EVOLUTION OF A CORONAL MASS EJECTION AND ITS MAGNETIC FIELD IN INTERPLANETARY SPACE

Following their first observation by a space-borne coronagraph (Tousey 1973), coronal mass ejections (CMEs) have received extensive scientific attention (e.g., Gosling et al. 1974; Howard et al. 1985; Hundhausen 1999). It is now understood that it is the magnetic field of CME ejecta impinging on Earth's magnetosphere that drives severe disturbances in the geospace plasma environment. Two major issues in solar–terrestrial physics are (1) the nature of the forces responsible for CME acceleration and subsequent propagation through the interplanetary medium and (2) the evolution of the magnetic field of a CME. We address these questions using both a theoretical model and data from the twin Solar Terrestrial Relations Observatory (STEREO) spacecraft.

Theoretical work in conjunction with observations from Large Angle and Spectrometric Coronagraph (LASCO) has shown that CMEs can be explained, both morphologically and quantitatively, as erupting flux ropes (Chen et al. 1997, 2000) as opposed to cone-like shell structures with rotational symmetry (Hundhausen 1999). Among theoretical models, the erupting flux-rope (EFR) model of Chen (1996) has been most extensively tested and validated using LASCO data (e.g., Wood et al. 1999; Krall et al. 2001). In this ideal magnetohydrodynamic (MHD) model, the CME dynamics and the magnetic field evolution are determined by a set of coupled equations representing the toroidal hoop force with momentum coupling to the ambient solar wind (SW) via drag (Chen 1989). Beyond the LASCO field of view (FOV), the dynamics of CMEs observed by the Solar Mass Ejection Imager have been found to be consistent with the continued driving by the hoop force competing with the drag force as predicted by the EFR model (Howard et al. 2007).

It has been conjectured that a CME ejecta is manifested as a magnetic cloud (MC) at 1 AU and beyond (Burlaga et al. 1981). Theoretically, the EFR model shows that a flux rope exhibiting CME-like dynamics evolves into an MC-like structure (Chen 1996; Krall et al. 2006). Numerical simulations of flux-rope CMEs in 2.5 and 3 dimensions also show similar structures at 1 AU (e.g., Wu et al. 1999; Manchester et al. 2004). No model, however, has been directly tested against the observed evolution of CMEs and their magnetic fields due to the absence of observation. The STEREO mission provides for the first time the ability to observe, with spatial and temporal continuity, a CME from the Sun to 1 AU and measure the magnetic field and plasma properties of the ejecta.

In this Letter, we present the first quantitative comparison of theory with the observed dynamics of a CME and the in situ magnetic field of the ejecta at 1 AU. We find that (1) the best-fit solution is in good agreement—within 1% of the data—with the measured CME position throughout the 1 AU STEREO-A FOV and (2) the model CME evolves under ideal MHD into an MC-like structure that is consistent with the in situ magnetometer and plasma data measured by STEREO-B.

The CME was observed on 2007 December 24 off the east limb near the equator (∼N4°) by the Sun–Earth Connection Coronal and Heliospheric Investigation (SECCHI) instruments (Howard et al. 2008) on board STEREO-A. Neither EUVI-A nor EUVI-B images revealed activities identifiable as the source on the disk. In SECCHI-A, the CME exhibited the typical flux-rope morphology with the toroidal axis inclined relative to the ecliptic (e.g., Chen et al. 2000). This is illustrated in Figure 1, with a dot ("Obs") marking STEREO-B. In the FOV of COR2-B, a faint nearly symmetric halo CME was observed. We tracked the leading edge (LE) of the CME through the SECCHI-A FOV (COR1, COR2, HI1, and HI2).

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The HI2-A images show that the LE of the CME reached the projected position of STEREO-B on December 29 at ∼2300 UT, approximately coinciding with the detection of the CME ejecta by IMPACT (Acuña et al. 2008) and PLASTIC (Galvin et al. 2008) on STEREO-B. Combined with the halo in COR2-B, we assume that the apex of the CME propagated along the radial direction at latitude 4°N and longitude 44°E in the FOV of SECCHI-A, the angular separation between the twin spacecraft. The measured elongation is converted to the "true" heliocentric position $\cal R$ using N4° and E44°. The SECCHI-A data for the LE position are plotted in Figure 2 (diamonds and circles).

