TEMPORAL AND PHYSICAL CONNECTION BETWEEN CORONAL MASS EJECTIONS AND FLARES

James Chen; Valbona Kunkel

doi:10.1088/0004-637X/717/2/1105

1. INTRODUCTION

Coronal mass ejections (CMEs) and solar flares are two of the most energetic eruptive phenomena observed on the Sun. The energy released in a single eruption is of the order of 10³¹–10³³ erg. CMEs appear as large-scale coherent density structures traveling away from the Sun, carrying upward of several times 10¹⁶ g in mass. They are presumed to be organized by magnetic field and are now understood to be the primary solar drivers of major geomagnetic storms. Flares are observed as localized brightening in the lower solar atmosphere in X-rays, Hα, ultraviolet (UV), and sometimes in white light, lasting tens of minutes to several hours.

Observationally, CMEs are known to be associated with flares, and the nature of the connection between flares and CMEs (and closely associated eruptive prominences—EPs) has been a long-standing question (see, for example, Hundhausen 1999 for a review). In early research, CMEs were conceptualized as the coronal response to the thermal energy released in flares (see, for example, reviews by Wu 1982; Dryer 1982). Another concept is that an initial arcade rises due to photospheric footpoint motion, leading to a CME and flare via magnetic reconnection (e.g., Hirayama 1974; Kopp & Pneuman 1976).

The precise connection between CMEs and flares is difficult to establish because coronagraphs occult the solar disk and innermost corona so that the initial dynamics of CMEs are typically not observed. Early attempts to establish causality between CMEs and flares focused on timing correlations. The CME initiation time was estimated by extrapolating the observed CME trajectories down to the solar surface assuming constant speeds. Extrapolating downward to the lower heights where complicated dynamics prevail is problematic, and comparisons of estimated CME launch times and the onset times of the associated flares yielded inconsistent correlations (e.g., Munro et al. 1979; Maxwell et al. 1985; Webb & Hundhausen 1987; Harrison 1995). Recently, based on a number of CMEs observed by the C1, C2, and C3 telescopes of Large Angle and Spectrometric Coronagraph (LASCO) for which the initial acceleration was adequately resolved, the "acceleration phase" of these CME events was empirically found to coincide with the rise phase of the associated flare X-ray emission (Zhang et al. 2001). This finding has been supported by recent work of Maričič et al. (2007) and Temmer et al. (2008). They, however, also discovered exceptions and considerable scatter in the timing correlation.

Theoretically, CMEs/EPs have been hypothesized as the eruption of toroidal magnetic flux ropes: some start with initial equilibrium flux ropes (Chen 1989; Vršnak 1990; Chen & Garren 1993; Gibson & Low 1998; Roussev et al. 2003) while others use flux ropes emerging from below the photosphere (Wu et al. 1997; Fan & Gibson 2003; Manchester et al. 2004). By a toroidal flux rope, we refer to a current-carrying structure that can be approximated as a partial torus with its footpoints anchored in the Sun (Figure 1). The first evidence of flux-rope CMEs with their legs connected to the Sun was found in LASCO data (Chen et al. 1997), supported by subsequent studies of CMEs (Wood et al. 1999; Dere et al. 1999; Chen et al. 2000; Plunkett et al. 2000; Krall et al. 2001; Thernisien et al. 2006). This flux-rope interpretation of CMEs is now standard (e.g., Krall & St. Cyr 2006; Thernisien et al. 2006) and is in contrast to the earlier notion that CMEs are shell- or cone-like structures with rotational symmetry (e.g., Hundhausen 1999).

**Figure 1.** Schematic of a toroidal flux rope, consisting of a current channel and its magnetic field *B_p* and *B_t*. The current channel has major radius R, minor radius a, and footpoint separation *S_f*. *B_p* is assumed to extend to r = 2a. Below the surface, the poloidal field (*B_p* *B_t*) is nonuniform and becomes increasingly incoherent as it propagates upward. Reproduced from Chen & Krall (2003).
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**Figure 1.** Schematic of a toroidal flux rope, consisting of a current channel and its magnetic field *B_p* and *B_t*. The current channel has major radius R, minor radius a, and footpoint separation *S_f*. *B_p* is assumed to extend to r = 2a. Below the surface, the poloidal field (*B_p* *B_t*) is nonuniform and becomes increasingly incoherent as it propagates upward. Reproduced from Chen & Krall (2003).
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The traditional concept is typically modeled using magnetohydrodynamic (MHD) simulations as an initial magnetic arcade that evolves via footpoint motion and coronal reconnection, resulting in a magnetic flux rope during the eruption (e.g., Antiochos et al. 1999; Chen & Shibata 2000; Amari et al. 2000; Linker et al. 2001; Cheng et al. 2003; Lynch et al. 2009). In some scenarios, reconnection occurs in conjunction with a "loss of equilibrium" (e.g., Forbes & Priest 1995). Using a resistive MHD simulation, Cheng et al. (2003) showed that a linear arcade can exhibit an acceleration–energy release relationship consistent with the CME–flare correlation discussed by Zhang et al. (2001). Reeves (2006) modeled flare X-ray light curves assuming magnetic reconnection in a linear arcade geometry. In these models, the reconnection results from numerical diffusion or assumed dissipation.

Among the existing CME models, the semianalytic erupting flux-rope (EFR) model of CMEs (Chen 1989) has been most extensively tested. It has been shown to correctly reproduce the observed dynamics from the initial acceleration to the outer edge of the LASCO field of view (FOV) of radius ~30 R_☉ (Chen et al. 1997, 2000, 2006; Wood et al. 1999; Krall et al. 2001, 2006). More recently, the model has been shown to quantitatively replicate the dynamics of a CME observed by the Sun–Earth Connection Coronal and Heliospheric Investigation (SECCHI) instruments (Howard et al. 2008) on board the Solar Terrestrial Relations Observatory (STEREO) spacecraft, with the calculated magnetic field and plasma properties in agreement with the in situ data at 1 AU observed by STEREO (Kunkel & Chen 2010). We will use this flux-rope model as the theoretical basis of the present work.

The underlying physics of the EFR model is the major radial Lorentz self-force (the Lorentz "hoop force") acting on coronal flux ropes (Chen 1989). Of the possible initiation mechanisms available to this model, we consider increasing the poloidal (locally azimuthal) magnetic flux Φ_p, a process referred to as poloidal "flux injection." For other possible mechanisms in the EFR model, see Krall et al. (2000). Mathematically, flux injection is represented by dΦ_p(t)/dt. The Lorentz hoop force was originally derived for axisymmetric tokamak equilibria (Shafranov 1966). While the application of Shafranov's work was initially regarded as a "laboratory technique" inapplicable to solar flux ropes, it has been shown to provide a demonstrably correct description of CME dynamics (see the references above), and this application of the Lorentz hoop force to CMEs has attracted renewed interest (e.g., Lin et al. 1998; Titov & Démoulin 1999; Kliem & Török 2006; Isenberg & Forbes 2007). Application of Shafranov's derivation to nonaxisymmetric solar flux ropes with stationary footpoints requires essential modifications (Chen 1989; Chen & Garren 1993).

In the present paper, the question regarding the physical connection between CMEs and associated flares is revisited from a theoretical point of view, focusing on the CME dynamics and soft X-ray (SXR; 1–8 Å) data. We discuss a new prediction of the EFR model arising from dΦ_p/dt ≠ 0 as a driver of CMEs, which produces a nonzero electromotive force (EMF), ${\cal E}(t) \equiv -(1/c)d\Phi _p(t)/dt$ . We posit that this EMF provides the electric fields to accelerate particles resulting in radiative energy release in flares. Although the mechanisms for particle acceleration and X-ray emission are not included in the theory, the temporal profile of ${\cal E}(t)$ , which is identical to that of dΦ_p(t)/dt, is a prediction of the model that can be directly compared to the observed SXR light curve. It is the purpose of the present paper to examine this prediction. We will show that the functional form of dΦ_p(t)/dt required to fit an observed CME trajectory exhibits close temporal coincidence with the observed SXR light curve of the associated flare regardless of the flare duration. Neither the equations of motion nor the CME height data contain any information on X-ray data. This temporal correlation, therefore, is nontrivial and physically significant. The results suggest a new theoretical framework of understanding the CME–flare connection: the poloidal flux injection produces an EMF to energize particles.

The organization of the paper is as follows. In Section 2, we briefly review the basic tenets of the EFR model relevant to the present paper. In particular, pertinent theoretical properties of flux-rope dynamics are summarized in Section 2.4. The main results of the paper are given in Section 3 where the application of the theoretical model to observed CME trajectories and comparison of the predicted dΦ_p(t)/dt profiles and GOES X-ray data are discussed. In Section 4, the new results are compared with those of models based on the arcade paradigm. Possible physical interpretations of the function dΦ_p(t)/dt are discussed in Section 5. Summary and conclusion are given in Section 6.

2. THE ERUPTING FLUX-ROPE (EFR) MODEL

2.1. Forces: Equations of Motion

The basic equations defining the EFR model and poloidal flux injection have been previously published (Chen 1989; Chen & Garren 1993). The current version of the model is given in Chen (1996). The model is formulated within the framework of ideal MHD and uses an integrated representation of the forces. In this section, we summarize the model equations and salient physics.

Figure 1, reproduced from Chen & Krall (2003), shows a schematic drawing of a toroidal flux rope embedded in the corona with pressure p_c and overlying magnetic field B_c. The electric current J is localized to a current channel of major radius R and minor radius a, with components J_t and J_p in the toroidal (locally axial) and poloidal (locally azimuthal) directions, respectively. The magnetic field B of the flux rope has poloidal component B_p and toroidal component B_t. The toroidal field is confined to the current channel (r ≤ a, where r is the minor radial coordinate), but the poloidal field can extend beyond r = a. Coronal magnetic field component B_c perpendicular to the flux rope and arising from currents external to it is shown. By a flux rope, we refer to the current and the magnetic field of the system, including the poloidal field B_p outside the current channel (but not B_c). Some representative field lines are illustrated. The EFR model defines the footpoints at the base of the corona and treats the dynamics of the flux rope in the corona.

In Figure 1, the centroid of the apex is at height Z measured from the photosphere, and the footpoint separation distance is S_f. Each footpoint has radius a_f. We will use a_a to denote the minor radius at the apex. The angular position θ along the flux rope is measured from a footpoint, denoted by θ = θ_f. The model assumes that a_f and S_f are constant. In relating this structure to CMEs and EPs, the outermost surface of a CME is taken to be at r = 2a where B_p becomes comparable to the overlying field B_c (Chen et al. 1997, 2000). That is, the height of the leading edge (LE) is at Z + 2a. The prominence is taken to be at the trailing edge of the current channel defined by the relatively strong magnetic field, i.e., Z − a. This identification has been found to be consistent with numerous observed CMEs and EPs (Krall et al. 2001; Chen et al. 2006). Theoretically, Krall & Chen (2005) demonstrated a physical mechanism that causes the density to peak at r sime 2a.

Given a flux rope, one can define the poloidal flux enclosed by the partial torus and the photosphere by

$\begin{equation} \Phi _p(t) \equiv cL(t) I_t(t), \end{equation} \tag{ 1 }$

where L(t) is the self-inductance of the current system and I_t is the net toroidal current. The toroidal flux is $\Phi _t = \pi a^2 \overline{B}_t$ where $\overline{B}_t$ is the toroidal field averaged over the minor radius. In ideal MHD, Φ_t= const. The poloidal magnetic energy of the flux rope above the photosphere is U_p(t) = (1/2)LI²_t = Φ²_p(t)/2c²L(t). If the poloidal flux is injected at the rate of dΦ_p/dt into a structure having Φ_p and L, the poloidal energy is injected at the rate of

$\begin{equation} {dU_p(t)\over {dt}}\Bigg |_{\rm inj} = {\Phi _p(t)\over c^2L(t)}{d\Phi _p(t)\over dt}. \end{equation} \tag{ 2 }$

Once injected, the poloidal flux is conserved while the magnetic field, the magnetic energy, and current I_t of the flux rope are not. The change in the magnetic energy, −(Φ²/2c²L²)dL/dt due to the resulting ideal MHD motion of the flux rope, does not alter the amount of the injected poloidal energy and does not enter this equation. The above equation shows that the injected poloidal magnetic energy δU_p corresponding to an amount of injected poloidal flux δΦ_p, or equivalently injected magnetic helicity δK ∝ Φ_tδΦ_p, depends on the inductance and poloidal flux of the existing structure into which the flux is injected.

