Propagation of Polarized Cosmic Maser Radiation in an Anisotropic Magnetized Plasma

The polarization plane of the cosmic maser radiation (CMR) can be rotated either in the space-time with a metric of the anisotropic Bianchi-I type or in a magnetized plasma. A unified treatment of these two phenomena is presented for cold anisotropic plasma. It is argued that the generalized express...

Повний опис

Збережено в:

Бібліографічні деталі
Дата:	2010
Автор:	Moskaliuk, S.S.
Формат:	Стаття
Мова:	English
Опубліковано:	Відділення фізики і астрономії НАН України 2010
Назва видання:	Український фізичний журнал
Теми:	Астрофізика і космологія
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/56208
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:	Propagation of Polarized Cosmic Maser Radiation in an Anisotropic Magnetized Plasma / S.S. Moskaliuk // Український фізичний журнал. — 2010. — Т. 55, № 5. — С. 636-643. — Бібліогр.: 31 назв. — англ.

Репозитарії

Digital Library of Periodicals of National Academy of Sciences of Ukraine

id	irk-123456789-56208
record_format	dspace
spelling	irk-123456789-562082014-02-14T03:12:04Z Propagation of Polarized Cosmic Maser Radiation in an Anisotropic Magnetized Plasma Moskaliuk, S.S. Астрофізика і космологія The polarization plane of the cosmic maser radiation (CMR) can be rotated either in the space-time with a metric of the anisotropic Bianchi-I type or in a magnetized plasma. A unified treatment of these two phenomena is presented for cold anisotropic plasma. It is argued that the generalized expressions derived in the present study may be relevant for direct searches of a possible rotation of the plane of polarization of the cosmic maser radiation. Площина поляризацiї випромiнювання космiчного мазера може обертатися як у просторi з анiзотропною метрикою типу Б’янки-I, так i в анiзотропнiй замагнiченiй плазмi. У випадку холодної плазми цi явища описуються в межах єдиного пiдходу. Показано, що отриманi в даному дослiдженнi узагальненi вирази можуть стати у нагодi при безпосереднiх вимiрюваннях величини обертання площини поляризацiї випромiнювання космiчного мазера. 2010 Article Propagation of Polarized Cosmic Maser Radiation in an Anisotropic Magnetized Plasma / S.S. Moskaliuk // Український фізичний журнал. — 2010. — Т. 55, № 5. — С. 636-643. — Бібліогр.: 31 назв. — англ. 2071-0194 PACS 95.30.Sf, 98.80.Jk, 98.80.Bp, 98.80.Cq http://dspace.nbuv.gov.ua/handle/123456789/56208 en Український фізичний журнал Відділення фізики і астрономії НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Астрофізика і космологія Астрофізика і космологія
spellingShingle	Астрофізика і космологія Астрофізика і космологія Moskaliuk, S.S. Propagation of Polarized Cosmic Maser Radiation in an Anisotropic Magnetized Plasma Український фізичний журнал
description	The polarization plane of the cosmic maser radiation (CMR) can be rotated either in the space-time with a metric of the anisotropic Bianchi-I type or in a magnetized plasma. A unified treatment of these two phenomena is presented for cold anisotropic plasma. It is argued that the generalized expressions derived in the present study may be relevant for direct searches of a possible rotation of the plane of polarization of the cosmic maser radiation.
format	Article
author	Moskaliuk, S.S.
author_facet	Moskaliuk, S.S.
author_sort	Moskaliuk, S.S.
title	Propagation of Polarized Cosmic Maser Radiation in an Anisotropic Magnetized Plasma
title_short	Propagation of Polarized Cosmic Maser Radiation in an Anisotropic Magnetized Plasma
title_full	Propagation of Polarized Cosmic Maser Radiation in an Anisotropic Magnetized Plasma
title_fullStr	Propagation of Polarized Cosmic Maser Radiation in an Anisotropic Magnetized Plasma
title_full_unstemmed	Propagation of Polarized Cosmic Maser Radiation in an Anisotropic Magnetized Plasma
title_sort	propagation of polarized cosmic maser radiation in an anisotropic magnetized plasma
publisher	Відділення фізики і астрономії НАН України
publishDate	2010
topic_facet	Астрофізика і космологія
url	http://dspace.nbuv.gov.ua/handle/123456789/56208
citation_txt	Propagation of Polarized Cosmic Maser Radiation in an Anisotropic Magnetized Plasma / S.S. Moskaliuk // Український фізичний журнал. — 2010. — Т. 55, № 5. — С. 636-643. — Бібліогр.: 31 назв. — англ.
