Analytical Solutions for the Dynamics of Planetary Nebulae with and without Common Envelope Evolution

TL;DR

This work provides analytical, self-similar solutions for planetary nebula expansion driven by a time-varying wind power, incorporating both isolated AGB progenitors and Common Envelope Evolution (CEE) scenarios. By modeling the hot shocked wind region and the swept-up shell with simple energy and momentum equations, closed-form expressions for the shell radius $R_s$ and expansion velocity $\dot R_s$ are derived as functions of time and a power-law index $\delta$ in the mechanical luminosity $L_w \propto t^{\delta}$. The results show that single AGB progenitors reproduce a narrow velocity range ($20-30$ km s$^{-1}$), while CEE configurations with different envelope-mass fractions can explain a broader range of observed PNe, including very fast and very slow cases. Fitting to observed PNe suggests a substantial fraction originate from CEE, with implications for envelope ejection efficiency, nebular morphology, and viewing geometry; the framework offers a practical tool to interpret PN kinematics and guides future morphokinematic studies.

Abstract

We present new analytical solutions for the dynamics of planetary nebulae. These equations consider the temporal variation of the mechanical luminosity as well as the common envelope evolution scenario. By comparing a database of nebulae with these solutions, a large portion of planetary nebulae can be better explained by the common envelope evolution scenario, especially the fast and slow ones. Single AGB stellar models can only reproduce nebulae with expansion velocities between 20 and 30 km/s.

Figures (5)

• Figure 1: Mechanical Luminosity as a function of time from models by Vas94. The numbers are the mass of the central star.
• Figure 2: Results for the single AGB scenario. The solid squares correspond to the data in Sabba84, while the open triangles to the data by Icker21.
• Figure 3: Same as Figure 2 for the case of a CEE scenario with 75 % of the envelope mass in the shell.
• Figure 4: Same as Figure 2 for the case of a CEE scenario with 25 % of the envelope mass in the shell.
• Figure 5: Same as Figure 2 for the case of a CEE scenario with 10 % of the envelope mass in the shell.