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Home All issues Volume 646 (February 2021) A&A, 646 (2021) A12 Full HTML
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A&A
Volume 646, February 2021
Article Number A12
Number of page(s) 7
Section The Sun and the Heliosphere
DOI https://doi.org/10.1051/0004-6361/202039013
Published online 29 January 2021

© ESO 2021

1. Introduction

Waves and oscillations are prevalent in the fully ionized solar corona, which has temperatures of several million Kelvin (MK; see Nakariakov & Verwichte 2005, and references therein). The periodic transverse displacements of coronal loops are usually considered as kink oscillations detected in the extreme ultraviolet (EUV) wavelengths (Andries et al. 2009; Ruderman & Erdélyi 2009). Standing fast kink-mode oscillations were initially discovered by the Transition Region and Coronal Explorer (TRACE; Handy et al. 1999) mission in 171 Å (Aschwanden et al. 1999; Nakariakov et al. 1999; Schrijver et al. 2002). Since the launch of the Solar Dynamics Observatory (SDO; Pesnell et al. 2012), kink oscillations of coronal loops observed by the Atmospheric Imaging Assembly (AIA; Lemen et al. 2012) on board the SDO have been extensively investigated (e.g., Aschwanden & Schrijver 2011; White & Verwichte 2012; Verwichte et al. 2013a,b; Pascoe et al. 2016; Nisticò et al. 2017; Duckenfield et al. 2018, 2019; Nechaeva et al. 2019).

The commencement of the loop oscillation usually coincides with a nearby eruption (flare, jet, or filament eruption) in the lower corona, which is considered the predominant mechanism for exciting kink oscillations (Zimovets & Nakariakov 2015). After the excitation, kink oscillations experience attenuation and last for several cycles in most cases (Goddard et al. 2016; Goddard & Nakariakov 2016). Resonant absorption, a result of resonance within a thin finite layer, is believed to play a key role in the rapid damping of fast-mode kink oscillations (Goossens et al. 2002; Ruderman & Roberts 2002). Phase mixing with anomalously high viscosity is also important in the dissipation of energy during loop oscillations (Ofman & Aschwanden 2002). Small-amplitude, transverse oscillations of coronal loops without significant damping have been noticed (Anfinogentov et al. 2013; Nisticò et al. 2013; Li et al. 2018a; Zhang 2020). The observed loop oscillations in combination with magnetohydrodynamics (MHD) wave theory provide an effective tool for determining the local physical parameters, such as the magnetic field strength and the Alfvén speed of the oscillating loops, which are difficult to measure directly (Edwin & Roberts 1983; Nakariakov & Ofman 2001; Aschwanden et al. 2002; Verwichte et al. 2006; Arregui et al. 2007; Goossens et al. 2008; Van Doorsselaere et al. 2008; Antolin & Verwichte 2011; Yuan & Van Doorsselaere 2016; Li et al. 2017).

Circular-ribbon flares (CRFs) are a special type of flares, whose short, inner ribbons are surrounded by circular or elliptical ribbons (Masson et al. 2009; Chen et al. 2019; Zhang et al. 2016b, 2019; Lee et al. 2020; Liu et al. 2020). Transverse loop oscillations excited by CRFs with periods of ≲4 min have been observed by SDO/AIA (Zhang et al. 2015; Li et al. 2018b). Recently, Zhang et al. (2020) investigated the transverse oscillations of an EUV loop excited by two successive CRFs on 2014 March 5. The oscillations were divided into two stages of development: The first-stage oscillation triggered by the C2.8 flare is decayless with lower amplitudes, and the second-stage oscillation triggered by the M1.0 flare decays with larger amplitudes. The authors also estimated the magnetic field and thickness of the inhomogeneous layer of the oscillating loop, which has a length of ∼130 Mm.

