The distribution and characteristics of magnetic flux features observed by SDO/HMI

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1 The distribution and characteristics of magnetic flux features observed by SDO/HMI Oliver Allanson University of St Andrews September 21, 2011 Abstract Using line-of-sight magnetograms from the Solar Dynamics Observatory/Helioseismic and Magnetic Imager we find the flux of all observable magnetic features on 8 separate dates over a 17 month period. A clumping algorithm is used to identify flux features and their characteristics. Fluxes that range over approximately 5 orders in magnitude are found. We find that the distribution of feature fluxes obeys a power law of mean slope This could suggest that there exist scale free mechanisms which create surface features. Other characteristics of flux features, including the distribution of feature areas and the latitudinal variation of features are also considered. 1 Introduction Sunspot observation has long been a pastime, with Chinese astronomer Gan De being the first to record them in 364 BC 1. To the naked eye, sunspots are seen as dark spots on the visible surface of the Sun (photosphere) and they can be seen without a telescope. In his short article Solar Observations during 1843, Heinrich Schwabe proposed a sunspot activity cycle (solar cycle) of length roughly 10 years. These cycles were later found by Rudolf Wolf to have an average period of 11.1 years 2. We are currently over years into solar cycle 24, solar cycle 1 dated from The most recent peaks and troughs were 2000 and 2008 respectively, hence activity is now on the rise. George Ellery Hale (1908) discovered that strong magnetic fields thread sunspot regions indicating that unlike the Earth with it s dipole field, the photosphere is threaded by multiple regions of outwardly directed (positive) and inwardly directed (negative) magnetic fields. We now know that the photosphere is covered by magnetic features or massifs over many scales (we call contiguous regions of like-polarity magnetic flux (Φ), with Φ greater than some lower bound, a magnetic feature or flux massif). The largest of which can have area (A) of order cm 2 and Φ of order Maxwells (Mx); the smallest have A of order cm 2 and Φ of order Mx 4,5. The magnetic field drives and influences solar processes and phenomena, with features such as sunspots being a manifestation of the 1

2 magnetic field. Figure 2a shows the Sun on 15th April 2011 in visible light, sunspots can be seen as dark spots. Figure 2b is a magnetogram from the same date, white and black represent positive and negative fields respectively. The dark spots on figure 2a correspond to the larger white and black clusters on figure 2b. Supergranules are photospheric structures of dimensions km and they dominate plasma convection processes. Flux builds at the boundaries between supergranule cells and higher concentrations form when more flux appears in the upwelling in the centre of the supergranule. The footprints of the field migrate to the cell boundaries and the field strength B can rise. Over hours or days, if sufficient flux adds to the structure, a sunspot may form. They are often in pairs and disappear within days, but large sunspots can last months 6. If a significant amount of flux is not added during the growth stage, then the structure is destined to be a smaller scale, shorter lived feature. These can coalesce to form larger ones, and larger features can fragment to form smaller ones. Solar flares are the most violent explosions in our solar system and they occur in active regions around sunspots, often accompanied by a coronal mass ejection. Parnell et al. (2009) analysed magnetograms from Solar and Heliospheric Observatory (SOHO)/Michelson Doppler Imager(MDI) and Hinode/Narrowband Filter Imager (NFI). They identified magnetic features and found a power law distribution of feature fluxes over more than five decades of flux 7. However, the two instruments have different observed fields of view, resolutions, time cadence and sensitivity and thus they observe very different sizes of magnetic features. The overlap of fluxes between them was very small, less than one decade -see figure 1. Hence, confirmation of this result is required. The Solar Dynamics Observatory (SDO)/Helioseismic and Magnetic Imager (HMI) instrument is a full disc, high resolution instrument and can therefore observe features between /10 17 Mx and /10 23 Mx. Thus, it can bridge the gap in fluxes left by the NFI and MDI full disc data. The aim of this project is to identify fluxes in the HMI data and their distribution. Then we compare the results to those of Parnell et al (2009). Having found the features we then analyse their characteristics to see what these tell us about photospheric magnetic features in general. 2 Preparation of data We used line-of-sight magnetograms from HMI - an example of which is shown in figure 2b - for a range of different dates (see Table 1). Firstly, a cos α correction is applied to convert line of sight fields ( B) into radial ones, where α is the angle between the radial unit vector and the line of sight. We then eliminate noise by only considering pixels with radial B of at least 17 Gauss (G). 17G is chosen because it is twice the width at half maximum (2σ) for the Gaussian fit distribution of individual pixel values of B. By only considering B > 2σ we eliminate the 2

