Observations and modeling of the early acceleration phase of erupting filaments involved in coronal mass ejections
aa r X i v : . [ a s t r o - ph ] O c t Observations and modeling of the early acceleration phase of erupting filamentsinvolved in coronal mass ejections
Carolus J. Schrijver , Christopher Elmore , Bernhard Kliem , , Tibor T¨or¨ok , , and Alan M.Title Lockheed Martin Advanced Technology Center, 3251 Hanover Street, Palo Alto, CA 94304,U.S.A; Astrophysical Institute Potsdam, An der Sternwarte 16, 14482 Potsdam, Germany; Kiepenheuer Institute for Solar Physics, Sch¨oneckstr. 6, 79104 Freiburg, Germany; University College London, Mullard Space Science Laboratory, Holmbury St. Mary, Dorking,Surrey RH5 6NT, United Kingdom; LESIA, Observatoire de Paris, CNRS. Univerit´e Paris Diderot, 5 Place Jules Janssen, 92190Meudon, France [email protected]; [email protected]; [email protected]; [email protected];[email protected]
November 5, 2018
ABSTRACT
We examine the early phases of two near-limb filament destabilizations involvedin coronal mass ejections on 16 June and 27 July 2005, using high-resolution, high-cadence observations made with the Transition Region and Coronal Explorer (TRACE),complemented by coronagraphic observations by Mauna Loa and the SOlar and Helio-spheric Observatory (SOHO). The filaments’ heights above the solar limb in their rapid-acceleration phases are best characterized by a height dependence h ( t ) ∝ t m with m near, or slightly above, 3 for both events. Such profiles are incompatible with publishedresults for breakout, MHD-instability, and catastrophe models. We show numericalsimulations of the torus instability that approximate this height evolution in case asubstantial initial velocity perturbation is applied to the developing instability. Weargue that the sensitivity of magnetic instabilities to initial and boundary conditionsrequires higher fidelity modeling of all proposed mechanisms if observations of rise pro-files are to be used to differentiate between them. The observations show no significantdelays between the motions of the filament and of overlying loops: the filaments seemto move as part of the overall coronal field until several minutes after the onset of therapid-acceleration phase. Subject headings:
Sun: coronal mass ejections (CMEs) – Sun: filaments 2 –
1. Introduction
Observations of the early rise phase of filaments and their overlying fields can in principle helpconstrain the mechanisms involved in the destabilization of the magnetic configuration throughcomparison with numerical simulations (e.g., Fan, 2005; T¨or¨ok and Kliem, 2005; Williams et al. ,2005; and references therein), because the detailed evolution depends sensitively on the modeldetails. For example, a power-law rise with an exponent m = 2 . t ) function (Kliem and T¨or¨ok, 2006) that is very similar to a pure exponential early on. TheCME rise in a breakout model simulation was well described by a parabolic profile (Lynch et al.,2004).The early rise phase of erupting filaments is best observed near the solar limb using high-resolution data, both in space and in time. Such data can be obtained by, for example, Big BearSolar Observatory H α observations (e.g., Kahler et al. , 1988), the Mauna Loa K-coronameter (e.g.,Gilbert et al. , 2000), the Nobeyama Radioheliograph (e.g., Gopalswamy et al. , 2003; Kundu et al. ,2004), and the Transition Region and Coronal Explorer, TRACE (e.g., Vrˇsnak, 2001; Gallagher et al. ,2003; Goff et al. , 2005; Sterling and Moore, 2004; Sterling and Moore, 2005; Williams et al. , 2005).In those few cases where observers had the field of view for an appropriate diagnostic to attemptto establish whether the high loops or the filaments were accelerated first, the temporal resolutionoften was not adequate (see, e.g., Sterling and Moore, 2004, who use the standard 12-min. cadenceof SOHO/EIT).These studies show that filaments that are about to erupt often – but not always – exhibit aslow initial rise during which both the filament and the overlying field expand with velocities inthe range of 1 −
15 km/s. Then follows a rapid-acceleration phase during which velocities increaseto a range of 100 km/s up to over 1000 km/s. The rapid-acceleration phase finally transitions intoa phase with a nearly constant velocity or even a deceleration into the heliosphere.The height evolution immediately following the onset of the rapid acceleration phase is of-ten approximated by either an exponential curve (e.g., Gallagher et al. , 2003; Goff et al. , 2005;Williams et al. , 2005 – who also show systematic deviations from that fit up to 2 σ in position –) or by a constant-acceleration curve (e.g., Kundu et al. , 2004; and Gilbert et al. , 2000 – whoshow one case in which a third-order curve improves the fit to the earliest evolution, and leaveothers for future analysis); Kahler et al. (1988) fit curves for the acceleration a = ct b to the first10 −
50 Mm for four erupting filaments, but do not list the best-fit values. Alexander et al. (2002)find a best fit for the height of the early phase of a CME observed in X-rays by YOHKOH’sSXT of the form h + v t + ct . ± . . For 184 prominence events observed by the Nobeyama Ra-dioheliograph, Gopalswamy et al. (2003) show that higher in the corona velocity profiles include 3 –decelerating, constant velocity, and accelerating ones for heights from ∼
