Relative Timing and Spectra of Solar Flare Hard X-ray Sources

Hard X-ray lightcurves, spectrograms, images, and spectra of three medium-sized flares observed by the Reuven Ramaty High-Energy Solar Spectroscopic Imager (RHESSI) are presented. Imaging spectroscopy of the 20 February 2002, 11:06 UT flare at 10′′ spatial resolution, comparable to the best previous hard X-ray imaging from Yohkoh, shows two footpoints with an ∼ 8 s delay of peak emission between footpoints. Subsequent imaging at le4′′ shows three sources consistent with two separate loops and simultaneous brightening in connected footpoints. Imaging for the simple two footpoint flare of 2 June 2002 also shows simultaneous footpoint brightening. The more complex 17 March 2002 flare shows at least four different sources during the main peak of the event, and it is difficult to clearly demonstrate simultaneous brightening of connected footpoints. Non-thermal power laws are observed down to ∼ 12–13 keV without flattening in all these events, indicating the energy content in energetic electrons may be significantly greater than previously estimated from assumed 25 keV low energy cutoff. Simultaneously brightening footpoints show similar spectra, at least in the three flares investigated. Double-power-law spectra with a relatively sharp break are often observed. Supplementary material to this paper is available in electronic form at http://dx.doi.org/10.1023/A:1022469902940     

Tracking of TRACE Ultraviolet Flare Footpoints

Solar flares produce bright, compact sources of UV emission in the lower atmosphere, identified as flare footpoints. Observed at high time cadence with the Transition Region and Coronal Explorer, groups of UV footpoints define flare `ribbons' which move as the flare progresses. We have developed a procedure to track individual bright kernels within flare ribbons, enabling us to study the motion of these sites of excitation through the solar chromosphere. We have applied this to a flare observed by TRACE in the 1600 Å passband at 2-s cadence. In this event, the footpoints have an average speed of 15 km s<sup>–1</sup>, with a superposed random `meandering' component, consistent with the footpoint magnetic field being anchored around the edges of granular cells. Examining the brightness as a function of time, we find that the timing of peaks in brightness is significantly correlated with the timing of peaks in the product of the footpoint speed with the line-of-sight magnetic field strength at the footpoint location; in other words with a measure of the coronal reconnection rate.     

Magnetic Twist and Writhe of δ Active Regions

We have selected 104 active regions with a δ magnetic configuration from 1996 to 2002 to study how important a role the kink instability plays in such active regions. In this study, we employ the systematic tilt angle of each active region as a proxy for the writhe of a flux tube and the force-free parameter, α_best, as a proxy for the magnetic field twist in the flux tube. It is found that 65–67% of the active regions have the same sign of twist and writhe. About 34% (32%) of the active regions violate (follow) the Hale-Nicholson and Joy's Laws (HNJL) but follow (violate) the hemispheric helicity rule (HHR). Sixty-one (61) of the 104 active regions studied each produced more than five large flares. Active regions violating HNJL, but following HHR, have a much stronger tendency to produce X-class flares and/or strong proton events. Comparing with previous studies for active regions with well-defined (simpler) bipolar magnetic configuration, it is found that the numbers following both HNJL and HHR are significantly lower in the δ-configuration case, while numbers violating one of the laws and the rule significantly increase with the increase of the magnetic complexity of the active regions. These results support the prediction for the presence of a kink instability, that the twist and writhe of the magnetic fields exhibit the same sign for δ active regions (Linton et al., Astrophys. J. 507, 40, 1998, Astrophys. J. 522, 1205, 1999; Fan et al., Astrophys. J. 521, 460, 1999). Finally, we analyze possible origins of the twist and writhe of the magnetic fields for the active regions studied.     

Evidence for Tether-Cutting Reconnection in a Quadrupole Magnetic Configuration in the April 9, 2001, M7.9 Flare

We studied the M7.9 flare on April 9, 2001 that occurred within a δ-sunspot of active region NOAA 9415. We used a multi-wavelength data set, which includes Yohkoh, TRACE, SOHO, and ACE spacecraft observations, Potsdam and Ondřejov radio data and Big Bear Solar Observatory (BBSO) images in order to study the large-scale structure of this two-ribbon flare that was accompanied by a very fast coronal mass ejection (CME). We analyzed light curves of the flare emission as well as the structure of the radio emission and report the following: the timing of the event, i.e., the fact that the initial brightenings, associated with the core magnetic field, occurred earlier than the remote brightening (RB), argue against the break-out model in the early phase of this event. We thus conclude that the M7.9 flare and the CME were triggered by a tether-cutting reconnection deep in the core field connecting the δ-spot and this reconnection formed an unstable flux rope. Further evolution of the erupted flux rope could be described either by the “standard“ flare model or a break-out type of the reconnection. The complex structure of flare emission in visible, X-ray, and radio spectral ranges point toward a scenario which involves multiple reconnection processes between extended closed magnetic structures.     

