On the synchrotron radiation of ultrarelativistic electrons moving along curved spiral trajectory
The synchrotron radiation of an ultrarelativistic charged particle moving along spiral trajectory winded on curved magnetic force line is considered. The radiation pattern has new properties on a comparison with the radiation in homogeneous magnetic field: there is a range of characteristic frequenc...
Збережено в:
| Опубліковано в: : | Вопросы атомной науки и техники |
|---|---|
| Дата: | 2003 |
| Автор: | |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2003
|
| Теми: | |
| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/111161 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | On the synchrotron radiation of ultrarelativistic electrons moving along curved spiral trajectory / Ya.M. Sobolev // Вопросы атомной науки и техники. — 2003. — № 4. — С. 197-202. — Бібліогр.: 12 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
nasplib_isofts_kiev_ua-123456789-111161 |
|---|---|
| record_format |
dspace |
| spelling |
Sobolev, Ya.M. 2017-01-08T16:43:44Z 2017-01-08T16:43:44Z 2003 On the synchrotron radiation of ultrarelativistic electrons moving along curved spiral trajectory / Ya.M. Sobolev // Вопросы атомной науки и техники. — 2003. — № 4. — С. 197-202. — Бібліогр.: 12 назв. — англ. 1562-6016 https://nasplib.isofts.kiev.ua/handle/123456789/111161 533.9 The synchrotron radiation of an ultrarelativistic charged particle moving along spiral trajectory winded on curved magnetic force line is considered. The radiation pattern has new properties on a comparison with the radiation in homogeneous magnetic field: there is a range of characteristic frequencies instead of one characteristic frequency, the peaks of the radiation pattern correspond to periodically repeated directions in space, which position depends on the frequency of radiation. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Космическая плазма On the synchrotron radiation of ultrarelativistic electrons moving along curved spiral trajectory Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
On the synchrotron radiation of ultrarelativistic electrons moving along curved spiral trajectory |
| spellingShingle |
On the synchrotron radiation of ultrarelativistic electrons moving along curved spiral trajectory Sobolev, Ya.M. Космическая плазма |
| title_short |
On the synchrotron radiation of ultrarelativistic electrons moving along curved spiral trajectory |
| title_full |
On the synchrotron radiation of ultrarelativistic electrons moving along curved spiral trajectory |
| title_fullStr |
On the synchrotron radiation of ultrarelativistic electrons moving along curved spiral trajectory |
| title_full_unstemmed |
On the synchrotron radiation of ultrarelativistic electrons moving along curved spiral trajectory |
| title_sort |
on the synchrotron radiation of ultrarelativistic electrons moving along curved spiral trajectory |
| author |
Sobolev, Ya.M. |
| author_facet |
Sobolev, Ya.M. |
| topic |
Космическая плазма |
| topic_facet |
Космическая плазма |
| publishDate |
2003 |
| language |
English |
| container_title |
Вопросы атомной науки и техники |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| format |
Article |
| description |
The synchrotron radiation of an ultrarelativistic charged particle moving along spiral trajectory winded on curved magnetic force line is considered. The radiation pattern has new properties on a comparison with the radiation in homogeneous magnetic field: there is a range of characteristic frequencies instead of one characteristic frequency, the peaks of the radiation pattern correspond to periodically repeated directions in space, which position depends on the frequency of radiation.
|
| issn |
1562-6016 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/111161 |
| citation_txt |
On the synchrotron radiation of ultrarelativistic electrons moving along curved spiral trajectory / Ya.M. Sobolev // Вопросы атомной науки и техники. — 2003. — № 4. — С. 197-202. — Бібліогр.: 12 назв. — англ. |
| work_keys_str_mv |
AT sobolevyam onthesynchrotronradiationofultrarelativisticelectronsmovingalongcurvedspiraltrajectory |
| first_indexed |
2025-11-24T11:37:43Z |
| last_indexed |
2025-11-24T11:37:43Z |
| _version_ |
1850845598830297088 |
| fulltext |
UDK 533.9
ON THE SYNCHROTRON RADIATION OF ULTRARELATIVISTIC
ELECTRONS MOVING ALONG CURVED SPIRAL TRAJECTORY
Ya.M. Sobolev
Institute of Radio Astronomy, NAS of Ukraine, Kharkov, Ukraine
sobolev@ira.kharkov.ua
The synchrotron radiation of an ultrarelativistic charged particle moving along spiral trajectory winded on
curved magnetic force line is considered. The radiation pattern has new properties on a comparison with the radia-
tion in homogeneous magnetic field: there is a range of characteristic frequencies instead of one characteristic fre-
quency, the peaks of the radiation pattern correspond to periodically repeated directions in space, which position de-
pends on the frequency of radiation.