50 Mm to 700 Mm abovethe solar surface.In many cases, the detailed study of the evolution of the early phase is hampered by insufficienttemporal coverage or by gaps between the fields of view of two complementing instruments thatcan be as large as a few hundred Mm. This results in substantial uncertainties in the heightevolution. Vrˇsnak (2001), for example, concludes that “[t]he main acceleration phase . . . is mostoften characterized by an exponential-like increase of the velocity”, but notes that polynomial orpower-law functions fit at comparable confidence levels.In this study, we examine two events displaying the early destabilization and acceleration ofring filaments leading to coronal mass ejections. The high cadence down to 20 s, and the highspatial resolution of 1 arcsec, for the early evolution result in relatively small uncertainties in theheight profiles. This enables a sensitive test of the height evolution against exponential, parabolic,and power-law fits. We find that a power-law with exponent near 3, or slightly higher, is statis-tically preferred in both cases. As no published model matches that profile, we experiment witha numerical model for the torus instability, and find that this model can indeed approximate theobservations provided that a sufficiently large initial velocity perturbation is applied (without whichan exponential-like profile would be found). This finding reminds us of the sensitivity of developinginstabilities to both initial and boundary conditions, and shows that the models, particularly theirparametric dependencies, need to be worked out in greater detail in order to use observations ofthe height-time observations to differentiate successfully between competing models.
2. Observations
Primary data for this study were collected by the Transition Region and Coronal Explorer(TRACE; see Handy et al. , 1999), and ancillary data by the Mauna Loa Solar Observatory MarkIV K-Coronameter (MLSO MK4) and the SOHO/LASCO C2 and C3 instruments (Brueckner et al. (1995)). The events we studied occurred on 16 June 2005 19:10 UT to 20:24 UT (emanating fromNOAA Active Region 10775), and on 27 July 2005 from 03:00 UT to 06:20 UT (from AR 10792).
This eruption in AR 10775 was associated with an M4.0 X-ray flare. TRACE data are examinedfrom 19:10:42 through 20:08:37 UT; MLSO MK4 data were available from 20:06:59 to 20:23:25 UTto characterize the later positions of the filament. SOHO’s LASCO did not observe at this time.A characteristic TRACE image is shown in Fig. 1, with a sampling of outlines for filament ridge,loops, and position tracks.Initial data were taken at 19:10:42, followed by a few frames beginning at 19:25:32 UT. There 4 –is a gap in the TRACE data from 19:29:34 to 19:47:35 UT as the spacecraft traversed a zone ofenhanced radiation in its orbit. Starting at 19:47:35 each available image was used for tracking,with a characteristic cadence of approximately 40 s, changing with exposure time and dependingon data gaps associated with orbital zones of enhanced background radiation.As no distinct features could be tracked in the filaments or in the overlying loop structures,we use outlines of the top segments of the filament and of some outstanding overlying loops asindicated in Fig. 1. We assign confidence intervals to these positions by estimating the range ofpixels that provides a reasonable approximation of a feature.The rising filament loses a traceable form mid-way through the acceleration. Once this occurs,short bright ’streaks’ of plasma parcels show up that are blurred by their motion during the expo-sures. The positions of the midpoints of these streaks were used to extend the position data for thefilament rise. The length of the objects was estimated by correcting for motion blur estimated fromtheir displacement from one exposure to the next, and then their average positions were obtained,complemented by an uncertainty estimate.The MLSO MK4 data do not provide the same clarity of features to track as do the TRACEdata, and their observations are at a lower spatial and temporal resolution. Thus, only an estimateof the filament position was tracked, and was chosen as the point furthest from the limb on theinnermost feature on each of the images.The displacement of the approximate outlines was tracked by fitting parabolas to sets of threeadjacent points on each outline. For each exposure, a vector was computed normal to the approx-imating parabola from the central point at time t i to where it intersects a subsequent parabolicfit for time t i +1 . That intersection point is then used as the central position for the next step inthe tracking algorithm, thus moving from beginning to end in the image sequence. The track ofthe filament ridge and of two overlying loops thus measured are identified in Fig. 1. The streaksobserved in the later phases were tracked as described above; their positions are also shown inFig. 1.The filament evolves through three stages (Fig. 2): 1) an initial slow rise phase at a near-constant velocity, followed by 2) a rapid-acceleration phase, and finally 3) a constant-velocity phasehigh in the corona beginning at about 1 R ⊙ above the surface.TRACE data for phase 1 up to 19:54:58 UT show the features to exhibit an approximatelyconstant velocity relative to the solar EUV limb. During this phase, the filament moves 11,500 kmat an average of 4.4 km/sec. We note that the contribution of the solar rotation to this is negligible:for a filament at geometric height H ( t ) above the photosphere, the apparent velocity ˙ h relative to thesolar limb induced by the perspective change as the Sun rotates is approximated by ˙ h ≈ ˙ H + R ⊙ α ˙ α, for a small angle α between limb direction and current longitude. For α ∼ ◦ , the apparent motiondue to rotation only would be no more than 0 . χ values for the fits.These three methods agreed in each case to within a tens of seconds. The position data were fitwith three different functional dependences of time: a parabolic fit a + bt + ct , a power law allowingfor an initial rise velocity a + bt + ct m , and an exponential a + c exp( dt ).The rapid acceleration phase begins at 19:54:58, at which time we note the initial appearanceof a brightening feature across the lower end of the central barb of the filament. This time is atthe beginning of a data gap from 19:54:58 to 19:57:36. The rapid acceleration phase continues atleast until the remnants of the filament leave the TRACE field of view at 20:09:15.We find that the rise of the left-hand segment of the filament is best fit by a power law. Thepower-law fit is superior to the exponential fit in the range 2 . ≤ m ≤ .