Interpreting Yohkoh hard and soft X-ray flare observations

A simple model is presented to account for theYohkoh flare observations of Feldmanet al. (1994), and Masuda (1994). Electrons accelerated by the flare are assumed to encounter the dense, small regions observed by Feldmanet al. at the tops of impulsively flaring coronal magnetic loops. The values of electron density and volume inferred by Feldmanet al. imply that these dense regions present an intermediate thick-thin target to the energised electrons. Specifically, they present a thick (thin) target to electrons with energy much less (greater) thanE _c , where 15 keV <E _c < 40 keV. The electrons are either stopped at the loop top or precipitate down the field lines of the loop to the footpoints. Collisional losses of the electrons at the loop top produce the heating observed by Feldmanet al. and also some hard X-rays. It is argued that this is the mechanism for the loop-top hard X-ray sources observed in limb flares by Masuda. Adopting a simple model for the energy losses of electrons traversing the dense region and the ambient loop plasma, hard X-ray spectra are derived for the loop-top source, the footpoint sources and the region between the loop top and footpoints. These spectra are compared with the observations of Masuda. The model spectra are found to qualitatively agree with the data, and in particular account for the observed steepening of the loop-top and footpoint spectra between 14 and 53 keV and the relative brightnesses of the loop-top and footpoint sources.     

Implications of rapid footpoint motions of photospheric flux tubes for coronal heating

Some recent observations at Pic-du-Midi (Mulleret al., 1992a) suggest that the photospheric footpoints of coronal magnetic field lines occasionally move rapidly with typical velocities of the order 3 km s^–1 for about 3 or 4 min. We argue that such occasional rapid footpoint motions could have a profound impact on the heating of the quiet corona. Qualitative estimates indicate that these occasional rapid motions can account for the entire energy flux needed to heat the quiet corona. We therefore carry out a mathematical analysis to study in detail the response of a vertical thin flux tube to photospheric footpoint motions in terms of a superposition of linear kink modes for an isothermal atmosphere. We find the resulting total energy that is asymptotically injected into an isothermal atmosphere (i.e., an atmosphere without any back reflection). By using typical parameter values for fast and slow footpoint motions, we show that, even if the footpoints spend only 2.5% of the time undergoing rapid motions, still these rapid motions could be more efficient in transporting energy to the corona than the slow motions that take place most of the time.     

Two-Dimensional Spectroscopy of Photospheric Shear Flows in a Small <Emphasis Type="Italic">δ</Emphasis> Spot

In recent high-resolution observations of complex active regions, long-lasting and well-defined regions of strong flows were identified in major flares and associated with bright kernels of visible, near-infrared, and X-ray radiation. These flows, which occurred in the proximity of the magnetic neutral line, significantly contributed to the generation of magnetic shear. Signatures of these shear flows are strongly curved penumbral filaments, which are almost tangential to sunspot umbrae rather than exhibiting the typical radial filamentary structure. Solar active region NOAA 10756 was a moderately complex β δ sunspot group, which provided an opportunity to extend previous studies of such shear flows to quieter settings. We conclude that shear flows are a common phenomenon in complex active regions and δ spots. However, they are not necessarily a prerequisite condition for flaring. Indeed, in the present observations, the photospheric shear flows along the magnetic neutral line are not related to any change of the local magnetic shear. We present high-resolution observations of NOAA 10756 obtained with the 65-cm vacuum reflector at Big Bear Solar Observatory (BBSO). Time series of speckle-reconstructed white-light images and two-dimensional spectroscopic data were combined to study the temporal evolution of the three-dimensional vector flow field in the β δ sunspot group. An hour-long data set of consistent high quality was obtained, which had a cadence of better than 30 seconds and subarcsecond spatial resolution.     

