Perspectives of Current-Layer Diagnostics in Solar Flares

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Fletcher 2004; soft al. therein). the et references and above (Emslie 2011 source others Hudson X-ray 2011; and hard top, loop a outflows, X-ray and rec inflows lines, coron field tion cusp-shaped reconnected evi newly flares: of indirect in shrinkage structures, reconnection provide magnetic (RHESSI), of Energy dence Imager High Ramaty Spectroscopic Reuven Solar and (SMM), Mission Yohko Maximum Hinotori, as such Spacecrafts, electrons. energies) uy1,2018 11, July srnm Astrophysics & Astronomy ihifrainaotslrflrscnb bandfrom obtained be can flares solar about information Rich u omn(03 ecietelm aeo 02April 2002 on flare limb the describe (2003) Holman & Sui oe fpam etn ytelayer. the by heating plasma of model future. Conclusions. Results. Methods. Aims. Context. r odrc bevtoso hs layers. these of observations direct no are urn ae rmobservations. from layer current di as Fe such lines features, observational the and dimensions, 2013 July 21 accepted 2012; September 7 Received e-mail: Mos Lomonosov M.V. Russia Institute, Astronomical Sternberg P.K. e words. Key interpret to s method. important high temperature very double with also or Å is single It 1.5–1.8 needed. range are X-ray (seconds) in Observations future. the h i forwr st nesadwyw culyd o dire not do actually we why understand to is work our of aim The [email protected] xxvi × eoncigcretlyri hat faslrflr,bec flare, solar a of ‘heart’ a is layer current reconnecting A h oesalwu osuyacrepnec ewe h mai the between correspondence a study to us allow models The h ehdi ae nasml ahmtclmdlo super- a of model mathematical simple a on based is method The 10 7 u:atvt u:flrs-Sn -as gamma-rays. X-rays, Sun: - flares Sun: - activity Sun: − 17 n .1Å n Ni and Å) 1.51 and (1.78 32 bevtoso HCsaedi are SHTCLs of Observations 10 rs srlae uigatpclflr;i is it flare; typical a during released is ergs, 8 )pam n ceeae u oMeV to (up accelerated and plasma K) esetvso urn-ae diagnostics current-layer of Perspectives aucitn.Oreshina no. manuscript xxvii 15 ) hsmto rvdsatertclbssfrdeter for basis theoretical a provides method This Å). (1.59 ffi ..Oehn n ..Oreshina I.V. and Oreshina A.V. ut eas h pcrlln neste r an,bti but faint, are intensities line spectral the because cult, nslrflares solar in ff rniladitga msinmaueo etdpam,in plasma, heated of measure emission integral and erential ,Solar h, onnec- tween while h bevtosuigamlitmeaueapoc inste approach multi-temperature a using observations the etc. , ABSTRACT al- r en- ve in 0; e- o tt nvriy nvriesi rset 3 Mosco 13, prospekt, Universitetskii University, State cow al e, h e eta eouin(etrta .1Å n ihtmoa re temporal high and Å) 0.01 than (better resolution pectral - - - . oecmatcretlyr htaeascae ihflrsa heights at flares corona with lower the associated in are formed are that that layers current compact more lsacno ergsee sn I.Tecaatrsi d is characteristic layer The the EIT. of using sity registered be cannot plasma Fe and (2 hbt 01 evse l 00 hne l 01.Tead- The 2011). al. possibil et e the various in Shen considering consists 2010; computations numerical al. & of Yokoyama et vantage 1999; Reeves al. et 2001; (Chen Shibata numerically Oreshina and 1966; 2000) 1963, Somov Syrovatskii 1963; Parker 1958; (Sweet fields. magnetic eishrcOsraoy(OO.Is15Åcanli sensi is (1 channel low-temperature Å to 195 tive Its Solar (SOHO). onboard Observatory (EIT) Heliospheric telescope observed imaging was It ultraviolet (CME). extreme ejection mass coronal a with sociated bv h upsae opa heights at loop cusp-shaped the above rmacle oreat source colder a from HSIosraino h aeo 02Jl 3 super-hot A report 23. (2010) July Lin 2002 & on Caspi at source flare flare 2008). elongated the the al. of in et observation observed (Liu RHESSI also 30 was April pattern 2002 similar on em A images. remains the (SHTCL), on layer turbulent-current super-hot the rsne yLue l 21) hydsrb rgtstruct bright a describe They (2010). length al. of et Liu by presented corona. the in higher located layer current a of concept uei sapaeo antcfil nryrlae oee t However release. energy magnetic-field of place a is it ause × oeso h urn aesaedvlpdanalytically developed are layers current the of Models h rtiaeo ag-cl eoncigcretlyri layer current reconnecting large-scale a of image first The tyosrecretlyr n htw edt oi nthe in it do to need we what and layers current observe ctly 10 7 xxiv o ( hot hrceitc ftelyr uha eprtr and temperature as such layer, the of characteristics n )pam eas ftepeec fteFe the of presence the of because plasma K) > 0 T 12Å eoac ie.Tesprht( super-hot The lines. resonance Å) (192 . 25 ∼ > R 10 ⊙ 8 ∼ n it (5 width and ∼ )truetcretlyr(HC)ada and (SHTCL) layer turbulent-current K) ff 10 4 cs etcnuto,pam oin ra- motion, plasma conduction, heat ects: × 6 ∼ − 10 . 2 6 10 7 × × sstae ihradseparately and higher situated is K 7 10 10 cm − stertclypsil in possible theoretically is t 7 10) 6 − iigprmtr fthe of parameters mining .I loare elwt a with well agrees also It K. )adt high-temperature to and K) 3 norwr,w consider we work, our In . × ∼ 10 (0 8 est fspectral of tensity . m hc ssituated is which cm, 78 do h usual the of ad ∼ − 0 ,119991, w, 1 . 05 .

