The results of VLA observations at frequencies 4.5, 8.0 and 15.1 GHz of a

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1 THE SOLAR ATMOSPHERE ABOVE SUNSPOT WITH RING MICROWAVE SOURCE E. Ya. Zlotnik Institute of Applied Physics, Uljanov St., 46, N.Novgorod, Russia S. M. White, M. R. Kundu Department of Astronomy, University of Maryland, College Park, USA Abstract The results of VLA observations at frequencies 4.5, 8.0 and 15.1 GHz of a source of microwave radiation, associated with the sunspot NOAA 7789 on October, 15, 1994, are presented. The ne structure of the source, which is a ring or horseshoe structure in intensity and polarization at frequencies 4.5 and 8.0 GHz, is discussed. It is shown that the features observed as well as weak source at 15.1 GHz can be explained by thermal cyclotron and bremsstrahlung mechanism if the magnetic eld is approximated by vertical dipole, buried under the photosphere, but the spatial distributions of kinetic temperature and electron density in the atmosphere above the sunspot considerably dier from the standard model. A two-dimensional source model (the dependencies of parameters on the height and distance from the center of the sunspot), which ts the observations at above frequencies, is evolved. The principal physical result is that the data observed are explainable by the presence of dense cool plasma in the atmosphere over the center of the umbra. 1. INTRODUCTION The microwave radiation of sunspot-associated sources (the so-called slowly varying component of solar radio emission or s-component), is being studied during more than thirty years. Up to now the bulky observational data on the sources, induced by magnetic elds of sunspots, such as frequency spectra, polarization, directivity, etc. (see, for example, [1-16] and references there), is available. All these data arm the idea [17-18], suggested in 196, that the enhanced radiation at the centimeter wavelengths appears mainly due to the cyclotron (magneto-bremsstrahlung, gyroresonance) radiation of thermal electrons in the atmosphere of an active region above the sunspot, and the bremsstrahlung mechanism can be ecient at the wavelengths shorter than -3 cm and longer than 0{30 cm. Naturally, the observed characteristics depend on physical conditions (three-dimensional distribution of the magnetic eld, kinetic temperature and electron density) and the location of a source on the solar disk. The characteristics of s-component for dierent models of a source have been calculated in many works (see, for example, [17-6,4-5,7,9-11,13-15] and references there), and the parameters of the atmosphere over an active region (extrapolated from the optical and UV observations of the photosphere and the chromosphere to the transitional region and lower corona) tted observational radio data rather well. On the other hand, the relative simplicity of the approximate expressions relating the frequency of the observed radio emission with the magnitude of magnetic eld and the brightness temperature with kinetic temperature of a source in the framework of the cyclotron mechanism, allows one to retrieve physical conditions in the 1

2 sunspot atmosphere from the observed characteristics. Thus, recording the sunspotassociated microwave radiation provides an easy diagnostics of the solar atmosphere above a sunspot, which is a region not observable by optical methods. Up to now the mentioned diagnostics was applied mainly to the magnetic eld [4,7,9,1-14,7-9]. The information about the distribution of electron density above a sunspot is not so easy to derive from the data on the s-component. As for the kinetic temperature, it is quite possible to nd out it from the observed frequency spectrum of radiation, since the dependence of the brightness temperature on wavelength copies in some scale (under the assumption of monotonous changing of parameters with height and optically thick gyroresonance layers) the dependence of the kinetic temperature on height. However, up to the moment there have been calculated in detail only models with the uniform temperature distribution over a source (plane parallel model), the observed radio brightness distributions being explained by the inhomogeneous distribution of the magnetic eld in the plane parallel to the solar surface. In this case a peculiar structure like a ring or a horseshoe which is one of the interesting and frequently recorded features of sunspot-associated sources (see,for example, [5-6,13-16] was stipulated by the dependence of the optical thickness of gyroresonance layers on the angle between the magnetic eld and the line-of-sight (enhanced transparency along the magnetic eld lines) [1,17-19,1-6]. The further study and comparison of ring-structure observations and model calculations showed that the only eect of the low opacity of gyroresonance layers can hardly explain all the varieties of recorded proles and polarization in microwave sources. Therefore as a second reason of brightness depression in the sunspot center, the presence of cool plasma above the sunspot umbra was suggested [5-7,10-16]. However in the cases described in literature, basing on only the radio data it was impossible to choose which of the two mechanisms is responsible for the brightness fall in the center of the source, so the observational data from other frequency bands (EUV or soft X-ray) were necessary to be attracted in order to understand the origin of ring-structure. Besides, distinguishing of two above eects is prevented as a rule by the complicated structure of an active region or its location near the limb. In this sense, the most appropriate sunspots are those which are isolated ones and are located close to the center of the solar disk. It is just such the source which is considered in the present paper. We give the results of comparison of high-dynamic range VLA observations of a ring structured sunspot-associated source with various atmospheric models. Studying of images at three frequencies in two opposite polarizations (with taking into account the marked symmetry of the source and its close-to-center location) allowed us to conclude that in this case the ring-structure cannot be explained only by the eect of enhanced transparency of gyroresonance layers along magnetic eld. Moreover, it is for the rst time, that basing only on radio data we can surely conclude that in this source the presence of more dense cool plasma above the center of the umbra, than in surrounding atmosphere, is necessary to provide observational features. In this connection, a two-dimensional electron density and temperature model (dependence on the height and the radial distance) of the atmosphere above the sunspot is developed.

