Hot bubbles of planetary nebulae with hydrogen-deficient winds - II. Analytical approximations with application to BD+30$^\circ$3639

The first high-resolution X-ray spectroscopy of the planetary nebula BD+30$^\circ$3639 allowed to study X-ray emitting "hot bubbles" (HBs) of planetary nebulae in unprecedented detail. We investigate (i) how diagnostic line ratios are affected by the HB thermal structure and chemical profile, (ii) if the HB chemical composition of BD+30$^\circ$3639 is consistent with the H-poor (H for hydrogen) composition of the stellar photosphere, and (iii) if H-rich nebular matter has been added to this HB by evaporation. We apply an analytical, 1D model for wind-blown HBs with temperature and density profiles based on self-similar solutions including thermal conduction. We construct heat-conduction HBs with chemical stratification. The X-ray emission is computed using the CHIANTI code. Our HB models are used to re-analyse the high-resolution X-ray spectrum of BD+30$^\circ$3639. Our models reproduce the observed line ratios much better than plasmas with single electron temperatures. All the temperature- and abundance-sensitive line ratios are consistent with BD+30$^\circ$3639 X-ray observations for (i) an intervening column density of neutral H, $N_{\rm H} = 0.20_{-0.10}^{+0.05}\times\!10^{22}\rm\ cm^{-2}$, (ii) a characteristic HB X-ray temperature of ${T_{\rm X} = 1.8\pm 0.1~ }$MK together with (iii) a very high neon mass fraction of about 0.05, virtually as high as that of oxygen. For lower values of $N_{\rm H}$, we cannot exclude that the HB of BD+30$^\circ$3639 contains a small amount of evaporated (or mixed) H-rich nebular matter. Given the possible range of $N_{\rm H}$, the fraction of evaporated H-rich matter cannot exceed 3% of the HB mass. The diffuse X-ray emission from BD+30$^\circ$3639 can well be explained by models of wind-blown HBs with thermal conduction and a chemical composition equal to that of the H-poor and carbon-, oxygen-, and neon-rich stellar surface.