Theory of gyroresonance and free-free emissions from non-Maxwellian quasi-steady-state electron distributions
aa r X i v : . [ a s t r o - ph . S R ] D ec Draft version October 10, 2018
Preprint typeset using L A TEX style emulateapj v. 5/2/11
THEORY OF GYRORESONANCE AND FREE-FREE EMISSIONS FROM NON-MAXWELLIANQUASI-STEADY-STATE ELECTRON DISTRIBUTIONS
Gregory D. Fleishman , Alexey A. Kuznetsov Draft version October 10, 2018
ABSTRACTCurrently there is a concern about ability of the classical thermal (Maxwellian) distributionto describe quasi-steady-state plasma in solar atmosphere including active regions. In partic-ular, other distributions have been proposed to better fit observations, for example, kappa-and n -distributions. If present, these distributions will generate radio emissions with differentobservable properties compared with the classical gyroresonance (GR) or free-free emission,which implies a way of remote detecting these non-Maxwellian distributions in the radio ob-servations. Here we present analytically derived GR and free-free emissivities and absorptioncoefficients for the kappa- and n -distributions and discuss their properties, which are in factremarkably different from each other and from the classical Maxwellian plasma. In particu-lar, the radio brightness temperature from a gyrolayer increases with the optical depth τ forkappa-distribution, but decreases with τ for n -distribution. This property has a remarkable con-sequence allowing a straightforward observational test: the gyroresonance radio emission fromthe non-Maxwellian distributions is supposed to be noticeably polarized even in the opticallythick case, where the emission would have strictly zero polarization in the case of Maxwellianplasma. This offers a way of remote probing the plasma distribution in astrophysical sourcesincluding solar active regions as a vivid example. Subject headings:
Sun: corona—Sun: magnetic fields—Sun: radio radiation INTRODUCTIONIn recent years a critical mass of observationally-driven concerns about the applicability of the classicalMaxwellian distribution to solar coronal plasma has accumulated. Perhaps, the most direct indication of theMaxwellian distributions insufficiency is routine detection of the kappa-distributions in the solar wind plasmaeven during the most quiet periods (i.e., in quasi-stationary conditions), which may imply the presence of suchkappa-distributions at the corona, where the solar wind is launched (Maksimovic et al. 1997). These kappa-distributions, characterized by a temperature T and index κ , have different indices in the slow and fast solarwind flows, originating in the normal corona and coronal holes respectively, implying that the “equilibrium”distributions in the normal corona and coronal holes can have accordingly different indices. In addition, a numberof coronal observations seem to require a kappa-like or n -distribution, e.g., some EUV line enhancements duringsolar flares (Dufton et al. 1984; Anderson et al. 1996; Pinfield et al. 1999; Dzifˇc´akov´a & Kulinov´a 2011). In fact,the temperature diagnostics based on the UV and X-ray lines depend on the temperature, the density (emissionmeasure), and the distribution type. Specifically, for the kappa-distribution (compared to the Maxwellian one)the filter responses to emission are broader functions of T , and their maxima are flatter, which may result ina systematic error in coronal temperature diagnostics (Dud´ık et al. 2009, 2012). Coronal hard X-ray (HXR)emission spectra are often well fit by a kappa-distribution of the flaring plasma (Kaˇsparov´a & Karlick´y 2009;Oka et al. 2013).Finally, the radio data on the GR emission from various active regions (e.g., Uralov et al. 2006, 2008; Nita et al.2011c) seem to favor harmonic numbers larger than the expected value of three (Lee 2007), which could implya deviation of the electron equilibrium distribution from a Maxwellian. Although the Maxwellian distributionis often assumed to be a true equilibrium distribution of the plasma particles, this is not necessarily the case.In terms of thermodynamics, the equilibrium distribution can easily be derived in the form of a Maxwellian ifone adopts the system to be extensive (e.g., the entropy of the system is the sum of entropies of its macroscopicparts). Microscopically, at the kinetic level, this same distribution is derived from the kinetic equation underthe assumption that the equilibrium is achieved via close binary (e.g., Coulomb) collisions. Stated another way,the Maxwellian distribution is a natural equilibrium state of a closed collisional system.In a non-extensive thermodynamical system (e.g., Tsallis 1988; Leubner 2002), however, the system entropyis no longer an additive measure and so is not equal to the partial entropy sum. Accordingly, the equilibriumdistributions are not unique, and under certain assumptions the kappa-distribution can represent the trueequilibrium solution. Microscopically, this non-extensivity means that far interactions (rather than close binarycollisions) play a dominant role in reaching the equilibrium distribution; a sustained heat or particle flux, if Center For Solar-Terrestrial Research, New Jersey Institute of Technology, Newark, NJ 07102 Ioffe Physico-Technical Institute, St. Petersburg 194021, Russia Central Astronomical Observatory at Pulkovo of RAS, Saint-Petersburg 196140, Russia Institute of Solar-Terrestrial Physics, Irkutsk 664033, Russia
Fig. 1.—