series	Український фізичний журнал
work_keys_str_mv	AT moskaliukss propagationofpolarizedcosmicmaserradiationinananisotropicmagnetizedplasma
first_indexed	2025-07-05T07:26:20Z
last_indexed	2025-07-05T07:26:20Z
_version_	1836790978277539840
fulltext	S.S. MOSKALIUK PROPAGATION OF POLARIZED COSMIC MASER RADIATION IN AN ANISOTROPIC MAGNETIZED PLASMA S.S. MOSKALIUK Bogolyubov Institute for Theoretical Physics, Nat. Acad. of Sci. of Ukraine (14b, Metrolohichna Str., Kyiv 03143, Ukraine) PACS 95.30.Sf, 98.80.Jk, 98.80.Bp, 98.80.Cq c©2010 The polarization plane of the cosmic maser radiation (CMR) can be rotated either in the space-time with a metric of the anisotropic Bianchi-I type or in a magnetized plasma. A unified treatment of these two phenomena is presented for cold anisotropic plasma. It is argued that the generalized expressions derived in the present study may be relevant for direct searches of a possible rotation of the plane of polarization of the cosmic maser radiation. 1. Introduction The presence of magnetic fields within astrophysical masers is believed to be a key ingredient in determining the observed polarization characteristics of astrophysical masers. The observation of the linear and circular po- larizations of maser radiation potentially provides infor- mation about the astronomical environments, in which masers occur. On the other hand, the maser polariza- tion behaves itself differently according to whether the Zeeman shift νB is larger or smaller than the linewidth ΔνD. The case νB � ΔνD is well understood. Sponta- neous decays occur in pure Δm transitions, producing the radiation that is fully polarized and centered at dif- ferent Zeeman frequencies in different transitions. Under these circumstances, the amplification process preserves the polarization. Therefore, thermal and maser polar- izations are the same. The only difference between the two cases is the disparity between the π and σ maser intensities, reflecting their different growth rates [1]. In the opposite limit, νB � ΔνD, the Zeeman compo- nents overlap, and the amplification mixes the different polarizations. The pumping processes produce radiation in three different modes of polarization with respect to the magnetic field (Δm = 0,±1), but only two inde- pendent combinations propagate in any direction, be- cause the electric field must be perpendicular to the wave vector. The elimination of the longitudinal component implies specific phase relations among the three pump- generated electric fields; only waves launched with these phase relations produce superpositions that are purely transverse, so that they can be amplified by propaga- tion in the inverted medium. The phase relations are transformed to the following ones for the polarization of maser radiation propagating at an angle θ to the mag- netic axis: Q I = −1 + 2 3 sin2 θ , V I = 16xxB 3 cos θ , (1) where x = (ν − ν0)/ΔνD and xB = νB/ΔνD. This so- lution, which was first derived by Goldreich, Keeley & Kwan [2] in the limit xB = 0 and extended by Elitzur [1] to finite xB < 1, follows also from the requirement that the four Stokes parameters produce fractional po- larizations that remain unaffected by the amplification process. How the unpolarized radiation produced in sponta- neous decays evolves into this stationary polarization so- lution remains an open problem. In this paper, a unified discussion of the rotation of a polarization plane in the cases of an anisotropic model of the Bianchi-I type and the Faraday rotation is presented for the cold plasma following works [3, 4]. 