Finding the Critical Decay Index in Solar Prominence Eruptions

Coronal mass ejections (CMEs) are considered to be the most geoeffective phenomena that happen in the solar atmosphere (Kahler 1992; Gosling 1993). These CMEs are frequently observed in association with filament or prominence structures (Gosling et al. 1974). Filaments or prominences are coronal structures that are two orders of magnitude more cool and dense than the surrounding coronal atmosphere. When these structures are observed against the solar disk, they are referred to as "filaments" and when they are observed at the solar limb, they are referred to as "prominences." The solar prominence eruptions (PEs) lead to ejection of large clouds of magnetized plasma observed in the lower and middle corona prior to the observations of CMEs in the upper corona. The topological structure of prominences is that they are supported by twisted magnetic field lines that wrap around an axial magnetic field called magnetic flux ropes (MFRs) such that the magnetic dips in their lower windings support the prominence plasma against gravity (Kippenhahn & Schlüter 1957; van Ballegooijen & Martens 1989; Aulanier & Demoulin 1998). This suggests that the MFRs play a key role in solar eruptions and hence almost all CME-initiation models assume the presence of the flux rope structure (Chen 2011; Xie et al. 2013; Vourlidas 2014).

Observations in Hα reveal that the filaments are visible for hours to several days before they erupt or disappear. This suggests that they are in equilibrium with the surrounding environment. Then the filament eruption is regarded as the loss of this equilibrium. In the flux-rope-based models (Kliem & Török 2006), the filament equilibrium is the balance between the inward-directed magnetic tension of the external overlying field that embeds the flux rope and the outward-directed magnetic pressure between the flux rope axis and the photospheric boundary. From the ideal MHD point of view, this equilibrium state will be ruptured in two ways. One is exceeding twist in the flux rope leading to kink instability and other is the torus instability arising when there is a rapid decline of the background field in the direction of the expansion of the flux rope. However, the sheared arcade model assumes that the filament/sigmoid is composed of the sheared and twisted core field and the reconnection of the shear field lines forms the flux rope and the subsequent eruption (Moore et al. 2001).

Kink instability can initiate the rise motion of the flux rope to a height from where the flux rope eruption is driven by the torus instability (Vemareddy & Zhang 2014). Even with exceeding critical twist, the flux rope cannot lead to a successful eruption when there is strong overlying field. A recent statistical study of 36 strong flare events (Jing et al. 2018) also confirms that kink instability plays a small role in the eruption of flux ropes. Therefore, the decrease of the overlying field with height (torus instability) plays a main role in deciding whether the instability leads to a confined event or to a CME. For example, the failed/confined eruption of an active region (AR) filament occurred on 2002 May 27 (Ji et al. 2003; Török & Kliem 2005). This AR filament started to rise rapidly and developed a clear helical shape. Eventually its rise motion got terminated after reaching a height of 80 Mm due to the strong strapping field and was just accompanied by a M2 flare without CME. However, there have been reports that the rotational motion of flux rope along with torus instability play significant roles in eruptivity of an event (Song et al. 2018b; Zhou et al. 2019).

The idea of torus instability was first proposed by Bateman (1978) in tokamaks, and first revisited for solar eruptions by Kliem & Török (2006). According to this instability, a current ring of major radius R is embedded in an external magnetic field. The ring experiences radially outward "hoop force" due to its curvature and this force decreases in magnitude if the ring expands. If the inwardly directed Lorentz force due to the external field decreases faster with major radius R than the hoop force, the system becomes unstable. Assuming an external magnetic field B_ex ∝ R⁻ⁿ, the decay index n is defined as n = −dlog(B_ex)/dlog(R). This means that when the decay index of the external field is equal to or higher than the critical decay index value n_c, the system becomes unstable and by any small disturbance to the current channel initiates its outward motion uninhibitedly. Titov & Démoulin (1999) and Török & Kliem (2007) have performed numerical simulations with a semicircular flux rope embedded in the external field. They found that the torus instability occurs when the flux rope axis reaches a height where the decay index of the external field is larger than n_c.