Examples of the electron distribution functions (Maxwellian, kappa- and n -distribution) used in this paper. All plotswere calculated for the plasma temperature of T = 10 K. a) Distribution functions in the momentum space (see Section 2); b) thesame distribution functions in the energy space. present, can further complicate the equilibrium established in such an open, non-extensive system. In particular,a moving plasma (e.g., in the presence of a mean DC electric field) can have a distribution similar to n -distribution (Karlick´y et al. 2012; Karlick´y 2012).The coronal plasma may be a good example of such open, potentially non-extensive plasma. Indeed, there isa sustained energy flux from lower layers of the solar atmosphere into the chromosphere and corona, and thecoronal plasma is indeed only weakly collisional, so that distant wave-particle interactions play a role that isoften much more important than the binary Coulomb collisions. The radio emission in general does depend (inaddition to the magnetic field) on the distant collisions and wave-particle interactions, and so offers a sensitiveprobe of such interactions. Therefore, remote sensing of the solar corona may be explicitly dependent on thisnew, fundamental physics of the collisionless plasma.Radio measurements, with their significant optical depth, are potentially most sensitive to the distribu-tion type, so it would be wise to use the radio measurements to address the fundamental question of theequilibrium/quasi-stationary distribution of the collisionless plasma of the solar corona. With the microwaveimaging spectroscopy available from Jansky Very Large Array (JVLA) and Expanded Owens Valley Solar Array(EOVSA), spatially resolved radio spectra of the requisite quality will be available for the first time to probethis question.In this paper we develop analytical theory of the GR and free-free emission from two, currently the most pop-ular, non-Maxwellian distributions—namely, the kappa- and n -distributions. Currently, the only available ele-ment of this theory is the free-free emission from kappa-distributions with integer κ (Chiuderi & Chiuderi Drago2004), which we further develop and incorporate into the general framework.In this study we do not take into account a possible moderate anisotropy of the distribution imposed bythe external magnetic field. Although in the presence of strong magnetic field the anisotropy of the plasmadistribution can be rather strong, it cannot be too strong in the quasi-stationary case discussed here; otherwise,a number of instabilities will develop and give rise to coherent radio emission easily detectable and recognizablein observations. DISTRIBUTION FUNCTIONSIn our study we will use the following distribution functions of the plasma particles, see Fig. 1.I. Maxwellian (thermal) distribution: F M ( p ) = 1(2 πmk B T ) / exp (cid:26) a p mk B T (cid:27) , (1)where a = − m is the electron mass, k B is the Boltzmann constant, T is the plasma temperature, and p isthe electron momentum vector.II. n − distribution (Hares et al. 1979; Seely et al. 1987; Kulinov´a et al. 2011; Karlick´y et al. 2012; Karlick´y2012): F n ( p ) = A n (2 πmk B T ) / (cid:18) p mk B T (cid:19) ( n − / exp (cid:26) − p mk B T (cid:27) , (2)where A n = √ π n/ . (3)Note that for all integer l = ( n − / n = 3 , , ... ) the n − distribution can be obtained fromthe Maxwellian distribution by differentiating the latter by parameter a and then accepting a = − F n ( p ) = A n d l da l F M ( p ) (cid:12)(cid:12)(cid:12) a = − . (4)Apparently, the n − distribution with n = 1 is equivalent to the Maxwellian one.III. Kappa-distribution (Vasyliunas 1968; Owocki & Scudder 1983; Maksimovic et al. 1997; Livadiotis & McComas2009; Pierrard & Lazar 2010): F κ ( p ) = A κ (2 πmk B T ) / (cid:18) p κ − / mk B T (cid:19) − κ − , (5)where A κ = Γ( κ + 1)Γ( κ − / κ − / / . (6) GENERAL APPROACHGeneral definitions for arbitrary incoherent emission process are j σf = Z I σ n ,f F ( p ) d p = Z I σf F ( p ) p d p, (7)where j σf is the volume emissivity of the wave-mode σ at the frequency f , I σ n ,f is the radiation power emittedby a single particle with a given momentum p per unit time, frequency, and element of the solid angle, I σf isthe same measure integrated over the full solid angle I σf = Z I σ n ,f dΩ , (8)and F ( p ) is the particle distribution function normalized by d p , as those defined in §
2. Note that the secondequality in Eq. (7) implies the distribution isotropy. Similarly, for the absorption coefficient we have κ σ = − c n σ f Z I σ n ,f v (cid:20) ∂F ( p ) ∂p + 1 − µ µp ∂F ( p ) ∂µ (cid:21) d p = − c n σ f Z I σf ∂F ( p ) v∂p p d p, (9)where n σ is the refraction index of the wave-mode σ , v is the electron velocity, β = v/c , and µ = cos α is thecosine of electron pitch-angle α . Again, the second equality in Eq. (9) implies the distribution isotropy.Gyrosynchrotron emission (magnetobremsstrahlung) from a single electron with arbitrary energy is describedby the formula (e.g., FT13, Eq 9.148) I σ n ,f = 2 πe c n σ f T σ ∞ X s = −∞ (cid:20) T σ (cos θ − n σ βµ ) + L σ sin θn σ sin θ J s ( λ ) + J ′ s ( λ ) β p − µ (cid:21) ×× δ (cid:20) f (1 − n σ βµ cos θ ) − sf Be γ (cid:21) = ∞ X s = −∞ I σ n ,f,s , (10)where e is the electron charge, f Be = eB/ (2 πmc ) is the electron cyclotron frequency, B is the magnetic fieldstrength, θ is the viewing angle (the angle between the wave vector and the magnetic field vector), γ is the Lorenzfactor, J s is the Bessel function, and the parameters T σ and L σ are the components of the wave polarizationvector. The argument of the Bessel functions is λ = ff Be γn σ β sin θ p − µ . (11)Classical theory of the GR radiation from a nonrelativistic plasma (where β ≪ λ ≪
1) employssmall argument expansion of the Bessel functions and their derivatives and keeping the first non-vanishing termsof this expansion only, which yields for any term s of the series I σ n ,f,s = 2 πe c n σ f T σ s s n s − σ β s sin s − θ (1 − µ ) s s ( s !) [ T σ cos θ + L σ sin θ + 1] × × δ (cid:20) n σ βµ cos θ − (cid:18) − sf Be f (cid:19)(cid:21) , (12)Regarding the free-free emission, the power of bremsstrahlung emitted by a single electron (FT13, Eq 9.267)is I σf = n σ πe Z n i vm c ln Λ C , (13)where n i is the number density of target ions with charge number Z and ln Λ C is the Coulomb logarithm. EMISSIONS PRODUCED BY N -DISTRIBUTIONSGiven that the n -distribution can be obtained from the Maxwellian using Eq. (4) we can first calculate theemissivities and absorption coefficients from the Maxwellian distribution written in form (1) and then determinethe wanted emissivities and absorption coefficients from the n -distribution differentiating the correspondingMaxwellian expressions over the a parameter ( n − / Gyroresonance emission from Maxwellian distributions
