Temperature of the Source Plasma in Gradual Solar Energetic Particle Events

Abstract

Scattering during interplanetary transport of particles during large, “gradual” solar energetic-particle (SEP) events can cause element abundance enhancements or suppressions that depend upon the mass-to-charge ratio [$A/Q$] of the ions as an increasing function early in events and a decreasing function of the residual scattered ions later. Since the $Q$-values for the ions depend upon the source plasma temperature [$T$], best fits of the power-law dependence of enhancements vs. $A/Q$ can determine $T$. These fits provide a fundamentally new method to determine the most probable value of $T$ for these events in the energy region $3\,\mbox{--}\,10~\mbox{MeV}\,\mbox{amu}^{-1}$. Complicated variations in the grouping of element enhancements or suppressions match similar variations in $A/Q$ at the best-fit temperature. We find that fits to the times of increasing and decreasing powers give similar values of $T$, in the range of 0.8 – 1.6 MK for 69 % of events, consistent with the acceleration of ambient coronal plasma by shock waves driven out from the Sun by coronal mass ejections (CMEs). However, 24 % of the SEP events studied showed plasma of 2.5 – 3.2 MK, typical of that previously determined for the smaller impulsive SEP events; these particles may be reaccelerated preferentially by quasi-perpendicular shock waves that require a high injection threshold that the impulsive-event ions exceed or simply by high intensities of impulsive suprathermal ions at the shock. The source-temperature distribution of ten higher-energy ground-level events (GLEs) in the sample is similar to that of the other gradual events, at least for SEPs in the energy-range of $3\,\mbox{--}\,10~\mbox{MeV}\,\mbox{amu}^{-1}$. Some events show evidence that a portion of the ions may have been further stripped of electrons before the shock acceleration; such events are smaller and tend to cluster late in the solar cycle.

Appendices

Appendix A: Scattering as a Power-Law in $A/Q$?

In this article we assumed that element enhancements or suppressions caused by scattering during transport have approximately a power-law dependence upon $A/Q$. How good is this approximation?

Given that the scattering mean free path [$\lambda_{\mathrm{X}}$] depends upon $(A_{\mathrm{x}} /Q_{\mathrm{x}})^{\alpha}$ but is independent of distance [$R$], we can use the expression for the solution to the diffusion equation (from Equation (5) in Tylka et al. 2012 or Equation (C3) in Ng, Reames, and Tylka 2003 based upon Parker 1963) to write the enhancement of element X relative to O as a function of time [$t$] as

$$ \mathrm{X}/\mathrm{O} = r^{-3/2} \exp \bigl\{ (1-1/r) \tau /t \bigr\} , $$

(1)

where $r = \lambda_{\mathrm{X}} / \lambda_{\mathrm{O}} = ( (A_{\mathrm{x}} /Q_{\mathrm{x}}) / (A_{\mathrm{O}} /Q_{\mathrm{O}}) )^{\alpha}$, and we have redefined the parameter $\tau$, factoring the $r$-dependence from it. In Figure 13 we plot $\mathrm{X}/\mathrm{O}$ vs. the relative value of $(A_{\mathrm{x}} /Q_{\mathrm{x}}) / (A_{\mathrm{O}} /Q_{\mathrm{O}})$ for several values of $\tau/t$. The value of $\alpha =0.6$ was used in this sample. This is nearly twice the value of $\alpha = 1/3$ for scattering by a Kolmogorov spectrum of waves. The nonlinearity increases with $\alpha$.

Figure 13

The enhancement of an element [$\mathrm{X}/\mathrm{O}$] is shown as a function of its value of $A_{\mathrm{x}} /Q_{\mathrm{x}}$ relative to that of the reference [O] for several values of the time variable $\tau/t$. A dashed fit line is plotted for comparison with the curve for $\tau/t = 8$.

For $\mathrm{X} = \mbox{Fe}$, experimentally observed enhancements rarely exceed ten, implying that $\tau/t \leq 8$, and $A/Q$ for Fe rarely exceeds that of O by a factor of four (see Figure 1). The $A/Q$ dependence at late times when $\tau/t = 0$ is a power law. A dashed fit line is plotted for comparison with the curve for $\tau/t = 8$. The discrepancy reaches ${\approx}\,20~\%$ at most and varies smoothly across the range. It cannot regroup elements with different values of $A/Q$ to compensate for the complex variation seen in Figure 1.

Formally, we can achieve a linear approximation if we remember the expansion of $\log x = (1-1/x) + (1-1/x)^{2} /2 +\cdots$ (for $x > 1/2$). Using only the first term to replace $1-1/r$ with log $r$ in Equation (1), we have

$$ \mathrm{X}/\mathrm{O} \approx r^{\tau/ t - 3/2} $$

(2)

for $r > 1/2$ as an expression for the power-law dependence of enhancements on $A/Q$ of species X. Since we can choose He or O as a reference, we can always ensure that $r \geq 1$.

Appendix B

Table 2 shows properties of the gradual SEP events for which we were able to determine source plasma temperatures. Successive columns show the source CME onset time, the end time of SEP accumulation, the CME speed, associated flare location, GLE? ($\mbox{true}=1$), stripped ions present? ($\mbox{true}=1$), and the derived source plasma temperature.

Table 2 Source plasma temperatures of gradual SEP events