[Paper Review] Photometric modal discrimination in Delta Scuti and Gamma Doradus stars

R. Garrido|arXiv (Cornell University)|Jan 5, 2000

Stellar, planetary, and galactic studies3 citations

TL;DR

This paper proposes using multi-band photometry, particularly Strömgren photometry, to discriminate between pulsation modes (l = 0, 1, 2) in δ Scuti and γ Doradus stars by analyzing phase lags and departure from adiabaticity. It demonstrates that phase lags near 0° in γ Dor stars and consistent radial mode identification in HADS stars validate the method’s potential for asteroseismology with future high-precision space missions.

ABSTRACT

The potential of photometric methods for the identification of l, the degree of spherical surface harmonic of a pulsating star, is investigated with special emphasis on Stromgren photometry applied to delta Scuti and gamma Dor variables. Limitations of actual model atmospheres when fine precision is required for the calculations of partial derivatives and integrals, which depend on limb darkening coefficients, are discussed. Two methods are discussed to calculate the phase lags, the angle between maximum temperature and minimum radius, and R, a parameter which describes departure from adiabaticity of the atmospheres of these pulsating stars. These quantities appear to be very dependent on the convection as parametrized by the mixing length theory. When one of the methods is applied to the gamma Dor stars gives phase lags close to 0 degrees, which are 90-180 degrees out of phase from typical delta Scuti stars. Examples are given for some High Amplitude Delta Scuti Stars (HADS) where the method can be easily applied and gives results consistent to interpret them as radial (l=0) pulsating stars. Other low amplitude delta Scuti stars could be oscillating in a non-radial (l=1, 2) mode. Multi-band photometry is concluded to be a very powerful tool for mode identification of delta Scuti and gamma Dor stars, specially with the more accurate photometry that will be achieved in the near future with the asteroseismological space missions now in progress.

Motivation & Objective

• To develop a photometric method for identifying the spherical harmonic degree l of pulsation modes in δ Scuti and γ Doradus stars.
• To assess the feasibility of using multi-band photometry, especially Strömgren filters, for mode identification when spectroscopic data are unavailable.
• To evaluate the impact of model atmosphere limitations—particularly in limb darkening and flux derivatives—on the accuracy of mode discrimination.
• To determine the required photometric precision for reliable mode identification, especially in preparation for future space missions.
• To explore the physical insights into convection and non-adiabatic effects by estimating phase lag ΨT and departure from adiabaticity R.

Proposed method

• Applies the linearized photometric variation equation (Watson 1988) to relate flux changes in Strömgren bands (u, v, b, y) to pulsation mode degree l.
• Calculates phase lags (ΨT) and adiabaticity parameter R using two distinct methods based on partial derivatives and limb darkening coefficients.
• Uses Kurucz and improved PHOENIX model atmospheres to compute flux derivatives and limb darkening, evaluating their consistency and smoothness.
• Analyzes observed multi-band light curves to extract phase lags and amplitudes, comparing results with theoretical expectations for radial (l=0) and non-radial (l=1,2) modes.
• Evaluates the sensitivity of the method to photometric precision, requiring signal-to-noise ratios of 20–30 for stable Fourier solutions.
• Proposes that future space missions (e.g., COROT) with 100–10 ppm photometric precision will make this method highly effective for asteroseismology.

Experimental results

Research questions

• RQ1 Can multi-band photometry alone reliably discriminate between radial (l=0) and non-radial (l=1,2) pulsation modes in δ Scuti stars?
• RQ2 How do uncertainties in model atmospheres—especially in limb darkening and flux derivatives—affect the accuracy of phase lag and adiabaticity parameter estimation?
• RQ3 What are the expected phase lag (ΨT) and departure from adiabaticity (R) values for γ Doradus stars, and how do they differ from δ Scuti stars?
• RQ4 To what extent can current photometric data (e.g., HADS stars) support radial mode identification using this method?
• RQ5 How will the photometric precision of upcoming space missions enable robust mode identification without spectroscopic data?

Key findings

• Phase lags (ΨT) for γ Doradus stars are estimated to be close to 0°, indicating they are 90°–180° out of phase with typical δ Scuti stars, suggesting a fundamental difference in pulsation physics.
• High Amplitude δ Scuti Stars (HADS) analyzed with this method are consistently identified as radial pulsators (l=0), consistent with their observed light curve behavior.
• For low-amplitude δ Scuti stars, the method suggests the presence of non-radial modes (l=1,2), indicating potential for mode identification beyond radial pulsation.
• The method provides physically meaningful estimates of ΨT and R, which are highly sensitive to convection parameters and could help constrain convection models in these stars.
• Photometric precision of ~1 mmag is sufficient for HADS, but signal-to-noise ratios of 20–30 are required for reliable Fourier analysis in lower-amplitude stars.
• Future space missions like COROT, with 100–10 ppm photometric precision, are expected to make this photometric mode discrimination method highly effective and widely applicable.