An ultra-long and quite thin coronal loop without significant expansion
Dong Li, Ding Yuan, Marcel Goossens, Tom Van Doorsselaere, Wei Su, Ya Wang, Yang Su, Zongjun Ning
aa r X i v : . [ a s t r o - ph . S R ] J un Astronomy & Astrophysicsmanuscript no. ms_r2 c (cid:13)
ESO 2020June 5, 2020
An ultra-long and quite thin coronal loop without significantexpansion
Dong Li , , Ding Yuan ⋆ , Marcel Goossens , Tom Van Doorsselaere , Wei Su , Ya Wang , Yang Su , , &Zongjun Ning , Key Laboratory of Dark Matter and Space Astronomy, Purple Mountain Observatory, CAS, Nanjing 210033, PR China e-mail: [email protected] State Key Laboratory of Space Weather, Chinese Academy of Sciences, Beijing 100190, PR China Institute of Space Science and Applied Technology, Harbin Institute of Technology, Shenzhen 518055, PR China e-mail: [email protected] Centre for mathematical Plasma Astrophysics, Department of Mathematics, KU Leuven, Celestijnenlaan 200B, 3001 Leuven,Belgium MOE Key Laboratory of Fundamental Physical Quantities Measurements, School of Physics, Huazhong University of Science andTechnology, Wuhan 430074, PR China School of Astronomy and Space Science, University of Science and Technology of China, Hefei 230026, PR ChinaReceived; accepted
ABSTRACT
Context.
Coronal loops are the basic building blocks of the solar corona, which are related to the mass supply and heating of solarplasmas in the corona. However, their fundamental magnetic structures are still not well understood. Most coronal loops do not expandsignificantly, whereas the diverging magnetic field would have an expansion factor of about 5-10 over one pressure scale height.
Aims.
In this study, we investigate a unique coronal loop with a roughly constant cross section, it is ultra long and quite thin. A coronalloop model with magnetic helicity is presented to explain the small expansion of the loop width.
Methods.
This coronal loop was predominantly detectable in the 171 Å channel of the Atmospheric Imaging Assembly (AIA). Then,the local magnetic field line was extrapolated by a Potential-Field-Source-Surface model. Finally, the di ff erential emission measureanalysis made from six AIA bandpasses was applied to obtain the thermal properties of this loop. Results.
This coronal loop has a projected length of roughly 130 Mm, a width of about 1 . ± . − . ± . Conclusions.
We use a thin twisted flux tube theory to construct a model for this non-expanding loop, and find that indeed withsu ffi cient twist a coronal loop can attain equilibrium. However, we can not rule out other possibilities such as footpoint heating bysmall-scale reconnection, elevated scale height by a steady flow along the loop etc. Key words.
Sun: corona — Sun: UV radiation — Sun: magnetic fields — Sun: activity
1. Introduction
Coronal loops are the basic structures in the solar corona.They can be detected everywhere on the Sun, such as thequiet region, the active region, or the solar limb, and theirsize scale could be ranging from sub-Megameter to hundredsof Megameters in the lower corona. These coronal loops of-ten confine plasmas at the temperature of Mega-Kelvin, so theyare prominently detectable in the extreme ultraviolet (EUV)and X-ray bandpasses (Bray et al. 1991; Reale 2014). More-over, the plasmas contained in a coronal loop may be eitherisothermal (e.g., Del Zanna, & Mason 2003; Tripathi et al. 2009;Gupta et al. 2019) or multithermal (e.g., Schmelz & Martens ⋆ Corresponding author