3 Figure 1: Histogram of flux massifs,parnell et al. (2009) (a) (b) Figure 2: (a)15th April 2011 at 450 nm - visible. (b)magnetogram for 15th April 2011 noise, considered to be 1σ. The noise could be due to the instrument, p-mode (pressure) oscillations on the photosphere (surface of Sun moving up and down) or even cosmic ray hits. The next task is to apply an area correction to the B values (detailed in Appendix 1). This is necessary since the equally sized pixels on a magnetogram correspond to nonequal areas on the photosphere. Pixels near the centre of the Sun and the magnetogram will represent much smaller areas than those pixels that project near to R S. However, at large latitudes the area correction yields very large results when applied, of which we are doubtful. Hence, we only consider data within ± 60 latitude. Once we have correctly found the photospheric area that each pixel corresponds to we can calculate the correct flux of each pixel since Φ = B A. Then a clumping algorithm is applied to identify magnetic features 8. This algorithm finds flux massifs - a collection of contiguous samesigned pixels with absolute values greater than a lower cutoff, such that each feature is the flux equivalent of a mountain massif 7. 3

4 Date N Range of Φ P Φ Φ Range of A P A Ā (Mx) (Mx) (Mx) ( cm 2 ) ( cm 2 ) ( cm 2 ) 12 May Jun Oct Nov Dec Apr May Jun Table 1: Characteristics of flux massifs. N is the number of flux features observed, A is the area of a flux feature with flux of magnitude Φ, Φ is the mean Φ of all features, Ā is the mean A of all features. Some characteristics of the flux massifs found in each month are detailed in Table 1. Note: My supervisor prepared the data as described above but we spent much time working together through it. I independently derived the area correction formula. We didn t identify reliable features until the 4 th week of the project and thus, beforehand, I spent considerable time checking the data for errors and inconsistencies that occurred in the translation between the raw data set and the identified features. For instance, the clumping algorithm might identify erroneously large flux features, often near R S. This suggested that B readings from beyond R S were inadvertently being counted as part of real features. Also, some clearly contiguous features were being identified as more than one feature by the algorithm. A particularly useful method I found to identify errors was to construct xy plots of the flux features, identified by the algorithm (see figure 3). The xy plot shows clear white space around the large flux features represented by red squares and orange crosses. This is good because it shows that the white area has been classified as one flux feature - the cross or square. Also, it was useful to compare the xy plot to figures 2a,b; then it was evident that the large features corresponded to the features on a magnetogram and even to those seen in visible light. 3 Results 3.1 Bridging the gap Parnell et al. (2009) found that flux massifs seem to follow a power law with power law index of when analysing data from NFI and MDI. Using SDO/HMI we can bridge the transition between the instrumental ranges of NFI and MDI. A log-log histogram showing the frequency of flux massifs found in the HMI magnetograms taken on 15th April 2011 is plotted in figure 4a. We can see a fall-off for low Φ, this is a result of the imposed lower cutoff Φ 0 = 17G. Hence, we ignore this tail when determining the true distribution. When plotted, the HMI data demonstrate a power law over roughly 5 orders of magnitude for each month detailed in the above table. To determine the distribution of fluxes from each data set we consider a probability 4

5 Figure 3: Image showing the classification of features identified in the magnetogram shown in Fig. 1. The active regions can be seen as orange and red, the more quiet regions as black and blue. The red band surrounding the plot represents the perimeter of the Sun. density function (PDF) of the form ( ) ( ) γ 1 Φ γ f(φ; γ) =, (1) Φ 0 where Φ are the fluxes greater than Φ 0 and γ is the index of the power law. We use maximum likelihood to estimate γ, and obtain a γ est, for a given minimum flux Φ 0 using the formula Φ 0 γ est = 1 N NlogΦ 0 N i=1 logφ i where N is the number of flux massifs and Φ i are observed fluxes. However, maximum likelihood does not test whether or not a power law is a good fit to the data, it just tells us the most probable power law that fits the given data. We can construct a model cumulative distribution function (CDF) and an empirical CDF for our data - detailed in Appendix 2. Then, from these we can use the Kolmogorov-Smirnov (K-S) goodness of fit test to determine the validity of our power law interpretation. The K-S statistic is determined by (2) KS = max F (Φ i ; γ est ) F emp (Φ i ) (3) i Here, F(Φ; γ est ) is the model CDF and F emp (Φ i ) is the empirical CDF. The smaller the value of KS, the better the power law fits the data. 5