3 . THE RESULTS OF OBSERVATIONS [to be written] The source was located above the bipolar group NOAA 7789 (October 15, 1994) with a big leading sunspot pretty distanced from the other part of the group. The coordinates of the group on the solar disk were 3 N and 11 W. Fig.1 shows an overview of the whole active region: soft X-ray image, H image and radio images at two frequencies (4.5 and 8.0 GHz) in two opposite circular polarizations. RCP and LCP (right and left circular polarizations) correspond to ordinary and extraordinary modes (o- and s). The bright radio source above the leading sunspot is under consideration. It is this source that is presented large scaled in Fig.. The sources at 4.5 GHz (a,b) and 8 GHz (c,d) have the form of an almost exact circle and are found to have a ne structure as a dark spot in the middle, surrounded by a more bright ring with two still brighter details. A horseshoe structure is seen in o-mode image at 8.0 GHz. High frequency source at 15.1 GHz shown in Fig. e,f, is of quite dierent structure: it is much weaker than those at less frequencies and seems to "ll in" the depression in the middle of the 4.5 GHz source. The radio emission of the whole source is polarized as an extraordinary wave, and the distribution of polarization over the sunspot also has a ring form: the region with a relatively high degree of polarization in the center of the sunspot is surrounded by a ring with a lower polarization, which in its turn is surrounded by a ring with a higher polarization at the outer edge of the sunspot. A remarkable feature of the whole source was its stability during many hours of observations. In order to get quantitative knowledge on the source we made two cuts of images(shown in Fig. a) which intersect maximum (cut I) and minimum (cut II) of the source brightness. One-dimensional brightness distributions along the chosen diameters are shown in Fig. 3; these distributions are compared with model computation in this paper. It should be pointed out that the source at 4.5 GHz frequency is brighter and larger, than the source at 8.0 GHz. At the same time, at each frequency the source of an extraordinary emission is larger and brighter than that of the ordinary one. It's interesting that the maximum brightness temperature of the radio emission of the ordinary wave, for example, at frequency 4.5 GHz, is markedly less than the minimum temperature of the extraordinary emission in the center of the sunspot (compare two points, marked with crosses in Fig.3a). This fact, as we'll see below, is the most important for choosing the model of the source and explaining depression of the brightness temperature in the center of the source at 4.5 and 8.0 GHz. Brightness distributions at 15.1 GHz (Fig.3 e,f) demonstrate quite dierent structure of the source, namely, location in the middle of the sunspot and the lower intensity. 3. THE MECHANISM OF RADIO EMISSION Bearing in mind the experience of numerous investigations (observational and theoretical) of sunspot-associated sources, it is easy to conclude that the 4.5 and 8.0 GHz source as a whole is a typical source of s-component and can be explained as a result of thermal cyclotron emission in the transitional region from the chromosphere to the corona. In the framework of approximate scheme of the cyclotron mechanism and the standard model of an active atmosphere region (with magnetic eld decreasing with the height, and kinetic temperature increasing with the height), when an ordinary wave at frequency f is generated, mainly, in a thin gyroresonance layer, where f = f B, and an extraordinary one occurs in the layer f = 3f B (f B = eb mc is the electron gyrofrequency, e and m are the mass and charge of electron, c is the 3

4 speed of light, B is the magnetic eld strength), one could explain the typical features of the source, such as polarization in the sense of an extraordinary mode, the brightness temperature, which is higher at frequency 4.5 GHz than at 8.0 GHz, size of the source increasing with decreasing frequency, etc. High frequency source at f = 15:1 GHz is expected to be of bremsstrahlung origin. Below both mechanisms are taken into account Cyclotron mechanism Since in the coronal plasma with temperature less than few million degrees the harmonics with numbers s > 4 are optically thin and cannot (according to many calculations) give noticeable contribution to escaping radiation, we consider the gyroresonance layers corresponding to s = 1; ; 3; 4 harmonics for the ordinary wave and s = ; 3; 4 for the extraordinary one. In the latter case the rst harmonic s = 1 should not be taken into account because the reecting point of an extraordinary wave v = 1 p u; (1) where v = f p =f = N ee mf ; u = f B =f () is located in the corona higher (i.e. closer to the observer), than the layer s = 1 (u = 1), while the ordinary mode is reected by the layer v = 1; (3) which is not connected with gyroresonance layers. In the model calculations below the following approximate expressions for optical depths of gyroresonance layers with numbers s = 4 are used [1-3,17-19,1]: where js ' ss e s s! mcf s T L B N e F js () ; (4) F js () = sin s (sin + s cos p sin 4 + 4s cos ) sin 4 + 4s cos sin p sin 4 + 4s cos ; (5) T = T 1 ; (6) mc is the angle between magnetic eld and line-of-sight, T and N e are kinetic temperature and electron density, L B = j( 1 db B dl )j 1 is the typical scale of the magnetic eld B along the line-of-sight, is the Boltzmann's constant. The upper and lower signs in (5) correspond to extraordinary (j = x) and ordinary (j = o) waves. The optical depth of an ordinary wave at the rst harmonic is described as follows [3]: o1 ' e m c 3 f NT L sin 4 (1 + cos ) B : (7) (1 + cos ) 3 As compared with more exact relations, given in [1,3], the expressions (5), (7) look simpler because of the used approximation v1, in which the refractive indices n j of the both modes are close to unity (the subsequent calculations proved the 4

