For CMEs observed near the solar limb with white-light coronagraphs (limb CMEs), the `location' of a CME feature is defined to be midway between its outer edges in a coronagraph image and the angular width is the angular distance between two legs of the CME. Both the location and the angular width can be directly measured from coronagraph images. The radial frontal speed and acceleration may also be estimated by tracking the bright front on a series of limb CME images. It has been reported that CME-associated flares or active regions are often located near one leg of CMEs, rather than near the center (Harrison, 1986). The observed locations of limb CMEs varies over the solar cycle in a similar manner to the variation in the locations of large-scale coronal features such as helmet streamers (Hundhausen, 1993). The average angular width of white-light CMEs is about 45 degrees, but some CME widths exceed 100 degrees (Howard et al., 1985; Hundhausen, 1993), and for many CMEs the angular widths remain nearly constant as a function of radial height (Webb et al., 1997). The average radial frontal speed is about 400 km/s with a very wide variation in speed around this average (Hundhausen, 1993). The majority of CMEs move with approximately constant speed through the LASCO field of view (St. Cyr et al., 1999). However, many CMEs gradually accelerate out to at least 30 solar radii, and some fast CMEs decelerate when they are far from the Sun (Sheeley et al., 1999).
The limb CMEs cannot hit the Earth and are not geoeffective in general. The `Halo' CME is assumed to originate from the center of the Sun within a circle of about 30 degrees (Brueckner, G.E., et al., 1998). It is necessary to determine the geometrical and kinematical properties, and in turn, the line-of-sight speed of the halo CMEs both in understanding the cause(s) and in predicting the geoeffectiveness of the front-side halo CMEs. The helo CME was first detected with the SOLWIND coronagraph on 1979 November 27. It was a bright cloud surrounding the entire Sun and propagating outward from it in all directions, and was interpreted as a broad shell or bubble of dense plasma ejected directly toward (or away from) the Earth (Howard et al., 1982). A broad interplanetary disturbance produced by this mass ejection was detected with the Helios photometers and the disturbance was found traveling along the Sun-Earth line towards the Earth and generating a small geomagnetic storm (Jackson, 1985). These observational results add to the plausibility of this interpretation. The angular size of the interplanetary plasma cloud is comparable to that of the coronal mass ejection near the Sun (Webb and Jackson, 1990), suggesting the angular size of the CME keeps constant while propagating through the corona, similar to that of limb CMEs mentioned above.
Halo CMEs, many of which are very faint, can be routinely observed now with LASCO due to LASCO's high sensitivity and wide dynamic range. The source regions of the halo CMEs can also be studied in detail with EIT observations of the solar disk. The combination of LASCO and EIT observations of halo CMEs makes it possible to determine whether or not a halo CME is on the front side (Plunkett et al., 2000). The close relationship of halo CMEs, magnetic clouds, and magnetic storms has been found recently (Webb et al., 2000).
This work develop a simple model to reproduce the expanding halo CMEs and to determine the location, the angular width, the acceleration and radial speed, and finally the line-of-sight speed of the front-side halo CMEs.
Because the angular widths of many limb CMEs remain constant while the CMEs propagate outward, as mentioned earlier, the boundary of emission observed by LASCO for these CMEs should form a cone of constant angular spread, and the apex of the cone is located at the center of the Sun.
Thus the expanding bright rings for a halo CME with a constant angular width may be reproduced when the angular width (\anw), the radial distance (\rrr), and the orientation (the latitude \lat and longitude \lon) of the central axis of the cone are given. Setting the apex of the cone to be at the original point of the right- hand coordinate system X'Y'Z', and the central axis is setting to be parallel to the X' axis, the radius (\rou) of the cross section at point x' depends on the angular width (\anw) and the radial distance (\rrr), \rou = \rrr SIN(\anw/2). The circular cross section can be transformed into the sky plane when the orientation of the central axis of the cone is given. Figure 1 shows how does the cross section of the cone projected on the sky plane changes in shape as the orientation and the radial distance change. The angular width for all panels is 45 degrees, the average value of angular widths for limb CMEs (see the top of each panel). The 3 solid circles denote, respectively, 1.0, 2.0 and 3.7 solar radii, corresponding to the radius of the solar disk, the C2 occulting disk, and the C3 occulting disk. The 6 dotted lines in each panel denote the 6 cross sections corresponding to the radial distances of 3.0, 6.0, 9.0, 12.0, 15.0, 18.0. The dotted lines in the panels from left (top) to right (bottom) are changing in shape due to the increase of the latitude (longitude). Figure 1 indicates that full bright rings occur when the orientation is within a small circle, say 5 degrees to the disk center. When the orientation of the central axis of the cone is located at a greater circle, say 20 degrees to the disk center, only partial bright rings can be observed. Figure 2 shows the effect of the angular width on the shape of the cross section of the cone on the sky plane. The solid and dotted lines here have the same meaning as in Figure 1. The angular width (the orientation) increases from left (top) to right (bottom). It indicates that the chance to form full halo CMEs increases as the angular width increases.
