The subterahertz solar cycle: Polar and equatorial radii derived from SST and ALMA
Fabian Menezes, Caius L. Selhorst, Carlos Guillermo Giménez de Castro, Adriana Valio
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The subterahertz solar cycle: Polar and equatorial radii derived from SST and ALMA
Fabian Menezes , Caius L. Selhorst , Carlos Guillermo Gim´enez de Castro ,
1, 3 and Adriana Valio Centro de R´adio Astronomia e Astrof´ısica Mackenzie (CRAAM), Universidade Presbiteriana Mackenzie, S˜ao Paulo, Brazil N´ucleo de Astrof´ısica, Universidade Cruzeiro do Sul / Universidade Cidade de S˜ao Paulo, S˜ao Paulo, SP, Brazil Instituto de Astronom´ıa y F´ısica del Espacio, UBA/CONICET, Buenos Aires, Argentina. (Received; Revised; Accepted)
Submitted to ApJABSTRACTAt subterahertz frequencies – i.e. , millimeter and submillimeter wavelengths – there is a gap of mea-surements of the solar radius as well as other parameters of the solar atmosphere. As the observationalwavelength changes, the radius varies because the altitude of the dominant electromagnetic radiationis produced at different heights in the solar atmosphere. Moreover, radius variations throughout longtime series are indicative of changes in the solar atmosphere that may be related to the solar cycle.Therefore, the solar radius is an important parameter for the calibration of solar atmospheric modelsenabling a better understanding of the atmospheric structure. In this work we use data from the So-lar Submillimeter-wave Telescope (SST) and from the Atacama Large Millimeter/submillimeter Array(ALMA), at the frequencies of 100, 212, 230, and 405 GHz, to measure the equatorial and polar radiiof the Sun. The radii measured with extensive data from the SST agree with the radius-vs-frequencytrend present in the literature. The radii derived from ALMA maps at 230 GHz also agree with theradius-vs-frequency trend, whereas the 100-GHz radii are slightly above the values reported by otherauthors. In addition, we analyze the equatorial and polar radius behavior over the years, by determin-ing the correlation coefficient between solar activity and subterahertz radii time series at 212 and 405GHz (SST). The variation of the SST-derived radii over 13 years are correlated to the solar activitywhen considering equatorial regions of the solar atmosphere, and anticorrelated when considering polarregions. The ALMA derived radii time series for 100 and 230 GHz show very similar behaviors withthose of SST.
Keywords:
Solar radius — Solar atmosphere — Solar radio emission INTRODUCTIONDue to technological limitations until some decades ago, only optical observations of the Sun were available. Ob-servations at radio wavelengths began to take place after 1950 (Coates 1958), and many authors had been usingmeasurements of the solar disk size – i.e. the center-to-limb distance – as ways to determine the solar disk radius(hereafter solar radius) at different radio wavelengths (Coates 1958; Wrixon 1970; Swanson 1973; Kislyakov et al. 1975;Labrum et al. 1978; F¨urst et al. 1979; Horne et al. 1981; Bachurin 1983; Pelyushenko & Chernyshev 1983; Wannieret al. 1983; Costa et al. 1986, 1999; Selhorst et al. 2004; Alissandrakis et al. 2017; Menezes & Valio 2017; Selhorst et al.2019b,a). There are different techniques to measure the solar radius at radio frequencies, such as the determinationfrom total solar eclipse observations (Kubo 1993; Kilcik et al. 2009), and from direct observations as the inflection pointmethod (Alissandrakis et al. 2017) and the the half-power method (Costa et al. 1999; Selhorst et al. 2011; Menezes &Valio 2017).
Corresponding author: Fabian [email protected] a r X i v : . [ a s t r o - ph . S R ] F e b Menezes et al.
Table 1.
