Resonant Amplitude Equations for Gravito-Inertial Modes: Non-linear mode coupling in a slowly pulsating B star

J. Van Beeck, T. Van Hoolst, J. Fuller, C. Aerts Resonant Amplitude Equations for Gravito-Inertial Modes Non-linear mode coupling in a slowly pulsating B star PARADISE C. Aerts, J. De Ridder, A. Tkachenko, T. Van Hoolst p. 1 Pushing AsteRoseismology to the next level with TESS, GaiAand the Sloan DIgital Sky SurvEy A 6-year KU Leuven research proposal written by: Prof. Dr. Conny Aerts, promoter and Director of the Host Institute Dr. Joris De Ridder, co-promoter Dr. Andrew Tkachenko, co-promoter Prof. Dr. Tim Van Hoolst, co-promoter Abstract The year 2018 marks a unique epoch in stellar astrophysics. The NASA mission TESS will be launched to measure millions of stars in the sky with ultra-precise uninterrupted high-cadence photometry, while the ongoing ESA mission Gaia will release its astrometric data of 1 billion stars in the Milky Way, among which the TESS targets. This Peta-byte sized set of data offers a PARADISE for asteroseismology and will revolutionise stellar and galactic physics. Once the 5th version of the Sloan Digital Sky Survey will become operational in 2020, we will be in a unique circumstance to combine all-sky time-resolved photometry, spectroscopy and astrometry of millions of single and binary stars. Thanks to their leading roles in these three major space and ground-based observing facilities in time-domain astronomy, the promoters of this C1 project PARADISE have the potential and ambition to push asteroseismology and stellar interiors to unprecedented levels. The ultimate goal of PARADISE is to develop a better and observationally-driven theory of stellar interiors. We focus on rotating stars born with a convective core and revealing non-radial gravity-mode oscillations in space photometry, i.e., stars born with mass M&1.4M. The improved theory resulting from PARADISE will be successful when it is able to 1. resolve the current discrepancies between theory and observations in angular momentum transport (two orders of magnitude improvement needed) and 2. deliver asteroseismically calibrated mixing profiles for stellar interiors (currently not available). This will be achieved for samples of thousands of single and binary stars with combined gravito-inertial asteroseismology, astrometry, and spectroscopy. The potential impact of PARADISE will be major, as all stellar evolution models in the covered mass range rely dominantly on the theory of stellar interiors. Our new prescriptions for mixing and angular momentum transport, calibrated by gravito-inertial modes, will be included in stellar evolution computations. Our coded versions of the new prescriptions will be offered in open access to the stellar physics community. Figure 1: Illustration of a typical PARADISE star at the start of its life. We focus on stars born with a fully mixed convective core in which nuclear fusion takes place (inner light blue region), surrounded by a zone of “convective core overshooting” (black zone with the white thin curly arrows). It is unknown how large this black zone is and how efficient the gas is mixed in it. Further out, the star has a radiative envelope in which energy is transported by diffusion of photons (thick white arrows) rather than convective motions. The transport of angular momentum and chemical species in the radiative envelope is subject to major uncertainty. Near the surface, a thin convective zone (outer light blue band) may occur, depending on the birth mass of the star and its evolutionary phase. In PARADISE, we will deduce the detailed physical properties of these stellar interior zones from the seismic behaviour of gravito-inertial waves detected in high-precision space photometry. Figure Courtesy of Dr. Pieter Degroote, KU Leuven. “The time of renaissance for nonlinear asteroseismology has come” — Zong et al. (2023) J. Van Beeck, T. Van Hoolst, J. Fuller, C. Aerts Resonant Amplitude Equations for Gravito-Inertial Modes Non-linear mode coupling in a slowly pulsating B star PARADISE C. Aerts, J. De Ridder, A. Tkachenko, T. Van Hoolst p. 1 Pushing AsteRoseismology to the next level with TESS, GaiAand the Sloan DIgital Sky SurvEy A 6-year KU Leuven research proposal written by: Prof. Dr. Conny Aerts, promoter and Director of the Host Institute Dr. Joris De Ridder, co-promoter Dr. Andrew Tkachenko, co-promoter Prof. Dr. Tim Van Hoolst, co-promoter Abstract The year 2018 marks a unique epoch in stellar astrophysics. The NASA mission TESS will be launched to measure millions of stars in the sky