In this paper, we report our multiwavelength observations of the transverse oscillation of a large-scale coronal loop excited by a blowout jet associated with the C4.2 CRF in active region (AR) 12434 on 2015 October 16. Zhang et al. (2016c) studied the explosive chromospheric evaporation at the inner and outer flare ribbons using spectroscopic observations. This work builds on the work of Zhang et al. (2016c, hereafter Paper I), and its main purpose is to estimate the magnetic field of the oscillating loop using two independent approaches: coronal seismology and magnetic field extrapolation. This paper is organized as follows. Observations and data analysis are presented in Sect. 2. The results are presented in Sect. 3. A brief summary and discussion are presented in Sect. 4.

2. Observations and data analysis

2.1. Instruments

The transverse oscillation of the coronal loop was observed by SDO/AIA, which has a spatial resolution of 1 . ″ $ {{\overset{\prime\prime}{.}}} $2 and a time cadence of 12 s in EUV wavelengths. The photospheric line-of-sight (LOS) magnetograms were observed by the Helioseismic and Magnetic Imager (HMI; Schou et al. 2012) on board the SDO with a spatial resolution of 1 . ″ $ {{\overset{\prime\prime}{.}}} $2 and a time cadence of 45 s. The level_1 data from AIA and HMI were calibrated using the standard Solar SoftWare (SSW) programs aia_prep.pro and hmi_prep.pro, respectively. Soft X-ray (SXR) light curves of the flare were recorded by the GOES spacecraft with a cadence of ∼2.05 s.

2.2. DEM analysis

Differential emission measure (DEM) analysis is a useful tool for performing temperature diagnostics. Several algorithms have been proposed and validated (e.g. Weber et al. 2004; Hannah & Kontar 2012; Aschwanden et al. 2013; Plowman et al. 2013; Cheung et al. 2015; Su et al. 2018b; Morgan & Pickering 2019). The observed flux Fi of each optically thin passband i is determined by:

F i = ∫ T 1 T 2 R i ( T ) DEM ( T ) d T , $$ \begin{aligned} F_{i} = \int _{T_1}^{T_2} R _{i} ( T )\mathrm{DEM} ( T ) \mathrm{d} T , \end{aligned} $$(1)

where Ri(T) is the temperature response function of passband i, DEM(T) represents the DEM of multi-thermal plasma as a function of temperature, and log T1 = 5.5 and log T2 = 7.5 stand for the lower and upper limits for the integral. To carry out the inversion of the DEM profile, we used the standard SSW program xrt_dem_iterative2.pro and six EUV passbands (94, 131, 171, 193, 211, and 335 Å). This method has been strictly justified and successfully applied to the temperature estimation of EUV hot channels as well as coronal jets (Cheng et al. 2012; Zhang & Ji 2014; Zhang et al. 2016a). We note that the background emissions should be removed before inversion.

The DEM-weighted average temperature T̄ is defined as (Cheng et al. 2012):

T ¯ = ∫ T 1 T 2 DEM ( T ) T d T ∫ T 1 T 2 DEM ( T ) d T . $$ \begin{aligned} \bar{T} = \frac{\int _{T_1}^{T_2} \mathrm{DEM} ( T ) {T\mathrm{d}T}}{\int _{T_1}^{T_2} \mathrm{DEM} ( T )\mathrm{d} T}. \end{aligned} $$(2)

Then, the total column emission measure (EM) along the LOS is expressed as:

EM = ∫ T 1 T 2 DEM ( T ) d T . $$ \begin{aligned} \mathrm{EM} = \int _{T_1}^{T_2} \mathrm{DEM} ( T )\mathrm{d} T. \end{aligned} $$(3)

2.3. Flux rope insertion method

We used the flux rope insertion method developed by van Ballegooijen (2004) to reconstruct the nonlinear force-free field (NLFFF) of AR 12434. The advantage of this method is that it can be applied to many different situations including both ARs (Su et al. 2009, 2011, 2018a) and the quiet Sun (Su et al. 2015), since no vector field observations are required and the magnetic field lines of the best-fit model match the observed coronal non-potential structures well. Su (2019) gives a detailed review of the application of the method. Reconstructing the coronal magnetic fields in the target AR requires four steps: (i) extrapolating the potential field based on the corresponding photospheric LOS magnetogram; (ii) according to observations, creating a cavity in the potential field model and inserting a magnetic flux rope along the selected paths; (iii) creating a grid of models by adjusting axial flux and poloidal flux of the inserted magnetic flux rope; and (iv) starting magneto-frictional relaxation (Yang et al. 1986) to drive the magnetic field toward a force-free state and comparing this with observations to find the best-fit model.