Article number, page 1 of 8 & Aproofs: manuscript no. ms_r2 suggested that the temperature variation along a coronal loopis highly sensitive to the heating mechanism (Priest et al. 1998;Warren et al. 2008). Therefore, to study the coronal loop in thecomplex magnetic environment could help us to better under-stand the fundamental problem in solar physics, i.e., coronalheating (e.g., Klimchuk 2000; Peter & Bingert 2012; Li et al.2015; Goddard et al. 2017).The coronal loop is expected to expand with height, sincethe coronal magnetic field is found to diverge strongly withthe height from the solar surface into the corona (Lionello et al.2013; Chen et al. 2014). The expansion of coronal loop could bediscovered on the active region (e.g., Malanushenko & Schrijver2013) or solar limb (e.g., Gupta et al. 2019). However, mostof coronal loops observed in X-ray and EUV images arefound to have roughly uniform widths in the plane of thesky, without significant expansions along their loop lengths,or only exhibit a small expansion from footpoints to the loopapex (e.g., Golub et al. 1990; Klimchuk et al. 1992; Klimchuk2000; Watko, & Klimchuk 2000; López Fuentes et al. 2006;Brooks et al. 2007; Kucera et al. 2019). The formation and ap-pearance of these loops in the complex magnetic environ-ment of the corona provides a pivotal test for a model ofthe coronal heating process (Klimchuk 2000; Petrie 2006;Peter & Bingert 2012). On the other hand, the loop cross sec-tion carries the information of magnetic fields and the spatialdistribution of corona heating, and the lower limit of the loopwidth is of fundamental importance to modern instrumentation,because it defines the spatial resolution of a space-borne orground-based telescope (Peter et al. 2013; Aschwanden & Peter2017). Moreover, the loop width variation is a proxy of theinter-coupling of plasma dynamics and magnetic fields, soit is believed to play a key role in coronal heating (e.g.,Vesecky et al. 1979; McTiernan & Petrosian 1990; Miki´c et al.2013; Chastain & Schmelz 2017). Aschwanden & Peter (2017)find that the loop widths are marginally resolved in AIA imagesbut are fully resolved in Hi-C images, their model predicts a mostfrequent value at about 0.55 Mm.The contradiction between the observed coronal loop witha roughly constant cross section and the extrapolated magneticfield with a strong expansion is still an open issue. In this paper,we investigated an ultra-long but quite thin coronal loop, whichcould be explained by a thin twisted flux tube model. The paperis organized as following: Section 2 introduces the data reductionand methods; and Section 3 describes properties of the coronalloop of interest; the conclusion and discussion are presented inSection 4.
2. Data reduction and methods
We combined the Atmospheric Imaging Assembly (AIA;Lemen et al. 2012) and the Helioseismic Magnetic Imager(HMI; Schou et al. 2012) on onboard the Solar Dynamic Ob-servatory (SDO; Pesnell et al. 2012) to observe the active re-gion NOAA 12524 near solar disk center (N20W04) on 2016March 23. A unique coronal loop was predominately observedin the AIA 171 Å channel, it is also vaguely simultaneouslydetectable in the AIA 193 Å and 211 Å channels, as shownin Figure 1 (a) − (c). This loop had an ultra long length and avery narrow width, which did not expand much radially alongits length. Moreover, this coronal loop retained this form forabout 90 minutes, as can be seen in the movie of anim.mp4.We then used the Potential-Field-Source-Surface (PFSS) model Fig. 1.
Overview of the coronal loop on 2016 March 23. Field-of-viewobserved at about 03:04 UT in AIA 171 Å (a), 193 Å (b), 211 Å (c) andHMI LOS magnetic magnetogram (d). The coronal loop of interest wasindicated by an arrow in each panel. Within the PFSS magnetic fieldextrapolation, the magnetic field line closely aligned with this loop wasoverlaid (cyan curve) with the HMI LOS magnetogram. The region ofinterest used in the DEM analysis was enclosed in a blue rectangle. Theevolution of this loop is shown in a movie of anim.mp4 (Schrijver & De Rosa 2003) to extrapolate the local magneticfield line. A cyan curve in Figure 1 (d) represents an open mag-netic field line derived in the PFSS model, which is closelyaligned with the coronal loop of interest.The SDO / AIA images used in this observation has a ca-dence of about 12 seconds, each pixel corresponds to about0.6 ′′ . The SDO / HMI observes the full-disk photospheric mag-netic fields. Both the AIA images and HMI magnetograms werecalibrated with the standard routines in the Solar SoftWare pack-age (Lemen et al. 2012; Schou et al. 2012).