Classical theory of the GR radiation from a Maxwellian plasma requires finding the emissivity and absorptioncoefficient defined by Eqs. (7) and (9) with the Maxwellian distribution, Eq. (1). We can easily perform thisstandard derivation for arbitrary negative a (the cylindrical coordinates p ⊥ , p k , ϕ are the most convenient touse here), which yields for the emissivity j M ,σf,s = √ πe n e fc (cid:18) k B Tmc (cid:19) s − / s s n s − σ sin s − θ s s !(1 + T σ ) | cos θ | [ T σ cos θ + L σ sin θ + 1] (cid:18) − a (cid:19) s +1 exp (cid:26) a mc k B T ( f − sf Be ) f n σ cos θ (cid:27) , (14)and absorption coefficient κ M ,σs = √ πe n e cf k B T (cid:18) k B Tmc (cid:19) s − / s s n s − σ sin s − θ s s !(1 + T σ ) | cos θ | [ T σ cos θ + L σ sin θ + 1] (cid:18) − a (cid:19) s exp (cid:26) a mc k B T ( f − sf Be ) f n σ cos θ (cid:27) . (15)It is straightforward to check that for a = − S σf,s = j σf,s / κ σs obeys Kirchhoff’s law asrequired: S σf,s = j σf,s κ σs = n σ f c k B T. (16)To obtain final formulae of the GR emission from a nonuniform source (nonuniform magnetic field at firstplace) we have yet to integrate the equations obtained over the resonance layer. To do so we expand the spatialdependence of the magnetic field around the resonance value: B ( z ) ≈ B (cid:18) zL B (cid:19) , (17)where B = 2 πf mc/ ( se ) is the resonant value of the magnetic field for the frequency f at the harmonic s , z isthe spatial coordinate along the line of sight with z = 0 at B = B , and L B = (cid:18) B ∂B∂z (cid:19) − . (18)With these definitions we get sf Be = f (1 + z/L B ), so the exponent readsexp (cid:26) a mc k B T ( f − sf Be ) f n σ cos θ (cid:27) ≈ exp (cid:26) a mc k B T z L n σ cos θ (cid:27) . (19)Now we can find the optical depth of the s -th gyrolayer by integrating the absorption coefficient along the lineof sight: τ M ,σs = ∞ Z −∞ κ M ,σs ( z ) dz = πe n e f mc (cid:18) k B Tmc (cid:19) s − s s n s − σ sin s − θ s − s !(1 + T σ ) L B [ T σ cos θ + L σ sin θ + 1] (cid:18) − a (cid:19) s +1 / (20)and, accordingly, the emissivity along the line of sight: J M ,σf,s = ∞ Z −∞ j M ,σf,s ( z ) dz = πe n e fc (cid:18) k B Tmc (cid:19) s s s n s − σ sin s − θ s − s !(1 + T σ ) L B [ T σ cos θ + L σ sin θ + 1] (cid:18) − a (cid:19) s +3 / . (21)Here we assume that the dependence of the emissivity and absorption coefficient on the coordinate z is onlycaused by exponent (19), while all other factors in expressions (4.1) and (4.1) are approximately constant withinthe gyrolayer. Apparently, the obtained expressions coincide with classical GR formulae (e.g., Zheleznyakov1970) for a = − J σf,s τ σs = j σf,s κ σs = S σf,s = n σ f c k B T. (22)Thus, the GR emission intensity from a given gyrolayer can be written down simply as J σf,s = S σf,s [1 − exp( − τ σs )] = J σf,s τ σs [1 − exp( − τ σs )] , (23)using the measures integrated over the gyrolayer, which simplifies the theory greatly.4.2. Gyroresonance emission from n -distributions Using Eq. (4) we can immediately write down the GR formulae for the n -distribution (for odd values of n ):Φ ( n ) = A n d l da l Φ (M) (cid:12)(cid:12)(cid:12)(cid:12) a = − , (24)where Φ ( n ) is any of j σf,s , κ σs , J σf,s , and τ σs for n -distribution and Φ (M) is the corresponding measure for theMaxwellian plasma. As a result, general expressions for the emissivity and absorption coefficient take the form j n,σf,s = A n j M ,σf,s l X q =0 l ! s ! ( s + l − q )!( l − q )! q ! (cid:18) ζ s (cid:19) q , (25) κ n,σs = A n κ M ,σs l X q =0 l !( s − s − l − q )!( l − q )! q ! (cid:18) ζ s (cid:19) q , (26)where l = ( n − / ζ s = mc k B T ( f − sf Be ) f n σ cos θ = β z β , (27) β z = f − sf Be f n σ | cos θ | , β = k B Tmc . (28)For example for n = 3 we obtain j (3) ,σf,s = A j M ,σf,s (cid:18) s + 1 + ζ s (cid:19) , κ (3) ,σs = A κ M ,σs (cid:18) s + ζ s (cid:19) . (29)Similarly, for n = 5 we obtain: j (5) ,σf,s = A j M ,σf,s (cid:20) ( s + 1)( s + 2) + ( s + 1) ζ s + ζ s (cid:21) , κ (5) ,σs = A κ M ,σs (cid:20) s ( s + 1) + sζ s + ζ s (cid:21) . (30)It is easy to see that, unlike the Maxwellian case, the source function, Eq. (16), does not take place anylonger for the n -distributions; moreover, in addition to the standard frequency dependence ∝ f , the sourcefunction also depends on ζ s and on the harmonic number s . Kirchhoff’s law recovers only for ζ s ≫
1, i.e.,outside the GR layer, where the GR emissivity and opacity are both exponentially small. Note that in spite ofthe positive derivative of the n -distribution over energy (or momentum modulus) the GR absorption coefficient(in the nonrelativistic approximation) is always positive, so no electron-cyclotron maser instability takes placefor the isotropic n -distributions.To obtain the optical depth and emissivity integrated along the line of sight, one can integrate expressions(25–26) in the way suggested by Eqs. (20) and (21). A more practical way, however, is to apply Eq. (24) to J σf,s , and τ σs directly, which yields J ( n ) ,σf,s = A n ( s + n/ s + n/ s + 1 / s + 1 / J M ,σf,s , (31) Fig. 2.—
Ratio of the optical depths of gyroresonance layers for the n -distribution to the Maxwellian ones. a) τ ( n ) ,σs /τ M ,σs vs. s for different n -indices. b) τ ( n ) ,σs /τ M ,σs vs. n for different harmonic numbers. τ ( n ) ,σs = A n Γ( s + n/ s + 1 / τ M ,σs , (32)which is here written in the form applicable to arbitrary n —not necessarily the odd integer numbers. TheKirchhoff’s law “generalization” to the GR emission from n -distributions reads: S ( n ) ,σf,s = J ( n ) ,σf,s τ ( n ) ,σs = ( s + n/ s + 1 / n σ f c k B T. (33)This equation converges to usual Kirchhoff’s law for n = 1 as required since n = 1 means the Maxwelliandistribution. Then, it also approaches the usual Kirchhoff’s law for large s ≫ n/
2. The reason is that thehigher gyroharmonics are produced by more energetic electrons from the distribution tails, which are similarfor both Maxwellian and n -distributions.We emphasize that the local source function, S σf,s = j σf,s / κ σs , which is now a function of the coordinate z ,is no longer equal to the averaged one S ( n ) ,σf,s unlike in the Maxwellian case. This further implies that the GRintensity from a given gyrolayer cannot be written in simple form (23), but requires more exact knowledge ofthe source function value at the level (inside the gyrolayer) making the dominant contribution to the intensity.Inspection of expressions (29) or (30) suggests that in the optically thick gyrolayer the radiation intensity will decrease with the optical depth increase . We return to this point later, in § n - and Maxwellian distribu-tions, according to Eq. (32); this ratio depends only on s and n . We should note that the parameter T does notplay a role of the effective energy for the n -distribution. The “pseudo-temperature” T ∗ = T ( n + 2) / T ∗ (instead of T ) isassumed to be the same for all distributions, which results in an additional correction factor of [3 / ( n + 2)] s − inthe right side of Eq. (32). We can see that the optical depth of a gyrolayer for the n -distributions, in general, is smaller than that for the Maxwellian distribution; the ratio of optical depths decreases with the increase of theharmonic number and/or the n -index. It is interesting to note that for the second gyrolayer, the optical depthis exactly the same for the Maxwellian and n -distributions with arbitrary n -index.4.3. Free-free emission from Maxwellian distributions