Quantitative analysis of hard X-ray ‘footpoint’ flares observed by the Solar Maximum Mission

We describe the instrumental corrections which have to be incorporated for reliable correction and deconvolution of images obtained in the 16–22 keV and 22–30 keV energy bands of the Hard X-Ray Imaging Spectrometer (HXIS) aboard the Solar Maximum Mission (SMM). These corrections include amplifier gain and collimator hole size variations across the field of view, amplifier/filter efficiency, variation in effective collimator hole size and angular response with photon energy, dead-time, and hard X-ray plate transmission. We also emphasise the substantial Poisson noise in these energy bands, and describe the maximum entropy deconvolution/correction routine we have developed to establish the spatial structure which can be reliably inferred from HXIS data.

Next we discuss the results of application of our routine to the three impulsive flare phases reported by Duijveman et al. (1982) as exhibiting hard X-ray ‘footpoints’, namely 1980, April 10, May 21, and November 5. Our main conclusions are:

1. (1)

   Maximum entropy smoothing and Poisson noise data perturbations do not remove the main footpoint features in 16–30 keV nor change their basic morphology. However the results emphasise the asymmetry in footpoint size in the May 21 flare and confirm its possible presence in April 10. They also reveal the 3rd weak distant footpoint in the May 21 flare at an earlier time than found by Duijveman et al.

When the 16–22 and 22–30 keV bands are analysed separately, however, it is found that the footpoints are much less visible above noise in the harder band - i.e. the footpoint spectra are steep. In the April 10 and November 5 flares they are steeper than either the spectrum of intervening pixels or the spectrum at higher energies measured for the whole flare by the SMM Hard X-Ray Burst Spectrometer (HXRBS).

1. (2)

   The footpoint contrast with surroundings is less than found by Duijveman et al., despite image deconvolution, because of the maximum entropy smoothing of noise.

2. (3)

   The 16–30keV HXIS footpoint fluxes in the three flares are respectively 28%, 17%, and 15% of the simultaneous HXRBS flare power-law spectrum extrapolated into this energy range.

3. (4)

   Where Poisson noise is taken into account we find, by cross-correlating pixel count rates, that footpoint synchronism was either not provable at all, or substantially less close than reported by Duijveman et al.

Next we considered the implications of these results for models of the footpoint emission. Contrary to Duijveman et al. we do not consider the HXIS ‘footpoint’ data as supporting a conventional thick target beam interpretation since:

1. (A)

   The footpoint photon (and electron) fluxes are much less than expected from HXRBS extrapolation. This result casts doubt on recent models of chromospheric heating by electron beams which usually assume all of the HXRBS emission to come from HXIS footpoints.

2. (B)

   The footpoint spectra for the April 10 and November 5 flares are much softer than the HXRBS spectrum and than the spectrum of intervening pixels, contrary to thick target predictions.

3. (C)

   Contrary to Duijveman et al. footpoint synchronism does not demand an unreasonable Alfvén speed and so does not require non-thermal particles.

In spite of these objections we also re-considered the constraints placed on the acceleration site conditions in a beam interpretation by return current stability and footpoint contrast in the summed 16–30 keV range. Using the smoothed maximum entropy contrast and taking explicit account of coronal thermal emission, we find maximum densities somewhat larger than Duijveman et al. estimated, and much higher maximum values of T _e /T _i .

Regarding thermal interpretations we found:

1. (a)

   Models involving continuous production of short-lived hot kernels in the arch top with Maxwellian tail electrons escaping to the footpoints could explain the 16–30 keV contrast with a rather higher energetic efficiency than a pure beam model. However, whatever the temperature distribution of hot kernel production, the model predicts footpoints harder than the arch summit, contrary to HXIS data.

2. (b)

   A model with hot kernels produced in one limb of an arch can explain the asymmetry in footpoint size observed in May 21, and probably April 10, and is energetically even more efficient than (a) but is also inconsistent with the spectral data.

3. (c)

   Finally we point out that HXIS footpoint data may be consistent with a purely geometric interpretation in an almost uniform arch filled with hot plasma.

