Miller–Bertolami Models in Post-AGB Evolution

• Miller–Bertolami Models are advanced evolutionary sequences for post-AGB stars that incorporate updated microphysics and produce faster, brighter central stars of planetary nebulae.
• They reduce post-AGB timescales by factors of 3–10 compared to classical models, ensuring timely nebula ionization and improved alignment with observational data.
• Enhanced convective mixing and efficient third dredge-up in these models lead to carbon enrichment, increasing hydrogen burning rates and elevating luminosities by 0.1–0.3 dex.

The term “Miller–Bertolami Models” refers to advanced evolutionary sequences for the post-asymptotic giant branch (post-AGB) phase of low- and intermediate-mass stars, with particular focus on central stars of planetary nebulae (CSPNe). These models, computed by Miller Bertolami (2016), integrate updated microphysics—including modern radiative opacities and nuclear reaction rates—and refined treatments of macrophysics such as convective boundary mixing and mass loss. The result is a new class of evolutionary tracks for CSPNe that are both substantially faster and brighter than their classical predecessors, with direct implications for the interpretation of planetary nebulae luminosity, white dwarf formation, and stellar evolutionary timescales.

1. Foundational Advances in Post-AGB Modeling

The Miller–Bertolami models were developed in response to persistent discrepancies between predictions of classical post-AGB tracks—principally those of Vassiliadis & Wood (1994) and Bloecker (1995)—and observational data concerning planetary nebulae and compact objects. Traditional tracks underestimated both the luminosity and speed of post-AGB evolution for remnants of a given mass across all metallicities. The inclusion of contemporary microphysics led to a marked recalibration of evolutionary rates and final star properties, effectively establishing a new benchmark for modeling CSPNe.

2. Evolutionary Timescales: Quantitative Acceleration

A principal outcome of the Miller–Bertolami revision is the significant reduction in post-AGB crossing timescales. The timescale $\tau_{\mathrm{cr}}$, defined as the interval during which the star moves from $T_{\mathrm{eff}} \simeq 10^4~\mathrm{K}$ to the temperature maximum on the Hertzsprung–Russell diagram, is observed to decrease by factors of $3$ to $10$ relative to classical calculations for all mass and metallicity regimes: $$\tau_{\mathrm{cr}} \propto \frac{M_{\mathrm{env}}^{(\mathrm{crit})}}{L_{\mathrm{burn}}}$$ Here, $M_{\mathrm{env}}^{(\mathrm{crit})}$ is the critical envelope mass at departure from the AGB, and $L_{\mathrm{burn}}$ is the shell burning luminosity. Enhanced carbon pollution of the envelope via the third dredge-up reduces $M_{\mathrm{env}}^{(\mathrm{crit})}$ while simultaneously raising $L_{\mathrm{burn}}$, thereby accelerating envelope consumption and post-AGB evolution. This rapid transit alters the conditions for nebula ionization and affects the predicted observable population of planetary nebulae.

3. Increased Luminosity: Impact and Mechanism

The models yield CSPNe with luminosities elevated by $0.1$–$0.3$ dex compared to previous tracks for identical remnant masses. This enhancement arises from improved convective boundary mixing treatments during AGB thermal pulses, which reinforce third dredge-up efficiency. The resultant carbon pollution of the H-rich envelope increases the hydrogen shell burning rate, directly impacting the luminosity. The mechanism is encapsulated in

$$M_{\mathrm{env}}^{(\mathrm{crit})} \propto f(L_{\mathrm{burn}}, \text{composition})$$

where envelope composition, especially carbon enrichment, is critical for both $L_{\mathrm{burn}}$ and the evolutionary endpoint.

4. Comparative Framework: Classical Versus Modern Models

The following table summarizes core contrasts between the Miller–Bertolami models and classical models:

Model	Evolutionary Timescale ($\tau_{\mathrm{cr}}$)	Luminosity (dex)
Classical (VW94/B95)	Baseline	Baseline
Miller–Bertolami	$3$–$10\times$ shorter	$+0.1$–$0.3$

Classical models assumed slower envelope depletion and lower maximum luminosity, in part due to less efficient mixing and older opacities. The newer tracks decouple the critical envelope mass from late-AGB mass loss rate, linking it instead to envelope composition and burning properties. This insensitivity to final AGB mass loss ($\dot{M}$) is a key outcome.

5. Astrophysical Implications

a) Planetary Nebula Formation and Ionization

Faster evolution enables central stars to ionize ejected nebular envelopes before dissipation, particularly for lower mass remnants ($\sim0.58~M_\odot$), bridging a gap in earlier model predictions.

b) Planetary Nebula Luminosity Function (PNLF)

Higher peak luminosities and shorter evolutionary phases render the bright end of the PNLF less dependent on progenitor age, allowing older populations to host bright planetary nebulae. Classical models struggled to account for such observational facts.

c) White Dwarf Population Synthesis

Remnant mass distributions from Miller–Bertolami tracks align more closely with observations, notably the Galactic Bulge’s white dwarf demographic, refining the initial–final mass relation calibration.

d) Hydrodynamics and Asteroseismology

Higher luminosities and faster evolution inform the interpretation of X-ray emission from planetary nebulae and are consistent with asteroseismological findings concerning central star absence and formation timing in select systems.

6. Conceptual Summary and Broader Context

The Miller–Bertolami models differentiate themselves by leveraging updated physical inputs to yield evolutionary tracks for low- and intermediate-mass stars that are demonstrably faster and brighter than precedents. The fundamental scaling

$$\tau_{\mathrm{cr}} \propto \frac{M_{\mathrm{env}}^{(\mathrm{crit})}}{L_{\mathrm{burn}}}$$

emphasizes the role of envelope composition and nuclear burning rates, recalibrating the transition from AGB stars to white dwarfs. By providing a more cohesive account of planetary nebula and white dwarf properties, these models impose new constraints on late-stage stellar mass loss and nucleosynthesis, impacting not only theoretical studies but also population synthesis and multiwavelength observations of evolving stellar systems.