The torus instability threshold depends on the geometry of the flux rope and is subjective to the case of particular study. For thin current distribution, the critical decay indices for straight and semicircular current channels are 1.0 and 1.5 respectively. For thick current channels, as expected in corona, the n_c lies in the range 1.1–1.3 if the cross-section increases during the eruption or 1.2–1.5 if the cross-section remains constant (Démoulin & Aulanier 2010). These thresholds are derived using the current-wire models, where the equilibrium properties are determined using only the momentum equation in terms of current distribution. On the other hand, many numerical MHD simulations (for example, Fan & Gibson 2007; Török & Kliem 2007; Démoulin & Aulanier 2010; Fan 2010) were conducted using the full set of MHD equations to validate the torus instability and they suggest the values of n_c in the range 1.4–1.9.

To determine the n_c in the actual observations, one needs to have the critical height of the flux rope from where it experiences rapid rise motion and the background magnetic field. Given a model of the three-dimensional (3D) background magnetic field, the critical height is still unknown due to projection effects. In the cases of the CME eruptions near the solar disk, Török & Kliem (2005) proposed to use a constant value of n_c = 1.5. Using this value, many studies have derived a critical height of 42 Mm being the dividing line between the confined and eruptive events (Liu 2008; Vemareddy & Demóulin 2018; Vasantharaju et al. 2018). However, a recent numerical simulation study by Zuccarello et al. (2015) clearly showed a slightly different value of 1.4 ± 0.1 at the onset of eruption.

Further, several observational studies were also made to determine the n_c. Filippov & Den (2001) performed a statistical study of 27 quiescent prominences and found that prominences are prone to erupt when they reach a critical height where the decay index of the external field is 1. Recently, McCauley et al. (2015) studied the kinematics of 106 PEs and found that the average decay index at the onset height of fast-rise phase is 1.1. Both these studies have not employed the STEREO observations to determine the true height of prominence features which means that the determined critical heights are subjective. Filippov (2013) employed different observational viewpoints provided by the twin STEREO and SDO spacecraft to study the quiescent filament eruption and found that n_c is  1.0. In this framework, Zuccarello et al. (2014) studied an AR filament eruption and concluded that the filament reaches a height where the decay index is in range 1.3–1.5. So, in general, n_c values determined by observational studies are a bit smaller than those of simulations and this apparent difference in n_c values is mainly due to the exact location where the torus instability criterion is evaluated (Zuccarello et al. 2016). They showed that in simulations the n_c is computed at the flux rope axis during the onset of the eruption whereas in observational studies the n_c is computed at the apex of the prominence. Owing to importance in space weather, the critical height and decay index have become the subject of many studies, including this article.

Motivated by the above studies, we studied the value of the n_c in 10 PE cases. We used simultaneous STEREO and SDO observations to derive the prominence kinematics based on the true height of the rising prominence. From the kinematics profile, the PE is characterized distinctly viz., slow-rise and fast-rise phases. Corresponding to the height at which the fast-rise motion commences, the n_c is obtained. Furthermore, we also paid attention to the acceleration mechanism of the flux rope. Now it is widely accepted that the flux rope instability triggers the prominence/CME eruption first, and then magnetic reconnection underneath provides further acceleration (Lin et al. 2003; Vršnak 2016). Past numerical studies demonstrated that both these mechanisms have comparable contributions to the prominence acceleration (Chen et al. 2007a, 2007b). This has been confirmed observationally by analyzing the relationship between kinematics and magnetic reconnection processes during an AR filament eruption (Song et al. 2015) and a quiescent filament eruption (Song et al. 2018a). Both these events are associated with X-ray flares. Nonetheless, the scenario might be different from event to event. If a good temporal correlation exists between the prominence kinematics and reconnection characteristics, then magnetic reconnection is important for the acceleration process, otherwise it is not (Song et al. 2013). A recent analytical study by Vršnak (2016) showed that magnetic reconnection is more dominant than MHD instability in impulsively accelerated events. In order to have more insights on the acceleration mechanism, we study a larger number of sample events to arrive at a solid conclusion. Details of the observational data and analysis procedure are given in Section 2. The analysis results are described in Sections 3. Summary and discussions are given in Section 4.

