Long-period oscillations of the line-of-sight velocities in and near sunspots at various levels in the photosphere

ISSN 1063-7729, Astronomy Reports, 2007, Vol. 51, No. 5, pp. 401410. c Pleiades Publishing, Ltd., 2007.Original Russian Text c V.I. Efremov, L.D. Parnenko, A.A. Solovev, 2007, published in Astronomicheski Zhurnal, 2007, Vol. 84, No. 5, pp. 450460.

Long-Period Oscillations of the Line-of-Sight Velocitiesin and near Sunspots at Various Levels in the Photosphere

V. I. Efremov, L. D. Parnenko, and A. A. SolovevPulkovo Observatory, Russian Academy of Sciences, Pulkovskoe shosse 65, St. Petersburg, 196140 Russia

Received August 21, 2006; in nal form, October 12, 2006

AbstractNew observational data on long-period oscillations of the line-of-sight velocities detected viathe Doppler shifts of spectral lines observed at various heights in and near sunspots are presented. Thesunspots and nearby magnetic elements oscillate with periods ranging from 40 to 80 min. The oscillationsin the line-of-sight velocities persist over the entire observation session (up to four hours). These resultssupport theoretical models in which this phenomenon represents natural long-period oscillations (verticalradial displacements) of entire magnetic elements (sunspots, pores, and magnetic knots) about some stableequilibrium positions.

PACS numbers : 96.60.qd, 96.60.MzDOI: 10.1134/S106377290705006X

1. INTRODUCTION

Studying oscillations in sunspots and the sur-rounding photosphere is of a fundamental interestfor solar physics, since these regions are occupiedby fairly strong (predominantly vertical) magneticelds. The wave and oscillatory properties of thesemedia dier considerably from those properties of theatmosphere, which is free from magnetic elds [1, 2].As a rule, attention has tended to be focused oncomparatively high-frequency MHD oscillations (pe-riods of 35 min) in sunspots. There is an extensiveliterature containing observational and theoreticalstudies of these oscillations (see, for example, [39]).In addition to these well-known oscillations, somelong-period (periods ranging from half an hour to sev-eral days) oscillations of physical parameters in andnear sunspots are known, which represent temporalvariations in the magnetic elds and line-of-sight(LOS) velocities of sunspots [1012], as well as theradio emission of certain sources located above thesunspots [1315]. These phenomena cannot be de-tected during short (1530 min) observing sessions,since suciently long, uniform time series in theplasma and magnetic eld parameters are required.Theoretical interpretation of long-period oscillationsin the magnetic elements of the solar atmosphere alsorequires approaches that are fundamentally dier fromthose used for short-period oscillations. Namely, anylocal magnetic elements (sunspots, pores, or knots)are fairly long-lived and mechanically stable, and areexposed to external perturbations, apparently causingthem to undergo oscillations about some equilib-rium positions. These oscillations have an integral

character in the sense that each magnetic elementmaintains its overall structure and geometry, whileits size and magnetic eld periodically vary. When anelement ascends in upper raried layers, the externalpressure decreases, so that the element expandsand its magnetic eld weakens. On the contrary,a magnetic element descending into deeper layersof the photosphere and convective zone becomescompressed, and its eld increases. Such naturaloscillations can be called vertical, or radial.

Theoretical sunspot models describing this phe-nomenon have been developed for a long time [16, 17].Recently, a model with small sunspots has becomefairly well developed [18, 19], and the eigenfrequencyfor this model has been calculated as a function ofthe magnetic eld, although the typical periods forthe oscillations of sunspots as the whole were shownto be 60300 min as early as 1992 (for magneticelds of 14 kG) [17, Table 2]. It is important that,in the latest version of this model [19], the period ofthese oscillations is independent of the transverse sizeof the element, and depends only on the magneticeld strength. Measurements of variations of sunspotmagnetic elds from 1.5 and 2.7 kG show that thecorresponding period of global natural oscillationsvaries from 40 to 200 min [12].