2. Polarization Effects of CMR in a Space-Time with a Metric of the Bianchi-I Type Let us consider the Maxwell equations for a free electro- magnetic field. In the metric ds2 = dt2 − 3∑ i=1 A2 i (t) ( dxi )2 , (2) they can be written as ∇µFµν = 0, ∇µ (∗F )µν = 0, where Fµν is the electromagnetic-field tensor and (∗F )αβ = 1√ −g [αβγη]Fγη is the quantity dual to it; here, 636 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 PROPAGATION OF POLARIZED COSMIC MASER RADIATION [αβγη] is the fully antisymmetric tensor specified by the condition [0123] = 1. The solutions of these equations can be represented in the form of the electric- and magnetic-field vectors [5–9] E(t,x) = = ∫ d3k [ Eθ(t,k)eθ(t,k) + Eϕ(t,k)eϕ(t,k) ] exp(ikx), H(t,x) = = ∫ d3k [ Hθ(t,k)eθ(t,k) +Hϕ(t,k)eϕ(t,k) ] exp(ikx), where eθ = cos θt cosϕt e1 A1 + cos θt sinϕt e2 A2 − sin θt e3 A3 , eϕ = − sinϕt e1 A1 + cosϕt e2 A2 are orthogonal vectors. Together with the vector ek = sin θt cosϕt e1 A1 + sin θt sinϕt e2 A2 + cos θt e3 a3 they form the vierbein unit basis in the momentum space. The angles θt and ϕt can be expressed in terms of spherical coordinates defined in the momentum space through the relation (k1, k2, k3) = k (sin θ cosϕ, sin θ sinϕ, cos θ) as follows (sin θt cosϕt, sin θt sinϕt, cosϕ) = = µ−1 ( sin θ cosϕ A1 , sin θ sinϕ A2 , cosϕ A3 ) . The coefficient µ is determined by equating the squares of both sides of this relation. The components Eθ, Eϕ, Hθ, and Hϕ can be written in the form Eθ(t,k)= 1√ 2(2π)3/2(−g)1/4 µ b1/2 ( σ+ + σ− ) , Eϕ(t,k)= 1√ 2(2π)3/2(−g)1/4 µ b1/2k d dt ( σ− + σ− ) , Hθ(t,k)=− 1√ 2(2π)3/2(−g)1/4 µ b1/2 ( σ− + σ− ) , Hϕ(t,k)=− 1√ 2(2π)3/2(−g)1/4 µ b1/2k d dt ( σ+ + σ− ) , (3) where b = 1√ −g ( A2 2 cos2 ϕ+A2 1 sin2 ϕ ) , and the functions σ± = σr satisfy the equation σ̈r − ḃ b σ̇r + [ k2µ2 + rkΔ ] σr = 0, (4) Δ = b d dt (a b ) , a = cos θ sin 2ϕ 2 √ −g (A2 2 −A2 1). (5) Following [10, 11], we describe the polarization effects of electromagnetic radiation with the aid of the polar- ization matrix defined in the plane orthogonal to the direction of propagation of the electromagnetic waves. This density matrix can be represented in the form Jab = Jab+ + Jab− , a, b = θ, ϕ, Jab = 1 2 E{a(t,k), Et∗b}(t,k), Jab− = 1 2 E [a(t,k), Et∗b](t,k), (6) where the symbol {, } ([, ]) denotes the symmetrization (antisymmetrization) with respect to the corresponding superscripts. The substitution of the electric-field components (2) into expression (5) yields Jθθ+ = ( 2(2π)3(−g)1/2b )−1 µ2\|Y +\|2, Jϕϕ+ = ( 2(2π)3(−g)1/2bk2 )−1 \|Ẏ −\|2, Jθϕ+ = −µ 2 ( 2(2π)3(−g)1/2bk )−1 2 Re Ẏ − ∗ Y +, Jθϕ− = i 2 µ 2(2π)3(−g)1/2bk 2 Im Ẏ − ∗ Y +, (7) where Y ± = σ+ ± σ−. The matrix Jab is expressed in terms of the Stokes parameters which describe the polarization properties of electromagnetic waves as follows: Jab+= 1 2 ( I +Q U U I −Q ) , Jab−= 1 2 ( 0 −iV iV 0 ) . (8) ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 637 S.S. MOSKALIUK Here, I is the total intensity of radiation, the parameters Q and U are related to the degree of linear polarization by the equation PL = √ U2 +Q2 I , and V determines the degree of circular polarization via the relation PC = V/I. Comparing relations (6) and (7), we find that the Stokes parameters are given by I = 1 2(2π)3(−g)1/2bk2 [ \|Ẏ −\|2 +K2 0 \|Y +\|2 ] , Q = − 1 2(2π)3(−g)1/2bk2 [ \|Ẏ −\|2 −K2 0 \|Y +\|2 ] , U = − µ (2π)3(−g)1/2bk Re Ẏ − ∗ Y +, V = − µ (2π)3(−g)1/2bk Im Ẏ − ∗ Y +, K0 = kµ. (9) To pursue the investigation of polarization effects fur- ther, we assume that the propagation of waves in an anisotropic space-time can be described in the short- wave approximation; that is, we retain the first terms of the asymptotic expansion of the solutions of Eq. (3) for σr in