5 validity of this approximation). However, we do not use quasi-longitudinal approach of the theory of radio wave propagation in the magneto-active plasma, though it is valid in the wide range of angles and essentially simplies calculations. The point is that for the angles > 60 the dierence between quasi-longitudinal approximation and more exact formulas (5), (7) can become considerable [30]. Note that relations (4), (7) are deduced by the integration along the line-ofsight in the approximation that plasma parameters are slowly varying inside the gyroresonance layer with geometrical thickness L js L B T cos : (8) The value L js is smaller than the typical scale of magnetic eld (L B ), electron density (L N ) and temperature (L T ) under usual conditions of the atmosphere over a sunspot, when L B < L N ; L T. However, if small scale inhomogeneities (for example, coronal loops, neutral current sheets, etc.) are present, the integration should be made more accurately [30-3]. The brightness temperature of the cyclotron emission escaping the corona without taking into account bremsstrahlung is determined by the sum of contribution of all layers including reabsorption in the every next layer: T jb = 4X s=m ft j;s 1 exp( js ) + T s [1 exp( js )]g ; (9) where m = 1 and m = for ordinary and extraordinary waves correspondingly, T o;0 = T x;1 = 0, T s is the kinetic temperature at the intersection point of the lineof-sight and a gyroresonance layer s. 3.. Bremsstrahlung mechanism Unlike the cyclotron radiation escaping from discrete layers, the bremsstrahlung occurs in the whole depth of solar atmosphere. Therefore when calculating the bremsstrahlung optical depth j and brightness temperature j it is necessary to take into account plasma non-uniformity: Z j = j dl; (10) Z j = T e j j dl: (11) Using appropriate relations from [3,1], bremsstrahlung absorption coecient j may be written in the form: s e 6 N Q j = 4 c(t m) 3= n j T 3= f g j(; u; v); (1) where dependence on magnetic eld is determined as follows: q g j (; u; v) = q u sin 4 + 4u(1 v) cos [u sin + (1 v) ] u sin 4 q u sin 4 + 4u(1 v) cos [(1 v) u sin u sin 4 + 4u(1 v) cos ] (13) Coulomb logarithm Q and refraction index n j are the following: ( ln 0T Q = ; N 1=3 if T < K; ln 103 T =3 ; N 1=3 if T > K; (14) 5

6 v(1 v) n j = [1 (1 v) u sin qu sin 4 + 4u(1 v) cos ]1= : (15) The equations (10)-(11) are computed along the line-of-sight in the limits between the reection points (1) or (3) (or photosphere if electron density is not enough to provide reection in the atmosphere) and the rst gyroresonance layer arising from under the photosphere, then in consecutive order between two adjacent layers and, at last, from the layer s=4 up to escaping from the corona Brightness temperature Transfer equation gives the following solution for the brightness temperature of the total (cyclotron and bremsstrahlung) radiation escaping from a gyroresonance layer of number s: T bj (s) = [T bj (s 1)e j (s 1;s) + j (s 1; s)]e js + T (s)(1 e js ); (16) where T bj (s 1) is brightness temperature of radiation escaping from the preceding layer of number s 1, the integrals j (s 1; s) and j (s 1; s) are taken between the levels s 1 and s, and kinetic temperature T (s) and optical depth js are calculated at the point of intersection of the line-of-sight and gyroresonance layer. If the layer s is the rst one above the photosphere, T bj (s 1) = 0 and the lower limit in the integrals (10)-(11) is the photospheric level. The resulting brightness temperature of the radiation escaping from the source is equal to: T bj = T bj (4)e (4;1) + (4; 1); (17) (4; 1) and (4; 1) give the input of bremsstrahlung in the corona above the last meaningful gyroresonance layer s = 4. Going ahead, we note that bremsstrahlung does not contribute remarkably into the radiation at 4.5 and 8.0 GHz, but it is a principle one at 15.1 GHz. Besides, in the models under consideration, at higher frequencies 8.0 and 15 GHz almost everywhere in the source strong inequality v 1 is valid, gyroresonance layer s = 1 is under the photosphere, so the approximation n j ' 1 for calculating js at these frequencies proves to be correct. As for f = 4:5 GHz, the reection points for x- mode are located in the atmosphere (between layers s = 1 and s = ), so in principle the dierence between n j and the unity must be taken into account, but the optical depth of gyroresonance layers s = ; 3, located higher in the corona is so great, that not accurately calculated cyclotron radiation from lower layers does not matter. Note also, that generally speaking it is not necessary to divide the solution of transfer equation into two separated inputs of discrete gyroresonance layers and distributed bremsstrahlung. Moreover, it seems to be more reasonable to count not from the photosphere but to begin in the rareed corona and move inside[15]. However, for our purpose to guess two-dimensional distributions of kinetic temperature and electron density (dependence on the height and the distance from the sunspot center) and to vary these distributions in order to t observational data, we must know separate contributions of both mechanisms, as well as relative inputs of dierent gyroresonance layers. 6

7 4. THE MODEL OF THE SOURCE 4.1. Magnetic eld The circle-shaped isolated source situated almost in the center of the solar disk, implies that its magnetic eld can be described as a eld of unipolar sunspot, induced by a vertical dipole! p buried under the photosphere:! B = 3(! p! R ) R 5! p R 3 (18) (! R is the radius-vector to a current point from the dipole). The system of coordinates convenient for modelling is chosen in the following way (Fig. 4a). The origin of coordinates O is in the sunspot center at the photosphere, the height axis h is oriented along the solar radius, the x and y axis are in the perpendicular to h axis surface, x and y being along corresponding parallel and meridian. The cylindrical system with axis % along a selected diameter of the sunspot, axis ' with angle ', counted from y axis and h axis is also used for description of cylindrically symmetrical magnetic eld. Substituting vectors! p (0; 0; p) and! R (r; 0; h + d) into (18), we obtain the following components of magnetic eld in a current point (x; y; h) or (%; '; h), created by dipole buried under the photosphere at a depth d = OP (see Fig. 4a): B r = B 0d 3 B h = B 0d 3 3%(h + d) [(h + d) + % ] 5=; (19) (h + d) % [(h + d) + % ] 5=; (0) where B 0 is magnetic eld in the center of the sunspot at the photosphere (% = 0,h = 0). Correspondingly, the value of magnetic eld is described by relation: q jbj = B 0d 3 4(h + d) + % [(h + d) + % ] ; (1) and gyroresonance layers f = sf B are dened in the following way: [(h + d) + % ] q 4(h + d) + % = sf B0d3 f () (f B0 = eb 0 =mc) and are shown in Fig. 5 for three frequencies under consideration. The values B 0 = 500 G and d = km are taken from optical observations [SGD]. In order to perform calculations along the line-of-sight, perpendicular to the surface of the Sun in the center of the solar disk, geometry of the source must be considered more thoroughly. Let us obtain the axis projections of the unit vector!! l = OL directed along the line-of-sight in the considered system. From Fig. 4a, where the hatched area is in the solar equator plane, the angles and are the sunspot longitude and latitude,! l -projections in the Cartesian coordinate system have a form: l x = cos sin ; l y = sin ; l z = cos cos (3) 7