This coronal loop was predominantly detectable in theAIA 171 Å channel, therefore we used the AIA 171 Å imagesto obtain its geometry. We created a two-dimension curvilinearcoordinate, one curve coordinate is chosen to be aligned with thespine of the coronal loop, the second coordinate is set to be nor-mal to the coronal loop. We made a bilinear interpolation of theemission intensity in the AIA 171 Å channel into the curvilinearcoordinate. Then each intensity profile across the loop was fitted
Article number, page 2 of 8ong Li et al.: An ultra-long and quite thin coronal loop without significant expansion
Fig. 2.
Estimation of the loop width. (a) Smaller FOV( ∼
76 Mm ×
149 Mm) of AIA 171 Å image. The loop is high-lighted by a green arrow. Nine sample cuts was marked by short linesand numbered from 1 to 9. (b) Intensity profiles along the cuts indicatedin (a), which are normalized by their maximum intensity, respectively.The color used in each curve is the same as used in the numbered shortlines in (a). Each profile is elevated progressively for visualizationpurpose. (c) Loop width variation along the loop length, the numbersmark the nine loop segments in panel (a). with a Gaussian function plus a linear background. In order toimprove the signal-to-noise ratio, we averaged three neighboringprofiles before fitting. The full width at half maximum (FWHM)of that Gaussian function was considered as the loop width ( w ).The fitting error was used as uncertainties of the loop width (e.g.,Aschwanden & Boerner 2011; Gupta et al. 2019). The error forthe AIA 171 Å intensity was estimated according to the methoddescribed by Yuan, & Nakariakov (2012).In Figure 2 (a), nine cross cuts were made along the coronalloop and were plotted with the short color lines. Each intensityprofile was scaled to a proper range and stacked in Figure 2 (b).The loop width was measured at locations perpendicular to theloop axis from the footpoint to the apparent top at a distance ofabout 130 Mm, as indicated in Figure 2 (c). We note that, at somepositions (such as the positions close to ‘8’ and ‘9’), the obtainedloop widths deviated significantly from those of their neighbors.The reason is that some random bright patchy background con-taminated those emission intensity profiles. On the other hand,at some positions of (i.e., ‘1’) the footpoint, the loop of interesthad overlap with some bright closed loops, which resulted intothe broad loop widths. In order to obtain the thermal properties of this loop, we focusedon a smaller FOV as marked in Figure 1 (blue rectangle) and per-formed Di ff erential emission measure (DEM) analysis. Observa-tions taken from six EUV channels of SDO / AIA (94 Å, 131 Å,171 Å, 193 Å, 211 Å, and 335 Å) were used to calculate theDEM(T) distribution for each pixel. We used an improved ver- sion (Su et al. 2018) of the sparse inversion code (Cheung et al.2015). The derived solutions could provide valuable informationby mapping the thermal plasma from 0.3 to 30 MK. The DEMuncertainties were estimated from Monte Carlo (MC) simulation(Su et al. 2018). Random noise of the observed emission inten-sity was added to the MC simulation and the inversion was re-peated for 100 times, then the standard deviations of the 100 MCsimulations were used as the uncertainties of DEM solutions.Figure 3 (a) − (d) draws the EM (Cheung et al. 2015; Su et al.2018) maps from 0.32 MK to 3.98 MK, within which coronalloops are normally detected. These EM maps are calculated froma set of six re-binned AIA narrow-band maps with a pixel sizeof 1.2 ′′ , in order to get clear view of the structures in di ff er-ent temperature ranges, whose emissions are accumulated alongthe line of sight (LOS) into the observed intensity. The coronalloop was clearly seen in the temperature range of 0.63 MK to1.12 MK (b) and to a weaker extent in the 1.26 MK − × ’ in panel b) in the coronal loop. It can be seen thatthe obtained DEM profiles exhibit two peaks at about 0.8 MKand 1.8 MK, respectively. However, the coronal loop of inter-est was most clearly seen in the DEM ranging from 0.63 MK to1.12 MK. So we suspect that the high-temperature peak at about1.8 MK could originate from the emission of the di ff use back-ground in the AIA 211 Å channel. For a comparison, we thentook the DEM profile of a reference point (‘0’) at the backgroundfor cross-validation, as marked by the magenta ‘ + ’ (Figure 3b).We noticed that the DEM profile (magenta) at the backgroundindeed only had a prominent peak at about 1.8 MK.The EM was calculated by integrating the DEM over temper-atures, EM = R DEM d T . We only use the temperature rangesbetween 0.32 − ff ective temperatureof the coronal loop of interest, as indicated by the grey regionin Figure 3 (e). The EM could be considered to the productof the square of the number density of electrons ( n e ) and LOSdepth, which could be approximated with the loop width ( w ).In this way, the electron number density could be calculatedwith n e = √ EM /w . Finally, a DEM-weighted mean tempera-ture (such as T = R DEM T d T / R DEM d T ) is used to estimatethe temperature of this coronal loop. The errors for the densityand temperature were also calculated from the 100 MC simu-lations. These steps were done for every pixel along the looplength. Then, using the obtained number density, plasma tem-perature and magnetic field, we calculated the plasma beta ( β )along the coronal loop.