The current work presents the results of test-ing the theoretical predictions of the small-sunspotmodel [1619]. The observations were carried outusing the solar telescope of the Main AstronomicalObservatory of the Russian Academy of Sciences inPulkovo. Oscillatory processes in and near sunspotswere studied using high-quality Doppler images. Our

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long observing sessions, which reach four hours witha sample step of 1530 s, make it possible to detectlow-frequency oscillations of magnetic elements withperiods ranging from 40 to 80 min.

Section 2 of the paper describes the observingtechnique and Section 3 the image processing, whileSection 4 presents and discusses our results. Ourmain ndings are summarized in the Conclusion.

2. OBSERVING TECHNIQUE

We present here data obtained in 2006 using a newtechnique. Instead of detecting the LOS velocitiesusing a CCD video camera mounted on a spectro-heliograph magnetograph [8, 9], we used direct mea-surements of the Doppler shifts in images of the solarspectrum obtained using a digital mirror CANONcamera. The camera array (CMOS sensor) is 22.2 14.8 mm in size, and is placed in the focal plane ofthe spectrograph of the horizontal solar telescope.The focal distance of the telescope is 17 m. The solardiameter at the spectrograph aperture is 161 mm, andwe have a scale of 11.9/mm. The isothermal four cellspectrograph of the solar telescope has a spectral dis-persion of about 3.7 mm/A in the fourth order for theH line, providing a spectral resolution of 0.15 nm.The total number of pixels is 8.2 million; we useda resolution of 3456 2304 pixels. The formats ofthe images obtained are JPEG or RAW (12 bit). Forsolar-spectrum observations, the illumination of thedigital array is sucient to enable exposures shorterthan 0.01 s at the sensitivity of the ISO 200.

The digital camera is controlled by computer viaa USB-2 interface. The camera provides automatedsnapshots of the solar spectrum every 1530 s duringthe entire observing session. To facilitate carrying outfast Fourier transforms, we obtained series of 512 im-ages. The observing sessions cover intervals from oneto four hours.

During the observations, the location of the re-gion selected by the spectrograph aperture must becarefully checked by the observer. The position ofthe region changes somewhat during long observingsessions due to the solar rotation and the annual vari-ation in the solar inclination. This checking is facil-itated by a mirror spectrograph aperture that reectsan enlarged aperture image onto a special screen, andan image of the spectrum displayed by an auxiliaryTV camera on a video monitor, which provides astraightforward means to check the position of imageelements on the spectrograph aperture.

The advantages of the digital mirror camera overstandard analog CCD video cameras are its largearray and the very high resolution of the digital images(this resolution is even excessive for our telescope

because resolution limits of 23 are imposed bythe atmosphere). Disadvantages include the sharpdecrease in the sensitivity of the CMOS sensor nearthe useful HeI 1083 nm infrared line and the specialsoftware required to derive the Doppler shifts from thedigital spectrum images. We studied the spectral re-gion between 649 and 650 nm. There are six lines withdierent formation heights [17]. Figure 1 presents oneof the spectrograms obtained for this spectral region.

3. IMAGE PROCESSING

It is well known that the LOS velocities in the solaratmosphere can be derived from the Doppler shifts ofspectral lines:

= 0 = vc0,

where is the detected line shift due to the motionof the source of emission relative to the observer (i.e.,the Doppler shift), and 0 are the observed andrest wavelengths of the source, v is the LOS velocity(LOS projection of the velocity), and c is the speed oflight.

The original data obtained by the telescope havethe form of sequential solar spectrograms (bit mapsin jpg format) for the wavelength interval 649.38649.97 nm. The time intervals between the spectro-grams range from 15 to 30 s, depending on the dura-tion of the observing session (from one to four hours).As a rule, we obtained series of 512 spectrograms.The data processing mainly consists of three stages:preparation, computation, and analysis of the mapsobtained.