05) c solution R S 2018 ESO xii R ⊙ ∼ > here ⊙ nstrong in 15Å) (195 n as- and 10 t of ity 8 and en- pty ure K) by nd & 1 s - A.V. Oreshina and I.V. Oreshina: Current-layer diagnostics in solar flares diative losses etc. It allows one to clarify qualitative role of the y effects but does not give real quantitative estimations. Modern v z E B0 numerical MHD algorithms cannot work with real transport co- 0 efficients replacing them with some fictive values. They always contain an artificial numerical viscosity and conductivity.As a a result, computations of fine structure of the layer, whose thick- x V ness is less than its width and length by 7 orders of magnitude, become impossible. Analytical models are free from the numer- b ical drawbacks. As they cannot simultaneously consider many effects, it is very important to determine the main one. Their ad- vantage consists in simplicity and clarity. Analytical models also Fig. 1. Reconnecting current layer and field lines. help to interpret the results of more complicated numerical mod- elling. They can serve as a test for these computations. Without doubt, both numerical and analytical approaches have to work a possible explanation, which consists in departing the emitting together, giving useful ideas to each other. plasma from the isothermal state. Thus, sometimes we can see As we have already noticed, we consider SHTCLs in large evidence of the multitemperature nonequilibrium plasma. These solar flares. These layers are expected to be significantly hot- events must be analyzed very accurately. ter, denser, and more compact than those associated with CME, The paper is organized as follows. In Sect. 2, we describe because they are situated at lower altitudes in the corona. an analytical model of the SHTCL and a model of plasma heat- According to the analytical self-consistent model of a SHTCL, ing by the layer. In Sect. 3, observational manifestations of the their temperatures are ∼> 108 K, and their thicknesses are ∼< 1m SHTCL (emission measure of heated plasma, intensities of some (Somov 2006b). Their emission measures are too low for di- spectral lines, etc.) are presented. In the last section, we formu- rect observations. Meanwhile, these observations are important, late our conclusions. because spatial and temporal averaging can lead to ambiguity and erroneous interpretation. For example, Doschek & Feldman (1987) demonstrate various plasma temperature structures lead- 2. Models of current layer and plasma heating ing to the same X-ray spectra; Li & Gan (2011) have shown 2.1. Current layer that the summation of two single-power X-ray continuum spectra from different sources looks like a broken-up spectrum. The Our study is based on the simple analytical model of the super- direct observations are necessary for understanding the basic hot turbulent-current layer (Oreshina & Somov 2000) with the physics and refinement theoretical models. heat flux by Manheimer & Klein (1975). They investigated the Our study is based on simple models of the SHTCL and heat transport in a plasma heated impulsively by a laser and ob- plasma heating by heat fluxes from it. We note that heat con- tained the following expression for the heat flux limitation: duction is not the only mechanism of plasma heating. Electrons, 3/2 accelerated in a current layer, can also make significant contribu- F = CF ne Te , ff ff tion. The relative role of each mechanism is di erent in di erent where flares (Veronig et al. 2005). Kobayashi et al. (2006) report about −10 −1 −3/2 purely thermal flare on 2002 May 24 that was observed in the CF = (0.22 − 0.78) × 10 erg cm s K . (1) 20–120 keV range by ballon-born hard X-ray (HXR) spectrom- −10 −1 −3/2 eter. Moreover, heat conduction seems to be dominant at pre- In this work, we take CF = 0.22 × 10 erg cm s K . impulsive phase; namely, it is a gradual rise phase that precedes Magnetic flux tubes with plasma move(from aboveand from the impulsive phase (Farnik & Savy 1998; Veronig et al. 2002; below) at low velocity v toward the SHTCL, penetrate into it, Battaglia et al. 2009). reconnect at the layer center, and then move along the x-axis Thus, our study is applicable primarily for the beginning of toward its edges, accelerating up to high velocity V (Fig. 1). Our the flare. This phase is of particular interest due to the absence model does not consider the internal structure of the layer. of accompanying processes like chromosphere evaporation and The input parameters are electron number density n0 outside others which can affect the reconnection region. They signifi- the layer, the gradient of the magnetic field h0 in the vicinity cantly complicate the overall picture, interpretation of observa- of separator, the inductive electric field E0, and the ratio ξ = tions, and theoretical models (for example, Liu et al. 2009). We By/Bx of the transverse and main components of magnetic field. would like to see ’pure’ current layers and their vicinities. The The mass, momentum, and energy conservation lows give the aim of our work is to clarify why we actually do not directly ob- following analytical expressions for the layer parameters: serve current layers in large solar flares and what we need to do for the half-thickness, it in the future. a = 1.7 × 106 × n−1/2, cm; As shown further, we need to consider multitemperature and 0 nonequilibrium effects in plasma. They are small; it is difficult to for the half-width, observe them. Often, it is difficult even to distinguish them from 1/4 −1 1/2 −1/2 instrumental effects (e.g. Holman et al. 2011,Kontar et al. 2011). b = 0.22 × n0 h0 E0 ξ , cm; Perhaps, this is one reason why many authors study mainly mean values of flare plasma. For example, Phillips et al. (2006) inves- for the electron temperature, tigated spectra obtained mostly during the flare decay phase “to T = 1.8 × 1012 × n−1/2 E ξ −1, K; minimize instrumental problems with high count rates and ef- s 0 0 fects associated with multitemperature and nonthermal spectral for the electron density, components”. Meanwhile, the authors note that the isothermal −3 model gives poor fits for flares near their maximum and propose ns = 7.9 × n0, cm ;

2 A.V. Oreshina and I.V. Oreshina: Current-layer diagnostics in solar flares for the magnetic field on the inflow sides of the current layer, Table 1. Two sets of parameters for the current layer model