9. Fits with χ ν ≤ . . . m . .
6, with a best-fit value of m = 3 . m = 3, we obtain a = 21 . ± . b = 1 . ± . − , and c = 0 . ± .
01 m s − , with χ ν = 1 .
05; if b ≡ m ≡
3, then χ ν = 1 .
07, only marginally worse than the best fit. The bestfit yields a constant jerk of 6 c = 1 . ± .
03 m s − . At the edge of the MLSO field of view, thevelocity approaches a terminal value of ∼
750 km s − .The above near-cubic fit characterizes the data better than the quadratic or exponential fits( χ ν of 4.7 and 2.2, respectively), and agrees better with the MLSO data for position and velocityneeded farther from the limb.The initial phase of the destabilization behaves as if the loops and filament are parts of arapidly-expanding volume with no discernible delays between the motions: the separations betweenthe filament ridge and two loops traced above it (lower dashed and upper solid curves in Fig. 1)appear to be essentially constant until the field is disrupted in the mass ejection (see Fig. 3):filament and high loops destabilize and begin moving at the same time, and the distance betweenthem stays close to constant. For both the higher and slightly lower loops discernible in the upperfield, their distance from the filament is almost unchanged until 20:01:41 for the outer loop and19:59:39 for the inner, lower loop. At this time, the aggregate distance increases, as the loops beginto move laterally to the primary motion of the expanding filament quickly. This indicates overallthat the high field is not evolving substantially to allow the filament through, as might be expectedin, e.g., the breakout process. TRACE data for the eruption associated with the 2005/07/27 M3.7 flare (Fig. 4) were analyzedfor 03:00:18 to 04:43:38 UT. LASCO C2/C3 data of the leading edge of the associated CME wereavailable from 04:56:37 to 06:18:05 UT to characterize the later phase. This eruption also exhibitsthree stages (Figs. 5 and 6): an initial constant velocity stage, a second rapid acceleration phase,and a final coasting phase at near-constant velocity. 6 –The initial slow rise lasts until 04:30:13 UT. This rise is already underway when TRACE datastart at 03:00:18 UT. The early data establish that the filament and the high field form one slowlyexpanding system. The early rise velocity is calculated to be 13.4 km/sec – considerably faster thanfor the event of 16 June – with the filament moving by 17,500 km prior to 04:30:13 when the rapidacceleration phase begins.The rapid acceleration phase lasts from 04:30:13 to at least 04:43:38 UT when the filamentleaves the TRACE field of view. The position data shown in Fig. 5 through 04:38:53 is for thefilament’s top ridge, and from 04:39:20 to 04:43:21 is for bright streaks similar to those seen in the16 June 2005 event.The data show this filament is also accelerating with a nearly constant jerk. The function h = a + bt + ct m fits the data very well for 2 . . m . .
7, with χ ν = 0 .
63 for m ≡ , b ≡
0. With m = 3, the fit yields a = 45 . ± . b = 4 . ± . − , and c = 0 . ± .
01 m s − , whichcorresponds to a constant jerk of 1 . ± .
06 m s − . The essentially cubic fit is also the only one ofour fits that reaches the appropriate height and velocity to follow the leading edge of the ejectionas observed with LASCO C2/C3. The exponential fit does not fit the acceleration phase as well( χ ν = 2 .
