The observations analyzed here were made with TIP at the German Vacuum Tower Telescope (VTT; Teide 1. INTRODUCTION 2. OBSERVATIONS

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1 THE ASTROPHYSICAL JOURNAL, 34:989È996, 2000 May 10 ( The American Astronomical Society. All rights reserved. Printed in U.S.A. OSCILLATIONS IN THE PHOTOSPHERE OF A SUNSPOT UMBRA FROM THE INVERSION OF INFRARED STOKES PROFILES L. R. BELLOT RUBIO, M.COLLADOS, B.RUIZ COBO, AND I. RODRI GUEZ HIDALGO Instituto de Astrof sica de Canarias, E-38200, La Laguna (Tenerife), Spain Received 1999 August 17; accepted 1999 December 23 ABSTRACT We report on the detection of magnetic Ðeld strength and velocity oscillations in the photosphere of a sunspot umbra. Our analysis is based on the inversion of the full Stokes vector of three Fe I lines at 160 A, from which the stratiðcation with optical depth of the di erent atmospheric parameters has been derived. This allows us to estimate the amplitude of the oscillations and the phase lag between the Ñuctuations in the line-of-sight velocity and Ðeld strength. Our results suggest that the inferred magnetic Ðeld oscillations are caused by opacity Ñuctuations that move upward and downward the region where the spectral lines are sensitive to magnetic Ðelds. Subject headings: Sun: magnetic Ðelds È Sun: oscillations È Sun: photosphere È sunspots 1. INTRODUCTION The detection of magnetic Ðeld oscillations in sunspot umbrae has been the subject of intense e orts during the last three decades (see Lites 1992; Staude 1999). However, their existence is still a matter of debate. Lites et al. (1998), for example, found oscillations of the magnetic Ðeld strength with rms amplitudes of 4 G from Milne-Eddington inversions of the pair of Fe I lines at 6302 A, but they consider these oscillations to be instrumental rather than solar in origin because of the small amplitudes inferred and phase lags that do not agree with theoretical predictions. Other groups favor the view that oscillations exist. Horn, Staude, & Landgraf (1997) report on magnetic Ðeld Ñuctuations at a marginal level from the analysis of time series of Fe I A Stokes I ^ V proðles. Their observations suggest a marked spatial Ðne structure and a nonstationary behavior. Ru edi et al. (1998) also Ðnd oscillations of the magnetic Ðeld strength with rms amplitudes of the order of 6 G from measurements obtained with the Michelson Doppler Imager (MDI) on board SOHO. Their data are not inñuenced by seeing Ñuctuations, which removes an important source of noise. However, instead of full Stokes proðles, MDI provides indicators of the magnetic Ðeld and velocity by means of Ðlters that sample the Ni I 6768 A line at Ðve wavelengths. Norton et al. (1999) have investigated the spatial distribution of the oscillations in a sunspot from MDI measurements. These authors detect signiðcant power in the magnetogram signal and phase di erences of D[90 between velocity and magnetic Ñux oscillations. More recently, Balthasar (1999) has reported on Ðeld strength oscillations with rms amplitudes of 40È0 G at the umbra/ penumbra boundary of a sunspot. This study, based on two-dimensional spectra of the Fe I line at 6843 A, suggests that enhanced power occurs in patches with sizes as small as 1A.. While velocity oscillations in sunspots have amplitudes that make their detection almost straightforward, magnetic Ðeld oscillations (should they exist) are much weaker. Theoretical models predict amplitudes of a few gauss at most (Lites et al. 1998). Noise arising from seeing Ñuctuations, errors in the data reduction, and limitations of the analysis techniques (which are unable to separate the e ects of the temperature, velocity, and magnetic Ðeld) contribute to the smearing of the subtle signatures that small changes of the 989 magnetic Ðeld would induce in the observed proðles. This is probably the reason that the existence of magnetic Ðeld oscillations has not yet been proved or disproved convincingly. On the other hand, the analysis techniques applied so far do not distinguish between di erent atmospheric layers, in which rather di erent conditions prevail. Our aim in this paper is to overcome some of these limitations. In order to increase the sensitivity to the magnetic Ðeld, the observations consist of time series of the four Stokes parameters of three Fe I lines at 160 A