The basic force in the system is described using standard ideal MHD, f = (1/c)J × B − ∇p. This force is integrated over the toroidal volume. Because of the major radial curvature, there is a net force in the major radial direction. Under the local curvature approximation (Chen 1989), the integrated force per unit length is

$\begin{eqnarray} F_R &=& {{\Phi _p^2}\over {c^4 L^2 R}} \left[\ln \left({{8 R}\over {a}}\right) + {1\over 2}\beta _p - {1\over 2}{{\overline{B}_t^2}\over {B_{\rm pa}^2}} \right.\nonumber\\ &&\left. + 2\left({R\over {a}}\right){{B_c}(Z)\over {B_{\rm pa}}} - 1 + {{\xi _i}\over {2}} \right] + F_g + F_d, \end{eqnarray} \tag{ 3 }$

where Φ_p is given by Equation (1), $\beta _p(t) \equiv 8\pi (\overline{p} - p_c)/B_{\rm pa}^2$ arises from ∇p with $\overline{p}$ the average internal pressure and p_c the ambient coronal pressure, $\overline{B}_t(t)$ is the average toroidal field inside the loop, B_pa(t) ≡ B_p(r = a) = 2I_t/ca, ξ_i ≡ 2∫rB²_p(r)dr/(a²B²_pa) is the internal inductance, and I_t is the total toroidal current I_t ≡ 2π∫J_t(r)rdr. The actual form of J_t(r) enters the analysis only through ξ_i and does not explicitly affect the other terms. Here, we have included the drag force (F_g) and gravitational force (F_g) per unit length. The sign of B_c is such that B_c < 0 yields downward (negative) Lorentz force J_tB_c, as shown in Figure 1. The local curvature approximation was rigorously verified for arbitrary nonaxisymmetric flux-rope configurations using the principle of virtual work by Garren & Chen (1994). The force equation has an error of ${\cal O}(a/R)$ (Chen 1989; Garren & Chen 1994).

The motion of the center of mass of the apex is governed by

$\begin{equation} M{{d^2Z}\over {dt^2}} = F_R = \left({{\Phi _p^2}\over {c^4 L^2 R}}\right)\,f_R + F_g + F_d, \end{equation} \tag{ 4 }$

where f_R is the quantities in the square brackets in Equation (3) and is a function of order unity near the Sun. Here, $M \equiv \pi a^2\overline{n}_T m_i$ is the mass per unit length.

The minor radius a(t) of the apex evolves according to

$\begin{equation} M{{d^2a}\over {dt^2}} = {{I_t^2}\over {c^2 a}} \left({\overline{B}_t^2 \over B_{\rm pa}^2} - 1 + \beta _p\right). \end{equation} \tag{ 5 }$

The dynamics of the minor radius per se will not be specifically discussed, but we emphasize that Equations (4) and (5) are coupled in the model. In the present paper, we use a(t) to denote the apex minor radius a_a(t) and use a_a only where it is necessary to distinguish it from a_f= const.

In this model, the major radial curvature 1/R(t) is assumed to be uniform along the flux rope. This geometrical simplification enters the equations of motion through the condition,

$\begin{equation} R(t) = {{Z^2(t) + S_f^2/4}\over 2Z(t)}. \end{equation} \tag{ 6 }$

The self-inductance L(t) depends on the flux-rope geometry and its dynamics and has been derived for a number of different profiles of a(θ), all increasing from a(θ_f) = a_f to a(θ_a) = a_a (Chen & Garren 1993; Garren & Chen 1994; Krall et al. 2001, 2006; Chen & Krall 2003). For the present paper, it is only necessary to note the scaling L(t) ∝ Rln(8R/a_f) for a flux rope with stationary footpoints. This arises from a_f a_a regardless of the detailed form of a(θ).

The identification that the LE of a CME is given by Z(t) + 2a(t) means that its speed is V(t) + 2w(t), where V = dZ/dt and w = da/dt. The momentum coupling between the flux rope and the ambient medium is modeled by the drag term F_d given by

$\begin{equation} F_d(t) =2 c_d a \rho _a [V_{\rm sw} - (V + 2w)]|V_{\rm sw} - (V + 2w)|, \end{equation} \tag{ 7 }$

where c_d is the drag coefficient. Equation (7) revises the expression, c_daρ_a(V_sw − V)|V_sw − V|, used in the original formulation (Chen 1989) and later in Cargill (2004). The difference between the two would not be apparent in the limited LASCO FOV but is significant when integrated over a wider FOV. As before, we treat c_d as a constant.

The gravity term is

$\begin{equation} F_g(t) = \pi a^2 m_i g(Z(t))(n_a - \bar{n}_p -\bar{n}), \end{equation} \tag{ 8 }$

where n_a(t) = n_a(Z(t)), and $\bar{n}(t)$ and $\bar{n}_p(t)$ are the average cavity and prominence density inside the flux rope at the apex, respectively. The prominence mass is "drained" out on a free-fall timescale, with 15% of the initial amount entrained as the flux rope expands. The initial value, $\bar{n}_{p0}$ , is an adjustable quantity. We do not alter these features of the original model. The gravity term is small in comparison with the other terms and will not be specifically discussed. We mention, however, that the ratio $\bar{n}_{p0}/\bar{n}_0$ is an important initial condition. A nonzero $\bar{n}_{p0}$ affects the initial force balance and the initial acceleration and is essential for replicating the structure at 1 AU (Chen 1996).

Equations (4) and (5), along with L(t) and several ancillary conditions such as Equations (1) and (6), are the basic equations of the model. The evolution of the hoop force and magnetic field is determined from the expansion of the flux rope, Z(t) and a(t). This then determines L(R, a|t), yielding I_t(t) = Φ_p(t)/cL(t), and B_p(t) = 2I_t/ca and $\overline{B}_t(t) = \overline{B}_{t0}a^2_0/a^2(t)$ , where R(t) is calculated from Equation (6). The notation L(R, a|t) signifies that the time dependence is implicit through R(t) and a(t). The system of equations is closed by the adiabatic energy equation with γ sime 1.2.

By stationary footpoints, i.e., S_f= const and a_f= const, we mean "quasi-stationary" in that they can vary in response to changes on the photospheric timescale much slower than the coronal timescale of eruption. This means that Φ_t is conserved, but $\overline{B}_t = \Phi _t/\pi a^2_f$ in the footpoints can vary on the slow timescale, which is neglected in the EFR formulation. This is distinct from imposing the ideal MHD line-tying condition.

2.2. Initialization of Model System

The model system has two categories of physical quantities to be specified: (1) for the corona and solar wind (SW) and (2) for the initial flux rope. The model corona is specified by an overlying field B_c(Z), ambient density n_a(Z) and temperature T_c(Z), and outward speed V_sw(Z), which is smoothly increased to the asymptotic value of V*_sw at about 25 R_☉. We use T₀ = T_c(Z₀) = 2 × 10⁶ K. The functional forms of n_a(Z), T_c(Z), and V_sw(Z) are motivated by a number of empirical models of the corona and SW. These corona/SW profiles are unchanged from those in Chen (1996), and the same functions are used for all of the solutions in the present paper. The values of V*_sw and B_c(Z₀) are adjustable parameters.

The geometry of the initial flux rope is defined by S_f, Z₀, and α₀ ≡ R₀/a₀. The internal pressure is specified in relation to the ambient coronal pressure by $\chi _p \equiv \overline{p}_0/p_{c0}$ , and the prominence material is specified by $\chi _n \equiv \overline{n}_{p0}/\overline{n}_0$ , where ${\overline{n}}_{0}$ is the average initial cavity density.

Once the initial flux-rope geometry is given, the initial magnetic field is calculated by requiring d²Z/dt² = 0 and d²a/dt² = 0, balancing the Lorentz force, pressure, and gravity. Thus, the initial magnetic field is determined within the model, not externally imposed.

2.3. Poloidal Flux Injection

The equilibrium flux rope is set into motion by increasing the poloidal flux, Φ_p(t). In the EFR model, the rate of poloidal flux injection dΦ_p(t)/dt is specified with a simple profile:

$\begin{eqnarray} &&{d\Phi _p(t)\over dt} \nonumber\\ &&= \left\lbrace \begin{array}{@{}l@{\quad }l@{}} Q_0 \equiv \left({d\Phi _p \over dt}\right)_{t=0}, & 0 \le t \le t_1 \cr Q_0 + Q_1 \left[ \hbox{sech}^2\left({t - t_2 \over \tau _1}\right) - \hbox{sech}^2\left({t_1 - t_2 \over \tau _1}\right) \right], &t_1 < t < t_2 \cr Q_0 + Q_1 \left[1 - \hbox{sech}^2\left({t_1 - t_2 \over \tau _1}\right)\right] \equiv \left({d\Phi _p \over dt}\right)_{\rm max}, &t_2 \le t \le t_3, \cr \left({d\Phi _p\over dt}\right)_{t = t_3} \hbox{sech}^2\left({t - t_2 \over \tau _2}\right), & t_3 < t,\cr \end{array}\right.\nonumber\\ \end{eqnarray} \tag{ 9 }$

where Q₀ ≡ (dΦ_p/dt)_t=0 and Q₁ are nonnegative constants and τ₁ (τ₂) is the ramp-up (ramp-down) timescale. Note that t₁, t₂, and t₃ are relative: they can be shifted by the same amount without altering the solution provided the period during which Q₀ is nonzero is also kept the same.

The above form of dΦ_p(t)/dt was first introduced by Chen & Garren (1993) and has remained unchanged. It is a simple generic pulse, and we do not attempt to fine tune the fit by modifying this basic form. It is noteworthy that this simple form is able to produce solutions replicating a wide range of observed CME trajectories as shown in previous works.

Physically, a nonzero dΦ_p(t)/dt produces an EMF ${\cal E}(t)$ along the flux rope and a path connecting one footpoint to the other, where

$\begin{equation} {\cal E}(t) \equiv \oint {\bf E}\cdot d{\bf l} = -{1 \over c} {d\Phi _p \over dt}. \end{equation} \tag{ 10 }$

This EMF was previously identified but was not studied in detail (Chen 1996). The importance of the EMF is that it produces an electric field around the closed loop. We define a characteristic electric field by

$\begin{equation} \hat{E}(t) \equiv {{\cal E}(t) \over S_f}. \end{equation} \tag{ 11 }$

The model does not determine the spatial distribution of ${\cal E}(t)$ . We conjecture that the EMF is predominantly between the footpoints in the lower atmosphere where the resistivity is greater, but we expect that a fraction of the EMF occurs along the expanding flux rope. Regardless of the spatial distribution, the temporal form of $\hat{E}(t)$ will be identical to that of ${\cal E}(t) \propto d\Phi _p(t)/dt$ . The particles accelerated by this electric field, therefore, gain energy and radiate while ${\cal E}(t)$ is significant. The minus sign in Equation (10) means that both $\cal E$ and $\hat{E}$ point oppositely to the direction of I_t.

2.4. CME Trajectories: Physical Scales

Equation (4) has a number of properties that are characteristic of nonaxisymmetric toroidal flux ropes with stationary footpoints. Under the action of the toroidal hoop force, a flux rope exhibits two—the main and residual—acceleration phases, distinguished by how the various force components compete (Chen & Krall 2003).

Equation (4) shows that the acceleration timescale is

$\begin{equation} \tau _R = R/V_{\rm Ap}, \end{equation} \tag{ 12 }$

where $V_{\rm Ap} \equiv B_{\rm pa}/(4\pi \overline{\rho })^{1/2}$ is the characteristic Alfvén speed of the flux rope. Here, τ_R depends on the intrinsic properties of the flux rope and increases as the flux rope expands.

Spatially, the physical length S_f governs the CME acceleration: the main acceleration phase is necessarily limited to the height range ΔZ ~ S_f, diminishing rapidly as the apex height reaches Z ~ (3/2)S_f. Observed CME and EP acceleration has been shown to obey this S_f-scaling law (Chen et al. 2006). Beyond the peak of the main acceleration, the Lorentz force continues to drive the flux rope but decreases with the scaling

$\begin{equation} {d^2Z \over {dt^2}} \propto \left(S_f \over R \right)^2 {1 \over [{\ln (8R/a_f)]^2}}, \end{equation} \tag{ 13 }$

where a_f is constant. This shows that S_f is also a dominant scale in the residual acceleration phase. Here, we have assumed that the total flux-rope mass (∝RM) is constant during the main acceleration phase. The factor [ln(8R/a_f)]⁻², rather than [ln(8R/a)]⁻², arises from the inductance of a nonaxisymmetric flux rope with stationary footpoints having a_f a_a. The difference between the two dependences is significant: the former scales as (ln R)⁻² whereas the latter is nearly constant because R/a is a slowly varying function of time except during the main acceleration phase. The maximum acceleration is influenced by the value and form of Φ_p(t), being the product of Φ²_p(t)/L²(t)f_R(t). In addition, the time constant τ₂ also affects how the Lorentz force decreases. Thus, the decay constant of flux injection τ₂, as well as S_f, determines the residual acceleration.

The derivation of τ_R and the S_f-scaling law (Chen & Krall 2003) show, however, that the duration (and time profile) of the main acceleration phase is insensitive to that of dΦ_p(t)/dt, being determined by certain intrinsic quantities: the rise phase is characterized by the intrinsic timescale τ_R and the decreasing phase of the main acceleration is governed by S_f. If there is an oscillatory component, τ_R provides the oscillatory timescale, and the envelope function is subject to the S_f-scaling law. The magnitude of acceleration depends on (dΦ_p/dt)_max, and the long-time dynamics (i.e., the residual acceleration phase) are influenced by τ₂. In comparing theory and data, it is useful to keep in mind that these characteristic quantities are manifested in the measured height data.