2) and makes for a much poorer transition to the LASCO data past 4:55 UT. The velocityfor the quadratic fit provides an even poorer fit to the acceleration phase observed by TRACE( χ ν = 9 . ∼ ,
250 km/sec.Although the leading edge always propagates much faster than the filament in a CME core, thedifference to the velocity at the last of the
TRACE data is substantial, and implies that the accel-eration continues at least part of the way out to the first C2 data at 1.41 R ⊙ . 7 –
3. Comparison with models
The two filament eruptions analyzed here are best fit by a power-law height evolution witha power-law index m near 3 or perhaps slightly higher ( χ ν values reach unity for values of m of 3.3 and 3.6, respectively; note that these values match the value of 3 . ± . − persists for about 10 −
15 min in both events. These phases were shown to be statisticallyinconsistent with either a constant acceleration or an exponential growth.The jerk values, d h/dt , of 1.4 and 1.9 m/s for the two filament eruptions studied here arevery similar. Estimated values using 6∆ h/ (∆ t ) based on erupting filaments up to ∼
200 Mm in thestudies referenced in § ∼ . (for an M6.5 event described by Hori et al. ,2005, and a C4 event observed by Mariˇci´c et al., 2004 and modeled by T¨or¨ok and Kliem, 2005) upto ∼
50 m/s (in an X2.5 event described by Williams et al. , 2005). There is no clear correlationbetween flare magnitude and jerk value for the small sample of events, other than that the largestoutlying flare shows the largest outlying value of jerk (we note that there is also no clear dependenceof eventual CME speed and flare magnitude – see Zhang and Golub (2003) – although the class offast CMEs has a 3 times higher maximum X-ray brightness than the class of slow CMEs). It thusremains unknown what determines the value of d h/dt , but the similarity of the values for the twocases studied here may be fortuitous.Our observations of two erupting filaments do not match the results of catastrophe, MHDinstability, or breakout models published thus far. The catastrophe model comes closest witha power-law rise with an index of 2.5, which is near, but significantly below, the range readilyallowed by the observations. The simplifying assumptions of a two-dimensional slender flux ropewith unrestricted reconnection below it may, of course, have modified the height evolution for themodel. Here we explore another effect, namely that of different initial conditions, specifically for thetorus instability (TI). The TI results if the outward pointing hoop force of a current ring decreasesmore slowly with increasing ring radius than the opposing Lorentz force due to an external magneticfield (Bateman, 1978): we investigate whether the instability can describe the rapid-accelerationphase of the two events and its transition to a nearly constant terminal velocity.The geometry of the two events appears compatible with a torus instability: the eruption on16 June 2005 exhibits an expanding main loop that approaches a toroidal shape within the rangeobserved by TRACE , and the eruption on 27 July 2005 is consistent with such a shape seen side-on.Neither shows indications of helical kinking. The sinh( t ) profile obtained analytically for the TI byKliem and T¨or¨ok (2006) relied on the simplifying assumption that the external poloidal field varieswith the major torus radius R as B ex ∝ R − n with a constant decay index n , and it is exact onlyas long as the displacement from the equilibrium position remains (infinitesimally) small.Allowing for a height dependence of the decay index n likely will cause the height evolutionin the model to differ even more from the observations: because n ( h ) is in reality an increasingfunction on the Sun (see, e.g., Fig. 2 in van Tend and Kuperus 1978), the acceleration profile would 8 –likely increase more steeply than the initially nearly exponential sinh( t ) function.We have performed numerical MHD simulations of the TI to study the evolution for finitedisplacements. For some parameter settings, the exponential expansion was found to hold up toseveral initial radii of the current ring, while for others a power-law-like expansion with exponentsscattering around m ∼ ≈ . π is chosen,which requires the flux rope to be relatively thick (the minor radius is 0.6 times the initial apexheight, yielding an aspect ratio of only 1.83). The approximation of a slender flux tube used inTD99 becomes relatively inaccurate for these settings, so that the simulations start with a shortphase of relaxation toward a numerical equilibrium, lasting about a dozen Alfv´en times ( τ A ).The TI is triggered by the motions set up in the relaxation phase, which may reach one tenthof the Alfv´en speed ( V A , measured at the flux rope apex in the initial configuration), depending onparameters. In a first set of simulations, we set the decay index of the external poloidal field at theinitial apex height to be n = 1 .
20, close to its critical value analytically derived to be 1 .
23 for theparameters given (see Eq. (5) in Kliem and T¨or¨ok 2006). The TI then develops very gradually, ina period of ∼ τ A , while the perturbations caused by the initial relaxation decay in ≈ τ A .This simulation yields a clearly exponential rise profile (solid lines in Fig. 7).In four subsequent runs in this set, an upward, linearly rising perturbation velocity is imposedat the flux rope apex of the same initial configuration at the start of the runs with an increasingduration (from 6 τ A up to 10 τ A ). Figure 7 shows the resulting transition from exponential topower-law-like rise profiles for these TI simulations.The fourth run (dashed lines) approaches a constant-jerk rise profile best. This best-fit runhas an initial velocity of 0 . V A at the onset of the TI-driven rise of the acceleration at t ≈ τ A and h = 1 .