recorded with the Tenerife Infrared Polarimeter (TIP; Mart nez Pillet et al. 1999). Infrared lines are particularly well suited to this analysis because of their large Zeeman splittings and because they sample deep photospheric layers, in which magnetic Ðeld oscillations are expected to be stronger. To separate the e ects of the di erent physical quantities, we use the SIR inversion code developed by Ruiz Cobo & del Toro Iniesta (1992). This will allow us to determine the temporal evolution of the various physical quantities at different optical depths in the photosphere. Recently, SIR has been applied to the study of the minute oscillations in the quiet Sun (Ruiz Cobo, Rodr guez Hidalgo, & Collados 1997). The results presented here are unique in that they are based on the inversion of three spectral lines particularly sensitive to magnetic Ðelds. However, they represent a Ðrst step toward the complete solution of the problem. Although we carry out the analysis in terms of optical depth, it is clear that possible Ñuctuations of the temperature and density might induce opacity Ñuctuations that would give rise to variations in the height of the region in which the spectral lines are formed. Because of the gradient db/dz of the magnetic Ðeld in sunspot umbrae, spurious oscillations of B may result if the line-forming region moves upward and downward by opacity Ñuctuations. Indeed, the oscillations of the di erent physical quantities as a function of optical depth show two e ects simultaneously: opacity variations and intrinsic changes at a Ðxed position in the atmosphere. Hence, the next step in our endeavor will be the analysis of the observations in terms of geometrical heights. 2. OBSERVATIONS The observations analyzed here were made with TIP at the German Vacuum Tower Telescope (VTT; Teide

2 990 BELLOT RUBIO ET AL. Vol. 34 Observatory) on 1998 November 24. The spectrograph slit was placed across the center of the main spot in NOAA group 8391 and included some portion of quiet Sun on the northern side of the spot. At the time of the observations, the active region was located slightly o disk center (k \ 0.89, 14 S, 22 W). The width of the slit was 100 km (0A.] 36.4 ma ), and its length covered 30A.3 (82 pixels) in the spatial direction. TIP was used to measure the four Stokes parameters of the Fe I lines at , 1648., and A along the slit. The observed wavelength range spans 7.1 A with a spectral sampling of 29.1 ma (24 pixels). The infrared polarimeter works as follows. Four alternate linear combinations of the Stokes parameters are produced at a frequency of 2.08 Hz. In contrast to optical CCD sensors, the NICMOS array of TIP is read in the so-called reset-read-read mode, whereby the exposure and readout is performed in several stages. First, the detector is cleaned with a reset signal. This operation is completed in about 20 ms. Next, an exposure of 0 ms is taken. The readout is nondestructive, implying that the charge at each pixel remains and is not cleared. In the third step, a second exposure of 0 ms is performed. The two exposures are subtracted by dedicated hardware to obtain the Ðnal image. The reason for this reading mode is that the sensor clearing is not very efficient and some random residual charge is always left. Since the di erential increase from the 0 ms to the 100 ms exposures is not sensitive to this e ect, the images are less noisy at the cost of halving the frame rate. The e ective integration time of the Ðnal image is 0 ms, the whole process requiring about 120 ms to be completed. Thus, a cycle of 4 ] 120 ms \ 480 ms (i.e., a frequency of 2.08 Hz) is needed in order to produce four di erent linear combinations of the Stokes parameters. This operation is repeated 10 times, adding in real time the images corresponding to the same linear combination. Finally, the accumulated images are displayed and stored on disk. The complete sequence (including some system control tasks) takes place in about.6 seconds. The observations consist of a time series of total length 22.1 minutes starting at 16:17 UT. During the observing interval, the seeing conditions were good and uniform, with an image blurring of D1A. To stabilize the image, the Instituto de Astrof sica de Canarias/Kiepenheuer-Institut fu r Sonnenphysik (IAC/KIS) correlation tracker (Ballesteros et al. 1996) was used. The measurements were