3.1. Data Preparation

The preparation stage mainly consists of choosingan appropriate approach to the preliminary reductionof the rst spectrogram and subsequently applying itto the entire series of images. As a rule, this approachincludes some standard programs, such as ResizeImage, Change to Grayscale, Negative Image, Ad-just Brightness/Contrast, Filter (blur more), etc. Thisenables us to chose an appropriate working region,remove local defects (scratches, damaged pixels), andincrease the contrast in order to determine the linecenters more precisely. The nal stage of the datapreparation is transforming the bit map into ASCIIcode. For this, we used special software that gener-ates a text le standard for the Pulkovo MFK-200photometric system [9].

ASTRONOMY REPORTS Vol. 51 No. 5 2007

OSCILLATIONS OF VELOCITIES IN AND NEAR SUNSPOTS 403

225 km 215 km 310 km 190 km 535 km 335 km

Ca 649.96 nmFe 649.89 nm

Ba 649.69 nmFe 649.65 nm

Fe 649.49 nm Ca 649.38 nm

Fig. 1. One of the spectrograms used. The height of the formation of each spectral line is indicated at the top, while thewavelength and the corresponding chemical element are indicated at the bottom.

3.2. Construction of Doppler Maps

Constructing the Doppler maps is the main stageof data processing that requires large computer re-sources and a long time. The construction of sevenDoppler maps (i.e., for seven spectral lines) from aseries of 512 spectrograms requires eight to ten hoursof computation time on an AMD Athlon XP 1700+processor.

For each line, we rst determine the boundariesof the spectral region, which are chosen taking intoaccount the halfwidth of the spectral line so as tomake the line prole in wings fairly at. This facilitatessuciently accurate determination of the position ofthe line maximum, and, consequently, the line shiftrelative to subsequent scans. The spectral lines arescanned numerically, i.e., each section yields a prole,with the central part of the prole approximated by asecond- or fourth-order polynomial. This procedureis necessary to eliminate any local prole surges,which can distort the real shift of the line center. Thesize of the approximated portion of the prole dependson the inclusion parameter, which is determined bythe decrease at a specied point in the prole, gener-ally 6080% of the central value.

The rst scan determines the position of the linecenter, which then becomes the reference point forthe subsequent scans. After scanning the line, we

obtain the shift vector, and then, after processing all512 spectrograms (four hours of observations), themap of Doppler shifts for the given spectral line. Tak-ing into account the dispersion for the given spectralregion, we nally obtain the map of Doppler velocities.

The nal stage of this phase of processing is lter-ing the maps, that is, eliminating trends associatedwith slight inclinations of the spectral lines or dis-tortions and removing various defects, such as moirepattern, ghosts, etc.

Further, we can proceed with the data analysis,since the spectrograph aperture selects a suitableregion on the Sun and the selected lines supply uswith the height distribution for the vertical velocities.Processing all the images for a given set results in atwo-dimensional array, with one coordinate showingthe calibrated velocity distribution along the spec-trograph aperture and the other the time evolutionof the velocities. This enables us to study oscillatoryprocesses simultaneously in all lines in the workinginterval for the array; that is, we can determine thevelocity phase shifts as a function of height in the solaratmosphere. Due to the shortness of the exposures,we obtain spectrograms with high spatial resolution,indicated by the zig-zag forms of the solar lines.


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Fig. 2. (a) Spectral interval 649.38649.96 nm with a sunspot. (b) Map of Doppler shifts calculated for the Fe 649.49 nmline (marked by an arrow in the top diagram). (c) Spectral power map (L diagram). The sunspot is No. 10 878, N14E22,May 3, 2006.

4. RESULTS AND DISCUSSION

Figure 2 presents the results for the Fe 649.49 nmline (the height of formation of the center of the lineprole is 500 km). Figure 2a (top) presents a spec-tral interval for a small sunspot. The horizontal axisshows the direction along the spectrograph aperture,with its height being 160. Figure 2b (middle) showsthe map of the Doppler shifts (dx map) for the sameFe 649.49 nm line obtained over the two-hour ob-servation (axis Y ); the calibrated velocity distributionalong the spectrograph aperture is shown along the

X axis. Figure 2c (bottom) presents the spectral mapof the oscillation power (L diagram); the Y axisshows the frequencies in milliHertz, while the X axiscorresponds to the direction along the spectrographaperture.