Solar radius and altitude values at different frequencies and wavelengths.Authors Wavelength Frequency Radius Altitude(arcsec) (10 m)F¨urst et al. (1979) 1 dm 3 GHz 1070 ±
17 80 ± ± ± ± ± ± ± ± ± ± ± . ± . . ± . . ± . . ± . ± ± ± ± ± ± ± ± . ± . . ± . . ± . . ± . . ± . . ± . ± ± ± ± ± ± . ± . . ± . ± . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . The study of the radio solar radius provides important information about the solar atmosphere and activity cycle(Swanson 1973; Costa et al. 1999; Menezes & Valio 2017; Selhorst et al. 2019b). Using radio measurements of thesolar radius derived from eclipse and direct observations one can probe the solar atmosphere, since these measurementsshow the height above the photosphere at which most of the emission at determined observation frequency is generated(Swanson 1973; Menezes & Valio 2017; Selhorst et al. 2019b). However, as the observation frequency changes, theheight changes as well (Selhorst et al. 2004, 2019b; Menezes & Valio 2017). To determine the height above thephotosphere of the radio emission, we consider the optical solar radius. At optical wavelengths the canonical value ofthe mean apparent solar radius is R θ =959 . (cid:48)(cid:48) corresponding to R (cid:12) = 6 . × m. This value has been widelyused in the literature, hence we adopt it as the reference value. In this work we focus on the solar disk radius atsubterahertz radio frequencies – i.e. , millimeter and submillimeter wavelengths.Therefore, with observations at several frequencies, different layers of the solar atmosphere can be observed andstudied. Furthermore, these parameters can be used to improve and calibrate solar atmosphere models as an inputparameter and boundary condition. In other words, solar radius at radio frequencies reflect the changes in the localdistribution of temperature and density of the solar atmosphere. In Table 1 and Figure 3 we compiled data of the solarradius at several radio frequencies from different authors. We note, however, that the different works use differentdefinitions and methods for determining the solar radius.Another aspect to be considered is that the Sun’s radius measured at the same radio frequency over time shows slightvariations. Temporal series of observations obtained over many years show that the radius can be modulated with the11-year activity cycle (mid-term variations) as well as longer periods (long term variations), as suggested by Rozelot he subterahertz solar cycle . (cid:48)(cid:48) ± . (cid:48)(cid:48) . In a period of 3 years (from 1990 to 1993), temporal variations wereobserved following the linear relation R = [1 . − . − R (cid:12) , (1)which yields a total decrease of 8 (cid:48)(cid:48) if extrapolated for a period of 5.5 yr (half a cycle). Considering this short period, thedata suggest that the radius decreases in phase with the monthly mean sunspot number and the soft X-ray flux fromGOES (Geostationary Operational Environmental Satellite). At 37 GHz using data from Mets¨ahovi Radio Observatoryfrom 1989 to 2015, Selhorst et al. (2019a) measured a radius of 979 (cid:48)(cid:48) ± (cid:48)(cid:48) and obtained a positive correlation coefficientof 0.44 between the monthly averages of the solar radius and the solar flux at 10.7 cm. In Selhorst et al. (2004), theaverage solar radius from NoRH (Nobeyama Radioheliograph) daily solar maps at 17 GHz is found to be 976 . (cid:48)(cid:48) ± . (cid:48)(cid:48) . Over 11 years (one solar cycle), from 1992 to 2003, the variation in the solar radius is correlated with the sunspotcycle with a coefficient ρ = 0 .
88. However, the polar radius – measurements above 60 ◦ N and below 60 ◦ S of the solardisk – is anticorrelated with the sunspot cycle. The anticorrelation between polar radius and sunspot number yieldsa coefficient ρ = − . F . , and mean magnetic field, | B | , of the Sun. We analyze the equatorial andpolar radius behavior over time, with qualitative analysis for 100 and 230 GHz (ALMA) from 2015 to 2018. Also, wedetermine the correlation coefficient between the solar activity proxies and equatorial and polar radii time series at 212and 405 GHz (SST), from 2007 to 2019. The methodology used for that is an improved version of the methodologypresented in Menezes & Valio (2017) with a new approach and different analysis. In this work we focused on thebehavior of polar and equatorial solar radius over time. OBSERVATIONS AND DATAThe data used for the determination of the solar radii were provided by daily solar observations of the So-lar Submillimeter-wave Telescope (Kaufmann et al. 2008) at 212, and 405 GHz, and the Atacama Large Millime-ter/submillimeter Array (Wootten & Thompson 2009) at 100 and 230 GHz. The SST radio telescope was conceivedto monitor continuously the submillimeter spectrum of the solar emission in quiescent and explosive conditions. Inoperation since 1999, the instrument