corrected for Ñat-Ðeld and dark current, following standard procedures and treating each step of the modulation scheme separately. The signal-to-noise ratio (S/N) in the reduced Stokes Q/I, U/I, and V /I proðles is typically between 40 and 600. c The c polarimeter c was calibrated by means of known polarization optics located at an appropriate position in the optical beam. The calibration optics allows us to remove most of the instrumental cross talk between the Stokes parameters, which, in the infrared, is less severe than in the visible (Collados 1999). Residual instrumental polarization I ] Q, U, V and V ] Q, U due to the telescope coelostat was also taken into account. The former correction was derived from the continuum level of Stokes Q, U, and V, while the latter was obtained from linear regressions between small-amplitude linear and large-amplitude circular proðles. The temporal variations of these correction factors show linear trends consistent with the slow change in relative orientation of the coelostat mirrors. No attempt was made to determine the Mueller matrix of the telescope at 160 A empirically, but we estimate that our procedure corrects for the instrument-induced cross talk below 0.1% of the continuum intensity. 3. DATA ANALYSIS We restrict the analysis to spatial points corresponding to the sunspot umbra, since the interpretation of oscillations in the penumbra is far more complex. Our Ðnal data set consists of six individual pixels along the slit (D2A.4) spanning the region between the northern umbraè penumbra boundary and a light bridge that was present in the center of the umbra. We have applied the SIR inversion code (Ruiz Cobo & del Toro Iniesta 1992) to the time series of the four Stokes parameters of the Fe I lines at , 1648., and A emerging from each of the six spatial points selected for treatment. The inversion code retrieves the temperature, macroscopic velocity, microturbulent velocity, and magnetic Ðeld strength, inclination, and azimuth stratiðcations along the line of sight. These stratiðcations are represented by 3, 2, 6, 2, 2, and 2 nodes, respectively. Additionally, we determine the macroturbulence and stray-light contamination needed in order to reproduce the observed proðles. In total, 19 parameters are sought during the inversion. The number of nodes for the di erent physical quantities has been selected according to our previous experience. The Stokes spectra are most sensitive to temperature, so it can be given more freedom than the other parameters (hence the adoption of three nodes). We have found that two nodes for the remaining physical quantities are enough to Ðt the observed proðles. For the microturbulence, we use six nodes, however. The reason is that allowing the microturbulence to increase with optical depth leads to marginally better Ðts, especially in the wings of the lines. Using one or two nodes for the microturbulence does not change the dynamical and magnetic parameters signiðcantly, but produces slightly poorer Ðts. The initial guess model needed in order to start the procedure is taken to be the cool umbral model of Collados et al. (1994) for the inversion of the proðles corresponding to the Ðrst time measurements. For subsequent measurements, the models resulting from the previous inversions are used. This strategy reduces the intrinsic noise of the inversion, since the initial guess models are very close to the Ðnal ones. Stokes Q, U, and V, being essentially una ected by Ñat-Ðeld errors within the achieved S/N, were assigned twice the weight of Stokes I in the inversion. The wavelength range observed by TIP includes four Fe I lines (Table 1) and several unidentiðed lines, some of which appear to be telluric. Only the three Fe I lines marked with an asterisk are strong enough to deserve consideration. Unfortunately, accurate atomic parameters for these highexcitation lines are not available. Nave et al. (1994), for example, determined their laboratory wavelengths to within ^12 ma. Since the availability of accurate atomic param- eters is crucial for the success of the inversion, we have used the two-component quiet-sun model of Bellot Rubio, Ruiz Cobo, & Collados (1999) to obtain better values of their oscillator strengths, enhancements to van der Waals damping, and central wavelengths. To this end, we have Ðtted the quiet-sun intensity proðles of these lines as recorded by the Fourier transform spectrometer at the McMath/Pierce telescope on Kitt Peak (Livingston & Wallace 1991) to the synthetic proðles emerging from our

