Determining CME parameters by fitting heliospheric observations: Numerical investigation of the accuracy of the methods

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1 Available online at Advances in Space Research 48 (2011) Determining CME parameters by fitting heliospheric observations: Numerical investigation of the accuracy of the methods Noé Lugaz a,, Ilia I. Roussev a, Tamas I. Gombosi b a Institute for Astronomy, University of Hawaii, 2680 Woodlawn, Dr. Honolulu, HI 96822, USA b Center for Space Environment Modeling, University of Michigan, 2455 Hayward St., Ann Arbor, MI 48109, USA Received 20 October 2010; received in revised form 8 March 2011; accepted 9 March 2011 Available online 16 March 2011 Abstract Transients in the heliosphere, including coronal mass ejections (CMEs) and corotating interaction regions can be imaged to large heliocentric distances by heliospheric imagers (HIs), such as the HIs onboard STEREO and SMEI onboard Coriolis. These observations can be analyzed using different techniques to derive the CME speed and direction. In this paper, we use a three-dimensional (3-D) magneto-hydrodynamic (MHD) numerical simulation to investigate one of these methods, the fitting method of Sheeley et al. (1999) and Rouillard et al. (2008). Because we use a 3-D simulation, we can determine with great accuracy the CME initial speed, its speed at 1 AU and its average transit speed as well as its size and direction of propagation. We are able to compare the results of the fitting method with the values from the simulation for different viewing angles between the CME direction of propagation and the Sun-spacecraft line. We focus on one simulation of a wide ( ) CME, whose initial speed is about 800 km s 1. For this case, we find that the best-fit speed is in good agreement with the speed of the CME at 1 AU, and this, independently of the viewing angle. The fitted direction of propagation is not in good agreement with the viewing angle in the simulation, although smaller viewing angles result in smaller fitted directions. This is due to the extremely wide nature of the ejection. A new fitting method, proposed to take into account the CME width, results in better agreement between fitted and actual directions for directions close to the Sun Earth line. For other directions, it gives results comparable to the fitting method of Sheeley et al. (1999). The CME deceleration has only a small effect on the fitted direction, resulting in fitted values about 1 4 higher than the actual values. Ó 2011 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Coronal mass ejections; Magneto-hydrodynamics (MHD); STEREO; Methods 1. Introduction Coronal mass ejections (CMEs) have been observed remotely by coronagraphs and by in-situ instruments since the 1960s. Until the launch of heliospheric imagers, CME speed was primarily derived from coronagraphic measurements (Hundhausen et al., 1994; Yurchyshyn et al., 2004) or from radio measurements (Reiner et al., 2003). CME direction was derived from forward fitting (Thernisien et al., 2006), from polarimetric measurements (Moran Corresponding author. Tel.: addresses: nlugaz@ifa.hawaii.edu (N. Lugaz), iroussev@ifa.hawaii.edu (I.I. Roussev), tamas@umich.edu (T.I. Gombosi). and Davila, 2004) or the CME was simply assumed to propagate radially outward from its source region (Cremades and Bothmer, 2004; Yurchyshyn et al., 2004). In the past five years, with the launch of spacecraft carrying heliospheric imagers (Coriolis and the solar-terrestrial relations observatory (STEREO)), CMEs are routinely observed to radial distances as far as 0.5 AU with the heliospheric imagers (HIs, see Eyles et al., 2009) and often up to Earth s orbit. The CME direction and speed can be estimated by visual fitting (Wood et al., 2009; Maloney et al., 2009), comparison to a family of pre-existing simulated ejections (Howard and Tappin, 2009), fitting to known functions of the speed and direction (Rouillard et al., 2008; Lugaz, 2010), or the analysis of stereoscopic /$36.00 Ó 2011 COSPAR. Published by Elsevier Ltd. All rights reserved. doi: /j.asr

2 N. Lugaz et al. / Advances in Space Research 48 (2011) measurements in the coronagraph COR fields-of-view (Mierla et al., 2008; Colaninno and Vourlidas, 2009; de Koning et al., 2009; Liewer et al., 2009; Temmer et al., 2009; Thernisien et al., 2009) and in the heliospheric imagers fields-of-view (Liu et al., 2010; Lugaz et al., 2010). Because these analysis methods are new and might be used in the future for space weather forecasting (Davis et al., 2011), it is important to test them and quantify their errors. For Earth-directed CMEs, it is possible to compare the speed estimated from remote observations with that measured in situ. However, this requires to make assumptions regarding the heliospheric evolution of the CME. For example, it is necessary to assume which part of the CME hits Earth and what is its speed with respect to the speed at the nose (or apex) of the CME (by distinguishing between radial and expansion speeds of CMEs, as discussed in Schwenn et al., 