is located at CASLEO observatory (lat.: 31 ◦ (cid:48) . (cid:48)(cid:48) S; lon.: 69 ◦ (cid:48) . (cid:48)(cid:48) W),at 2552 m elevation, in the Argentine Andes. From its six receivers, we used data from one radiometer at 405 GHzand three at 212 GHz, which have nominal half-power beam widths, HPBW, of 2 (cid:48) and 4 (cid:48) (arcminutes), respectively(Kaufmann et al. 2008).The SST radiometers were upgraded in 2006 to improve bandwidth, noise and performance, and in 2007 the SSTreflector was repaired to provide better antenna efficiency, according to Kaufmann et al. (2008). Therefore, we useddata from 2007 onward. From 2007 to 2019 there were 3093 days of solar observation, with an average of approximately17 solar maps per day considering all receivers, resulting in an extensive data set of 36 034 maps (27 109 at 212 GHzand 8925 at 405 GHz). Most of the maps are obtained from azimuth and elevation scans from a 60 (cid:48) × (cid:48) area, witha 2 (cid:48) separation between scans and tracking speed in the range of 0 . ◦ and 0 . ◦ per second (Gim´enez de Castro et al.2020). For an integration of 0.04 second, for example, it results in rectangular pixels of 0 . (cid:48) × (cid:48) which are theninterpolated to obtain a square matrix (600 × ◦ (cid:48) . (cid:48)(cid:48) S;lon.: 67 ◦ (cid:48) . (cid:48)(cid:48) W), at 5000 m elevation. From four solar single-dish observation campaigns between 2015 and 2018,we use 196 fast-scan maps (125 at 100 and 71 at 230 GHz). These maps are derived from full-disk solar observations,which consist of a circular field of view of diameter 2400 (cid:48)(cid:48) using a “double-circle” scanning pattern (White et al. 2017).The nominal spatial resolutions, HPBW, are 58 (cid:48)(cid:48) and 25 (cid:48)(cid:48) at 100 (Band 3) and 230 GHz (Band 6), respectively. Figure1, bottom panels, shows examples of the ALMA maps obtained at 100 GHz (left panel) and 230 GHz (right panel),where the color represents the brightness temperature.
Menezes et al.
Figure 1.
SST maps (upper panels) obtained on 2008-02-09 and ALMA maps (lower panels) obtained on 2015-12-17. (a) 212GHz; (b) 405 GHz; (c) 100 GHz (green lines and crosses are the constraints of the polar region and in blue, the equatorial ones);(d) 230 GHz.
Radius Determination
Two widely used methods for measuring the radio solar radius are the inflection point method (Alissandrakis et al.2017) and the the half-power method (Costa et al. 1999; Selhorst et al. 2011; Menezes & Valio 2017). In Menezes et al.(2021), both methods are compared and it is shown how the combination of limb brightening of the solar disk withthe radio-telescope beam width and shape can affect the radius determination depending on the method. Menezeset al. (2021) showed that the inflection point method is less susceptible to the irregularities of the telescope beams andto the variations of the brightness temperature profiles of the Sun ( e. g. limb brightening level and active regions).Thus, the inflection point method brings a low bias to the calculation of the solar radius and, therefore, we use thismethod to determine the solar radius.The first step is to extract the solar limb coordinates from each map which are defined as the maximum and minimumpoints of the numerical differentiation red curve in Figure 2-a of each scan that the telescope makes on the solar disk.All solar maps are rotated so that the position of the solar North points upwards, and the coordinates are correctedaccording to the eccentricity of the Earth’s orbit – the apparent radius of the Sun varies between 975 . (cid:48)(cid:48) (perihelion)and 943 . (cid:48)(cid:48) (aphelion) during the year. During the limb points extraction, some criteria are adopted to avoid extractinglimb points associated with active regions, instrumental errors or high atmospheric opacity, which may increase thecalculated local radius in that region and hence the average radius. With this radius determination filter, only pointswith a center–to–limb distance between 815 (cid:48)(cid:48) (0 . R θ ) and 1100 (cid:48)(cid:48) (1 . R θ ) are considered.The second step is to calculate the average radius of each map. The solar limb coordinates are fit by a circle and theradius is calculated as the average of the center-to-limb distances. Successive circle fits are made until certain conditionsare met. For each fit, the points with center-to-limb distance (obtained from the fitting) outside the interval ( ¯ R − (cid:48)(cid:48) ,¯ R +10 (cid:48)(cid:48) ) are discarded, and then a new circle fit is performed with the remaining points. This process is repeated untilthere are at least 35% of the points, usually 6 out of 16 depending on the map and the latitude region – polar orequatorial. If there are fewer points remaining, the entire map is discarded; otherwise, the radius is calculated. If the he subterahertz solar cycle Figure 2.