the limit k → ∞(λ → 0). Such an expan- sion was constructed by Sagnotti and Zwiebach [12–15]. From their results, it follows that, in the leading approx- imation in k−1 for k →∞, the required solutions can be represented as Y + 0 = ( bµ0 b0µ )1/2 ( C+ 0 e iλ + C−0 e −iλ) eiΩ, Y −0 = ( bµ0 b0µ )1/2 ( C+ 0 e iλ − C−0 e−iλ ) eiΩ, Ẏ + = iK0Y +, Ẏ − = iK0Y −, (10) where λ = t∫ t0 Δ 2µ dt′, Ω = t∫ t0 kµdt′, t0 corresponds to an arbitrary initial instant of propa- gation, and C±0 are the values of the functions σ±(t) at the point t0. The substitution of the WKB solutions (9) into ex- pressions (8) for the Stokes parameters yields I = µµ0√ −g(2π)3b0 [ \|C+ 0 \|2 + \|C−0 \|2 ] , Q = µµ0 (2π)3 √ −gb0 2Re ( C+ 0 ∗ C−0 e 2iλ ) , U = µµ0 (2π)3 √ −gb0 2Im ( C+ 0 C − 0 e 2iλ ) , V = − µµ0√ −gb0 [ \|C+ 0 \|2 − \|C − 0 \|2 ] . Eliminating the constants from these relations, we can find the Stokes parameters as functions of time. These are given by I(t) = µ(t) µ(t0) √ −g(t0) −g(t) I(t0), Q(t) = = µ(t) µ(t0) √ −g(t0) −g(t) [Q(t0) cos 2λ(t)− U(t0) sin 2λ(t)] , U(t) = = µ(t) µ(t0) √ −g(t0) −g(t) [Q(t0) sin 2λ(t) + U(t0) cos 2λ(t)] , V (t) = µ(t) µ(t0) √ −g(t0) −g(t) V (t1). It immediately follows that PL(t) = PL(t0), PC(t) = PC(t0); that is, the degree of linear polarization and the degree of circular polarization do not vary with time as the electro- magnetic wave propagates in an anisotropic space-time (this result is in the perfect agreement with analogous conclusions drawn in [16–19] by different methods). 638 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 PROPAGATION OF POLARIZED COSMIC MASER RADIATION Under such conditions, the rotation of the polarization plane is the only nontrivial polarization effect. The angle τ of this rotation in the plane orthogonal to the direction of wave propagation is determined by the relation tan 2τ = U Q . It follows that tan 2τ(t) = sin 2λ+ tan 2τ(t0) cos 2λ cos 2λ− tan 2τ(t0) sin 2λ . Differentiating this last relation, we find that the variable y ≡ tan 2τ satisfies the equation d dt y = 2λ(1 + y2). Solving this equation, we obtain τ(t) = λ(t) + τ0 or Δτ = t∫ t0 Δ 2µ dt′. Taking expression (4) for Δ into account and using the linear approximation in the anisotropy parameter ΔĀ = A2 − A1 [the latter is reduced to setting b = µ = 1/A, where A = (A1A2A3)1/3], we obtain Δτ(t) = 1 2 cos θ sin 2ϕ (A(t)ΔA(t)−A(t0)ΔA(t0)) . We can see that the polarization plane undergoes rota- tion only if the wave propagates in a direction other than that specified by the coordinate values θ = π 2 (2k+1) and ϕ = π 2 k (k = 0, 1, 2, . . .) and if ΔA 6= 0, that is, if the anisotropic model under study is not axially symmetric [9]. 3. Cosmic Maser Radiation Polarization Effects in a Magnetized Plasma with Bianchi-I type Anisotropy If linearly polarized CMR passes through the cold plasma containing a magnetic field, the polarization plane of a wave can be rotated since the two circular polarizations (forming the linearly polarized beam) are travelling at different speeds. This effect has been stud- ied in a variety of different frameworks even in relativistic QED plasmas (see, for instance, [3, 20]). The CMR has a degree of linear polarization. If CMR is linearly polarized, then its polarization plane also can be rotated provided a sufficiently strong magnetic field is present [21]. In this section, a unified discussion of rotation of the polarization plane in the case of an anisotropic model of the Bianchi-I type and the Faraday rotation will be presented for the cold plasma following works [3, 4]. A related aspect of the present analysis will be to study the range of validity of the Faraday rotation