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$\begin{figure} \resizebox{3.25in}{!}{\includegraphics[clip]{8346fig1.eps}} \end{figure}$ Figure 1: Baryon mass vs. equatorial radius for hydrostatic equilibrium configurations calculated for three types of EOSs of dense matter, described in the text. Solid line - stable; dotted line - unstable configurations. Thick lines correspond to the non-rotating models, thin lines to the rigidly rotating configurations with a fixed total stellar angular momentum $J=1.2\times GM^2_\odot /c$ .
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In the text

$\begin{figure} \resizebox{3.25in}{!}{\includegraphics[clip]{8346fig2.eps}} \end{figure}$ Figure 2: Zoomed fragment of Fig. 1, in the vicinity of the phase transition. For other explanations see the text.
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In the text

$\begin{figure} \resizebox{2.75in}{!}{\includegraphics[clip]{8346fig3.eps}} \end{figure}$ Figure 3: Transition from a one-phase configuration ${\cal C}$ with a meta-stable core of radius $r_{\rm _N}$ to a two-phase configuration ${\cal C}^\star$ with a S-phase core with a radius $r_{\rm _S}$ . These two configurations have the same baryon mass $M_{\rm b} =M_{\rm b}^\star$ and total angular momentum $J=J^\star$ . Notice that in contrast to the case of a first order phase transition with $\lambda <\lambda _{\rm crit}$ studied in Zdunik et al. (2007, see Fig. 2 of that paper), here $r_{\rm _N}$ is much smaller than $r_{\rm _S}$ (in the limiting case $r_{\rm _N}=0$ while $r_{\rm _S}$ is sizeable).
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In the text

$\begin{figure} \par\resizebox{3.25in}{!}{\includegraphics[clip]{8346fig4.eps}} \end{figure}$ Figure 4: Total baryon mass $M_{\rm b}$ of hydrostatic stellar configuration, versus central pressure $P_{\rm c}$ , at fixed stellar angular momentum J. The solid line denotes stable states, dash-dot line - the states that are metastable with respect to the N $\longrightarrow$ S transition. For a central pressure $P_{\rm nucl}$ the S-phase nucleates in the super-compressed core of configuration ${\cal C}$ , and this results in a transition ${\cal C}\longrightarrow {\cal C}^*$ into a stable configuration with a S-phase core and central pressure $P^*_{\rm c}$ . Both configurations ${\cal C}$ and ${\cal C}^*$ have the same baryon mass $M_{\rm b}$ .
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In the text

$\begin{figure} \resizebox{8.8cm}{!}{\includegraphics[angle=0]{8346fig5.eps}} \end{figure}$ Figure 5: (Color online) The energy release due to the core-quake of a rotating neutron star as a function of the dimensionless equatorial radius of the S-phase core, ${\bar r}_{\rm _S}$ . Calculations are performed for the EOS PolS1 from Table 1. Different curves correspond to the different values of the total angular momentum of rotating star, fixed along each curve, $J=\widetilde{J}\;GM^2_\odot/c=(0, 0.1,~\ldots,~1.3)\times GM^2_\odot/c$ , from the right-most curve to the left-most curve.
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In the text

$\begin{figure} \par\resizebox{8.8cm}{!}{\includegraphics[angle=0]{8346fig6.eps}} \end{figure}$ Figure 6: (Color online) The energy release due to the mini-collapse of a rotating neutron star as a function of the overpressure $\delta \overline {P}$ of the N phase of the matter in the center of the star, for three EOSs from Table 1. The bottom curve is calculated for the PolW EOS with a weak first-order phase transition, so that $E_{\rm rel}(0)=0$ . The middle curve and the top curve are for the PolS1 and PolS2 EOSs, respectively, with strong first-order phase transitions. The points of different color correspond to the different values of total angular momentum of the rotating star, $J=\widetilde{J}\;GM^2_\odot/c=(0, 0.1,\ldots,1.3)\times GM^2_\odot/c$ . For a given EOS, results for all rotating configurations can be very well approximated by a single curve, independent of J.
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In the text

$\begin{figure} \par\resizebox{8.8cm}{!}{\includegraphics[angle=0]{8346fig7.eps}} \end{figure}$ Figure 7: (Color online) The difference between the energy release due to the mini-collapse of a rotating neutron star and a non-rotating one as a function of the overpressure $\delta \overline {P}$ for the PolS1 EOS (the middle curve in Fig. 6). The points of different color correspond to the different values of total angular momentum of the rotating star, $J=\widetilde{J}\;GM^2_\odot/c=(0, 0.1,\ldots,1.3)\times GM^2_\odot/c$ from bottom to top. Note the two orders of magnitude difference in the scale of the vertical axis in this figure and Fig. 6.
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In the text