The main and residual acceleration phases roughly correspond, respectively, to the acceleration and propagation phases discussed by Zhang et al. (2004). The S_f-scaling law is reminiscent of the finding of Vršnak (1990) that the height at which a flux-rope prominence becomes eruptively unstable is comparable to half the footpoint separation distance. The two results are sometimes presented as similar (e.g., Vršnak 2008). The S_f-scaling law, however, refers to the height where maximum acceleration occurs rather than to a stability threshold, two distinct physical processes (also see Filippov & Koutchmy 2008).

3. APPLICATION OF THE EFR MODEL TO CME–FLARE EVENTS

3.1. Methodology

For each event, we treat the CME height data as the observational input to the EFR model. We seek to determine the solution that best fits the observed CME trajectory and examine the physical quantities given, i.e., predicted, by the solution. We focus on dΦ_p(t)/dt and S_f for the present study. The measure of goodness of fit is defined below. The temporal form of output dΦ_p(t)/dt is compared with that of the associated SXR (1–8 Å) light curve. Where the dimensions of the initial flux rope can be estimated from the data, the predicted S_f will be compared with the data. For each event, for comparison, we also obtain one solution using the estimated value of S_f as an additional constraint.

3.2. Height Measurement

All measured positions in the paper refer to "true" or "deprojected" positions except as noted otherwise. For this purpose, we identify the candidate source region and the pre-eruption structure, typically using Hα, Extreme Ultraviolet Imaging Telescope (EIT) on board the Solar and Heliospheric Observatory (SOHO), Extreme Ultraviolet Imager (EUVI)/STEREO, or Michelson Doppler Imager (MDI)/SOHO images. If the source is at longitude phgr and latitude θ, we assume that the CME LE moves along the radial line originating at phgr and θ. The "true" distance $\cal R$ from Sun center of the LE at elongation α is given by a simple trigonometric relationship

$\begin{equation} {\cal R} = R_{\rm obs} {\sin \alpha \over {\sin (\alpha + \mu)}}, \end{equation} \tag{ 14 }$

where the observer–Sun distance is taken to be R_obs= 214 R_☉ and μ is defined by

$\begin{equation} \mu \equiv \cos ^{-1}(\cos \phi \cos \theta). \end{equation} \tag{ 15 }$

Here, the solar disk center is at phgr = 0 and θ = 0 in the given FOV. The height coordinate Z is measured from the photosphere (Figure 1) and is related to $\cal R$ by $Z = {\cal R} - R_{\odot }$ . Equation (14) assumes a localized "object." The density in a CME, however, is distributed within a three-dimensional (3D) volume, and a line-of-sight (LOS) integrated density feature of a CME can gradually shift relative to the overall structure (Morrill et al. 2009). This effect is not included.

We obtain the "true" speed by first applying a three-point smoothing algorithm to the deprojected position data and then calculating the speed using two successive smoothed data points. Acceleration is determined in a similar way using the speed data.

3.3. Goodness of Fit

In judging the goodness of fit of a solution to data, we define the following measure:

$\begin{equation} {\cal D} \equiv {1 \over T} \sum _{i=1}^N {|{{\cal R}_{\rm data}(t_i) - {\cal R}_{\rm th}(t_i)}| \over \Delta {\cal R}_i}\delta t_i, \end{equation} \tag{ 16 }$

where ${\cal R}_{\rm data}(t_i)$ is the measured trajectory, ${\cal R}_{\rm th}(t_i)$ is the theoretical solution, $\Delta {\cal R}_i$ is the uncertainty at t = t_i, N is the number of data points, δt_i = t_i+1 − t_i, and T = ∑_iδt_i. Thus, $\cal D$ measures the average deviation from the height data normalized to $\Delta {\cal R}_i$ . If the deviations are equal on average to the error bars, we obtain ${\cal D} =$ 1. If ${\cal D} <$ 1, the solution fits the data within the error bars.

The reason for using the above definition instead of the more common χ² is as follows. In tracking the CME LE, we treat the uncertainty $\Delta {\cal R}_i$ as the range within which the actual position of the LE is located with equal likelihood; $\Delta {\cal R}_i$ does not represent the width of a Gaussian distribution of measurements. The quantity ${\cal D}$ is simply a measure of the distance between two discrete functions, $|{\cal R}_{\rm data} - {\cal R}_{\rm th}|$ , normalized to the uncertainty $\Delta {\cal R}_i$ .

We have found that the measurement uncertainties are approximately 1%–2% of measured height. We will use 2% as the canonical uncertainty throughout the paper. For finding the best-fit solutions, only the relative value of $\cal D$ is relevant. We use $\Delta V_i = (\Delta {\cal R}_i^2 +\Delta {\cal R}_{i+1}^2)^{1/2}/(t_{i+1} - t_i)$ to propagate errors.

3.4. Theory–Data Comparison

We have analyzed five CME–flare events that are sufficiently well observed. Table 1 lists the dates, source locations of the CMEs, and the peak power of the associated SXR emissions. The first event will be discussed in detail to illustrate both the methodology of the study and the physical interpretation of the results.

Table 1. CME–Flare Event List

Event	Date	Time^a	Location^b	Flare^c	Data^d
ID		(UT)	(deg)	(W m⁻²)
A	2000 Sep 12	1200	W14, S20	1.0 × 10⁻⁵ (M)	L
B	2008 Apr 26	1400	W36, N8	3.9 × 10⁻⁷ (B)	SA
C	2008 Mar 25	1900	E108, S19	1.7 × 10⁻⁵ (M)	SA
D	1998 Jun 11	1000	E97, N20	1.4 × 10⁻⁵ (M)	L
E	1999 Aug 17	1500	E38, N31	4.4 × 10⁻⁶ (C)	L

Notes. ^aNearest UT hour of time of eruption. ^bLongitude and latitude ( phgr , θ) of the candidate initial structure measured from solar disk center in the observing telescope FOV. phgr > 90° is behind the limb. ^cPeak intensity of the disk-integrated GOES X-ray in the 1–8 Å band and the corresponding flare classification in parentheses. ^dHeight data sources: L = LASCO; SA = SECCHI-A.

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3.4.1. Event A: 2000 September 12

Observation and data. This eruption was observed as an EP–CME–flare event by the Kanzelhöhe Solar Observatory (KSO) Hα telescope, EIT, and LASCO C2 and C3 instruments on SOHO and has been discussed in the literature (e.g., Wang et al. 2003; Vršnak et al. 2003; Schuck et al. 2004; Qiu et al. 2004; Chen et al. 2006). The main acceleration phase of this event was observed by KSO Hα (1 minute cadence) and EIT (12 minute cadence). The acceleration and the dynamics of the prominence were sufficiently well resolved in the EIT–C2–C3 FOV whereas the CME was not observed below the C2 FOV. We therefore tracked the motion of the prominence. The eruption was accompanied by a two-ribbon flare. GOES-8 X-ray data in the 1–8 Å band indicate that this was an M1 class flare (Table 1). The flare lasted approximately 3 hr, with an FWHM duration of ΔT_SXR= 105 ± 2 minutes.

Figure 2(a) shows the erupting prominence observed by EIT at 195 Å during the main acceleration phase. The inferred footpoints of the prominence, F1 and F2, are indicated. They are identified by tracing the two legs of the prominence to quasi-stationary features during the eruption. The midpoint between the two footpoints is marked by O, and the apex of the erupting prominence is denoted by P. Point O is at phgr = 14°W and θ= 20°S measured from the solar disk center. Connecting F1 and F2 is a relatively dark filament channel. On either side of this channel is a bright extreme-ultraviolet (EUV) ribbon, corresponding to the two-ribbon flare in the KSO Hα data. The deprojected linear distance between F1 and F2 is S_p sime 3.5 × 10⁵ km, where S_p refers to the prominence footpoint separation. We refer to point O as the "source" and assume that the prominence apex moves along the solar radial direction at this point. The source coordinates are listed in Table 1 and are used in Equation (14) to convert elongation to height. Based on this source location, we estimate the pre-eruption apex height of the prominence to be Z_p0 sime 1.6 × 10⁵ km.

The overall magnetic structure of the CME is not evident in the EIT data, but the relationship between the prominence and the associated flux-rope CME becomes clear in C2 (Figure 2(b)). This image shows that the white-light counterpart of the prominence coincides with the trailing edge of the CME flux rope. This spatial relationship between the prominence and CME remains unchanged throughout the C2–C3 FOV. The observed CME–EP structure is consistent with the inverse polarity prominence morphology (Kuperus & Raadu, 1974; Low & Hundhausen 1995) and the 3D flux-rope CME concept (e.g., Chen 1996, Figure 2; Filippov & Koutchmy 2008, Figure 5). Based on this relationship, the underlying flux-rope quantities—S_f and Z₀—can be calculated from S_p and Z_p0, viz., Z₀ = Z_p0 + a₀ and S_f = S_p + 2a_f, with a_f = a₀ at t = 0 (Chen et al. 2006). Using the above S_p0 and Z_p0, we obtain S_f = 5 × 10⁵ km and Z₀ = 2 × 10⁵ km assuming R₀/a₀ = 2, with an uncertainty of ± 25% or perhaps greater.

The observed position of the prominence apex is plotted in Figure 3(a) using open diamonds (EIT), solid circles (C2), and open circles (C3). A few 2% error bars are plotted, but they are comparable to or smaller than the symbols in the plot. The speed and acceleration data are plotted (open diamonds) in Figures 3(b) and (c), respectively. The prominence accelerated through the C2 FOV and attained a speed of ~1600 km s⁻¹ in the C3 FOV. The measured peak acceleration is ~700 m s⁻², with an estimated uncertainty of approximately ±200 m s⁻² in the C2 FOV and ±300 m s⁻² in the C3 FOV. Error bars are not shown in Figure 3(c) to simplify the plot. The various color-coded curves in these panels represent several solutions of the model to be discussed below.

$ {\cal R} = Z -a + R_\odot$ — **Figure 3.** Event A. (a) Prominence apex height measured from Sun center. Data: EIT (open diamonds), C2 (solid circles), and C3 (open circles). Solid curve 1 shows the "best-fit" theoretical solution ( ${\cal R} = Z -a + R_\odot$ ). Error bars are 2% of height (smaller than the symbols). (b) Speed. The model SW speed is shown. (c) Acceleration. The dashed curve is the time derivative of the SXR data, scaled to the maximum acceleration. (d) Predicted poloidal flux injection function dΦ_p(t)/dt. The dashed curve is the *GOES* X-ray (1–8 Å) data normalized as indicated. Curves 1–6 are solutions 1–6 in Tables 2, 3, and 6.
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The black dashed curves in Figures 3(c) and (d) are the time derivative of the GOES SXR intensity, "d(Xray)/dt," and the intensity itself, respectively. We see that the peaks in acceleration and "d(Xray)/dt" coincide within 10 minutes. Brightenings in the EIT images suggest that the disk-integrated SXR flux of interest primarily comes from the site of the eruption. The X-ray data are derived from the 3 s data averaged over 3 minutes.

Best-fit solution. In relating theoretical solutions to the data, we identify Z(t) − a(t) as the prominence height, consistent with the CME–EP image in Figure 2(b) (also Chen et al. 2006). For the first solution, we initialize a flux rope using the above estimates, S_f = 5 × 10⁵ km and Z₀ = 2 × 10⁵ km. The minor radius could not be identified and we use α₀ = 2 based on previous model-data comparisons. For concreteness, we start with B_c(Z₀) = −1 G, χ_p= 1, and χ_n= 1, where n_c(Z₀) = 2 × 10⁸ cm⁻³ and p_c(Z₀)= 0.1 dyn cm⁻² according to the model corona. The initial flux rope is described in Table 2, labeled "0." We use "0" to designate the solution using the observationally estimated value of S_f. The asymptotic SW speed is set to V*_sw= 400 km s⁻¹ with the profile shown in Figure 3(b). The drag coefficient is c_d= 1.