The 1997 May 12 halo CME showed a series of expanding bright rings centered about the occulting disk. The CME was first visible in a C2 image recorded at 06:30 UT and in a C3 image recorded at 08:06 UT, respectively. The CME followed an eruptive event observed by EIT in the lower corona at approximately 04:35 UT. The eruptive event was centered on active region 8038 at N23 W07 (Plunkett et al., 1998; Thompson et al., 1998). Thus this is a front-side halo CME. Figure 3 shows development of the 1997 May 12 halo CME between 08:06:05 Ut and 15:37:54 UT, as seen by the LASCO/C3 coronagraph. The images shown in Figures 3 and 4 are differences from a pre-event base image.
To reproduce the 6 observed expanding bright rings, as shown in Figure 3, the initial values for the angular width, the latitude, and the longitude are taken to be 45, 5 and 5 degrees, respectively. Then these values are adjusted reiteratively to best match the outer edges of the 1997 may 12 halo CME observed at different times. The black circles in Figure 4 are the predicted cross sections of a cone with its angular width, latitude and longitude shown on the top of panels. The circle in each panel corresponding to a specified radial distance matches the bright ring observed at an appropriate time. The specified radial distance and the appropriate time is shown on the top of each panel.
Using the 6 pairs of values for the time and radial distance, a scatter plot of time v.s. distance can be obtained (see the top panel of Figure 5). The dotted line in the top panel is the least square fitting to the scatter points, representing a homogeneous speed, and the solid curve is the Downhill Simplex fitting, representing an acceleration. The bottom panel of Figure 5 displays the inferred speed at various heliocentric distances. The solid curve in the top panel matchs the scatter points better than the dotted line, suggesting it is more likely that the 1997 May 12 CME was accelerating between 5.60 and 17.60 solar radii and between 08:06:05 and 14:51:05 UT. It was accelerated from 120 km/s at 08:06:05 UT and 5.6 solar radii to 600 km/s at 14:51:05 UT and 17.6 solar radii. The corresponding line-of-sight speed can be easily calculated using the inferred latitude and longitude of the central axis of the cone, being 119.7 and 598.6 km/s respectively at 5.6 and 17.6 solar radii. It appears the acceleration continues beyond 17.6 solar radii.
Observations of limb CMEs showed that the angular widths for many limb CMEs remain nearly constant as a function of radial height, and that most of CMEs propagate almost radially through the C2 and C3 fields of view, though some show clear non-radial motion in the first few solar radii. Based on these properties of limb CMEs, the cone model is developed mathematically. The free parameters in the cone model are the angular size of the cone and the orientation of the central axis of the cone.
It is found on the basis of the prediction of the cone model that the central axis of the cone for a full halo CME with the average angular width of 45 degrees is located in a circle centered at the solar disk center, of which the radius is much less than 30 degrees.
The cone model can be used to reproduce the expanding halo CME observed by LASCO/C3 coronagraph, suggesting that the 1997 May 12 halo CME was propagating radially in the C3 field of view and that the angular width of 55.0 degrees remains constant during its propagation through the corona. The halo CME was most likely accelerating from 120 km/s at 5.6 solar radii to 600 km/s at 17.6 solar radii in 5.3 hours. The inferred orientation of the central axis of the cone is the latitude of 3.7 and longitude of 1.1 degrees, located in between the solar disk center and active region 8038 (23N 7W). Since it is located in a little bit north to the equator the halo CME would be expected to be brighter to the north (where it is closer to the sky plane) than to the south. The CME does appear to be somewhat brighter to the north than to the south (see Figure 2). Since the inferred location is very close the disk center, the inferred line-of-sight speed is only slightly different from the radial speed for this CME.
There is uncertainty in identifying the outer edges of rather fuzzy rings of halo CMEs. It may introduce some error in determining the free parameters of the cone model, and thus, the acceleration and speed. It is needed to find a rather objective criteria to locate the expanding bright rings of halo CMEs.
Not all CMEs have constant angular width and fit a symmetrical cone model. It is also likely that the origin of CMEs is not near a point on the surface but up into the corona and involves the evacuation of a large volume of coronal material, such as evidenced in the dimmings or Gopal's EUV brightenings. Some CMEs clearly involve the blowout of a p-e streamer. More samples are needed to be examined in order to see what fraction of halo CMEs can be reproduced with the cone model.
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