2005). It is also possible to compare different methods with each other. Such a comparison between methods was recently performed by Davis et al. (2010) between the visual fitting of COR images by Thernisien et al. (2009) and the analytical fitting to a constant direction and velocity by Sheeley et al. (1999). In Lugaz (2010), we compared two different fitting methods and two different methods based on stereoscopic HI measurements. These comparisons are interesting and can reveal some bias in the methods (see, Lugaz, 2010, for more details). However, they are limited inasmuch as both methods which are compared have some errors and it is hard to distinguish the error associated with one method with that from the other method. Numerical simulations can be used to test methods, because the speed and direction of the CME is known as an input of the model (or at least, it can be controlled and measured accurately). It is particularly true for models which can reproduce coronagraphic and heliospheric observations. In Lugaz et al. (2005), for example, we tested the error associated with the derivation of the mass and kinematics of CMEs from single coronagraphic observations. In the present article, we quantify the error of the fitting methods of Sheeley et al. (1999) and Lugaz (2010) for one simulation of a fast and wide CME. In Section 2, we give an overview of the models used and of the CME evolution. In Section 3, we derive the CME speed and direction from synthetic line-of-sight observations and quantify the error of the methods. We also give an estimate of the error associated with the CME width as well as that associated with the CME deceleration. In Section 4, we discuss our results and conclude. 2. Models 2.1. Solar wind and CME models The simulation is done using the space weather modeling framework (SWMF) with the solar corona (SC) and the inner heliosphere (IH) components (for a description of the SWMF, see: Tóth et al., 2005). The SC domain is resolved with 2.9 millions cells ranging in size from 1/50 R at the inner boundary (corona) to 0.62 R. The IH domain is resolved with 6.8 millions cells ranging in size from 0.43 to 3.4 R. In both domains, the heliospheric current sheet has been refined in order to better capture the density gradients there. The solar wind and coronal magnetic field have been produced using the model developed by Cohen et al. (2007). This model makes use of the Wang Sheeley Arge (WSA) model (Wang et al., 1990) for the asymptotic solar wind speed at 1 AU. The solar magnetic field is a set-up as a dipole, resulting in a maximum intensity at the pole on the solar surface of ±4 G. To model the CME, we have used a semi-circular flux rope prescribed by a given total toroidal current, as in the models by Titov and Démoulin (1999) and Roussev et al. (2003). A more complete description of this implementation of the flux rope model can be found in Lugaz et al. (2007). The flux rope solution once superimposed on the background magnetic field leads to an immediate eruption because of the force imbalance with the ambient magnetic field. Note that this model is not aimed at reproducing the complexity of the flux rope s formation. Our main goal here is to compare the speed and direction of the CME as derived from the fitting method of Sheeley et al. (1999) with that from the MHD simulation CME evolution We initiate the CME at time t = 0 by adding a flux rope across the polarity inversion line with a low inclination with respect to the solar ecliptic (see left panel of Fig. 1). The current inside the flux rope is chosen so that the maximum magnetic field inside the flux rope is about 30 G and the speed of the eruption after 4 h in the current sheet is about 850 km s 1 (a view of the CME 2 h after the initiation is shown in the right panel of Fig. 1). We follow the CME as it propagates towards 1 AU in the heliosphere. 3-D views of the CME after 12 h are shown in Fig. 2. By this time, the CME has expanded to be about wide in the azimuthal direction in the ecliptic plane. We track the position of the density maximum at the nose of the CME in the ecliptic plane. The average transit speed of the CME is about 750 km s 1 and its final speed (at 0.9 AU) is about 680 km s 1. The kinematics are consistent with an average deceleration of about 1 ms 2 from the Sun to the Earth Line-of-sight Images, J-maps and fitting method The synthetic line-of-sight routine was introduced in the SWMF for LASCO-like coronagraphs (Manchester et al., 2004), and it was modified by Lugaz et al. (2005) to simulate wide-angle observations by HIs. In Lugaz et al. (2008), we compared synthetic SECCHI/HI images with real images for the two successive eruptions of January 24 and 25, In numerical simulations, the densest region in a fast CME is the dense sheath ahead of the magnetic ejecta. For fast CMEs, this dense sheath is composed of