8 The geometry of the sunspot location on the disk is shown in Fig. 4b. The longitude of the source is = 11 (W ), the latitude = 3 (N); the angles between the chosen cuts and solar parallel (y axis) are ' I = 15 ; ' II = 45. An angle between magnetic eld! B and line-of-sight! l is found from relation (! B ;! l ) = B cos : cos = B r B (sin 'l x+cos 'l y )+ B hl h B = 3%(h + d)(sin ' cos sin cos ' sin ) + [(h + d) % ] cos cos q [4(h + d) + % ][(h + d) + % ] (4) The scale L B of magnetic eld change along the line-of-sight (a term included in gyroresonance optical depth (4),(7)) is also found using (0),(): L B = j 1 B db dl j 1 = [4(h + d) + % ][(h + d) + % ] 3%[% + 5(h + d) ](sin ' cos sin cos ' sin ) + 1(h + d) 3 cos cos (5) An integration along the line-of-sight in (10)-(11) is reduced to integration over the height h with replacing dl by dh= cos cos and substituting a current coordinate % in the form: tan cos ' % = r + h cos where r is the distance from the center of the sunspot to the point of intersection of the line-of-sight and the chosen cut at the photosphere (% axis). Correspondingly, the coordinates (% s ; h s ) of the point of intersection of the line-of-sight with gyroresonance layers (where the optical depth js (4),(7) and the brightness temperature T b (9) of cyclotron radiation are calculated) are dened by Eq. () and the following relation: % s = r h s tan cos ' cos Relations (3)-(7) are valid for arbitrary location of a sunspot on the solar disk (angles ; ') and chosen cut (angle '). In case of our group the latitude = 3 N is so small, that taking into account displacement of the sunspot from the solar equator results in negligible corrections, so below we put sin = 0; cos ' 1. At the same time, for quantitative tting the observational proles and understanding the relative part of two eects resulting in ring-structure, the shift of the source from the central meridian plane to the west in 11 must be taken into account. 4.. Kinetic temperature and electron density The bulk of the model distributions of kinetic temperature T and electron density N e in the atmosphere of an active region, suggested earlier for interpretation of s- component, were uniform over the sunspot (one-dimensional), i.e. T and N e were dependent only on the height h above the photosphere. A nonuniform distribution of the brightness temperature over the spot, including structures of a ring or a horseshoe kind, were explained by the two-dimensional dependence (on % and h) of the magnetic eld and the angle between the magnetic eld and the line-ofsight. From (5) we see that the optical depth drops when decreases, so in the center of a unipolar sunspot with uniform distribution of the temperature and the electron density over its surface we could expect (with the reasonable parameters) the depression of the brightness temperature (so-called windows of transparency [1]). 8 (6) (7)

9 The second possibility to explain the depression in the source center is that the plasma above the sunspot umbra is cooler than at its periphery [5,7,9,1-15]. In this case the brightness temperature distribution, which follows to some extent the kinetic temperature distribution, also may look like a ring with minimum in the center. It should be emphasized, that, as a rule, in cases described in the literature, basing only on radio data it was dicult to decide whether the ring-like structure is a result of the enhanced transparency of the gyroresonance layers or of the presence of a relatively cool plasma above the sunspot [5,13]. Both eects are considered below to explain the observed source Uniform distribution. First, let us try model observational proles by uniform distribution of the electron density and kinetic temperature over the source surface, that is assume N e and T to be dependent only on the height. As an example, we will consider the tting of one-dimensional radio brightness distributions along the cut I. The dependence of the kinetic temperature T on h will be approximated with the analytical function: T = T ch + (T c T ch ) exp h h ch h 1 + exp h h ch h ; (8) according to which T increases from the chromospheric T ch to the coronal T c value in the transitional layer, with thickness h and the height h ch. The electron density gradient is assumed to be described by an exponential law: N e = N 0 10 ( h=h N ) : (9) Relations (8) and (9) demonstrate well known behavior of the electron temperature and density in the transitional layer from the chromosphere to the corona. The search of six parameters T ch, T c, h ch, h ch, N 0 and h N is a process of selection a model to t observational data. As a rst step the following typical set of parameters is discussed: T ch = 10 4 K; T c = ; K; h ch = ; cm; h = 1; cm; (30) N 0 = cm 3 ; h N = cm: (31) The corresponding dependencies are shown in Figs.6 a,b. The calculated distributions of the cyclotron radiation brightness temperature for the both normal waves at frequency 4.5 GHz are presented in Fig.7, where r is the projection of the line-of-sight on axis % along the chosen cut (see Fig.5). The calculated proles shown by dashed lines, are compared with observational ones shown by solid lines in Fig.7a, and with kinetic temperature distributions T s along the gyroresonance layers s = 1 4. The plots of x- and o-mode optical depths along gyroresonance layers are presented in Fig.7c,d. It can be seen from Fig.7a that the chosen model qualitatively ts observations in general: it provides maximum value of brightness temperature, observed size of the source (and the dierence in sizes for x- and o-modes), depression in the middle of the source. It is these properties which were taken into account when choosing the values (30)-(31). Fig.7b-d allow us to judge about a relative contribution of dierent 9