3 No. 2, 2000 SUNSPOT UMBRA OSCILLATIONS 991 TABLE 1 ATOMIC PARAMETERS OF THE FE ILINES OBSERVED BY TENERIFE INFRARED POLARIMETER AT 1.6 km j s 0 e (A ) Damping (ev) log gf Transition (*) [0.9 f 7D 1 Ès6D 6pP 2 o (*) [0.67 e7d 1 Èn7D 1 o vd 1 o ÈeS (*) [0.06 f 7D Ès6D 4. 4f [3.] 4 o NOTES.ÈHere j stands for the laboratory wavelength of the line; 0 enhancements to van der Waals broadening are listed under Damping ÏÏ; s indicates the excitation potential of the lower level; log gf represents the e logarithm of the degeneracy of the lower level times the oscillator strength of the line. All the parameters, except s, have been derived from the quiet e Sun model of Bellot Rubio et al. (1999), adopting an iron abundance of 7.46 on the usual logarithmic scale. Only the three Fe I lines marked with an asterisk are strong enough to deserve consideration. two-component model. The rms residual of the Ðt, using the atomic parameters listed in Table 1, is 1.7 ] 10~3 in units of the continuum intensity. Another important problem concerns the Zeeman pattern of the line at A. While LScoupling is valid for Fe I and A, the Fe I line at A is best described in JK coupling. The lower level of the transition is usually expressed in LS coupling, which would result in a Lande factor g \ 1.6. However, Sugar & Corliss 1 (198) assign g \ 1.1 to this level. For the upper level, 1 Johansson & Learner (1990) Ðnd g \ in JK coup- 2 ling. Using these nominal values, the inversion code is unable to Ðt the three lines simultaneously in the sunspot umbra. Therefore, we have determined the experimental Lande factors for Fe I A that produce the best Ðts. The inversions presented in this work have been carried out adopting g \ g \ 1.4 (i.e., the line behaves as a normal 1 2 Zeeman triplet), but it is important to stress that we were unable to Ðnd a combination of Lande factors leading to satisfactory Ðts for the intensity proðles of this line. In Figure 1 we show a typical Ðt to the four Stokes parameters emerging from an individual pixel in the sunspot umbra. Usually, the di erences between observed and best-ðt proðles resulting from the inversion are of the order of 4 ] 10~3 in units of the continuum intensity. The S/N of the observed proðles is poorer in Stokes I than in Stokes Q, U, and V because of strong interference fringes that cannot be removed appropriately. This, however, does not introduce signiðcant errors in the retrieved model atmospheres. Indeed, inverting the same proðles with di erent weights for Stokes I leads to Ðnal model atmospheres that are the same within the error bars. Figure 2 displays the average umbral model determined from the inversion of the 237 individual Stokes proðles emerging from the pixel nearest to the center of the umbra. The average magnetic Ðeld stratiðcation has a gradient of about 3.8 G km~1 in the deepest layers. For comparison, Collados et al. (1994) and Westendorp Plaza et al. (2000) determined gradients of about 4Gkm~1 at the bottom of the photosphere from the inversion of a number of visible Fe I lines. Gradients as large as 4 G km~1 in such deep layers are not unreasonable; in fact, the values of db/dz usually quoted refer to higher layers, in which the magnetic Ðeld gradient is smaller according to the results of these authors. The average microturbulence varies from D1.6 km s~1 at log q \[2.0 to D.0 km s~1 at log q \ 0.. The reason for this increase with depth is not completely clear, although we note that the same behavior has been inferred repeatedly from the analysis of other structures of the solar photosphere. The average macroturbulence turns out to be 0kms~1 with a standard deviation much smaller than the formal error in the retrieved values (0.37 km s~1). Again, the reason for this is not clear. FIG. 1.ÈStokes proðles recorded by TIP (dots) in the center of the umbra and best-ðt proðles (solid lines)