3 294 N. Lugaz et al. / Advances in Space Research 48 (2011) Fig. 1. Left: Initial configuration at time t = 0 showing the flux rope added onto the solar surface. The Sun is color-coded with the radial magnetic field B R, the black line shows the polarity inversion line. 3-D magnetic field lines are drawn in white. Right: CME at time t = 2 h in the meridional x, z plane. The contours show the radial velocity, the black streamlines are 2-D projection of the magnetic field lines. Fig D view of the CME at time t = 12 h from 2 different viewpoints. The white translucent isosurface corresponds to an increase of 150 km s 1 in the radial velocity over the background solar wind (taken from the pre-event steady-state solution), and it illustrates the approximate extent of the dense sheath in front of the magnetic ejecta. Magnetic field lines are color-coded with the radial velocity. The large sphere is centered at the Sun and is colorcoded with the azimuthal angle to show the extent of the CME (about ). shocked solar wind material as well as swept-up mass (Manchester et al., 2004). It is what is typically imaged in synthetic line-of-sight images, especially in the heliosphere (Lugaz et al., 2005). In the field-of-view of an heliospheric imager, the position is measured as the angle between the observing spacecraft, the Sun and the density structure, and it is commonly referred as the elongation angle, a. In Lugaz et al. (2009), we presented synthetic time-elongation maps (J-maps) of simulated CMEs, which we compared to the actual J-maps. J-maps are one of the methods to study the evolution of density enhancements (Sheeley et al., 2008; Rouillard et al., 2008; Davies et al., 2009). Such maps allow for the tracking of CMEs to large elongation angles and sample points following a bright feature can be easily extracted from such a plot. In our simulation, we create J-maps from different perspective by placing in the 3-D simulation STEREO-A-like spacecraft with angular separations 15, 30, 45, 60 and 75 away from the direction of propagation of the CME. We create HI-1 and HI-2 images with cadences of 20 min and 1 h, respectively, and we create the J-map by doing running difference with the typical 20 min and 2 h cadence of the STEREO observations.

4 N. Lugaz et al. / Advances in Space Research 48 (2011) In general, the elongation angle is a complex function of the CME heliocentric distance, its shape and its angle of propagation with respect to the observing spacecraft. Therefore, the shape of the elongation vs. time curve depends at least on the speed and direction of propagation of a transient (Sheeley et al., 1999). For all but the narrowest ejections, the CME shape also influences the elongation vs time curves (see, Webb et al., 2009; for example Howard and Tappin, 2009). When CMEs are observed to large elongation angles (up to 40 and beyond), the elongation vs. time profile can be fitted to analytical functions and the average speed and average direction of the CME can be derived under certain assumptions. Such an analytical formula can be derived for a single plasma element propagating with a constant speed, V, on a radial trajectory making an angle b with the Sun-spacecraft line (Sheeley et al., 1999; Rouillard et al., 2008), as: a ¼ arctan Vt sin b d ST Vt cos b where d ST is the heliocentric distance of the observing spacecraft (hereafter STEREO) and t the time since the launch at the Sun of the observed plasma element. We refer to this method as the Sheeley Rouillard (SR) fitting method. A different formula was recently proposed in Lugaz (2010). It assumes a spherical CME front whose center propagates with constant speed, V, and on a radial trajectory making an angle b with the Sun-spacecraft line. This CME is further assumed to be anchored at the Sun, resulting in its diameter to be given by the harmonic mean approximation (Lugaz et al., 2009). This relation can be inverted, as: Vt sin b a ¼ arcsin þ arctan Vt ; ð2þ d d with qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d ¼ ð2d ST Vt cos bþ 2 þðvt sin bþ 2 : We refer to this fit as the Lugaz (L) fitting method. A time series of sample points can be fitted with such theoretical formulae, in the following manner. For a given value of the velocity, V, and the CME direction, b, we calculate the standard error r (standard deviation of the residue) between the observed profile and the theoretical profile as follows: r 2 ¼ 1 N X N k¼1 ða observed ðt k Þ a theoretical ðt k ÞÞ 2 where a observed is the observed elongation angle at time t k and a theoretical is the elongation angle calculated with Eq. (1) or (2) for the same time. t k is the sample time of the kth point on the J-map track. The procedure is repeated for values of the speed between 100 and 1000 km s 1 by 1kms 1 increment and for values of the direction between 1 and 100 with 1 increment. This way, we obtain an ð1þ ð3þ error map giving the value of r for all possible combinations of V and b. The best-fit values of (V,b) is that for which r is at its minimum. We give the uncertainties in the fitting quantities corresponding to the value of (V, b) for which r =2r min, corresponding to a 95% certainty. The typical error associated with the manual selection of the sample points has recently been addressed in Williams et al. (2009). They estimated the error in the direction to be typically 2 5 for CMEs observed up to 45 elongation and beyond. A more detailed explanation of the fitting procedure can be found in Rouillard et al. (2010) and Davis et al. (2010). 