Interpretation of the expansion law of planetary nebulae

We reproduce the expansion velocity--radius ($V_{\rm{exp}}$--$R_{\rm{n}}$) relation in planetary nebulae by considering a simple dynamical model, in order to investigate the dynamical evolution and formation of planetary nebulae. In our model, the planetary nebula is formed and evolving by interaction of a fast wind from the central star with a slow wind from its progenitor, the AGB star. In particular, taking account of the mass loss history of the AGB star makes us succeed in the reproduction of the observed $V_{\rm{exp}}$-$R_{\rm{n}}$ sequence. As a result, examining the ensemble of the observational and theoretical evolution models of PNe, we find that if the AGB star pulsates and its mass loss rate changes with time (from $\sim 10^{-6.4}M_{\odot}$ yr$^{-1}$ to $\sim 10^{-5}M_{\odot}$ yr$^{-1}$), the model agrees with the observations. In terms of observation, we suggest that there are few planetary nebulae with larger expansion velocity and smaller radius because the evolutionary time-scale of such nebulae is so short and the size of nebulae is so compact that it is difficult for us to observe them