1/4 1/2 −1/2 B0 = 0.22 × n0 E0 ξ , G; Set 1 Set 2 Input parameters for the inflow velocity, −3 10 10 n0, cm 10 10 h , G cm−1 1.2 × 10−6 1.2 × 10−6 v = 1.4 × 1011 × n−1/4 E1/2 ξ 1/2, cm s−1; 0 0 0 E0, CGSE 0.1 0.0425 ξ = B /B 0.005 0.005 for the outflow velocity, y x Output parameters a, cm 17 17 V = 1.7 × 1010 × n−1/4 E1/2 ξ −1/2, cm s−1; 0 0 b, cm 2.6 × 108 1.7 × 108 8 8 and for the heating time, which is the time for a given magnetic Ts, K 3.5 × 10 1.5 × 10 −3 × 10 × 10 tube to be connected to the SHTCL, ns, cm 7.9 10 7.9 10 B0, G 312 204 v, cm s−1 9.6 × 106 6.3 × 106 2b −11 1/2 −1 th = = 2.6 × 10 × n h , s. −1 × 8 × 8 V 0 0 V, cm s 2.4 10 1.6 10 th, s 2 2 Table 1 presents two sets of input parameters, which are specific to solar flares, and correspondingoutput values. We consider two input sets to emphasize the diagnostic capabilities of our models. The properties of these thermal waves are complex because In the Set 2, only the electric field is decreased when compared of hydrodynamic flows of emitting plasma and kinetic phenom- with the Set 1. This leads to decreasing the temperature Ts, while ena in it (Somov 1992, Ch. 2). Here, we consider the simpli- the heating time th is not changed. Both these parameters are fied model to emphasize the main effect, heat conduction. This important for our further computations of plasma heating by the process can be described by a simple equation (Somov 1992, SHTCL. p. 169): The input and output parameters agree with the following ∂ε observational and theoretical results. The electron number den- = − div F, (2) sity is consistent with statistical data from the overview of flares ∂ t by (Caspi 2010): (0.8 − 30) × 1010 cm−3. Fig. 5.5 of that work where F is the thermal energy flow density, and ε is the internal (page 70, right panel) presents densities of super-hot plasma ver- energy of unit plasma volume, which is the energy of chaotic sus maximumRHESSI temperaturesthat can be as high as 45-50 motion of particles. We consider only the electron heating be- MK. We note that they are estimated using an isothermal ap- cause the timescale of the thermal wave propagation is too short proach. The multitemperature method gives higher temperatures to heat ions. Indeed, the time of collisions for electrons is (e.g. of some portion of plasma. Somov 2006a, Sect. 8.31, p. 142) The electric field agrees with estimations by Somov et al. −1 2 3/2 (2008), Lin (2011): E ∼ 10 − 30 Vcm ≈ 0.03 − 0.1 CGSE. m (3 kB Te/me) 1 0 τ = e , ξ ee 4 The ratio belongs to the range of values where the current layer π ee ne (8lnΛ) 0.714 is stable (Somov & Verneta 1993). Magnetic fields up to 350 G are expected in coronal sources where me, ee, ne, and Te are the respective electron mass, charge, (Caspi 2010, Fletcher et al. 2011). We assume that these fields density, and temperature; kB is the Boltzmann constant; and 6 1/2 are in the magnetic-reconnection region, which is in the vicinity lnΛ = ln 9.44 × 10 Te/ne is the Coulomb logarithm for of a separator at heights up to ∼ 4 × 109 cm. At present, there 5 Te > 5.8 × 10 K. The ion collision time is are no reliable direct measurements of the coronal magnetic 2 3/2 fields. Our theoretical estimations show that magnetic fields mi (3 kB Ti/mi) 1 τii = , can achieve even higher values, up to 900 G, at the separator π e4 n (8lnΛ) 0.714 (Oreshina & Somov, 2009). i i We note that the thickness of the layer is very small, and where mi, ei, ni, and Ti are the respective ion mass, charge, hence, magnetohydrodynamics (MHD) is not a good approxi- density, and temperature. The characteristic time of temperature mation. We should solve the kinetic equation, but this signifi- equalizing between the electron and ion components in plasma cantly complicates the model. Instead, we use anomalous trans- is port coefficients, thus considering the deviation from the MHD. m m (3 k (T /m + T /m )3/2 Relaxation heat flux (Eq. 4) is also due to the deviation from the τ = e i B e e i i . ei 1/2 2 2 Maxwell distribution function. (6 π) ee ei ni (8lnΛ)

Therefore, we get τee ∼ 1 s, τii ∼ 50 s, and τei ∼ 1100 s for 2.2. Plasma heating the electron and proton components of plasma with temperature 8 10 −3 Let us now consider heating the surrounding plasma by the ∼ 10 K and density ∼ 10 cm . Thus, Coulomb collisions SHTCL. Due to conversion of magnetic-field energy, the layer between electrons in the vicinity of the SHTCL are much more temperature is very high, of about 3 × 108 K (Table 1). When a frequent than between protons; energy exchange between elec- magnetic tube comes into contact with the layer, plasma inside is trons and protons is very slow. The thermodynamic equilibrium heated. While the tube moves from the center to the edge of the is achieved first for electrons, then for protons, and finally for layer, a large amplitude heat wave travels along it. After the dis- the whole plasma. connection from the heating source, the tube continues to move Thus, the internal energy in Eq. (2) is in the corona. Now, only the energy redistribution occurs inside 3 it. ε = nekB T , (3) 2

