Models and Formulas

This appendix collects the relations used to compute the quantities reported by the application. Heights $r$ are heliocentric distances in solar radii ($R_\odot = 696\,340$ km), electron densities $n_e$ are in cm$^{-3}$ and frequencies $f$ in MHz.

Intensity scales

CALLISTO digits to dB

the CALLISTO receiver’s logarithmic detector maps 2500 mV onto 256 digits with a slope of 25.4 mV/dB, so that $\Delta I_\mathrm{dB} = \Delta I_\mathrm{digits}\times 2500/(256\times25.4)\approx 0.384$ dB per digit. The Plotutil median-dB recipe used by Median (dB) and the batch processor uses $2500/(255\times25.4)$.

ARTEMIS-IV

$4096$ ADC counts over a 70 dB range: 58.51 counts per dB.

GOES flare class

from the peak 0.1–0.8 nm flux $F$: A for $F<10^{-7}$, B for $10^{-7}\le F<10^{-6}$, C for $10^{-6}\le F<10^{-5}$, M for $10^{-5}\le F<10^{-4}$ and X for $F\ge10^{-4}$ W m$^{-2}$; the number is $F$ divided by the lower limit of the class ($10^{-8}$ for A up to $10^{-4}$ for X), for example M2.5 for $F = 2.5\times10^{-5}$ W m$^{-2}$.

Radio burst analysis

Plasma frequency.

The fundamental emission frequency is the local plasma frequency $$\begin{equation} f_p = 8.98\times10^{-3}\sqrt{n_e}\quad\text{MHz}, \end{equation}$$ so $n_e = (f/8.98\times10^{-3})^2$. For harmonic emission $f = 2f_p$; the application divides the observed frequencies and drift rates by two before computing heights and speeds.

Coronal density models.

Each model is multiplied by the fold number $N$ ($n_e \to N\,n_e$).

For the Newkirk model the emission height has the closed form $$\begin{equation} r = \frac{4.32\,\ln 10}{\ln\!\left[n_e/(N\times4.2\times10^{4})\right]}. \end{equation}$$ For the other models the height is found numerically between $1\,R_\odot$ and 215 $R_\odot$ (1 AU); frequencies outside this range have no height.

Power-law backbone and drift rate.

$$\begin{equation} f(t) = a\,(t-t_0)^{-b},\qquad \frac{\mathrm{d}f}{\mathrm{d}t} = -a\,b\,(t-t_0)^{-b-1}. \end{equation}$$ The goodness of fit is reported as $R^2 = 1 - \sum(f_i-\hat f_i)^2/\sum(f_i-\bar f)^2$ and $\mathrm{RMSE} = \sqrt{\overline{(f_i-\hat f_i)^2}}$.

Shock speed.

$$\begin{equation} v = R_\odot\left|\frac{\mathrm{d}f}{\mathrm{d}t}\right|\left|\frac{\mathrm{d}r}{\mathrm{d}f}\right|, \qquad \frac{\mathrm{d}r}{\mathrm{d}f} = \frac{2\,n_e}{f\,\left|\mathrm{d}n_e/\mathrm{d}r\right|}. \end{equation}$$ The initial values are evaluated at the starting frequency, the 90th percentile of the fitted frequencies; the average values are means along the backbone with their standard errors.

Band splitting

(Vršnak et al. 2002). $$\begin{gather} X = \left(\frac{f_u}{f_l}\right)^2,\qquad M_A = \sqrt{\frac{X\,(X+5)}{2\,(4-X)}},\qquad V_A = \frac{V_s}{M_A},\\ B = V_A\sqrt{\mu_0\,m_p\,n_e}\quad(\text{SI units; reported in gauss}). \end{gather}$$

Image measurements

Circle fit.

For a sphere of fitted radius $r$ resting on the solar surface, the height of its top is $h = 1\,R_\odot + 2r$.

Height–time fits.

$$\begin{align} \text{linear:}\quad & h = h_0 + v\,t,\\ \text{quadratic:}\quad & h = h_0 + v_0\,t + \tfrac12 a\,t^2,\\ \text{cubic:}\quad & h = h_0 + v_0\,t + \tfrac12 a_0\,t^2 + \tfrac16 j\,t^3. \end{align}$$ Uncertainties are propagated from the covariance of the fitted coefficients.

Three-dimensional CME models

GCS

(Thernisien et al. 2006; Thernisien 2011). Parameters: Stonyhurst longitude and latitude of the propagation direction, tilt, leading-edge height $h$, half-angle $\alpha$ and aspect ratio $\kappa$. Intrinsic widths: $$\begin{equation} w_\text{face-on} = 2\left[\alpha + \arcsin\kappa\right],\qquad w_\text{edge-on} = 2\arcsin\kappa. \end{equation}$$

Shock spheroid and ellipsoid

(Kouloumvakos et al. 2022). With the radial semi-axis $a$ and lateral semi-axes $b$ and $c$: $$\begin{equation} h = r_\text{center} + a,\qquad \kappa = \frac{b}{h - 1\,R_\odot},\qquad \varepsilon = \begin{cases} +\sqrt{1-(b/a)^2} & a > b,\\ -\sqrt{1-(a/b)^2} & a < b,\end{cases} \qquad \alpha = \frac{b}{c}. \end{equation}$$ A spheroid has $c = b$ ($\alpha = 1$, no tilt).