estimates. Let us start by discussing the typical scales involved in the problem. The plasma is globally neutral, and the ion density equals the electron density, i.e. ni ' ne = n0, where n0 stands for the common electron–ion number density in the corresponding area of the Universe. The global neutrality of the plasma occurs for typical length scales L� λD, where λD = √ kBTei 8πe2n0 , (11) is the Debye screening length, and kB is Boltzmann con- stant. If the plasma is not magnetized, the only relevant fre- quency scales of the problem are the plasma frequencies which can be constructed from the electron and ion den- sities, i.e. ωpe = √ 4πe2n0 me , ωpi = √ 4πe2n0 mi . (12) The frequencies given in Eq. (12) enter the dispersion relations determining the group velocity of an electro- magnetic signal in the plasma. The plasma frequencies for both electrons and ions are much larger than the collision frequencies constructed from the inverse of the mean free paths. Then the plasma can be described, to a very good approximation, within a two-fluid framework [26, 27]. If the plasma is magnetized, two new frequency scales arise in the problem, namely the electron and ion gy- rofrequencies, i.e. ωBe = eB0 mec , ωBi = eB0 mic , (13) where B0 is the magnetic field strength in the corre- sponding area of the Universe, and c is the speed of light in vacuo. The electron and ion gyrofrequencies, together with the plasma frequencies of Eq. (12), affect the dispersion relations in the case of a magnetized plasma. Assuming that, at the time moment tin on the back- ground of the initially homogeneous and isotropic gravi- tational field in the Universe with the Friedman metric, ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 639 S.S. MOSKALIUK there arises the homogeneous anisotropic perturbation so that, as a consequence, the metric can be represented as (2). Let us assume also that, at t < tin, the state of the EM field can be described with the density matrix with the non-zero occupation number of photons in the mode n0(ν0) corresponding to the black-body radiation. The latter is strictly constant at t < tin and constant in the zeroth approximation in the anisotropy parameters at t > tin: ∂ ∂t n0(ν0) = 0. The frequency ν0 is considered to be independent of time and equal to the radiation frequency in the cur- rent epoch. With the frequency at any time moment t, it is related as follows: ν0A(t0) = ν(t)A(t), (14) where A(t) is the scale factor in the Friedman model at t < tin, and A3 = (A1A2A3) at t > tin. Defining the appropriately rescaled electron and ion densities, ne = A3ñe and ni = A3ñi in a conformally flat geometry of the Bianchi-I type background characterized by a scale factor A(η) and by the line element (2), the continuity equations for the charge densities read n′e + 3weHne + (we + 1) div (neve) = 0, (15) n′i + 3wiHni + (wi + 1) div (nivi) = 0, (16) where H = A′/A; the prime denotes a derivation with respect to the conformal time coordinate η; w is the barotropic index for the electron or ion fluid. In the cold plasma, both electrons and ions are non- relativistic. Hence, the barotropic index w will be close to zero to a good approximation. For instance, the en- ergy density of an ideal electronic gas is given by ρe = ne ( mec 2 + 3 2 kBTe ) , (17) and since we,i = kBTe,i/me,ic 2, we,i � 1 as far as kBTe,i � m,iec 2. In the cold-plasma approximation, the temperature of ions and electrons vanishes. In the warm-plasma ap- proximation, the temperature of the two charged species may be very small but non-vanishing. The warm-plasma treatment will lead, in practice, only to an effective cor- rection of