary elec-
tron gamma-factor. If CR dominates then the estimation
of the height at which the energy losses due to CR
almost compensate acceleration process, is given by [7]
CRz
7/327/12
32
3
Ωω
≈
cH
НЗ
e
c
CR z
cR
r
R
z ,
where . For pulsar parame-
ters s, G, cm we have
cm, the numerical solving of the equa-
tion (2.1) gives the dependence , shown in
Fig. 1 (curve 1). The estimation of the maximum
gamma-factor of primary electron is
mceBmcer sHe /,/ 22 =ω=
1.0= 12103.1 ⋅=sB cz
4102.1 ⋅
prΓ
P
≈
4102 ⋅≈
(zprΓ=
CRz
)
( ) 73/12max 10~)2/(3 CRecpr zrR≈Γ .
Fig. 1. The dependence of primary particle gamma-
factor on height z for various mechanisms of losses:
1−without ICS [7]; 2−with ICS ( u );
3−with ICS ( u )
3710 erg/cm=
39 /10 cmerg=
Let us consider the ICS acting on primary particle
acceleration with various values of . From (2.2) we
obtain the value of gamma-factor Γ at which the
energy losses due to CR and ICS become comparable
u
(eq
pr
)
euR Tc
eq
pr /2)( σ=Γ . (2.4) /=
If then the CR energy losses dominate, in
opposite case ICS losses are significant and for the last
case we have from (2.1): then the
gamma-factor of primary particle is given by
)(eq
prpr Γ>>Γ
)()( 2
|| zuzeE prT Γσ≈
ICSTpr zzzumcz ~),/()( 2 σ≈Γ , (2.5) e
where is a height at which the particle almost
compensated by ICS. For we obtain an equation
ICSz
ICSz
)/()()( 222
|| umczzeE TICSICS σ≈ . (2.6)
For the Arons’ field (2.3) in case of z we ob-
tain the estimation for
cICS z<<
ICSz
3/122 )(
σΩ
≈
uzBe
cRmcz
Tcs
НЗ
ICS
, (2.7) Kx=Φ
The maximum value of gamma-factor is then
3/22max )/( −∝σ≈Γ uzumc ICSTpr , (2.8)
the estimations yield in this case
cm, and
with low-energy radiation density
> u . The numerical integra-
tion of (2.1) with various values of gives the depend-
ences , shown in Fig. 1 (curves 2 ,3).
44 105.1...10~ ⋅ICSz
1010 3erg/cm > 810
)(zprpr Γ=Γ
6max 103~ ⋅Γpr
u
7105.1... ⋅
3erg/cm
3. PAIR PRODUCTION BY CURVATURE
RADIATION AND ICS PHOTONS
Let us consider the process of the electron-positron
plasma production in the region above the pulsar vac-
uum gap. Distribution functions of photons N and pro-
duced particles satisfy the kinetic equations (KE) f
ICSCR
c
qqNkwN
R
Bkc
z
Nc ++ε−=
ε∂
∂′
+
∂
∂
γ ),(
4
3 , (3.1)
),(),( zQzQ
z
fc CRICS Γ+Γ=
∂
∂ , (3.2)
where , is an angle between the photon
momentum and the magnetic field strength,
is a dimensionless photon energy,
, , and w is a
probability of pair production by a photon [8]. Func-
tions , q , ,Q are sources for the photons
and secondary plasma particles:
2/ mck ω=
4/sinβ′B
crBB / Bcr =
CRq ICS CRQ
β
2m
3=ε k
B =′ )/(3 ec
ICS
),( kεγ
∫ ΓΓΓ= prprCRprprCR dkPFq ),()( , (3.3)
∫ ΓΓ= 1
3
1)()( kddwkNFq prICSsoftprprICS , (3.4)
(∫ β−Γδ= γ kdNwQ CRCR
3sin/1 )
)
, (3.5)
(∫ β−Γδ= γ kdNwQ ICSICS
3sin/1 , (3.6)
2/3,
2
3 /2
2/5
2/1
Be
k
c
w f ′≥εε
α
= ε−
γ
, (3.7)
( ). The condition in (3.7) corresponds to the
pair production threshold . is the dis-
