Post-AGB Stellar Evolution Models

• Post-AGB stellar evolution models are detailed frameworks describing rapid stellar evolution after envelope loss, bridging the AGB and white dwarf stages.
• They integrate updated microphysics, including modern opacities, nuclear reaction rates, convective overshooting, and calibrated mass-loss prescriptions.
• These models predict shorter crossing times, revised core mass–luminosity relations, and improved insights into planetary nebula formation and dust-driven winds.

Post-AGB stellar evolution models describe the rapid evolution of low- and intermediate-mass stars ($\sim$0.8–8 $M_\odot$) after they lose their envelopes at the tip of the Asymptotic Giant Branch (AGB), on their way to becoming white dwarfs. The models quantify the crossing timescales, luminosity evolution, nucleosynthetic yields, and mass-loss prescriptions for stars in this phase, predicting observable signatures that connect the final stellar core properties to planetary nebula formation, dust return, and nebular emission in diverse galactic environments.

1. Input Microphysics and Model Ingredients

Post-AGB grids now employ significantly updated microphysics, yielding markedly different evolutionary outcomes than classical calculations. Essential ingredients include:

• Equation of State (EOS): Post-AGB models use the OPAL EOS_2005 for H–He mixtures from the H-rich envelope to the white dwarf cooling track, with an additive–volume approximation outside this region (Bertolami, 2014).
• Radiative Opacities: High-temperature OPAL opacities (Iglesias & Rogers 1996) are combined with low-temperature molecular opacities from Ferguson et al. (2005). C-rich molecular opacities for C/O>1 envelopes are sourced from pre-tabulated Weiss & Ferguson (2009) tables. For dust and dusty outflows, composition-dependent opacities and detailed dust-formation chemistry are also used, e.g. the Ferrarotti & Gail formalism for grain growth and opacity (Dell'Agli et al., 2022).
• Nuclear Network: Updated reaction rates, especially for the slowest CNO-cycle reaction ($^{14}\mathrm{N}(p,\gamma)^{15}\mathrm{O}$) based on recent LUNA measurements (Imbriani et al. 2005), directly impact core growth, third dredge-up, and hot-bottom burning (HBB) efficiency (Bertolami, 2014, Kamath et al., 2021).
• Convection and Overshooting: Mixing-length theory is adopted, typically with solar-calibrated $\alpha_{\rm MLT}\sim 1.7$–1.9. Convective overshooting is treated by a diffusive, exponentially-decaying prescription with parameters tuned to reproduce AGB nucleosynthetic yields and PG1159 surface abundances (e.g., $f=0.0174$ for H-burning core, $f^{\rm PDCZ}=0.005$ for the pulse-driven convective zone, $f^{\rm CE}=0.10$ at the convective envelope base) (Bertolami, 2014, Kamath et al., 2021).
• Mass-Loss Prescriptions: Recent work employs a combination of:
  • Schröder & Cuntz (2005) for RGB/pre-dust AGB winds
  • Blöcker-type superwind for the TP-AGB (e.g., $$\dot M = 4.83 \times 10^{-9}\,\eta_B\, (L/L_\odot)^{2.7}\, (M/M_\odot)^{-2.1}$$), with $\eta_B\sim0.02$–$0.05$;
  • Groenewegen et al. (1998, 2009) period-dependent prescriptions for O-rich and C-rich AGB mass loss;
  • Vassiliadis & Wood (1993) for long-period Mira superwinds;
  • Radiatively-driven CSPN winds for $T_{\rm eff} \gtrsim 10^4$ K using Pauldrach et al. (2004): $$\dot{M} = 9.8 \times 10^{-15} (L/L_\odot)^{1.674} M_\odot\,\rm yr^{-1}$$ (Bertolami, 2014, Dell'Agli et al., 2022, Bertolami, 2015, Kamath et al., 2021).

The combination of these physical updates, in particular better opacities, new nuclear rates, stronger and more physically motivated convective boundary mixing, and revised mass-loss laws, is central to the revision of post-AGB evolutionary tracks.