3. Properties of the coronal loop
The coronal loop under study was very thin and ultra long. It wasdetectable for a projected length of about 130 Mm. We note thisas the lower limit, since it become di ff use and invisible in thebackground. The loop width was about 1 Mm at the footpointand expands to about 1.5 Mm to 2.0 Mm at the visible end. Theloop expansion ratio was about 1.5 to 2.0. In this dataset of about2 hours, we observed the distinctive coronal loop to fade outeventually, however, it had almost the constant width during itslifetime (see the movie of anim.mp4).In the PFSS extrapolation model, we traced a magnetic fieldline that was closely aligned with the coronal loop of interest,as indicated by a cyan curve in Figure 1 (d). This magnetic fieldline was connected to a patch of negative polarity and extends to Article number, page 3 of 8 & Aproofs: manuscript no. ms_r2
Fig. 3.
DEM results of the target coronal loop. (a) − (d) Narrow bandEM maps integrated in the temperature ranges of 0 .
32 MK − .
56 MK,0 .
63 MK − .
12 MK, 1 .
26 MK − .
24 MK, and 2 .
51 MK − .
98 MK, re-spectively. (e) DEM profiles at seven selected positions (‘1’ − ‘7’) alongthe loop and one location (‘0’) away from the loop, the color corre-sponds to the positions labeled in (b). For clarity, panel (e) only drawsthe error bars at the loop position ‘4’. The grey region indicates the EMintegrated range. the outer space, therefore, this coronal loop could be regarded asan open structure. It has an inclination in the range of 40 ◦ -80 ◦ based on the estimation in the PFSS model. The polarity at theloop footpoint has an average line-of-sight (LOS) magnetic fieldcomponent of about 100 Gauss. Along the field line, the strengthof the magnetic field is about 10 Gauss on average, whereas themaximum field strength could reach 60 Gauss. So we used 10Gauss as the coronal loop’s field strength. Figure 3 presents the DEM results to the coronal loop of inter-est. It is apparent that this loop is most clearly identifiable inthe EM map at 0.63 MK − ff use background in the 211 Å channel(Figure 1c). Therefore, we estimated that the coronal loop con-sidered here had a temperature of about 0.8 MK.In Figure 4, we present the quantitative estimation of thephysical parameters of the coronal loop. Figure 4 (a) drawsthe AIA 171 Å intensity (Yuan, & Nakariakov 2012) and EMvariations at the selected positions along the loop length. Theyfirst decreased quickly with the loop length and then becameroughly stable. We notice that the AIA 171 Å intensities aremuch stronger at the beginning, which could be due to the factthat some random bright patchy or closed loops contaminated atthe base of the coronal loop (see also Figures 1 and 2). Fig. 4.