6 (a) 15 th April 2011 (γ = 1.916) (b) Without correction factor (c) With correction factor Figure 4: (a) Distribution of flux massifs on 15 th April 2011 identified from SDO/HMI. (b) distributions from Parnell et al (2009) with the distributions I identified from SDO/HMI overplotted. (c) Same distributions as (b), but with a global factor of 2.5 applied to those identified by SDO/HMI. Figure 5 shows how the K-S statistic varies with the choice of cutoff (Φ 0 ) and then there is a plot of F (Φ; γ est ) against F emp (Φ) with Φ 0 = Mx and y = x overplotted. If the two lines in figure 4b perfectly overlap, then the K-S statistic is 0 and the data is a perfect power law. From figure 6a it is evident that our calculation of γ est clearly depends on our choice of Φ 0. Ideally, we would choose γ est by minimising KS. But, as is seen from figure 6b this may mean that large, important portions of data are neglected and we get a skewed result. Figure 6b shows how the number of data points depends on Φ 0. What might seem at first glance a quite trivial change in Φ 0 will actually make a big difference in the proportion of original data points left. Hence, when determining the best γ est, local minima were sometimes chosen rather than the global minimum when it seemed prudent. The mean γ est value was found to be with standard deviation This is greater than the value found by Parnell et al. (2009) of -1.87; but within the error bars 6

7 (a) K-S Statistic (b) Model CDF vs Empirical CDF Figure 5: 15 th April 2011, Statistical Plots. (a) K-S statistic as a function of Φ 0 and (b) Model CDF against the empirical CDF for a Φ 0 = Mx. (a) γ vs Φ 0 (b) N/N 0 vs Φ 0 Figure 6: 15 th April 2011, (a) Power law index γ against Φ 0 and (b) N/N 0 - where N is the number of features with Φ > Φ 0 and N 0 is the total number of features observed - versus Φ 0 of ± Figure 4b is a plot of all histograms from HMI data and MDI/NFI data. The slopes of each histogram are evidently similar but the HMI frequencies are globally lower than the MDI frequency and the NFI frequency. SDO is a mission in it s infancy, having only been launched in February 2010, hence it is quite possible that there may be calibration issues yet to be resolved. If we assume a global correction of 2.5 to our data then the histograms match up very well - see figure 4c. Either way, figure 4 strongly supports a power law distribution across 5 decades of flux, Φ.The crucial fact being that the region between Φ = Mx and Φ = Mx has been shown to adhere to a power law interpretation. 7

8 3.2 Characteristics of flux features From Table 1 we can see that N barely changes from the most quiet Sun (May 10) to the most active Sun (June 11) that we observed. N decreases by 2% whereas Φ increases by an order of magnitude. Here, we are measuring the relative activity of the Sun by Φ. This is surprising as it has been thought that during active times - when there are many sunspots - N decreases. Also, Φ varies with time and is proportional to Φ. This suggests that Φ and Φ are equivalent measures of activity. It is also clear that Ā is proportional to the activity of each date. However, the maximum Φ identified does not seem to be proportional to the maximum A identified on each date. This inconsistency suggests that measuring the activity by using maxima is inappropriate. Figure 7: Distribution for A on 15 th April 2011 Figure 7 is the area distribution of flux massifs on 15 th April It demonstrates a power law over roughly 4 decades of area. Similar distributions were observed for the other 7 dates. This would appear to lend more credence to the scale free mechanism argument. Figure 8a identifies the percentage of total flux observed on 4th June 2011 with respect to the latitude. The distribution is peaked at two distinct locations, but comparitively minimal elsewhere. This behaviour is also evident in each of the other dates that we observed, sometimes one peak only is evident, as in the case for 15th April 2011 where there was a particularly high flux feature concentration near to the equator. These peaks are due to sunspots which have very high flux concentrations. Sunspots are observed to originate near ±40 at the solar minimum (the beginning of the solar cycle) and as the solar cycle heads towards maximum the sunspot concentrations move towards ±5. See figure 9 for a butterfly diagram detailing this progression. 4 th June 2011 is the date observed to have the largest Φ and figure 8a shows the 8