3 A.V. Oreshina and I.V. Oreshina: Current-layer diagnostics in solar flares where T is the electron temperature. This state may not be solar flares. In the following computations, we use τ0 = 1.3 × reached in the conditions under consideration. We use Eq. 3 to 10−11 sK−3/2. define the effective temperature. In the frame of kinetic theory, it Thus, we solve both the heat conduction equation (2) and is a measure of the width of the electron distribution function in the equation for heat flux (4), which determine the tempera- velocity space (Somov 2006a, p. 171). ture T(r, t) and the heat flux F(r, t). For simplicity, we take 10 −3 Various types of heat flux in the SHTCL vicinity have been ne = const = 10 cm . We also assume that magnetic tubes are considered in detail by Oreshina & Somov (2011). The classi- straight and have a constant cross-sectional area. Equations (2) cal Fourier expression F = −κ∇T is not valid here, because it is and (4) are rewritten in one-dimensional form: derived for plasma in a state close to local thermodynamic equilibrium. The latter suggests that the characteristic timescale of 3 ∂ T ∂ F nekB = − , (6) the process is much longer than the electron collision time τee 2 ∂ t ∂ l and the characteristic length scale exceeds greatly the electron 5/2 ∂T 3/2 ∂F mean free path λee. These conditions are violated at the temper- F = − κ0 T − τ0 T , (7) ature of about 108 K. The thermal wave is too fast and has too ∂l ∂t steep front, so that the characteristic time of the wave propaga- where l is a coordinate measured from the layer along the tube. tion t = T is less than τ , and the characteristic length scale T ∂T/∂t ee We note that the SHTCL is very thin: the ratio of its thickness T −8 lT = ∂T/∂l is less than λee by 1 − 3 orders of magnitude. The to the width is less 3 × 10 (Table 1). Thus, we can neglect its corresponding plots are presented in the paper by Oreshina & thickness and solve Eqs. (6) and (7) for the following boundary Somov (2011). conditions. < < We also have showed that there is a generalization of the 1) For the time interval 0 = t = th, one end of the tube is classical Fourier law, which is better suited to solar flares. It is connected to the SHTCL with a temperature Ts, and the other described by Moses & Duderstadt (1977), who applied it to a end is far away and is maintained at a coronal temperature T0 = laser heated plasma. A general consideration of the problem can 106 K: be found in (Shkarovskii et al. 1969; Golant et al. 1977). The = = method is developed in a fundamental framework and is physi- T(l, t) l=0 Ts , T(l, t) l→∞ T0 . (8) cally substantiated. It is not restricted to slow time variations in the distribution function and thus better suited to the treatment 2) For t > th, the tube is disconnected from the layer and of rapidly varying thermal processes. The technique is based on does not receive any heat from it: the Grad 13 moment equations. The electron distribution func- F(l, t) = 0 , T(l, t) = T , (9) tion is expanded in terms of Hermite-Chebyshev polynomials. l=0 l→∞ 0

The thirteen moments of the distribution function have a clear ∞ physical meaning – namely, the density, velocity, temperature, Q = T(l, t)dl = const . (10) pressure tensor, and heat flux. The system of equations for them Z is derived from the Boltzmann equation and includes the mass, 0 momentum, and energy conservation laws, the equation for the pressure tensor, and, finally, the equation for heat flux: Equation (10) is the law of thermal-energy conservation in the tube after the separation from the layer. ∂F The problem was solved numerically. The obtained temper- F = −κ ∇T − τ . (4) ∂t ature distributions at consequent moments of time are shown in Fig. 2 by the black curves. The grey curves indicate the cor- Here, κ is the heat conductivity coefficient, τ is some character- responding distributions obtained for the classical Fourier law, istic relaxation time. without the relaxation, τ0 = 0. The top panel presents the ther- According to Braginskii (1963, v. 1, p. 183), mal wave for the first SHTCL model (see Table 1) and the bot- 5/2 3/2 tom panel for the second one. They differ in the temperature κ = κ0 T , τ = τ0 T , (5) 8 of the current layer: Ts = 3.5 × 10 K in the first case and −5 T = 1.5 × 108 K in the second one. We stop computations when where κ0 = 1.84 × 10 / ln Λ . s The technique for calculating the constant τ is described by the heat wave passes from the layer toward the chromosphere at 0 10 Oreshina & Somov (2011). the distance about 1.9 × 10 cm. Thus, the computations were interrupted at t = 9 s for the first model and at t = 20 s for the 2 k κ0 second one. For more details, see Appendix. τ = B , 0 2 3 ne C Comparison between the relaxation and classical heat con- F duction allows us to conclude the following. The relaxation lim- where CF is determined by equality (1). For the solar flare con- its the heat flux, making it more realistic. The amount of ther- ditions, we estimate mal energy Q, supplied to the tube during its contact with the

−11 −3/2 SHTCL, decreases by a factor of 10. The thermal wave slows τ0 ≈ (0.11 − 1.3) × 10 sK . down as compared to the classical heat conduction.For example, Fig. 2(a) shows that the front velocity is about 3 × 109 cm s−1, The corresponding relaxation time at the temperature of about whereas it is more than the speed of light in the first second of 108 K is tube contact with the SHTCL in the classical case. The wave = 3/2 ∼ − shape is also changed after the tube disconnection from the τ τ0 T T∼108 1 10 s . SHTCL. In the relaxation case, the temperature maximum is at These values are comparable to the time of tube contact with the the wave front and moves along the tube; a steep front is fol- SHTCL, th ≈ 2 s (Table 1). Therefore, the collisional relaxation lowed by a smooth decrease. In the classical case, the temper- of heat flux must be considered in models of plasma heating in ature is at a maximum at the point l = 0. The relaxation effect

4 A.V. Oreshina and I.V. Oreshina: Current-layer diagnostics in solar ﬂares y 2 s a 3 MT t1 l t2 K

6 01 , 01 8 a tn 2 x

T CL 1 Lz 9 z 2 6 9 Fig. 3. Schematic representation of the radiative source. CL – 0 current layer, MT – magnetic tubes, t1, t2 ... tn – time passed from 1 2 3 4 5 6 7 the tube penetration into the CL, l - coordinate along a tube, α – l, 1010 cm tube inclination to the CL plane.

bold lines in Fig. 3. The result is also applicable to the other 2 three quarters if we assume the process to be symmetrical. Let 2 s b all the tubes be in the (x, y)-plane and parallel planes within the layer length Lz. For simplicity, only the tubes in the (x, y)-plane are shown in Fig. 3. In the layer vicinity, all the tubes are mod- K 01 , 01 K eled by straight lines, which are inclined at the same angle α to

8 8 1 the (x, z)-plane. In the frames of this model, the magnetic tubes diﬀer from each other only by the time (t1, t2 ... tn) passed from