To compute the free-free emission from the Maxwellian distribution is even easier than the GR emissionconsidered above. We can discard the weak dependence of the Coulomb logarithm on the particle energy tothe first approximation while taking integrals in Eqs. (7) and (9). These integrations are straightforward; theyyield well-known results for the emissivity j M ,σf, ff = − e n σ n e n i ln Λ C √ π ( mc ) / ( k B T ) / a ; a = − , (34)and absorption coefficient κ M ,σ ff = 8 e n e n i ln Λ C √ πn σ cf ( mk B T ) / . (35) Fig. 3.—
Intensity spectra (a) and brightness temperatures (b) of the free-free emission from the Maxwellian, kappa- and n -distributions. The emission parameters were computed for a fully ionized unmagnetized hydrogen plasma with the density of n = 10 cm − (that corresponds to the plasma frequency of about 0.9 GHz) and temperature of T = 10 K; the source size alongthe line-of-sight is L = 10 cm. Evidently, these expressions obey Kirchhoff’s law as needed for the thermal emissions.4.4.
Free-free emission from n -distributions Now the theory of the free-free emission from the n -distributions is derived from that for the Maxwelliandistribution by consecutive differentiating the obtained emission and absorbtion coefficients over a parameter.For the emissivity we obtain j ( n ) ,σf, ff = A n l ! 8 e n σ n e n i ln Λ C √ π ( mc ) / ( k B T ) / ; l = ( n − / , (36)which allows a straightforward analytical continuation to a non-integer l : j ( n ) ,σf, ff = √ π Γ( n/ / n/ e n σ n e n i ln Λ C √ π ( mc ) / ( k B T ) / , (37)where expression (3) for the normalization constant A n has been taken into account. It is easy to estimate thatthe free-free emissivity slightly decreases compared with the Maxwellian distribution with the same T parameteras n increases.Unlike the emissivity, the absorption coefficient described by Eq. (35) does not depend on a ; thus, all deriva-tives of the absorption coefficients over this parameter are zeros, which means no free-free absorption by electronswith the n -distribution. This happens because the positive contribution to the absorption coefficient from thenegative slope of this distribution at high velocities is fully compensated by the negative contribution (amplifi-cation) from the positive slope at low velocities. This corresponds to a marginal stability state when a non-zeroemissivity is accompanied by zero absorption coefficient. No analogy to Kirchhoff’s law can be formulated inthis case; arbitrarily deep plasma with such a distribution remains optically thin as is clearly seen from Fig. 3. EMISSIONS PRODUCED BY KAPPA-DISTRIBUTION5.1.
Gyroresonance emission from kappa-distribution
Integrals (7) and (9) with the cyclotron radiation power, Eq. (3), are convenient to take in the cylindricalcoordinates: integration over the azimuth angle results in 2 π factor, while the integral over dp k is taken with the δ -function. The remaining single integration over dp ⊥ is a tabular integral of a rational fraction, which yieldsthe emissivity j κ,σf,s = √ πe n e fc ( κ − / s − / Γ( κ − s )Γ( κ − / (cid:18) k B Tmc (cid:19) s − / s s n s − σ sin s − θ s s !(1 + T σ ) | cos θ | × [ T σ cos θ + L σ sin θ + 1] h ζ s κ − / i κ − s , κ > s, (38)and the absorption coefficient κ κ,σs = √ πe n e cf k B T ( κ − / s − / ( κ − s )Γ( κ − s )Γ( κ − / (cid:18) k B Tmc (cid:19) s − / s s n s − σ sin s − θ s s !(1 + T σ ) | cos θ | × [ T σ cos θ + L σ sin θ + 1] h ζ s κ − / i κ − s +1 , κ > s − , (39)where the parameter ζ s is defined by Eq. (27). The source function S κ,σf,s = j κ,σf,s κ κ,σs = ( κ − / κ − s ) n σ f c k B T (cid:20) ζ s κ − / (cid:21) (40)depends on the ζ s parameter that complicates the GR theory significantly for the same reason that has beenexplained for the n -distribution. However, unlike n -distribution, here the GR intensity from an optically thickgyrolayer increases as the optical depth increases.Now we can find the optical depth of the s -th gyrolayer by integrating the absorption coefficient in the linearlychanging magnetic field, Eq. (17), along the line of sight: τ κ,σs = ∞ Z −∞ κ κ,σs ( z ) d z = πe n e f mc ( κ − / s − Γ( κ − s + 1 / κ − / (cid:18) k B Tmc (cid:19) s − s s n s − σ sin s − θ s − s !(1 + T σ ) L B × [ T σ cos θ + L σ sin θ + 1] , κ > s − / , (41)and, accordingly, the emissivity along the line of sight: J κ,σf,s = ∞ Z −∞ j κ,σf,s ( z ) d z = πe n e fc ( κ − / s Γ( κ − s − / κ − / (cid:18) k B Tmc (cid:19) s s s n s − σ sin s − θ s − s !(1 + T σ ) L B × [ T σ cos θ + L σ sin θ + 1] , κ > s + 1 / . (42)The ratio of these two expressions yields the effective source function (again, different from the Maxwellian’sone): S κ,σf,s = J κ,σf,s τ κ,σs = ( κ − / κ − s − / n σ f c k B T. (43)Since kappa-distribution (5–6) converges to the Maxwellian one when κ → ∞ , the GR emission parameters forlarge κ -indices ( κ ≫ s ) approach those for the Maxwellian distribution; in particular, relation (43) approachesthe usual Kirchhoff’s law.Note that the above equations are only valid for relatively small gyroharmonics (otherwise, the correspondingintegrals diverge), so that the derived here GR theory for the kappa-distribution may only be applicable at s < κ − /
2. For higher harmonics, s > κ − /
