DETERMINING THE AZIMUTHAL PROPERTIES OF CORONAL MASS EJECTIONS FROM MULTI-SPACECRAFT REMOTE-SENSING OBSERVATIONS WITH STEREO SECCHI

CMEs propagating between the STEREO-A and B spacecraft, i.e., CMEs propagating approximatively toward Earth, can be imaged to large elongation angles by the HIs onboard both STEREO spacecraft. Taking into account the difference in spacecraft heliocentric distances (d_A∼ 0.95 AU, d_B∼ 1.05 AU), the fact that the HIs observe the same CME at the same time at different elongation angles can give us information about the CME central position. Such information cannot be deduced using the Point-P approximation, because it assumes a too simplistic CME geometry. Direct triangulation can be done (Liu et al. 2010), but only under the assumption that both STEREO spacecraft observe the exact same plasma element, which is unlikely to be the case for most CMEs at large elongation angles. For some CMEs, direct triangulation might not give realistic results. We suspect that it is the case for the 2008 April 26 CME, because it appears as a halo CME in STEREO-B. The best fit of the Rouillard et al. (2008) method for this CME gives ϕ_A = −335 ± 180 and ϕ_B = 21 ± 65 for STEREO-A and B, respectively; these two results are not consistent with each other. However, one can use the expanding-propagating bubble approximation of Howard & Tappin (2009) and Lugaz et al. (2009a) with two parameters to analyze the data from the two spacecraft simultaneously; for each pair of position angles (for example, 90 and 270), the derived parameters are the diameter of the circular front and its direction of propagation (see the left panel of Figure 1). The main assumptions of this model are that the CME cross section is circular and that the CME is anchored at the Sun.

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Assuming simultaneous observations by STEREO-A and STEREO-B, one can write an expression for the diameter of the sphere for each of the spacecraft following Lugaz et al. (2009a):

$$\begin{equation} R_A = 2 d_A \frac{\sin (\epsilon _A)}{1 + \sin (\beta _A - \phi + \epsilon _A)}, \end{equation} \tag{ 1 }$$

where ϕ is the direction of propagation of the CME away from the Sun–Earth line (defined positive from B to A) and β_A is the angular separation (defined as a positive number) of the spacecraft with the Sun–Earth line. The same relation applies for spacecraft B by replacing ϕ by −ϕ. Solving for R_A = R_B and regrouping the terms in ϕ gives

$$\begin{equation} \phi = \arcsin \left(\frac{P-1}{Q}\right) + \alpha, \end{equation} \tag{ 2 }$$

with

$$\begin{eqnarray*} P & = & d_B \sin \epsilon _B / (d_A\sin \epsilon _A),\\ Q & = & \sqrt{P^2 + 2P \cos (\beta _B + \beta _A + \epsilon _B + \epsilon _A) +1}, \mathrm{ and}\\ \tan \alpha & = & \frac{P \sin (\beta _A + \epsilon _A) - \sin (\beta _B + \epsilon _B)}{P \cos (\beta _A + \epsilon _A) + \cos (\beta _B + \epsilon _B)}.\\ \end{eqnarray*}$$

The diameter of the sphere can then be simply derived from Equation (1).

Another totally independent way to explain the different elongation measurements of the two spacecraft is to consider that it is due to the expansion of the CME front (see the right panel of Figure 1). In this model, the CME is not anchored at the Sun and its radius is one of the derived parameters. To compute the CME radius, we have to consider the direction of propagation, ϕ, as a known quantity. It can be, for example, obtained from the COR-2 fitting done by Thernisien et al. (2009). The main assumptions of this model are the circular cross section and the absence of heliospheric deflection.

Noting R₁ as the radius of the CME, R₂ as the heliocentric distance of the center of the CME, R_C as the distance of the center of the CME to the observer, and _A − γ as the angle between the Sun, STEREO-A, and the center of the CME, the law of cosines and the law of sines give, respectively,

$$\begin{eqnarray*} && R_C^2 = d_A^2 + R_2^2 - 2 d_A R_2 \cos (\beta _A - \phi)\\ && R_2 \sin (\beta _A - \phi) = R_C \sin (\epsilon _A - \gamma). \end{eqnarray*}$$