T their penetrating into the layer. 14 By definition, the emission measure of the radiative source 20 is 14 20 EM = n2 dV , (11) 0 Z e V 1 2 3 10 where V is the volume of the heated plasma. Assuming the tubes l, 10 cm are distributed homogeneously along the z-axis, Eq. (11) can be rewritten for x, y, and z coordinates: Fig. 2. Temperature distributions along a magnetic tube at consequent moments of time, which are obtained with (black curves) 2 2 EM = n dx dy dz = Lz n dx dy . (12) and without (grey curves) heat-flux relaxation. (a) Ts = 3.5 × Z Z Z e Z Z e 8 8 10 K, (b) Ts = 1.5×10 K. The numbers near the curves denote x y z x y the time (seconds) passed after the beginning of the tube contact The length Lz of the SHTCL is a new input parameter. In the with the SHTCL. 9 following computations, we use Lz = 10 cm. We replace coordinates x and y by t and l; t defines a tube and l is the point on it. decreases in time and the wave shape becomes almost classical These are related by the equalities (Fig. 2(b)). = + Comparison between the cases with different temperature of x V t l cos α, (13) the layer shows that the amount of heat Q in the tube is less by a y = l sin α, (14) factor of 2 in the second model. This result is expected because the temperature of the layer in the second model is 1.5 × 108 K where V is the velocity of tube movement along the x-axis. The instead of 3.5 × 108 K in the first model. We note that the tem- Jacobian of this transformation is perature behind the wave in the first seconds is close to the back- ∂x ∂x V cos α ground, and we have a solitary wave. In course of time, the tem- ∂t ∂l perature behind the wave is increased, and the wave becomes = = V sin α, ∂y ∂y similar to the classic wave about 20 seconds after the discon- 0 sin α nection from the layer (Fig. 2(b)). The role of the relaxation is ∂t ∂l decreased in time. and the expression (12) for the emission measure takes the form:

t lf 3. Observational manifestations of the current layer EM = L n2 (V sin α)dl dt = z Z Z e 3.1. Emission measure of the heated plasma 0 0 The set of all the magnetic tubes, heated by the SHTCL, is a t lf source of radiation. To compute the emission measure of this = L V sin α n2 dl dt . z Z Z e heated plasma, we consider a quarter of the tubes, shown by 0 0

5 A.V. Oreshina and I.V. Oreshina: Current-layer diagnostics in solar ﬂares

Here, lf is a coordinate of the thermal wave front. We note that 39 we have a unique relationship between coordinate l and temper- 10 a

ature T(t, l) for every ﬁxed time t (see Sec. 2). Hence, -1

t T(t,lf )

2 1 -3 EM = LzV sin α  ne dT dt = 9 Z  Z ∂T(t, l)/∂l  38 0 T(t,0) 

  K , cm 10   6 t  T(t,lf )  = LzV sin α  dem(t, T)dT dt. (15) 4 Z  Z  0 T(t,0)      DEM(T) 37 Function 10 2 1 2 dem(t, T) ≡ ne ∂T(t, l)/∂l l=l(t,T) 7 8 10 10 is the differential emission measure of a single magnetic tube of T, K unit cross-section. It is determined from the temperature distributions obtained in Sec. 2. The differential emission measure of the heated plasma is 20 t d EM b DEM(t, T) = = 4 L V sin α dem(t, T)dt. (16) -1 39 14 d T z Z 10 0 -3 Here, the multiplier 4 considers the entire volume of the heated ff 38 plasma. The di erential emission measure for models 1 and 2 K , cm 8 (Table 1) are presented in Fig. 4 (a) and (b). 10 The behavior of the function DEM(T) depends on the parameters of the layer and hence can serve as a diagnostic tool for it. In the first model, we see two local maxima. The narrow 37

DEM(T) 10 maximum at the layer temperature T ≈ 3.5 × 108 K is dueto the super-hot current layer and exists for every moment. The wide 2 maximum at T ≈ (1 − 3) × 107 K is gradually formed due to a smooth temperature drop behind the wave front after the tube 7 8 10 10 disconnection from the layer. In the second model, a remark- T, K able maximum is formed at T ≈ (1 − 3) × 107 K, too. We note that these temperature estimations are usually obtained from ob- Fig. 4. Differential emission measure of the plasma heated by the servations of super-hot plasma in solar flares (Caspi 2010). The SHTCL (a) for model 1 and (b) for model 2 (see Table 1). The local narrow maximum at the layer temperature 2 × 108 K is al- numbers near the curves denote time (seconds) after the flare most invisible, which makes direct observations of current layers start. difficult. Differential emission measure of high-temperature plasma Here, R (cm) is the distance from the Sun to the Earth, and in the M3.5 class flare was determined by Battaglia & Kontar −3 −1 (2012). The authors used the observations of the Solar Dynamic Pλ(T) (ergcm s ) is the line power, which is the energy ra- diated by a unit volume of plasma per one second. Observatory (SDO) in 94, 171, 211, 335, and 193 Å spec- 2 tral channels. Thus, they cannot see the super-hot plasma with The relative power Pλ/ne of 2131 spectral lines in X-rays T ∼ 108 K. They consider only the plasma along the line of (1.29–300 Å) are computed over the temperature range (3 × 2 −5 −1 104) − 109 K by Mewe et al. (1985). As we are interested in ob- sight: DEM = ne dl/dT, cm K . Nevertheless, the maximum of DEM(T) around (1 − 3) × 107 K seems to be common serving super-hot plasma, we have selected lines with a power > 8 xxvi for our both methods. maximum at T = 10 K. The most powerful lines are Fe Fig. 5 presents the integral emission measure as a function (1.78 Å and 1.51 Å) and Ni xxvii (1.59 Å). They are listed in of time: Table 2 (lines 1–3). The classification of the transitions is given 2 in usual form. (Pλ/ne)max is the maximum power value, reached EM(t) = DEM(t, T)dT . Z at the temperature Tpeak. Figure 6 presents the relative power of the lines as a function of temperature. The super-hot lines 1–3 Our estimations are consistent by an order of magnitude with are shown by black curves. RHESSI observations at the beginning of a flare (Liu et al. 2008; There are many other lines in the spectral vicinity of the Caspi & Lin 2010). super-hot lines 1–3, which are close in power but radiate at 8 the lower temperatures, Tpeak < 10 K (lines 4–10 in Table 2, 3.2. Spectral line intensities grey curves in Fig. 6). For example, the difference between the strongest lines of the two groups is λ1 − λ4 = 0.071 Å. Thus, The obtained differential emission measure allows us to compute it would be very valuable to have spectral resolution that allows intensities of spectral lines: one to separate these lines from each other. For fruitful SHTCL 1 P (T) observations, the spectral resolution must be better 0.01 Å. The I (t) = λ DEM(t, T)dT, photons cm−2 s−1. (17) λ 2 2 spacecraft-borne instruments, such as X-ray crystal spectrome- 4 π R Z ne

6 A.V. Oreshina and I.V. Oreshina: Current-layer diagnostics in solar ﬂares

Table 2. Spectral lines for super-hot plasma observations.