with ultra-precise uninterrupted high-cadence photometry, while the ongoing ESA mission Gaia will release its astrometric data of 1 billion stars in the Milky Way, among which the TESS targets. This Peta-byte sized set of data offers a PARADISE for asteroseismology and will revolutionise stellar and galactic physics. Once the 5th version of the Sloan Digital Sky Survey will become operational in 2020, we will be in a unique circumstance to combine all-sky time-resolved photometry, spectroscopy and astrometry of millions of single and binary stars. Thanks to their leading roles in these three major space and ground-based observing facilities in time-domain astronomy, the promoters of this C1 project PARADISE have the potential and ambition to push asteroseismology and stellar interiors to unprecedented levels. The ultimate goal of PARADISE is to develop a better and observationally-driven theory of stellar interiors. We focus on rotating stars born with a convective core and revealing non-radial gravity-mode oscillations in space photometry, i.e., stars born with mass M&1.4M. The improved theory resulting from PARADISE will be successful when it is able to 1. resolve the current discrepancies between theory and observations in angular momentum transport (two orders of magnitude improvement needed) and 2. deliver asteroseismically calibrated mixing profiles for stellar interiors (currently not available). This will be achieved for samples of thousands of single and binary stars with combined gravito-inertial asteroseismology, astrometry, and spectroscopy. The potential impact of PARADISE will be major, as all stellar evolution models in the covered mass range rely dominantly on the theory of stellar interiors. Our new prescriptions for mixing and angular momentum transport, calibrated by gravito-inertial modes, will be included in stellar evolution computations. Our coded versions of the new prescriptions will be offered in open access to the stellar physics community. Figure 1: Illustration of a typical PARADISE star at the start of its life. We focus on stars born with a fully mixed convective core in which nuclear fusion takes place (inner light blue region), surrounded by a zone of “convective core overshooting” (black zone with the white thin curly arrows). It is unknown how large this black zone is and how efficient the gas is mixed in it. Further out, the star has a radiative envelope in which energy is transported by diffusion of photons (thick white arrows) rather than convective motions. The transport of angular momentum and chemical species in the radiative envelope is subject to major uncertainty. Near the surface, a thin convective zone (outer light blue band) may occur, depending on the birth mass of the star and its evolutionary phase. In PARADISE, we will deduce the detailed physical properties of these stellar interior zones from the seismic behaviour of gravito-inertial waves detected in high-precision space photometry. Figure Courtesy of Dr. Pieter Degroote, KU Leuven. Slowly pulsating B (SPB) stars Slowly pulsating B (SPB) stars KIC 7760680 Pápics et al. (2015), Bowman and Michielsen (2021) Slowly pulsating B (SPB) stars KIC 7760680 Pápics et al. (2015), Bowman and Michielsen (2021) Slope —> Rot. Rate KIC 7760680 Slowly pulsating B (SPB) stars Pápics et al. (2015), Bowman and Michielsen (2021) KIC 7760680 Slope —> Rot. Rate Gravito-Inertial Modes (GIMs) ω≈ Ωrot KIC 7760680 Slowly pulsating B (SPB) stars Pápics et al. (2015), Bowman and Michielsen (2021) KIC 7760680 Slope —> Rot. Rate Gravito-Inertial Modes (GIMs) ω≈ Ωrot Perturbation Theory KIC 7760680 Slowly pulsating B (SPB) stars Gravito-Inertial Modes (GIMs) Pápics et al. (2015), Bowman and Michielsen (2021) KIC 7760680 ω≈ Ωrot Slope —> Rot. Rate Perturbation Theory Traditional Approximation (TAR) (e.g., Bildsten et al. 1996, Lee & Saio 1997) ∂2ξ ∂t2+B[∂ξ ∂t]+C[ξ]= 0 Equation of Motion Non-linear mode interactions? •Uniform rotation •No centrifugal deformation Assumptions: TAR Lag. Displacement Coriolis operator forces not dependent on mode frequencies ξ: B[] : C[] : ∂2ξ ∂t2+B[∂ξ ∂t]+C[ξ]=a(2)(ξ,ξ)+O(ξ3) Equation of Motion Coupled-Mode Equations •Uniform rotation •No centrifugal deformation Assumptions: TAR Non-linear mode interactions mode energy exchange Lag. Displacement Coriolis operator forces not dependent on mode frequencies ξ: B[] : C[] : ∂2ξ ∂t2+B[∂ξ ∂t]+C[ξ]=a(2)(ξ,ξ)+O(ξ3) Resonant amplitude equations ? Focus on strongest interactions ω εϑ (i) ω εϑ (ii) ω εϑ (iii) ω1≈ω2+ω3 (e.g., Van Beeck et al. (2024)) Equation of Motion Coupled-Mode