4 992 BELLOT RUBIO ET AL. Vol. 34 FIG. 2.ÈAverage umbral model determined from the individual Stokes proðles emerging from the pixel nearest to the umbra center. The average stray-light contamination is 20.7%. The macroturbulent velocity inferred from the inversion is 0 km s~1. The shaded areas represent the standard deviations of the Ñuctuations in the physical quantities. As stray light, we use the average intensity proðle emerging from nearby quiet regions. The thermal stratiðcation of the atmosphere producing the stray light is inferred not from the inversion; only the stray-light is. This strategy keeps the number of free parameters within reasonable limits and does not introduce errors, since the actual contamination by scattered light is taken into account. The average stray-light factor is 20.8% with a standard deviation of 1.6%. 4. RESULTS Similarly to Lites et al. (1998), our power spectra for the di erent physical quantities are obtained by Ðrst computing the discrete Fourier transform of the time series for the individual pixels and then averaging the power spectra. While this procedure diminishes the noise, it has the disadvantage that the computed power represents the oscillations of the umbra as a whole, although it is clear that the amplitude of the oscillations varies from one pixel to the next. Our power spectra have a frequency resolution of 0.7 mhz and a Nyquist frequency of 89 mhz. It is important to remark that the Nyquist frequency in analyses based on MDI data is 8.3 mhz. Therefore, they lack information about the high-frequency range of the spectrum, where the noise is best estimated and where short-period oscillations may be present. The response functions (Ruiz Cobo & del Toro Iniesta 1994) of the emergent Stokes spectrum to the various atmospheric parameters show that the three Fe I lines employed in this analysis are formed very deep in the photosphere of the umbra. As can be seen in Figure 3, the Stokes spectrum turns out to be sensitive to the physical conditions of the atmosphere between log q D ]1.0 and log q D [2.. Higher layers have little inñuence on the emergent proðles because of their low temperatures, which cause the lower atomic levels to be less populated. We therefore select two representative optical depths (log q \ 0.0 and log q \ [1.0) for further discussion. It is important to emphasize, however, that the radiative transfer equation was integrated from log q \ 1.2 to log q \[4.0 in order to take proper account of the whole atmosphere. Figure 4 displays the Ñuctuations in the magnetic Ðeld strength B (top) and line-of-sight velocity v (bottom) at log q \[1.0, as derived from the inversion. The time series corresponding to the six adjacent pixels along the slit that sample the umbra have been put together for ease of visualization. At log q \[1.0, the velocity shows a marked oscillatory behavior. Note the strong spatial coherence of the oscillations of v, which keep the same phase in the di erent spatial points. The Ñuctuations in the Ðeld strength are much noisier, but nevertheless it is possible to infer the existence of an oscillatory behavior. Again, there is some spatial coherence. The amplitude of the oscillations of the magnetic Ðeld is seen to decrease as the umbra/ penumbra boundary is approached. In contrast, the amplitude of the oscillations of v increases toward the penumbra, in accordance with the Ðndings of Lites et al. (1998). This particular behavior of db and v is very sug- gestive that the oscillations of db are not an artifact of the inversion produced by cross talk from real Doppler velocities into magnetic Ðelds. Indeed, one would intuitively expect larger amplitudes of db the larger the amplitude of v, should this cross talk mechanism be important. The left panels of Figure display the power spectra for the Ñuctuations in db (top) and v (bottom) atlog q \ 0.0 (dashed lines) and log q \[1.0 (solid lines) averaged over the six individual pixels. Here we see the Ñuctuations in these quantities more clearly. The average spectrum for the magnetic Ðeld strength is characterized by signiðcant power