3. Results

3.1. Transverse coronal loop oscillation

Figure 1 shows the EUV images observed by the AIA in 171, 131, 193, and 211 Å before flare. The northeastern footpoint of the large-scale coronal loop (dashed yellow line) is rooted in AR 12434 and is very close to the C4.2 CRF, which is indicated by the red arrow. As described in Paper I, the flare brightened from ∼13:36:30 UT. The accompanying blowout jet started to rise at ∼13:39 UT and propagated in the southeastern direction at a speed of ∼300 km s−1. Transverse oscillation of the long loop was excited by the jet and lasted for a few cycles (see the online movie).

thumbnail Fig. 1.

EUV images of the large-scale coronal loop (dashed yellow line) observed by SDO/AIA in 171, 131, 193, and 211 Å before flare. The straight solid lines (S1, S2, and S3) are used to analyze the transverse oscillation. The red arrows point to the locations of the CRF and blowout coronal jet. In panel b, the three tiny boxes along the loop mark the locations for DEM analysis. (a) AIA 171 Å16–Oct-2015 13:36:22 UT, (b) AIA 131 Å16–Oct–2015 13:36:31 UT, (c) AIA 193 Å16–Oct–2015 13:36:32 UT, (d) AIA 211 Å16–Oct–2015 13:36:22 UT.

Figure 2 shows the SXR light curves of the flare in 1–8 Å (red line) and 0.5–4 Å (blue line). The SXR emissions started to increase at ∼13:36:30 UT and reached peak values at ∼13:42:30 UT before declining gradually until ∼13:51 UT (see also Fig. 5 in Paper I). The loop oscillation between 13:39 UT and 14:05 UT is indicated by the yellow area. It is found that the start time of transverse loop oscillation coincides with the fast ejection of the jet (see also Fig. 4 in Paper I), and the oscillation covers part of the impulsive phase as well as the whole decay phase of the flare, lasting for ∼26 min. The excitation of loop oscillation in our study can be interpreted by a schematic cartoon (Zimovets & Nakariakov 2015, see their Fig. 2).

thumbnail Fig. 2.

SXR light curves of the flare in 1–8 Å (red line) and 0.5–4 Å (blue line). The yellow area represents the time of loop oscillation between 13:39 UT and 14:05 UT.

To investigate the transverse loop oscillation, we chose three points along the loop: The first is near the loop top, the third is close to the southwestern footpoint, and the second is located in between. We placed three artificial slices (S1, S2, and S3) across the points and just perpendicular to the loop, which are drawn with solid white lines in Fig. 1. The corresponding time-distance diagrams in 171 Å are plotted in the top three panels of Fig. 3. The loop, excited by the jet, deviated from the equilibrium state and began to move coherently southward at ∼13:39 UT. Then, the loop moved backward and oscillated with decaying amplitude for more than three cycles. The almost identical phases of the transverse loop oscillation at different positions along the loop indicate that the oscillation belongs to the standing fast kink mode.

thumbnail Fig. 3.

Time-distance diagrams of the three slices (S1, S2, and S3) in 171 and 131 Å. The dashed magenta lines outline the transverse loop oscillation between 13:39 UT and 14:05 UT. On the y-axis, s = 0 and s = 43.5 Mm denote the southern and northern endpoints of the slices, respectively.