1. INTRODUCTION
The formulae for the synchrotron radiation mecha-
nism in a homogeneous magnetic field [1-3] are widely
applied in various branches of science and engineering
[4]. However, the formulae for synchrotron radiation in
straight magnetic field lines may be insufficient to de-
scribe radiation of ultrarelativistic electrons moving
along dipolar field lines in the magnetosphere of a pul-
sar [5]; or the radiation emitted by runaway electrons in
tokamaks [6]. It is necessary to take into account the
curvature of the magnetic force line [5, 6].
Since synchrotron radiation comes from a small
length along the trajectory, the curved magnetic field
lines are approximated by circular force lines and the ra-
diation from relativistic electrons moving with small
pitch angles along spiral trajectory is considered. The
radiation formulae have been calculated by various
methods in the papers [5 - 8]. In [5] the radiation was
called as synchrotron curvature. In [5, 7] the spectrum
and polarization characteristics of radiation was ob-
tained, in [6] the radiation spectrum was found. Expres-
sions for the spectrum obtained in the papers [5 - 7]
have various forms. The comparison of the formulae [5]
and [7] is carried out in [8]. The limit of an undulator
radiation, when the contribution to radiation occurs
from a lot of cyclotron rotations, has been considered in
[9, 10].
At the same time the spectral angular distribution of
synchrotron radiation emitted by ultrarelativistic
charged particles moving along curved spiral trajectory
is not investigated. In the present paper such spectral an-
gular distribution is studied and the comparison of ex-
pressions for the radiation spectrum obtained in the pa-
pers [6, 7] will be also carried out.
2. TRAJECTORY OF PARTICLE
Let us assume that magnetic force lines look like a
circle, and the magnitude of magnetic field is B0. Select
a system of Cartesian coordinates with (x, y)-axes in the
plane of magnetic field lines, and z-axis coinciding with
the axis of cylindrical magnetic surface. The magnetic
field vector can be expressed as
)cos(sin0 jiB ϕϕ −= B , (1)
where φ is the polar angle in (x, y)-plane, i,j are the ba-
sis vectors of Cartesian frame. The particle with the
Lorentz-factor 1)/1( 2/122 > >−= −cvγ is moving along
magnetic force lines with the velocity close to speed of
light. The angular velocity Ω corresponding this mo-
tion ( R/||v≡Ω , where ||v is the velocity of the guiding
center along the magnetic line with curvature radius R)
is much less than the frequency of rotation around mag-
netic force line Bω , || Bω< <Ω . The radius of Larmor
circle Br is much less than R , R< <Br .
Equations of motion of a charged particle in the
magnetic field (1) are integrated in quadratures. The so-
lution expresses through elliptic integrals of the 1-st and
3-rd kind. The asymptotic expansion of the position
vector of trajectory, in which the terms proportional
1)/ 2
B < <R(r are dropped, has the form [7, 8]
[ ) ]
[ ( ) ]
[ ] ,sin
coscossinsin2
sincos(cossin2
BBD
BBBB
BBBB
k
j
ir
trt
ttrRttr
ttrRttr
ω
ωωδ
ωωδ
−+
+Ω++Ω+
+Ω++Ω−=
v
(2)
where γω αα cmBe /0B = , 1/ B < <Ω= ωδ ,
B
2
D /ωRΩ−=v is the drift velocity, αe and αm is the
charge and mass of a particle of a sort α , i, j, k are the
basis vectors of the Cartesian frame.
In contrast to known expressions of drift theory, the
terms, which are proportional tBsin2 ωδ , is taken into
account. It is necessary to reduce evaluations in the
Cartesian frame to evaluations in the frame of natural
trihedral [7, 8].