Equations Lag. Displacement Coriolis operator forces not dependent on mode frequencies ξ: B[] : C[] : Resonant Amplitude Equations ∂A1 ∂t1 =γ1A1+ 2 η1A2A3ω1sin(Υ) (similar for modes 2 and 3 and mode phases ) ϕi Mode amplitude Linear damping/driving rate coupling coefficient : combination phase Ai: γi: η1: Υ Resonant Amplitude Equations (e.g., Van Beeck et al. (2021)) Observations of SPBs Many time-invariant amplitudes (similar for modes 2 and 3 and mode phases ) ϕi ∂A1 ∂t1 =γ1A1+ 2 η1A2A3ω1sin(Υ) Mode amplitude Linear damping/driving rate coupling coefficient : combination phase Ai: γi: η1: Υ Resonant Amplitude Equations Observations of SPBs Many time-invariant amplitudes (e.g., Van Beeck et al. (2021)) Stationary Solutions (As 2 As 1)2 =γ1ω2 γ2ω1 (similar for modes 2 and 3 and mode phases ) ϕi ∂A1 ∂t1 =γ1A1+ 2 η1A2A3ω1sin(Υ) Mode amplitude Linear damping/driving rate coupling coefficient : combination phase Ai: γi: η1: Υ Resonant Amplitude Equations Observations of SPBs Many time-invariant amplitudes (e.g., Van Beeck et al. (2021)) (similar for modes 2 and 3 and mode phases ) ϕi ∂A1 ∂t1 =γ1A1+ 2 η1A2A3ω1sin(Υ) Stationary Solutions (As 2 As 1)2 =γ1ω2 γ2ω1 η1? Mode amplitude Linear damping/driving rate coupling coefficient : combination phase Ai: γi: η1: Υ Resonant Amplitude Equations η1 Stability Selection rules Location Onset (similar for modes 2 and 3 and mode phases ) ϕi For more information, see Van Beeck et al. (2024) ∂A1 ∂t1 =γ1A1+ 2 η1A2A3ω1sin(Υ) Mode amplitude Linear damping/driving rate coupling coefficient : combination phase Ai: γi: η1: Υ Location ∝Evolutionary Stage of SPB Less evolved SPB More evolved SPB η1(r) 0 N2(Hz2) 0 > 0 > 0 < 0< 0 coupling coefficient : squared Brunt-Väisälä/buoyancy frequency η1: N2 What can we do with this theory? OBSERVATIONS Typical SPB Amplitude Ratios Predictions vs. Observations Paxton et al. (2011,2013,2015,2016,2018,2019) Jermyn et al. (2023) Townsend and Teitler (2013) Townsend et al. (2018) Goldstein and Townsend (2020) Van Beeck et al. (2024) Van Beeck et al. (2021) SPB Model Grid Kepler Time Series Typical SPB Amplitude Ratios Predictions vs. Observations Observed Predicted Lowest frequency in triad Corresponding Period Observed Amplitude Ratio Predicted Amplitude Ratio Based on data of Van Beeck et al. (2021) and Van Beeck et al. (2024) Lowest frequency in triad Typical SPB Amplitude Ratios Predictions vs. Observations Observed Predicted Lowest frequency in triad Corresponding Period Observed Amplitude Ratio Predicted Amplitude Ratio Based on data of Van Beeck et al. (2021) and Van Beeck et al. (2024) Lowest frequency in triad Typical SPB Amplitude Ratios Predictions vs. Observations Observed Predicted Lowest frequency in triad Corresponding Period Observed Amplitude Ratio Predicted Amplitude Ratio Based on data of Van Beeck et al. (2021) and Van Beeck et al. (2024) Lowest frequency in triad Predictions of amplitude ratios provide additional modeling constraints Conclusions •Resonant amplitude equations describe the lowest-order non-linear saturation of SPB gravito-inertial mode amplitudes. •Strong contributions to mode coupling occur in the -gradient region of SPBs and this region increasingly determines the mode coupling for more evolved models. •Stationary amplitude ratios can be estimated from linear frequencies and driving/damping rates. •Estimation of the coupling coefficient is necessary to determine the stability of this stationary solution. •Predicted stationary amplitude ratios provide additional constraints on non-linear saturation and linear driving/damping processes. •Need more work on linear excitation, higher-order non-linear effects, surface simulations, … μ Want more? And/or contact me using: Check out the published articles: Van Beeck et al. (2021) Van Beeck et al. (2024) [email protected] And/or ask me about my computational frameworks: The time of renaissance for nonlinear asteroseismology has come Resonant Amplitude Equations (similar for modes 2 and 3 and mode phases ) ∂A1 ∂t1 =γ1A1+ 2 η1A2A3ω1sin(Υ) Multiple (time) scales perturbation theory (e.g., Nayfeh (1973)) Ai=Ai(t1, …) Υ=Υ(t1, …) ∂Υ ∂t1 =(ω2+ω3−ω1)+ ∂ ∂t1 (ϕ1−ϕ2−ϕ3)≡ − δω + ∂Φ ∂t1 ϕi Mode amplitude Linear damping/driving rate coupling coefficient : combination phase Ai: γi: η1: Υ Typical SPB Amplitude Ratios Predictions vs. Observations Observed Predicted Lowest frequency in triad Observational Pseudoclass Satisfied Checks Corresponding Period Observed Amplitude Ratio Predicted Amplitude Ratio Based on data of Van Beeck et al. (2021) and Van Beeck et al. (2024) Lowest frequency in triad (Eu=A2 u)