2 No. Ion λ, Å Transition (Pλ/ne )max, Tpeak, 10−25erg cm3 s−1 108 K 1 Fe xxvi 1.780 1s g2S 1/2 − 2p 2P1/2, 3/2 2.9 1.58 2 Fe xxvi 1.510 1s g2S 1/2 − 3p 2P1/2, 3/2 0.42 1.58 3 Ni xxvii 1.590 1s2 g1S 0 − 1s2p 1P1 0.19 1.00 4 Fe xxv 1.851 1s2 g1S 0 − 1s2p 1P1 3.5 0.63 5 Fe xxv 1.858 1s2 g1S 0 − 1s2p 3P2, 1 1.23 0.63 6 Fe xxiv 1.864 1s22p − 1s2p2, 1s2s2 1.10 0.34 7 Fe xxv 1.869 1s2 g1S 0 − 1s2s 3S 1 1.00 0.63 8 Fe xxv 1.570 1s2 g1S 0 − 1s3p 1P1 0.52 0.79 9 Fe xxv 1.510 1s2 g1S 0 − 1s4p 1P1 0.20 0.79 10 Fe xxv 1.790 1s2l − 2p2l (l = s, p) 0.18 0.79

5 4 1 a - 4 s 3 3- m 4 m 1

c 3 c 01 , 01 c gre 64 3

52 2 - 2 01 5 7 M 6 , 2 E E e 1 1 n/P 8 l 2 3 0 0 7 8 9 0 2 4 6 8 10 10 10 t, s T, K 8 s

b 3 1- 0.2

m 9

6 c 3 3- m gre c 01 , 01 c 52- 64 4 0.1 01 , 2 e n/P M 2

E E 10 l 0 0 7 8 9 0 10 10 10 5 10 15 20 T, K t, s Fig. 6. Relative line power as a function of temperature, drawn Fig. 5. Integral emission measure of the heated plasma (a) for using the table by Mewe et al. 1985.The numbersnear the curves model 1 and (b) for model 2. denote the spectral lines according to Table 2. The black curves 8 correspond to the super-hot spectral lines with Tpeak ≥ 10 K and the grey curves to the lower-temperature ones with Tpeak < 108 K. The weakest lines 3, 9, and 10 are also shown separately ters onboard the NASA Solar Maximum Mission and Japanese on the bottom panel. Hinotori spacecraft, have even better resolution, allowing one to 2 2 resolve Lyα lines from hydrogenic iron, 1s S 1/2 − 2p P3/2 and 2 2 1s S 1/2 − 2p P1/2 that occur at 1.778 and 1.783 Å. − − Moreover, the sensitivity of the detectors should be suﬃcient 2 photonscm 2 s 1 whereas line 3, for example, is characterized −2 −1 to record weak photon ﬂuxes. Figure 7 shows the intensities of by Iλ ∼ 0.01 − 0.1 photons cm s . the super-hot lines 1−3 (black curves) computed using Eq. (17). We note that the line power Pλ was calculated by Mewe et They are faint. Only the resonance line 1 has intensity Iλ ∼ 0.1 − al. (1985) under the assumption of ionization equilibrium; that

7 A.V. Oreshina and I.V. Oreshina: Current-layer diagnostics in solar ﬂares a 1.0 1 1.0 -2 1-

0.1 2 1 0.1 10 3 /P P

photons cm s cm photons 0.01

, 10

l 0.01 I 0.001 0 2 4 6 8 7 8 9 t, s 10 10 10 T, K 1.0 b 1 Fig. 8. Ratio of powers of the satellite line Fe xxv, 1.790 Å, (No. 10 in Table 2) to the resonance line Fe xxvi,1.780Å, (No. 1 -2 1- in Table 2) as a function of temperature, computed using data by Mewe et al. (1982). 0.1 2

(1.780 Å) is independent of ionization balance but is a function

photons cm s cm photons of temperature (Fig. 8). The role of the low-temperature satellite , xxv xxvi l line Fe relative to the the super-hot resonance line Fe I 0.01 3 decreases signiﬁcantly in increasing T. 10 Observations provide us intensities of spectral lines. Their interpretation is not easy, because it depends on the DEM(T) 0 5 10 15 20 (Eq.17). Veryoften the emission source is assumed to be isother- t, s mal. Then the intensity ratio equals the power ratio:

Fig. 7. Dependence of intensities of the super-hot spectral lines I10 P10 = = f (T) . on time (a) for set 1 of SHTCL parameters and (b) for set 2. The I1 P1 numbers near the curves denote the spectral lines, according to Table 2. Knowing the intensity ratio from observations, we can determine the plasma temperature using Fig. 8. Let us consider the results obtained in the frame of this is the ionization rate from the stage j to j + 1 is equal to the isothermal approach on the basis of our theoretical intensities. recombination rate from j + 1 to j: Figure 9 demonstrates computed Fe xxv/Fe xxvi intensity ratio for models 1 (a) and 2 (b). The plasma is very hot, which leads = α , ne N j Q j ne N j+1 j+1 to a low intensity ratio ≈ 0.022−0.042for the first model. Using Fig. 8, we obtain the temperature T ≈ 1.5 × 108. Itis a goodap- where Q j and α j are the rate coefficients of ionization and re- proximation, but the difficulty consists in the faint intensity I10 combination from the ionization stage j. The ionization state N j −2 −1 (fig. 7, grey curves): I10 ∼< 0.1 photonscm s . of an ion does not depend on the ion temperature Ti, but it is a function of the electron temperature T . We can estimate the In model 2, the plasma is relatively cooler and the ratio I10/I1 e is higher: 0.04 − 0.1. It corresponds to the temperature change characteristic times of ionization and recombination using the 8 7 formulas by Phillips (2004): from 1.5 × 10 K at the first few seconds to 5 × 10 K at t ≈ 20 s. At the very beginning of a flare, it is very difficult to register −2 −1 1 1 I10, because its intensity is less than 0.01 photonscm s . For τion = , τrec = . Q j ne α j ne example, Yohkoh observed these lines had integration time pe- riods that were too long, from 24 seconds to 20 minutes (Pike Q j and α j were computed using the method by Arnaud & et al. 1996). However, the inferred temperature at 20 s is about Raymond (1992). We do not present here the formulas, because 5×107 K and decreases in time. Thus, we do not obtain tempera- they are very cumbersome. For ions Fe xxv and Fe xxvi at the tures that are more than 108 K, even when they are present in the 8 10 −3 temperature Te ∼ 10 K and density ne ∼ 10 cm , we ob- current layer. Theoretically, we can observe these lines, but we tain τion ∼ τrec ∼ 100 s. Thus, the ionization equilibrium is not need a better telescope. Rough estimations are the following. To achieved during rapid plasma heating in the vicinity of SHTCL. detect at least one photon per second from the satellite line with −2 −1 That is why the intensities in Fig. 7 can be considered only as a the intensity of about I10 ∼ 0.01 photonscm s , we need the zero-order approximation, an upper limit. effective area of a telescope of about 100 cm2. This is problem- Meanwhile, Dubau et al. (1981) and Pike et al. (1996) show atic. We note, however, that even a comparison between data of that the Fe xxv satellites can be very useful despite they are Yohkoh and a new instrument having better (not the best) char- faint. They are formed at 1.79 Å due to transitions 1s2l − 2p2l acteristics (effective area, sensitivity of detectors, etc.) can give (l = s, p). The ratio of Fe xxv satellites to Fe xxvi Lyα lines the right trend for detecting the higher temperatures in flares: the

8 A.V. Oreshina and I.V. Oreshina: Current-layer diagnostics in solar ﬂares

nent is low, ∼< 0.05 − 0.1. This conclusion may be considered as a an indirect conﬁrmation for a SHTCL. 0.04 1

I 4. Conclusions / / We describe an analytical model of high-temperature turbulent- 10 I current layer. It is shown that the thickness of the layer is very 0.03 small, of about a few tens of centimeters, whereas the width and the length are about 109 cm. Its temperature can be as high as 3 × 108 K. This model allows us to compute the temperature distribution in the vicinity of the SHTCL at the beginning of a solar flare. The model considers only heat conduction, the main 0.02 ff 4 e ect. Its analysis shows that the classical collisional approach 0 2 t, s 6 8 is not valid under conditions of solar flare, because we deal with processes that are too fast and wave fronts that are too steep. 0.1 For a more accurate description, we considered the heat-flux relaxation (Oreshina & Somov 2011). It significantly changes the b form of the thermal wave, slows down the front velocity, and decreases the heat input in a magnetic tube during its contact with 0.08 the SHTCL. 1

I Using the obtained temperature distributions in the vicinity

/ / of the layer, we compute the observable characteristics of the 10 I plasma in solar ﬂares: diﬀerential and integral emission measure

0.06 and intensity of some spectral lines. These values are computed for two sets of input parameters for the SHTCL models. The difference between the results demonstrates the diagnostic capabilities of our method, which creates a theoretical basis for de- 0.04 termining characteristic values of the current layer from future observations. 0 5 10 15 20 t, s Particular attention is given to lines of high-charged ions Fe xxvi (λ = 1.780 Å), Fe xxvi (λ = 1.510 Å), and Ni xxvii > 8 Fig. 9. Ratio of intensities of the satellite line Fe xxv, 1.790 Å, (λ = 1.590 Å), which are formed at T = 10 K. Their obser- (No. 10 in Table 2) to the resonance line Fe xxvi,1.780Å, (No. 1 vations can be very useful for detecting SHTCLs at the pre- in Table 2) (a) for model 1 and (b) for model 2. impulsive stage, when the heat conduction is usually a dominant process. The later effects, such as plasma heating by accelerated particles, magnetic traps, chromospheric evaporation, etc., can significantly complicate the data and their interpreta- better characteristics, the higher inferred temperatures in large tion. However, the difficulty consists in the weak intensities of flares. spectral lines at first seconds. Thus, observation of multitemperature and nonequilibrium The ratio of the Fe xxv satellite line (λ = 1.790 Å) to the effects in flare plasma are difficult at present, because they are Fe xxvi resonance line (λ = 1.780 Å) is of particular interest, very small. Sometimes, they are not distinguished from instru- because it does not depend on the ionization balance. This ra- mental effects (e.g. Holman et al. 2011, Kontar et al. 2011). tio is often used to determine the plasma temperature using an To eliminate this ambiguity or because of different aims of the isothermal approach. We show that this technique gives high work, the majority of observers study mean plasma character- temperatures (> 108 K) only at the very beginning of a flare istics, which are much more reliable. For example, Phillips et (about seconds) and then the inferred temperature rapidly deal. (2006) mostly analyze the flare decay phase, when the tem- creases. Meanwhile, early observations are difficult because of a perature is slowly varying “to minimize instrumental problems very weak Fe xxv satellite line. We expect one photon per cm2 with high count rates and effects associated with multitemper- during a few tens of seconds. The integration time of actual de- ature and nonthermal spectral components”. The authors, how- tectors is about some tens of seconds and more. It can be one of ever, note that the isothermal model gives poor spectral fits near the causes why the inferredtemperaturesdo not exceed 5×107 K, the flare maximum. They write that it can be due to deviation whereas plasma with T ∼> 108 K is present in a flare. of the emitting plasma from the isothermal state. Thus, we can The other reason why these temperatures are not reported is sometimes see evidence of the multitemperature nonequilibrium that the isothermal approach for interpreting observations (e.g., plasma, which should be studied very carefully. Phillips 2004, Phillips et al. 2006, Holman et al. 2011, Kontar A more accurate approach than an isothermal one is de- et al. 2011) gives volume-averaged temperatures. It is a good scribed by Doschek & Feldman (1987). They determine tem- method to determine the mean characteristics of flare plasma. perature structure of the radiation source using 16 spectral lines Thus, it gives maximum temperatures of about 30 − 50 MK. of ions S xv, Ca xix, Ca xix, Fe xx, Fe xxv, Fe xxvi, and Ni xxvii. However, Phillips (2004), for example, writes at the end of The analysis demonstrates the possible existence of a super-hot Sect. 4: “An isothermal plasma is implicit in equations (1), (2), component that extends up to several hundred million degrees. and (4) and in the discussion so far, but more realistically the The fraction of the emission measure of the super-hot compo- plasma has a nonthermal temperature structure, describablebya 2 nent to the emission measure of the lower temperature compo- differential emission measure (DEM), φ(Te) = Ne dV/dTe... .”