The magnitude of particle velocity remains constant
and is given by expression
2/12
B
2
B
2
D
22 )( rR ω++Ω= vv . (3)
The curvature radius of trajectory (2) is equal to
2/1
B
221
C )cos21(/ tqqRkr ω++Ω== 2− v , (4)
where k is the curvature, the parameter
)/( 2
B
2
B Rrq Ω= ω is equal to the ratio of the Larmor ve-
locity BBL || rω=v to the magnitude of drift velocity.
Further we shall consider the case, for which the
projection of a particle velocity on magnetic lines is
close to speed of light, c→||v . Thus the Lorentz-factor
corresponding to motion along magnetic field lines,
1)/1( 2/122
|| > >−= −
| | cvγ .
If the magnitude of magnetic field depends on radi-
al coordinate )(0 rBB → , the corresponding drift veloci-
ty )/()2/1( B
2 Rg ω⊥= vv , (where ⊥v is the velocity
transverse to magnetic field lines) is smaller than the ve-
locity of centrifugal drift )/()2/1( B
2
|| RD ωvv = . There-
fore, we shall consider the constant magnetic field ap-
proximation.
It is known that radiation of a relativistic charged
particle occurs from the small part of trajectory and con-
centrates within the angle γ/1~ at apex of cone along
particle’s velocity [4, 11]. Thus, the instantaneous angle
γ/1~ of the radiation beam should be less than the an-
gle between the particle velocity and drift trajectory.
From this requirement, definition of Lorentz-factor γ ,
and inequality 1> >| |γ follows that the limit of syn-
chrotron radiation takes place, if
2
| |
2 > > γγ (5)
Suppose that the inequality (5) is fulfilled.
3. SPECTRAL ANGULAR DISTRIBUTION
OF SYNCHROTRON RADIATION
The energy E emitted by a charged particle in the
solid angle between ο and οο d+ , and the interval of
frequencies between γ and ωω d+ is given by [11]
ωοω
π
dd
cR
d 2
2
2
0 )(
4
E=E , (6)
where the Fourier integral representation of an electrical
field is
.)}/(exp{]],[[)(
}{
0
0
∫
+ ∞
∞−
−
−
= dtctie
cR
ei c
Ri
nrβnnE ω
ω
ω
ω
α (7)
Here R0 is the distance up to the observer, n is the unit
vector pointing to the observer, c/v=β , v is the veloc-
ity, r is the particle position vector (2).
To calculate integral (7), we use a frame of natural
trihedral at time 0t . Denote by v/)( 0tv=τ , )( 0tν ,
)( 0tb the tangent, normal, and binorma, respectively.
By definition, the instant 0t is found from requirement
that the vector n belongs to (τ, b)-plane, i. e., the
equation 0)( 0 =tnν being satisfied
bτn χχ sincos += , 0)( 0 =tnν , (8)
where χ is the angle between the vectors τ and n.
The polarization unit vectors πe , σe on the plane
that is perpendicular to line of sight are
,νe =σ
,cossin bτe χχπ −= ],[ neσ
. (9)
Expanding the position vector r(t) into a Taylor se-
ries about )( 0tt − , and then substituting in (7), we ob-
tain [8]
{ }
( )( )[{
( ) ( ) ]} ,6sincos
cosexp
/)(
sin
exp(
3
0
332
0
0
2
0
dtcttkk
tti
cttk
i
cR
ei
Ei
−−+
+−−1×
×
−−
Φ
−
=) ∫
∞
∞−
χχ
χβω
χβω
ω α
vv
v
κ
(10)
where σπ ,=i , the top string is related to π- polariza-
tion, and ]/)(/ 000 cttcR nr−+[=Φ ω is the constant
phase.
As shown in [8], it is possible to neglect the term
χsin3kvκ , and we have
( ) )(1
3
2
3/1422
2
2
2
ηψψ
γ
β
π
ω
ωο
απ K
kc
e
dd
d 22
2
+=
v
E
, (11)
( ) )(1
3
2
3/2
2
422
2
2
2
ηψ
γ
β
π
ω
ωο
ασ K
kc
e
dd
d 2
2
+=
v
E
, (12)
where γ χψ = , 2/3)1)(/(21 2+)/(= ψωωη c ,
vkc
3)2/3( γω = , )(3/1 xK , )(3/2 xK are the modified
Bessel functions.