To determine the parameters of kink oscillation along different slices, we manually marked the positions of the loop, which are connected with dashed magenta lines in Fig. 3. The kink oscillation was fitted with an exponentially decaying sine function (Nakariakov et al. 1999; Zhang et al. 2020) using the standard SSW program mpfit.pro:

A ( t ) = A 0 sin ( 2 π t P + ψ ) e − t τ + A 1 t + A 2 , $$ \begin{aligned} A(t) = A_{0}\sin \bigg (\frac{2\pi t}{P} + \psi \bigg )e^{-\frac{t}{\tau }} + A_{1}t + A_{2}, \end{aligned} $$(4)

where A0 is the initial amplitude, P is the period, τ is the damping time, ψ is the initial phase, and A1t + A2 is a linear term of the equilibrium position of the loop.

In Fig. 4, the green crosses represent the positions of the loop along the three slices in 171 and 131 Å, and the results of curve fitting are plotted with blue lines. It is obvious that the curve fitting that uses Eq. (4) is satisfactory. The fitted parameters are listed in Table 1. In 171 Å, the initial amplitude decreases from ∼13.6 Mm near the loop top to ∼8.9 Mm near the footpoint. The period ranges from ∼440 s to ∼480 s, with an average value of ∼462 s. The damping time ranges from ∼710 s to ∼1190 s, with an average value of ∼976 s. The quality factor (τ/P) lies in the range of 1.5–2.5. The oscillation decays much faster near the loop top than the loop leg. The parameters in 131 Å are close to those in 171 Å.

thumbnail Fig. 4.

Positions of the oscillating loop along the three slices (green crosses) and the results of curve fitting (blue lines) using Eq. (4).

Table 1.

Fitted parameters of the coronal loop oscillation.

3.2. Magnetic field estimated from coronal seismology

As mentioned in Sect. 1, estimation of the magnetic field strength of oscillating loops is an important application of coronal seismology. To estimate the magnetic field of the large-scale coronal loop in Fig. 1, we considered the loop as a straight cylinder with the magnetic field lines frozen. The period of standing kink-mode oscillation depends on the loop length (L) and phase speed (Ck) (Nakariakov et al. 1999):

P = 2 L C k , C k = 2 1 + ρ e / ρ i C A , $$ \begin{aligned} P=\frac{2L}{C_{k}}, \ \ \ C_{k}=\sqrt{\frac{2}{1+\rho _{e}/\rho _{i}}}C_{A}, \end{aligned} $$(5)

where CA is the Alfvén speed in the loop, and ρe and ρi stand for the external and internal plasma densities.

In Fig. 1, the apparent distance between the footpoints of the loop is ∼256″, while the real distance becomes ∼268″ after correcting for the projection effect. Since there were no stereoscopic observations from the STEREO (Kaiser 2005) spacecrafts, the true geometry of the loop could not be inferred by stereoscopy. We assumed a semielliptical shape of the loop initially in the xz-plane, which is determined by the major axis (2a) and minor axis (2b). The minor axis was taken to be the length of the loop baseline in spherical coordinates, while the major axis was varied. The loop was sequentially rotated around the x, y, and z axes by angles of θ, α, and β, respectively. The projected loop in the xy-plane was then translated to compare with the observed loop in Fig. 1. By varying continuously the values of 2a and the three rotation angles, we can find the best values of the loop parameters when the average distance between the projected loop and the observed loop is minimized using mpfit.pro.

Figure 5 shows the three-dimensional (3D) geometry of the semielliptical loop, where a = 175″, b = 132″, θ = −60°, α ≈ −9°, and β ≈ 40°. In Fig. 6, the projected loop in the xy-plane is drawn with the solid blue line and the observed loop is drawn with the dashed red line. It is clear that the projected loop fits well with the observed loop, suggesting that the semielliptical shape can satisfactorily represent the true geometry of the oscillating loop. The length of the loop was calculated to be ∼377 Mm, which is ∼1.2 times longer than the value when assuming a semicircular shape. Therefore, Ck was estimated to be ∼1630 km s−1 by adopting an average period of oscillation (P ≈ 462 s). The CA was estimated to be ∼1210 km s−1 by assuming that the density ratio ρe/ρi is equal to ∼0.1 (Nakariakov et al. 1999; Nakariakov & Ofman 2001).

thumbnail Fig. 5.