2, the quasi-continuum gyrosynchrotron contribution fromthe power-law tail of the kappa-distribution, where the non-relativistic expansions used above are invalid,dominates over the contribution from the nonrelativistic core of the distribution. If needed, this contributioncan be computed in a usual way (Fleishman & Kuznetsov 2010).Figure 4 demonstrates the ratio of optical depths of gyroresonance layers for the kappa- and Maxwelliandistributions, according to Eqs. (5.1) and (20); this ratio depends only on s and κ . We can see that the opticaldepth of a gyrolayer for the kappa-distributions, in general, is larger than that for the Maxwellian distribution;the ratio of optical depths increases with the increase of the harmonic number and/or decrease of the kappa-index. The optical depth of the second gyrolayer is the same for all considered distributions—the Maxwellian, n - and kappa-distributions. The reason for this equivalence is that all these optical depths are proportional to T s − ; thus, linearly proportional to T for s = 2. This means that the optical depths of the second gyrolayer aredefined by the second moment of the given distribution only, that is the mean electron energy, which is adoptedthe same for all these distributions.5.2. Free-free emission from kappa-distribution
Chiuderi & Chiuderi Drago (2004) developed analytical theory of free-free emission from kappa-distributionswith integer indices κ . It is straightforward, however, to extend this theory to arbitrary real index κ . To do sowe consider again Eqs. (7) and (9) with the free-free radiation power, Eq. (13), but with kappa-distribution (5)instead of the Maxwellian one. Neglecting the (weak) energy dependence of the Coulomb logarithm as before,we can easily take the remaining integrals which yields for the emissivity j κ,σf, ff = A κ κ − / κ e n σ n e n i ln Λ C √ π ( mc ) / ( k B T ) / , (44) Fig. 4.—
Ratio of the optical depths of gyroresonance layers for the kappa-distribution to the Maxwellian ones. Only finite-rangedata is presented because for each finite κ there is a highest harmonic number, s < κ − /
2, up to which the developed theory isvalid. a) τ κ,σs /τ M ,σs vs. s for different κ -indices. b) τ κ,σs /τ M ,σs vs. κ for different harmonic numbers. and absorption coefficient κ κ,σ ff = A κ e n e n i ln Λ C √ πn σ cf ( mk B T ) / . (45)Chiuderi & Chiuderi Drago (2004) took into account the energy dependence of the Coulomb logarithm, whichallowed them to obtain the results in the closed form for integer κ only. This results in small corrections tothe Coulomb logarithm, slightly different for the emissivity and absorption coefficient. With these corrections,which we interpolated with the parenthetical expressions below, we can write j κ,σf, ff = A κ κ − / κ e n σ n e n i ln Λ C √ π ( mc ) / ( k B T ) / (cid:20) − . /κ ) . ln Λ C (cid:21) , (46)and κ κ,σ ff = A κ e n e n i ln Λ C √ πn σ cf ( mk B T ) / (cid:20) − . /κ ) . ln Λ C (cid:21) . (47)Therefore, Kirchhoff’s law extension to the free-free emission from the kappa-distribution reads S κ,σf, ff = j κ,σf, ff κ κ,σ ff ≈ κ − / κ n σ f c k B T, (48)where we discarded the ratio of two parenthetical expressions entering Eqs. (46) and (47), which are both close toone, for brevity. Eq. (48) implies that the effective temperature from a plasma volume with kappa-distribution islower than that for the Maxwellian plasma with the same temperature T ; (see Chiuderi & Chiuderi Drago 2004,for greater detail); the same statement is valid for the brightness temperature in the optically thick case. Incontrast, in the optically thin regime the brightness temperature here is slightly larger than for the Maxwellianplasma with the same T ; see Fig. 3. RADIATION TRANSFER THROUGH A GYROLAYER IN THE NON-MAXWELLIAN PLASMASAs has been noted at the end of § S σf,s does not depend on coordinates (for aconstant T ) in a Maxwellian plasma, which simplifies the theory greatly. In particular, the GR emission intensityfrom a given gyrolayer is described by Eq. (23) regardless of the actual value of the optical depth τ . This is nolonger valid for the non-Maxwellian distributions as their source functions do depend on the coordinates withina gyrolayer. This calls for explicit consideration of the radiation transfer through the gyrolayer.Generation and propagation of emission in a self-absorbing medium is described by the radiation transferequation (e.g., Fleishman & Toptygin 2013) d J σf d z = j σf − κ σ J σf , (49)where we neglect refraction and scattering and assume that the emission modes propagate independently.GR emissivity and absorption coefficient strongly increase at a gyrolayer where f ≃ sf Be . Therefore, in aninhomogeneous magnetic field, emission and absorption of radiation occur primarily within such GR layers. We0assume that a GR layer is narrow (in practice, this implies a somewhat low plasma temperature, T . K,see § f − sf Be f = zL B , (50)the adjacent GR layers do not overlap, and all other source parameters (except the magnetic field) are approx-imately constant within the GR layer. In this case, for the Maxwellian distribution, the intensity of emission after passage the GR layer is given by J M ,σ, out f,s = J M ,σ, in f,s exp (cid:0) − τ M ,σs (cid:1) + S M ,σf,s (cid:2) − exp (cid:0) − τ M ,σs (cid:1)(cid:3) , (51)where J M ,σ, in f,s is the intensity of emission incident on the gyrolayer from below and S M ,σf,s is the source functiondescribed by Eq. (22).For non-Maxwellian distributions, the radiation transfer equation (49) cannot analytically be solved even forthe narrow layer approximation adopted, because the corresponding source function vary in space at the GRlayers. However, we can write its solution in a form similar to (51), namely, J σ, out f,s = J σ, in f,s exp ( − τ σs ) + R σs S σf,s [1 − exp ( − τ σs )] , (52)where S σf,s = J σf,s /τ σs is the effective source function (described by Eq. (33) for n -distribution and Eq. (43)for kappa-distribution) and the factor R σs is introduced to describe the deviation from the Kirchhoff law. Theadvantage of this solution form is that the R σs -factor can be computed once and then used together with theadopted form of the radiation transfer solution, Eq. (52). Evidently, this factor approaches unity in the opticallythin limit and when the distribution function approaches the Maxwellian one. In general, the factor R σs hasto be found numerically. One can note that the first term in Eq. (52) (describing the GR absorption of theemission produced in the deeper regions) is exactly the same as in Eq. (51) because the absorption of theincident radiation is only determined by the total optical depth of the gyrolayer.By substituting GR emissivity and absorption coefficient (5.1–5.1) for the kappa-distribution into radiationtransfer equation (49), introducing a new dimensionless integration variable t = ζ s / √ κ − ∝ z and having inmind that the solution of the resulting equation should have general form (52) at t → ∞ (or z → ∞ ), we canwrite the factor R κ,σs for the kappa-distribution as R κ,σs ( τ κ,σs , κ − s ) = τ κ,σs − exp( − τ κ,σs ) u ∞ ( τ κ,σs , κ − s ) √ π Γ( κ − s )Γ( κ − s − / , (53)where u ∞ is a solution (at t → ∞ ) of the differential equationd u ( t )d t = 1(1 + t ) κ − s − α (1 + t ) κ − s +1 u ( t ) , (54) α = τ κ,σs √ π Γ( κ − s + 1)Γ( κ − s + 1 /
