A Statistical Model of CME Acceleration

Abstract

An algorithm for automatic approximation of the time dependence x(t) is built for the observed coordinate of the coronal mass ejection (CME) front from the admissible starting point to the first appearance in the field of view of the LASCO coronagraph and further, up to a heliocentric distance of ~25 solar radii (R_S). In the region from the starting point to the first appearance of the CME, two sections are assumed, with uniform (impulsive) acceleration and with uniform motion; then, the motion is approximated by observations. At the beginning of the approximation, either the CME start time is found through the appearance of certain frequencies of radio emissions (RSTN data) and type II and IV radio emissions (sequence characteristics are determined by machine learning), or the start time is determined by averaging over the allowable takeoff area; then the polynomial-ballistic model is optimized. The first and second derivatives x(t) determine the speed and acceleration of the CME at any point of its trajectory. Such an algorithm is necessary to obtain the most accurate kinematic characteristics of CMEs, which can allow one to study the physical, spatial, and temporal relationships between flares and CMEs in all their diversity. Widely used approximation techniques simplify the real CME trajectories x(t), thereby possibly discarding important features of the CME kinematics and flare development in the posteruptive phase. The algorithm was trained and tested on 17 solar flares and associated CMEs, which are known for their powerful proton events with proton energies greater than 300 MeV. The rate of the first occurrence of CMEs turned out to be different from the average rate given in the LASCO catalog, which is important for estimating the energy of flares and CMEs. In 7 out of 17 events, there was acceleration only in the impulsive phase (and then deceleration), while acceleration in the impulsive and posteruptive phases occurred in 10 events. In 4 out of 17 events, CME velocities greater than 2000 km/s were reached at a distance of 20R_S. The accuracy of determining the kinematic characteristics of CMEs can be improved by using additional observations, for example, SDO AIA in the September 10, 2017 event.

Appendices

APPENDIX 1

1.1 INTERVAL ESTIMATION OF THE CME START TIME

Let us consider the procedure for finding an admissible take-off area. There is a standard zero zone configuration, i.e., the triple $\left\{ {{{t}_{1}},{{x}_{1}},{{U}_{1}}} \right\}$, which is the time of the first observation, the coordinate of the first observation and the speed 1–2. The pair $\left\{ {{{t}_{0}},{{r}_{0}}} \right\}$ is the starting time and starting radius, let us call it the starting point. Then, the inequalities following from natural physical constraints $1\,{{{\text{km}}} \mathord{\left/ {\vphantom {{{\text{km}}} {{{{\sec }}^{2}}}}} \right. \kern-0em} {{{{\sec }}^{2}}}} = A_{0}^{{\min }} \leqslant {{A}_{0}} \leqslant$ $A_{0}^{{\max }} = 12\,{{{\text{km}}} \mathord{\left/ {\vphantom {{{\text{km}}} {{{{\sec }}^{2}}}}} \right. \kern-0em} {{{{\sec }}^{2}}}}$ and ${{t}_{0}} \leqslant \xi \leqslant {{t}_{1}}$ are added to the relations

$$\left\{ \begin{gathered} {{A}_{0}} = \frac{{{{U}_{1}}}}{{2\left( {{{t}_{1}} - {{t}_{0}} - \frac{{{{x}_{1}} - {{r}_{0}}}}{{{{U}_{1}}}}} \right)}} \hfill \\ \xi = {{t}_{0}} + \frac{{{{U}_{1}}}}{{A{}_{0}}} \hfill \\ \end{gathered} \right.$$

(App.1)

from (App. 1) it follows that

$$\begin{gathered} A_{0}^{{\min }} \leqslant \frac{{{{U}_{1}}}}{{2\left( {{{t}_{1}} - {{t}_{0}} - \frac{{{{x}_{1}} - {{r}_{0}}}}{{{{U}_{1}}}}} \right)}} \leqslant A_{0}^{{\max }}, \\ {{t}_{1}} - \frac{{{{U}_{1}}}}{{2A_{0}^{{\min }}}} - \frac{{{{x}_{1}} - {{r}_{0}}}}{{{{U}_{1}}}} \leqslant {{t}_{0}} \leqslant {{t}_{1}} - \frac{{{{U}_{1}}}}{{2A_{0}^{{\max }}}} - \frac{{{{x}_{1}} - {{r}_{0}}}}{{{{U}_{1}}}} \\ \end{gathered} $$

and thus,

$${{t}_{1}} - 2\frac{{{{x}_{1}} - {{r}_{0}}}}{{{{U}_{1}}}} \leqslant {{t}_{0}} \leqslant {{t}_{1}} - \frac{{{{x}_{1}} - {{r}_{0}}}}{{{{U}_{1}}}}. $$