Table 2. Event A: Best-fit Initial Flux-rope Parameters^a

No.	${\cal D}$ ^b	S_f	M_T^c	B_pa0^c	B_t0^c	τ_R	Specified
		(10⁵ km)	(10¹⁶ g)	(G)	(G)	(minutes)	Quantity^d
0	0.88	5.0	2.43	2.52	2.52	15.7	⋅⋅⋅
1	0.66	4.25	1.56	2.46	2.46	13.3	⋅⋅⋅
2	3.94	4.25	1.56	2.46	2.46	13.3	(dΦ_p/dt)_max^e
3	2.18	4.25	1.56	2.46	2.46	13.3	(dΦ_p/dt)_max^f
4	1.53	7.0	7.44	2.50	2.50	23.2	⋅⋅⋅
5	0.74	4.25	1.56	2.46	2.46	13.3	τ₁= 10 minutes
6	1.21	4.25	1.56	2.46	2.46	13.3	τ₁= 40 minutes
7	0.77	4.75	2.10	2.50	2.50	14.9	⋅⋅⋅
8	0.69	4.5	1.81	2.48	2.48	14.1	⋅⋅⋅
9	0.67	4.0	1.34	2.44	2.44	12.6	⋅⋅⋅
10	0.74	3.5	1.00	2.41	2.41	11.3	⋅⋅⋅
11	0.90	2.5	0.58	2.36	2.36	9.1	⋅⋅⋅
12	0.69	5.0	1.55	2.70	2.70	14.6	α₀= 2.5

Notes. ^aBest-fit values for all solutions: $\chi _p = \bar{p}_0/p_{c0} = 1$ , $\chi _n = \bar{n}_{p0}/\bar{n}_0 = 1$ , B_c = −1 G α₀ = 2.0, and V*_sw= 400 km s⁻¹. ^b $\cal D$ , Equation (16), is minimized by adjusting dΦ_p(t)/dt except as noted in the last column. Measurement uncertainty is 2%. ^cCalculated using the equilibrium force-balance conditions. ^dParameters that are prescribed. ^e(dΦ_p/dt)_max = 1.21 × 10¹⁹ Mx s⁻¹, approximately 10% greater than for solution 1. Other parameters of dΦ_p(t)/dt are unchanged. ${\cal D}$ is not minimized. ^f(dΦ_p/dt)_max = 1.50 × 10¹⁹ Mx s⁻¹. Other parameters of dΦ_p(t)/dt are adjusted to minimize ${\cal D}$ .

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The force-balance conditions at t = 0 yield $B_{\rm pa0} = \overline{B}_{t0} =$ 2.52 G with I_t0 = 1.3 × 10¹¹ A. The equality $B_{\rm pa0} = \overline{B}_{t0}$ is particular to the choice $\bar{p}_0/p_{p0} =$ 1. The calculated total initial mass at t = 0 is M_T0 = 2.4 × 10¹⁶ g.

After adjusting dΦ_p/dt to minimize $\cal D$ , the best-fit solution is found to have $\cal D =$ 0.88, meaning that the average deviation is 1.76% in the EIT–LASCO FOV. The FWHM duration of the required dΦ_p(t)/dt is ΔT_p= 90 minutes, which is ~15 minutes shorter than the FWHM duration of the observed SXR light curve, ΔT_SXR= 105 minutes. The output quantities are given in Table 3, including (dΦ_p/dt)_max, $\hat{E}_{\rm max}$ , maximum acceleration, ΔΦ_p ≡ ∫(dΦ_p(t)/dt)dt, and ΔU_p ≡ ∫(dU_p(t)/dt)dt. "Y/N" in the last column indicates that there is/is not a first-order normal-mode oscillation component, which is most apparent in the acceleration.

Table 3. Event A: Poloidal Flux Injection and Electromotive Force

No.	ΔT_p^a	(dΦ_p/dt)_max	$\hat{E}_{\rm max}$	ΔΦ_p	ΔU_p	Accel	Osc^b
	(minutes)	(10¹⁹ Mx s⁻¹)	(V cm⁻¹)	(10²² Mx)	(10³² erg)	(m s⁻²)
ΔT_SXR = 105 minutes
0	90	1.3	2.56	7.8	7.5	622	N
1^c	95	1.1	2.56	6.8	5.3	573	N
2	95	1.2	2.84	7.5	5.8	609	N
3	58	1.5	3.53	5.7	5.5	679	N
4	90	1.8	2.56	11.0	18.0	637	Y
5	79	1.2	2.90	6.6	5.8	1205	Y
6	103	1.0	2.39	6.8	5.0	538	Y
7	91	1.2	2.56	7.4	6.7	582	N
8	93	1.1	2.53	7.1	5.9	531	N
9	92	1.0	2.60	6.5	4.8	626	N
10	92	1.0	2.71	5.9	3.9	675	N
11	92	0.8	3.23	5.0	2.7	980	N
12	91	1.1	2.26	6.9	5.6	530	N

Notes. ^aThe FWHM duration ΔT_p of dΦ_p(t)/dt for each solution shown in Figure 3(d) is to be compared with that of the GOES X-ray light curve ΔT_SXR= 105 minutes. The uncertainty in ΔT_p and ΔT_SXR is roughly ± 2 minutes. ^bA major radial oscillation is judged to be present and marked "Y" if two distinct peaks are present in addition to the main acceleration maximum. Otherwise, a solution is marked "N." ^cThe overall best-fit solution. The initial magnetic flux and energy: Φ_p0 = 2.3 × 10²¹ Mx, Φ_t0 = 8.8 × 10²⁰ Mx, U_p0 = 1.5 × 10³¹ erg, and U_t0 = 5.5 × 10³⁰ erg.

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We have obtained numerous solutions unconstrained by the estimated S_f. That is, S_f is treated as a free parameter. For each set of initial values, the minimum- $\cal D$ solution is obtained by adjusting dΦ_p/dt and minimizing $\cal D$ constrained only by the position–time data including ${\cal R}_0 = Z_0 + R_\odot$ . Solution 1 provides the overall best fit and is shown in Figure 3 (thick solid black curve). It has $\cal D =$ 0.66, i.e., the average deviation is 1.3%, a measurably better fit than solution 0. In appearance, however, solutions 0 and 1 are similar and are not separately shown. Solution 2 (thin black curve) is obtained by changing only the maximum value of dΦ_p/dt holding all else unchanged from solution 1. Curves 1–6 in Figure 3 provide a representative sample of solutions with minimized $\cal D$ in the range of ~0.6–2. The values of $\cal D$ and the model quantities given by a range of solutions are tabulated in Tables 2 and 3, where those quantities that are specified to be different from solution 1 are shown in boldface. The initial magnetic flux and energy predicted by solution 1 are given in note c of Table 3.

Figure 3(d) shows that the rise phase of dΦ_p/dt leads that of the flare light curve by ~15 minutes for solutions 1–6. The FWHM durations of dΦ_p/dt required to produce good fits are in the range of ΔT_p= 90–95 minutes, which are roughly 10–15 minutes shorter than ΔT_SXR. The exceptions are solutions 3, 5, and 6 for which the flux injection profiles are required to be different from the optimum form in some aspects as indicated in Table 2. For these solutions, the average deviation from the height data is significantly greater. The X-ray light curve contains effects of radiative decay and possibly additional X-ray emissions that are not included in the model calculation.

In Section 2.4, we have identified S_f as the important scale length for CME dynamics. Solutions 0, 4, and 7–11 provide the best-fit solutions for S_f in the range of (2.5–7) ×10⁵ km. (For this event, Z₀ = 2 × 10⁵ km is determined from measurement and is not adjustable.) The output quantities given by the model are tabulated in Tables 2 and 3. The results show that the EFR equations favor S_f in the neighborhood of (4.0–4.5)×10⁵ km, having comparably good fit. The individual minimum- $\cal D$ solutions increasingly deviate from the observed height data as S_f is varied outside this range. Keep in mind that solution 1 and therefore all the physical quantities given by it are predictions constrained by height data alone. For this event, the predicted best-fit value, S_f = 4.25 × 10⁵ km, can be directly compared with the observational estimate of 5.0 × 10⁵ km. The two values are well within the estimated uncertainty of 25%.

We have found that χ_p= 1, χ_n= 1, and B_c(Z₀) = −1 G are the best-fit values demanded by the observed trajectory (see the Appendix). The calculated value of $\overline{B}_{t0} =$ 2.5 G implies B= 7.5 G on the flux-rope axis (see Equation (30), Chen 1996). This is consistent with quiescent prominence fields estimated using the Hanley effect (Rust 1967; Leroy et al. 1983). We judge B= 7.5 G to be consistent with the currently available observational knowledge, but unlike the approximate dimensions of the initial structure, B cannot be measured in the corona to allow theory–data comparison. For a CME whose ejecta is detected at 1 AU, the dynamically evolved quantities may provide a consistency check on the evolutionary end product of the predicted initial field (Kunkel & Chen 2010). For the SW speed, V*_sw= 300–600 km s⁻¹ yields comparably good fits. The relative insensitivity is due to the limited LASCO FOV. The complete parameterization of dΦ_p(t)/dt for solutions 1–12 is tabulated in the Appendix, which also provides the results of an additional sensitivity test for solutions 1, 5, and 6.

The EIT and LASCO data in Figure 3(c) indicate that the main acceleration phase ceased early (at about 1200 UT) in the eruption process, followed by the residual acceleration phase. The best-fit solution correctly captures the varying timescales of acceleration. In particular, Equation (12) gives the acceleration timescale of τ_R = R/V_Ap sime 13 minutes, which increases as the flux rope expands and the magnetic field decreases. Note that τ_R depends on the initial geometry and calculated equilibrium magnetic field. Thus, the comparison of τ_R provides an internal consistency check of the initial magnetic field and geometry.

The nonzero dΦ_p(t)/dt produces an EMF given by Equation (10). For this event, the magnitude of the peak EMF for solution 1 is ${\cal E}_{\rm max} \simeq 1.4 \times 10^{11}$ V. The characteristic electric field associated with this EMF has $\hat{E}_{\rm max} =$ 2.6 V cm⁻¹. The temporal form of ${\cal E}(t)$ and $\hat{E}(t)$ is obviously the same as that of dΦ_p(t)/dt.

We emphasize that the temporal relationship predicted by the model is between the duration of the calculated dΦ_p(t)/dt and that of the associated SXR light curve. This is to be distinguished from the empirical CME–flare correlation discussed by Zhang et al. (2001), Maričič et al. (2007), and Temmer et al. (2008), where it is the observed CME acceleration peak and the rise phase of the soft SXR emissions that are compared. Nevertheless, it is interesting that the calculated acceleration peak occurs within about 5–6 minutes of the peak of the time derivative of the observed X-ray light curve. If a solution exhibits oscillations, this conclusion applies to the envelope function. This offset in the temporal correlation is consistent with the range of scatter noted by Maričič et al. (2007). The physical basis of this apparent agreement cannot be addressed by the model as it stands because it has no mechanism of X-ray emission.

3.4.2. Event B: 2008 April 26

The CME was observed by both STEREO-A and STEREO-B. The eruption site was closer to the limb viewed from STEREO-A than from STEREO-B. We used STEREO-A data to measure the CME position and STEREO-B data to examine the source region. Using EUVI-B images, we identified as the source a structure consisting of a bright loop encircling a relatively dark cavity and an EP. The footpoints of the prominence are at (N17°, W15°) and (N13°, W6°), from which we estimate S_f ≈ 2 × 10⁵ km. In the STEREO-A FOV, the source coordinates are estimated to be θ = 8°N and phgr = 36°W, which are used to calculate distance from elongation.

In EUVI-A, the cavity became visible around 1315 UT, and we identified the LE of this cavity as the LE of the CME. The eruption was visibly more rapid by 1353 UT. The white-light counterpart of the LE was tracked through the COR1–COR2–HI1–HI2 FOV. There are uncertainties in tracking LOS density features from EUV images to white-light images and across different telescopes. Nevertheless, we did not find significant inconsistencies in the measured height data from telescope to telescope.

We estimate the initial height of the flux-rope LE to be Z₀ = 2 × 10⁵ km. The measured CME LE position, speed, and acceleration of the CME in the STEREO-A FOV are given in Figures 4 and 5. Figures 4(a)–(c) show the CME dynamics during the first 3 hr recorded by EUVI, COR1, and COR2. Figure 5 shows the position and speed data including HI1 data (solid diamonds) extending to about 100 R_☉. We tracked this CME through the HI2-A FOV to 1 AU, but the HI2 height data are not shown here because they would crowd the plot and are not essential for the present paper. The main acceleration profile is not resolved in detail, but it is evident that the peak acceleration lasted less than 10 minutes. Figure 4(d) shows the GOES SXR light curve (dashed). Its time derivative is shown in Figure 4(c) (dashed curve).