3. J-maps of simulated CME and analysis 3.1. Simulated CME In the top panels of Fig. 3, we show two J-maps corresponding to a spacecraft of 15 and 60 away from the direction of propagation of the nose of the CME. Theses J-maps are produced using a sequence of running differences sampled in a bin of 2 centered at the position angle (PA) 90, i.e. along the solar equator. This PA was chosen because it corresponds to the central PA of the CME and also the axis of symmetry of our axisymmetric solar wind. Comparing the two J-maps, it is clear that the track corresponding to the case of the CME propagating close to the Sun-spacecraft line shows an apparent acceleration, which is what is expected from theoretical consideration (Rouillard et al., 2008). In contrast, for the spacecraft separation of 60, the track is more linear or exhibits a small deceleration. In the bottom panel of Fig. 3, we compare the time-elongation plots of the CME as seen from the five viewpoints. One obvious result is that there is almost no difference between the five tracks up to elongation angle 30 35, all the tracks are approximately linear. This is consistent with the fact that the Point-P approximation, which does not take into consideration the direction of propagation of the CME is known to work well until about 30 (Lugaz et al., 2009; Howard and Tappin, 2009; Webb et al., 2009). Next, we fit the five tracks with the SR fitting method as explained in the previous section. The results are summarized in Table 1. It should be noted that, in general, larger directions correspond to larger speeds. For example for viewing angle 15, if the fitted speed is 650 km s 1 (resp. 590 km s 1 ), the best-fit direction is 38 ±6 (resp. 28 ± 5 ). Overall, the best-fit speed gives a good estimate of the final speed of the CME (about 680 km s 1 at 0.9 AU). While the best-fit direction increases for increasing viewing angles, as expected, the two values do not correspond to each other very well. The error bars increase with increasing viewing direction, and, except for the spacecraft 15 away from the direction of propagation of the CME, the actual direction is within the 1-r interval. We also fit the five tracks with the L fitting method. The results, summarized in Table 2, are very similar to that

5 296 N. Lugaz et al. / Advances in Space Research 48 (2011) o 30 o 45 o 60 o 75 o Fig. 3. Top: Jmaps (time-elongation maps) at PA 90 for the simulated CME as seen from a spacecraft 15 (left) and 60 (right) away from the direction of propagation of the CME. The x and y axes show the time in hours and the elongation angle in degrees, respectively. The running difference of the whitelight signal is plotted with a cadence of 20 min (HI-1 up to 18 ) and 2 hours (HI-2, thereafter). Bottom: Time-elongation measurements for the CME as observed 15, 30, 45, 60 and 75 away from its direction of propagation. Table 1 Speed and direction of the CME as observed by the five STEREO-like spacecraft and determined from the SR fitting method. Viewing angle Best-fit speed (km s 1 ) Best-fit direction 1-r interval ± ± ± ± ± from the fitting with the SR method. There are two main differences: for CMEs propagating close to the Sun-Earth line (15 and 30 ), the results from the fitting method given by Eq. (2) are in better agreement with the actual direction of propagation. This is consistent with what was found in Lugaz (2010), where CMEs propagating outside of 40 ±20 are best fitted with the fitting method given by Table 2 Speed and direction of the CME as observed by the five STEREO-like spacecraft and determined from the L fitting method. Viewing angle Best-fit speed (km s 1 ) Best-fit direction 1-r interval ± ± ± ± No upper limit Eq. (2). It is because this method takes into account the fact that the same part of the CME is not imaged at all times. The second difference is the uncertainty interval which is larger for the fit with Eq. (2) than that with Eq. (1). Here, it is mainly due to the fact that the L fit with

6 N. Lugaz et al. / Advances in Space Research 48 (2011) Eq. (2) has a larger residual error than the fit with Eq. (1): the time elongation profiles derived from the numerical simulations are best fitted with the classical SR fitting method Analytical considerations It is important to try to separate two possible source of errors: the large width of the CME and its deceleration. To do this, we fit the time-elongation profile given by analytical functions using the fixed-u and the Point-P approximations. The Point-P approximation does not take into consideration the CME direction of propagation. However, it assumes an extremely wide CME: a sphere centered at the Sun; it further assumes that the signal originates from the intersection of this spherical CME with the Thomson surface (Vourlidas and Howard, 2006). Here, we assume that the CME has a constant velocity