A simple visual analysis of the map of shifts(Fig. 2b) shows that the oscillation amplitude isstrongly suppressed inside the sunspot. This is espe-cially clear in the spectral power map (Fig. 2c). Thisrepresentation is quite convenient, and facilitates theeasy readability of the data, since we can see both



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Fig. 3. L diagram for the oscillation amplitudes obtained simultaneously in two spectral lines Fe 649.499 nm (top) andFe 649.647 nm (bottom). The sunspot is No. 10 878, N14W36, May 7, 2006.

the spatial power distributionthe spatial locationof the powerand the temporal distributiontheperiodicity of the process.

Figure 2 presents the Doppler shifts obtainedin a strong spectral line formed at high altitudes(500 km) in the photosphere. Since both oscillationmodes are inhibited in the sunspot compared withthe surrounding photosphere, the total spectral map(Fig. 2c) shows weak oscillations in the sunspot. Acomparative analysis of the height distributions of theamplitudes is considered below (see Fig. 3).

It turns out that, for all the spectral lines chosen,i.e., for virtually all photospheric heights from about190 to 500 km, the spectral power of the verticalvelocity oscillations is concentrated in two spectralbands, namely, 0.10.4 and 34.5 mHz, with theircentral frequencies corresponding to periods of 80 and5 min, respectively.

There is a wide gap with virtually no oscillationsbetween these well dened high-frequency and low-frequency bands. Figure 6 (lower map) shows thata similar spectral map constructed for a telluric linells the entire spectral power space (uniformly in thiscase), as should be the case for noise.

It is an important general property of the oscilla-tory processes studied that there is a very clear spatiallocalization of both modes; the sizes of the spatial

islands of power of the vertical oscillations are com-parable to each other, and are excited synchronouslyat all the photospheric heights studied using all theselected spectral lines.

However, comparing the spectral maps (L dia-grams) derived from weak and strong lines formed atdierent heights, we see that the height distributionsof the oscillatory processes are dierent, in spite of thegeneral features indicated above.

Figure 3 presents the spectral maps obtained inthe Fe 649.49 nm (upper) and Fe 649.65 nm (lower)lines, formed at photospheric heights of 500 and190 km, respectively. Both these maps display the twomain spectral bands of the oscillations, at relativelyhigh (3.24 mHz, 5 min) and low (0.20.3 mHz, 6080 min) frequencies. However, in the upper map, the5-minute component dominates around the sunspotand is virtually absent inside the sunspot, whereas,in the weak line formed in the lower photosphere,the amplitudes of the 5-minute and 80-minute os-cillations become comparable. Moreover, the low-frequency mode clearly dominates inside the sunspot,compared to the surrounding region. At these depths,the oscillations of the sunspot as a whole becomedominant.

Thus, the low-frequency oscillations are fairlyconcentrated in lower layers of the sunspot atmo-sphere, and the oscillation power decreases with


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Fig. 4. Local sections of the spectral map in the zone where the oscillation is excitated (scan 145) and between zones(scan 510). Plot A shows the original process, plot B its Fourier transform (power density spectrum), and diagram C itswavelet spectrum (the real Morley wavelet spectrum with various frequency resolutions). The sunspot is No. 10 878, N14W36,May 7, 2006.



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Fig. 5. Complex Morley wavelet spectrum for scan 145. A zone of regularity (wave train) is clearly visible; the wave trainoccupies 2530 min for the 5-min oscillations (ordinate 10), while the zone of regularity for the low-frequency oscillations(ordinate200), occupies the entire series of duration4 hr. The right window presents the global wavelet spectrum (the sumof amplitudes over the entire observation period for a given frequency).

height. It is possible that the dierent behaviorsof the low-frequency mode in the sunspot and themagnetic elements accompanying the sunspot is dueto the fact that the sunspot atmosphere is subjectto much stronger changes than the atmospheres ofsmall magnetic elements (which probably do not diermuch from the unperturbed photosphere). Additionaldetailed observations of various sunspots are neces-sary to clarify this point.