Steps of solar radius measurement; (a) minimum and maximum points of the numerical differentiation of a scan (redcurve) corresponding to the limb coordinates (black crosses); (b) limb coordinates (black crosses) extracted from a solar mapwith a circle fit (red dashed line). radius value is between 800 (cid:48)(cid:48) and 1300 (cid:48)(cid:48) and the standard deviation is below 20 (cid:48)(cid:48) , then the calculated radius is storedand the next map is submitted to this process. The same method is applied to both telescopes, so that a comparisonbetween the 212 GHz radius of SST and the 230 GHz radius of ALMA could be made.As mentioned in Section 1, both the equatorial and polar radii are calculated. First, the radius of each map isdetermined using only the equatorial latitudes of the solar disk – points between 30 ◦ N and 30 ◦ S. Then, the polarradius is determined considering only points above 60 ◦ N and below 60 ◦ S. These latitude boundaries are depicted inFigure 1-c, in dashed blue (equatorial) and green (polar) lines. Finally, the visible solar radius is subtracted from thethe mean subterahertz radii to determine the altitudes in the atmosphere at which the 100, 212, 230, and 405 GHzemissions are predominantly produced.Even with the strict criteria adopted in the mean radius determination, there is still a large scattering in thedistribution of SST radius values. Thus, we applied a sigma clipping on the distribution, subtracting from thedistribution a running mean of 300 points, and then discarding values that are outside the ± . σ range. From theremaining values, we obtain the average radii.Gim´enez de Castro et al. (2020) have shown the influence of the SST irregular beams in the study of quiescent solarstructures. To assess the quality of the radius determinations, here we carry out a series of simulations convolving 2-Dantenna beam matrix representations with a solar disk with uniform temperature. The results show that azimuth-elevation maps increase the uncertainty of radius determination in the direction perpendicular to scans. Since SST hasan altazimuthal mount and maps are obtained at different times of the day, the uncertainty is uniformly distributedin the data set. 2.2. Observational Time Series
To analyze the radius variation over the solar activity cycle, we use the average taken every 6 months from thesolar radii and solar proxies. We use radius daily averages, smoothed over a 100-day period to build time series atradio frequencies. As solar activity proxies, we used the 10.7-centimeter solar flux, F . , and the intensity of themean photospheric magnetic field, | B | , both smoothed by a 396-day running mean (13 months) to avoid the influenceof annual modulations. Daily flux values of the 10.7-centimeter solar radio emission (Dominion Radio AstrophysicalObservatory, DRAO), given in solar flux units (1 SFU = 10 − Wm − Hz − ), is a very good proxy for solar activitycycle, as it is always measured by the same instruments, and has a smaller intrinsic scatter than the sunspot number.Another proxy used was the mean solar magnetic field, given in µT , provided by The Wilcox Solar Observatory (WSO;Scherrer et al. 1977). RESULTSWe used 29 088 SST maps (from a total of 36 034 maps) to calculate the radii at 212 GHz (24 186 maps) and 405GHz (4902 maps), and 196 ALMA maps to calculate the radii at 100 GHz (125 maps) and 230 GHz (71) maps, withthe method described in Section 2. 3.1.
Subterahertz Radii
Menezes et al.
Table 2.
Measured average radii and altitudes at subterahertz frequencies and radii derived from SSC modelFrequency Latitude Radius Radius Altitude SSC radius(GHz) (arcsec) (10 km) (10 km) (arcsec)100 Equatorial 968 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± Figure 3.
Solar radius as a function of frequency or wavelength. The dashed line represents the exponential trend of the radius.The black crosses are previous measurements from other authors (listed in Table 1), the blue (212 GHz) and red crosses (405GHz) are the present radius values derived from SST maps, the green (100 GHz) and yellow (230 GHz) crosses are the radiiderived from ALMA.