We selected 10 AR prominences located within the longitudinal belt of 40°–80°. Note that our selection of prominence source regions are not located at the solar limb so that the computation of background magnetic field is not significantly affected by the discontinuity of magnetic field on the solar limb in synoptic maps. At the same time, in order to minimize the projection effects in the determination of height of the prominence features, the regions located beyond the ±40° longitude from central meridian are selected. These prominence events, named P1–P10, are listed in the first column of Table 1. The filaments in all our sample events are lying above the polarity inversion line (PIL) of source ARs except for P2, P7, and P8 filaments, which are located above the neutral region between two adjacent extended bipolar regions (EBRs). The distinct PIL or neutral region beneath the eruption will be used to compute the decay index as described in the following Section 3.2. The second, third, and fourth columns in Table 1 give the basic information of events, viz., source NOAA AR number, date, and location of eruptions respectively. All PEs in our sample are characterized by distinct slow- and rapid-acceleration phases. The critical time (T_c) of eruptions is defined as the onset time of the fast acceleration phase of PEs. T_c for all the sample events are listed in the fifth column of Table 1. The true prominence apex height from the photospheric surface at the critical time refers to critical height (H_c) and is listed in the sixth column of Table 1. The decay index at the critical height of PE is termed as n_c, which are tabulated in the seventh column of Table 1. The Geostationary Operational Environmental Satellite (GOES) provides the soft X-ray (SXR) flux integrated from the full solar disk, which are used to characterize the magnitude, onset, and peak times of solar flares. The eighth column gives the associated GOES X-ray class flares and the ninth column gives the CME linear speed (obtained from the SOHO LASCO CME catalog: https://cdaw.gsfc.nasa.gov/CME_list/). In the following section, we describe the procedures for computing T_c, H_c, n_c, and kinematics of our sample events.

The true height of the prominence feature is determined by the tie-pointing method (Thompson 2009). This method is computerized as the IDL routine scc_measure.pro available in SolarSoftware (SSW) distribution. For this, we use simultaneous observations in 304 Å waveband of the Atmospheric Imaging Assembly (AIA; Lemen et al. 2012) on board SDO and Extreme UltraViolet Imager (EUVI; Wuelser et al. 2004) on board STEREO. AIA images the solar corona with a pixel size of 06 and a high cadence of 12 s whereas EUVI images from STEREO have a pixel size of 16 and obtained at a cadence of 10 minutes. Using this 3D coordinate measuring tool, the precise 3D positions of the prominence feature is determined (see Section 3.2).

Further the decay index, n(z) of the background field is computed by reconstructing the coronal magnetic field using potential field source surface model (PFSS; Schrijver & De Rosa 2003). PFSS approximates the coronal magnetic field as the potential field between the photosphere and spherical surface at 2.5R_⊙ and the magnetic field on the spherical surface is radial. This model is implemented in the PFSS package available in Solarsoftware (SSW). The model requires the radial magnetic field at the photosphere as a boundary condition, which is the daily-updated synoptic chart of the photospheric radial magnetic field observations of the Helioseismic and Magnetic Imager (HMI; Schou et al. 2012) on board SDO. Each daily-updated synoptic chart is composed of two parts (Sun 2018). The update part is a 120° wide band from a 4 hr average of the remapped magnetograms centered at the central meridian time of interest. The 120° updated region provides data in longitude from the left-edge toward the right. The remainder of the map comes from the standard Carrington synoptic chart(s) that makes up the rest of the Carrington rotation.

3.1. Overview of Sample Events

3.2. Critical Heights and Decay Indices

As exemplary cases, we present events P1 (flare associated) and P3 (flareless) to illustrate the procedure of determining the critical heights of erupting prominences and the corresponding decay indices. The P1 erupted on 2011 March 7, at 19:33 UT from NOAA AR 11164 located near the western limb (N24 W62). This strong eruption happened before the M-class flare, which started at 19:42 UT and peaked at 20:12 UT according to GOES. Also during this time (19:42–20:58 UT) clear arcs are seen at the footpoints of the flare in 304 Å image. The P1 eruption appears to have a light bulb-shaped structure with the twisted filament in the middle and eventually leads to a halo CME observed in LASCO C2/C3 FOV. A detailed study of P1 is presented in Cheng et al. (2013a). The P3 erupted on 2011 June 12, at 13:34 UT from AR NOAA 11232 located near the western limb (N08W74). The P3 eruption displayed a large kink associated with no observable X-ray flare as recorded by GOES and mass leaving the surface leads to CME observed in LASCO C2's field of view. The enhancement of the SXR profile is observed only after 40 minutes of the P3 eruption (Figure 2(d)) and this small SXR intensity enhancement of B-class level accounts for the brightening caused by some of the erupted mass when it falls back to the surface.