2) (55)with the initial condition u ( −∞ ) = 0. Note that this factor depends on two parameters only, since the index κ and the harmonic number s enter the corresponding expressions in a combination of κ − s , but not separately.Equation (54) has a finite solution at κ − s > / R κ,σs -factor as a function of τ κ,σs and κ − s as computed numerically is given in Figs. 5a,b bysolid curves. It is easy to show that the asymptotes of this factor are R κ,σs ≈ τ κ,σs < R κ,σs ≈ (cid:20) τ κ,σs κ − s ) . (cid:21) κ − s +0 . for τ κ,σs ≫
1. With the use of these two asymptotes one can construct an analyticalformula, which correctly describes the R κ,σs -factor in the entire range of interest. A quantitatively accurateapproximation is R κ,σs ≈ , for τ κ,σs < κ − s, (cid:20) τ κ,σs κ − s ) . (cid:21) κ − s +0 . + 684 + (4 + τ κ,σs ) , for τ κ,σs > κ − s ; (56)the corresponding curves are given by dashed lines in the same figures, which explicitly confirm validity of theapproximation. For the parameter range of 0 < τ κ,σs < and 1 < κ − s <
100 (which covers most cases ofinterest for solar radio astronomy), a relative error of the analytical approximation (56) does not exceed 8%.Thus, with the described modification including the analytical form of the R κ,σs -factor, the gyroresonant theoryfrom the kappa-distribution turns to become almost as simple as that for the Maxwellian distribution.1 Fig. 5.—
Dependence of the correction factor for the kappa-distribution R κ,σs ( τ κ,σs , κ − s ) on its parameters. Solid lines: exactvalues given by Eq. (53); dashed lines: asymptotical approximation given by Eq. (56). a) R κ,σs vs. the optical depth τ κ,σs fordifferent values of κ − s ; b) R κ,σs vs. the difference κ − s for different values of τ κ,σs . The presence of the non-unitary R κ,σs -factor implies that the brightness temperature of the GR emission froma gyrolayer will depend now on the total optical depth of the gyrolayer. Figure 6 displays this dependence forthe kappa-distribution with different indices. In contrast with the Maxwellian plasma, for which the brightnesstemperature is just equal to the plasma kinetic temperature for τ ≫
1, the brightness temperature of theGR emission from a kappa plasma continues to grow with the optical depth τ . Not surprisingly,this growth is more pronounced for smaller kappa-indices, i.e., for stronger departure of the plasma from theMaxwellian distribution. The brightness temperature can exceed the kinetic temperature of the kappa plasmaby an order of magnitude or even more for a realistic set of parameters.For the n -distribution, the factor R n,σs has the form similar to (53): R n,σs ( τ n,σs , n, s ) = τ n,σs − exp( − τ n,σs ) u ∞ ( τ n,σs , n, s ) √ π Γ( s + 3 / s + 1 + n/ , (57)but the differential equation for u ∞ is more cumbersome:d u ( t )d t = " e − t l X q =0 l ! s ! ( s + l − q )!( l − q )! q ! t q − " αe − t l X q =0 l !( s − s − l − q )!( l − q )! q ! t q u ( t ) , (58)with α = τ n,σs √ π Γ( s + 1 / s + n/
2) (59)and l = ( n − /
2. For l = 0 ( n = 1, the Maxwellian distribution), as expected, we obtain R (1) ,σs ≡
1. Asin this case the R n,σs -factor is a function of three (rather than two) parameters, it is more convenient here togenerate a look-up table of its values, rather than introduce an analytical interpolation, which is more difficultto reliably test in the 3D parameter domain. Such table (providing the relative computation error of less than2 × − for 0 < τ < , 2 ≤ s ≤
20 and 1 ≤ n ≤
15) is included into our numerical code (see below); thevalues of the factor R n,σs for some subset of the mentioned parameter range are presented in Fig. 7.In the case of n -distribution the brightness temperature of the GR emission also deviates from the parameter T . However, the dependence of T eff on τ is nonmonotonic here: T eff reaches a peak at τ ∼ T eff from the T parameter does not exceed a factor of 2-3 for a realistic set ofparameters; see Fig. 8a. For a constant value of T , the brightness temperature (both in the optically thickand thin modes) for n -distributions is always higher than for the Maxwellian one; it increases with increasing n . However, this is caused by the already mentioned fact that the parameter T does not play a role of theeffective energy for the n -distribution and higher n -indices actually correspond to higher average energies of theelectrons; as has been said above, the “pseudo-temperature” T ∗ = T ( n + 2) / n -distributions with different n -values and the Maxwellian distribution. As can be seen inFig. 8b, for a constant value of T ∗ , the brightness temperature for n -distributions is always lower than for theMaxwellian one and decreases with increasing n . APPLICATION TO ACTIVE REGIONS2
Fig. 6.—
Dependence of the brightness temperature of the gyroresonance emission on the optical depth of the gyrolayer for theMaxwellian distribution ( κ → ∞ ) and kappa-distributions with different κ . The plasma temperature is T = 10 K, the cyclotronharmonic number is s = 3 and the refraction index is n σ → Fig. 7.—
Dependence of the correction factor for the n -distribution R n,σs ( τ n,σs , n, s ) (57) on the optical depth τ n,σs for differentvalues of n and s . Different line types correspond to different harmonic numbers: s = 2 (solid), s = 3 (dotted), s = 4 (dashed), s = 5 (dash-dotted) and s = 6 (dash-triple-dotted). Fig. 8.—
Dependence of the brightness temperature of the gyroresonance emission on the optical depth of the gyrolayer for theMaxwellian distribution ( n = 1) and n -distributions with different temperatures and n -indices. The cyclotron harmonic numberis s = 3 and the refraction index is n σ →
1. a) Plots for the constant parameter T ( T = 10 K); b) plots for the constant“pseudo-temperature” T ∗ ( T ∗ = 10 K).