74. It approximates constant jerk up to t ∼ τ A (i.e., nearly until the peak accelerationis reached) and h ∼ . n = 2 .
85, close to the asymptotic value for a dipolefield ( n →
3) in the TD99 equilibrium (Figure 9 shows a rendering of this simulation). On theother hand, with the depth of the torus center chosen to be 3/8 of the initial apex height, the linetying has a stronger stabilizing effect. Except for a somewhat larger aspect ratio of 2.3, the other 9 –parameters are identical to those of the runs shown in Fig. 7. The initial velocity at the onset ofthe TI-driven rise of the acceleration is, again, approximately 0 . V A . This velocity results fromthe initial, more vigorous relaxation towards a numerical equilibrium and from the early onset ofmagnetic reconnection in the vertical current sheet, which is formed below the flux rope similarto the simulation shown in Kliem et al. (2004). By the end of the relaxation ( t ≈ τ A ), bothupward and downward reconnection outflow jets from the current sheet are formed and the upwardjet reaches 0 . V A , reducing the marked decrease of the upward perturbation velocities observedin the first run in Fig. 7. During the whole phase of nearly constant-jerk rise, the TI-driven rise ofthe flux rope apex and the upward reconnection outflow jet grow synchronously, reaching similarvelocities.Such close coupling between the ideal instability and reconnection can obviously support apower-law rise of the unstable flux rope, but it is neither a necessary nor a sufficient conditionfor its occurrence, as the comparison with Run 4 in Fig. 7 and with the CME simulation inT¨or¨ok and Kliem (2005) shows. Run 4 exhibits a nearly power-law rise, but reconnection outflowjets from the vertical current sheet develop here only after the acceleration of the flux rope haspassed its peak ( t > τ A ). The CME simulation in T¨or¨ok and Kliem (2005) showed a similarcoupling between the ideal MHD instability (the helical kink in this case) and reconnection as therun shown in Fig. 8, but with an initial velocity of ≈ . V A the rise was clearly exponential.While all data in Fig. 7 and the solid line in Fig. 8 monitor the apex of the magnetic axis ofthe flux rope, the dashed lines in Fig. 8 show the rise of a fluid element near the bottom of the fluxrope, which is a likely location for the formation of filaments. Lying initially at 0 . h , it belongsto an outer flux surface of the rope. Although the flux rope in the simulation expands during therise, both the axis and the bottom part show an approximately constant jerk, and no significanttiming differences between the acceleration profiles. Figure 8 presents a scaling of the simulation data to the rise profile of the 2005/06/16 eruption,determined in three steps. First, the time of the velocity minimum near 10 τ A in the simulationis associated with the onset time, t , of the rapid-acceleration phase, 19:54:58 UT, as obtained inSect. 2. Second, the time t of maximum simulated velocity is associated with the time halfwaybetween the final MLSO data points, which yields a substantially better match between the acceler-ation profiles than assuming that the acceleration ceased at or after the final MLSO data. These twochoices yield τ A = 32 . t , resultingin a length unit for the simulation of h = 44 . V A = h /τ A = 1370 km/s,and a normalization value for the acceleration of a = V A /τ A . Figure 8 shows the observed heightson a linear scale, with derived velocity and acceleration data (based on central differences, with a7-point boxcar averaging to smooth the heights and velocities, and a 5-point boxcar averaging forthe accelerations). 10 –Both the rise of the magnetic axis of the flux rope (solid line in Figure 8) and the rise of afluid element originally below the magnetic axis (dashed line) are scaled to the data. The lowerfluid element yields the best match, and is shown in Fig. 8.We note that a correction of the observed heights for perspective foreshortening may improvethe fit of model to observations. The TRACE images suggest that the direction of ascent may havebeen inclined from the vertical direction by ∼ ◦ at the onset of the accelerated rise, and it isplausible to assume that it had become vertical by the time of the final MLSO data point. Sucha correction brings all height data points even closer to the dashed line in Fig. 8. However, sincesuch a correction introduces a degree of uncertainty while the effects are relatively minor, we donot attempt to apply such a correction.Not only is the overall match between the observations and the scaled simulation quite satis-factory, the scaling also yields plausible values for the Alfv´en velocity and the footpoint spacing ofthe model flux rope, D foot = 98 Mm. The latter agrees well with the observed value, which Fig. 1suggests to be ≈
94 Mm (from x ≈
230 to x ≈ ∼ . V A required in the simulations of Figs. 7 and 8 forthe transition from an exponential to a nearly cubic height-time profile. The observed velocities at t even exceeded the initial velocity of 0 . V A of the run shown dotted in Fig. 7, which developedan intermediate rise profile quite close to the observed profile. We infer from this that the initialvelocity is a parameter which helps control the detailed properties of the rise profile.The observations of the eruption on 2005/07/27 do not constrain the scaling of the simulationas well as the 2005/06/16 data. The LASCO data at large distances refer to the leading edge ofthe CME, i.e., to a different part of the ejection than the TRACE data, and the two sets do notjoin to form as nearly a continuous h ( t ) profile as the 2005/06/16 data. Only the TRACE datacan be used for the scaling, leaving more ambiguity in the scaling for this event. The best matchbetween the simulation and the data is obtained when the final
