Динамика релятивистских частиц в поле сильно намагниченного вращающегося шара
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UDC 530. 1- 539. 1
DYNAMICS OF RELATIVISTIC PARTICLES IN THE FIELD OF HIGHLY MAGNETIZED ROTATING
M. A. Masterova
Department of Theoretical Physics, Tomsk State Pedagogical University, ul. Kievskaya, 60, 634 061 Tomsk, Russia. Tomsk State University of Control Systems and Radioelectronics, pr. Lenina, 40, 634 050 Tomsk, Russia.
E-mail: Masterova@tspu. edu. ru
The dynamics of a charged relativistic particle in electromagnetic field of a rotating magnetized celestial body with the magnetic axis inclined to the axis of rotation is studied. The covariant Lagrangian function in the rotating reference frame is found. Effective potential energy is defined on the base of the first integral of motion. The structure of the equipotential surfaces for a relativistic charged particle moving in strong magnetic field is studied and depicted. Behavior of the stationary points of the effective potential energy near the light cylinder is discussed.
Keywords: St0rmer'-s problem, oblique rotator, potential energy, trapping zones.
Motion of the charged particles in the field of a magnetized rotating celestial body is of large practical significance for astrophysics. Magnetic field of such objects in good approximation can be described as the field of an inclined rotating magnetized sphere or & quot-oblique rotator& quot- .
Theoretical study of the field of an oblique rotator has a long history. Deutsch  described a non-relativistic rotating magnetized star as a perfectly conducting sphere in rigid rotation in vacuo. In order to introduce a relativistic model of the field source Be-linsky et al. [3,4] considered an infinitely thin permanent magnet of finite length. This model is acceptable for calculation of the field at large distances from the source, but it can not be used for the near field calculations. In paper  has been found an exact special relativistic solution for the electromagnetic field in the interior and exterior of rapidly rotating perfectly conducting magnetized sphere. The calculation of the field is made as generalization of the field of slowly rotating magnetized neutron star, which was studied in  under consideration of general relativistic effects. The field of a rotating magnetized sphere which is neither a conductor nor a dielectric was calculated by Kabu-raki . There is a great variety of other papers which present calculations of the electromagnetic field of rotating magnetized sphere. The results differs essentially dependent on the used model of the magnetized sphere and its speed of rotation.
Allowed and forbidden regions of the motion of charged particles in such field was studied by Katsiaris and Psillakis . Dynamics of a charged particle near the force-free surface of a rotating magnetized sphere was explored in [9,10]. Some issues of charged particle dynamics within the electromagetic vacuum fields of
an inclined rotator have already been discussed in the papers [11,12].
Effective potential energy for a non-relativistic particle in the field of inclined rotating dipole was investigated in details in recent paper . The calculations were made for the near region, i. e. for distances much less then the radius of the light cylinder. In paper  we studied the structure of the effective potential energy for a relativistic particle in the field of inclined rotating magnetized sphere at the distances up to the light cylinder. In the present paper we study the structure of the effective potential energy for a relativistic particle moving in strong magnetic field of rotating celestial body. We investigate behaviour of the stationary points of the effective potential energy near the light cylinder.
2 The electromagnetic field of a rotating magnetized sphere
In this section we analyse the field of rotating magnetized sphere.
Let us consider the expression for the exterior electromagnetic field obtained by . We expand these
equations in powers of a — -, where w is the anc
gular speed of rotation, r0 is the radius of the sphere, and c is the speed of light. But we keep terms like r0/r which are sufficient near the surface of the sphere. Up to the first order of a we receive the next equations for the electric (E) and magnetic (H) field vectors in a spherical coordinate system r, 0, & lt-p (axis Z is directed along the vector of angular velocity w):
Er — -
^ 4 [cos a (3cos20 + 1) + sin a sin 20(3C — p2 cos A)],
M. A. Masterova. Dynamics of relativistic particles in the Geld of highly magnetized rotating sphere
ufc3 T (a2
Eg =--TT C sin a (1--k cos 20
P2 L V P2
±-t: cos a sin 20
r Pk3 e • / a2
E" =-S sin a cos 0 1--^
(cos a cos 0 + C sin a sin 0),
(3C sin 20 sin a + cos a (3cos20 + 1)),
A2 = - 4 S sin a,
r3 sin 0
(cos a sin 0 — C sin a cos 0).