But R_Csin γ = R₁ and $R_C \cos \gamma = \sqrt{R_C^2-R_1^2}$, and the second equation can be rewritten as

$$$ R_2 \sin (\beta _A - \phi) = \sqrt{R_C^2 - R_1^2} \sin \epsilon _A - R_1 \cos \epsilon _A, $$$

which can be combined with the first equation into the following simple equation:

$$\begin{eqnarray} & & \hspace{-2pc}R_2 \sin (\beta _A - \phi) + R_1\cos \epsilon _A \nonumber \\ &=& \sin \epsilon _A \sqrt{d_A^2 + R_2^2 - R_1^2 - 2d_A R_2 \cos (\beta _A -\phi) }, \end{eqnarray} \tag{ 3 }$$

with the equation for spacecraft B obtained by replacing ϕ by −ϕ. By solving Equation (3) for STEREO-A and STEREO-B simultaneously, R₁ and R₂ can be uniquely derived. The CME half-angle is given by θ = arctan(R₁/R₂). It should be noted that for CMEs propagating directly toward Earth (ϕ ∼ 0), this model always predicts that the HI instruments onboard the two STEREO spacecraft will observe the exact same time-elongation profile, which is a direct consequence of the assumption of a circular front. Also, if _A < _B at one time but _B < _A at another time, there is no solution with R₁ > 0 and R₂ > 0, and the model cannot be applied. In these cases, the CME deformation, deflection, and the deviation from circular of its cross section must be taken into account.

The main hypothesis of these two models is that the HI instruments onboard STEREO-A and B do not observe the same plasma element, as is likely to be the case in the COR fields of view and as is assumed in Liu et al. (2010). The assumption here is that the CME front is locally circular and that it is "projected" onto the HI fields of view at the elongation angle corresponding to the tangent to this front. Similar assumptions have been made by Wood et al. (2009) and Howard & Tappin (2009) to derive CME positions from observations at large elongation angles from one spacecraft.

For these new methods to be applied, the studied CMEs have to propagate between the two STEREO spacecraft, and they may appear as halos, partial halos, or wide CMEs in Large Angle and Spectromeric Coronagraph Experiment (LASCO) field of view. To find potential CMEs to study, we look at all instances of CMEs with an apparent width larger than 100° in LASCO field of view from 2008 January when the spacecraft separation reached 45° until 2009 May. We found 17 CMEs, 5 of which were observed simultaneously by the HI instruments onboard STEREO-A and B; they are the 2008 April 26, June 2, August 30, November 3, and December 12 CMEs. The November 3 is associated with an instance of CME–CME interaction, and we exclude it from the current study (Kilpua et al. 2009). Although the August 30 CME also interacts with a preceding ejection, the preceding ejection appears to be small enough in size so that it did not influence too strongly the propagation of the overtaking CME and, therefore, we keep this CME in this study. The December 12 CME had two bright fronts observed in both spacecraft, and we analyzed three different data sets for the April 26 CME, resulting in a total of seven analyzed pairs of data sets. We first apply the two models to a detailed case study for the 2008 April 26 CME, before presenting the results for the other CMEs.

On 2008 April 26 at 1425UT, SECCHI-A/COR-1 observed an eastern limb CME. This event was studied by Thernisien et al. (2009) and Colaninno & Vourlidas (2009) in the COR-1 and COR-2 fields of view, and by Wood & Howard (2009) in the entire SECCHI field of view. Images of the CME in COR2-A, HI1-A, and HI1-B are shown in Figure 2. To do our analysis, we used elongation measurements for PA 90 and 270 obtained from the Naval Research Laboratory (hereafter, NRL measurements) from J-maps including COR and HI observations and from the Rutherford Appleton Laboratory (hereafter, RAL measurements) from J-maps derived from HI-only observations along the ecliptic (Davies et al. 2009). The time-elongation plots are shown on the bottom left panel of Figure 3. The tracked front corresponds to the top of the ejecta for the RAL and the first of the NRL measurements and to the center of the ejecta for a second set of NRL measurements. Two difficulties in applying the models described above to real data are that the elongation measurements must be derived accurately and that the boundary of the same structure must be tracked in STEREO-A as well as in STEREO-B. This last condition may not always be fulfilled in HI-2 where the cause of bright fronts is sometimes hard to establish (e.g., see Lugaz et al. 2008).

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We analyzed the time-elongation data for STEREO-A and B for this CME with the two methods proposed above. Using data from two different groups allows us to quantify the errors associated with the manual determination of the elongation angles. The results of our analysis are shown in Figure 3. In the bottom right panel of this figure, we show the error bars for model 1 (black) and model 2 (red) assuming the elongation measurements are made with a precision of 15 pixels corresponding to uncertainties of ±015 and ±05 in HI-1 and HI-2 fields of view, respectively. The error in the direction (model 1) is typically ±2° in HI-1 as well as HI-2 fields of view, and the error in the CME radius (model 2) is ±1 R_☉ in HI-1 and increases to ±2.5 R_☉ when the CME is observed in both HI-2 simultaneously.