Panels in Figures 1((a) and (b)) and 2((a) and (b)) show the HMI line-of-sight (LOS) magnetic field and AIA 304 Å observations of the PEs of P1 and P3 respectively. The slits were placed on the respective AIA 304 Å images to characterize the overall trajectory of erupting prominences. Using these slits, spacetime plots were generated (Figures 1(c) and 2(c)) and green asterisk symbols in these panels represent the leading edge of ascending prominences. We applied the height correction using the 3D coordinates obtained from the tie-pointing method. The SDO, STEREO-A, and STEREO-B spacecraft positions during the P1 event were obtained using the STEREO science center website (https://stereo-ssc.nascom.nasa.gov/cgi-bin/make_where_gif). We used near simultaneous observations of SDO/AIA and STEREO-A in a wavelength passband of 304 Å in the scc_measure.pro routine and then, with the aid of the graphical interface of the SECCHI 3D coordinate tool, we are allowed to select a feature on one image (see left panel of Figure 3, "+" sign on STEREO-A image), then an epipolar line displayed on the SDO image passing through the same feature as shown in right panel of Figure 3. After the user identifies the intersection between the projected line of sight and the feature of interest, the program triangulates the feature's three-dimensional (3D) location. Using this 3D coordinate of the selected feature, the correction is added/subtracted to the projected height.

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The deprojected or corrected height–time plots of P1 and P3 are shown in the Figure 1(d) and 2(d) respectively. The corrected height–time curve consists of two distinct profiles, a slow-rise phase having almost constant velocity (e.g., Sterling & Moore 2005) and a fast-rise phase with rapid acceleration approximated by an exponential curve (e.g., Goff et al. 2005). We used a model containing the linear term to treat the slow-rise phase and the exponential term to account for the rapid-acceleration phase as described in Cheng et al. (2013b) and it is given by

$$\begin{eqnarray}&&h(t)={C}_{0}{e}^{(t-{t}_{0})/\tau }+{C}_{1}(t-{t}_{0})+{C}_{2},\end{eqnarray} \tag{ 1 }$$

where h(t) is height at a given time t, and τ, t₀, C₀, C₁, and C₂ are free coefficients. This model has two distinct advantages: (1) a single function describes the two phases of eruption effectively and (2) it provides a convenient method to determine the time of onset of rapid-acceleration phase (T_c). Critical time, T_c, is defined as the time at which the exponential component of velocity equals its linear component as T_c = τ ln(C₁τ/C₀) + t₀. Using this equation, T_c for P1 and P3 events are computed to be 19:33 UT (19.55 hr) and 13:34 UT (13.57 hr) respectively. These timings are marked by blue horizontal dashed lines in Figures 4(a) and 5(b). We used mpfit.pro to fit the corrected height–time data by model function and the fit is shown as a blue solid curve. From this fit, the critical height (H_c) is determined corresponding to T_c. The H_c for P1 and P3 events are determined to be 0.035R_⊙ and 0.028R_⊙, respectively, and they are represented by black vertical dashed lines in Figures 4(a) and 5(b).