Three-dimensional (3D) geometry of the semielliptical loop. The major and minor axes are 350″ and 264″, respectively. The rotation angles of the initial loop around the x, y, and z axes are −60°, −9°, and 40°, respectively. The colors along the loop represent the heights, and the thin light blue lines represent the projections of the loop onto the three planes.

thumbnail Fig. 6.

The observed loop (dashed red line) in Fig. 1 and projected semielliptical loop in the xy-plane (solid blue line).

The Alfvén speed was determined by the magnetic field strength and mass density of the plasma. Consequently, we were able to estimate the magnetic field strength in the loop (Nakariakov & Ofman 2001):

B = 4 π ρ i C A , $$ \begin{aligned} B=\sqrt{4\pi \rho _i}C_A, \end{aligned} $$(6)

where ρ i = m p n i ≈ m p n e = m p EM / H $ \rho_{i}=m_{p}n_i \approx m_{p}n_e=m_{p}\sqrt{\mathrm{EM}/H} $, mp is the proton mass, and H is the LOS depth of the coronal loop. In Fig. 1b, three tiny boxes, representing the loop top, loop leg, and loop footpoint, are used to perform the DEM analysis described in Sect. 2.2. The inverted DEM profiles are displayed in Figs. 7a–c, with the calculated values of T̄ and EM labeled. The average temperature of the oscillating loop is ≤2 MK, which is consistent with the fact that the oscillation is observed in 131, 171, 193, and 211 Å. The background-subtracted intensity distribution of a short line (9.1 Mm in length) across the middle box is shown in Fig. 7d. Single-Gauss fitting of the profile is used to derive the full width at half maximum (∼3.4 Mm), which is considered to be the width or LOS depth of the loop. Hence, the number density (ni) and corresponding mass density (ρi) of the loop decrease from 6 × 109 cm−3 and 1 × 10−14 g cm−3 near the footpoint to 2.9 × 109 cm−3 and 4.8 × 10−15 g cm−3 near the loop top. The magnetic field strength of the loop (B) falls in the range of 30–43 G, according to Eq. (6).

thumbnail Fig. 7.

Panels a–c: DEM profiles of the three tiny boxes in Fig. 1b. The average temperature and total EM are labeled. Panel d: intensity distribution of a short line across the middle box, which is fitted with a single-Gauss function. The full width at half maximum (FWHM) representing the width or LOS depth of the loop is labeled.

3.3. Magnetic field determined by NLFFF modeling

To validate the magnetic field strength of the oscillating loop inferred from coronal seismology, we carryed out NLFFF modeling to construct magnetic field models by using the flux rope insertion method (van Ballegooijen 2004). We briefly introduced the method in Sect. 2.3, and more details can be found in Bobra et al. (2008) and Su et al. (2009, 2011).

The boundary condition for the high-resolution region is derived from the LOS magnetogram taken by the HMI at 13:36 UT on 2015 October 16. The longitude-latitude map of the radial component of the magnetic field in the high-resolution region is presented in Fig. 8b. Three flux ropes with the same poloidal flux (0 Mx cm−1) are inserted. However, the axial fluxes are different, namely, 8 × 1020 Mx, 1 × 1020 Mx, and 4 × 1020 Mx. In Fig. 8c, selected model field lines matching the observed non-potential coronal loops are overlaid on the AIA 171 Å image. In Fig. 8d, the observed loop is traced manually and marked with the red line. In order to find the field line that best fits the observed loop, we first measured the distance between a point on the observed loop and the closest point on the projected field line on the image plane. The distances for various points along the observed loop were then averaged, which is defined as the “average deviation” (Su et al. 2009). The manually selected 3D field line that can minimize the deviation is considered to be the line that best fits the observed loop (pink line), which is overlaid on the AIA 171 Å image. The length of the pink line (∼354 Mm) accounts for ∼94% of the loop length, assuming a semielliptical shape. We obtained the magnetic field strength at several locations along the model field line, which lies in the range of 21–23 G and is on the same order of magnitude as the result of coronal seismology.

thumbnail Fig. 8.