TRACE height measurement isassumed to lie slightly past the time of peak acceleration, by 2–5 τ A . Equating the simulated andobserved heights at this time gives a match of comparable quality to the one in Fig. 8 for both themagnetic axis and the lower fluid element. We present the former in Fig. 10, which yields the scaledparameters τ A = 26 sec, V A = 940 km/s, and D foot = 55 Mm. Scaling the rise of the lower fluidelement to the observations yields τ A = 29 sec, V A = 1500 km/s, and D foot = 99 Mm instead. Aswith the 2005/06/16 data, the observed velocity closely approaches the scaled simulation velocityshortly after the estimated onset time of the fast rise (within ∼ τ A ).The scalings support the hypothesis that the torus instability of a flux rope has been a possibledriver of both eruptive filaments in their rapid-acceleration phase. We note that the only parameter 11 –that was adjusted particularly to fit the observations is the decay index for the overlying field( n = 2 . n & Figure 9 shows that field lines that initially pass over the legs of the flux rope, lean stronglysideways during the rope’s rapid acceleration phase, similar to the motion of the observed overlyingloops. Their lateral motions in Figs. 3 and 6 commence with little or no delay to the beginning rapidacceleration of the filament (except for a much weaker lateral motion of the left overlying loop inthe slow rise phase of the 2005/07/27 event), and they combine with the vertical motions such thatthe total distance between loop apex and filament apex varies only little in the first ≈ ∼ τ A ), but increases rapidly thereafter.We emphasize that the observations of the two events do not permit us to determine thedelay between the start of the displacement of the overlying loops relative to the filament’s rapidacceleration to better than an Alfv´en travel time: the Alfv´en velocities of order 1,000 km/s and theinstrument cadence mean that signals can propagate between the overlying loops and the filamentwithin 1 to 2 imaging intervals. Consequently, we can only conclude that the data are compatiblewith a delay of at most one Alfv´en travel time.Figure 11 plots the distances for a set of loops in a format similar to Figs. 3 and 6. These loopswere selected such that their apex points have equally-spaced initial distances on a straight linefrom the origin, inclined by 25 ◦ from the vertical. The second lowest of these loops is marked by anasterisk in Fig. 9. We find that the model’s horizontal and vertical distances combine to a slowlyvarying total distance for about 10 τ A after TI onset (at t ≈ τ A ), followed by a rapid increase ofthe total distance, as in the observations. This behavior occurs in an angular range between thevertical and the initial origin-apex line of, roughly, 20–35 ◦ . For larger inclinations of the overlyingloop the initial ratio of vertical and horizontal distance is smaller than observed, and for smallerinclinations the horizontal motion commences too late.Figure 11 also reveals two types of perturbations in this simulation. The first is an initial phaseof relaxation from the analytical TD99 field to a nearby, numerically nearly potential-field state,which occurs in the whole surrounding field of the flux rope and is of nearly uniform duration of2–3 τ A . The second is a wave-like perturbation, launched by the (more vigorous) initial relaxationof the current-carrying flux rope, of duration ∼ τ A , and propagating outward trough the wholebox at about the Alfv´en speed. The motion of the overlying loops is seen to commence with thepassage of the second perturbation, i.e., with a delay of only one Alfv´en travel time, and to continuesmoothly after its passage (similar to the behavior of the flux rope, whose instability develops outof the initial relaxation). A delay this short is consistent with the observations.The feature of an initially only slowly varying total distance occurs in a substantial height 12 –range, so that one cannot conclude that the observed overlying loops give a good indication of theedge of the flux rope in the two events considered. However, with increasing initial height of theloops, the phase of rapid increase of the distance to the rope occurs progressively delayed. Thescalings place the observed transition between the two phases at t ∼ τ A , in agreement with thelowest two or three loops included in Fig. 11, indicating that the overlying loops were located inthe range between the surface of the flux rope and about three minor radii from its axis.
4. Conclusions
We study two well-observed filament eruptions, and find that their rapid acceleration phases arewell fit by a cubic height-time curve that implies a nearly constant jerk for 10 −
15 minutes, followedby a transition to a terminal velocity of ∼
750 km/s and ∼ ∼ ,
000 km/s, thepropagation of a perturbation over the separation of ∼ ,
000 km would require only ∼ . m
13 –slightly exceeding 3), our TI model requires an initial perturbation velocity that is in agreementwith the observed rise velocity at the onset of the rapid-acceleration phase. If a nearly exact cubicrise were to be matched, however, initial velocities moderately exceeding the observed ones, by afactor ≈ .