2. Crossing Timescales and Core Mass–Luminosity Relations

A pivotal outcome of modern post-AGB modeling is the dramatic reduction in predicted post-AGB crossing timescales and an upward revision of the mass–luminosity relation:

• Crossing Timescale ($\tau_{\rm PAGB}$): Defined as the time for the remnant to evolve from $\log T_{\rm eff}=4.0$ to its maximum $T_{\rm eff}$ ("knee" of the HR track). In updated models,

$$\tau_{\rm PAGB}(M_f,Z) \simeq 1.5 \times 10^6\,M_f^{-7.0}\,Z^{0.10} \quad \text{(yr)}$$

valid for $0.52\leq M_f/M_\odot\leq0.80$, $0.001\leq Z \leq 0.02$. This is up to 3–10 times shorter than older predictions from Vassiliadis & Wood (1994) or Blöcker (1995), where $\tau_{\rm PAGB}$ exceeded $10^4$ yr by factors of 3–5 at $M_f\sim0.6\,M_\odot$ (Bertolami, 2014, Gesicki et al., 2016, Bertolami, 2016, Bertolami, 2015).

• Core Mass–Luminosity Relation: Post-AGB plateau luminosities at $\log T_{\rm eff}=4$ obey

$L/L_\odot = 5.0 \times 10^5\,(M_f/M_\odot)^7$

with negligible metallicity dependence. This relation is notably steeper than classical grids (e.g., $$L_{\rm old}\approx4\times 10^4\,(M_f/M_\odot)^{3.5}$$), and the new models are $\sim$0.2 dex more luminous at fixed $M_f$ (Bertolami, 2014, Bertolami, 2016).

A summary table of predicted crossing times and luminosities for selected core masses and metallicities:

$M_f$ ($M_\odot$)	$L_{\log T_{\rm eff}=4}$ ($L_\odot$)	$\tau_{\rm cross}$ (yr)
0.518	2,966	26,407
0.528	3,396–3,396	22,660–25,000
0.560	5,674–6,624	5,319–2,936
0.580	6,800	4,000
0.635	10,524	708
0.797	18,270	≃67

The weak metallicity dependence ($\propto Z^{0.10}$ for $\tau$, $Z^{0.02}$ for $L$) means that at fixed core mass, post-AGB tracks are only modestly affected by $Z$ (Bertolami, 2014).

3. Post-AGB Evolutionary Pathways and Nucleosynthesis

Modern grids track the detailed physics of the envelope ejection, core growth, and late thermal pulses:

• Sequence Phases: Each track traverses (a) envelope contraction at nearly constant luminosity; (b) rapid heating to high $T_{\rm eff}$ over $10^2$–$10^4$ yr; (c) transition across the HRD "knee;" and (d) descent to the WD cooling curve (Gesicki et al., 2016).
• Initial–Final Mass Relation (IFMR): Modern models capture a pronounced IFMR plateau (0.56–0.58 $M_\odot$ for initial masses 1.25–2.0 $M_\odot$), reflecting mass loss, convective boundary mixing, and pre-TP-AGB core growth. This plateau aligns with the dominant peak in the Galactic white dwarf mass distribution (Gesicki et al., 2016).
• Thermal Pulses (LTP, VLTP, AFTP): Final thermal pulses can dramatically alter surface abundances and evolutionary timescales:
  • AFTP (AGB Final TP): Occurs with $M_{\rm env}\sim10^{-2} M_\odot$ while still on the AGB; post-pulse H fraction $X_{\rm H}\sim0.1$–$0.2$.
  • LTP (Late TP): Post-AGB, $X_{\rm H}\lesssim10^{-2}$; transient excursions in $L$ and $T_{\rm eff}$ over decades–centuries (Lawlor, 2023).
  • VLTP: After entry to the WD track, resulting in almost complete H destruction ($X_{\rm H}\sim0$) (Löbling et al., 2019).
• Surface Chemistry: Third dredge-up and HBB strongly impact C, N, s-process enrichment, and the yields of Li, Na, and Al. The most C- and s-process-rich post-AGB stars are those with $1 \lesssim M_{\rm ini} \lesssim 3\,M_\odot$ (Kamath et al., 2021).