$\begin{figure} \par\resizebox{8.8cm}{!}{\includegraphics[]{8346fig8.eps}} \end{figure}$ Figure 8: Top panel: the ratio of polar radial coordinate to the equatorial radial coordinate ratio. Bottom panel: the ratio of the kinetic energy, T and the absolute value of the potential energy, W, for the reference stellar configurations (central pressure P₀, Table 1) consisting of the N phase of dense matter, described by the polytropic EOS, PolN, of Table 1. Large dots correspond to the values of the total stellar angular momentum, $J=\widetilde{J}GM^2_\odot/c= (0.1,\ldots,1.5)\times GM^2_\odot/c$ , which were used in Figs. 5, 6.
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In the text

$\begin{figure} \par\resizebox{8.8cm}{!}{\includegraphics{8346fig9.eps}} \end{figure}$ Figure 9: EOS with first-order phase transition due to kaon condensation described and used in Sect. 3.2. in the present paper. The phase transition occurs at $P_0=4.604\times 10^{34}~{\rm erg~cm^{-3}}$ . On the N-phase side, $n_{\rm _N}=0.3375~{\rm fm^{-3}}$ and $\rho_{\rm _N}=5.973\times 10^{14}~{\rm g~cm^{-3}}$ . On the kaon-condensed S-side, $n_{\rm _S}=0.6122~{\rm fm^{-3}}$ and $\rho_{\rm _N}=1.125\times 10^{15}~{\rm g~cm^{-3}}$ . Therefore, $\lambda=\rho_{\rm _S}/\rho_{\rm _N}=1.884$ , which is greater than $\lambda _{\rm crit}= {3\over 2}(1+P_0/\rho _{_{\rm N}}c^2)=1.629$ .
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In the text

$\begin{figure} \resizebox{8.8cm}{!}{\includegraphics[angle=0]{8346fig10.eps}} \end{figure}$ Figure 10: (Color online) The total energy release, $E_{\rm rel}$ , due to the collapse of a rotating neutron star, implied by kaon condensation, as a function of the metastability (overpressure) of the normal phase of the matter in the center of the star for the EOS in Fig. 9. The points of different color correspond to the different values of total angular momentum of the rotating star. The results for all rotating configurations can be very well approximated by one curve. Solid lines: kinetic energy increase, $\Delta T=T^\star - T$ , for fixed values of $J=\widetilde{J}GM^2_\odot/c= (0.1,\ldots,0.8)\times GM^2_\odot/c$ . The bottom line is for $\widetilde{J}=0.1$ , and top line for $\widetilde{J}=0.8$ . Note the different definition of the sign of $E_{\rm rel}=(M-M^\star )c^2$ and $\Delta T=T^\star - T$ .
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In the text

$\begin{figure} \resizebox{3.25in}{!} {\includegraphics[clip]{8346fig11.eps}} \end{figure}$ Figure 11: Three families of neutron stars in the $M_{\rm b}-R_{\rm eq}$ plane and the transition ${\cal C}_1\longrightarrow {\cal C}_1^\star$ via overcoming the energy barrier.
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In the text

$\begin{figure} \resizebox{3.25in}{!} {\includegraphics[clip]{8346fig12.eps}} \end{figure}$ Figure 12: (Color online) Energy release, Eq. (1), versus overpressure $\delta \overline {P}=(P_{\rm c}-P_0)/P_0$ , for transitions ${\cal C}_1\longrightarrow {\cal C}_1^\star$ and ${\cal C}_1\longrightarrow {\cal C}_1^{\star \prime }$ visualized in the $M_{\rm b}-R$ plane in Fig. 11. Points of different color correspond to transitions with different angular momentum $J=\widetilde{J}GM_\odot^2/c^2=(0, 0.1, \dots, 0.9)\times GM_\odot^2/c^2$ . Solid line - results for J=0(non-rotating stars). Point A corresponds to ${\cal C}_0\longrightarrow {\cal C}_0^\star$ . Point B corresponds to ${\cal C}_0\longrightarrow {\cal C}^{\star \prime }_0= {\cal C}_0$ , point D to ${\cal C}_{\rm min}\longrightarrow {\cal C}_{\rm min}^\star$ , and point C to ${\cal C}_2\longrightarrow {\cal C}_2^\star$ .
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In the text

$\begin{figure} \resizebox{3.25in}{!}{\includegraphics[clip]{8346fig2.eps}} \end{figure}$	Figure 2: Zoomed fragment of Fig. 1, in the vicinity of the phase transition. For other explanations see the text.
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$\begin{figure} \resizebox{3.25in}{!} {\includegraphics[clip]{8346fig11.eps}} \end{figure}$	Figure 11: Three families of neutron stars in the $M_{\rm b}-R_{\rm eq}$ plane and the transition ${\cal C}_1\longrightarrow {\cal C}_1^\star$ via overcoming the energy barrier.
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