5, were required. In any case, our modeling is consistent with the observed velocitiesafter the first few minutes of the eruption.Having established that the model for the TI instability is very sensitive to the initial condi-tions, we should of course also acknowledge that it depends sensitively on the model details itself.These include the details of the external field and of the rates and locations of the reconnectionthat occurs behind the erupting filament. That such reconnection occurs in reality is suggestedfor both events by the occurrence of brightenings mainly at the bottom side of the filaments atthe onset of the rapid-acceleration phase. These brightenings develop later into the streaks usedfor position determination in Sect. 2. The onset of reconnection even before the rapid-accelerationphase of the filament eruption on 27 July 2005 is strongly suggested by precursor soft and hardX-ray emission during about 04:00–04:30 UT, whose analysis revealed heating to 15 MK and theacceleration of non-thermal electrons to energies >
10 keV (Chifor et al., 2006).The observed rise velocity early in the filament eruption may be an underestimate of the trueexpansion velocity of the hoop formed by the flux rope: the filament channel in the pre-eruptionphase of AR 10775 is strongly curved, and one of the two possible channels in AR 10792 is too(ambiguity exists here because the eruptions occurred very near the limb, so that the configurationsof the filament channels can only be observed some days before and after the events, respectively).If the initial expansion of the flux rope would have a strong component in the general direction ofthe inclined plane of the curved filament channel rather than be purely normal to the solar surface,projection effects could cause us to underestimate the expansion velocity in particular early in theevolution. In addition to that, we must realize that the TI model assumes a flux rope that standsnormal to the solar surface and that erupts radially. Future more detailed modeling will have toshow how deviations from that affect the evolution of the eruption.The fact that the torus-instability model yields qualitatively different rise profiles (exponentialvs. power law) in different parts of parameter space, cautions against expectations that precisemeasurements of the rise profile of filament eruptions by themselves permit a determination of thedriving process: the non-linearities in the eruption models clearly require high-fidelity modelingif such observations are to be used to differentiate successfully between competing models. Ourinitial modeling discussed here suggests that the torus instability is a viable candidate mechanismfor at least some filament eruptions in coronal mass ejections. Given the dependence of nonlinearmodels on the details of boundary and initial conditions, it will be necessary to investigate how othermodels for erupting filaments compare to the data, as well as how the fidelity of our modeling of thetorus instability can be improved before we can reach definitive conclusions about the mechanism(s)responsible for filament eruptions in general.We thank Joan Burkepile for providing us with MLSO MK4 observations, and Terry Forbes for 14 –helpful discussions. We are grateful to the referee for constructive and helpful comments that led usto pursue the model-observation parallels in this study in more detail. This work was supported byNASA under the TRACE contract NAS5-38099 with NASA Goddard Space Flight Center, by NSFgrant ATM 0518218 to the University of New Hampshire, by the European Commission throughthe SOLAIRE Network (MTRN-CT-2006-035484), and by the Deutsche Forschungsgemeinschaft.
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This preprint was prepared with the AAS L A TEX macros v5.2.
16 –
Figure captions:
Figure 1: TRACE 171 ˚A image taken at 2005/06/16 19:25:32 UT. Sample outlines of the topedge of the rising filament over time and of two overlying loop structures are shown for the timeinterval from 19:25 UT to 20:04 UT. The positions for which the heights are shown in Fig. 2 aremarked.Figure 2: Distances from the solar EUV limb for the rising and erupting filament on 2005/06/16(see Fig. 1 for the tracked positions). The bottom panel shows the central phase with rapid filamentacceleration in detail. Both panels show several fits to the data (see legend). The top panel alsoshows positions derived from the MLSO coronagraphic data for the later phase when the eruptionturns into a proper mass ejection.Figure 3: Distances between the tracked ridge of the filament shown in Fig. 1 and the lower(top panel) and upper (bottom panel) overlying loops. The total distance is shown by the solidline; distances in the figure’s x and y directions are shown separately by dashed and dashed-dottedlines, respectively.Figure 4: TRACE 171 ˚A image taken at 2005/07/27 03:00:08 UT. This figure, similar to Fig. 1,identifies segments of overlying loops on the left and right side of the rising filament.Figure 5: As Fig. 2 for the event observed on 2005/07/27. Note that the exponential andquadratic fits are shown offset by +2 min. in the top panel to reduce overlap, but shown properlyplaced in time in the lower panel.Figure 6: Distances between the tracked ridge of the filament shown in Fig. 4 and the left (toppanel) and right (bottom panel) overlying loops.Figure 7: Transition from exponential to approximately power-law rise profile with increasinginitial velocity for a torus-unstable flux rope equilibrium with an external field decay index of n ≥ . h ( t ), velocity u ( t ), acceleration a ( t ),and jerk j ( t ) = da/dt are normalized using the initial apex height h , the Alfv´en speed V A , andthe corresponding derived quantities. Time is normalized by τ A = h /V A . Solid lines show theunperturbed run, i.e., the development of the instability from rest. For the further runs of theseries a velocity perturbation at the apex is linearly ramped up until 6, 8, 9.25, and 10 τ A (dashed-dotted, dotted, dashed, dashed-triple dotted, respectively).Figure 8: Nearly constant-jerk rise profile for an unperturbed torus-unstable flux rope equilib-rium with steeper field decrease above the flux rope than in Fig. 7; the field decay index in this caseis n ≥ .