10 gyroresonance layers into resulting radio emission. For the layer s = is optically thick everywhere but is visible only in the center of the source where the layers s = 3 4 are transparent; the layer s = 3 can be seen at intermediate distances from the center, while s = 4 becomes optically thick at the outer edge. As for o- mode, it originates mainly from the layers s = 3 at the periphery of the source, but in its center all the layers s = 1 4 are optically thin, so there is a deep fall of brightness around r = 0. Consideration of model calculation at frequency 8.0 GHz (Fig.8a) shows the similar picture: observed and calculated proles qualitatively coincide, and so do the sizes of the source for two modes. However at frequency 15.1 GHz there is a great dierence between observed and calculated proles (Fig.8b): expected brightness temperature exceeds observed one in an order of magnitude, while the intensity of o-mode is too low. Note that in the model considered the input of bremsstrahlung (including that at 15.1 GHz) is negligibly small because of relatively low electron density. At the rst glance, it seems possible to get better t for observations when using some modications of T and N e height dependencies. For example, calculated brightness temperature at 15.1 HGz could be lowered down to observed values, if kinetic temperature T is decreased and electron density N e is increased at low heights ( 1 thsd.km); it wouldn't eect remarkably on 8.0 and 4.5 GHz proles. However it would result in still lower level of radiation in o-mode, that cannot be reconciled with observations. Moreover, there is not much sense in trying to improve the above model, because there are some source features, which in essence could not be provided by the uniform over the sunspot model. One of the most important features of the uniform model, which contradicts observations, is a big dierence of the calculated brightness depression for ordinary and extraordinary waves in the middle of the sunspot (compared with the observing values), stipulated by a considerable distinction of optical depths of the gyroresonance layers when angle is small. From Fig.6b,7a it follows that depression at 4.5 GHz can be made wider and deeper (closer to the observed one) if the electron density at the proper heights (and hence the value of optical depth which is proportional to N e ) is lowered down (judging by Fig.7c, N e at the height of the s = layer must be decreased in 1- orders of magnitude). But in this case the o-mode window of transparency (which is too great as it is) would become still more dierent from the observed one (the curves in Fig.7d would move downwards). Moreover, it is clear from Fig.7c,d that raising the o-mode optical depth and brightness temperature in the center of depression up to observed values is possible if electron density is increased in few orders of magnitude. It is exactly this dierence (in our model the minimum values of the optical depth for an extraordinary wave exceed the same values for an ordinary wave for four-ve orders with s = ; 3) which was the main obstacle in explaining the ring structure in [6] by the eect of the gyroresonance layer transparency along the magnetic eld. The other argument, which indicates that the observational data are not explainable in the framework of the uniform over the sunspot model, is that, unlike observations, at any height dependence of electron density and temperature the calculated maximum at the right (when r < 0) will always be higher than at the left (when r > 0), because the sunspot is located in the western part of the disk, and rays, coming approximately along the magnetic eld, are projected on the left part of the source, resulting in the window of transparency. Under the approximation of 10

11 a radial direction of the magnetic eld axis and uniform distribution of the brightness temperature across the sunspot it is impossible to explain the inverse observing maximum ratio (see Fig.7a,8a). One more important argument against the uniform model is that the observed maximum brightness temperature of an ordinary emission at frequency 4.5 GHz exceeds the minimum brightness temperature of an extraordinary emission in the same cut (see crosses in Fig.3a). However, when the distribution of kinetic temperature over the sunspot is uniform, this relation cannot be realized. Indeed, in this case the only reason for the brightness depression in the center could be the transparency of the third and fourth layers in directions close to the sunspot axis; thus, the minimum temperature in the depression will be close to the kinetic temperature at the level of the layer s = (it is hard to expect lower values, because e 1 or 1 for any reasonable parameters). Since with moving o the center this layer goes down to cooler regions and if the o-mode escapes from the same layer s =, then its maximum brightness temperature achieved at a certain distance from the center must be lower than minimum brightness temperature (see crosses in Fig.7a). So, in order to t the observed ratio, designated in Fig.3a, 7a, the higher o-mode brightness temperature at the distance R 10 arcsec from the center should be provided by the layer s >. However, as the calculations show, in the wide range of values N e and T the optical depth o3 at intermediate does not exceed e3 in the sunspot center (it is conrmed also by Fig.7c,d). Thus, if the electron density is chosen in the way that the optical depth e3 in the sunspot center is suciently small, the layers s = 3; 4 for an ordinary wave are optically thin everywhere, i.e. cannot provide higher brightness temperature as is required by observations. All the arguments mentioned above testify to the observed ratio of brightness temperatures of ordinary and extraordinary waves and their distributions over the sunspot are not explainable by the uniform model. The similar consideration of the distributions at f = 8 GHz, as well as models proposed in [7,1,15,0-6] yields the same conclusion Non-uniform distribution. The circle-shaped source and qualitative coincidence of recorded sizes and gyroresonance layers for the sunspot NOAA 7879 (including frequency dependence of the source size) imply the magnetic eld to be approximated as a dipole rather well. The most reasonable way to try to t observational data is to assume kinetic temperature and electron density to be non-uniform over the source surface and to search for the distributions T (h; %) and N(h; %) which could t observations. We follow this way and by numerous trials have chosen the following model. The distribution of the kinetic temperature along the height is dened as before by relation (7), but now the parameters depend on the distance %, and this dependence should vary with height. The starting point of the research is that the extraordinary prole at frequency 4.5 GHz approximately copies the varying of the kinetic temperature distribution over the gyroresonance layer s = 3. The condition necessary for deriving the observing distribution is that the coronal temperature should be lower in the center, and the intensity of depression should be increased with height. Moreover, in view of the asymmetry of the observed distributions it is necessary for the values T c to be dierent in the right and left parts of the source. The consideration of many cases has led to the following optimal set of parameters, which determines the two-dimensional temperature model (the dependence on 11