the plasma frequency. Since the cold-plasma results turn out to be, a posteriori, rather accurate, the discussion will be presented in terms of the cold-plasma description. To have a self-consistent set of two-fluid equations, Eqs. (15) and (16) will be supplemented by the evolution equations of the velocity fields and of the electromagnetic field, namely, ρe[v′e + Hve + (vae∇a)ve] = −nee ( E + ve c ×B ) , (18) ρi[v′i + Hvi + (vbi∇b)vi] = nie ( E + vi c ×B ) , (19) where E and B are the conformally rescaled electro- magnetic fields obeying the following set of generalized Maxwell equations [4]: div E = 4πe(ni − ne), (20) div B = 0, curlE = −1 c B′, (21) curlB = 1 c E′ + 4πe c (nivi − neve), (22) Eqs. (20)–(22) are the usual two-fluid equations [22]. Equations (15) and (16) together with Eqs. (18), (19), and (20)–(22) can then be linearized in the presence of the weak background magnetic field B0, i.e. ne, i(η,x) = n0 + δne, i(η,x), B(η,x) = B0 + δB(η,x), ve, i(η,x) = δve, i(η,x), E(η,x) = δE(η,x). (23) Using Eq. (23), the system of equations (15)–(19) and (20)–(22) can be written as δn′e + n0 div (δve) = 0, δn′i + n0 div (δvi) = 0, (24) δv′e + Hδve = − e me [ δE + δve c ×B0 ] , δv′i + Hδvi = e mi [ δE + δvi c ×B0 ] , (25) curl (δE) = −1 c δB′, div (δE) = 4πe(δni − δne), (26) curl (δB) = 1 c δE′ + 4π en0 e (δvi − δve). (27) 640 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 PROPAGATION OF POLARIZED COSMIC MASER RADIATION From Eqs. (24)–(26), the relevant dispersion relations and the associated refraction indices can be obtained by treating separately the motions in parallel and perpen- dicularly to the magnetic field direction. Defining the current direction parallel to the magnetic field as j‖ = n0e(δvi, ‖ − δve, ‖), Eqs. (25) yield j′‖ + Hj‖ = 1 4π (ω2 p, i + ω2 e, i) δE‖. (28) Since a variation of the geometry is slow with respect to the typical frequencies of plasma oscillations, the follow- ing adiabatic expansions can be used: j‖(η,x) = j‖,ω(x)e−i ∫ η dη′ω(η′), δE‖(η,x) = δE‖,ω(x)e−i ∫ η dη′ω(η′). (29) Thus, defining α = iH/ω � 1, Eq. (28) implies that j‖,ω = i 4π ω2 p, i + ω2 p, e ω(1 + α) δE‖,ω. (30) Inserting Eq. (30) into the parallel component of Eq. (27), the following equation can be obtained: curl (δBω)‖) = −iω c ε‖(ω, α)δE‖,ω , (31) where the parallel dielectric constant is ε‖(ω, α) = 1− ω2 p, i ω2(1 + α) − ω2 p, e ω2(1 + α) . (32) With a similar procedure, the equation of motion in the plane orthogonal to the magnetic field direction can be solved as well, and the evolution equations of the electric and magnetic fluctuations can then be written, in compact notation, as curl (δEω) = i ω c δB, (33) curl (δBω) = −i ω c ε(α, ω)δEω , (34) where δEω and δBω have to be understood as column matrices containing, in each row, the components of the electric and magnetic fields in each of the three spatial directions, while ε(ω, α) is a 3× 3 matrix given by ε(ω, α) =  ε1(ω, α) iε2(ω, α) 0 −iε2(ω, α) ε1(ω, α) 0 0 0 ε‖(ω, α)  , (35) where ε‖(ω, α) is defined by Eq. (32); ε1,2(ω, α) are in- stead ε1(ω, α) = 1− ω2 p i(α+ 1) ω2(α+ 1)2 − ω2 B i − ω2 p e(α+ 1) ω2(α+ 1)2 − ω2 B e , (36) ε2(ω, α) = ωB e ω ω2 p e ω2(α+ 1)2 − ω2 B e − ωB i ω ω2 p i ω2(α+ 1)2 − ω2 B i . (37) The coordinate system can be fixed by setting kx = 0 and ky = k sin θ, kz = k cos θ with B0 oriented along the ẑ direction. Since Eqs. (33) and (34) yield curl curl (δBω) = ω2 c2 ε(ω, α)δBω , (38) the Fourier transformation of Eq. (38) in the coordi- nate system selected previously leads to the generalized Appleton–Hartree equation [4]: A δBk,ω =  [ 1− ε1 n2 ] −i [ ε2 n2 ] 0 i [ ε2 n2 ] [ c2 − ε1 n2 ] −sc 0 −sc [ s2 − ε‖(ω,α) n2 ] × × δBk,ω,xδBk,ω,y δBk,ω,z  = 0, (39) where the refraction index n = c/v has been introduced so as to eliminate the comoving momentum in such a way that k = ω/v = nω/c; we have written c(θ) = cos θ and s(θ) = sin θ. From Eq. (39), we get A† = A, where the dagger denotes the transposition and complex conjugation of a given matrix. The non-trivial solutions of the system of algebraic (homogeneous) equations given by formula (39) come from setting the determinant of the coefficients equal to zero, i.e. det A = 0. It was found that the determinant vanishes if [4] s2(θ) {( 1 ε‖ − 1 n2 )[ 1 n2 − 1 2 ( 1 εL + 1 εR )] − −c2(θ) [( 1 n2 − 1 εL )( 1 n2 − 1 εR )] = 0, (40) ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 641 S.S. MOSKALIUK where the right-handed and left-handed dielectric con- stants have been defined as εR = ε1 + ε2 = = 1− ω2 p i ω[ω(α+ 1)− ωB i] − ω2 p e ω[ω(α+ 1) + ωB e] , (41) εL = ε1 − ε2 = = 1− ω2 p e ω[ω(α+ 1)− ωB e] − ω2 p i ω[ω(α+ 1) + ωB i] . (42) Equation (40) reduces exactly to the Appleton– Hartree equation known from the two-fluid plasma the- ory [26] with a minor difference that the leading depen- dence upon the background geometry appears in εR,L through the function α. The dispersion relations for a wave propagating in parallel and perpendicularly to the magnetic field direction can be obtained by setting, re- spectively, θ = 0 and θ = π/2 in Eq. (40). Consequently, the relevant equations determining the refraction index are, in this case, (n2 − εR)(n2 − εL) = 0, θ = 0, (43) (n2 − ε‖)[n2(εL + εR)− 2εLεR] = 0, θ = π 2 . (44) Equation (43) gives the usual dispersion relations for the two circular polarizations of the electromagnetic wave, i.e. n2 = εR and n2 = εL, while Eq. (44) gives those for the “ordinary” (i.e. n2 = ε‖) and “extraordinary” (i.e. n2 = 2εRεL/(εR + εL)) plasma waves [26, 27]. From Eq. (43), the generalized Faraday rotation ex- perienced by the linearly polarized CMR travelling in parallel to the magnetic field can be obtained as ΔΦ = ω 2c [ √ εR − √ εL ] ΔL, (45) where ΔL is the distance travelled by the signal in the direction parallel to the magnetic field. According to Eqs. (12) and (13), the leading contribu- tion to the generalized Faraday rotation arises as follows:( ωBe ωCMR )( ωpe ωCMR )2 . (46) In a complementary perspective, when analyzing the possible rotation of the CMR polarization, it seems then preferable to adopt the generalized formulae derived in the present study. In summary, the contributions to the observed CMR polarization from the magnetized plasma and the anisotropic space-time with a metric of the Bianchi-I type are believed to be as follows: • the degree of linear polarization and the degree of circular polarization do not vary with time as the electromagnetic wave propagates in an anisotropic space-time of Bianchi-I type; • the polarized CMR propagates in a Bianchi-I space-time undergoes a rotation of the polarization plane without any change in the degree of polar- ization; • the magnetized plasma contribution has a fre- quency dependence that may allow us to disentan- gle their relative weight, since the magnetic con- tribution vanishes for ω > ωCMR as 1/ω2. The author is grateful to Georgij Rudnitskij for help- ful discussions and useful recommendations leading to a better structure of the present article. The author is also especially grateful to the Austrian Academy of Sciences and the Russian Foundation for Fundamental Research which in the framework of the collaboration with the Na- tional Academy of Sciences of Ukraine co-financed this research. 1. M. Elitzur, ApJ 457, 415 (1996). 2. P. Goldreich, D.A. Keeley, and J.Y. Kwan, ApJ 179, 111 (1973). 