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Furthermore, to determine the decay index corresponding to the critical height, we used HMI daily-updated synoptic maps as the boundary conditions in potential field approximation (PFSS). Then the horizontal component of the background magnetic field (B_h) as a function of height is obtained at 8 to 10 points along the main polarity inversion line (PIL), and an average of n is derived. Errors of n are mainly from the uncertainties in height, which are regarded as the standard deviations of the number of measurements. Then using the decay index curve, the decay index corresponding to critical height (H_c) is determined and considered as n_c. The critical decay indices of P1 and P3 events are found to be 1.02 and 0.8, respectively, and are represented by black horizontal dashed lines in Figures 4(a) and 5(b). We followed the same procedure to derive parameters like T_c, H_c, and n_c for all 10 events in our sample, which are tabulated in Table 1. The average critical decay indices of our sample of 10 events is ≈1.05. This result is consistent with past studies, like those of Filippov & Den (2001) and McCauley et al. (2015; see 1). But both of these studies involve the errors induced by the projection effects on the determination of prominence positions. To account for these errors, Filippov (2013) used three vantage point observations to study the quiescent filament eruption and found that n_c is ≈1.0. Thus, generally, n_c for both quiescent and AR prominences are almost equal to 1 and its value prominently depends on the strength of coronal background magnetic field confinement of an individual event.

We observed two types of decay index curves in our sample events. Four (P1, P4, P5, and P10) out of 10 events exhibit a gradual increase of decay index with height and are shown in Figure 4. For the remaining six events, the decay index curves exhibit the unusual "bump" in low corona and then increase gradually with height. These events are shown in Figure 5. The critical heights of our sample events are less than 40 Mm (height above the photosphere) except for the P8 event, which has a critical height at about 50 Mm. The obtained critical heights are in agreement with the recent study of Vasantharaju et al. (2018) where the majority of critical heights of eruptive events are less than 42 Mm. Corresponding to the critical height (H_c), the n_c of our sample events is in the range of 0.8–1.3.

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Thus the background magnetic field in the lower corona (< 50 Mm) decreases fast enough over all the eruption sites of our sample events to facilitate the eruptions irrespective of type of decay index profiles. However, in the events where the decay index curves exhibit a bump in the lower corona, the background transverse magnetic field decreases very rapidly over the sites where decay index curves show gradual increase. For example, the events like P2, P3, and P7 (except for P9) exhibit a bump in decay index curves in lower corona (i.e., within 50 Mm), the maximum decay index values reached are in the range of 1.5–2.0 (Figure 5) but for events like P1, P4 , P5, and P10 that do not show bump in decay index curves, the maximum decay index values attained are smaller and in the range of 1.1–1.6 (Figure 4). Though the P9 event exhibits a bump in the decay index curve (Figure 5(d)) in lower corona, the maximum decay index value attained is just about 1.2 at the height of about 20 Mm. Also, the P9 eruption was initiated at about 20 Mm (blue curve in Figure 5(d)). This strongly suggests that enough weaker transverse magnetic field strength at about 20 Mm facilitates the eruption or initiates the fast-rise motion of an erupting structure. After its eruption the decay index value decreases and then gradually increases with height. The decay index variations with height for P6, P8 (both figures are not shown), and P7 (Figure 5(c)) events are similar and they exhibit a large bump in the decay index curves representing large decay index values (1.5–2.5) up to the height of more than 120 Mm. This very rapid decay of background field in the lower corona initiates the eruptions of flux ropes/erupting structures in these events. Then the decay index values decrease slowly with height and again start to rise after about 300 Mm.

3.3. Kinematics of Prominences

All our sample events exhibit distinct slow- and fast-rise phases, which are well-fitted by the linear-cum-exponential model proposed by Cheng et al. (2013b). The two representative events P1 and P3 used in the previous section will also be considered here along with P10 and P8 to illustrate the kinematics of our sample events. P1 and P10 are flare accompanied PEs whereas P3 and P8 are flareless events. The velocity and acceleration profiles are obtained by taking time derivatives of the model function fit to the corrected height–time data. In Figure 6, the top panels present typical examples (P1 and P10) of kinematics of flare-associated events and bottom panels present the kinematics of flareless PE events (P3 and P8). The solid black curve indicates the fit to corrected height–time data points (black asterisks in Figure 6) and the derived acceleration profiles are overplotted in blue in all four panels. The critical time obtained from the fit bifurcates the prominence rise motion into slow- and fast-rise (rapid-acceleration) phases. Thus, the slow-rise phase for P1 and P10 is observed until 19:33 UT on 2011 March 7 and 01:01 UT on 2015 May 9 with average velocities of  7 km s⁻¹ and  9 km s⁻¹ respectively. For P3 and P8 events, the slow-rise phase observed until 13:34 UT on 2011 June 12 and 13:38 UT on 2013 March 16 with the average velocities of  4 km s⁻¹ and  9 km s⁻¹ respectively. After that, the fast-rise phase starts for flare-associated events P1 and P10 with an average acceleration of 1541 m s⁻² and 578 m s⁻² respectively. For flareless events P3 and P8, the average acceleration during the rapid-acceleration phase is 72 m s⁻² and 128 m s⁻² respectively. Similarly, kinematic details of other events are tabulated in Table 2. In our sample, we found that in the rapid-acceleration phase, the flare-associated events have average acceleration in the range of 400–1550 m s⁻² and flareless events (P3, P6 and P8) have average acceleration well below 200 m s⁻². These observational results imply that the flare magnetic reconnection that occurred during the fast-rise phase is responsible for the acceleration of flare-associated prominences significantly compared to that of flareless counterparts.