The sunspot boundaries are observed between the320th and 400th scans. The spatial distributions ofboth modes correspond to the scales of the mesogran-ulation (1012). It is obvious that the magneticeld is responsible for the clear spatial localizationof the power of the oscillation modes; i.e., the os-cillations are excited in locations where local mag-netic elements are observed. This close relationshipbetween the spatially localized oscillations of the LOSvelocities and the magnetic eld will be discussed indetail in a separate article using extensive recentlyaccumulated observational material, including bothDoppler images and high-resolution magnetograms.Our experience enables us to identify the places wherethe power of the LOS velocities are located as be-ing coincident with individual magnetic elements,suggesting that the oscillations of the low-frequencymode indicated by the LOS velocities are manifes-tations of vertical/radial oscillations of these mag-netic elements about their equilibrium positions in thephotosphere, in good agreement with the theoreticalmodel [16].

An important property of the oscillatory process isits stability, i.e., the duration of the oscillation excita-tion, or, in other words, the duration of the observed

wave trains. Figures 4 and 5 present the results of awavelet analysis examining this property. Since theoscillations are spatially localized, we used both thescan for the excitation zone (scan 145) and the scanbetween excitation zones (scan 510).

There are both modes in the power spectrum inscan 145 shown in Fig. 4, while the low-frequencymode is extremely weak in scan 510. The appear-ance of the 5-minute mode results from the spatialoverlap of the excitation zones for the correspondingfrequencies. There are virtually no scans without thismode, while the low-frequency mode is more spatiallylocalized, probably in association with the magneticeld in these zones. This suggests these two modesmay have dierent excitation mechanisms.

Figures 4 and 5 clearly show that the durations ofthe wave trains for the 5-minute mode are 30 min.At the same time, the low-frequency mode persistsboth in and near the sunspot during the entire ob-servation period (the individual sunspot No. 10 878,N14W36, May 7, 2006, four hours of observations).This again provides evidence for real natural oscilla-tions of the magnetic element as a whole, which iscontinually excited by external sources in the turbu-lent convective zone.

Figure 6 presents maps of the Doppler shiftsin the solar Ca 649.37 nm line (Fig. 6a) and theH2O 649.33 nm telluric line (Fig. 6b). Both the shiftmap and the spectral power map for the telluric line(Fig. 6d) show only white noise, whereas the calciumline clearly reveals oscillatory processes with thetypical space and frequency localizations. In this case,there is no low-frequency band of oscillations in the


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Fig. 6. Shift maps for the (a) Ca 649.379 nm solar line and (b) H2O 649.325 nm telluric line obtained simultaneously via asingle technique, together with their spectral shift maps (c) and (d).

upper map due to the short duration of the observingsession (about one hour).

Figure 7 presents the two observations of the cen-ter of the quiescent Sun on August 7 and 9, 2006. Theduration of each observation reaches four hours.

A comparative analysis of the power maps con-structed for the same spectral lines (Fig. 1) and mapsobtained for active regions near sunspots indicates

dierences only in the low-frequency portion of thespectrum, i.e., an absence of any signicant powerfor periods of 6080 min. Note that the signal-to-noise ratio decreases to ve for lines formed in thelower photosphere (h 150200 km), while this ratioreaches ten for higher lines (h 400500 km).This increase in the noise is explained by the decrease



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Fig. 7. Power maps constructed using the Doppler shift maps for the Fe 649.49 nm (upper, h 535 km) and Fe 649.89 nm(lower, h 215 km) lines.

in the accuracy of the Doppler shifts derived from theproles of weak lines.

Thus, no low-frequency oscillatons (periods of6080 min) were detected in the observations of thecentral region of the quiescent Sun. We associatethese oscillations to activity in the region observed,in particular, to magnetic islands in this region.

5. CONCLUSIONS

(1) Low-frequency oscillations of the line-of-sightvelocities with periods ranging from 40 to 80 minhave been detected in sunspots and at the locationsof magnetic elements accompanying the sunspots.The spatial distribution of the low-frequency modecorresponds to the scales of the mesogranulation(1012).