5 No. 2, 2000 SUNSPOT UMBRA OSCILLATIONS 993 FIG. 3.ÈResponse functions of Stokes I (top) and V (bottom) to changes in the magnetic Ðeld strength (left) and velocity (right) evaluated for the umbral model of Fig. 2. in the minute band (2È4. mhz). The maximum power is reached at l D 3.7 mhz, with smaller peaks at about 10, 12, 1, and 19 mhz. Beyond l \ 20 mhz, the spectrum is featureless. The horizontal lines, representing 99% con- Ðdence levels, have been computed from the average power in the frequency interval 20È80 mhz (the threshold power corresponding to a 99% conðdence detection is 17 times larger than the average noise when six individual spectra are considered; see Groth 197). From these values, one would conclude that the smaller peaks are not statistically signiðcant. However, we note that they appear systematically in the individual spectra, suggesting that they might represent real oscillations. The amplitude of the oscillation of db at l \ 3.7 mhz is found to decrease toward the upper layers: at log q \ 0.0 we obtain amplitudes of 11 G, while at log q \[1.0 the amplitude is 8 G. This trend is consistent with theoretical predictions (see Fig. 13 in Lites et al. 1998) and may explain why Milne-Eddington (ME) inversions of the visible Fe I lines at 6302 A do not reveal Ñuctuations in db larger than 4 FIG. 4.ÈTime series of db (top) and (bottom) at as derived from the inversion of the Stokes proðles emerging from the six adjacent v log q \[1.0, pixels considered in this work. Positive velocities indicate downñows. The time series corresponding to each of the individual pixels have a total duration of 22.1 minutes. To make the oscillations more clearly visible, we have created an artiðcial time series by putting together the individual time series. The vertical dotted lines indicate where the series corresponding to a new pixel starts. The farther from the left, the larger the distance to the center of the umbra.

6 994 BELLOT RUBIO ET AL. Vol. 34 FIG..ÈL eft: Average power spectra of magnetic Ðeld (top) and velocity (bottom) Ñuctuations in a sunspot umbra at log q \ 0.0 (solid lines) and log q \[1.0 (dotted lines). Right: Average power spectra of temperature (top) and magnetic Ðeld inclination (bottom) Ñuctuations at the same optical depths. The horizontal lines represent 99% conðdence levels according to Groth (197). G (rms). Indeed, the magnetic Ðeld determined from ME inversions of the Fe I lines at 6302 A is representative of layers with log q D [2.0, in which the Ñuctuations in db have small amplitudes (D G according to our results). The decrease of the amplitude of db toward higher layers is also seen at other frequencies. As for the oscillations of v, our analysis reveals strong power in the minute band (2È4. mhz) and signiðcant power in the 3 minute band (4.È7 mhz), outside of which no peaks are present. The amplitude of the Ñuctuations in v is found to increase toward the upper atmospheric layers, consistent with the exponential drop of the density in the atmosphere. At log q \ 0.0, the amplitude of the oscil- lation with l \ 3.7 mhz is 70 m s~1, while at log q \ [1.0, the corresponding amplitude is 84 m s~1. Atl \ 6.7 mhz, where the peak power in the 3 minute band is reached, the amplitude of v is 19 m s~1 (log q \ 0.0) and 34ms~1 (log q \[1.0). Except for the magnetic Ðeld strength and velocity, we do not Ðnd signiðcant Ñuctuations in the other atmospheric parameters. The upper right panel of Figure shows what could be a marginal detection of Ñuctuations in temperature (T ) at log q \[1.0, but the peaks are only slight- ly above the noise. At l \ 3.7 mhz and l \ 6.72 mhz, the amplitudes of these oscillations are 23 and 26 K, respectively. At log q \ 0.0, dt has a maximum amplitude of 11 K, right at the limit of statistical signiðcance. As pointed out before, the oscillations of T result from opacity Ñuctuations coupled with intrinsic changes in the temperature, so the actual behavior of T in a Ðxed Eulerian reference frame will remain unknown until the variations in the physical parameters are expressed in terms of geometrical heights. Additionally, we want to remark that the maximum amplitude of the oscillation of the magnetic Ðeld inclination (c) at log q \[1.0 is 0.6 (see the lower right panel of Fig. ). Even if the variation of c were statistically signiðcant, it could not produce the observed db Ñuctuations, because we determine the three components of the magnetic Ðeld vector simultaneously. No information on the depth stratiðcation of the phase angle of the Ñuctuations in db and v can be obtained from the inversion, because two nodes are used to describe the run with depth of these quantities. However, it is possible to determine a representative phase lag between the velocity and the Ðeld strength throughout the atmosphere. From the cross-correlation of the individual time series, we Ðnd that, on average, v leads db by some 77 ^ 22 s. This