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Table 2.  Kinematics of Prominence Eruption Events

P No.	Tc	Tf	Vavg (km s−1)	Vmax (km s−1)	aini (m s−2)	amax (m s−2)	aavg (m s−2)	apf (m s−2)	aip (m s−2)
P1	19:33	19:42	7.1	1347.1	30.1	6441.2	1540.8	299.2	2970 .3
P2	02:16	02:23	30.2	713.5	109.2	1273.1	532.6	235.8	726.3
P3	13:33	⋯	3.9	205.6	4.5	264.6	71.6	⋯	⋯
P4	12:57	13:04	18.8	670.8	54.6	911.4	397.1	90.3	499.8
P5	08:16	08:27	10.8	617.3	37.8	2389.1	649.7	201.6	1301.3
P6	17:02	⋯	30.3	315.3	67.7	616.5	198.6	⋯	⋯
P7	18:48	18:55	24.4	696.7	90.6	1676.3	591.2	191.6	808.9
P8	13:38	⋯	8.6	359.3	13.1	422.8	128.2	⋯	⋯
P9	16:04	16:06	52.4	982.6	220.4	1683.2	839.7	247.4	928.6
P10	01:01	01:11	8.9	635.6	47.2	2023.3	577.2	166.7	1020.3

Note. T_c: onset time of the fast-rise phase; T_f: onset time of flare accompanied; V_avg: average velocity of the slow-rise phase; V_max: maximum velocity in field of view (FOV) of the slice; a_ini: acceleration at the onset (T_c) of the fast-rise phase; a_max: maximum acceleration in FOV of slice; a_avg: average acceleration of the fast-rise phase; a_pf: average acceleration of the preflare phase; a_ip: Average acceleration of the impulsive phase.

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Furthermore, the GOES SXR derivative can be used as a proxy of hard X-ray according to the Neupert effect (Neupert 1968) and is overplotted in red in all panels of Figure 6 to identify the time of onset of magnetic reconnection (flare). The times of onset of the flare during P1 and P10 events are 19:42 UT and 01:11 UT, respectively, which are about 9–10 minutes later than the time of onset of the fast-rise phases (i.e., 19:33 UT and 01:01UT) of P1 and P10 respectively. It is worth noting that in all our flare-associated events, the onset time of the fast-rise phase is earlier than the onset time of the flare by 2–10 minutes (see Table 2). After careful inspection of temporal correlations between the velocity and SXR flux as well as the acceleration and hard X-ray flux, we noticed that the flux rope instability triggers and accelerates the prominences first, and the magnetic reconnection is induced subsequently to provide further acceleration as suggested in past studies (Priest & Forbes 2002; Lin et al. 2003; Temmer et al. 2010; Vršnak 2016).