Quantitative estimation of the coronal loop parameters. The EM(a), the emission intensity in the AIA 171 Å bandpass (a), the plasmatemperature (b), and the number density of electrons (b), as well as theplasma beta ( β ) as a function of projected length along the loop. Theblue line represents the best fitted result for the number density. Figure 4 (b) plots the plasma temperature variation along theloop length. It shows that this coronal loop had a very uniformtemperature at (0 . ± .
2) MK, this is consistent with our previousestimations. Since this loop was very thin, we cannot obtain thetemperature distribution across the loop. Figure 4 (b) also showsthe estimated number density of electrons along the loop length,with a mean number density of (8 ± × cm − . The electronnumber density was about 1 . × cm − at the footpoint, anddropped o ff exponentially to around 5 × cm − at the end ofthe loop. This was a pattern of stratification. Therefore, we fit-ted an exponential function to the density profile and obtaineda density scale height of about (38 ±
13) Mm. This scale heightonly incorporate a reduced gravity, because this corona loop isinclined with respect to the solar radial. The theoretical densityscale height is (22 . ± .
6) Mm for a plasma with a tempera-ture of (0 . ± .
2) MK. With the ratio of theoretical and fittedscale heights, we estimated that the loop deviated from the grav-ity vector on average by an angle of about 54 ◦ ± ◦ . This value isconsistent with the estimation of the extrapolated magnetic field,i.e., 40 ◦ − ◦ .Figure 4 (c) draws the plasma beta parameter ( β ) as a func-tion of the loop length. It can be seen that the plasma beta in-creased from about 0.02 at the footpoint to roughly 0.1 at itsvisible end. The average plasma beta of this loop is estimated tobe ∼ ±
4. Conclusion and Discussion
In this study, we used the SDO / AIA data to observe an opencoronal loop associated with AR 12524. This coronal loopswas clearly detected in AIA 171 Å and to an weaked extentin the AIA 193 Å and 211 Å channels. This loop was ultra-long and had a small width, its lifetime was about 90 min-
Article number, page 4 of 8ong Li et al.: An ultra-long and quite thin coronal loop without significant expansion utes. The loop width was about 1.5 Mm, and a projected lengthabout 130 Mm. The coronal loop investigated in this study isthinner and longer than those reported earlier. For instance,Aschwanden & Boerner (2011) reported coronal loops of about2 − −
40 Mm long. Moreover, the most loopshas a lifetime of about 20 −
30 minutes (e.g., Peter & Bingert2012), whereas in our case, the loop survived for over one hour.The coronal loop had a plasma temperature of (0 . ± .
2) MK,no signification variation of temperature was detected alongthe loop. This loop is relatively cold (e.g., < . × cm − , and itdropped o ff exponentially to about 0 . × cm − at the visibleend of the loop. This was a pattern of stratification, and then thedensity scale height was measured at roughly (38 ±
13) Mm. Theplasma beta increases from about 0.02 to roughly 0.1, it meansthat the gas pressure decreases with height by a smaller amountthan the magnetic pressure. More observational, geometrical andphysical parameters of this loop are listed in table 1.It should be mentioned that the little expansion was found inmany coronal loops at SXR / EUV wavelengths since the eras ofTRACE and earlier (e.g., Klimchuk et al. 1992; Klimchuk 2000;Watko, & Klimchuk 2000; Peter & Bingert 2012; Kucera et al.2019). These coronal loops were closed structures and oftenexhibited weak expansion from double footpoints to the loop-apex (López Fuentes et al. 2006; Brooks et al. 2007). Althoughan open coronal loop of about 280 Mm long was reported byGupta et al. (2019), but its loop width expanded from 20 Mmat the footpoint to 80 Mm at the loop top. Here, we investigatean ultra-long ( ∼