Fig. 9.—
Active region model n , magnetic field strength B and viewing angle θ along the chosen line of sight. Plasma temperature is 10 K everywhere.
Let us consider now how the properties of the radio emission from an active region (Alissandrakis & Kundu1984; Akhmedov et al. 1986; Gary & Hurford 1994; Kaltman et al. 1998; Gary & Hurford 2004; Gary & Keller2004; Peterova et al. 2006; Lee 2007; Bogod & Yasnov 2009; Tun et al. 2011; Nita et al. 2011c; Kaltman et al.2012) filled with the non-Maxwellian plasmas differ from those in the classical Maxwellian case. To do so weadopt a line-of-sight distribution of all relevant parameters taken from a 3D model we built with our modelingtool, GX Simulator (Nita et al. 2011a,b, 2012), for a different purpose, and compute the expected emissionassuming various energy distribution types of the radiating plasma. At this point we do not address any partic-ular observation but only need a reasonably inhomogeneous distributions of the magnetic field, thermal density,and temperature along the line of sight, implying some complexity of the radio spectrum and polarization.Specifically, we selected two sets of the line-of-sight distributions of the parameters, which are given in Figs. 9and 11. As can be seen in the figures, all model parameters were defined on a regular grid along the line-of-sight;however, if necessary (e.g., to find the GR layers), a linear interpolation between the grid nodes was used in thesimulations.The first example given in Fig. 9 includes a realistic distribution of the magnetic field obtained from anextrapolation of the photospheric magnetic field data, but a simplified hydrostatic distribution of the thermalplasma with a single temperature T = 1 MK. Figure 10 displays the radiation spectra and polarization forthe Maxwellian, kappa-, and n -distribution with various indices (for the n -distributions, the temperaturescorresponding to the constant “pseudo-temperature” of T ∗ = 1 MK were used). Both GR and free-free processesare included. Not surprisingly, the intensity of the GR emission increases as the kappa-index decreases, which issimply an indication of stronger contribution from the more numerous high-energy electrons from the tail of the4 Fig. 10.—
Gyroresonance emission spectra (a, c) and polarization (b, d) for the source model shown in Fig. 9. Visible source areais 1” × κ → ∞ ) and kappa-distributions with different κ . c-d) Maxwellian distribution ( n = 1)and n -distributions with different n . Fig. 11.—
Active region model n , plasma temperature T , magnetic field strength B and viewing angle θ along the chosen line of sight. The abscissa axis is logarithmic to demonstrate better the active regionstructure at low heights. The vertical dotted lines correspond to the formation layer of the narrowband spectral peaks visible inFigs. 12a,c. kappa-distribution with smaller indices. However, the shape of the spectrum from the kappa-distribution witha given T is difficult to distinguish from that from a Maxwellian plasma with somewhat higher T . In contrast,the intensity of the GR emission from the n -distributions decreases with increasing n -index, but, again, thespectrum shape remains almost the same.Remarkably, the polarization behavior is distinctly different for the cases of the Maxwellian, kappa- and5 Fig. 12.—
Gyroresonance emission spectra (a, c) and polarization (b, d) for the source model shown in Fig. 11. Visible sourcearea is 1” × κ → ∞ ) and kappa-distributions with different κ . c-d) Maxwellian distribution( n = 1) and n -distributions with different n . n -distributions. Indeed, at the frequencies where the GR emission is optically thick (below 8 GHz in ourexample) the degree of polarization from the Maxwellian plasma is zero, see Fig. 10b. This follows from thewell known fact that the brightness temperature of the optically thick emission produced by thermal plasma isequal to the kinetic temperature of the plasma in both ordinary and extraordinary wave-modes, which resultsin a non-polarized emission. The situation is distinctly different for the plasma with the kappa-distribution.As has been shown in §
6, the brightness temperature of the GR emission from a kappa plasma increases asthe optical depth of the gyrolayer increases. It is easy to see that for a given gyrolayer the optical depth ofthe ordinary mode emission is noticeably smaller than that of the extraordinary mode; thus, the brightnesstemperature of the extraordinary mode emission is stronger than that of the ordinary mode, which results ina noticeably polarized emission in the sense of extraordinary mode as seen from Fig. 10b. This offers quite asensitive tool of distinguishing GR emission from kappa- or Maxwellian distributions. A similar effect takesplace for n -distribution, but with the opposite (ordinary) sense of polarization (Fig. 10d), because the emissionintensity in the optically thick regime decreases with the optical depth; the degree of polarization is slightlylower than for kappa-distribution.A more realistic inhomogeneous distributions of the plasma density and temperature along the line of sight aredemonstrated in Fig. 11. In this case, the chromospheric part of the active region (with the plasma density of upto 10 cm − and temperature of 3500 K) is included; the coronal part of the active region contains a number ofnarrow flux tubes, filled with the thermal plasma according to a nanoflare heating model (Klimchuk et al. 2008,2010), which makes the height profiles of all parameters non-monotonic. In addition, the sign of the projectionof the magnetic field vector on the line-of-sight experiences a reversal within the active region. As one mightexpect, the radiation spectra and polarization (see Fig. 12) become now more diverse and structured. There is apolarization reversal at the frequency of about 8 GHz, caused by the frequency-dependent mode coupling at thelayer with the transverse magnetic field (roughly, at the level of 10 000 km above the photosphere, see Fig. 11b).Furthermore, there are several sharp narrowband peaks at the harmonically-related frequencies in the intensityspectra. These peaks are produced at the bottom of the corona, at the layer where magnetic field reaches itsmaximum along the line of sight ( B ≃ Fig. 13.—