Since the instantaneous curvature radius changes as
the particle moves from one to another trajectory points,
the synchrotron radiation mechanism in circular mag-
netic field differs from synchrotron radiation in straight
magnetic lines. Let us consider the radiation pattern.
As it is known, the radiation of an ultrarelativistic
charge is concentrated into a cone along the particle ve-
locity. When the charge drives along trajectory (2), the
instantaneous direction of the radiation beam changes.
As a result, the radiation will be concentrated in the
neighborhood of a surface (design it by S), which gener-
ating lines coincide with the velocity vectors. The inter-
section of the surface S with the unit sphere gives a line
L. Points at line L correspond to directions at different
instants of time 0t .
In the plane, which is perpendicular to the line L,
the form of radiation pattern is described by equations
(11), (12) with the curvature radius at time 0t being tak-
en as the circle radius. For σ - polarization the radiation
has maximum in directions at the line L, 0=χ ; the ra-
diation in π - polarization has peaks for angles
γχ /1|| = and tends to zero at line L, ( 0=χ ).
Let us consider the radiation pattern, assuming that
the direction of emission passes near to x-axis. Intro-
duce angular coordinates yθ and zθ , where yθ is the
angle between n and (x, z)-plane, and zθ is the angle
between x-axis and the projection of the vector n onto
(x, z)-plane.
The projection of line L on (y, z)-plane in the case
of small angles 1< <yθ , 1< <zθ is given by equations
( )00 sin/ tqt BB
B
yy ωω
ω
θ +Ω−=≈ vv
( )0cos1/ tq B
B
zz ω
ω
θ +Ω−=≈ vv . (13)
The form of the radiation pattern depends on the re-
lation between the velocity of centrifugal drift || Dv and
Larmor velocity, BL r|||| Β= ωv , |/| DLq vv≡ .
In case 1> >q the radiation pattern resembles the
radiation cone (with the apex angle | |γ/1~ and angular
width γ/1~ for the cone wall) in straight magnetic
field. In the case 1< <q we have the limit of curvature
radiation. In both cases 1> >q and 1< <q , the curva-
ture radius does not depend on time 0t so that the pro-
files of radiation pattern remain constant.
In case 1~q the spectral angular distribution of ra-
diation has specific features as compared with the syn-
chrotron radiation mechanism in homogeneous magnet-
ic field. For 1~q the drift velocity is approximately
equal to Larmor velocity BD r|||| Β≅ ωv )2/( | |≅ γc so
that the curvature radius varies with time 0t . There ap-
pears the range of characteristic frequencies from
|q−1|Ω∼ 3γω up to )( q+1Ω∼ 3γω instead of one
characteristic frequency ( Ω∼ 3 qγω for an emission in
straight magnetic field lines, or Ω∼ 3γω for a curvature
radiation).
In Fig.1 the radiation patterns (for σ -+π -polariza-
tion) at frequencies corresponding to minimal (Fig.1a)
and maximum (Fig.1,b) curvature radius of trajectory
(2) at 2,1=q , 1 5=γ δ , πωπ ≤≤− 0tB are represented.
The picture is periodically repeated with time along yθ -
axis, Fig.2,a. If the frequency of radiation corresponds
to the maximum characteristic frequency
)1(8338,0 qc +=/ ωω , the radiation pattern has peaks
for directions corresponding to trajectory points with
minimal curvature radius, ny π δθ 2= , ...,1,0 ±=n ;
δθ )1( qz +−= . Denote these directions by A. At higher
frequencies, the radiation is more concentrated in the
neighborhood of directions A.
At the minimal characteristic frequency
|1|8338,0 qc −=/ ωω , Fig. 1,b, the peaks of the radia-
tion pattern correspond to the trajectory points with
maximal curvature radius, ny π δπ δθ 2+−= ,
...,1,0 ±=n ; δθ )1( qz +−= , Fig. 2,b. For lower fre-
quencies the radiation is even more concentrated in the
neighborhoods of these points. When the direction of
light biases to A, the section of radiation pattern be-
comes two-humped because of increasing the relative
contribution of π -polarization component in the total (
π - + σ -) radiation beam, Fig. 1,b.