130 Mm) and very thin ( ∼ ± − e − ), the cross-section of this loopshould expand by a factor of about 19. However, this is not sup-ported by the observation, therefore, to explain the nearly con-stant cross-section of this coronal loop, there must be unresolvedfeatures (e.g., Klimchuk et al. 2000; Petrie 2008). This steadycoronal loop has a lifetime much longer than the timescales ofthe radiative cooling and thermal conduction, therefore, we useda magnetostatic model (appendix A) to explain the small ra-dial expansion of this loop. We used a thin flux tube approxi-mation and assumed the magnetic field has a twist term (e.g.,Ferriz-Mas & Schüssler 1989). Only when the twist term waslarge enough, then the external pressure became negative. Thus,we could find an upper limit for the twist, the detailed derivationprocess was seen in the appendix B. We then constrained theexpansion factor to be 1.5 and obtained the solution for the ex-ternal pressure. We could see that to constrain the thin magneticflux tube, the twist has to be smaller than about 0.65 (see, Fig-ure B.1). Therefore, with a su ffi ciently twisted magnetic field,this coronal loop could be constrained at magnetic equilibriumstate.If we go beyond a magnetostatic equilibrium, this loop hasto be supplied with steady flows of mass and energy, as wedid not observe any features propagating emission intensity orbrightening. This flow must have very long lifetime or shouldoccur intermittently with short time scales. Evidence has col-lected that the footpoint of a coronal loop could have an un-resolved minority-polarity, an miniature bipolar field emerging into a constant background field. This configuration favors smallscale magnetic reconnection (Wang et al. 2019). Repetitive re-connections provide continuous mass and energy to the coronalloops (Chitta et al. 2018). Unfortunately we did not observe fea-tures to support this scenario, however we shall note that it couldbe that the AIA instrument is not sensitive enough to capturethese features.At last, we stress that the loop widths are measured as theFWHM of the cross-sectional profiles of the coronal loop de-tected in the SDO / AIA 171 Å channel. One AIA pixel cor-responds to 0.44 Mm (Lemen et al. 2012). A loop width ofabout 1.5 to 2.0 Mm spans about 4-5 pixels. This measure-ment is obtained by fitting a intensity profile with about 12pixels, this practice could reach a higher accuracy than thepixel scale, and has been used by many other researchers(e.g., Aschwanden & Boerner 2011; Anfinogentov et al. 2013;Anfinogentov & Nakariakov 2019; Reale 2014).
Acknowledgements.
We acknowledge the anonymous referee for his / hervaluable comments. This study is supported by NSFC under grant11973092, 11803008, 11873095, 11790300, 11790302, 11729301, 11773079,the Youth Fund of Jiangsu No. BK20171108 and BK20191108. TheLaboratory No. 2010DP173032. D.L. is also supported by the Special-ized Research Fund for State Key Laboratories. D.Y. is supported bythe NSFC grant 11803005, 11911530690, Shenzhen Technology Project(JCYJ20180306172239618), and Shenzhen Science and Technology program(group No. KQTD20180410161218820). T.V.D. is supported by the EuropeanResearch Council (ERC) under the European Union Horizon 2020 research andinnovation programme (grant agreement No 724326) and the C1 grant TRACEs-pace of Internal Funds KU Leuven (number C14 / / References