Furthermore, to compare the mechanisms contributing to the acceleration process during the fast-rise phase of flare-associated events (impulsive events) in AIA FOV, the fast-rise phase can be divided into two phases viz. a preflare and flare-impulsive phases, based on the reference time of onset of flare during the events. Here, the preflare phase is assumed to be the duration of the fast-rise phase from the critical time of eruption to the time of onset of flare and rest of the fast-rise phase, i.e., from the time of onset of the flare until the prominence structure leaves the AIA FOV is considered to be an impulsive phase. Though the preflare phase is actually a combination of the slow-rise phase and the fast-rise phase until the flare onset, we have excluded the slow-rise phase due to the fact that the acceleration process in the slow-rise phase is contributed by both kink instability and/or quasi-separatrix-layer (QSL) reconnection (Cheng et al. 2013b), i.e., reconnection at the interface between the filament and its surrounding corona. As it is difficult to disentangle the contributory mechanisms to the acceleration process in the slow-rise phase, we deliberately excluded it in considering from preflare phase. So basically, we are concentrating on the mechanisms contributing to the acceleration process from the prominence's eruption until it leaves the AIA FOV. In Figure 6, the enhancement of the SXR flux derivative as indicated by the red vertical dashed lines mark the onset time of flare magnetic reconnection in top panels and steady behavior of SXR flux derivative in the bottom panels indicate the absence of flare during PEs. The enhancement of SXR flux derivative for P1 (Figure 6(a)) and P10 (Figure 6(b)) events occurred at 19:42 UT and 01:11 UT, respectively, indicates the time of onset of the impulsive phase. The average acceleration in the preflare phase of the P1 event is about 299 m s⁻² and in the impulsive phase is about 2970 m s⁻². For the P10 event, the average acceleration during the preflare phase and impulsive phase is  167 m s⁻² and  1020 ms⁻² respectively. The average acceleration in the impulsive phase is almost 10 times larger than that of the preflare phase for the P1 event and in the same way for the P10 event it is almost 6 times greater. Similar to P1 and P10 events, the remaining flare-associated events are more highly accelerated in their impulsive phases than in preflare phases and their kinematic parameters are listed in Table 2. These results suggest that magnetic reconnection is a greater contributor to the acceleration process than the torus instability in impulsive phases of PEs. Also, by considering the whole acceleration process during the fast-rise phase, the contribution of magnetic reconnection to the acceleration of the flux rope is more dominant than the flux rope instability and these two mechanisms do not have comparable contributions to the acceleration process in impulsive events. This result is in agreement with the recent analytical study of Vršnak (2016).

The background field is assumed to play a prime role in erupting structures like prominences. In the flux rope models, the n_c is a measure of vertical gradient of the background magnetic field and an important dimensionless parameter determining the eruptive nature of prominences. Owing to difficulties in obtaining the critical height of the background field, a typical value of n_c = 1.5 is adopted from the numerical studies (e.g., Török & Kliem 2005). In this study, we investigated the critical decay indices of 10 PE events, by estimating the critical height from two vantage point observations of the erupting prominence.

Ideal MHD instabilities responsible for the flux rope eruption include the torus instability and helical kink instability. For a flux rope of exceeding magnetic twist, the kink instability may drive the eruption up to a point of torus regime. Kink instability is not necessarily a trigger in all events, it can be tether-cutting reconnection. When the flux rope reaches a height of steep gradient of horizontal field strength, both of the instabilities may contribute to the impulsive acceleration of the prominence simultaneously, even along with the reconnection. At a height of critical point, the downward force is dominated by the hoop force, leading to the commencement of flux rope fast-rise motion. Therefore, we argue that the critical height of the steep field strength gradient is the height of onset of the fast-rise motion of the prominence.

Our sample of events exhibits linear slow- and exponential fast-rise phases during the eruption, and the decay indices are determined at the onset height of the fast-rise phase. The assumption, as justified earlier, is that at the time the prominence commences the fast-rise motion, it (the apex) is at a critical height of background field. From the critical height, the erupting structure is allowed to expand exponentially provided there is no strapping background field. In this scenario, the height–time profile of the erupting structure exhibits a turning point from slow to fast-rise motion. We use a fitting model to determine this turning point as critical height. The background field is obtained from the PFSS model by using an HMI synoptic magnetic map.