According to model 1, the CME is deflected toward the east (i.e., away from Earth) reaching a near-constant central longitude of −35° ± 2° at about 50 R_☉, the CME continues to be deflected toward the east with a rate of about −35 day⁻¹ from this distance on (see the top left panel of Figure 3). The initial angle is ∼−20° comparable to the angle of −21° derived by Thernisien et al. (2009) based on COR2 data. The central position of the CME using only STEREO-A data and the Fixed-ϕ procedure of Rouillard et al. (2008) can be estimated at −335 ± 18°, while Wood & Howard (2009) reports a best-fit value of −265. Our value of −35° ± 2° appears in relatively good agreement with these values.

For model 2, we assume the fixed direction of propagation ϕ = −21° from Thernisien et al. (2009). Then, the CME radius monotonically increases from a value of 8 ± 3 R_☉ at 20 R_☉ to 26 ± 1 R_☉ at 80 R_☉, and to about 37 R_☉ at 140 R_☉. The corresponding CME half-angle is equal to 25° ± 2° from 60 R_☉ to 120 R_☉ with a general shrinking rate of 4° day⁻¹ until the end of the measurements (see the top right panel of Figure 3).

In the first 40 R_☉, the results using the different data sets are inconsistent with each other (see top panels of Figure 3). This is because the CME appears in STEREO-B as a halo and in STEREO-A as a limb CME, and we are not able to identify the same front in the observations from the two spacecraft. In fact, the CME is first visible in HI-B (at ∼4°) when it reaches a distance of about 32 R_☉ from the Sun, corresponding to observations around 8° elongation in HI-A (see the bottom left panel of Figure 3). Before this time, since there is no measurement of the CME in HI-B, we use the elongation angles measured by STEREO-B in COR-2 for the NRL data. We find it impossible to separate the top of the ejecta from the shock front and piled-up mass in COR2-B and early on in HI1-B. Therefore, we are not able to identify the same front between STEREO-A and STEREO-B when the CME is within 40 R_☉ from the Sun, and the results are inconsistent between the RAL and NRL measurements. For the center of the ejecta, we are able to identify the common feature in both STEREO-A and B, and the models, in this case, give a more realistic evolution of ϕ and R₁. This is because in STEREO-B, where the CME appears as a halo, it is still relatively easy to separate the center of the ejecta from the other density features. It should be noted that the center of the ejecta might be of relatively small angular extent in the azimuthal direction, in which case, using direct triangulation as in Liu et al. (2010) is more adapted.

We compared the CME distance as derived with these two models, with the HM model proposed in Lugaz et al. (2009a) using only STEREO-A data. All methods yield the same distances for the CME within 10%. However, from the two new models, it is possible to predict a hit/miss at different spacecraft's positions in the heliosphere based on the azimuthal properties of the CME. Based on the height–time profile in the heliosphere using the second model, we fit the data to a CME with a constant speed of 534 km s⁻¹—to compare to the final speed of 543 km s⁻¹ derived by Wood & Howard (2009). Additionally, we calculate the CME half-angle, θ, with the fixed rate of −4° day⁻¹, using the best-fit formula of θ = 25°–4° × (t − t₀), where t is the time in day since t₀ = April 27 1200UT. With these parameters, the model correctly predicts that the CME does not hit ACE but it predicts that the CME hits STEREO-B at 0300UT on 04/30. In situ observations by ACE and the two STEREO spacecraft show that only STEREO-B detected the passage of an iCME from 1530UT on April 29 to 0700UT on April 30, which translates into an error of about 11 hr for the arrival time for our model.

The 2008 June 2 CME was included in the study by Thernisien et al. (2009) and studied in Robbrecht et al. (2009), while the 2008 December 12 CME has been analyzed by Davis et al. (2009) and Liu et al. (2010). For our study, we used the data available from the Rutherford Appleton Laboratory Web site. In the left panel of Figure 4, we show the variation of the central longitude of the CMEs following model 1 for the four data sets. In Table 1, we compare our results with information available from the flare location, with the direction of propagation as calculated with the procedure of Rouillard et al. (2008) and with other published analyses. It should be noted that the method of Liu et al. (2010) assumes that d_A = d_B, whereas we use the real values of the spacecraft heliocentric distances (which we assume constant over the duration of an event). In our experience, doing so is required to obtain consistent values, especially at large elongation angles. For example, analyzing the last pair of elongation angles from our measurements for the second front of the December 12 event, the method of Liu et al. (2010) gives −5° with d_A = d_B, but using the exact values of d_A and d_B shifts the derived longitude from −5° to 13°. This corrected value is closer to the last value of 27° obtained with model 1 than the value of −5° reported by Liu et al. (2010).