implies a phase di erence of 10 ^ 30 for the main oscillation frequency l \ 3.7 mhz. In computing these values, we adopt the convention that positive velocities are downñows and that the time dependence of the oscillations is of the form exp (iut). Such a phase di erence between v and db agrees with theoretical predictions (Lites et al. 1998), although it is somewhat larger than expected. The sign is also consistent with the results of Lites et al. (1998), who determined positive phase lags of D20 between v and db from their Advanced Stokes Polarime- ter observations. Apparently, our results do not conðrm the phase di erences of D[90 derived from MDI data by Ru edi et al. (1998) and Norton et al. (1999), who came to the conclusion that the magnetic Ðeld leads the velocity by about a quarter of a cycle. However, Ru edi et al. (1999) have

7 No. 2, 2000 SUNSPOT UMBRA OSCILLATIONS 99 shown that the relative phase between velocity and Ðeld strength oscillations determined from MDI Ðlter measurements has opposite sign to those obtained from spectropolarimetric observations, presumably because of a sign reversal of the magnetic Ðeld provided by MDI.. DISCUSSION AND CONCLUSIONS The analysis of three infrared Fe I lines with a sophisticated inversion technique has led us to the detection of oscillations of the magnetic Ðeld strength in the photosphere of a sunspot umbra. In spite of our sensitivity to the magnetic Ðeld, the inferred Ñuctuations in db are only slightly above the noise. This demonstrates that the oscillations at a Ðxed optical depth are indeed very small. Analyses making use of lines in the visible, where the Zeeman splitting is much smaller, would hardly detect such oscillations with conðdence. We believe that the inferred oscillations are not the result of cross talk from velocity Ñuctuations. To verify this, we have synthesized the Stokes proðles of the three lines emerging from models that are the same as those resulting from the inversion except for their Ðeld strength stratiðcations, which were kept Ðxed to the average stratiðcation. The resulting model atmospheres harbor oscillations of the velocity but no oscillations of the Ðeld strength. After adding noise with the S/N of the original spectra, the synthetic proðles were inverted in the same manner as the observed proðles. This test demonstrates that the magnetic Ðeld retrieved from the inversion is a ected by some cross talk from velocity (see Fig. 6). However, the amplitude of the oscillation of db induced by v is small (2. G at log q \ 0.0 and 2.0 G at log q \[1.0). We therefore con- clude that this cross talk mechanism cannot explain the observed magnetic Ðeld oscillation. As we already pointed out, additional evidence that no signiðcant cross talk from velocity occurs is provided by the fact that the amplitude of db decreases in those umbral points at which the amplitude of v increases, while one would expect the opposite behavior if cross talk were producing the oscillation of db. On the other hand, phase di erences of 0 or 180 between v and db should be expected in this case, whereas we observe a phase di erence of D10. In our opinion, the inferred oscillations of db are not produced by Ñat-Ðeld errors either. Lites et al. (1998) have suggested that spurious oscillations of db may occur if there are small errors in the Ñat Ðeld that alter the shape of the Stokes proðles, because the proðles shift back and forth across the calibration anomaly. While this might happen, we believe that our procedure successfully handles such small calibration problems. On the one hand, the shift of the proðles caused by an oscillation of v of amplitude 300 m s~1 (such as the one represented in Fig. 4) is only D1.1 pixels, whereas each of the lines spans D100 pixels in the spectral direction. Thus, the calibration anomaly would have to a ect a large portion of the chip systematically in order for it to be able to modify the entire shape of the three lines, which is unlikely. On the other hand, we Ðt the full shape of the proðles, not single wavelength samples. Even if the Stokes parameters at some wavelengths are a ected by small Ñat-Ðeld errors, these cannot modify the stratiðcations of the physical quantities signiðcantly, because such modiðcations would produce misðts at other wavelengths. In this respect, we recall that Zeeman splitting of these FIG. 6.