9 A.V. Oreshina and I.V. Oreshina: Current-layer diagnostics in solar ﬂares

We emphasizethat the aim of our work is to find theevidence Somov, B.V. 1992, Physical Processes in Solar Flares (Kluwer Acad. Publ., of the SHTCLs. It is an effect of the second order as compared Dordrecht) to the mean plasma characteristics, because a percentage of the Somov B.V. 2006a, Plasma Astrophysics, Part I, Fundamentals and Practice > 8 (Springer Science + Business Media, LLC, New York) super-hot (T ∼ 10 K) plasma in flares is small. As we can see Somov, B.V. 2006b, Plasma Astrophysics, Part II, Reconnection and Flares from Fig. 4 of our paper, the differential emission measure has (Springer Science + Business Media, LLC, New York) a main maximum at the temperatures of about 10-30 MK. That Somov, B.V., & Verneta A.I. 1993, Space Sci. Rev., 65, 253 is, there is much more plasma in flares at the temperature of Somov, B.V., Oreshina, I.V., & Kovalenko, I.A. 2008, Astronomy Letters, 34, 10-30 MK. These values agree well with the isothermal estima- No. 5, 327, DOI: 10.1134/S1063773708050058 8 Sui, L., & Holman, G.D. 2003, ApJ, 596, L251 tions. Only a small peak of DEM(T) at T ∼> 10 K is due to the Sweet, P.A. 1958, Nuovo Cimento Suppl., 8, Ser. X, 188 SHTCL. Finding it from observations is not an easy task. The Syrovatskii, S.I. 1963, Soviet Ast., 6, 768 isothermal approach for interpreting observations is very fruit- Syrovatskii, S.I. 1966, Soviet Ast., 10, 270 ful to determine the general, mean characteristics of the flare Veronig, A., Vrsnak, B., Temmer, M., & Hanslmeier, A., 2002, Sol. Phys., 208, 297 plasma, but it is not appropriate for our purposes. We have to Veronig, A., Brown, J.C., Dennis, B.R. et al. 2005, ApJ, 621, 482 consider namely multitemperature and nonequilibrium effects. Wang, T., Sui, L., & Qiu, J., 2007, ApJ, 661, L207 Therefore, for the SHTCL diagnostic, the observations in the Yokoyama, T., & Shibata, K., 2001, ApJ, 549, 1160 X-ray range of 1.5–1.8 Å with high spectral resolution (better than 0.01 Å) and high temporal resolution (seconds) are needed. It is also very importantto interpret the observationsusingamul- Appendix A: Length of magnetic-field tubes titemperature approach. The maximal length of magnetic-field tubes (1.9 × 1010 cm) is Acknowledgements. The authors thank an anonymous referee for interesting estimated by the following manner. Let us consider a Bastille questions and comments, which have improved the text. Day Flare as an example. A magnetic-chargesmodel of this flare was described by Oreshina & Somov (2009). References Fig. A.1 presents MDI/SOHO magnetograms for AR NOAA 9077, model magnetograms for the same AR, and a separator. Arnaud, M., & Raymond, J. 1992, ApJ, 398, 394 The energy release could be explained by reconnection on two Battaglia, M., & Kontar, E.P. 2012, ApJ, 760, 142 Battaglia, M., Fletcher, L., & Benz, A.O. 2009, A&A, 498, 891 separators: one connects the null points X1 and X2, while the Braginskii, S.I. 1963, Questions of the Plasma Theory (Atomizdat, Moscow) other connects the null points X1 and X3 in the plane of the (in Russian) charges. Caspi, A., 2010, PhD thesis (University of Califorinia, Berkley) Both separators have approximately the same length. As it http://arxiv.org/ftp/arxiv/papers/1105/1105.1889.pdf Caspi, A., & Lin, R.P. 2010, ApJ, 725, L161 can be seen from the figure, the distance between the endpoints Chen, P.F., Fang, C., Tang, Y.H., & Ding, M.D. 1999, ApJ, 513, 516 of each separator is about 56 pixels of MDI, that corresponds Doschek, G.A., & Feldman, U. 1987, ApJ, 313, 883 to 8.2 × 109 cm. 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10 A.V. Oreshina and I.V. Oreshina: Current-layer diagnostics in solar ﬂares

MDI 14.07.2000 08:00 UT 22N01W a) G 1800 1400 1000 600 200 -200 -600 -1000 -1400 -1800 -2200 -2600 -3000 40 ” b)

X1 60 n2 n1 s1 s3 y 40

20 s2 X2 X3 0 0 20 40 60 80 100 120 140 x

c) MFL SEP

X1 X 80 2 120 140 60 100 80 y 40 FR 60 x 20 40 20 0 0 Fig. A.1. (a) MDI/SOHO magnetograms for AR NOAA 9077; the magnetic field strength on the isolines is shown in gausses. (b) Model magnetograms for the same AR with the same isolines: n1 and n2 are the north-polarity sources; s1, s2, and s3 are the south-polarity sources, and X1, X2, and X3 are the null points. (c) Separator X1X3 (SEP, dashed black line), magnetic field lines in its vicinity (MFL, black lines), and flare ribbons (FR).