Fig. 1 Radiation pattern at the given frequency ω :
a) ω is equal to the maximal characteristic frequency;
b) ω equals minimal characteristic frequency;
max
)(
+≡
ωοωο
σπσ
dd
EEE d
dd
dF
Thus the form of radiation pattern depends on the
frequency of radiation while the particle moves along
trajectory (2).
Polarization properties of radiation are described by
equations (9), (10). The ort σe coincides with the tan-
gent to line L, Fig. 2, πe is perpendicular to σe and n.
For the directions at line L the radiation has linear polar-
ization ( 0≠σE , 0=πE ). The sense of elliptical polar-
ization coincides with the sense of particle rotation
around n.
Fig.2. The width of radiation pattern. (solid line) is
the contour at level of one half of maximal value; (+),(-)
denote the sense of elliptical polarization
In the case of exact equality 1=q the trajectory (2)
has points at which the curvature becomes equal to zero.
At these point equation (10) is failed. In [8] it was found
that the approximation (10) is correct, if
2/1
B ))/(|(||1| Ω>− γωq
2/1)/( γγ | |= . The case 1=q
needs a special study. This will be the object of another
paper.
4. RADIATION SPECTRUM
To find the radiation power per unit frequency, we
shall integrate equations (11), (12) over the radiation an-
gle and then divide it by the time interval of radiation.
Let µ and χ be angular coordinates. The variable µ
describes directions that correspond to segments of line
L. The angle χ corresponds to arcs of the great circle,
which is perpendicular to line L. v|vv /|/ 0dtdd
••
== rµ
0dtkv= , and the element of solid angle has the form
0dtkdddd vχµχο == . (14)
Dividing (11), (12) by |2 Bωπ |/ and integrating
over a solid angle, we obtain
∫ ∫
∞2
=
π
π
ω
γ
β
πω 0
3/5
B
3 )()()|(|
4
33
y
xdxKy
kc
kWtd
d
dP
, (15)
where cy ωω /= , 22)/)(3/2()( vkcekW 42= γα . The ex-
pression after the first integral sign in (15) can be inter-
preted as a spectral power of radiation for a charged par-
ticle moving in a circular orbit with the instantaneous
radius )(1
C tkr −= .
Let us derive formula (15) without using expression
(14). At first we integrate over solid angle in (6).
4.1. SCHWINGER’S FORMULA
Substituting (7) in equation (6), we obtain
,)]}(exp{
)])(([
4
1212
21
2
21
rrn
nn
−−−[×
×−= ∫
∞
∞−
2
2
tti
c
e
dtdt
dod
d
ω
π
ω
ω
α
21 ββββE
(16)
where )( ii trr = , )( iii tββ = , 2,1=i .
Integrating by parts the second term in (16), then
integrating in οd , and introducing the variable
12 tt −=τ , we obtain (see also [12])
.
|)()
/|)()sin
cos(1 2
2
tt
ctt
c
ttddt
e
d
d
rr
rr
vv
−+(|
−+(|×
×
)()+−−= ∫
∞
∞−
τ
τω
ω τττ
π
ω
ω
αE
(17)
From expression (17) follows the expression for a
spectral power at the time t
.
|)()
/|)()sin
cos(1)(
2
2
tt
ctt
c
ttd
e
d
tdP
rr
rr
vv
−+(|
−+(|×
×
)()+−−= ∫
∞
∞−
τ
τω
ω τττ
π
ω
ω
α
(18)
It is the formula (I.37) obtained by Schwinger in [3]. He
considered the rate at which the electron does work on
the radiation field. Equation (18) is the starter formula
in [6]. Let us now show that (15) is also followed from
equation. (18). Using the Frenet formulae, we obtain
−1=−+(|
2
22
24
|)() vktt τττ rr , (19)
22
2 2
/1(1 vk
c
tt 2
2 +≅)()+− τγτ vv . (20)
Substituting equations (19), (20) into (18), we re-
duce expression (18) to
+−
+×
×
+−=
∫
∫
∞
2
2
2
2
2
∞
2
2
2
0
222
0
22
2
sin
2
1
12
1
2
sin
2
1)(
ω τγτ
τ
τγτ
γ
ω τ
γτ
τ
τ
π γ
ω
ω
α
vv
v
v
2 kdk
kde
d
tdP
(21)
where )(tkk = .
