During these phases of propagation a given CME traverses the respective distances: we have. Below, we present various methods for its determination. In the article we consider different methods to determine the velocities of the CMEs. These velocities can be correlated with the TTs.
Fitting curves to TT—velocity points we built theoretical models that can be used to predict the TT in the future. For individual CMEs, we can determine the error in the estimation of TT as the difference between the TT determined on the basis of the model and the actual observations.
Having these errors for a given model and entire populations of considered CMEs, we can determine the average absolute and maximal errors. This means that the maximal error for a given model is the maximal error from the distribution. Our study concentrates on the ascending phase of Solar Cycle These events are the basis of our study. Additionally, in our research, for each CME included in their list, we conducted the analysis which was described in the previous section.
Having the profiles of instantaneous velocities, we estimated the time and distance when the CMEs reached their maximal speeds. All of these data are shown in Table 1. The onset date and time of CME ejection are in columns two and three, respectively.
These data are from Syed Ibrahim, Manoharan, and Shanmugaraju In the respective columns we find that the average velocity from a linear fit to all data points, the maximal velocity from linear fit to five successive height—time points, distance, and the time when a given CME reaches the maximal velocity. The results are presented in the following sections. In the previous sections, we presented methods for determining the initial speeds of ejections.
Dashed lines are linear fits to data points. Formulas representing linear fits are placed in the lower-right corners of each panel. It can be seen that the average ejection velocities are strongly correlated with their maximal velocities Panels a and b , regardless of the instrument used for their determination.
Formulas representing these linear fits are placed in the lower-right corners of each panel. Open diamond symbols are for non-interacting and star symbols are for interacting CME, respectively. Correlations between the speeds for these two instruments are slightly smaller. The correlation coefficients are 0. Depending on the source location on the solar disk and the position of the spacecraft, the determined speeds are subject to different projection effects Bronarska and Michalek This effect, among others, is the reason that the determined speeds may be different for each of the telescopes.
This is due to projection effects. In Figure 2 , the relationships between the initial speeds and speeds of interplanetary coronal mass ejections ICMEs obtained from in-situ measurements are shown. Formulas presenting these linear fits are placed in the lower-left corner of each panel. Open diamond symbols are for non-interacting and star symbols are for interacting CMEs, respectively.
For the initial speed determined in the SOHO images, the correlation coefficients are less than 0. From the point of view of space weather this is a new and very important result. These relations for speeds obtained from the STEREO telescopes allow for a more precise estimation of ICME velocities in the vicinity of the Earth and thus the prediction of their impact on the Earth becomes more accurate.
The third-order polynomial relationship between the observed TT and the speed is indicated by a dashed curve. This fitting can be considered as an empirical model to predict the TT. The difference between the observed and predicted TTs for a given CME speed can be considered as an error by the model. The lower panels show the distribution of errors in determining the TT for this empirical model. These errors are the maximum of the distribution of errors.
Maximum errors were determined as the maximum difference in time between the theoretical model and the observational data. These errors were determined only for non-interacting CMEs. This means that the introduction of the maximal velocity of CME has no effect on a more accurate TT prediction. Figures 3 and 4 and the above discussion refer to SOHO observations.
The values of the average and maximal errors are presented in the panels. As shown in the figure, the TT is significantly related to the average speed. The data points are not scattered around the empirical model represented by a third-order polynomial fit.
This fit is represented by a dashed curve. However, a more significant fact is that in this case the maximal errors are much lower than in the previously presented models Gopalswamy et al. Results for these considerations are shown in Figure 6. Open diamond symbols are for non-interacting and star symbols for interacting CMEs, respectively.
The values of average and maximal errors are presented in the panels. For these events, the average errors are below five hours and the maximal error does not exceed ten hours. Having determined the different initial velocities maximal or average , we are able to calculate TTs directly using kinematic equations of motion. For this purpose, we assume that once the CME reaches the initial velocity it moves with a uniform negative acceleration.
This movement is controlled by drag force due to interaction with the solar wind. Deceleration stops when the CME speed reaches that of the solar wind i. Knowing the acceleration and using the above assumptions, we can easily calculate TT.
The data we have allows us to determine the effective accelerations of CMEs Gopalswamy et al. It can be expressed by the equation. The quadratic relationship between the effective accelerations and the respective speeds are indicated by a solid line. These curves represent empirical models of the effective acceleration of CMEs depending on their initial speeds. They proposed to introduce the effective acceleration of a CME which is computed using a linear fit to two extreme samples of CMEs.
The slowest events have no acceleration and the fastest events are accelerated up to 1 AU. The dashed lines in Figure 7 are linear fits to these groups of CMEs only. The open diamond symbols are for non-interacting and star symbols for interacting CMEs. The solid curves are quadratic fits to the data points. The dashed lines are linear fits for the three slowest and the three fastest events from the entire sample of CMEs.
With the help of different CME acceleration empirical models, we are able to determine the TT using the general equations of motion. The results are displayed in Figures 8 and 9. The scatter plots below display the observed versus predicted TT for the respective acceleration models displayed in Figure 7 , which also shows a comparison of TTs for SOHO observations.
Unfortunately, the results are not very accurate. Average errors for models obtained from the average and maximal speeds are above 13 and 15 hours, respectively. Scatter plots presenting comparison between observed and predicted TT based on acceleration profiles obtained from SOHO observations.
In the respective panels we have results for effective acceleration obtained from a quadratic fit Panels a and b and from a linear fit Panels c and d. Solid lines show the theoretical model when observed and predicted times are identical.
Solid lines show the theoretical model when observed and predicted TT are identical. As seen from the figure, the results seem to be better for effective accelerations determined from linear fits Panels b and d in comparison to those received from quadratic fits Panels a and c. For the effective acceleration models obtained from the average velocities, data points are symmetrically scattered around the solid line, which shows the ideal situation when both times are equal.
In the case of the maximal velocities the data points are mostly placed below this line. This means that on average the predicted TT are lower than observed TT. In this case the results are more promising. However, the best results are for the effective acceleration obtained from the average velocities Panels a and b. As seen from the figures, the average and maximal errors are smallest and the symmetric scatter of the data points around the solid line represents the ideal situation when the observed and predicted TT are equal.
Using these two models the predicted TT are subject to minimal 9. More intense levels of geomagnetic storming are favored when the CME enhanced IMF becomes more pronounced and prolonged in a south-directed orientation.
Some CMEs show predominantly one direction of the magnetic field during its passage, while most exhibit changing field directions as the CME passes over Earth. SWPC forecasters discuss analysis and geomagnetic storm potential of CMEs in the forecast discussion and predict levels of geomagnetic storming in the 3-day forecast.
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