4. Mass Loss, Dust Formation, and Outflow Properties

Comprehensive grids now embed time-dependent mass-loss histories and dust chemistry:

• Mass-Loss Regimes: Early AGB and RGB winds are modeled by Reimers (1975), while TP-AGB superwinds use Blöcker (1995) or Vassiliadis & Wood (1993). Dust-driven mass loss is computed self-consistently, with separate prescriptions for O-rich and C-rich stars as a function of L, $T_{\rm eff}$, core mass, and pulsation period (Dell'Agli et al., 2022, Bertolami, 2015).
• Dust Formation: The Ferrarotti & Gail (2006) framework calculates grain growth rates, condensation chemistry, and resulting dust yields. For O-rich outflows, alumina and silicates dominate; for C-rich, amorphous carbon and SiC are central (Dell'Agli et al., 2022, Tosi et al., 2022).
• Wind Dynamics: Dust-driven wind velocities are related to $L$, $\dot M$, and the dust-to-gas ratio; high-mass stars drive faster winds ($v_{\rm exp}\sim15$–20 km s⁻¹) and higher dust production than low-mass stars ($v_{\rm exp}\sim2$–5 km s⁻¹) (Dell'Agli et al., 2022).
• Envelope Ejection: Departure from the AGB is typically defined at $M_{\rm env}\sim0.01$–$0.02\,M_\odot$. Post-AGB shell ejection times and dust condensation radii drive the observable double-peaked SEDs and set infrared excess (Dell'Agli et al., 2022, Tosi et al., 2022).

5. Applications: Planetary Nebulae and Population Synthesis

The revised models have direct consequences for planetary nebula (PN) formation, the planetary nebula luminosity function (PNLF), and interpretations of stellar/galactic populations:

• PN Visibility: Shorter crossing times imply a higher fraction of faint or missing PNe in old populations, addressing longstanding discrepancies between predicted and observed PN counts in, e.g., globular clusters (Bertolami, 2014, Bertolami, 2015).
• PNLF Cutoff: The steep mass–luminosity relation implies the PNLF cutoff is dominated by a narrow mass range, explaining the universality of the cutoff across diverse galactic environments (Bertolami, 2014, Bertolami, 2016).
• Nebular Emission in ETGs: Post-AGB stars are the dominant source of ionizing photons in quiescent systems. Population synthesis models reproduce observed equivalent widths (EW$_{\rm H\alpha}=0.1$–2.5 Å), LIER-like emission line ratios, and UV-optical colors with typical post-AGB parameters (Byler et al., 2019).
• Binary and Circumbinary Disk Effects: In systems with circumbinary disks, disk-fed accretion can stall post-AGB evolution, extend lifetimes by factors of $\sim3$–10, and drive surface chemical depletion patterns via reaccretion of refractory-poor gas. However, these effects generally do not produce crossing times exceeding PN visibility timescales (Martin et al., 7 Feb 2025).

6. Model Uncertainties, Limitations, and Future Work

Despite substantial progress, major uncertainties persist:

• Convective Boundary Mixing: The efficiency and physical implementation of overshoot, especially at the base of the pulse-driven convective zone, remains a primary source of uncertainty. It affects third dredge-up, intershell composition, and timescales for LTP/VLTP events. Discrepancies between predicted and observed oxygen in AFTP stars underscore this issue (Löbling et al., 2019).
• Mass-Loss Prescriptions: The precise functional form and calibration of mass loss at the end of the AGB affect the envelope mass at departure and transition times. Current approaches combine empirical and semi-empirical laws (Reimers, Blöcker, VW93), with $\eta_{\rm B}$ and $\eta_{\rm R}$ free parameters.
• Dust/Wind Coupling: For both single and binary post-AGB stars, the coupling of dust formation, wind acceleration, and radiative transfer is a frontier. High-mass and high-luminosity stars show more efficient dust production and outflow acceleration.
• Binary Effects and Disk Accretion: Stalled evolution and chemical surface depletion from disk accretion are confirmed in some post-AGB binaries but require further parameter-space exploration for comprehensive understanding (Martin et al., 7 Feb 2025).

Adoption of improved microphysics, physically motivated mixing schemes, and self-consistent wind/dust coupling is key for future modeling. 3D hydrodynamical simulations and high-resolution spectroscopy will further constrain mixing and surface enrichment. Addressing these aspects will refine predictions of observable properties (surface abundances, PN emission, PNLF shape) and the integrated light from evolved populations.