The average equatorial and polar radii are calculated at 100, 212, 230, and 405 GHz. Moreover, the correspondingheight with respect to the photosphere is deduced from the angular radii. In Table 2 the radii are listed by frequencyand latitude.Our results are plotted with those from other authors (listed in Table 1) in Figure 3. To guide the eye, an exponentialcurve (dashed line) is over plotted to show the trend of the radius as a function of the observing frequency or wavelength,indicating that the radius decreases exponentially at radio frequencies. Note that the trend curve is just a least-squareexponential fit, not a physical model. Our results are shown with green (100 GHz), blue (212 GHz), yellow (230 GHz)and red (405 GHz) crosses. SST and ALMA radii seem to agree within uncertainties with the trend of Figure 3 andthe solar atmospheric model predictions.Next, we used a 2-D solar atmospheric model developed by Selhorst et al. (2005) (hereafter referred to as the SSCmodel) to generate profiles of temperature brightness, T B , at 100, 212, 230, and 405 GHz, which yield radii of 964 . (cid:48)(cid:48) ,963 . (cid:48)(cid:48) , 963 . (cid:48)(cid:48) , and 962 . (cid:48)(cid:48) respectively (also listed in the last column of Table 2). Our results at 212, 230 and 405 he subterahertz solar cycle Figure 4.
Solar radius time series, R θ , at 212 and 405 GHz from January 2007 to December 2019. The top panel (a) show timeseries for equatorial (dark blue) and polar (light blue) radii at 212 GHz. The bottom panel (b) show time series for equatorial(red) and polar (pink) radii at 405 GHz. The orange and gray lines represent F . and | B | , respectively. Every box representsdata for a 6-month period: the horizontal line inside the box is the period median and the crosses represent the outliers of thedistribution. GHz are very close to the radii derived from the model, whereas the 100-GHz radii are about 4 (cid:48)(cid:48) bigger, not agreeingwith the model. 3.2.
Correlation with Solar Activity
We investigated the temporal variation of the solar radius and its relationship with the 11-year solar cycle. Thesubterahertz radius time series was analyzed from 2007 to 2016 using radii derived from the whole solar disk only(Menezes & Valio 2017). Here, we analyze a 13-year period – from January 2007 to December 2019 – using radiiderived from equatorial and polar latitude regions. As solar activity proxies, we used the 10.7-cm solar flux, F . , andthe mean solar magnetic field intensity, | B | . The results are plotted in Figure 4.We compared the equatorial and polar radius time series at 212 GHz (SST) with the time series of the equatorialand polar radii at 230 GHz (ALMA), with 6-month averages. The radii derived from SST’s maps are found to haveboth average values and behavior over time very close to ALMA’s, which can be seen in Figure 5. Moreover, besidesthe lower frequency and higher radius values, equatorial and polar radii at 100 GHz seem to have similar behaviorwith those of 212 and 230 GHz time series.The comparison of the solar radii in time with the solar proxies is summarized in Table 3, where the calculated cor-relation coefficients, ρ , are organized by frequency, solar latitude regions and solar proxies. The correlation coefficientsbetween equatorial radius, R equat. , and solar proxies are very low (0.05 and 0.16, respectively). However, for R polar the coefficients indicate weak anticorrelation (-0.36 for F . and -0.23 for | B | ). R equat. is moderately correlated with F . (0.64) and | B | (0.50), while R polar is weakly anticorrelated with F . (-0.39) and | B | (-0.23). In summary, theradii – R equat. , R polar , R equat. , and R polar – have stronger correlation with | B | than with F . . Menezes et al.
Figure 5.
Boxplot time series for equatorial R θ at 100 (green), 212 (dark blue) and 230 GHz (orange), and for polar R θ at100 (light green), 212 (light blue) and 230 GHz (yellow), from January 2014 to December 2019. Every box represents data fora 6-month period: the horizontal line inside the box is the period median. The dots represent the outliers of the distribution.The red line represents F . and the gray line represents | B | . Table 3.
Linear correlation coefficients, ρ , between solar radii and solar proxies.Frequency Latitude ρ F . ρ | B |