of 650 km s 1 and that it is tracked for 48 h (up to elongation 50 ). The Point-P formula gives a = arcsin(vt/d ST ). We fit this time-elongation profile with the SR fitting procedure. In this case, the best-fit speed is 612 ± 10 km s 1 with a direction of 48 ± 5. We repeat this analysis for different speed and the results are summarized in Table 3. We can see that the effect of an extremely wide CME front is to mimic the properties of a CME going approximatively away from the observing spacecraft, and this independently of the CME actual direction. It also results in a lower speed that the actual CME speed. Next, we consider the influence of the CME deceleration on the fitted parameters. We start from the time-distance values for the nose of the simulated CME and we derive the elongation angle using the fixed-u formula (Kahler and Vourlidas, 2005), therefore removing the influence of the CME width: a ¼ arctan R sin b d ST R cos b ; where b is the direction with respect to the Sun-spacecraft line, which we fix here to be the viewing angle. With this procedure, the variation of the elongation angle with time depends on the kinematics of the simulated CME but not on its width. The results of the fit with the SR fitting method are summarized in Table 4. They can be easily understood as follows: the physical (real) deceleration of the CME intensifies the geometrical deceleration. It results in Table 3 Speed and direction of a transient given by the Point-P approximation as observed by the five STEREO-like spacecraft and determined from the fitting method of Sheeley et al. (1999). Actual speed (km s 1 ) Best-fit speed (km s 1 ) Best-fit direction ± ± ± ± ± ± ± ± ± ±13 Table 4 Speed and direction of a transient at the position of the CME nose but analyzed with the fixed-u approximation as observed by the five STEREO-like spacecraft and determined from the SR fitting method (see text for details). Viewing angle Best-fit speed (km s 1 ) Best-fit direction ± ± ± ± ± ± ± ± ± ±10 the fitted direction being systematically greater than the actual direction. However, this effect is small, resulting in deviation by only 1 4. There is also an effect in the fitted speed, with CMEs propagated towards the spacecraft having slower fitted speed than CMEs propagating away from the spacecraft. In all cases, the fitted speed is close to the average speed of propagation of the CME. 4. Discussions and conclusions We have produced five J-maps of the same simulated CME as seen from different directions from head-on (15 ) to close to the limb (75 ). We have analyzed timeelongation measurements from the five maps using the SR fitting technique, which is readily used to derive and predict CME direction from observations by one heliospheric imager. In all cases, the best-fit speed is in good agreement with the speed of the CME as it reaches 0.9 AU (the end of our study), with errors of about 10%. Concerning the CME direction, we have found that the smaller observing directions correspond to the smaller fitted directions and that, in all but the head-on case, the CME actual direction is within the 1-r error interval of the best-fit direction. However, the best-fit direction is generally in relatively poor agreement with the actual CME direction. We also analyzed the same five datasets with the L fitting method. This method was devised to take into account the CME width and the fact an heliospheric imager does not track the same part of the CME front as the CME propagates. For each of the five CME tracks, the residual error is larger for the L method than that for the SR method. The best-fit speed is also in fairly good agreement with the speed of the CME as it reaches 0.9 AU, being about 5 10% larger than the bestfit speed with the SR fit. For small viewing angles (15 and 30 ), the best-fit angle from the L fit is in better agreement with the actual CME speed than the SR fit, but there is no noticeable difference for the larger viewing angles. Overall, the fitted direction is also in poor agreement with the real direction. In addition to the error associated with the manual selection of the sample in the J-maps, there are three main factors which can account for the poor agreement between the fitted direction and the actual direction of propagation of the CME : the extremely wide nature of the simulated

7 298 N. Lugaz et al. / Advances in Space Research 48 (2011) CME, the deceleration of the CME and the influence of the Thomson surface. We have quantified the errors associated with the first two factors and have found that the CME deceleration has a small effect on the fitted direction, comparable to the error associated with the manual selection of points along the CME track (1 5 ). The fact that the simulated CME was about wide in the azimuthal direction is probably the dominant source of errors in the fit, especially for large viewing angles. Finally, since fitting methods neglect the effect of the Thomson surface, it is possible that this causes additional errors, which we have not quantified here. 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