12 % and h) giving a good t for observations. The temperature of the chromosphere T ch and the height of transition region from the chromosphere to the corona (h c h and h) remains the same, as before(see (30)), but the coronal temperature (instead of T c = : K for the uniform model) is described by the relation: T c = T 0 + (T c0 T 0 ) exp jrj r 0 r 1 + exp jrj r 0 r ; (3) according to which T c changes from the value T 0 in the center of the source (when jrj r 0 r) to the values T c0 at its edges (when jrj r 0 r). The parameter r 0 characterizes the width of the depression region (which, in its turn, should depend on height), and r refers to the size of the transitional region. The dependence of the electron density on the height is chosen similar to the uniform model (9) with the same gradient h N (31), but the value N 0 instead of being constant in (31), is now dependent on %: N 0 = (1 + N 1 e % =r 0 e j%j r )(1 N ) (33) 1 + e j%j r According to (33), the value N 0 increases in about (1 + N 1 ) times in the central part of the source (% < r 1 ) and decreases at its periphery (% > r 1 ), if N 1. Some of the mentioned above parameters are the same for two considered cuts of the source image, shown in Fig.3. They are the following: T ch = 10 4 K; T 0 = 0: K; (34) r = 0:7; h ch = :5; r 0 = 3:0 + :5 1 + e e3(h 3) e(h 3) (h 3); (35) N 1 = : (36) 1 + e3(h 3) (All the distances in (36)-(37) and below are expressed in units of 10 8 cm for the sake of simplicity.) Other parameters dier for two considered cuts. Moreover, some of them are not the same for the left and the right sides of the proles. Below they are: a) cut I through brightness temperature maximum: T c0 = ( : K; if % < 0; 1: K; if % > 0; (37) r 1 = 3; (38) N 1 = 0:995 (39) ( 10; if % < 0; r = (40) 8; if % < 0; b) cut II through brightness temperature minimum: ( 1: K; if % < 0; T c0 = 0: (41) K; if % > 0; 1

13 r 1 = 5; (4) ( 0:99; if % < 0; N 1 = (43) 0:90; if % < 0; r = ( 10; if % < 0; 1; if % < 0; (44) The models described are shown in Fig.9 for cut I and in Fig.10 for cut II. The kinetic temperature distributions are presented depending on a)the distance % from the center of the sunspot at dierent heights h and b)the height h at dierent distances % from the center. The plots of electron density distribution are also given in Fig.9-10 as functions of c)n e (%) at dierent heights h and d) N e (h) at dierent distances % from the center. The results of calculations of the total emission containing both cyclotron radiation and bremsstrahlung are shown in Fig.11 at f = 4:5 GHz and in Fig.1 at f = 8:0 GHz and f = 15:1 GHz for the cut I. The same is given in Fig.13 and 14 for the cut II. 5. Discussion. First, let us consider in detail the cut I through brightness temperature maximum (Figs.9,11,1). It follows from Fig.9 that the model for this cut suggests depression of kinetic temperature in the center of the sunspot, the relative deepness of depression being increased with height, and at each point of the source the temperature increases with height, but with its own gradient. Coronal temperatures are dierent on the left and right sides of the source. Electron density is enhanced in the center of the source and decreases when moving out of the spot. Decreasing of electron density with height is monotonous only beginning with a certain radial distance, but in the center of the source electron density height dependence has a maximum at the height near 4 thsd.km. Calculations show that 4.5 GHz proles are due to the cyclotron mechanism, and the input of bremsstrahlung is negligibly small. It is seen from Fig.11b, that brightness temperature follows kinetic temperature distribution along the gyroresonance layer s = 3 with transition to the layer s = in the very center of the source (it is conrmed by Fig.9c, according to which the optical depth of the layer s = 3 is greater than unity everywhere except for the center of the spot). This prole coincides with the observational one quite well. As for o-mode at 4.5 GHz, calculated distribution ts observations satisfactorily at a distance from the center (the value and location of maxima, the size of the source), but expected window of transparency is wider and deeper than the recorded one. Thus, consideration of calculation results in Fig.11 shows that at f = 4:5 GHz depression is associated with lower kinetic temperature at the level of -3 harmonics in the center of the sunspot than at its edges, while brightness temperature fall in o-mode is due mainly to eect of low cyclotron radiation along magnetic eld. The shift of the depression center to the left is due to location of the sunspot in the west part of the solar disk and eect of projection of the line-of-sight on the considered diameter. It should be noted that the "wings" of recorded proles and their dierence for two modes are modelled rather well. It is due to the right choice of the temperature gradient (corresponding dierence at the levels of layers s = and s = 3 when they are sinking into the chromosphere) and rather low values of electron density (at 13