9 (a) Percentage of total flux (b) Number of massifs with Φ > Φ i Figure 8: Latitude plots for 4 th June (a) percentage of Φ identified at particular latitudes. (b) plot of the number of features found with Φ > Φ i in each latitude band 5 wide. Φ i increases on a logarithmic scale as histograms go from purple to orange. Figure 9: Butterfly diagram; The latitudinal progression of sunspots with respect to solar cycles since peak flux concentration near to the equator. This is surprising as it suggests that we may be near solar maximum. For the same date, figure 8b plots the number of features with flux identified above some Φ i, with Φ i increasing on a logarithmic scale. As Φ i increases - purples to oranges - the distribution sinks whilst maintaining a rectangular shape. But, once Φ i is sufficiently large, the distribution begins to peak at latitudes common to figure 8a. Hence, it is clear that relatively small scale flux features can, and do, appear at any latitude on the photosphere. But, the largest scale features are localised to specific bands, this varying in time as described above. Figures 8a,b do not show a uniform distribution for the non-peaked latitudes, we can observe a fall-off at high latitudes. I suspect that this is due to what is effectively a lower instrumental resolution. Since one pixel near the centre of a magnetogram projects to a smaller area on the photospherethe area distribution than a pixel further from the centre, it is possible that we pick up fewer features at these 9

10 locations, hence the fall-off. 4 Conclusion and discussion I have identified a power law distribution over all observable scales (5 decades) of flux for flux massifs, bridging the gap for Parnell et al (2009). This strongly suggests that the mechanism generating features of all scales is the same. See Parnell et al. (2009) for a discussion. It also seems evident that there exists a power law over roughly 4 decades in area for flux massifs When creating histograms we have to choose a bin size. This is to some degree an arbitrary choice and has an effect on the power law index γ. For consistency, I used the same bin size for the flux histograms as in Parnell (2009). Another choice made is that of the cutoff flux Φ 0. This certainly has an effect on γ but considerable effort was made to choose an appropriate Φ 0 for each frame analysed (as discussed previously). In essence, both the choice of Φ 0 and the bin size can change γ, but variations of the scale that were used would not fundamentally change the inherent distribution. Something of note is a flick observed in the large flux tail of figure 4a. This could be a real flick in the distribution, or it could be down to our choice of dates to observe. We purposefully chose dates that were quite active and thus had many sunspots. But, the lifetime of sunspots can be days or even hours whilst the lifetime of smaller scale features can be minutes. Therefore, since we identified months with a greater than average number of sunspots and did not observe them over their whole lifetime, we may have got a flick. The area distribution does not have such a noticeable flick, which could suggest that the one found in the flux distribution is real. It is not my intention that the correction factor discussed in Section 3 be interpreted as a fix for SDO/HMI data. It is meant to demonstrate how different methods of data preparation between MDI/NFI and SDO/HMI could explain why the histograms in this paper do not overlap with those from Parnell et al (2009) as discussed previously. The crucial fact is that a power law relationship (with very similar index, γ) over 5 decades has been identified, regardless of this (possible) discrepency. 4.1 Future Work In 1993, C.J. Schrijver and K.L. Harvey purported a straight line relationship between the flux of a magnetic feature and it s area using 122 data points. I have looked into the relationship between Φ and the area with our much larger data set of massifs: collating all the data into one plot. A straight line fit is not evident; a polynomial fit seems more likely. This could be a topic for further investigation. I would also like to investigate whether or not the gradient of Φ as a function of area is itself a function of the relative activity of the Sun. Linked to this is the flick, identified 10

11 in the flux distributions. If the distributions were cut just before they all flicked, it would be useful to know whether or not the characteristics in Table 1 would become less variable. In other words, are the characteristics of flux features on the photosphere constant once the largest, highest concentration features are disregarded? If so, this would imply that high solar activity only implies the existence of more numerous larger scale features, not the maturation of existing small scale features. Also, we did not track features in time - from one frame to the next - for this project. Tracking could also constitute further work as this would enable the observation of a sunspot over it s entire lifespan. I have investigated the distributions for both Φ and A and found power laws, this would suggest a power law also exists for the mean field, B of flux massifs. This would also be interesting to investigate. 5 Appendix Area Correction The area of a magnetogram pixel corresponds to a projected area on the surface of the Sun. This projected area is dependent on θ and φ. Here I calculate the area correction (or projection ) factor for each pixel. A projection factor is required so that we can correctly calculate the flux that each pixel of the magnetogram holds Area of a pixel projection on the surface of the Sun Let da pp be an infinitesimal area projected on to the surface of the Sun by an infinitesimal area on a 2D grid described by θ 1, θ 2, φ 1, φ 2. Let φ be the azimuthal angle with range (0,π) and θ be the inclination angle with range (0,π).Then which gives da pp = Let θ 2 = θ 1 + δθ and φ 2 = φ 1 + δφ. φ2 θ2 φ 1 θ 1 r 2 sin θ dθ dφ da pp = r 2 (φ 2 φ 1 )[ cos θ] θ 2 θ 1 This gives da pp = r 2 δφ(cos θ 1 cos θ 1 cos δθ+sin θ 1 sin δθ) using trigonometric formulae. If we let δθ,δφ 0 then cos δθ 1 and sin δθ δθ and hence da pp = r 2 δθδφ sin θ 1. (4) Area of pixel as a function of θ, φ 1 We now calculate an infinitesimal area on the planar magnetogram, as a function of θ and φ. 11