(App.2)

Therefore, the admissible take-off area, whose horizontal cut is given by the double inequality (App. 2) is a trapezoid whose right border has the same slope as the straight line 1–2. However, the left boundary of a trapezoid can be defined as the inequality

$${{t}_{1}} - \frac{{{{U}_{1}}}}{{2A_{0}^{{\min }}}} - \frac{{{{x}_{1}} - {{r}_{0}}}}{{{{U}_{1}}}} \leqslant {{t}_{0}}$$

as well as the inequality

$${{t}_{1}} - 2\frac{{{{x}_{1}} - {{r}_{0}}}}{{{{U}_{1}}}} \leqslant {{t}_{0}}.$$

The last relation specifies the slope of the left side of the trapezoid ${{{{U}_{1}}} \mathord{\left/ {\vphantom {{{{U}_{1}}} 2}} \right. \kern-0em} 2}$; this is two times less than the slope of the straight line 1–2, so the trapezoid can be “oblique.” Taking for the range of admissible starting heights 1.1R_S < r₀ < 1.3R_S, we obtain an inequality for the start time t₀:

$$\begin{gathered} \max \left[ {{{t}_{1}} - 2\frac{{{{X}_{1}} - {{r}_{0}}}}{{{{U}_{1}}}};\,\,\,\,{{t}_{1}} - \frac{{{{U}_{1}}}}{{2A_{0}^{{\min }}}} - \frac{{{{X}_{1}} - {{r}_{0}}}}{{{{U}_{1}}}}} \right] \\ \leqslant {{t}_{0}} \leqslant {{t}_{1}} - \frac{{{{U}_{1}}}}{{2A_{0}^{{\max }}}} - \frac{{{{X}_{1}} - {{r}_{0}}}}{{{{U}_{1}}}}. \\ \end{gathered} $$

APPENDIX 2

1.1 A FORMAL DESCRIPTION OF THE ALGORITHM FOR DETERMINING THE CME START TIME

We will look for the indicator function of the fuzzy set of estimated CME start times in the form of a linear combination of elementary Gaussians:

$$\begin{array}{*{20}{c}} {{{\varphi }_{i}}\left( t \right) \equiv \exp \left( { - {{{\left( {\frac{{t - {{\tau }_{i}} - {{D}_{i}}}}{{\delta t}}} \right)}}^{2}}} \right),} \\ {\Phi \left( t \right) = \sum\limits_{i{\kern 1pt} = {\kern 1pt} {\kern 1pt} 1}^m {{{L}_{i}}{{\varphi }_{i}}\left( t \right).} } \end{array} $$

(App.3)

The dependency (App. 3) reflects the “contribution” of time to the time of the ith driver in the plausibility of the occurrence of CME at the point in time t: this contribution is greatest if the time of takeoff of the CME coincides with ${{\tau }_{i}} + {{D}_{i}}$. The coefficient L_i is the weight of the driver and the one for which the sum (App. 3) takes the largest value.

Having ℵ = 17 interesting CMEs, for which it was noted that the flare plasma was heated in flares before they started that emitted SXRs (Soft X-Rays), without a visible increase in the emission level, but the time of the start of the kth CME $t_{0}^{k}$ that we can consider as the beginning of the growth of the derivative of the emissions, we had the task of optimizing the unknown coefficients D and L based on the average approximation of the indicator functions estimated by the maximum (App. 3) starting points of those already found:

$$ \times \,\,\begin{array}{*{20}{c}} {\left[ {{\mathbf{D}}{\mathbf{,L}}} \right] = \arg \min } \\ {\sum\limits_{k{\kern 1pt} = {\kern 1pt} {\kern 1pt} 1}^\aleph {\left| {t_{0}^{k} - \arg \mathop {\max }\limits_{t_{1}^{k} - 60{\kern 1pt} \leqslant {\kern 1pt} t{\kern 1pt} \leqslant {\kern 1pt} t_{1}^{k}} \sum\limits_{i{\kern 1pt} = {\kern 1pt} {\kern 1pt} 1}^{{{N}_{k}}} {{{L}_{i}}{{e}^{{ - {{{\left( {\frac{{t - \tau _{k}^{i} - {{D}_{i}}}}{{\delta t}}} \right)}}^{2}}}}}} } \right|} } \end{array} $$

(App.4)

Here, N_k is the number of drivers preceding the time $t_{1}^{k}$ of the appearance of the first frame of the kth CME. Due to the smallness of the training sample, we had to aggregate the predictor weights L in (app. 4), making them equal within the same group.