tribution functions of photons of low-energy radiation in
the gap,
mc
2sin ≥βk softN
( ))(zprpr Γ−
ICS
)( nepr Γδ=Γ
CRP w
Fpr
n
is the distribution
function of the primary electrons with the concentration
. The values и are probabilities of the cur-
vature radiation and ICS as functions of the energy of
final photon [3, 9]
3
2
3,
2
3
),( pr
c
c
c
pr
c
f
prCR R
k
k
k
kR
c
kP Γ=
Φ
Γ
π
α
=Γ
)cos1(
cos1
cos1
)(2
5
1
1
222
2
ζ−
ζ−
α−
−δ
Γ
=
pr
pr
pr
pr
e
ICS vkk
v
v
kk
mcV
rw , (3.8)
with , dyyx
x
∫
∞
)()( 3/5 Γ−Γ= /12
prprv , and
is an angle between primary electron and initial soft
α
313
photon momenta, is an angle between primary elec-
tron and final photon momenta, V is a volume of re-
gion in which ICS is considered. Taking into account
(3.8) one can rewrite (3.4) in a form
ζ
k )1
−
−
z
ne
)(
−
+
cos
cos
sin2
/1
2
||v
ICS
CRN
c
B′
(Γ
cR
B′
(ICS
N
Φ
k
Ndk
kV
rq soft
pr
e
ICS
1(5
12
2
∝
Γ
π
= ∫ . (3.9)
In equations (3.5) and (3.6) δ functions mean that
a gamma-factor of secondary particle after synchrotron
radiation is determinated only by the angle β of photon
which produces the particle. Indeed, let be the elec-
tron (positron) gamma-factor before synchrotron emis-
sion. In pulsar magnetic field the relation [10]
takes place; let us calculate the particle gamma-factor
after synchrotron emission. May the pair components
are produced at Landau levels and n with angles
and to the magnetic field direction. As it is
shown in [10], the pair production probability has a
maximum under θ and .
Then from conservation of the momentum projection to
the field direction
γ
+
=
2/k=γ
1>>= n
n
−θ +θ
θ=θ= + −n +n
β
ω
=θ+θ +−− coscos
c
pp
the relation between angles and β arises θ
cp /cos2 βω=θ , ( ) 2222/ cmcp e−ω= ,
or in an equivalent form
)4/()4(sin 222 −−β=θ kk . (3.10)
k sin
Let us come to the frame , where the particle motion
along magnetic field is absent, the velocity of this frame
is
K′
2
|| 1cos γ−θ=v . After synchrotron emission the
particle gamma-factor in this frame is to be Γ , re-
turning to the initial frame we have
1=′
222 /cossin/11/ γθ+θ=−Γ′=Γ ,
then, using (3.10), we obtain . β=Γ sin/1
We find the solution of KEs (3.1) and (3.2) in a form
CR NNN += , , (3.11) ICSCR fff +=
and distributions satisfy equations ,, ICSN ICSCR ff ,
CRCR
CRCR qNkwN
R
ck
z
Nc +ε−=
ε∂
∂
+
∂
∂
γ ),(
4
3 ,
), zQ
z
fc CR
CR =
∂
∂ , (3.12)
ICSICS
ICSICS qNkwNck
z
Nc +ε−=
ε∂
∂
+
∂
∂
γ ),(
4
3 ,
), zQ
z
f
c ICS Γ=
∂
∂
. (3.13) =r
The solutions for were obtained before [3]
with the low-energy asymptotics [7]
CRCR f,
Bk
Rzk
k
k
k
constN c
c
CR ′
ε
≥εφ
=
3
4,),(2 ,
′≥ε
εϕ−
′<ε
=εφ
2/3,)(exp
2/3,1
),(
2
2
* B
k
k
B
k , (3.14)
B
R
k cf
′
α
=
62
2
*
, ,)(
2/3
/2 dxex
B
x∫
ε
′
−=εϕ
2
−ΓΘΓ
η
=Γ
Γ
Γ
−
−
z
Renf c
CR
CR
CR
0
3/2)( , (3.15)
Bc
m eRk ′
−α
ΛΒ′≈Γ 3
4
0 )(4.0 , (3.16)
where is the concentration of the particles being
produced by CR-photons and
CRn
( )∫
Γ
Γ
−ΓΓ− ΓΓ=η
max
min
2
0 3/2/ deCR
(3.17)
is normalizing factor.