212 GHz Equatorial 0.05 0.16212 GHz Polar -0.36 -0.23405 GHz Equatorial 0.64 0.50405 GHz Polar -0.39 -0.234.
DISCUSSION AND CONCLUSIONSFrom the extensive SST and the ALMA data set we determined the polar and equatorial radii of the Sun at 100,212, 230, and 405 GHz. The average radii are in agreement with the radius-vs-frequency trend (Fig. 3), however thevalues obtained for the ALMA maps are higher than those obtained by Selhorst et al. (2019b) and Alissandrakis et al.(2017). The 100-GHz average radius is about 4 (cid:48)(cid:48) larger then that measured by Alissandrakis et al. (2017), and about2 (cid:48)(cid:48) larger then those measured by Labrum et al. (1978) and Selhorst et al. (2019b). Nevertheless, the 100-GHz radiuswas measured by Alissandrakis et al. (2017) and Selhorst et al. (2019b) using maps observed only in December 2015,and by Labrum et al. (1978) using observations of the total eclipse of 1976 October 23. Also there is a difference ofroughly 4 (cid:48)(cid:48) between our 100-GHz measurements and the radius derived from the SSC model. Menezes et al. (2021)showed that the radius increases as a function of the limb brightening intensity. By convolving the ALMA beam withthe SSC model (limb brightening level 33.6% above the quiet Sun level), they estimated an increase of 2 . (cid:48)(cid:48) on theradius at 100 GHz. The limb brightening levels could be increasing over time as the solar activity decreases, andtherefore affecting the solar radius.Moreover, we have analyzed the subterahertz solar radius time series for more than a solar cycle, over 13 years (2007– 2019), at 212 and 405 GHz. The radii time series are not strongly correlated with the solar proxies, however onecan observe particular behaviors of these time series, with the polar radius being anticorrelated and the equatorialradius being correlated with | B | . This is a similar behavior of the radio radius presented in the literature for lowerfrequencies Costa et al. (1999); Selhorst et al. (2004, 2019a), which are expected to be correlated to the solar cycle –positively for the average radius and negatively for the polar radius. The equatorial radii time series are expected to bepositively correlated to the solar cycle, since the equatorial regions are more affected by the increase of active regionsduring solar maxima, making the solar atmosphere warmer in these regions. On the other hand, the anticorrelationbetween polar radius time series and the solar activity proxies could be explained by a possible increase of polar limb he subterahertz solar cycle | B | . The polar radiitime series just increases from 2015 to 2018, which is the opposite of | B | . As the subterahertz radiation is influencedby Bremsstrahlung emission, the observed variations are expected since the solar atmosphere’s density, temperature,and magnetic field change with time. Longer periods of future observations at these frequencies will reveal the polarand equatorial trends of the solar atmosphere.Measuring the solar radius at subterahertz frequencies is not an easy task as well as to compare the values fromdifferent studies, since different instruments observe at different band widths. Our results are important to testatmospheric models, and better understand the solar cycle, since they probe directly different layers of the solaratmosphere over time. More studies of such kind at other frequencies and for longer periods of time are needed toachieve this goal. ACKNOWLEDGMENTSThe authors thank J. Valle, P. J. A. Sim˜oes, and D. Cornejo for fruitful discussions. F. M. thanks MackPesquisaand CAPES for the scholarship.We acknowledge the financial support for operation of the Solar Submillimeter Telescope (SST) from S˜ao PauloResearch Foundation (FAPESP) Proc. Complejo Astron´omico El Leoncito , operated under agreement between the
Consejo Nacional de Investigaciones Cient´ıficas y T´ecnicas de la Rep´ublica Argentina and the
National Universities of La Plata , C´ordoba and
San Ju´an .This paper makes use of the following ALMA data: ADS/JAO.ALMA
Alissandrakis, C. E., Patsourakos, S., Nindos, A., &Bastian, T. S. 2017, A&A, 605, A78,doi: 10.1051/0004-6361/201730953 Bachurin, A. F. 1983, Izvestiya Ordena TrudovogoKrasnogo Znameni Krymskoj AstrofizicheskojObservatorii, 68, 68 Menezes et al.
Coates, R. J. 1958, ApJ, 128, 83, doi: 10.1086/146518Costa, J. E. R., Homor, J. L., & Kaufmann, P. 1986, inSolar Flares and Coronal Physics Using P/OF as aResearch Tool, ed. E. Tandberg, R. M. Wilson, & R. M.HudsonCosta, J. E. R., Silva, A. V. R., Makhmutov, V. S., et al.1999, ApJL, 520, L63, doi: 10.1086/312132F¨urst, E., Hirth, W., & Lantos, P. 1979, SoPh, 63, 257,doi: 10.1007/BF00174532Gim´enez de Castro, C. G., Pereira, A. L. G., Valle Silva,J. F., et al. 2020, arXiv e-prints, arXiv:2009.03445.https://arxiv.org/abs/2009.03445Horne, K., Hurford, G. J., Zirin, H., & de Graauw, T. 1981,ApJ, 244, 340, doi: 10.1086/158711Kaufmann, P., Levato, H., Cassiano, M. M., et al. 2008, in