Anfinogentov, S., Nisticò, G., & Nakariakov, V. M. 2013, A&A, 560, A107Anfinogentov, S. A., & Nakariakov, V. M. 2019, ApJ, 884, L40Aschwanden, M. J., & Boerner, P. 2011, ApJ, 732, 81Aschwanden, M. J., & Peter, H. 2017, ApJ, 840, 4Bray, R. J., Cram, L. E., Durrant, C., et al. 1991, Plasma Loops in the SolarCoronaBrooks, D. H., Warren, H. P., Ugarte-Urra, I., et al. 2007, PASJ, 59, S691Chastain, S. I., & Schmelz, J. T. 2017, arXiv e-prints, arXiv:1705.06776Chen, F., Peter, H., Bingert, S., et al. 2014, A&A, 564, A12Cheung, M. C. M., Boerner, P., Schrijver, C. J., et al. 2015, ApJ, 807, 143Chitta, L. P., Peter, H., & Solanki, S. K. 2018, A&A, 615, L9Del Zanna, G., & Mason, H. E. 2003, A&A, 406, 1089Dudík, J., Dzifˇcáková, E., & Cirtain, J. W. 2014, ApJ, 796, 20Feng, L., Inhester, B., Solanki, S. K., et al. 2007, ApJ, 671, L205Ferriz-Mas, A., & Schüssler, M. 1989, Geophysical and Astrophysical Fluid Dy-namics, 48, 217Golub, L., Herant, M., Kalata, K., et al. 1990, Nature, 344, 842Goddard, C. R., Pascoe, D. J., Anfinogentov, S., et al. 2017, A&A, 605, A65Gupta, G. R., Del Zanna, G., & Mason, H. E. 2019, A&A, 627, A62Klimchuk, J. A., Lemen, J. R., Feldman, U., et al. 1992, PASJ, 44, L181Klimchuk, J. A. 2000, Sol. Phys., 193, 53Klimchuk, J. A., Antiochos, S. K., & Norton, D. 2000, ApJ, 542, 504Kucera, T. A., Young, P. R., Klimchuk, J. A., et al. 2019, ApJ, 885, 7Lemen, J. R., Title, A. M., Akin, D. J., et al. 2012, Sol. Phys., 275, 17Li, L. P., Peter, H., Chen, F., et al. 2015, A&A, 583, A109Lionello, R., Winebarger, A. R., Mok, Y., et al. 2013, ApJ, 773, 134López Fuentes, M. C., Klimchuk, J. A., & Démoulin, P. 2006, ApJ, 639, 459Malanushenko, A., & Schrijver, C. J. 2013, ApJ, 775, 120McTiernan, J. M., & Petrosian, V. 1990, ApJ, 359, 524Miki´c, Z., Lionello, R., Mok, Y., et al. 2013, ApJ, 773, 94Peter, H., & Bingert, S. 2012, A&A, 548, A1Peter, H., Bingert, S., Klimchuk, J. A., et al. 2013, A&A, 556, A104Petrie, G. J. D. 2006, ApJ, 649, 1078Petrie, G. J. D. 2008, ApJ, 681, 1660Pesnell, W. D., Thompson, B. J., & Chamberlin, P. C. 2012, Sol. Phys., 275, 3Poletto, G., Vaiana, G. S., Zombeck, M. V., et al. 1975, Sol. Phys., 44, 83Priest, E. R., Foley, C. R., Heyvaerts, J., et al. 1998, Nature, 393, 545Reale, F. 2014, Living Reviews in Solar Physics, 11, 4Schmelz, J. T., & Martens, P. C. H. 2006, ApJ, 636, L49
Article number, page 5 of 8 & Aproofs: manuscript no. ms_r2
Table 1.
Observational, geometrical and physical parameters of the analyzed coronal loop.
Parameter ValueDate of observation 2016 March 23Active region NOAA 12524Projected length ∼
130 MmLoop width 1.5 ± − ± ± × cm − Plasma beta 0.056 ± ◦ − ◦ Inclination angle inferred from stratified plasma 54 ◦ ± ◦ Fitted density scale height 38 ±
13 MmTheoretical density scale height 22.8 ± Schou, J., Scherrer, P. H., Bush, R. I., et al. 2012, Sol. Phys., 275, 229Schrijver, C. J., & De Rosa, M. L. 2003, Sol. Phys., 212, 165Su, Y., Veronig, A. M., Hannah, I. G., et al. 2018, ApJ, 856, L17Tripathi, D., Mason, H. E., Dwivedi, B. N., et al. 2009, ApJ, 694, 1256Vasheghani Farahani, S., Nakariakov, V. M., & van Doorsselaere, T. 2010, A&A,517, A29Vesecky, J. F., Antiochos, S. K., & Underwood, J. H. 1979, ApJ, 233, 987Watko, J. A., & Klimchuk, J. A. 2000, Sol. Phys., 193, 77Warren, H. P., Ugarte-Urra, I., Doschek, G. A., et al. 2008, ApJ, 686, L131Wang, Y.-M., Ugarte-Urra, I., & Reep, J. W. 2019, ApJ, 885, 34Yuan, D., & Nakariakov, V. M. 2012, A&A, 543, A9