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Figure 3(a) shows the magnetometer data from IMPACT on STEREO-B in the Radial-Tangential-Normal (RTN) coordinates (Figure 1): B_R (green), B_T (blue), B_N (magenta), and the magnitude B_tot ≡ (B²_R + B²_T + B²_N)^1/2 (red), where +R points from the Sun to STEREO-B and the R–N plane contains the solar rotation axis. The original 125 ms data have been averaged over 3 minutes. The maximum value of B_tot is 12.0 nT. We show in black the rotation angle in the plane of the sky, θ ≡ sin⁻¹(B_N/B) (scaled to fit the plot), where B ≡ (B²_T + B²_N)^1/2. The two vertical lines show the extrema of θ, indicating the beginning and ending of the magnetic field rotation (Burlaga et al. 1981). The SW plasma data from PLASTIC/STEREO-B show that the proton temperature was T_p≃ (3–4) ×10⁴ K between the extrema of θ (the ejecta) and was higher at T_p ≃ 6 × 10⁴ K outside. The SW density was mostly 5–10 cm⁻³ during the period shown but was 2–5 cm⁻³ immediately ahead of the leading θ extremum. We conclude that the ejecta between the two vertical lines was an MC according to the criteria of Burlaga et al. (1981). The speed of the MC was approximately 340 km s⁻¹ at maximum B_tot. The fact that B_tot decreases slowly behind the MC suggests that the trailing edge deviated significantly from a simple flux rope.

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In this work, the measured LE position ${\cal R}_{{\rm data}}(t_i)$ is used as the observational input to the EFR model, and the best-fit solution provides the model output. Here, t_i is the measurement time.

The average deviation of a solution ${\cal R}_{{\rm theory}}(t)$ from the data relative to the error bars is given by $\cal D$ (Chen & Kunkel 2010)

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

where T = ∑_iδt_i, $\Delta {\cal R}_i$ is the error bar at t_i, and δt_i = t_i+1 − t_i. For this event, we use $\Delta {\cal R}_i/{\cal R}_{{\rm data}}(t_i) =$ 0.01 (1% error) for all i, consistent with the typical measurement uncertainties.

To find a solution, the initial flux-rope geometry is specified by the centroid height of the apex Z₀, footpoint separation S_f, and minor radius a₀ of the current channel of the flux rope. We use $Z \equiv {\cal R}-\hbox{\it R}_{\odot }$ to denote the height measured from the photosphere. The model corona is given by pressure p_c(Z) and density n_c(Z), where p_c(Z) = 2n_c(Z)kT_c(Z) and k is Boltzmann's constant. The overlying coronal field perpendicular to the flux rope is B_c(Z). To initialize a flux rope, we specify S_f, Z₀, and B_c0 = B_c(Z₀) with p_c0 = p_c(Z₀) and n_c0 = n_c(Z₀) given by the model corona. The requirement that the initial flux rope be in equilibrium—d²Z/dt² = 0 and d²a/dt² = 0—then yields the initial magnetic field, B_pa0 and $\bar{B}_{t0}$ at Z₀, where B_pa ≡ B_p(r = a) and $\bar{B}_t$ refer to the poloidal and average toroidal components of the flux-rope magnetic field. The EFR model evolves B_pa(t) and $\overline{B}_t(t)$ according to B_pa(t) = 2I_t/ca(t) and $\overline{B}_t(t) =\overline{B}_{t0}a^2_0/a^2(t)$.

The momentum of the flux rope is coupled via drag to the model ambient medium. The drag coefficient is taken to be c_d= 1, but c_d in the range of 1–3 yields comparable solutions. The SW speed is smoothly increased to its asymptotic value V*_sw ≡ V_c(Z → ∞) at about 25 $\hbox{\it R}_{\odot }$ with V*_sw treated as an adjustable parameter. The coronal and SW speed, density, and temperature—V_c(Z), n_c(Z), and T_c(Z)—are specified according to empirical models. The above corona-SW model has not been altered from the original formulation (Chen 1996). The drag force term, however, has been improved (Chen & Kunkel 2010, Section 2.1)

The initial flux rope is set into motion by increasing the poloidal flux Φ_p according to dΦ_p(t)/dt. For each set of initial values, this function is adjusted to obtain the solution that minimizes $\cal D$, and the initial values can be varied to obtain a range of solutions. The overall best-fit solution is given by the smallest $\cal D$. The properties of the solution—Z(t) and a(t)—including S_f dΦ_p(t)/dt, and B_pa(t) and $\overline{B}_t(t)$ are the model predictions constrained by ${\cal R}_{{\rm data}}(t_i)$.