12 From figures 10a,b it can be seen that. y = r cos θ 2 r cos θ 1 for all values of φ and x = sin θ 2 (r sin φ 2 r sin φ 1 ). (a) y derivation (b) x derivation Figure 10: Figure 10a shows a vertical cut through the Sun from a side-on view, aiding the derivation of y. Figure 10b shows a horizontal cut at θ = π 2 derivation of x Using trigonometric formulae and then taking limits as before gives by equation (4). from a top-down view, aiding the x y = r 2 δθδφ sin θ 1 sin θ 2 cos φ 1 = sin θ 2 cos φ 1 da pp (5) If we describe our pixels by this method then the area of a pixel is x y. Note: This is an approximation. We are assuming that our pixels are very small in comparison to the Sun. Using equation (5), we see that our projection factor is sin θ 2 cos φ 1. But, sin θ 2 = sin(θ 1 + δθ) = sin θ 1 cos δθ + sin δθ cos θ 1. Therefore, sin θ 2 sin θ 1, for small δθ. Hence, our projection factor, pf, is pf = sin θ 1 cos φ 1 (6) Projection factor It would be useful to have pf in terms of x,y coordinates. If we associate (x 1, y 1 ) with the top-left corner of a pixel. Then and y 1 = r cos θ 1 x 1 = r sin θ 1 sin φ 1 = (r 2 y1) 2 1/2 sin φ 1. ( ) 1/2. Hence, cos φ 1 = 1 x2 1 r 2 y This means that we can now express our pf in terms 1 2 of x 1 and y 1 as ( ) ( ) pf = + 1 y2 1 r 2 1 x2 1 1/2 r 2 y1 2 (7) We use modulus brackets to prevent taking the square root of a negative. 12

13 6 Appendix CDF Mathematics A probability density function (PDF) is a function of a random variable f(φ) such that P (Φ 1 < Φ < Φ 2 ) = Φ2 Φ 1 f(φ) dφ. We are interested in a power law of the form 0 0 Φ < Φ f(φ) = ( ) ( ) 0 γ γ 1 Φ (8) Φ 0 Φ 0 Φ Φ 0 This is a normalised PDF - proof of normalisation performed but omitted. If we wish to know the probability that Φ X then we consider a (right) cumulative distribution function (CDF) F R (Φ), such that F R (X) = X ( ) X 1 γ f(φ)dφ =. This is our model CDF. Since, γ > 1, clearly F R (Φ 0 ) = 1 and F R 0 as X. If we order our data set according to size Φ 1,Φ 2,...,Φ N where Φ i Φ 0 - and we believe that the set adheres to a PDF of the form above - then we can create an empirical CDF of the form Fi = 1 i 1 2 N where i=1,...,n. This is a monotonically decreasing function with F1 1 and F N 0 for large N, i.e. a large data set. If we then plot the model CDF against the empirical CDF we can graphically assess the goodness-of-fit of our model. A straight y = x line implies a perfect fit 10. Φ 0 7 Comments I would like very much to thank my supervisor - Professor Clare Parnell - for giving up so much of her time to steer an initially IDL/Latex/Linux illiterate through his project. I would also like to express my gratitude to the RSE for awarding my Scholarship. 8 References 1. NRICH: (University of Cambridge) [ 2. St Andrews History of Maths Website [ 3. [ 4. Schrijver,C.J,& Zwaan,C.2000,Solar and Stellar Magnetic Activity (Cambridge:Cambridge Univ.Press). 5. Solanki,S.K.,Inhester,B., & Schusssler, M.2006, Rep. Prog. Phys., 69, Priest,E.R.1982,Solar Magneto-hydrodynamics(Dordrecht:Reidel) 13

14 7. Parnell,C.E., et al. 2009,ApJ,698, Parnell,C.E.,2002,MNRAS,335, Schrijver,C.J. and Harvey,K.L.,1994,Solar Physics,150, Parnell,C.E. and Jupp,P.E.,1999,ApJ,529,