[Paper Review] The envelope of the power spectra of over a thousand \delta Scuti stars. The $\bar{T}_{eff}$-$ u_{max}$ scaling relation

S. Barceló Forteza, T. Roca Cortés+1 more|arXiv (Cornell University)|Feb 23, 2018

Stellar, planetary, and galactic studies41 references9 citations

TL;DR

This study establishes a linear scaling relation between the frequency at maximum power (ν_max) and effective temperature (T_eff) in over 1,000 δ Scuti stars observed by Kepler and CoRoT. The relation, explained by the κ-mechanism and gravity-darkening effects, enables direct estimation of stellar temperature from power spectrum envelopes, independent of rotation or viewing angle, and helps constrain rotation rates and inclinations—especially for extreme rotators like HADS stars.

ABSTRACT

CoRoT and Kepler high-precision photometric data allowed the detection and characterization of the oscillation parameters in stars other than the Sun. Moreover, thanks to the scaling relations, it is possible to estimate masses and radii for thousands of solar-type oscillating stars. Recently, a \Delta u - ho relation has been found for \delta Scuti stars. Now, analyzing several hundreds of this kind of stars observed with CoRoT and Kepler, we present an empiric relation between their frequency at maximum power of their oscillation spectra and their effective temperature. Such a relation can be explained with the help of the \kappa-mechanism and the observed dispersion of the residuals is compatible with they being caused by the gravity-darkening effect.

Motivation & Objective

• To characterize the envelope of oscillation power spectra in a large sample of δ Scuti stars.
• To identify empirical scaling relations between ν_max and stellar parameters such as T_eff.
• To investigate the origin of scatter in the ν_max–T_eff relation, particularly due to gravity darkening and oblateness.
• To assess the applicability of the scaling relation to extreme rotators, including High-Amplitude δ Scuti stars (HADS).
• To enable improved seismic characterization using ν_max, Δν, and rotation rate as key indices.

Proposed method

• Analyzed 1,055 δ Scuti stars from Kepler Long Cadence and 8 from CoRoT sismo-channel using high-precision photometry.
• Applied a three-stage iterative method to interpolate light curves, reduce gap effects, and improve mode parameter accuracy (frequencies, amplitudes, phases).
• Used super-Nyquist analysis (up to 1132 µHz) to resolve high-frequency modes beyond the Nyquist limit of Kepler LC data.
• Defined the power spectrum envelope using energy-based peak selection (Ei ≳0.1% of total signal energy), identifying Nenv peaks.
• Calculated ν_max as the amplitude-weighted mean frequency of the envelope peaks using ν_max = Σ(Aiνi)/Σ(Ai).
• Quantified asymmetry using α = (ν_max − (ν_l + ν_h)/2)/δν_env, where δν_env = ν_h − ν_l.

Experimental results

Research questions

• RQ1 Is there a systematic scaling relation between the frequency at maximum power (ν_max) and effective temperature (T_eff) in δ Scuti stars?
• RQ2 What physical mechanisms explain the observed scatter in the ν_max–T_eff relation?
• RQ3 Can the ν_max–T_eff relation be used to infer stellar rotation rate and inclination, especially for rapidly rotating stars?
• RQ4 How do gravity-darkening effects and stellar oblateness influence the observed ν_max–T_eff dispersion?
• RQ5 To what extent does the scaling relation hold for extreme rotators, such as High-Amplitude δ Scuti stars (HADS)?

Key findings

• A linear ν_max–T_eff scaling relation was found across 700 δ Scuti stars, with ν_max increasing with T_eff, consistent with excitation of higher radial order modes at higher temperatures.
• The observed scatter in the ν_max–T_eff relation is well explained by gravity-darkening effects due to stellar oblateness and viewing inclination, with 99.3% of stars lying within predicted dispersion limits when oblateness is considered.
• For HADS stars, the relation helps constrain rotation and inclination: KIC 9408694 is inferred to be an equator-on system with Ω ≳ 0.86 Ω_K and i ≳ 76°, possibly showing twice the rotational frequency.
• The relation holds for both Kepler and CoRoT data, with CoRoT’s higher cadence and precision supporting the ν_max–T_eff trend.
• The scaling relation enables direct estimation of T_eff from ν_max without relying on rotation or viewing geometry, offering a new seismic diagnostic for δ Scuti stars.
• The relation is robust across hybrid stars (δ Sct/γ Dor) and is consistent with theoretical expectations from the κ-mechanism and mode trapping effects.