(2) The low-frequency amplitude in the sunspotrapidly decreases with height, being clearly visible ata level of 200 km but becoming barely perceptible at aheight of 500 km.

(3) The oscillations of the low-frequency modepersisted during an entire observing session (for upto four hours).

(4) Our results support the theoretical model [19],which describes global natural oscillations of mag-netic elements as a whole (verticalradial oscilla-tions) that move about a stable equilibrium position.

ACKNOWLEDGMENTS

This work was supported by the Program of thePresidium of the Russian Academy of Sciences So-lar Activity and Physical Processes in the SolarTerrestrial System and the Russian Foundation forBasic Research (project code 04-07-90254).

REFERENCES1. E. R. Priest, Solar Magnetohydrodynamics (Reidel,

Dordrecht, 1982; Mir, Moscow, 1985).2. V. N. Obridko, Solar Spots and Activity Complexes

(Nuak, Moscow, 1985) [in Russian].3. J. H. Thomas, L. E. Cram, and A. H. Ney, Nature 297,

485 (1982).4. T. J. Bogdan, Sol. Phys. 192, 373 (2000).5. Y. D. Zugzda, J. Staude, and V.A. Locans, Sol. Phys.

91, 219 (1984).6. V. I. Efremov and L. D. Parnenko, Astron. Zh. 73,

103 (1996) [Astron. Rep. 40, 89 (1996)].


410 EFREMOV et al.

7. V.I. Zhukov, Astron. Astrophys. 433, 1127 (2005).8. L.D. Parnenko, Sol. Phys. 213, 291 (2003).9. V. I. Efremov, R. N. Ikhsanov, and L. D. Parnenko, in

Proceedings of the Conference Solar Activity asFactor of Cosmic Weather, St. Petersburg, 2005,p. 643.

10. V. V. Borzov, G. F. Vialshin, and Yu. A. Nagovitsyn,Contrib. Astron. Obs. Skalnate Pleso 15, 75 (1986).

11. Yu. A. Nagovitsyn and G. F. Vyalshin, Astron. Tsirk.,No. 1533, 1 (1992).

12. A. A. Solovev and Yu. A. Nagovitsyn, in Proceedingsof the Conference Solar Activity as Factor of Cos-mic Weather, St. Petersburg, 2005, p. 593.

13. G. B. Gelfreikh, K. Shibasaki, E. Yu. Nagovitsyna,and Yu. A. Nagovitsyn, in IAU Symposium No. 223:Multi-Wavelength Investigations of Solar Activ-ity, Ed. by A. V. Stepanov, E. E. Benevolenskaya, andA. G. Kosovichev (Cambridge, Cambridge UniversityPress, 2004), p. 245.

14. G. B. Gelfreikh, Yu. A. Nagovitsyn, and E. Yu. Nagov-itsyna, Publ. Astron. Soc. Jpn. 58, 29 (2006).

15. G. B. Gelfreikh et al., in Proceedings of theVII Pulkovo Conference on Solar Physics. Climaticand Ecological Aspects of Solar Activity (GAORAN, 2003), p. 111.

16. A. A. Solovev, Astron. Zh. 61, 764 (1984) [Sov. As-tron. 28, 447 (1984)].

17. A. A. Solovev, Doctoral Dissertation in MathematicalPhysics (IZMIRAN, Moscow, 1992).

18. A. A. Solovev, in Proceedings of IX Pulkovo Con-ference on Solar Physics (GAO RAN, St. Peters-burg, 2005), p. 577.

19. A. A. Solovev and E. A. Kirichek, in Proceedingsof X Pulkovo Conference on Solar Physics (GAORAN, St. Petersburg, 2006), p. 496.

20. E. Wiehr and F. Kneer, Astron. Astrophys. 195, 310(1988).

Translated by V. Badin


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Long-period oscillations of the line-of-sight velocities in and near sunspots at various levels in the photosphere