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. 
Hiroko Matsumoto, Tsubasa Fukue, and Hideyuki Kamaya