After introducing in (21) the new integration vari-
able vkx τ γ= and employing the formula from [3],
∫∫
∞∞
=−++
y
xdxK
x
dxxxyx )(
3
1
2
)
3
(
2
3sin)21( 3/5
3
2
0
π
,
we obtain expression (15).
4.2. GENERALIZATION OF RADIATION SPEC-
TRUM
To integrate with respect to 0|| tBω in (15), we in-
troduce the variable 0
2 cos21 tqqz Bω++= and change
the order of integration. Then [8]
),( C
C
C qyfP
d
dP
ωω
= , (22)
,
2
/1
arcsin)(1
)(
8
39),(
|1|
1
22
C
2
3/5
|q1|
3/5CC
C
C
C
−+
+
2
+
+
=
∫
∫
−
+
∞
−
q
y
q
y
y
q
xyq
xdxK
xdxKyqyf
π
π
π
where 22
| |
4 Ω= βγα
c
eP
2
C 3
2 is the total power emitted by a
charged particle moving with velocity ||v along a circu-
lar orbit of radius R, Cy ωω /C = .
Thus, the universal function of synchrotron radia-
tion for a relativistic electron moving in circular orbit
[2, 3, 12]
f y y dxK x
y
( ) ( )/=
∞
∫9 3
8 5 3π (23)
is replaced by expression (22) for a relativistic electron
moving along the spiral trajectory in circular magnetic
field. In [6], the radiation spectrum, which form is dif-
ferent from (22), was obtained from the Schwinger for-
mula (18). As it has shown above, spectrum (22) also
follows from (18). The radiation spectrum obtained in
[6] is given in Appendix. Thus (22) and the correspond-
ing formula in [6] are two different representations of
the radiation spectrum.
Integrating in (22) with respect to frequency, we
obtain the total emitted power
( )21
3
2 q
c
eP +Ω= 22
| |
4
2
βγα . (24)
The same form has the expression for power losses of a
relativistic electron moving along the circular trajectory
of effective radius 21/ qR + .
Equation (22) at 1< <q и 1> >q reduces to formu-
lae of the curvature radiation and synchrotron radiation
for spiral trajectory in straight magnetic field, respec-
tively. In these cases, the first integral in (22) is smaller
than the second one. The most essential difference from
the case of synchrotron radiation in straight magnetic
field arises, if 1~q , and then the second integral in
(22) is larger than the first.
Fig.3. Universal functions of synchrotron radia-
tion: (solid line) is the spectrum (23); (dotted line) the
spectrum in straight magnetic field lines; (dot-and-dash
line) the curvature radiation spectrum; (dashed line)
synchrotron radiation for an electron having the circu-
lar trajectory with effective radius 21/ qR +
Let us compare exact expressions for spectrum (22)
and total energy losses (24) with approximate expres-
sions (usually used at interpretation of experimental
data), in which formula (23) is used. In Fig. 3 we com-
pare different radiation mechanisms when 1~q . a) The
curvature of magnetic force lines is not taken into ac-
count, and the formula for synchrotron radiation of an
electron moving with the pitch angle crBBP /sin ωψ = in
straight magnetic field is used. In this case
][= 3
PBcy ψωγω sin||)2/3(/ and the spectrum is de-
scribed by the first integral with the lower limit of inte-
gration qyc / in (22), (dotted line in Fig.3). The total
emitted power is proportional to 2q . b) We neglect
pitch angles and consider the curvature radiation for an
electron moving along the circular magnetic line with
curvature radius R. This spectrum is described by the
first integral, which has the lower limit of integration
cy , in (22). The spectrum of curvature radiation is plot-
ted by the dot-and-dash curve. The total power losses is
described by the first term in (24). c) As it was already
mentioned above, the total power loss for particles in
curved magnetic field (24) coincides with the power
loss of a relativistic electron having circular trajectory
of radius 21/ qR + (dashed line in Fig.2).