**Figure 4.** Event B. Trajectory of the LE of the CME of 2008 March 25 and the *GOES* X-ray light curve. EUVI-A, COR1-A, and COR2-A data points are shown in the same format as in Figure 1. The time axis starts at 13:00 UT, 2008 April 26. In each panel, the solid curve is the solution that best fits the entire trajectory from EUVI-A through HI1-A. Solutions 1–4 are tabulated in Tables 4 and 5.
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**Figure 5.** Event B: entire trajectory of the LE including the HI1-A data. COR2-A and HI1-A data points are indicated. The time axis is shown from 1200 UT, April 26. V*_sw= 400 km s⁻¹. The curves are solutions shown in Figure 4.
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In calculating the trajectory for the CME LE, we identify Z(t) + 2a_a(t) as the LE height above the photosphere. Measured from Sun center, the LE position is ${\cal R} = Z + 2a + R_\odot$ . As before, we first obtained the best-fit solution (solution 0) using the observationally estimated S_f ≈ 2 × 10⁵ km and Z₀ = 2 × 10⁵ km with α₀= 2 assumed. It has $\cal D =$ 0.96, and its output quantities are given in Tables 4 and 5. We have obtained solutions unconstrained by the estimated S_f for a range of initial values. The overall best fit is solution 1, which has $\cal D =$ 0.46, and is shown in Figure 4 (solid curves). This solution is within 1% of the height data in the entire 100 R_☉ EUVI–HI1 FOV (and remains in good agreement with the HI2-A data to 1 AU). The predicted value S_f = 1.8 × 10⁵ km is in good agreement with the estimated initial value. For this event, α₀= 2.5 is preferred rather than α₀= 2. The duration of flux injection is ΔT_p= 41 minutes, slightly shorter than ΔT_SXR= 45 minutes. The initial magnetic flux and energy given by solution 1 (and for other events) are noted in Table 5.

Table 4. Best-fit Initial Flux-rope Parameters

No.	$\cal D$	S_f	Z₀	α₀	χ_p	χ_n	B_c0	M_T	B_pa0	B_t0	Osc^a
		(10⁵ km)	(10⁵ km)				(G)	(10¹⁵ g)	(G)	(G)
Event B
0	0.96	2.0	2.0	2.0	1.0	1.0	−2.0	2.3	4.38	4.38	N
1	0.46	2.0	1.8	2.5	0.5	1.0	−2.0	1.4	4.93	5.09	N
2^b	0.43	2.0	1.8	2.5	0.5	1.0	−2.0	1.4	4.93	5.09	Y
3^c	0.68	2.0	1.8	2.5	0.5	1.0	−2.0	1.4	4.93	5.09	Y
4^d	2.12	2.0	1.8	2.5	0.5	1.0	−2.0	1.4	4.93	5.09	N
5	1.13	4.0	1.8	2.5	0.5	1.0	−2.0	4.7	4.93	5.09	N
6	0.54	1.0	1.8	2.5	0.5	1.0	−2.0	0.9	4.93	5.09	N
Event C
0	1.88	1.4	0.6	2.0	1.0	1.0	−2.5	1.9	5.58	5.58	N
1	0.88	1.2	0.6	2.5	0.5	2.0	−3.0	0.6	7.55	7.88	N
2	1.01	1.2	0.6	2.5	1.0	3.0	−2.5	1.7	6.44	6.44	N
3	1.33	1.2	0.6	2.0	0.5	2.0	−3.0	0.6	7.55	7.88	Y
Event D
1	0.87	3.0	0.8	2.0	1.0	1.0	−2.5	17.7	5.79	5.79	N
2	0.89	2.7	0.8	2.0	1.0	1.0	−2.5	12.0	5.73	5.73	Y
Event E
0	0.72	5.5	2.2	2.0	1.0	1.0	−0.5	28.2	1.62	1.62	Y
1	0.48	5.1	2.0	2.0	1.0	3.0	−0.5	51.5	2.34	2.34	Y
2	0.59	2.5	2.0	2.0	1.0	3.0	−0.5	11.6	1.90	1.90	Y
3	0.56	4.6	2.0	2.0	1.0	3.0	−0.5	38.3	2.24	2.24	Y
4	0.52	5.6	2.0	2.0	1.0	3.0	−0.5	68.8	2.44	2.44	N
5	0.88	7.8	2.0	2.0	1.0	3.0	−0.5	22.4	2.93	2.93	Y
6	0.96	10.0	2.0	2.0	1.0	3.0	−0.5	62.0	3.45	3.45	N

Notes. ^aA major radial oscillation is judged to be present and marked "Y" if two distinct peaks are present in addition to the main acceleration maximum. Otherwise, a solution is marked "N." ^bBest-fit solution with c_d = 3. ^cBest-fit solution with τ_R is prescribed to be 50 minutes. All other initial parameters remain unchanged from solution 1. ^dτ₂ is set to τ₂ = 36.3 minutes, 10% greater than that for solution 1. All other parameters remain unchanged.

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Table 5. Poloidal Flux Injection and Electromotive Force

No.	ΔT_p	(dΦ_p/dt)_max	$\hat{E}_{\rm max}$	ΔΦ_p	ΔU_p	Accel	(dΦ_p/dt)₀	τ_R0
	(minutes)	(10¹⁹ Mx s⁻¹)	(V cm⁻¹)	(10²² Mx)	(10³² erg)	(m s⁻²)	(10¹⁶ Mx s⁻¹)	(minutes)
Event B: ΔT_SXR= 45 minutes
0	41	0.93	4.67	2.6	3.5	3282	0.0	4.4
1^a	41	0.90	4.49	2.5	2.6	3762	0.0	2.8
2	41	1.30	6.62	3.7	4.1	2019	0.0	2.8
3	71	0.59	2.94	2.5	2.3	2358	0.0	2.8
4	44	0.90	4.49	2.7	2.7	3762	0.0	2.8
5	42	1.20	2.94	3.4	4.8	1366	0.0	4.8
6	38	0.88	8.81	2.2	2.3	7567	0.0	2.3
Event C: ΔT_SXR= 26 minutes
0	34	1.10	7.72	2.5	1.5	1587	0.0	3.7
1^b	38	0.96	8.03	2.6	1.3	2292	0.0	2.0
2	36	1.00	8.64	2.5	1.2	2019	0.0	3.8
3	27	1.30	10.4	2.3	1.3	2358	0.0	2.0
Event D: ΔT_SXR= 75 minutes
1^c	53	1.00	3.36	3.5	4.7	812	0.28	8.1
2	53	0.93	3.46	3.3	3.7	871	2.52	7.0
Event E: ΔT_SXR= 168 minutes
0	140	0.65	1.18	5.9	4.0	406	8.69	25.1
1^d	139	0.76	1.48	6.9	5.4	363	18.1	24.5
2	138	1.00	1.99	9.2	8.3	345	18.1	24.5
3	141	0.69	1.49	6.4	4.3	408	15.4	22.6
4	141	0.82	1.47	7.6	6.8	313	12.1	26.4
5	126	1.20	1.50	9.7	16.0	257	23.6	35.8
6	135	1.40	1.40	12.0	34.0	256	17.3	45.8

Notes. Initial magnetic flux and energy given by the best-fit solutions. ^aΦ_p0 = 1.9 × 10²¹ Mx, Φ_t0 = 3.6 × 10²⁰ Mx, U_p0 = 1.1 × 10³¹ erg, U_t0 = 3.6 × 10³⁰ erg. ^bΦ_p0 = 5.4 × 10²⁰ Mx, Φ_t0 = 1.4 × 10²⁰ Mx, U_p0 = 2.5 × 10³⁰ erg, U_t0 = 8.4 × 10²⁹ erg. ^cΦ_p0 = 2.5 × 10²¹ Mx, Φ_t0 = 1.5 × 10²¹ Mx, U_p0 = 3.3 × 10³¹ erg, U_t0 = 1.2 × 10³¹ erg. ^dΦ_p0 = 2.9 × 10²¹ Mx, Φ_t0 = 1.3 × 10²¹ Mx, U_p0 = 2.3 × 10³¹ erg, U_t0 = 8.2 × 10³⁰ erg.

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Solution 2 is obtained for the identical initial flux rope but with c_d= 3. The minimized average deviation is $\cal D=$ 0.43. For this solution, (dΦ_p/dt)_max is greater than for solution 1, but ΔT_p remains unchanged. In Figure 4(a), solutions 1 and 2 are not distinguishable. For the acceleration curve, this solution (blue dashed curve) has a lower peak value but is otherwise nearly indistinguishable from solution 1. In particular, solution 2 exhibits comparable timescales. Although solution 2 yields a slightly better fit, in the interest of not adjusting c_d, we use solution 1 as the best-fit solution. The values c_d= 1–3 are consistent with estimates given by an MHD simulation study (Cargill et al. 1996).

We have imposed a different profile of the driving function dΦ_p(t)/dt on the identical initial structure. Specifically, we set τ₁= 50 minutes in Equation (9), and adjust the other parameters in dΦ_p(t)/dt to minimize $\cal D$ . This results in solution 3 with $\cal D =$ 0.68. This is significantly worse than solution 1 but is still approximately within 1.4% of the data points within the 100 R_☉ FOV. The total injected poloidal flux and energy are not significantly different from those of solution 1 (Table 5). The resulting acceleration profile is virtually unchanged, except that the maximum value is somewhat less, reducing the maximum speed. Figure 4(c) shows that the temporal profile of the main acceleration phase is nearly the same for the range of dΦ_p(t)/dt shown here. Physically, these profiles of best-fit dΦ_p(t)/dt yield similar values of B_p during the main acceleration phase, and the values of τ_R are similar. The width of the main acceleration profile is determined by intrinsic properties of the flux rope—S_f and the flux-rope magnetic field through τ_R—and is insensitive to the temporal form of dΦ_p/dt (the Appendix and Section 2.4.) The maximum acceleration depends on the quantities (t₂ − t₁)/τ₁ and Q₁. To test the sensitivity to τ₂, we have taken solution 1 and increased τ₂ by 10%. This is shown as solution 4 (dotted curves) with $\cal D =$ 2.12. Figures 4 and 5 show that the solutions are sensitive to τ₂: even though flux injection is brief relative to the entire trajectory, any difference in speed ΔV at the end of the main acceleration phase is amplified linearly in time as (ΔV)t. Thus, the long-time CME trajectory is a strong constraint on τ₂ and therefore ΔT_p.

Comparing events A and B, we see that the best-fit solutions yield S_f and the duration ΔT_M of the main acceleration phase that are remarkably consistent with those of the respective events exhibiting a wide range of scales. Thus, these solutions are able to capture both spatial (S_f) and temporal (τ_R and τ₂) scales. The key feature in the height data that allows the model equations to "extract" this information is the height range ΔZ over which the curvature in the Z(t) curve is maximum because the S_f-scaling law requires ΔZ sime S_f (Section 2.4). For event A, Table 2 shows that "incorrect" choices of S_f (solutions 0, 4, 7, 10, and 11) produce noticeably worse fits because the curvature in the calculated trajectory does not fit the data. Similarly, for event B, the best-fit solution for S_f = 4.0 × 10⁵ km with all other initial values held unchanged yields a significantly worse fit. Thus, adjusting dΦ_p/dt cannot compensate for the "incorrect" choice of S_f.

Note that the main acceleration phase of event B was not resolved, but it is clear that this phase was shorter in duration than the gap between EUVI and COR1 observations. Thus, the height range corresponding to this gap is the upper bound for S_f of the actual eruptive structure. Solutions for S_f greater than this value should produce greater deviations measured by $\cal D$ . Solution 5 exemplifies this property. In contrast, the value of $\cal D$ should be relatively insensitive to the choice of S_f < 2 × 10⁵ km because the height data contain no information for ΔZ below this limit. We have obtained several solutions for S_f < 2 × 10⁵ km. For S_f = 1.5 × 10⁵ km, we find $\cal D =$ 0.50, and for S_f = 1 × 10⁵ km, we obtain $\cal D =$ 0.54. The latter is given as solution 6 in Tables 4 and 5.

The observed duration of maximum curvature in Z(t) also constrains the flux-rope magnetic field through the timescale τ_R. For event A, we find τ_R= 13.3 minutes for the best fit, compared to τ_R= 2.8 minutes for event B. This requires R to be longer, V_Ap to be slower, or both for event A. For these two events, the difference in R is not sufficient to account for the difference in τ_R. The magnetic field, therefore, must be weaker for event A, implying a weaker B_c0. This prediction cannot be tested against measured coronal field but is consistent with the fact that event A corresponds to a larger structure at a higher initial position, where the coronal field would be weaker. Thus, the height–time data contain information on B_c through the observed timescale of the main acceleration phase. The key physics is the initial force-balance conditions, d²Z/dt² = 0 and d²a/dt² = 0 for the initial flux rope. This is an important point because τ_R is calculated self-consistently in the EFR model. That is, minimizing $\cal D$ constrains B_p to yield the most appropriate acceleration time τ_R ∝ R/B_p.

Finally, the duration of flux injection ΔT_p is constrained by the residual acceleration manifested in the height data. The calculated long-time trajectory is sensitive to the value of τ₂ in Equation (9) because its effect on the calculated trajectory— Equation (4) twice integrated in time—is cumulative. Even though the predicted dΦ_p(t)/dt for solution 3 in Figure 4(c) has a significantly different rise phase and duration (71 minutes versus 41 minutes), the ramp-down time constant, τ₂= 39 minutes, is comparable to τ₂= 33 minutes for solution 1. Solution 3 produces a slightly longer τ₂ because its maximum speed is lower, requiring a slightly slower rate of decrease in dΦ_p/dt to fit the residual acceleration through the HI1 FOV. Thus, the best-fit value of τ₂ and therefore ΔT_p are strongly constrained by the long-time dynamics.