APPENDIX 3

1.1 THE INFLUENCE OF THE CME DELAY TIME ON THE WIDTH OF THE INTERVAL ESTIMATION OF THE START TIME

Let us show how the late appearance of a high-speed CME in the LASCO field affects the accuracy of the interval estimation of the launch time t₀. We will consider the size of the admissible set of points {t₀, r₀} (trapezoid), for each value 1.1R_S < r₀ < 1.3R_S defined by THE double inequality (2), from which it trivially follows that for a given takeoff height r₀ THE start time uncertainty indicator $\delta {{t}_{0}}$ equals

$$\begin{gathered} \begin{array}{*{20}{c}} {\delta {{t}_{0}}{\kern 1pt} {\kern 1pt} = {\kern 1pt} {\kern 1pt} \min {\kern 1pt} \left[ {{{t}_{1}} - \frac{{{{U}_{1}}}}{{2A_{0}^{{\max }}}} - \frac{{{{X}_{1}} - {{r}_{0}}}}{{{{U}_{1}}}} - {{t}_{1}} + 2\frac{{{{X}_{1}} - {{r}_{0}}}}{{{{U}_{1}}}}{\kern 1pt} ;} \right.} \\ {\left. {{{t}_{1}} - \frac{{{{U}_{1}}}}{{2A_{0}^{{\max }}}} - \frac{{{{X}_{1}} - {{r}_{0}}}}{{{{U}_{1}}}} - {{t}_{1}} + \frac{{{{U}_{1}}}}{{2A_{0}^{{\min }}}} + \frac{{{{X}_{1}} - {{r}_{0}}}}{{{{U}_{1}}}}} \right]...} \\ { = \min \left[ {\frac{{{{X}_{1}} - {{r}_{0}}}}{{{{U}_{1}}}} - \frac{{{{U}_{1}}}}{{2A_{0}^{{\max }}}};\frac{{{{U}_{1}}}}{{2A_{0}^{{\min }}}} - \frac{{{{U}_{1}}}}{{2A_{0}^{{\max }}}}} \right]} \end{array} \\ = \min \left[ {\frac{{{{X}_{1}} - {{r}_{0}}}}{{{{U}_{1}}}} - \frac{1}{2}{{\xi }^{{\min }}};\frac{1}{2}\left( {{{\xi }^{{\max }}} - {{\xi }^{{\min }}}} \right)} \right].\,\,\,\,\,\,\,\,\,\,\,\, \\ \end{gathered} $$

(App.5)

${{\xi }^{{\min }}}$ and ${{\xi }^{{\max }}}$ in the formula (App. 5) mean, respectively, the minimum and maximum possible time to reach the speed U₁, based on the physical limitations on acceleration. Substituting the characteristic high speed of 3000 km/s, we obtain that

$$\left\{ \begin{gathered} {{\xi }^{{\min }}} = 4.16\,\,\min \hfill \\ {{\xi }^{{\max }}} = 50\,\,\min . \hfill \\ \end{gathered} \right.$$

Obviously, the characteristic value of the term ${{X}_{1}} - {{{{r}_{0}}} \mathord{\left/ {\vphantom {{{{r}_{0}}} {{{U}_{1}}}}} \right. \kern-0em} {{{U}_{1}}}}$ in the first component of the minimum (13) has a value of about 10 min, then half the difference between the maximum and minimum acceleration times is about half an hour. Thus, the size of the uncertainty can be estimated as

$$\delta {{t}_{0}} = \frac{{{{X}_{1}} - {{r}_{0}}}}{{{{U}_{1}}}} - \frac{1}{2}{{\xi }_{{\min }}}.$$

Since it follows from (4), in particular, that

$$\begin{gathered} {{t}_{1}} - 2\frac{{{{X}_{1}} - {{r}_{0}}}}{{{{U}_{1}}}} \leqslant {{t}_{0}}, \\ {{t}_{1}} - {{t}_{0}} \leqslant 2\frac{{{{X}_{1}} - {{r}_{0}}}}{{{{U}_{1}}}}, \\ \frac{{{{X}_{1}} - {{r}_{0}}}}{{{{U}_{1}}}} \geqslant \frac{{{{t}_{1}} - {{t}_{0}}}}{2}, \\ \end{gathered} $$

then we finally obtain that

$$\delta {{t}_{0}} \geqslant \frac{{{{t}_{1}} - {{t}_{0}}}}{2} - \frac{1}{2}{{\xi }^{{\min }}} = \frac{1}{2}\left( {{{t}_{1}} - {{t}_{0}} - {{\xi }^{{\min }}}} \right).$$

That is, the uncertainty of the interval estimate of the start time is not less than half the difference between the delay of the first observation and the minimum acceleration time, and it is clear that with an increase in the delay time the uncertainty will grow linearly.