conserved. The Lagrangian for a charged particle with mass m and charge e in rotating reference frame is
uv = (ct, r, 0, -?A),
T m v'-, e
L = - u uv'- ± uv'- A 2 c
where uv is the four-dimensional velocity, prime shows that the quantity relates to the rotating reference frame, and the dot denotes derivative with respect to the proper time t. As stated above, the time component po'- of the generalized 4-momentum is an integral of motion: dL
Eg =-r- [cos a sin 0 — sin a cos 0(C — p2 cos A) l, (2) P3 ufc3
=-5- sin a (S — p2 sin A), P3
S = sin A — p cos A, C = cos A + p sin A,
^ is the dipole moment vector, p = A = p + y — wt, p = rw/c, k = w/c, and a is the angle between the vectors ^ and w. We have also expanded:
sin (A — fl) «sin A — fl cos A, cos (A — fl) «cos A + fl sin A.
The magnetic field (2) is the field of rotating pointlike magnetic dipole, while the electric field (1) is a superposition of dipole and quadrupole fields. The quadrupole part is presented by terms proportional to a2/p2 and decreases with distance as p-4. At great distances p ^ a this part vanishes and the electromagnetic field becomes that of rotating magnetic dipole.
The fields (2) and (1) can be represented by 4-dimensional vector potential Av. In the spherical coordinate system xv = (ct, r, 0, y) it is
du0'- = mM0'- + cAo'-.
This means that the energy of the particle in the coro-tating frame defined as E'- = cp0'- is conserved.
The total energy E'- of a particle in curved space can be expressed as follows :
e'- = cpo'- =
mc7 ffo'-0'- v/1-?2
where fl = v/c, and v is the particle velocity. As mc2^g0'-0'- is the energy of the particle at rest, we can define the kinetic energy as
T = mc2^go'-0'-
Then, the potential energy U can be introduced as U = cp0'- - T, which gives
Vg0'-0'- + eAo
Let us find the potential energy of a particle in the rotating magnetized sphere. Transformation of the potential (3) into rotating reference frame leads to
pw f sin a sin 20
Ao'- = - ---(cos? + p sin ?)
cos a sin
pw 2 a2
rc 3 r2
where? = p + -. Substituting this into Eq. (8) and introducing a dimensionless potential energy V = U/mc2 we obtain
V = - p2 sin2 0 +(^ sin 20 (cos? + p sin ?)
3 Effective potential energy
Let us consider an arbitrary electromagnetic field
rotating with angular velocity w. The four-dimensional
potential of such field in the inertial spherical coordi-
nate system xv = (ct, r, 0, y), v = 0,1, 2, 3 is defined as
Av = Av (r, 0, y — wt + p).
In the corotating reference frame xv = (ct, r, 0, -),
with — = y — wt, the field does not depend on time.
Hence, the corresponding generalized momentum is
--11 sin2 0
1 ^ 2N 01
PV 3P P2.
N. = N sin a,
N = N cos a, N
Notice, that all physical parameters are gathered in one dimensionless parameter N. For example, for electrons the value of N for pulsar in Crab Nebula is 5 • 1010, Jupiter — 0. 03, Earth — 3 • 10−7. In the present paper we consider equipotential surfaces for large N.
4 Equipotential surfaces
In this section we present the profiles of potential energy defined by Eq. (10) for large N. The shape of the profiles in case of large N does not depend on N. Indeed, if N ^ 1, we can neglect the first term in Eq. (10) and N becomes just a scale factor.
Due to the argument? — p + - the equipotential surfaces take a form of surfaces twisted around the Z-axis. If we & quot-twist back& quot- the whole picture, introducing coordinate n — - + p — with
sin a =
cos a =
we find out that the potential energy becomes symmetric with respect to the plane n — 0, n which contains vectors ^ and w:
/sin20 V/N = J ---sin a cos n
sin2 в P
^ ам _2
p V 3 p3
terms with a2 in Eq. (10). Equations -- = 0 give
Solution a) In the equatorial plane 6 = -, we have two stationary points:
1 +1 +
1 — 1 +
П = 0, n.
Solution b) For the off-equatorial area we obtain two equations for p and 0 of the stationary points:
P3 sin в
eN i cos в
+ Л , — Ni sin в = 0,
— P2 sin2 в + P2
P3 sin2e «i-r-
/ «: — 2eN± cos 2в^ 1+ p2
V1 — P2 sin2 в
+ 2Nn sin 2 В = 0,
At the plots below we show the sections of the equipotential energy surfaces by plane n = const. The equipotential surfaces are marked by numbers equal to the energy level C = const.