3.4.3. Event C: 2008 March 25

This CME was observed by STEREO-A and B spacecraft. Judging by EUVI-B data, the source region is estimated to be at approximately S19° and 18° behind the east limb relative to the STEREO-A FOV. The onset of the CME first became apparent in EUVI-B data at 1843 UT, when the LE appeared to be already in the FOV of EUVI-A. We tracked the CME LE using SECCHI-A data. Figure 6 shows the CME and GOES-10 X-ray data for the first 3 hr of the event in the same format as before. The HI1 images record the CME dynamics past 90 R_☉ (projected). When plotted together with EUVI and COR data, the CME dynamics in the EUVI–COR1/2–HI1 FOV looks similar to what is shown in Figure 5 (except for the timescale) and therefore the HI1 points are not shown. The theoretical solutions discussed below, however, are obtained by minimizing $\cal D$ with respect to the CME trajectory in the entire EUVI–HI1 FOV.

**Figure 6.** Event C. Trajectory of the LE of the CME of 2008 March 25 and the *GOES* X-ray light curve of the associated flare, shown starting from 1800 UT on 2008 March 25.
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The properties of solution 0 ( $\cal D =$ 1.88) obtained for the observationally estimated S_f and Z₀ are given in Table 4. Solution 1, unconstrained by S_f, is the overall best fit to the EUVI–COR1/2–HI1 data. It has $\cal D=$ 0.88 and is plotted in Figure 6. As before, the theoretically predicted value of S_f is consistent with the observationally determined value. Tables 4 and 5 show a number of solutions for different parameters. Compared to events A and B, the best-fit value |B_c0| = 3 G is larger, consistent with an initial flux rope at a lower initial height. Unlike event A observed in the more limited LASCO FOV, the solutions subject to HI1 data are more sensitive to the value of V*_sw, and the fit is measurably better for V*_sw= 500 km s⁻¹ than for 400 km s⁻¹ or 600 km s⁻¹. The predicted dΦ_p(t)/dt function has an FWHM duration of ΔT_p= 38 minutes, which is longer than the FWHM duration of the SXR light curve, ΔT_SXR= 26 minutes. The peak of acceleration coincides with the peak of the "d(Xray)/dt" curve.

Solution 1 yields the initial Alfvén time τ_R= 2 minutes. The relatively short S_f causes the main acceleration to cease rapidly, in the height range ΔZ sime S_f (Section 2.4). Estimating the average speed to be ~600 km s⁻¹ during the decreasing phase, we conclude that the main acceleration phase should cease in about 3 minutes after reaching the peak value. This is reflected in the calculated acceleration profile and is consistent with the observed data. After the completion of the main acceleration phase, the calculated acceleration exhibits a small amplitude normal-mode oscillation in the major radial speed. The oscillation period is ~τ_R, which is about 3–4 minutes at first and increases as the flux rope expands.

3.4.4. Event D: 1998 June 11

The CME was observed off the solar northeast limb by EIT and LASCO C1, C2, and C3 and has been previously described (Raymond et al. 2000; Zhang et al. 2004). In C1, C2, and C3 images, the CME exhibits the morphology of a flux-rope CME whose axis is oblique with respect to the local north–south direction (Chen et al. 2000; Thernisien et al. 2006). EIT images show that a post-eruption arcade appeared from behind the limb shortly after the eruption, with the full arcade appearing on the limb approximately 12 hr after the onset. We take the source region to be half a day behind the northeast limb at phgr = 97°E and θ = 20°N. Subsequent images show that the arcade appeared to be more aligned north–south than east–west, also suggesting an obliquely oriented flux rope. The slowly rising LE of the CME can be seen in EIT as early as 00:00 UT on 1998 June 11, but the CME onset was shortly before 10:00 UT. We tracked the LE through the EIT and LASCO FOVs. Figure 7 shows the CME LE dynamics in the same format as before. The GOES X-ray light curve is given in Figure 7(d) and its time derivative in Figure 7(c).

The initial flux rope for this event was not observed, but based on the limb EIT images prior to the eruption and the estimated source location, we estimate Z₀ = 8 × 10⁴ km. Using the post-eruption arcade and the neutral line length in an active region at ~N20° visible in the MDI magnetograms obtained on 1998 June 16–18 as a proxy for S_f, we infer S_f = 3 × 10⁵ km. Since the initial structure was not observed, we do not regard this estimate as an adequate constraint and use it only for a consistency check. Accordingly, we have treated all parameters as adjustable. The best-fit solution (solution 1) has $\cal D =$ 0.87 and is shown in Figure 7. Its output quantities are given in Table 4. It is satisfying that the best-fit solution yields S_f = 3 × 10⁵ km, with α₀= 2.0, and B_c0 = −2.5 G. Nevertheless, we regard the seemingly "exact" agreement with the estimated S_f only as "consistent." We have obtained the best-fit solution for S_f = 2.7 × 10⁵ km, designated as solution 2. With $\cal D=$ 0.89, it is comparable to solution 1. They are virtually indistinguishable in height (Figure 7(a)) but are perceptibly different in speed and acceleration, exhibiting a first-order normal-mode oscillatory component for the slightly shorter length and greater curvature of the initial flux rope of solution 2. The initial oscillation period is τ_R= 7 minutes, which increases as the flux rope expands.

The two solutions yield nearly identical flux injection profiles (Figure 7(d)), both having ΔT_p= 53 minutes (Table 5). At ΔT_SXR= 75 minutes, the flare is significantly longer, which is mostly due to the secondary peak at about 1240 UT, indicating the presence of additional X-ray brightening. Examination of the EIT images shows that there was no significant EUV brightening elsewhere on the disk during this period. However, the lower part of the post-eruption arcade either expanded or rotated into view around 1100 UT. If the secondary peak is due to the additional X-ray emissions previously occulted by the limb coming into view, the calculated duration ΔT_p= 53 minutes is consistent with the actual X-ray emission profile of the source region had it not been occulted by the limb.

For this event, all the solutions require a small (dΦ_p/dt)₀≠ 0. This is due to a period (~30 minutes) of slow rise prior to 1000 UT, most evident in the speed data. This is in contrast to the preceding three CME–flare events, all of which require (dΦ_p/dt)₀= 0 for best fit. Note that it is not the value of speed per se but the functional form of increase that matters. For (dΦ_p/dt)₀= 0 (Φ_p = Q₀ until t = t₁), the initial rise is nearly exponential (Chen 1989, 1996). If (dΦ_p/dt)₀≠ 0, the initial dynamics given by Equation (4) is governed by Φ_p(t) = Q₀t, introducing a factor of Φ²_p(t) ∝ t² in the multiplicative factor in Equation (4) and significantly altering the dependence of the right-hand side on t and therefore the form of the solution Z(t) during the initial acceleration. Evidently, the EFR equations can discern the difference in the form of Z_data(t) and demand a zero or nonzero value for Q₀. The empirically defined initiation phase (Zhang et al. 2004) may correspond to the case where (dΦ_p/dt)₀≠ 0 prior to the actual eruption. (In this particular limb event, however, the initial rise may be partly due to the solar rotation. Either way, the EFR equations evidently can discern the different functional form of Z_data.)

3.4.5. Event E: 1999 August 17

This CME was observed by the EIT/LASCO C2 and C3 telescopes on 1999 August 17, first appearing off the east limb in the C2 FOV at 15:30 UT and was accompanied by a long-duration flare in GOES X-ray data. Lacking C1 data, the main acceleration phase of this CME is not resolved. The EIT data, however, show the associated prominence eruption, yielding sufficient information on the approximate onset time and the initial flux-rope geometry for comparison with the theoretical results. We include this event in the present study for the long duration of the associated flare. The EIT images show that an arch, the EUV counterpart of a filament, became visible at 12:48 UT, extending from a footpoint at about longitude E33° and latitude N13° northward to a footpoint at longitude E43° and latitude N48°. The source coordinates are taken to be at (E38°, N31°). We tracked the position of the CME LE. The C2 (solid circles) and C3 (open circles) data as well as the GOES SXR data are shown in Figure 8 in the same format as before.

A brightening occurred at 12:36 UT along the leg of this prominence structure. At 13:26 UT, relatively rapid changes became visible near the southern footpoint, and the prominence lifted off at 14:48 UT. We identify the prominence with the trailing edge of the underlying flux-rope structure. Using these measurements, we estimate the prominence legs to be approximately S_p = 4.5 × 10⁵ km apart. The inferred separation distance between the flux-rope footpoints is S_f = 6 × 10⁵ km with a ±25% uncertainty. The initial estimated height of the centroid of the flux-rope apex is Z₀ = 2.2 × 10⁵ km above the photosphere.

Using the above observational estimates of S_f and Z₀, solution 0 is obtained, which has $\cal D =$ 0.72. The overall best fit is solution 1 in Figure 8 (solid black curves). It has $\cal D =$ 0.48. Tables 4 and 5 give the calculated output quantities. For solution 1, we made no a priori assumption regarding the initial flux rope and demanded only that $\cal D$ be minimized, yielding S_f = 5.1 × 10⁵ km and Z₀ = 2.0 × 10⁵ km in good agreement with the estimated values. We have also found that χ_n in the range of 1–6 leads to comparably good fits for the same initial structure, with χ_n= 3 yielding the best fit.

We note that the height data do not capture the main acceleration phase where the length scale S_f is most prominently manifested. Nevertheless, the residual acceleration is still determined by S_f as shown by Equation (13). We have examined solutions with S_f in the range (2.5–10.0) ×10⁵ km, holding all other initial quantities unchanged and adjusting dΦ_p/dt to minimize $\cal D$ . Tables 4 and 5 tabulate the output properties of the solutions. Of these, solutions 0–4 with S_f~ (2.5–6)×10⁵ km have comparable values of $\cal D$ . In contrast, solutions 5 and 6 show that greater values of S_f lead to increasing deviations from the height data. In Figure 8, we have plotted solutions 1, 5, and 6. The three solutions differ only slightly in Z(t). The speed and acceleration curves, however, are noticeably different, with solution 5 having a significant first-order oscillatory component.

Figure 8(d) shows the temporal profile of the predicted dΦ_p(t)/dt (solid curve) compared with that of the GOES X-ray light curve (dotted). Before 14:00 UT, there was already a post-eruption arcade to the west of the CME under study, indicating that there was an earlier eruption. We interpret the X-ray emission during this period to comprise contributions from both eruptions. We attribute, by fiat, 80% of the X-ray intensity at 1400 UT to the earlier eruption and subtracted this value from the X-ray data and use the result as the "corrected" X-ray intensity attributable to the eruption of the CME under study. This is represented by the dashed curve and has ΔT_SXR= 168 minutes. Overall, solution 1 yields a dΦ_p/dt that leads the X-ray flare light curve by about 20 minutes and has ΔT_p= 139 minutes. The difference is consistent with the presence of the secondary peak in the X-ray data.

It is noteworthy that the lack of height data during the maximum acceleration period does not prevent the EFR equations from extracting the correct S_f and ΔT_p. The reason is given in Section 2.4 and parallels the discussion given in connection with events A and B (Section 3.4.2). This also supports the hypothesis that the flare energy release is physically determined by dΦ_p(t)/dt that can be derived from the height data alone.

4. COMPARISON WITH OTHER MODELS

4.1. The Electromotive Force and Flares

Physically, the present findings suggest a hypothesis that the EMF due to poloidal flux injection provides a mechanism to accelerate particles responsible for the X-ray emissions. The EMF and electric field ( $\hat{E}$ ) are calculated by constraining the solution with height data alone. The procedure includes no information on X-rays. Here, we compare these results with those based on the hypothesis that flare energization is due to the reconnection electric field in the corona. Using the apparent motion of Hα kernels of flares, Jing et al. (2005) inferred fields of 0.2–5 V cm⁻¹ and reconnection rates of (0.5–6) × 10¹⁸ Mx s⁻¹. Qiu et al. (2007) inferred reconnected flux of 10²⁰–10²² Mx for several CMEs. Using a 2.5D simulation with current-dependent resistivity, Cheng et al. (2003) modeled an X-class flare and obtained a reconnection field of ~10 V cm⁻¹.

For the five events studies here, the characteristic electric field $\hat{E}(t)$ has $\hat{E}_{\rm max} =$ 2–8 V cm⁻¹ (Tables 3 and 5). These values are similar to the above values based on the reconnection interpretation. In the present work, we have focused on CMEs with associated flares. The EFR model also admits eruptive solutions with no flux injection, dΦ_p(t)/dt= 0. These scenarios can produce eruptive behaviors with varying degrees of reality and have been discussed by Krall et al. (2000). The arcade-to-flux-rope scenario generally requires reconnection on the timescale of eruption so that X-ray emission at some level must accompany such events.