In relating the solution to observation, we make the identification that the outermost flux surface of the flux rope at ${\cal R}_{{\rm theory}} = {\cal R}(t) + 2a(t)$ is the CME LE position (Chen et al. 1997; Krall et al. 2001). Table 1 gives several representative minimum-$\cal D$ solutions and their predicted physical properties. Solution 1 provides the overall best fit and is shown in Figure 2(a) (solid curve), with the initial segment shown in Figures 2(b)–(d) (solid curves). Panels (b) and (c) show that there is a small-amplitude first-order oscillation. The acceleration of the LE and the predicted dΦ_p(t)/dt (dashed curve) are shown in panel (d). Two representative error bars in speed (1% in position) are shown. The acceleration after the main acceleration phase is due to the increasing ambient SW speed imparting momentum to the flux rope via drag. This solution has $\cal D =$ 0.68, i.e., the average deviation from the data is 0.68% in the 1 AU FOV. This demonstrates that the EFR equations correctly capture the physics of the macroscopic forces acting on CMEs.

Table 1. Output Quantities of Minimum-$\cal D$ Solutions^a

No.	${\cal D}$	Sf	α0	MTb	${\overline{B}_{t0}}$b	ΔTp	(dΦp/dt)max	V*sw	B1AUc	$\overline{T}_{1 {\rm AU}}$c
		(105 km)	(G)	(1015 g)	(G)	(minutes)	(1018 Mx s−1)	(km s−1)	(nT)	(104 K)
1d	0.68	1.8	1.8	3.3	2.39	70	1.0	441	11.8	4.30
2	1.10	1.5	2.0	1.7	2.41	66	0.9	418	11.9	4.25
3	1.28	2.5	2.0	7.3	2.60	68	1.4	486	12.0	4.38
4	0.80	1.8	1.6	4.2	2.32	70	1.1	442	11.8	4.31
5	0.79	1.8	2.0	2.7	2.46	68	1.0	441	11.9	4.28
6e	0.78	1.8	2.0	3.3	2.63	64	1.1	441	12.3	4.26
7f	0.72	1.8	1.8	3.3	3.31	72	1.1	440	13.6	4.16
8g	0.71	1.8	1.8	3.3	2.39	70	1.0	442	12.6	6.52

Notes. ^aFor each solution, S_f and α₀ are specified initial values. V*_sw and dΦ_p(t)/dt are varied to minimize $\cal D$. Z₀ = 8 × 10⁴ km is estimated from the data. $\chi _p \equiv \bar{p}_0/p_{c0} = 1$, $\chi _n \equiv \bar{n}_{p0}/\bar{n}_0 =$ 0.6, B_c0=−1 G, and c_d= 1 are used for all solutions except as noted for solutions 6 and 7. The equality $B_{pa0} = \overline{B}_{t0}$ results from χ_p = 1. Here, subscript "0" refers to initial quantities. ^bCalculated using the equilibrium force-balance conditions; see Chen (1996). ^cB_1AU is the field strength on the flux-rope axis. $\overline{T}_{1 {\rm AU}}$ is the average temperature in the flux rope. T_sw ≃ 6.8 × 10⁴ K in the ambient SW. T₀= 1 MK at the initial height Z₀. ^dTotal poloidal energy injected is ΔU_p = 2.0 × 10³¹ erg. Use ΔU_p ∝ (dΦ_p/dt)²_maxΔT_p to scale to other solutions. ^eχ_n = 1.0. ^fB_c0 = −1.5 G. ^gThe model coronal temperature at Z₀ is set to T₀ = 1.5 × 10⁶ K. The initial structure is the same as for solution 1. The calculated 1 AU flux-rope density is $\overline{n} =$ 5.7 cm⁻³, with n_sw = 5.2 cm⁻³.

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We have varied S_f and α₀ ≡ R₀/a₀, where R₀ is the initial major radius calculated from S_f and Z₀ (Chen 1996), and obtained several minimum-$\cal D$ solutions (Table 1). Solutions 2 and 3 bracket S_f = 1.8 × 10⁵ km, showing solution 1 yields a significantly better fit. Solutions 4 and 5 show influence of the aspect ratio α₀. We have also varied two selected ambient coronal parameters (solutions 6 and 7) as indicated in Table 1. The value of V*_sw required to minimize $\cal D$ is in the range of V*_sw= 440 km s⁻¹, except for solutions 2 and 3 with significantly greater values of $\cal D$. Solutions with ${\cal D} =$ 0.6–0.8 are nearly indistinguishable in appearance. For each solution, the predicted dΦ_p(t)/dt is characterized by its peak value and the FWHM duration ΔT_p. For solution 1, the initial flux rope has Φ_p0 = 1.9 × 10²⁰ Mx and Φ_t0 = 4.0 × 10²⁰ Mx, with a total of ΔΦ_p = 4.8 × 10²¹ Mx injected.