     

The evolution of line-tied coronal arcades including a converging footpoint motion

It has been demonstrated in the past that single, two-dimensional coronal arcades are very unlikely driven unstable by a simple shear of the photospheric footpoints of the magnetic field lines. By means of two-dimensional, time-dependent MHD simulations, we present evidence that a resistive instability can result if in addition to the footpoint shear a slow motion of the footpoints towards the photospheric neutral line is included. Unlike the model recently proposed by van Ballegooijen and Martens (1989), the photospheric footpoint velocity in our model is nonsingular and the shear dominates everywhere. Starting from a planar potential field geometry for the arcade, we find that after some time a current sheet is formed which is unstable with respect to the tearing instability. The time of its onset scales with the logarithm of the magnetic diffusivity assumed in our calculation. In its nonlinear phase, a quasi-stationary situation arises in the vicinity of the x-line with an almost constant reconnection rate. The height of the x-line above the photosphere and the distance of the separatrix footpoints remain almost constant in this phase, while the helical flux tube, formed above the neutral line, continuously grows in size.     

Helicity Patterns of the Active Regions Connected by Transequatorial Loops

Using photospheric vector magnetograms of the Huairou Solar Observing Station and coronal X-ray images from the Yohkoh Soft X-Ray Telescope, we calculate the helicity patterns of 43 pairs of active regions and the chirality of 50 pairs of opposite magnetic polarity regions that are connected by transequatorial loops (TLs). To make the results more convincing, two helicity proxies including the local current helicity h _c and the force-free factor α _best are computed. The results, which are similar for both parameters, are as follows: (1) Current helicity of the active regions pairs connected by transequatorial loops have no obvious regularity: About 50% of the active region pairs carry the same current helicity sign and about 50% of them have the opposite. (2) If we consider the magnetic polarity pairs connected by the TLs, the result is almost the same as that for the active region pairs, with a little more than half of them showing the same chirality. We also make linear force-free extrapolations for 33 TLs and determine their force-free parameter α by comparing extrapolated field lines to X-ray images of the TLs. Out of the 19 cases when the footpoints of the TLs have the same current helicity sign, we find that the sign of α of the TLs is the same as the sign of the current helicity in the footpoints in 12 cases, whereas it is of opposite sign in 4 cases, and in 3 cases the TLs were found to be potential.     

The Magnetic Structure and Generation of EUV Flare Ribbons

The `ribbons' of two-ribbon flares show complicated patterns reflecting the linkages of coronal magnetic field lines through the lower solar atmosphere. We describe the morphology of the EUV ribbons of the July 14, 2000 flare, as seen in SOHO, TRACE, and Yohkoh data, from this point of view. A successful co-alignment of the TRACE, SOHO/MDI and Yohkoh/HXT data has allowed us to locate the EUV ribbon positions on the underlying field to within ∼ 2′′, and thus to investigate the relationship between the ribbons and the field, and also the sites of electron precipitation. We have also made a determination of the longitudinal magnetic flux involved in the flare reconnection event, an important parameter in flare energetic considerations. There are several respects in which the observations differ from what would be expected in the commonly-adopted models for flares. Firstly, the flare ribbons differ in fine structure from the (line-of-sight) magnetic field patterns underlying them, apparently propagating through regions of very weak and probably mixed polarity. Secondly, the ribbons split or bifurcate. Thirdly, the amount of line-of-sight flux passed over by the ribbons in the negative and positive fields is not equal. Fourthly, the strongest hard X-ray sources are observed to originate in stronger field regions. Based on a comparison between HXT and EUV time-profiles we suggest that emission in the EUV ribbons is caused by electron bombardment of the lower atmosphere, supporting the hypothesis that flare ribbons map out the chromospheric footpoints of magnetic field lines newly linked by reconnection. We describe the interpretation of our observations within the standard model, and the implications for the distribution of magnetic fields in this active region.     

Multiresolution Analysis of Active Region Magnetic Structure and its Correlation with the Mount Wilson Classification and Flaring Activity

Two different multiresolution analyses are used to decompose the structure of active-region magnetic flux into concentrations of different size scales. Lines separating these opposite polarity regions of flux at each size scale are found. These lines are used as a mask on a map of the magnetic field gradient to sample the local gradient between opposite polarity regions of given scale sizes. It is shown that the maximum, average, and standard deviation of the magnetic flux gradient for α,β,β γ, and β γ δ active-regions increase in the order listed, and that the order is maintained over all length scales. Since magnetic flux gradient is strongly linked to active-region activity, such as flares, this study demonstrates that, on average, the Mt. Wilson classification encodes the notion of activity over all length scales in the active-region, and not just those length scales at which the strongest flux gradients are found. Further, it is also shown that the average gradients in the field, and the average length-scale at which they occur, also increase in the same order. Finally, there are significant differences in the gradient distribution, between flaring and non-flaring active regions, which are maintained over all length scales. It is also shown that the average gradient content of active-regions that have large flares (GOES class “M” and above) is larger than that for active regions containing flares of all flare sizes; this difference is also maintained at all length scales. All of the reported results are independent of the multiresolution transform used. The implications for the Mt. Wilson classification of active-regions in relation to the multiresolution gradient content and flaring activity are discussed.     