It is interesting that the calculated main acceleration phase is correlated with the observed SXR rise phase for the five events in this study. The SXR rise phase in turn is correlated with the hard X-ray emissions in flares (Neupert 1968). This correlation is distinct from the prediction of the EFR model, which does not contain the hard X-rays. We conjecture that hard X-rays are produced by intense transient electric fields generated in response to the rapid changes resulting from initiation of the injection of poloidal flux and acceleration of the flux rope. Because the EFR model contains no particle acceleration or X-ray emission mechanisms, we are not able to go beyond this conjecture.

4.2. Driving Forces in CME Models

The expression in the square brackets in Equation (3) was originally derived by Shafranov (1966) for axisymmetric toroidal equilibria. This work was applied to coronal loop equilibria (Xue & Chen 1983) and extended to the dynamics of nonaxisymmetric solar flux ropes with quasi-stationary footpoints (Chen 1989). The fundamental assumption is that the force is governed by the local curvature, which was rigorously demonstrated by Garren & Chen (1994) for R/a 1. Essential in this extension are the length S_f, the constraint (6), nonuniform a(θ), and momentum (drag) coupling to the ambient medium. Equation (5) is coupled to Equation (4) and is necessary to correctly describe CME dynamics. During the main acceleration phase, R(t)/a(t) can vary significantly with time, but in the residual phase, R(t)/a(t) varies only slowly. The nonuniform a(θ) with a_a a_f, which leads to the important scaling L ~ Rln(8R/a_f) rather than Rln(8R/a), was incorporated into the model (Chen & Garren 1993); the scaling L ~ Rln(8R/a_f) is necessary for quantitatively matching the CME trajectories near the Sun as well as in interplanetary space. The inclusion of prominence mass was found to be important for the ability to account for the full range of observed magnetic clouds (Chen 1996). Equation (3) is general, and the profile of J_t(r) only affects ξ_i.

The flux-rope inductance L ~ Rln(8R/a_f) arises from the minor radius increasing from the footpoint (a_f) to the apex (a_a a_f) as the flux rope expands. It is not sensitive to the precise form of a(θ): it can be analytically shown (Chen & Krall 2003) that the difference in L(R, a) between a linear and exponential dependence of a(θ) is an additive term of the order to 20%–30% and that the scaling dependence on R and a is not altered. More recently, Žic et al. (2007) numerically extended the calculation of L to thick (R/a < 2) axisymmetric toroidal flux ropes. They found that the inductance approximately scales as L ~ Rln(8R/a) as in a slender torus (their Figure 5(a)) but may be reduced by a factor of nearly 2 at R/a sime 1.8. Self-similar expansion R/a = const is assumed so that L ~ R. In this axisymmetric result, the scale a_f does not exist, and the overall scaling differs from the nonaxisymmetric inductance L ~ Rln(8R/a_f).

Some early CME models assumed that the bright rim of a CME (our LE) was a thin flux rope and calculated the force acting on such a system (Mouschovias & Poland 1978; Anzer 1978). The former work derived a force equation based on the difference between the field on the inward side and that on the outward side of the curved flux rope. This heuristic derivation, which has also been used more recently by Vršnak (2008), does not recover the logarithmic term Rln(8R/a) characteristic of the Lorentz hoop force. Anzer (1978) used the well-known force for a metallic current loop with no minor radial dynamics. Shafranov's (1966) derivation provides the full treatment of the Lorentz and pressure forces acting on a plasma system. These differences have been previously discussed (Chen 1989).

Recently, the major radial force, Equation (3), has been used with particular forms of J_t(r) in a number of axisymmetric flux-rope models (e.g., Lin et al. 1998; Titov & Démoulin 1999; Kliem & Török 2006) and a partial torus with fixed footpoints and uniform minor radius (Isenberg & Forbes 2007). The "torus instability" (Kliem & Török 2006) is an axisymmetric expansion under Equation (4) with β_p = 0 and R/a= const and parameterized by the function B_c(R) ∝ R⁻ⁿ. As such, it is a particular and limiting case of the more general EFR Equation (3) in which these assumptions are not made. In the EFR model, B_c(Z) is a free function, and the choice of Kliem & Török (2006) is a valid possibility within the EFR theory. Axisymmetric models, however, lack the important length S_f and minor radius a_f distinct from a_a. This means that while the equations for the major radius have similar appearances to the expression in the square brackets of Equation (3), the dynamics of an axisymmetric flux rope are governed by different physical scales and constraints, and the solutions, therefore, are necessarily different from those of the nonaxisymmetric equations.⁴ Nevertheless, axisymmetric models (e.g., Titov & Démoulin 1999; Kliem & Török 2006) are sometimes invoked as the theoretical basis of numerical simulation of flux-rope dynamics subject to stationary footpoints (e.g., Roussev et al. 2003; Török & Kliem 2007). These simulation models solve the flux-rope equations of motion containing characteristic length and timescales that are absent in the axisymmetric models. Thus, the simulated flux-rope dynamics are more correctly described by the nonaxisymmetric equations of Chen (1996) subject to quasi-stationary footpoints. The lack of the length S_f was recognized as problematic by Kliem & Török (2006), who imposed I_t= const to mimic the effect of S_f. This, combined with the flux conservation assumption, leads to mathematical inconsistencies (Chen 2007).

Isenberg & Forbes (2007) derive the major radial force equation imposing the ideal MHD line-tying condition on the flux-rope footpoints. For this purpose, they use an image current below the photosphere, which also yields B_z= const in the photosphere in a dynamic situation. They note (see their Section 3.3) that their derivation is mathematically equivalent to that of Garren & Chen (1994). The Garren–Chen derivation, which uses the principle of virtual work for arbitrary and nonuniform R/a, does not require an image current. The EFR formulation treats Φ_t, not B_z in the footpoints, as a specifiable quantity (Φ_t= const). The Garren–Chen results show that subphotospheric current is unimportant unless it is near the surface. Another difference is that Isenberg & Forbes (2007) use a particular form of J_t(r) whereas Garren & Chen (1994) do not, with the specific profile of J_t accounted for by the internal inductance ξ_i. Although the equations are similar in their basic structure, the requirement of an image current and the boundary property B_z= const versus Φ_t= const may lead to different behaviors in the dynamics of flux ropes. The effects of these differences are currently unknown.

5. PHYSICAL INTERPRETATIONS OF POLOIDAL FLUX INJECTION

The model output of main interest, dΦ_p(t)/dt, is a mathematical function that admits two distinct physical interpretations (e.g., Chen 1990; Chen & Krall 2003): one derives the poloidal flux from the existing coronal field and the other from a subphotospheric source. The EFR model describes the flux-rope dynamics in response to the injection of Φ_p but is independent of the specific mechanisms that may produce dΦ_p/dt ≠ 0.

The former interpretation, which is the prevailing paradigm, is modeled as an arcade that evolves into a flux rope via magnetic reconnection. Several MHD simulation models provide numerical realizations of CME initiation starting with arcades (see the references in Section 1). In the mathematical representation of these models, reconnection determines dΦ_p(t)/dt, the rate at which reconnection converts the coronal field into the flux-rope field and increases Φ_p. For example, the recent MHD simulation study of the breakout CME model shows such a process resulting in a constant Φ_t and an increasing Φ_p(t) (Lynch et al. 2009, Figure 8), both of which are in agreement with the predictions of the theoretical EFR model. Vršnak (1990) proposed a semianalytic flux-rope model allowing dΦ_p(t)/dt ≠ 0 (in our notation) to represent reconnection during the eruption. In these numerical models, to the extent a flux rope is formed with an increasing Φ_p(t), the EFR equations should be applicable to the simulated flux-rope dynamics.

Observationally, coronal magnetic fields cannot be measured. As a result, magnetic reconnection in the corona has not been directly observed, although phenomenological evidence/interpretation supporting reconnection has been discussed (e.g., Shibata 1999). Theoretically, reconnection physics on the scale of solar eruptions is not fully understood (e.g., Birn et al. 2001). Currently, reconnection realized in MHD simulations is governed by numerical diffusion dependent on the simulation grids or ad hoc current-dependent resistivity.

In the latter interpretation (Chen 1989; Chen & Garren 1993), the form of dΦ_p(t)/dt is implicitly assumed to be determined by the subphotospheric plasma dynamics, and the poloidal energy U_p increases according to Equation (2). This process does not affect the toroidal flux, which is the axial (vertical) component in the photosphere, and Φ_t (not B_t) is assumed to be constant. The EFR model describes the motion of the flux rope in response to this increase. This concept does not require nonideal MHD dissipation and is favorable if the physical coronal dissipation—classical or nonclassical—does not provide the required reconnection in the arcade paradigm.

In terms of observations, the poloidal field of the legs at the base of the corona is predicted to increase by 25%–50% for a few tens of minutes, decreasing thereafter (Chen 1996). This is suggestive of the apparent increase in "twist" observed in the legs of EPs, also decreasing after a few tens of minutes (Engvold et al. 1976; Vršnak et al. 1993). These observations are possible evidence of poloidal flux injection but typically are not interpreted as such and have not been independently studied.

In the photosphere, no realistic calculations of possible signatures of flux injection exist. Nevertheless, it is possible to deduce some basic properties of potential observables. In the EFR model, the poloidal magnetic field is implicitly assumed to be generated by the solar dynamo deep down in the convection zone. As the poloidal field rises, it becomes increasingly incoherent (Chen 2001). This is represented in Figure 1 by the poloidal field lines that become wavier as they rise. The poloidal field is weaker than the toroidal (locally vertical) field in the photosphere and is characterized by β>1. In addition, the observed magnetic field is a superposition of contributions from all sources of magnetic field, including any horizontal component of the toroidal field. It is difficult, if not impossible, to observationally identify injected B_p of the specific flux rope. To calculate observational photospheric signatures of the subphotospheric flux injection process, it is necessary to account for the incoherence in the poloidal field in space and time resulting from the subphotospheric plasma and magnetic field dynamics, and the rapid change in the opacity in any quantitative predictions (also pointed out in Chen & Krall 2003). The spatial and temporal scales of incoherence in the emerging poloidal field are not quantitatively understood and may not be fully resolved in the current measurements.

One aspect of the source of the poloidal flux requires clarification. The process of poloidal flux injection in the EFR model has been mischaracterized as a "photospheric dynamo" that requires fast and unobserved photospheric flows to drive eruptions (Forbes 2000). The dynamo implicit in the EFR concept is the solar dynamo. The dynamo-generated energy ultimately reaches the photosphere and the corona, both of which respond at the local characteristic (i.e., magnetosonic) time and velocity. In this concept, the photosphere does not drive coronal dynamics, and an "eruption" is an Alfvénic relaxation process in the corona in response to the increasing Φ_p (Chen 2001); the photosphere is an optical boundary that responds to energy transport from below, as does the solar atmosphere above it. In the arcade models, the ultimate source of poloidal flux is the horizontal photospheric fluid motion, a quasi-static photospheric dynamo process.

An open question is the actual timescale of magnetic energy transport through the photosphere ("flux emergence"). The FWHM duration of the predicted dΦ_p(t)/dt is approximately 1–3 hr for the events in this study. This is the timescale of the poloidal flux injection into a given flux rope through the photosphere and is comparable to that of the appearance of loop structures in observed emerging flux events. For example, an event observed by the Transition Region and Coronal Explorer in 171 Å on 1998 June 8 showed the first indications of emergence before 0200–0230 UT and a well-established rising system of hot and cool coronal loops by 0330–0400 UT (Schrijver et al. 1999, Section 10). The region continued to evolve with new emerging loops for approximately 2 days as is typical of such events, but assuming that the loops are magnetically organized, this observation indicates that a significant amount of magnetic energy associated with individual loops emerged on a timescale of approximately 1 hr or possibly shorter. This has already been pointed out (Chen 2001).

While the two distinct paradigms have a common mathematical description in dΦ_p/dt, there are potentially observable differences. One is that as a flux rope develops via reconnection, new flux surfaces and mass are added to the flux rope so that its minor radius a(t) increases. This is in addition to the dynamics given by Equation (5), which assumes that all flux surfaces are topologically preserved. Another difference is that in the arcade-to-flux-rope transformation scenario, footpoints of the newly created flux rope may evolve during the eruption. In contrast, if the pre-eruptive structure is a flux rope, the main acceleration is governed by the S_f-scaling law as previously reported. The influence of the arcade-to-flux-rope transformation in arcade models on the S_f-scaling observed in CME acceleration is not known. The relative importance of the two scenarios in CMEs and flares is currently not understood.

6. SUMMARY AND CONCLUSION

We have studied the observed trajectories of five CMEs and the SXR (1–8 Å) light curves of associated flares. For each event, we have used the height data as the sole observational input to the EFR model and obtained the best-fit solution by minimizing $\cal D$ , defined by Equation (16). The best-fit solution yields S_f and dΦ_p(t)/dt as the main predictions of the EFR model for comparison with data. It is suggested that the EMF, i.e., ${\cal E}(t) = -(1/c)d\Phi _p(t)/dt$ , accelerates particles responsible for the flare SXR emissions, thereby producing a light curve with a comparable duration to that of dΦ_p(t)/dt. An extensive sensitivity test has been carried out, quantifying the behavior of the model in the parameter space near the best-fit EFR solutions (Tables 2–8). The main findings of the study are as follows.