14 greater N e the layer s = 3 could contribute into the o-mode, and then the dierence between wings would appear to be less than observed one). However, too wide and deep fall in o-mode brightness proles can hardly been eliminated in the framework of considered model: at given T and N e distributions the o-mode optical depth of the layers s = 1 4 is so small in the center (see Fig.11d), that change of parameters under reasonable limits cannot result in required decrease of depression (in spite of electron density in the center is remarkably higher that at the periphery). Studying of proles for cut I at frequencies 8.0 and 15.1 GHz (Fig.1) leads to the following conclusions. Bremsstrahlung input at 8 GHz is insignicant: the dierence between the total radiation (dotted lines) and cyclotron radiation is detectable only in the center of depression. In general features (existence and location of maxima, ratio of brightness temperatures for two modes) the prole coincides, but some details dier: calculated ones don't contain maximum in the very center, and the values of maxima dier. As at f = 4:5 GHz, here depression is due to the lowered temperature in the center of the spot, but o-mode brightness distribution with appropriate ratio of maximum temperatures (less at the left than at the right, unlike kinetic temperature) is a result of enhanced transparency along magnetic eld. Radio emission at f = 15:1 GHz (Fig.1b) is mainly of bremsstrahlung origin: cyclotron input from s = 4 is shown by dashed line for and is not detectable in o-mode, while bremsstrahlung curves (dotted lines) are quite reconciled with observed ones. Note, that the calculated shift of brightness temperature peak to the west (coinciding with observed) is due to the dependence of bremsstrahlung absorption coecient on the angle between magnetic eld and the line-of-sight. It is the relation between brightness temperatures at 8.0 and 15.1 GHz that determines non-monotonous height dependence of electron density at the distances close to the center of the sunspot: if the electron density at the heights 1- thsd.km had been greater that suggested by Fig.9c,d (for example, the upper curves in Fig.9d have been continued from greater heights to less ones), then the 15.1 GHz brightness would have been unjustied high (cf. Fig.8b); on the contrary, if the electron density would have been lower at h 4 thsd.km than suggested by the model (in case when the same curves would have been continued from less to greater heights), then 8.0 GHz radiation would have appeared to be considerably lower than observed one. However, at the periphery of the source monotonous decrease of N e with height fully satises observations. Note that existence of cool and dense plasma above the sunspot doesn't contradict optical observations [33-36]. Let us consider now (but less in detail) modelling the proles of cut II through brightness minimum. In view of cooler plasma here, the input of bremsstrahlung becomes more signicant. Extraordinary emission at f = 4:5 GHz as follows from Fig.13a,b, is mainly of cyclotron origin (escapes from the layer s = 3), calculated and recorded proles being coincided rather well. Calculated o-mode radiation at the same frequency, as well as that for cut I, diers from observed one in the center having too large window of transparency due to propagation eect (Fig.13). At frequency 8.0 GHz (Fig.14a) the chosen cut in gives not ring-structure, but brightness temperature maximum in the center (associated, probably, with some inhomogeneities at the proper heights, for example, laments which are not stipulated by the model), while the theoretical curve gives brightness distribution with fall in the center. Thus, radiation at 8.0 GHz is described only qualitatively (the value of brightness temperature, the size of the source), but not in details. The 14

15 o-mode radiation at 8.0 GHz is tted by the model much better (Fig.14a), and the greater value of brightness temperature at the right maximum than at the left one proves the proles (both calculated and observed) with depression in the center to be a result of low opacity, but not cooler plasma. Fig.14b for f = 15:1 GHz also demonstrates quite good result of modelling: calculated and observed proles have close brightness temperatures for both modes, similar shift of the peaks to the west and sizes of the source. It means that the kinetic temperature and electron density distributions at the appropriate heights are retrieved right. It should be noted that both kinetic temperature and electron density for both cuts are dierent at the left (% < 0) and right (% > 0) halves of our cuts. It is provoked by clearly visible dependence of the brightness temperature on the azimuthal angle ' (see images in Figs.1,). The size and orientation of dark spots on 4.5 and 8.0 GHz images as well as the form of 15.1 GHz bremsstrahlung source, show that more dense and cool plasma has a form of a strip elongated approximately from south-east to north-west, the hottest plasma being concentrated at north-east. Thus, the calculated radio brightness distributions t the observed proles in both polarizations rather well. The suggested model provides essential features of the source (not explainable by the uniform model), like the ratio of the maximum and minimum of the brightness temperature and their location, the existence of the depression of the needed width and intensity in the center, the dierence between the brightness temperatures of ordinary and extraordinary emission (i.e. the polarization pattern), the depression itself being implied by two eects - the low opacity of gyroresonance layers along magnetic eld and the presence of more cool plasma above the sunspot center. As is shown above, the observed characteristics could not be a result of only the rst eect. The important peculiarity of the modelling in this paper is that six brightness distributions (for three frequencies and two polarizations), each of them being generated at dierent (but some at close) heights, must be tted. It makes dicult to change one parameter maintaining the others and not breaking conformity with observations. For example, increase of T at the height of the layer s = at f = 4:5 GHz in order to enhance o-mode brightness temperature (as is required by measurements; see Fig.11) immediately will result in the undesirable enhancing of radiation at 8.0 GHz escaping from the layer s = 3 located at approximately the same heights as the layer s = responsible for 4.5 GHz o-mode (see Fig.5). Moreover, the varying of one of the parameters, for instance, the height temperature gradient will lead to necessity of changing other characteristics, say, the height gradient of magnetic eld and many other parameters. It means that the distributions B(h; %), T (h; %) and N e (h; %) are consistent with each other. So the real behavior of electron density and temperature cannot deviate considerably from that described above (though detail devergencies are certainly possible). Apparently, the approximation of the kinetic temperature and electron emission distributions with the above analytical functions is really conditional one. Shown in Figs.9-10 could be dened by some other way, including numerical one. In this case one could get empirical model giving much better t for observations: for instance, input of some "laments" or other inhomogeneities at certain heights could result in better coincidence of 8.0 GHz proles. Besides, in principle, having experience in choosing a model for four radii, we would create a three-dimensional model (that is to set up dependence not only on the height and radial distance but on azimuthal angle '), then to compute two-dimensional images of the source and compare them 15