Repeated Flaring from Loop–Loop Interaction

A series of solar flares was observed near the same location in NOAA active region 8996 on 18–20 May 2000. A detailed analysis of one of these flares is presented where the emitting structures in soft and hard X-rays, EUV, H, and radio at centimeter wavelengths are compared. Hard X-rays and radio emission were observed at two separate loop footpoints, while soft X-rays and EUV emission were observed mainly above the nearby positive polarity region. The flare was confined although the observed type III bursts at the time of the flare maximum indicate that some field lines were open to the corona. No flux emergence was evident but moving magnetic features were observed around the sunspot region and within the positive polarity (plage) region. We suggest that the flaring was due to loop–loop interactions over the positive polarity region, where accelerated electrons gained access to the two separate loop systems. The repeated radio flaring at the footpoint of one loop was visible because of the strong magnetic fields near the large sunspot region while at the footpoint of the other loop the electrons could precipitate and emit in hard X-rays. The simultaneous emission and fluctuations in radio and X-rays – in two different loop ends – further support the idea of a single acceleration site at the loop intersection.     

Energy-Dependent Timing of Thermal Emission in Solar Flares

We report solar flare plasma to be multi-thermal in nature based on the theoretical model and study of the energy-dependent timing of thermal emission in ten M-class flares. We employ high-resolution X-ray spectra observed by the Si detector of the “Solar X-ray Spectrometer” (SOXS). The SOXS onboard the Indian GSAT-2 spacecraft was launched by the GSLV-D2 rocket on 8 May 2003. Firstly we model the spectral evolution of the X-ray line and continuum emission flux F(ε) from the flare by integrating a series of isothermal plasma flux. We find that the multi-temperature integrated flux F(ε) is a power-law function of ε with a spectral index (γ)≈−4.65. Next, based on spectral-temporal evolution of the flares we find that the emission in the energy range E=4 – 15 keV is dominated by temperatures of T=12 – 50 MK, while the multi-thermal power-law DEM index (δ) varies in the range of −4.4 and −5.7. The temporal evolution of the X-ray flux F(ε,t) assuming a multi-temperature plasma governed by thermal conduction cooling reveals that the temperature-dependent cooling time varies between 296 and 4640 s and the electron density (n _e) varies in the range of n _e=(1.77 – 29.3)×10¹⁰ cm⁻³. Employing temporal evolution technique in the current study as an alternative method for separating thermal from nonthermal components in the energy spectra, we measure the break-energy point, ranging between 14 and 21±1.0 keV.     

X-ray Jets in Interconnecting Loops

We present examples of X-ray jets, observed by the Soft X-ray Telescope on board Yohkoh, which followed trajectories of transequatorial interconnecting loops (TILs). All these TILs were preexisting, seen some time before, but were mostly invisible at the time of the onset of the jet which often made them bright along their total length. With few exceptions, these TIL-associated jets have properties very similar to other jets ejected inside active regions or along open field lines (footpoints in X-ray bright points, recurrence, strong collimation, average speed close to 350 km s⁻¹), but may reach larger lengths, in our examples up to 450 000 km. Exceptions are one jet that moved slower and one that had no brightened area at its supposed source region at the time of its origin (an X-ray bright point appeared there only 3 hours later). It appears that quite a high number of X-ray jets may be of this TIL-associated kind. Supplementary material to this paper is available in electronic form at http://dx.doi.org/10.1023/A:1014963812437     

Multiwavelength Observations of the 5 January 1992 Flare

The 5 January 1992 flare around 13:16 UT was observed in H, H, and Ca ii H with the imaging spectrographs at Locarno-Monti, Switzerland and in soft and hard X-rays by the Yohkoh satellite. In this paper we discuss the analysis of the temporal and spatial evolution of this flare well observed at chromospheric and coronal layers. We find that the strongest footpoint emission in the optical lines does not coincide with the sites of non-thermal electron injection and show that these footpoints are mainly heated by thermal conduction. The chromospheric electron density, determined from the H line profiles, shows several temporally well correlated rises with the hard X-ray intensity at the electron injection sites. Two of the flare loops clearly are associated with strong chromospheric evaporation, while very weak evaporation is observed in the loop with the strongest footpoint emission in the optical lines.     