Considering the graphs such as represented in Fig.3
at various values of parameter q, we find that the differ-
ences between spectrum (22) and the spectrum of syn-
chrotron radiation in straight magnetic field are essential
if q belongs to the interval, 52,0 << q . Thus when
1~q , it is necessary to use formula (22). The derived
formulae, strictly speaking, are obtained when condition
[8] 2/1
B ))/(|(||1| Ω>− γωq 2/1)/( γγ | |= is taken place.
These conditions can be fulfilled both in astrophysical,
and in laboratory plasma.
APPENDIX
Averaging (21) over time |2 BT ωπ |/= and by tak-
ing expansion
∑
∞
∞−
= )(intsin zJee n
tiz ,
where )(zJ n are Bessel functions, we obtain
∫ ==
π
ωπ
ω
ω
2
0
)(
2
||
d
tdPtd
d
dP B
×
+Ω+−= ∫
∞ 222
2
0
2
2
)1(
2
1 qde τγ
τ
τ
π γ
ωα
v
( ) −
+Ω+Ω×
222
2
32
2
0 1
12
1
2
sin)
12
( qqJ τγ
γ
ω ττω
∫
∞
222 ×′Ω−
0
3
0 )( yqxJqd τγ
τ
τ
.)1(
12
1cos 2
2
−
+Ω+
2
×
222
2
πτγ
γ
ω τ q
Replacing the variable of integration / 2Ω= τγx and
introducing )2/3( Ω)/ (= 3γωy , we obtain expression
(14) from the paper [6].
REFERENCES
1. V.V. Vladimirskij. On influence Earth’s magnetic
field at the Auger showers // JETPh. 1948. V.18, #
4, p.392 – 401.
2. D.D. Ivanenko, A.A. Sokolov. On the theory of
‘lighting’ electron // DAN SSSR. 1948. V..59, # 9,
p.1551 – 1554.
3. J. Schwinger. On the classical radiation of
accelerated electrons // Phys.Rev. 1949. V.75, # 4,
p.1912 – 1925.
4. I.M. Ternov, V.V. Mikhaylin. Synchrotron radia-
tion. Theory and experiment. M.: “Energoatomiz-
dat”, 1986, 296 p.
5. K.S. Cheng, J.L. Zhang. General radiation formulae
for a relativistic charged particle moving in curved
magnetic field lined: The synchrocurvature radia-
tion mechanism // Astrophys. J. 1996. V.463, #1
p.271–283.
6. I.M. Pankratov. Towards analyses of runaway
electrons synchrotron radiation spectra // Fizika
Plazmi. 1999. V. 25, # 2, p. 165 – 168.
7. Ya.M. Sobolev. Drift trajectory and synchrotron
radiation of an ultrarelativistic electron moving in
magnetic field with curved force lines // Voprosi
Atomnoj Nauki i Tekhniki. Ser. “Plasma electronics
and new acceleration methods”. 2000. # 1, p. 27 –
30.
8. Ya.M. Sobolev. Influence of magnetic line
curvature on spectrum and polarization of
synchrotron radiation of a charged particle // Radio
Physics and Radio Astronomy. 2001. V. 6, # 4, p.
277 – 290.
9. V. Epp, T.G. Mitrofanova. Radiation of relativistic
particles in a quasi-homogeneous magnetic field //
Proceedings of the ninth Lomonosov conference on
elementary physics, Particle physics at the star of
the new millennium, 20-26 September 1999,
Moscow: World Scientific, 2001.
10. Ya.M. Sobolev. New radiation formulae of rela-
tivistic electrons in curved magnetic field lines. //
Abstracts of IAU Symposium 1999, The Universe at
Low Radio Frequencies, Nov.30 - Dec.4, 1999,
Pune, India, p.136; Ya.M. Sobolev. Towards radia-
tion theory of a relativistic charged particle in
curved magnetic field // Radio Physics and Radio
Astronomy. 2000. V.5, #2, p.137 - 147.
11. L.D. Landau, E.M. Lifshitz. Teoriya polya. M.:
“Nauka”, 1973. 504 p.
1. 12. N.L. Laskin, A.S. Mazmanishvili
N.N. Nasonov, V.F. Shulga. Contribution to theory
of radiation emission by relativistic particles in
amorphous and crystalline media // JETPh. 1985.
V. 89, # 3, p.763–780.
References
|