Simulated emission spectra for the Maxwellian distributions with different temperatures. Solid lines: results obtainedusing the gyroresonance approach described in this paper; dashed lines: results obtained using precise relativistic gyrosynchrotronformulae. The temperatures (in Kelvins) are indicated by numbers near the lines; other source parameters are given in the text. third, fourth, and fifth gyroharmonic, respectively), the plasma density is relatively high ( n ≃ cm − ), whilethe plasma temperature ( T ≃ T ≃ . ∼
100 G) to producethe gyroemission above 1 GHz; however, they could be distinguished at lower frequencies. The mentionedfine spectral structures can be potentially used as a very precise tool for measuring magnetic fields in suchfluxtubes; however, this requires observations with high spectral and angular resolutions (since the peaks in thespatially integrated spectra can be smoothed due to the source inhomogeneity across the line-of-sight). Likein the previous model, the intensity spectra for different electron distributions have similar shapes, although insome frequency ranges (e.g., at f &
10 GHz in Fig. 12a) they can demonstrate noticeably different slopes; thepolarization remains much more sensitive to the electron distribution type and parameters than the radiationintensity. APPLICABILITY OF THE GR APPROXIMATIONLet us address now the applicability of the considered here GR approximation. For this purpose, we havecompared the numerical results obtained using the approximate (gyroresonance) and exact (gyrosynchrotron)formulae (see Figs. 13–15). The simulations were performed for a model emission source with homogeneousplasma density and temperature and constant magnetic field direction; in all simulations we used the plasmadensity of n = 10 cm − and viewing angle of θ = 60 ◦ . The magnetic field strength varied linearly with thedistance along the line-of-sight (from 1000 to 300 G over the source depth of 10 000 km, which correspondsto the inhomogeneity scale of L B ≃ km .We can see that for the Maxwellian distribution (Fig. 13), the GR approximation ensures excellent accuracyup to the plasma temperature of about 10 MK. At higher temperatures ( &
30 MK) the approximation correctlyreproduces the optically thick part of the spectrum, but tends to overestimate the optically thin emissionintensity; in addition, the exact optically-thin spectra are smoother than the approximate ones. At very hightemperatures and/or frequencies (e.g., at f &
30 GHz for T = 100 MK), the GR approximation completelybreaks down, because the condition λ ≪ Fig. 14.—
Same as in Fig. 13, for the n -distributions with different temperatures and n -indices. The numbers near the linesindicate the effective “pseudo-temperatures” T ∗ . Fig. 15.—
Same as in Fig. 13, for the kappa-distributions with different temperatures and kappa-indices. The gyroresonanceapproach (shown by solid lines) is available only at s < κ − /
810 MK, respectively. Since the gyroemission intensity (for the typical coronal temperatures) at those frequenciesis negligible, the GR approximation seems to be well sufficient for solar applications.Similar conclusions can be drawn for the n -distributions (see Fig. 14). The applicability range of the GRapproximation is now even wider than for the Maxwellian distribution, because for the same average energiesof the electrons (characterized by the “pseudo-temperature” T ∗ ) the n -distributions have lower T parameters,which implies a steeper decrease of the distribution function at high energies.The situation is somewhat different for kappa-distribution as it has a power-law tail at high energies; this iswhy the GR approximation can only be valid up to certain low harmonics determined by the κ index, as hasalready been noted earlier. Figure 15 gives a good idea about the applicability region of the GR approximationfor the kappa-distribution where both the temperature and kappa-index are important. In general, the regionof the GR approximation applicability is narrower for the kappa- than for the Maxwellian distribution. For veryenergetic plasmas (e.g., at κ . T &
30 MK), even the optically thick emission is badly reproduced. Asexpected, the smaller the kappa-index the smaller the highest temperature at which the GR formulae can beused. Another limitation is the frequency: although the GR approximation breaks formally down only above f /f Be ≃ s = Integer[ κ + 1 / κ &
7, which can be expected in a steady-state plasma of solar active regions. For evenhigher kappa-indices, the distribution properties approach those of the Maxwellian distribution, and for κ > DISCUSSIONThis paper has developed analytical theory of the GR and free-free emission from plasmas characterized bynon-Maxwellian isotropic kappa- or n -distributions. In particular, we demonstrated that the free-free emissionfrom n -distribution is always optically thin, while the brightness temperature of the optically thick free-freeemission from a plasma with kappa-distribution is lower than that for the Maxwellian plasma with the same T . We emphasize that the use of these new formulae is needed any time when an emission from such non-Maxwellian plasma is modelled. For example, if one considers the gyrosynchrotron or plasma emission from aplasma with kappa-distribution, then the free-free emission and absorption must although be computed for thesame kappa-distribution; the use of the standard free-free formulae can lead to inconsistent results.We note that for some solar flares the coronal X-ray emission can equally well be fitted by either kappa- ora Maxwellian core plus a power-law distribution (Kaˇsparov´a & Karlick´y 2009; Oka et al. 2013), which can alsobe presented in the form of thermal-nonthermal (TNT) distribution (Holman & Benka 1992; Benka & Holman1994). This implies that the results obtained here for kappa distribution can, to some extent, apply to the TNTdistribution. However, this analogy is limited for the following two reasons. First, the TNT distribution is onlysimilar to the kappa distribution if the power-law tail of the TNT distribution contains a significant fractionof the total number of the electrons at the source. And second, during a flare the plasma temperature is oftenhigh, larger than 10 MK, and the nonthermal tail spectra are relatively hard, so the GR approximation breaksdown (see § REFERENCESAkhmedov, S. B., et al. 1986, ApJ, 301, 460Alissandrakis, C. E., & Kundu, M. R. 1984, A&A, 139, 271Anderson, S. W., Raymond, J. C., & van Ballegooijen, A.1996, ApJ, 457, 939 Benka, S. G., & Holman, G. D. 1994, ApJ, 435, 469Bogod, V. M., & Yasnov, L. V. 2009, Astrophysical Bulletin,64, 372Chiuderi, C., & Chiuderi Drago, F. 2004, A&A, 422, 3319