The result depends on the proportion of X-rays creating a charge pattern with i pixels called $\alpha_i$ in Ballet (1999). Here $\alpha_i = [0.708;0.233;0.021;0.038]$ (from the data, averaged over energy).

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We note that a power-law ejecta density structure running into a circumstellar medium would also lead to a temperature slope compatible with the observations but with a too low temperature to account for the Fe K line (Decourchelle & Ballet 1994). So this case is rejected and no more considered in the following.

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... parameters

$E_{{\rm SN}}(M_{\odot}/M_{{\rm ej}})^{2/3} = 0.53 \times 10^{51}$ ergs and $\rho_0 = 0.42$ amu cm^-3 for a power-law index of the initial outer density profile in the ejecta n=9 and a flat ambient medium.

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...)

We grouped the channels of these spectra until we have a signal-to-noise ratio greater than $3 \sigma$ in each bin.

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... abundance

For solar abundances, $\sum_i z_{i} Ab_i Ab_{\odot,i} \simeq 1.21$ .

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... shock

For Kepler's SNR, we set $\theta_{\rm s} = 120 \hbox{$^{\prime\prime}$ }$ from the 4-6 keV continuum image by choosing the same coordinates center as in Hughes (1999). This angular radius corresponds to $r_{\rm s} = 2.8 \pm 0.8$ pc.

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...)

Using Kepler's SNR parameters in the region A1 $D = 4.8 \pm 1.4 \; {\rm kpc}$ , $\theta_{\rm s} = 120 \hbox{$^{\prime\prime}$ }$ , $\theta_{{\rm A1}} = 14 \hbox{$^{\prime\prime}$ }$ , $t_{\rm s} = 400 \; {\rm yr}$ , $v_{\rm s} = 4550 \; {\rm km~s^{-1}}$ , $\alpha = 0.008$ , $\Omega = 4$ leads to the lower limit $\xi_{{\rm min}} \simeq 0.43 \: \varepsilon \pm 0.13$ .

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