Two types of decay indexes are observed in our 10 sample events. Four events exhibit the gradual increase of decay index with height, and the remaining six events exhibit a "bump" in decay index curves in lower corona. This unusual bump in decay index curves was first reported in Cheng et al. (2011) and claimed to be mostly seen in the eruptive flare cases. However, we observed a similar bump for flareless PEs (P6 and P8) as well. These bumps in the lower corona represent a weaker transverse magnetic field or a very rapid decrease of background magnetic field facilitating the eruptions (initiating the fast-rise motion of eruptive structure) at critical heights.

The n_c is not a constant value and varies from event to event as the background field configuration depends on the field distribution in the source region. The n_c of our sample events ranges from 0.8 to 1.3 and the average value of n_c of our sample is found to be 1.05. This value is in agreement with past observational studies (Filippov 2013; McCauley et al. 2015). Any differences in critical decay indices obtained from theoretical and observational studies is just an apparent and this is mainly due to the exact location where the torus instability criterion is evaluated. Numerical simulations by Zuccarello et al. (2016) showed that in curved flux rope geometry, the height of the flux rope axis is larger than the apex of prominence structure. Due to this, the n_c obtained at the height of the flux rope axis in simulations and the apparent n_c obtained at the top of the dipped structure using observations are found to be 1.4 ± 0.1 and 1.1 ± 0.1 respectively.

The critical times are the onset time of fast-rise (rapid-acceleration) phases, which are obtained using the linear-exponential model of Cheng et al. (2013b). In all our sample events, the onset time of flares is 2–10 minutes later than the onset time of rapid-acceleration phases. We observed that prominence events associated with flares are highly accelerated compared to flareless counterparts.

To study the comparison of mechanisms contributing to the acceleration process within AIA FOV of flare-associated (impulsive) events, the fast-rise phase is further separated into two subphases based on the onset time of the flares viz. a preflare phase (excluding slow-rise motion) and an impulsive phase. We inferred from temporal correlations between the velocity and SXR flux, as well as the acceleration and hard X-ray flux, that the flux rope instability holds a major contribution to the initial phase of acceleration in the preflare phase and the flare magnetic reconnection was dominant in the second phase, i.e., the impulsive phase. This analysis further leads us to infer that the magnetic reconnection is a more dominant contributor to the acceleration process than the MHD instability within AIA FOV of impulsively accelerated prominence events. The weak magnetic reconnection in the preflare phase might be insufficient to accelerate the MFR as it cannot weaken the tension force of the overlying magnetic loops fast enough, and may even lead to the MFR deceleration as calculated by Lin & Forbes (2000). However, the increase in the GOES SXR flux derivative (proxy of hard X-rays) indicates the increased flare reconnection rate that may lead to rapid decrease in the magnetic tension of overlying loops which in turn leads to the enhanced outward acceleration of prominence structure. Also, the analytical study of Vršnak (2016) shows that magnetic reconnection not only reduces the tension of overlying magnetic loops and increases the magnetic pressure below the ejecting flux rope but also supplies the additional poloidal flux to the flux rope and increases its hoop force. These factors enhance and prolong the flux rope acceleration significantly. Whereas for three flareless events (P3, P6, and P8) in our sample, the ideal MHD instabilities appear to be the dominant contributors to the acceleration process of erupting prominence structures. A numerical study by Chen et al. (2007a) showed that the ideal MHD instability process alone can produce fast CMEs but not faster than impulsive events. They further showed that if the magnetic reconnection sets in then it enhances the CME acceleration significantly, and both ideal MHD instability and magnetic reconnection have comparable contributions to the acceleration process in impulsive events. This notion is supported by observational studies of filament eruptions from both the AR (Song et al. 2015) and the quiet region (Song et al. 2018a). However, the scenario is different from event to event. For example, in a statistical study of CME kinematics of 22 events (Maričić et al. 2007), a quarter of the sample of events exhibits the weak-synchronization of CME kinematics and magnetic reconnection characteristics. In such events, the ideal MHD instability would be the major contributor to the CME acceleration.

SDO is a mission of NASA's Living With a Star Program. Full-disk EUVI images are supplied courtesy of the STEREO Sun Earth Connection Coronal and Heliospheric Investigation (SECCHI) team. N.V. is a CSIR-SRF, and gratefully acknowledges the funding from CSIR-HRDG, New Delhi. We thank the referee for the encouraging comments and suggestions.