Panel a: AIA 171 Å image (gray scale) overlaid with positive (red contours) and negative (green contours) polarities of the photospheric magnetic field taken by the HMI. Panel b: zoomed-in view of the longitude-latitude map of the radial component of the photospheric magnetic field by SDO/HMI in the high-resolution region. The blue curves (1, 2, and 3) with circles at the two ends refers to the three paths along which we inserted the flux ropes. Panel c: AIA 171 Å image and selected field lines from the NLFFF model matching the observed non-potential coronal loops. Panel d: AIA 171 Å image and comparison of the observed coronal loop (red line) traced manually and the best-fit modeling field line (pink line).

4. Summary and discussion

In this work, we report our multiwavelength observations of the transverse oscillation of a large-scale coronal loop induced by a blowout jet related to a C4.2 CRF in AR 12434 on 2015 October 16. The oscillation is most pronounced in AIA 171 and 131 Å. The oscillation is almost in phase along the loop with a peak initial amplitude of ∼13.6 Mm, meaning that the oscillation belongs to the fast standing kink mode. The oscillation lasts for ∼3.5 cycles with an average period of ∼462 s and an average damping time of ∼976 s. The values of τ/P lie in the range of 1.5–2.5. Based on coronal seismology, the Alfvén speed in the oscillating loop is estimated to be ∼1210 km s−1. Two independent approaches were applied to calculate the magnetic field strength of the loop, resulting in 30–43 G using coronal seismology and 21–23 G using NLFFF modeling. The results of the two methods are on the same order of magnitude, which confirms the reliability of coronal seismology in diagnosing coronal magnetic fields.

As mentioned in Sect. 1, transverse loop oscillations triggered by CRFs have been observed. Zhang et al. (2015) analyzed an M6.7 flare as a result of partial filament eruption on 2011 September 8. Kink oscillation was induced in an adjacent coronal loop within the same AR (see their Fig. 7). The oscillation that had a small amplitude (∼1.6 Mm) lasted for ∼2 cycles without significant damping. The estimated parameters are listed and compared with the parameters of this study in Table 2. It is found that the values of Ck and CA are similar for the two events. Both the loop length and the period in this study are ≥2 times larger than those for the event in 2011. However, the density and magnetic field of the loop in 2011 are much larger than those in our study. Zhang et al. (2020) investigated the decayless and decaying kink oscillations of an EUV loop on 2014 March 5. For comparison, the parameters, including the shortest loop length and period among the three events, are listed in the last row of Table 2.

Table 2.

Parameters of the oscillating coronal loops observed by SDO/AIA on 2011 September 8, 2015 October 16, and 2014 March 5.

Aschwanden & Schrijver (2011) compared the magnetic field of an oscillating loop determined by coronal seismology with the result of magnetic extrapolation based on potential field source surface (PFSS) modeling using the photospheric magnetogram. They found that the average extrapolated magnetic field strength exceeded the seismologically determined value (∼4 G) by a factor of two. After improving the physical parameters estimation method by taking the effect of density and magnetic stratification into account, the extrapolated magnetic field is optimized (Verwichte et al. 2013a). Guo et al. (2015) reported the kink oscillation of a coronal loop with a total length of ∼204 Mm, which was excited by the global fast magneto-acoustic wave as a result of flux rope eruption and the associated eruptive flare on 2013 April 11. Based on coronal seismology, they derived the spatial distribution of the magnetic field (∼8 G) along the loop, which matches with that derived by a potential field model. Long et al. (2017) estimated the magnetic field of a trans-equatorial loop system using two independent techniques. They found that the magnetic field strength (∼5.5 G) estimated by the two approaches are roughly equal.