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Table 1. Direction of Propagation for the Four CMEs from this Work Compared to Other Methods

CME	Method and Instruments	ϕ-
	(Reference)	Estimate
Apr 26	Flare	−9°
Apr 26	Visual COR2s (1)	−21°
Apr 26	Visual STEREOs (2)	−28°
Apr 26	Mass COR2s (3)	−48°
Apr 26	Fixed-ϕ HI-A	−335 ± 18°
Apr 26	Fixed-ϕ HI-B	21 ± 65
Apr 26	Model 1	−11° (at 20 R☉) to −375 (at 130 R☉)
Jun 2	Visual COR2s (1)	−33°
Jun 2	Fixed-ϕ HI-A	−222 ± 5°
Jun 2	Fixed-ϕ HI-B	209 ± 115
Jun 2	Model 1	−11° ± 27
Aug 30	Fixed-ϕ HI-A	−11° ± 17°
Aug 30	Fixed-ϕ HI-B	192 ± 105
Aug 30	Model 1	10° (at 20 R☉) to −22° (at 140 R☉)
Dec 12 Front 1	Fixed-ϕ HI-A	−117 ± 13°
Dec 12 Front 1	Fixed-ϕ HI-B	126 ± 65
Dec 12 Front 1	Triangulation (4)	0° ± 5°
Dec 12 Front 1	Model 1	10° ± 10°
Dec 12 Front 2	Fixed-ϕ HI-A	83 ± 45
Dec 12 Front 2	Fixed-ϕ HI-B	−15 ± 7°
Dec 12 Front 2	Triangulation (4)	5° ± 3° (up to 70 R☉) −3° ± 5° (after)
Dec 12 Front 2	Model 1	20° ± 7°

References. (1) Thernisien et al. 2009; (2) Wood & Howard 2009; (3) Colaninno & Vourlidas 2009; and (4) Liu et al. 2010.

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We find that the June 2 CME and the two fronts from the December 12 CME propagate close to radially outward until about 140 R_☉ (0.65 AU), while the August 30 CME shows a deflection toward the east of about 30° during its propagation and crosses the Sun–Earth line. As noted in Section 2.2, the second model, because it assumes a fixed direction of propagation, cannot be used for CMEs propagating exactly toward Earth (ϕ ∼ 0°) nor for CMEs which cross the Sun–Earth line during their propagation; therefore, we only applied this model with the measurements from the June 2 CME as well as the second front of the December 12 CME, which appear to propagate away from the Sun–Earth line. We used directions of propagation of −11°and 20° for these features, respectively. From model 1, the June 2 CME appears to move on a quasi-radial trajectory at about −15°, while the second front of the December 12, CME appears to propagate on a direction of about 20° with respect of the Sun–Earth line; hence, we choose these numbers for the fixed directions of propagation. The results of the second model are shown in the right panel of Figure 4.

The June 2 observations can be explained, using model 2, by a CME which propagates on a fixed radial trajectory and whose radius in the ecliptic increases more slowly than self-similarly, the CME half-angle decreasing from 45° to 35° (see Figure 4). Alternatively, it can be explained by model 1 as a CME whose direction of propagation is deflected by about 5° toward the east in about 0.45 AU. Both these explanations appear physically realistic, involving a propagation close to radially outward and an evolution close to self-similar, and it is likely that the evolution of the June 2 CME is a combination of these two results, with a limited eastward deflection and a small shrinking of the CME front. The observations from the August 30 CME cannot be explained by model 2, because the CME appears first farthest in STEREO-B before appearing farthest in A (as noted in Section 2.2, this causes model 2 to be inapplicable). Using model 1, it corresponds to a CME being deflected by 30° toward the east in 0.5 AU. The two features from the December 12 CME appear to propagate close to the Sun–Earth line on near-radial trajectories but they cannot be simply analyzed with model 2, either because the fronts propagate too close from the Sun–Earth line or because the model's assumption of observing the tangents to a circular CME cross section is not correct for this CME. It appears from both our study and that by Liu et al. (2010) that the first front of this CME propagates about 5°–10° east of the second front.