1.
For each CME trajectory, the best-fit EFR solution agrees with the height data to within 1%–2% throughout the FOV of up to 100 R_☉.
2.
For each CME–flare event, the predicted dΦ_p(t)/dt has an FWHM duration ΔT_p comparable to that of the associated X-ray light curve, ΔT_SXR. Thus, ΔT_SXR scales with ΔT_p (i.e., ΔT_SXR ~ ΔT_p) regardless of the flare duration. The temporal form of dΦ_p(t)/dt closely coincides with that of the associated GOES SXR light curves.
3.
The value of ΔT_p calculated by minimizing $\cal D$ is strongly constrained by the long-time CME trajectory (the residual acceleration phase).
4.
The predicted S_f is consistent with the observational estimates of the candidate initial flux rope where identifiable. The best-fit S_f for a given trajectory is determined by the observed height range ΔZ of maximum curvature in Z(t).
5.
The FWHM duration of the main acceleration phase, ΔT_p, depends on the intrinsic flux-rope quantities τ_R and S_f and is insensitive to the temporal profile of dΦ_p(t)/dt. The value of B_c(Z₀), in conjunction with the initial geometry and force-balance conditions, determines B_p0 and $\overline{B}_{t0}$ and hence τ_R, the acceleration timescale. The declining phase of the main acceleration is governed by S_f.

It is significant that neither the observational input to the EFR model, i.e., the observed CME trajectories, nor the equations of motion contain any information regarding X-ray data. Thus, the good temporal correlation between the best-fit dΦ_p(t)/dt and the X-ray light curve (excepting the flare decay phase) is physically significant. We conclude that the function dΦ_p/dt provides a physical connection between CMEs and associated flares. The good temporal correlation between dΦ_p(t)/dt and the X-ray light curves, in particular, ΔT_SXR ~ ΔT_p, implies that the temporal form of the SXR light curve can be inferred from the scales S_f and τ_R manifested in the height–time data. In other words, scaled to S_f, the main acceleration profiles of CMEs as functions of Z have an invariant form, and scaled to the calculated ΔT_p, the observed flare SXR light curves also have an invariant form from event to event. We hypothesize that it is the EMF given by Equation (10) that accelerates particles to produce the X-ray emissions. Physically, the source of dΦ_p/dt ≠ 0 may be of subphotospheric or coronal origin (Section 5), but our conclusions are independent of the specific mechanisms, which are outside the EFR model.

We thank the anonymous referee for careful readings of the paper and insightful comments, which led to significant improvement of the paper. We acknowledge valuable discussions with R. A. Howard and J. Zhang. Our gratitude goes to N. Rich and L. Simpson for their assistance in analyzing LASCO and STEREO data. The STEREO/SECCHI data used in this work are produced by an international consortium of the NRL (USA), LMSAL (USA), NASA/GSFC (USA), RAL (UK), UB-HAM (UK), MPS (Germany), CSL (Belgium), IOTA (France) and IAS (France). LASCO and EIT are a project of international cooperation between ESA and HASA. This work was supported by the Office of Naval Research.

APPENDIX

In this appendix, we provide more detailed information concerning the solutions discussed in the text for completeness. Specifically, Table 6 shows the time constants of dΦ_p(t)/dt, Equation (9), required by each of the solutions for the 2000 September 12 CME–flare event (Table 2). These constants are obtained by adjusting them until $\cal D$ is minimized for each set of initial values. For solutions 5 and 6, the values of τ₁ are shown in boldface, indicating that they are prescribed. Other values are adjusted to obtain the best-fit solutions. In Tables 7 and 8, we present the results of a more extensive parameter study for the same event. Solutions 1, 5, and 6 are tabulated in Table 2 and are used for reference. The other solutions are obtained by varying the parameters as indicated (boldface). All solutions have S_f = 4.25 × 10⁵ km.

Table 6. Time Constants^a of Best-fit dΦ_p(t)/dt: Event A

No.	t₁	t₂	t₃	τ₁	τ₂	Q₁
0	55.3	113.1	113.1	20.5	81.6	5.21
1	48.8	111.5	111.5	20.9	83.7	5.95
2	48.8	111.5	111.5	20.9	83.7	6.60
3	46.0	118.0	118.0	22.0	41.7	8.60
4	48.8	111.7	111.7	20.9	82.6	3.57
5	47.2	104.2	104.2	10.0	78.6	3.20
6	49.8	125.0	125.0	40.0	76.8	11.59
7	55.4	112.7	112.7	20.5	81.8	5.45
8	48.8	111.7	111.7	20.9	83.9	5.67
9	48.2	111.6	111.6	20.9	83.7	6.24
10	48.8	110.0	110.0	20.9	83.4	6.84
11	49.4	110.2	110.2	20.9	82.6	8.08
12	55.3	112.6	112.6	20.5	81.5	8.08

Notes. ^aAll time constants are in units of minutes. Q₀ = 0 for all solutions. An arbitrary time offset of 9.92 hr has been applied to all solutions. This offset shifts t₁, t₂, and t₃ but does not affect the solution (because Q₀ = 0) or the values of τ₁ and τ₂. The best-fit solutions consistently require t₂ = t₃ for this event.

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Table 7. Best-fit Initial Flux-rope Parameters: Event A

No.	$\cal D$	α₀	χ_p	χ_n	B_c0	M_T	B_pa0	B_t0	Specified
					(G)	(10¹⁶ g)	(G)	(G)	Quantity
1	0.66	2.0	1.0	1.0	−1.0	1.56	2.46	2.46	⋅⋅⋅
1.1	0.90	2.0	1.0	1.0	−1.0	1.56	2.46	2.46	c_d= 3
1.2	0.89	2.0	0.5	1.0	−1.0	0.78	2.44	2.72	⋅⋅⋅
1.3	0.67	2.0	1.0	2.0	−1.0	2.34	2.73	2.73	⋅⋅⋅
1.4	0.78	2.0	0.5	2.0	−1.0	1.17	2.58	2.84	⋅⋅⋅
1.5	0.75	2.0	1.0	1.0	−0.75	1.56	2.0	2.0	⋅⋅⋅
1.6	0.68	2.0	1.0	1.0	−1.25	1.56	2.94	2.94	⋅⋅⋅
1.7	0.80	2.0	0.5	2.0	−1.25	1.17	3.05	3.27	⋅⋅⋅
1.8	0.71	2.0	1.0	1.0	−1.5	1.56	3.44	3.44	⋅⋅⋅
1.9	0.82	2.5	1.0	1.0	−1.0	1.00	2.66	2.66	⋅⋅⋅
1.10	0.94	1.5	1.0	1.0	−1.0	2.77	2.31	2.31	⋅⋅⋅
1.11	0.70	2.0	1.0	1.0	−1.0	1.56	2.46	2.46	V*_sw= 600
1.12	0.69	2.0	1.0	1.0	−1.0	1.56	2.66	2.66	V*_sw= 300
5	0.74	2.5	1.0	1.0	−1.0	1.56	2.6	2.6	τ₁= 10
5.1	0.89	2.5	1.0	1.0	−1.0	1.56	2.6	2.6	τ₁= 10, c_d= 3
6	1.21	2.5	1.0	1.0	−1.0	1.54	3.0	2.9	τ₁= 40
6.1	1.47	2.5	1.0	1.0	−1.0	1.54	3.0	2.9	τ₁= 40, c_d= 3

Notes. This table is shown in the same format as Table 2. V*_sw= 400 km s⁻¹ and c_d = 1 except as noted. S_f = 4.25 × 10⁵ km for all solutions. Initial values specified differently from solution 1 are shown in boldface.

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Table 8. Poloidal Flux Injection and Electromotive Force: Event A

No.	ΔT_p	(dΦ_p/dt)_max	$\hat{E}_{\rm max}$	ΔΦ_p	ΔU_p	Accel	Osc
	(minutes)	(10¹⁹ Mx s⁻¹)	(V cm⁻¹)	(10²² Mx)	(10³² erg)	(m s⁻²)
ΔT_SXR= 105 minutes
1	95	1.1	2.56	6.8	5.3	573	N
1.1	94	1.5	3.46	9.3	8.6	555	Y
1.2	94	1.1	2.55	6.8	5.2	561	Y
1.3	94	1.2	2.71	7.2	6.1	528	N
1.4	97	1.1	2.59	7.1	5.6	601	N
1.5	96	1.0	2.40	6.4	4.4	530	N
1.6	95	1.1	2.56	7.2	6.2	647	N
1.7	99	1.2	2.71	7.5	6.4	632	N
1.8	95	1.2	2.86	7.6	7.3	709	N
1.9	91	1.0	2.28	6.1	4.1	606	N
1.10	91	1.3	2.77	8.0	7.7	533	N
1.11	88	1.1	2.54	6.4	5.1	564	N
1.12	96	1.1	2.59	7.0	5.4	545	N
5	79	1.2	2.90	6.6	5.8	1205	Y
5.1	82	1.7	3.97	9.0	9.5	1735	Y
6	103	1.0	2.44	6.8	5.0	538	Y
6.1	103	1.4	3.26	9.4	8.2	858	Y

Note. This table is shown in the same format as Table 3.

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Taken together with Tables 2 and 3, these tables show the region of the parameter space where the solutions have $\cal D$ values in the neighborhood of the overall minimum value. In this region, the predicted values of S_f and ΔT_p are insensitive to the parameter values. Outside this region, $\cal D$ increases noticeably with increasing deviations in parameter values.

We return to the sensitivity of the solutions to the temporal profile of dΦ_p(t)/dt briefly discussed in connection with Figure 4. Referring to the four solutions obtained by minimizing $\cal D$ , we see that the temporal shape of the main acceleration profile is essentially the same, with τ₁ ranging from 13.6 minutes for solution 1 to 40.0 minutes for solution 3. In particular, the FWHM duration of the main acceleration, ΔT_M, is approximately 6 minutes, with the peak value occurring at about 14:09 UT. Obviously, the equations of motion possess infinitely many unconstrained solutions, most of which do not yield small enough values of $\cal D$ . If we replace the value τ₁ for solution 1 with τ₁= 40 minutes without changing any other quantity, the unconstrained solution has ${\cal D} =$ 25.0, indicating that it does not fit the data at all. Nevertheless, this solution has a main acceleration phase with an FWHM duration of ΔT_M sime 7 minutes. The peak acceleration, however, occurs at approximately 14:30 UT, shifted by about 21 minutes from that of solution 1. This solution has a maximum acceleration of 1989 m s⁻² and (dΦ_p/dt)_max = 0.23 × 10¹⁹ Mx s⁻¹ and ΔU_p = 1.3 × 10³² erg. If τ₁ is increased to 50 minutes holding all other parameters unchanged, the unconstrained solution is oscillatory with a period of about 8 minutes. The net acceleration for this solution is limited in duration and magnitude (~15 m s⁻²). The same exercise can be carried out for other events, and ΔT_M is found to be insensitive to τ₁ to the same extent as for this event.

Physically, the main acceleration phase is determined by τ_R and S_f (Section 2.4). Varying τ₁ only changes the rate at which B_p is changed during the initial stage. As soon as the flux rope is sufficiently out of equilibrium, it accelerates on the timescale of τ_R, and B_p begins to decrease. As a result, τ_R during the main acceleration phase does not significantly vary from the initial equilibrium value and is not sensitive to the ramp-up timescale τ₁. The decay phase of the main acceleration phase is governed by S_f. Thus, ΔT_M is not strongly affected by τ₁. Rather, ΔT_M is more sensitive to B_c0 because B_pa0 ∝ −(R₀/a₀)B_c0 for the initial equilibrium (Chen 1996; Section 2.2). This means that τ_R ∝ a/B_c0 at t = 0.

TEMPORAL AND PHYSICAL CONNECTION BETWEEN CORONAL MASS EJECTIONS AND FLARES

Abstract

1. INTRODUCTION

2. THE ERUPTING FLUX-ROPE (EFR) MODEL

2.1. Forces: Equations of Motion

2.2. Initialization of Model System

2.3. Poloidal Flux Injection

2.4. CME Trajectories: Physical Scales

3. APPLICATION OF THE EFR MODEL TO CME–FLARE EVENTS

3.1. Methodology

3.2. Height Measurement

3.3. Goodness of Fit

3.4. Theory–Data Comparison

3.4.1. Event A: 2000 September 12

3.4.2. Event B: 2008 April 26

3.4.3. Event C: 2008 March 25

3.4.4. Event D: 1998 June 11

3.4.5. Event E: 1999 August 17

4. COMPARISON WITH OTHER MODELS

4.1. The Electromotive Force and Flares

4.2. Driving Forces in CME Models

5. PHYSICAL INTERPRETATIONS OF POLOIDAL FLUX INJECTION

6. SUMMARY AND CONCLUSION

APPENDIX

Footnotes

References

Citations