Article number, page 6 of 8ong Li et al.: An ultra-long and quite thin coronal loop without significant expansion
Appendix A: Thin flux tube model
In this study, we use a magnetostatic plasma to model the coro-nal loop of interest. This model accounts for plasma stratifi-cation and a static magnetic field. In our case, the loop hasa lifetime significantly greater than the timescales of thermalconduction and radiative cooling, we resort to a magnetostaticplasma. This loop have large aspect ratio (ratio between theloop length and radius), we use thin-flux tube approximation(e.g., Ferriz-Mas & Schüssler 1989) to model a twisted coro-nal loop. We expand the MHD quantities for a thin cylindri-cal flux tube with cylindrical coordinates ( r , ϕ, z ). We consideran axisymmetric magnetohydrostatic equilibrium solution to theMHD equations. We assume the velocity u = ϕ − derivatives to 0. In particular, we use the following thinflux tube expansion for the loop parameters, B r ( r , z ) = rB r ( z ) (A.1) B ϕ ( r , z ) = rB ϕ ( z ) (A.2) B z ( r , z ) = B z ( z ) + r B z ( z ) (A.3) p ( r , z ) = p ( z ) + r p ( z ) (A.4) ρ ( r , z ) = ρ ( z ) + r ρ ( z ) (A.5) T ( r , z ) = T ( z ) + r T ( z ) , (A.6)where B = ( B r , B ϕ , B z ) is the magnetic field, p is the pressure, ρ is the density and T is the temperature.In this study, we derive the key equations for a thin andtwisted flux. We expand the magnetohydrostatic equations in theradial component, and balance the forces in the flux tube at a cer-tain distance R ( z ) with an external pressure force p e ( R ( z )). Thelatter relationship is their closure relationship for the (otherwise)infinite system of equations. The resulting equations for the thinand twisted flux tube are R T p ′ + g p = B r + B ′ z = p + µ (cid:16) − B z B ′ r + B z B z + B ϕ (cid:17) = B z B ′ ϕ − B ′ z B ϕ = p ′ − p ′ p p − T T ! + µ (cid:16) B r B ′ r − B r B z + B ϕ B ′ ϕ (cid:17) = p + B z µ + R p + B r + B ϕ µ + B z B z µ = p e , (A.12)where primes denote the derivative with respective to z , µ is themagnetic permeability, g is the gravity pointing in the negative z -direction, and R is the universal gas constant divided by themolar mass.Since the radial expansion is of particular importance in ourproblem, let us relate it to the magnetic field variables in theproblem. The general equation for the flux surface in the ( r , z )-plane is drdz = B r B z = B r B z r + r B z B z , (A.13) This is not a separable equation. However, when we take B z = drdz = B r B z r , (A.14)Using Eq. A.8, we find as solution R = R ⋆ s B ⋆ z B z . (A.15)where we use the same ⋆ notation to indicate the value at a ref-erence height z =
0. Indeed, this equation expresses the con-servation of magnetic flux in a flux tube: as the magnetic fielddecreases, the radius of the flux tube must increase quadratically.
Appendix B: Solution for an expanding loop
We assume that the loop radius expands exponentially with theheight (see also Dudík et al. 2014): R = R ⋆ exp ( z / L ) , (B.1)For an expansion factor R ( z top ) / R ⋆ = η , we find L = z top / ln η . Inparticular, we can consider η = η = . z top =
130 Mm, then L =
186 Mm or 320 Mm, respectively.From the conservation of magnetic flux (Eq. A.15), we thenfind B z = B ⋆ z exp ( − z / L ) , (B.2)Eq. A.8 can be used to find the radial component B r = B ⋆ z L exp ( − z / L ) , (B.3)and Eq. A.10 can be integrated (following byFerriz-Mas & Schüssler (1989)) to determine the ϕ -componentof the magnetic field: B ϕ = B ⋆ϕ exp ( − z / L ) , (B.4)From Eq. A.9, we obtain p = − B z µ L + B ⋆ϕ B ⋆ z exp ( − z / L ) , (B.5)This equation shows that p <