On the shape of the ascending branch of the light curves of Long Period Variables

1. Problem Statement

Long Period Variable (LPV) stars, particularly Mira variables on the Asymptotic Giant Branch (AGB), exhibit complex light curves that encode information about their internal physics and evolutionary state. Previous studies (Paper I and Paper II by the same authors) established correlations between light curve properties (period, amplitude, width, irregularity) and stellar evolution parameters (effective temperature, mass loss, isotopic ratios). However, these studies primarily focused on the width of the light pulse and the overall asymmetry of the cycle, leaving the detailed shape of the ascending branch (the rise from minimum to maximum brightness) largely unexplored.

The central problem addressed is the lack of a quantitative description of the ascending branch's morphology and its potential to reveal specific evolutionary stages, such as the occurrence of Third Dredge-Up (TDU) events, the transition from oxygen-rich (M-type) to carbon-rich (C-type) stars, and the nature of pulsation instabilities.

2. Methodology

Sample Selection

The authors analyzed a refined sample of 93 Long Period Variables selected from a larger dataset of 116. The selection criteria required:

• High-quality, dense observational data suitable for detailed profile analysis.
• Well-defined spectral types, specifically distinguishing between:
  • Mno: Oxygen-rich stars with no detectable Technetium (Tc) in their spectra (indicating no strong TDU events).
  • Myes: Oxygen-rich stars with detectable Tc (indicating strong TDU events).
  • S: S-type stars (intermediate chemistry).
  • C: Carbon-rich stars.
• Special attention was given to stars with periods between 400–450 days to fill gaps in previous samples.

Parameterization of the Ascending Branch

To characterize the shape of the ascending branch, the authors introduced a new two-parameter model:

1. Normalization: Light curves were normalized in time ($x$, from 0 at minimum to 1 at maximum) and magnitude ($y$, from 0 to 1).
2. Baseline Model: A standard sinusoidal profile was defined as $f(x) = 0.5(1 - \cos(\pi x))$.
3. Deviation Terms: The actual profile $S(x)$ was modeled as a deviation from the sine wave using two orthogonal functions:
   • $g(x) = x(x-1)\sin(2\pi x)$ (Antisymmetric term, affecting the slope).
   • $h(x) = \cos(2\pi x) - 1$ (Symmetric term, affecting the curvature/humps).
4. New Parameters:
   • $p$: Measures the deviation in the slope (derivative) at the midpoint ($x=0.5$).
   • $q$: Measures the deviation in the value (magnitude) at the midpoint ($x=0.5$).
   • The fitting function is $S(x) = f(x) + p \cdot g(x) + q \cdot h(x)$.

Analysis Procedure

• The authors performed $\chi^2$ minimization fits for all 93 curves to determine optimal $p$ and $q$ values.
• They investigated the robustness of the fit by allowing small shifts in the timing of minima and maxima ($\epsilon_1, \epsilon_2$), finding that while individual shifts occurred, the relative positions of stars in the parameter space remained stable.
• The study correlated the new parameters ($p, q$) with existing stellar parameters: Period ($P$), Amplitude ($A$), Width ($W$), Irregularity ($\Delta$), Color Index ($C$), and Spectral Type.

3. Key Contributions

1. Novel Parameterization: The introduction of parameters $p$ and $q$ provides a compact, quantitative description of the ascending branch shape, specifically capturing features like "humps" (abrupt changes in slope) that were previously difficult to categorize.
2. Evolutionary Tracking via $q$: The paper identifies $q$ as the primary tracer of stellar evolution along the AGB. It demonstrates a strong, continuous correlation between $q$ and fundamental stellar properties (period, temperature, mass loss), suggesting that $q$ evolves monotonically as the star ages on the AGB.
3. Resolution of "Outliers": The study identifies a specific group of 8 Mno stars (e.g., R Dra, R UMa) that were previously unexplained. These stars share similar periods and temperatures to standard Mno stars but possess distinct ascending branch shapes (outliers in the $p$ vs. $q$ plane), suggesting a specific sub-population or evolutionary branch.
4. Hump Phenomenon: The parameterization successfully describes the presence of "humps" on the ascending branch not as anomalies, but as expected episodes in the continuous evolution of the pulse profile, analogous to the Hertzsprung progression in Cepheids.

4. Key Results

Evolution of Mno Stars (No TDU)

• Trend: As Mno stars evolve (period increases, temperature decreases), the parameter $q$ increases.
• Correlations: Strong positive correlations exist between $q$ and Period ($P$), Color Index ($C$), and Irregularity ($\Delta$). Conversely, $q$ is negatively correlated with the Width parameter ($W$).
• Outliers: A distinct group of 8 Mno stars deviates from the main trend in the $p$-$q$ plane, characterized by specific ascending branch shapes, though their physical differences remain unclear.

Transition to Myes and S Stars (With TDU)

• Amplitude Jump: The transition from Mno to Myes (detectable Tc) is marked by a sudden, significant increase in oscillation amplitude (by >1 magnitude).
• Evolutionary Paths:
  • Group My1/S1: Stars that transitioned from Mno at a later evolutionary stage (higher $q$). They retain high amplitudes and occupy regions distinct from the warm Mno stars.
  • Group My2/S2: Stars that transitioned early, remaining close to the warm Mno region in parameter space but with higher periods.
  • Group S3: A distinct group of S-type stars with very low $q$ values and low color indices, suggesting a unique evolutionary path possibly linked to lower mass-loss rates or specific dust properties.

Carbon-Rich (C) Stars

• Transition: The shift from S to C spectral types involves a drastic drop in amplitude (~2 magnitudes) and an increase in the width parameter $W$.
• Correlations: For C stars, $q$ correlates with effective temperature and mass loss rates.
  • High $q$ (Positive): Associated with cooler stars, high mass loss, and high isotopic ratios ($^{12}C/^{13}C$).
  • Low $q$ (Negative): Associated with warmer stars, low mass loss, and lower isotopic ratios.
• Continuity: The data suggests a potential evolutionary continuity where S3 stars may evolve into low-$q$ Carbon stars, while S1/S2 stars evolve into high-$q$ Carbon stars.

Special Cases

• Hot Bottom Burners (HBB): Stars like R Cen and R Nor (Mno type but massive) occupy a unique region in the $p$-$q$ plane (very low $p$), distinct from standard AGB stars.
• Irregularity: The study distinguishes between magnitude irregularity ($\Delta$) and shape irregularity ($\delta$, the RMS deviation of the profile). Stars like T Cas show massive shape changes (humps, double maxima) that are not captured by traditional irregularity parameters.

5. Significance and Conclusion

This work significantly advances the understanding of LPV evolution by linking the morphology of the light curve's ascending branch directly to the internal physics of the star.

• Physical Insight: The strong correlation between the shape parameter $q$ and stellar evolution (period, temperature, mass loss) implies that the ascending branch shape is a sensitive probe of the star's internal structure and the convective envelope's properties.
• Evolutionary Map: The authors propose a unified evolutionary picture in the Period vs. $q$ plane. Stars evolve from warm Mno states (low $q$) along different trajectories depending on their initial mass and the timing of TDU events, eventually transitioning to S and C types.
• Limitations: The authors acknowledge that while correlations are strong, the underlying physical mechanisms governing the transition between stable and unstable pulsation regimes (instability strips in the HR diagram) remain poorly understood. The "outlier" stars and specific irregularities (like those in T Cas) highlight gaps in current theoretical models.

In summary, the paper provides a robust empirical framework using the $p$ and $q$ parameters to classify and track the evolution of AGB stars, offering new constraints for theoretical models of stellar pulsation and mass loss.