Locations of Footpoints of Transequatorial Interconnecting Loops

We discuss footpoints of loops seen by Yohkoh in soft X-rays that connect active regions across the equator (transequatorial interconnecting loops – TILs). While most TILs are rooted in moderately strong fields at peripheries of active regions, there are also cases when these loops are anchored in very weak or very strong fields, ranging from < 30 G to several hundred gauss. Some have their footpoints near sunspot penumbrae, creating `X-ray fountains' in a combination with active region loops. But TILs are never rooted in sunspots. The most likely explanation is that magnetic field lines leave spots almost vertically so that TILs rooted in them extend high into the corona and density in them is below the limit of visibility in X-rays. The fact that in force-free modeling some TILs are rooted in sunspots is most probably due to the difference between field-line connections in `vacuum' and in the highly conductive plasma on the Sun. Some TILs end before they reach active regions which sometimes may indicate the real situation, but mostly this `gap' is probably due to a temperature decrease near the loop footpoints which makes them invisible in X-rays. In that case the fact that these cool lowest parts of TILs are never found in TRACE or SOHO EIT images indicates that plasma density in TILs must be very low. Still, the total absence of any counterparts of X-ray TILs in TRACE and EIT images is puzzling and, therefore, other possible interpretations of the `gap' origin are also briefly mentioned.     

A Quantitative Study on Magnetic Configuration for Active Regions

As compared with the Mount Wilson Magnetic Classification (MWMC), effective distance (d _E) is a useful parameter because it gives a quantitative measure of magnetic configuration in active regions. We have analyzed magnetograms of 24 active regions of different types with MWMC. We have studied the evolution of magnetic fields of five active regions using d _E, total flux (F _t) and tilt angle (Tilt) quantitatively. Furthermore, 43 flare-associated and 25 CME-associated active regions have been studied to investigate and quantify the statistical correlation between flares/CMEs and the three parameters. The main results are as follows: (1) There is a basic agreement between d _E and MWMC. (2) The evolution of magnetic fields can be described in three aspects quantitatively and accurately by the three parameters, in particular by d _E on the analysis of δ-type active regions. (3) The high correlation between d _E and flares/CMEs means that d _E could be a promising measure to predict the flare-CME activity of active regions.     

Diagnostics of a flare on EQ Peg B from optical pulsations

Based on the methods of coronal seismology, we have investigated the ten-second quasi-periodic pulsations of the optical flare emission from the active red dwarf EQ Peg B detected with the William Herschel Telescope on La Palma. We propose and analyze a model in which they could be produced by sausage oscillations of a coronal flare loop. The amplitude and phase relations between the displacement components of the radial oscillations and the conditions for their excitation in loops with footpoints frozen into the photosphere are considered. The temperature (≈6 × 10⁷ K), plasma density (≈2.7 × 10¹¹ cm⁻³), and magnetic field strength (≈540 G) in the region of energy release have been determined. Our estimate of the flare loop length (≈0.4R _⋆) provides evidence for the existence of extended coronae on red dwarf stars.     

A Statistical Study of Transequatorial Loops

We identify 356 transequatorial loops (TLs) from the data set of Yohkoh Soft X-Ray Telescope (SXT) in the period of solar cycles 22 and 23. The classification of the TLs can be made on two bases. One is according to the magnetic polarities of the footpoints of TLs, and the other is according to the number of TLs in the same region. Based on the first criterion, TLs fall into two categories: PTLs in which the magnetic polarities of the footpoints are the same as the preceding polarities and FTLs in which the footpoint polarities are the same as the following polarities of active regions, respectively. It is found that PTLs have a preference; about 66% of the TLs are PTLs, and this preference of PTLs is independent of the solar cycle. The percentage of FTLs is about 34%. Based on the second criterion, TLs are also divided into two categories: the number of TLs in a region is either single (STLs) or multiple (MTLs). In addition, we find that the number of TLs, PTLs, and FTLs have good correlations with solar cycle indices. By comparing the number of TLs and the number of active regions in each year, we obtain the ratio between them. The separation of footpoints and their yearly variations are calculated, and we find that our result is consistent with spörer's law.