16 with those shown in Fig.. However we didn't do such a cumbersome computing, because it doesn't give much physical sense, but calculations performed for the models shown in Figs.9,10 and described by analytical formulas (8)-(9), (3)-(44) are enough to get the following important conclusions about the physical conditions in the observed sunspot atmosphere: 1. The temperature distribution over the source surface should have a depression in the center, intensity and width being increased with the height and the limiting values of coronal temperature being dependent on the azimuthal angle '; otherwise it would be impossible to explain quantitatively the observed fall in brightness temperature in two polarizations.. The electron density should be signicantly greater in the center of the sunspot than at its edges; otherwise it would be impossible to explain the dierence in the ordinary and extraordinary emission at the sunspot edges (the outer ring of the enhanced polarization) and provide the observed value of depression in the center at 4.5 and 8.0 GHz as well as the structure of the 15.1 GHz source. The last remark concerns magnetic eld model. We have considered a classic model of unipolar sunspot as a vertical dipole oriented along solar radius and buried under the photosphere. We have chosen such a model because the microwave source as well as H source has a form of a sharp circle and because using optical data (SGD) about the sunspot (B 0 = 500G and d = km) we obtain the sizes of o- and s at 4.5 and 8.0 GHz to be equal to radial distances at which corresponding gyroresonance layers go down under the photosphere. We were successful in such modelling. It is not excluded, however, that real magnetic eld diers from dipole one in small details. In particular, failing in decreasing of calculated "window of transparency" in o-mode in the framework of dipole model (with temperature and density varying in the reasonable limits) may be an evidence of the fact that in the center of the sun 6. CONCLUSION In the present paper an empirical two-dimensional model of the atmosphere above the sunspot, which includes the dependence of the magnetic eld, kinetic temperature and electron density on the height and the distance from the sunspot center, is proposed. The thermal electron cyclotron emission and bremsstrahlung from the non-uniform plasma with the mentioned characteristics ts well the pro- les of the radio brightness over selected cuts of the source, obtained by VLA at three frequencies 4.5, 8.0 and 15.1 GHz in two circular polarizations. It is shown that specic radio brightness distribution over the source, looking like a ring or a horseshoe, and the observed ratio of maximuma and minima of brightness temperature of ordinary and extraordinary radiation can't be explained only by the eect of enhanced transparency of gyroresonance layers along magnetic eld within the framework of one-dimensional model (height dependent electron density and temperature) of active region atmosphere (as is usually accepted for explaining structures of such kind): the details observed testify to non-uniform distribution of kinetic temperature and electron density over the source. In our case horseshoe structure the eect of propagation along magnetic eld appears to be more signicant. It should be noted that the proposed distributions are obtained by trying to adjust the model with the observed characteristics of the microwave radiation. We 16

17 have taken into account the well-known facts about the behavior of T and N e in the chromosphere, the corona and the transitional region, but neglected some other constraints, for example, the hydrostatic equilibrium and constancy of conductive ux. However, the calculations of the radiation for a plane-parallel, constant conductive ux and hydrostatic equilibrium model, accepted, for example, in [7,4], give the results that are far from the present observations. Our goal was to determine the main features of the physical parameters distributions in the observed source. The principle result is that basing only on radio observations we can arm the presence of the dense cool plasma above the sunspot center. Evidently, successful modelling of the observed microwave source by no means signies that there is a temperature depression in the atmosphere of any sunspot. A great variety of observed microwave and optical sunspot characteristics testies to dierent physical conditions in the plasma above sunspots. Unlike the above source in some cases density decit [9,11], strong density gradients or sharp temperature discontinuity [7], excess of temperature [6] may be present above the umbra. Similarly, magnetic eld in the atmosphere above sunspots often cannot be described by dipole approximation. In many sources the loop structure can play a signicant part [6,8]. To conclude we would like to stress the importance of multi-frequency high resolution microwave observations of the Sun for retrieving the physical conditions in the solar corona. Indeed, obtained conclusions about density and temperature distributions in the sunspot atmosphere turned out to be possible only due to VLA observations at three frequencies in two opposite polarizations. The absence of if only one frequency would decrease information so remarkably (to say nothing about low spatial resolution), that creation of two dimensional model hardly would be possible. Note also, that it would be extremely useful to follow center-to-limb variation of the source characteristics at multiple frequencies; in this case consequences of two competing eects resulting in ring structure would be divided much easier and retrieving of physical parameter distribution would be more reliable. All this would provide more deep insight to the nature of active regions. The authors are grateful to V. V. Zheleznyakov for useful discussions and remarks. The work of E.Z. was supported by the Russian Foundation of Basic Research (Grant N a). References [1] Zheleznyakov V. V. 1970, Radio Emission of the Sun and Planets, Pergamon Press, Oxford. [] Kundu M. R. 1965, Solar Radio Astronomy, Interscience Publ., New York. [3] Zheleznyakov V. V. 1996, Radiation in Astrophysical Plasmas, Kluwer Acad. Publ., Astrophys. and Space Science Library, 04. [4] Akhmedov Sh. B., Gelfreikh G. B., Bogod V. M., and Korzhavin A. N., 198, Sol. Phys., 79, 41. [5] Strong K. T., Alissandrakis C. E., and Kundu M. R. 1984, Astrophys.J., 77,