3. S.S. Moskaliuk, Phys. Rev. D 68, 084023 (2003). 4. M. Giovannini, Phys. Rev. D 71, 021301 (2005). 5. A.A. Grib and A.V. Nesteruk, Yad. Fiz. 38, 1357 (1983). 6. S.S. Moskaliuk, A.V. Nesteruk, and S.A. Pritomanov, Preprints of Inst. for Theoretical Physics, National Acad. Sci. of Ukraine, Kiev, No. 90-40R, 90-79R (1990). 7. G.I. Kuznetsov, S.S. Moskalyuk, Yu.F. Smirnov, and V.P. Shelest, Graphical Theory of Representations of Orthog- onal and Unitary Groups and their Physical Applications (Naukova Dumka, Kiev, 1992) (in Russian). 8. G.S. Bisnovatyi-Kogan, Stellar Physics (Springer, Berlin, 2001). 9. S.S. Moskaliuk and A.V. Nesteruk, Phys. At. Nucl. 58, 1744 (1995). 10. Ya.B. Zel’dovich and I.D. Novikov, Structure and Evolu- tion of the Universe (Nauka, Moscow, 1975) (in Russian). 642 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 PROPAGATION OF POLARIZED COSMIC MASER RADIATION 11. A.Z. Dolginov, Yu.N. Gnedin and N.A. Silant’ev, Prop- agation and Polarization of Radiation in Cosmic Media (Nauka, Moscow, 1979) (in Russian). 12. B.W. Tolman and R.A. Matzner, Proc. R. Soc., London 392, 391 (1984). 13. C.H. Brans, Astrophys. J. 197, 1 (1975). 14. N. Cademi, R. Fabbri, B. Melchiorri et al., Phys. Rev. D: Part. Fields 17, 1901 (1978). 15. A. Sagnotti and B. Zwiebach, Phys. Rev. D: Part. Fields 24, 305 (1981). 16. A.M. Cruise, Mon. Not. R. Astron. Soc. 204, 485 (1983). 17. U. Rees, Astrophys. J. 153, LI (1968). 18. J. Negroponte and J. Silk, Phys. Rev. Lett. 44, 1433 (1980). 19. M.M. Basko and A.G. Polnarev, Astron. Zh. 57, 465 (1980). 20. A.K. Ganguly, S. Konar and P.B. Pal, Phys. Rev. D 60, 105014 (1999); J. C. D’Olivo, J.F. Nieves and S. Sahu, Phys. Rev. D 67, 025018 (2003); A.K. Ganguly and R. Parthasarathy, Phys. Rev. D 68, 106005 (2003). 21. A. Kosowsky and A. Loeb, Astrophys. J. 469, 1 (1996); M. Giovannini, Phys. Rev. D 56, 3198 (1997). 22. M. Giovannini, Int. J. Mod. Phys. D 13, 391 (2004); G. Piccinelli and A. Ayala, Lect. Notes in Phys. 646, 293 (2004). 23. J.A. Frieman, C.T. Hill, A. Stebbins and I. Waga, Phys. Rev. Lett. 75, 2077 (1995). 24. S.M. Carroll, G.B. Field and R. Jackiw, Phys. Rev. D 41, 1231 (1990); S.M. Carroll, Phys. Rev. Lett. 81, 3067 (1998). 25. K.R.S. Balaji, R.H. Brandenberger and D.A. Easson, JCAP 0312 008 (2003). 26. T.J.M. Boyd and J.J. Sanderson, The Physics of Plasmas (Cambridge Univ. Press, Cambridge, 2003). 27. H.T. Stix, The Theory of Plasma Waves (McGraw-Hill, New York, 1962). 28. M. Giovannini, Phys. Rev. D 61, 063004 (2000); Phys. Rev. D 61, 063502 (2000); M. Giovannini and M.E. Sha- poshnikov, Phys. Rev. Lett. 80, 22 (1998). 29. B. Feng, H. Li, M. z. Li, and X. m. Zhang, arXiv:hep- ph/0406269. 30. V.A. Kostelecky, R. Lehnert, and M.J. Perry, Phys. Rev. D 68, 123511 (2003); O. Bertolami, R. Lehnert, R. Pot- ting, and A. Ribeiro, Phys. Rev. D 69, 083513 (2004). 31. J. Skilling, Phys. of Fluids 14, 2523 (1971). Received 26.02.09 ПОШИРЕННЯ ПОЛЯРИЗОВАНОГО ВИПРОМIНЮВАННЯ КОСМIЧНОГО МАЗЕРА В АНIЗОТРОПНIЙ ЗАМАГНIЧЕНIЙ ПЛАЗМI С.С. Москалюк Р е з ю м е Площина поляризацiї випромiнювання космiчного мазера мо- же обертатися як у просторi з анiзотропною метрикою типу Б’янки-I, так i в анiзотропнiй замагнiченiй плазмi. У випадку холодної плазми цi явища описуються в межах єдиного пiдхо- ду. Показано, що отриманi в даному дослiдженнi узагальненi вирази можуть стати у нагодi при безпосереднiх вимiрюваннях величини обертання площини поляризацiї випромiнювання ко- смiчного мазера. ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 643

Propagation of Polarized Cosmic Maser Radiation in an Anisotropic Magnetized Plasma

Репозитарії

Схожі ресурси