Sections of the potential surfaces for N = 100 are plotted in Fig. 1 for positive, and in Fig. 2 for negative charges. We have plotted the equipotential surfaces for a = 0.1. One can see in Fig. 1 that the energy levels form a potential valley in a shape of a torus around the centre of the field. There is a significant difference between the structure of equipotential surfaces for positive and negative charges, though they share a number of traits. As we change the sign of the charge in expression for potential energy, the & quot-potential hills& quot- become & quot-potential valley& quot- and vice versa. The trapping regions in this case have a form of two symmetric dumb-bell shaped figures, as shown in Fig. 2.
5 Stationary points of the relativistic potential energy
where e — 1 for n — 0 and e — -1 for n — n.
If we expand the equation (14) in powers of 1/N, we obtain:
p =l1 — N*).
The coordinate p in Eq. (17) asymptotically tends to unity as N|| -y tt.
If one considers the stationary points, which satisfy the second solution, one notes that the first term in Eqs (15), (16) for the strong magnetic fields should be of order N. Or /1 — p2 sin2 0 ~ N-1. Hence, psin 0 «1 — O (N-2). This means that in case of large N, the stationary points are laying close to the light cylinder. Figs. 3 and 4 show the sections of the equipo-tential surfaces in the vicinity of stationary points for the positive and negative charges for N — 100. One can see that the stationary points are actually the saddle points.
The power of potential formulation of the problem is the possibility to find the & quot-potential valleys& quot- where the charged particles can be trapped. And the slope of the & quot-valley"- shows the force exerted on the particle. Having this in mind, we find the stationary points of the potential energy, i.e. the points satisfying the dV
set of equations: -- - 0, where q^ - p, 0, -. Taking dqi
the derivatives, we can not neglect the first term in Eq. (10) because its derivatives with respect to p and 0 tend to infinity as p2 sin2 0 — 1. Besides, it is shown in  that in case of large N the stationary points lie near the light cylinder. Hence, we can neglect the
We have studied the structure of the effective potential energy for a relativistic particle in the field of inclined rotating magnetized sphere at the distances up to the light cylinder in case of large N. We have presented the relativistic Lagrange function for a charged particle in the field of inclined rotating magnetized sphere. Existence of the integral of motion gives the possibility to introduce effective potential energy which allows studying some general features of the particle motion without solving the equations of motion. We considered the case of a celestial body which has a
M. A. Masterova. Dynamics of relativistic particles in the field of highly magnetized rotating sphere
strong magnetic field. For this case the equipoten-tial surfaces have been demonstrated in pictures. Behaviour of the stationary points of the effective potential energy near the light cylinder has been discussed.
The work is supported by the RFBR grant for LRSS project No. 88. 2014.2.
Figure 1. Equipotential profiles for
q = 1, n = 00, a = 600
Figure 3. Equipotential profiles for
q = l, n = n, a = 600- l) — 82. 304- 2) — 82. 303- 3) — 82. 301- 4) — 82. 3008- 5) — 82. 3007
Figure 2. Equipotential profiles for
q = -1, n = 00, a = 60°
Figure 4. Equipotential profiles for
q = -1, n = 0, a = 600- 1) — 37. 1- 2) — 37. 3- 3) — 37. 6- 4) — 37. 4912
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Received 09. 11. 2014
M. A. Macmepoea
ДИНАМИКА РЕЛЯТИВИСТСКИХ ЧАСТИЦ В ПОЛЕ СИЛЬНО НАМАГНИЧЕННОГО
Изучается динамика заряженной частицы в электромагнитном поле вращающегося намагниченного небесного тела с магнитной осью, наклоненной к оси вращения. Найдена функция Лагранжа во вращающейся системе отсчета в кова-риантной форме. На основе первого интеграла движения найдена эффективная потенциальная энергия. Построена структура эквипотенциальной поверхности для релятивистской частицы, движущейся в сильном магнитном поле. Исследовано поведение стационарных точек эффективной потенциальной энергии вблизи светового цилиндра.
Ключевые слова: Проблема Штёрмера, наклонный ротатор, потенциальная энергия, области захвата.
Томский государственный педагогический университет.
Ул. Киевская, 60, 634 061 Томск, Россия. E-mail: Masterova@tspu. edu. ru