ÈAverage power spectra of magnetic Ðeld (top) and velocity (bottom) Ñuctuations as resulting from the inversion of the synthetic pro- Ðles generated from the retrieved model atmospheres without magnetic Ðeld Ñuctuations. A small cross talk from velocity into magnetic Ðeld is apparent. The power spectra for the velocity, however, is remarkably similar to that displayed in Fig.. infrared lines occurs in the strong-ðeld regime. Indeed, the inversion code retrieves the magnetic Ðeld strength from the separation of the p components (not from their amplitudes). Therefore, any calibration error would have to produce a coherent modiðcation of the polarization proðles at several di erent wavelengths leading to larger or smaller splittings of the p components to be able to modify the inferred magnetic Ðeld strengths. Again, we consider this very unlikely. Another possible origin of the oscillations of db is seeing Ñuctuations that cause a magnetic feature to jitter on and o of the spectrograph slit. Clearly, this would generate spurious magnetic Ðeld and Doppler velocity signals. However, such seeing Ñuctuations are high-frequency phenomena, so they can hardly create oscillations with the typical periods found here. Our analysis reconciles the apparently contradictory results presented by di erent authors. Oscillations of the magnetic Ðeld strength appear to exist, but their amplitudes decrease toward higher layers. This conclusion is drawn

8 996 BELLOT RUBIO ET AL. from the stratiðcation of the oscillations with optical depth, which has been determined for the Ðrst time. The phase di erence of D10 between the velocity and magnetic Ðeld Ñuctuations found here suggests that the oscillation of B is mainly the result of opacity Ñuctuations due to temperature and density changes. Velocity oscillations in sunspot umbrae are expected to produce density (and perhaps temperature) variations, very much the same as in nonmagnetized regions. If the magnetic Ðeld remains Ðxed in geometrical height and gas motion occurs only along the lines of force, the line-forming region will move up and down in response to the oscillating atmosphere. Even if the Ðeld strength does not change, oscillations of B would result as a consequence of the gradient db/dz. In this case, the magnetic Ðeld Ñuctuation is expected to lag the vertical displacement by 180 and thus the vertical velocity by 90 (see Lites et al. 1998). The agreement with our value of 10 ^ 30 is remarkable. Not only phase considerations, but also amplitude considerations, support the view that this simpliðed scenario is appropriate. With gradients of D3.8Gkm~1 in the deepest layers, vertical displacements of a gas parcel of D3 kmat log q \ 0.0 and D2 km at log q \[1.0 would be suffi- cient to account for the observed amplitudes of db. Such displacements can be produced easily by the velocities retrieved from the inversion. In any case, a deðnite physical interpretation of the oscillations of B will not be available until the stratiðcation of the di erent atmospheric parameters is expressed in terms of geometrical heights. This is not trivial, since the transformation from optical depths to geometrical heights requires the determination of a height origin that will also be oscillating owing to opacity changes. The question of whether magnetic Ðeld Ñuctuations may be stronger than those found here still needs to be addressed. We certainly did not detect large amplitudes, but this might not be the general case. There is some evidence that the oscillations are very localized within the umbra (Ru edi et al. 1998; Balthasar 1999) or even intermittent (Ru edi & Solanki 1999). If so, strong oscillations can be missed very easily unless the spectrograph slit is placed at exactly the right position within the umbra. In addition, recent model calculations by Zhugzhda, Balthasar, & Staude (2000) suggest that di erent spots behave di erently, the smallest producing larger amplitudes and vice versa. New insight into this problem will be gained in the future through the inversion of two-dimensional spectropolarimetric measurements. We would like to express our gratitude to Tom Bogdan for his collaboration, criticism, and hospitality. We also thank Roberto Casini for his help and Terry Mahoney for a careful revision of the manuscript. L. R. 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