The centroid of the flux rope given by solution 1 reaches 1 AU with V≃ 330 km s⁻¹ and a temperature of $\overline{T} \simeq 4.3 \times 10^4$ K while the model SW has T_c ≃ 6.8 × 10⁴ K at 1 AU. The density is ∼5 cm⁻³ for both the flux rope and model SW. These values, obtained as the predictions of the model with no reference to any SW data, are consistent with the PLASTIC data discussed above. Here, we have taken the ambient coronal temperature at the initial apex height (Z₀) to be T₀ = 1 × 10⁶ K, and an adiabatic energy equation is assumed with γ≃ 1.2. Note that the model uses a profile of the form $T_{{c}}(Z) = T_0 ({\cal R}/R_\odot)^{-\alpha }$ (recall ${\cal R} = Z + R_\odot$) so that the SW temperature at 1 AU directly scales with T₀. If we choose T₀ = 1.5 × 10⁶ K for the initial flux rope of solution 1, we obtain $\overline{T} \simeq 6.5 \times 10^4$ K and T_c ≃ 10⁵ K (solution 8).

The best-fit B_pa and $\overline{B}_t$ for this solution (Table 1) can be used to construct a field profile—B_p(r) and B_t(r). We use the non-force-free model of Chen (1996) for this purpose. Figure 3(b) shows the magnitude B_tot (red solid curve) and the rotation angle θ (black solid curve) of the model as would be measured by a stationary observer at 1 AU based on the simplified geometry in Figure 1. In constructing the synthetic field, the expansion of the minor radius a(t) during the observation is accounted for, but no distortion, major radial curvature, or ambient SW magnetic field is included. The individual components of B(t) depend on the flux-rope orientation. Because it is not possible to unambiguously reconstruct the actual CME magnetic field in three dimensions, we use physical properties of the field that can be calculated without a detailed knowledge of the flux-rope orientation: the maximum value of B_tot and the extrema of the rotation angle (θ) of the field. The model field has a maximum value of B_tot= 11.8 nT, which is consistent with 12.0 nT indicated by the data. This is the lower bound because STEREO-B did not necessarily pass through the center of the MC. The temporal profiles of B_tot(t) and θ(t) do depend on the angle between the flux-rope axis and the observer path through the structure. The solid curves in Figure 3(b) assume that the observer intersects the flux rope perpendicularly to the axis of symmetry. If the same field is measured at an intersection angle of 55°, the profiles represented by dashed curves in Figure 3(b) result, which provide a better fit to the IMPACT magnetometer data.

The most notable discrepancy between the MC and the model ejecta is the predicted SW speed V*_sw∼ 440 km s⁻¹. The PLASTIC plasma data show V_sw∼ 350 km s⁻¹ prior to the arrival of the MC. The actual speed of the SW with which the CME interacted along the way, however, is not known. The more detailed dependence of the predicted V*_sw on other parameters will be explored in the future. Nevertheless, noting that V*_sw is predicted by the CME position data only, with no reference to any SW data, we judge the prediction to be remarkably consistent with the estimated SW speed.

We have presented the results of the application of the theoretical EFR model to the observed dynamics of a CME and the evolution of the CME magnetic field. The CME was continuously tracked by SECCHI-A and the magnetic field and the plasma parameters of the ejecta were measured by IMPACT and PLASTIC, respectively, on STEREO-B. All model output quantities are constrained by the observed CME trajectory alone. Our main findings are the following.

• 1.  
  The minimum-$\cal D$ (best-fit) initial-value solution of the ideal MHD EFR model agrees with the measured CME trajectory to within 1% in the 1 AU FOV of STEREO-A.
• 2.  
  The predicted magnetic field B(t) at 1 AU, which is evolved from the initial field given by the initial force balance conditions and constrained only by the CME position data, is consistent with that of the MC detected by the IMPACT magnetometer aboard STEREO-B.
• 3.  
  The temperature and the density of the model MC at 1 AU are consistent with the PLASTIC plasma data.
• 4.  
  The best-fit value of V*_sw is faster than the measured SW speed immediately preceding the observed MC.
• 5.  
  The propagation of the model CME is determined by the superposition of the Lorentz force and drag force.

These results demonstrate that the EFR model provides a good approximation of the net driving force on the CME and represent the first theoretical model defined by a single set of initial-value equations to replicate the dynamics of a CME from the Sun to 1 AU and the measured in situ magnetic field and plasma parameters at 1 AU, an important step toward satisfying the requirement that any proposed CME model be able to produce initial-value solutions to correctly describe both the early-time and late-time dynamics.

We acknowledge beneficial discussions with R. A. Howard. We thank N. Rich and L. Simpson for their gracious assistance with the analysis of the STEREO data. The STEREO/SECCHI data 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). The work of V.K. was supported by NASA and that of J.C. by ONR.