From (3.14) we see that under k the CR-photon
spectrum is a power-law: . The expo-
nential decreasing of the photon distribution (3.14) in
case of k ( ε ) is explained by the
resulting photon deficit due to intensive pair production.
According to [3], the pair production process becomes
the most effective when
ck<
/1~) k 3/5(kNCR
2/3B′2sin >β ≥
=Λ
Λ′
≈β
k
kk
kB
*ln)(,
)(3
4 . (3.18)
In similar way we obtain the solutions for the distri-
butions , (ICS contribution). From (3.13) we
have for the ICS-photon distribution
ICSICS fN ,
2/),(const kkN ICS εφ⋅= . (3.19)
Substituting the formulae (3.6), (3.19) to the equa-
tion for and integrating, we obtain for the secon-
dary plasma distribution
ICSf
Λ′
Γ
=
Λ
Γ
=
∂
∂
02 ;,
3
44 zz
B
kN
Rz
f
ICS
c
ICS . (3.20)
Here is a height of photon emission point
and is a photon path length. If β we have sim-
ple relation . It can be obtained by writing the
length of the interval between photon emission and pair
production points and using the magnetic force line
equation [2]: , ,
where the parameter of the magnetic force line is intro-
duced . For the photon emission
point we have . In case of ϑ
and tiny changing of ξ (when one has ) we have
, the angle difference is related to β .
Actually, having written the equations of tangent (using
coordinates , ) in the emission
and the absorption points in a form ,
γ−= szz0
γ
≈γs
ξ= /RA L
0
ϑ∆ϑA2
= rz
s
=
1<<
2)r∆
A ≈
ϑ∆
ϑ
=z
βcR
2s =γ
,2 RL
A
cos
( )20 (rr ≈−
Ω= /c
0
2
0 sin ϑ
ϑ = sinry
ϑ= 2sinAr
1, 0 <<ϑ
0A
const0 +yP
∆r
314
and , with , , we
obtain tg . From here we have
a relation β . Thus we obtain
const+= yPz
( 0P=β≈β
2/3 ϑ∆=
ϑ∆=∆≈γ rrs /2
)3/(4 ϑ= r
Λ
00 3/2 ϑ=P
)1 0PP+
β=ϑβ cR)3/(
ϑ= 3/2P
/()P−
=ϑ r4
Γ
Γ=′ eBRcf
Λ
Γ
Γ
2
2
0 Be ′
−
3
4
,
Rc being a curvature radius of the dipolar
field force line.
Equations (3.18) and (3.20) lead to the following re-
lation between and
6323 ΛαΛ , (3.21)
hence the factor weakly (logarithmically) depends on
. It can be seen from (3.14) and (3.16) that the low-
energy behavior of the produced particle distributions is
determinated by an exponential factor
Γ
−exp , . (3.22) c
m
Rk α
ΛΒ′≈Γ0 )(4.0
From (3.20) we obtain
)()(
2
0
Γ−ΓΘ
−ΓΘ
Γη
=Γ
Γ
Γ
−
ICS
c
ICS
ICS
ICS z
Renf , (3.23)
with being the concentration (in a point with coor-
dinate ) of the particles produced by ICS photons,
ICSn
z
( )∫
Γ
Γ
−ΓΓ− ΓΓ=η
ICS
deICS
min
2
0 1/ (3.24)
is a normalizing factor and is a high-energy end of
the spectrum, for which one can write
ICSΓ
)(~),(
4
3 2max zkkkkB prsoftICSICSICSICS ΓΛ′=Γ , (3.25)
where is a maximum value of in the spectrum
of low-energy initial photons.
max
softk k
4. THE RESULT DISCUSSION
We have shown that taking into account pair produc-
tion by ICS photons leads to the appearance of the addi-
tional part in the spectrum of the secondary plasma par-
ticles of 1-st generation (3.23). According to (3.19) ICS-
photons have a power-law spectrum ,
that is why the power-law asymptotic of the distribution
function of particles produced by ICS photons is given
by . The fast falling-off of the distribu-
tion function with decreasing (but still
) is described by the same absorption coeffi-
cient (3.14), that in case of curvature radiation.