Movie

Movie of Fig. 1 Access here

Acknowledgments

The authors are grateful to the referee for valuable suggestions. The authors also appreciate Drs. Z. J. Ning, D. Li, and F. Chen for valuable discussions. SDO is a mission of NASA’s Living With a Star Program. AIA and HMI data are courtesy of the NASA/SDO science teams. Q. M. Z. is supported by the Science and Technology Development Fund of Macau (275/2017/A), CAS Key Laboratory of Solar Activity, National Astronomical Observatories (KLSA202006), Youth Innovation Promotion Association CAS, and the International Cooperation and Interchange Program (11961131002). This work is funded by NSFC Grants (No. 11790302, 11790300, 11773079, 41761134088, 11473071, 12073081), and the Strategic Priority Research Program on Space Science, CAS (XDA15052200, XDA15320301).

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All Tables

Table 1.

Fitted parameters of the coronal loop oscillation.

In the text
Table 2.

Parameters of the oscillating coronal loops observed by SDO/AIA on 2011 September 8, 2015 October 16, and 2014 March 5.

In the text

All Figures

thumbnail Fig. 1.

EUV images of the large-scale coronal loop (dashed yellow line) observed by SDO/AIA in 171, 131, 193, and 211 Å before flare. The straight solid lines (S1, S2, and S3) are used to analyze the transverse oscillation. The red arrows point to the locations of the CRF and blowout coronal jet. In panel b, the three tiny boxes along the loop mark the locations for DEM analysis. (a) AIA 171 Å16–Oct-2015 13:36:22 UT, (b) AIA 131 Å16–Oct–2015 13:36:31 UT, (c) AIA 193 Å16–Oct–2015 13:36:32 UT, (d) AIA 211 Å16–Oct–2015 13:36:22 UT.
In the text
thumbnail Fig. 2.

SXR light curves of the flare in 1–8 Å (red line) and 0.5–4 Å (blue line). The yellow area represents the time of loop oscillation between 13:39 UT and 14:05 UT.
In the text
thumbnail Fig. 3.

Time-distance diagrams of the three slices (S1, S2, and S3) in 171 and 131 Å. The dashed magenta lines outline the transverse loop oscillation between 13:39 UT and 14:05 UT. On the y-axis, s = 0 and s = 43.5 Mm denote the southern and northern endpoints of the slices, respectively.
In the text
thumbnail Fig. 4.

Positions of the oscillating loop along the three slices (green crosses) and the results of curve fitting (blue lines) using Eq. (4).
In the text
thumbnail Fig. 5.

Three-dimensional (3D) geometry of the semielliptical loop. The major and minor axes are 350″ and 264″, respectively. The rotation angles of the initial loop around the x, y, and z axes are −60°, −9°, and 40°, respectively. The colors along the loop represent the heights, and the thin light blue lines represent the projections of the loop onto the three planes.
In the text
thumbnail Fig. 6.

The observed loop (dashed red line) in Fig. 1 and projected semielliptical loop in the xy-plane (solid blue line).
In the text
thumbnail Fig. 7.

Panels a–c: DEM profiles of the three tiny boxes in Fig. 1b. The average temperature and total EM are labeled. Panel d: intensity distribution of a short line across the middle box, which is fitted with a single-Gauss function. The full width at half maximum (FWHM) representing the width or LOS depth of the loop is labeled.
In the text
thumbnail Fig. 8.

Panel a: AIA 171 Å image (gray scale) overlaid with positive (red contours) and negative (green contours) polarities of the photospheric magnetic field taken by the HMI. Panel b: zoomed-in view of the longitude-latitude map of the radial component of the photospheric magnetic field by SDO/HMI in the high-resolution region. The blue curves (1, 2, and 3) with circles at the two ends refers to the three paths along which we inserted the flux ropes. Panel c: AIA 171 Å image and selected field lines from the NLFFF model matching the observed non-potential coronal loops. Panel d: AIA 171 Å image and comparison of the observed coronal loop (red line) traced manually and the best-fit modeling field line (pink line).
In the text

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