85, i.e., near the value for the far field in the dipolar case (see text for other parameterdifferences for aspect ratio and initial torus depth). Solid lines show the rise profile of the apexpoint of the magnetic axis as in Fig. 7, dashed lines show the rise profile of a fluid element belowthe apex, initially at h = 0 . h . The simulation data for this lower fluid element are scaled tothe rise profile of the 2005/06/16 filament eruption, and the resulting Alfv´en time, Alfv´en speed, 17 –and footpoint distance are given.Figure 9: Side view of a torus instability simulation (see Fig. 8). The field lines of the torusare shown lying in a flux surface at half the minor torus radius. Sample field lines for the overlyingfield are also shown. The starting points in the bottom plane for the traced field lines are the samefor all panels. The times (expressed in Alfv´en crossing times, as in Figs. 7–11) are = 0 , ,
30, and40, respectively. The motion of the loop apex marked by an asterisk is shown in Fig. 11.Figure 10: Scaling of the simulation data from Fig. 8 to the rise profile of the 2005/07/27filament eruption; here the rise of the magnetic axis’ apex point (solid line) is scaled.Figure 11: Distances of the apex point of representative loops, initially overlying the flux ropeat an angle of 25 ◦ from the vertical, to the lower fluid element of the simulation shown in Figs. 8and 10 (dashed line in these figures). The format is similar to Figs. 3 and 6. For clarity, horizontaland vertical distances are included only for the lowest and highest of the selected loops. The secondlowest of these loops is marked by an asterisk in Fig. 9. 18 –Fig. 1.— 19 – D i s t an c e F r o m L i m b ( l og ( k m )) Time (HH:mm) 20:2020:1020:0019:5019:4019:3019:20 19:54:58 Beginning of Acceleration phase
Distance From Limb For Left Filament and Streaks For 16 June 2005 Event a + bt Fit of MLSO dataa + bt + ct^2 Fit (Reduced Chi-Square = 4.74)a(e^bt) + c Fit (Reduced Chi-Square = 2.21)a + bt^3 (Reduced Chi-Square = 1.07)a + bt Fit of Early PhaseObserved Position (MLSO MK4)Observed Position (TRACE) D i s t an c e F r o m L i m b ( k m ) Time (HH:mm) 20:07:3020:05:0020:02:3020:00:0019:57:3019:55:00
Distance From Limb For Left Filament During Acceleration Phase For 16 June 2005 Event a + bt + ct^2 Fit (Reduced Chi-Square = 4.74)a(e^bt) + c Fit (Reduced Chi-Square = 2.21)a + bt^3 Fit (Reduced Chi-Square = 1.07)Observed Position (TRACE)
Fig. 2.— 20 – D i s t an c e ( p i x ) Distance Between Filament and Upper Loop of High Field For 16 June 2005 Event
Y DistanceX DistanceAggregate Distance D i s t an c e ( p i x ) Distance Between Filament and Inner Loop of High Field For 16 June 2005 Event
Y DistanceX DistanceAggregate Distance
Fig. 3.— 21 –Fig. 4.— 22 – D i s t an c e F r o m L i m b ( l og ( k m )) Time (H:mm) 6:005:305:004:304:003:3004:30:13 Beginning of Acceleration Phase
Distance From Limb For Filament During Acceleration Phase For 27 July 2005 Event a + bt Fit of Late Phasea + bt + ct^2 Fit (Reduced Chi-Squared = 9.07)a(e^bt) + c Fit (Reduced Chi-Squared = 2.19)a + b(t-t0)^3 Fit (Reduced Chi-Squared = 0.629)a + bt Fit of Early PhaseObserved Position (LASCO)Observed Position (TRACE) D i s t an c e F r o m L i m b ( k m ) Time (H:mm) 4:424:404:384:364:344:324:30
Distance From Limb For Filament During Acceleration Phase For 27 July 2005 Event a + bt + ct^2 Fit (Reduced Chi-Squared = 9.07)a(e^bt) + c Fit (Reduced Chi-Squared = 2.20)a + b(t - t0)^3 Fit (Reduced Chi-Squared = 0.629)Observed Position (TRACE)
Fig. 5.— 23 – D i s t an c e ( p i x ) Distance Between Filament and Left Loop of High Field For 27 July 2005 Event
Y DistanceX DistanceAggregate Distance D i s t an c e ( p i x ) Distance Between Filament and Right Loop of High Field For 27 July 2005 Event
Y DistanceX DistanceAggregate Distance
Fig. 6.— 24 – h ( t ) u ( t ) a ( t ) t j ( t ) Fig. 7.— 25 – h ( t ) h ( t ) h [ Mm ] Simul. rope magn. axisSimul. lower flux surfaceTRACE data 16-06-2005MLSO MK4 data u ( t ) u [ k m s - ] τ A = 32.5 sV A = 1366 km/sD foot = 98 Mm a ( t ) t [ τ A ] 00.010.02 a ( t ) a [ k m s - ] Fig. 8.— 26 –Fig. 9.— 27 – h ( t ) h ( t ) h [ Mm ] Simul. rope magn. axisSimul. lower flux surfaceTRACE data 27-07-2005 u ( t ) u [ k m s - ] τ A = 26.3 sV A = 939 km/sD foot = 55 Mm a ( t ) t [ τ A ] 00.010.02 a ( t ) a [ k m s - ] Fig. 10.— 28 – t [ τ A ]024681012 aggregate distancevertical distancehorizontal distanceaggregate distancevertical distancehorizontal distance