There is an exponential factor
2)( −∝ kkN ICS
k
1)( −Γ∝ΓICSf
N
2sin ≥β
),( kεφ
ICS
k
( )22
0 /Γexp Γ−
z
which re-
sults from coefficient in the spectrum (3.23),
describing the decreasing of the distribution function
in the low-energy region. The function (3.23)
is a distribution function in a point (point O in Fig. 2)
and it is determinated by all ICS-photons arriving to this
point with various anglesβ (from different magnetic
force lines).
),( kεφ
)(ΓICSf
Γ
Fig. 2. A scheme demonstrating the secondary parti-
cle distribution in the point O, are photon emis-
sion points
41...PP
In case of high values of energy density of low-
energy radiation (which enters our model as a parame-
ter) the particle production by ICS-photons may pre-
dominate over the pair production by CR-photons. In
this case a transition when between ICS- and
CR-part of the secondary particle spectrum is expected
and after it remains only the part related to the curvature
radiation. The distribution function of secondary plasma
particle (with low-energy, ICS- and CR-asymptotics)
normalized by the concentration is shown in Fig. 3.
ICSΓ=
Fig. 3. The distribution function of the 1-st genera-
tion particles (in a point with 2000 m). The low-
energy falling-off, CR- and ICS-asymptotics are shown
=z
5. CONCLUSIONS
We have considered the ICS acting on the primary
particle acceleration in the pulsar vacuum gap and the
pair production by ICS-photons. It has been shown that
315
in case of high values of the energy density of the soft
radiation in the gap ICS energy losses may predominate
over CR-losses and determinate the maximum value of
primary particle gamma-factor.
The low-energy behavior of the distribution func-
tions of the 1-st generation particles produced by curva-
ture radiation and ICS photons has been considered in
this work. It has been shown that the distributions con-
tain exponential factor ( )22
0 /exp ΓΓ− , describing the
fast distribution decreasing in the region of small
gamma-factors. The expression for the distribution
function maximum position has been obtained. We have
obtained the ICS-photon distribution function and the
distribution of the particle produced by ICS-photons.
It is important to note that the distribution functions
of secondary plasma obtained in the work with low-
energy falling-off may lead to arising of the flux insta-
bility and the wave generation that may be important for
the pulsar radio emission theory.
The authors thank V.S. Beskin for useful discussion
of the problem setup.
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О ГЕНЕРАЦИИ ВТОРИЧНОЙ ЭЛЕКТРОННО-ПОЗИТРОННОЙ ПЛАЗМЫ
В МАГНИТОСФЕРЕ ПУЛЬСАРА В ОБЛАСТИ МАЛЫХ ЭНЕРГИЙ
В.М. Конторович, А.Б. Фланчик
Предложено аналитическое описание низкоэнергетического завала функций распределения электронов и
позитронов, порождаемых в магнитосфере пульсара, с учетом влияния обратного комптоновского рассеяния
на ускорение частиц и рождение пар комптоновскими фотонами. Возникающее распределение частиц вто-
ричной плазмы с завалом в области малых энергий может приводить к развитию неустойчивости.
ПРО ГЕНЕРАЦІЮ ВТОРИННОЇ ЕЛЕКТРОННО-ПОЗІТРОННОЇ ПЛАЗМИ
У МАГНІТОСФЕРІ ПУЛЬСАРУ В ОБЛАСТІ МАЛИХ ЕНЕРГІЙ
В.М. Конторович, О.Б. Фланчик
Пропонується аналітичний опис низькоенергетичного завалу функцій розподілу електронів та позитро-
нів, які породжуються у магнітосфері пульсару. Враховується вплив зворотного комптонівського розсію-
вання на прискорення часток та народження пар комптонівськими фотонами. Виникаючий розподіл часток
вторинної плазми с завалом в області малих енергій може приводити до розвитку нестійкості.
316
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