Resonant amplitude equations for gravito-inertial modes: mode coupling in a slowly pulsating B star

Accompanying Zenodo page for the Poster "Resonant Amplitude Equations for Gravito-inertial modes: Mode Coupling in a Slowly Pulsating B star" by Jordan Van Beeck (corresponding author), Tim Van Hoolst, Jim Fuller and Conny Aerts created for the KASC13/TASC6 workshop/conference in Leuven, Belgium, July 10-15, 2022.

Linear asteroseismic studies of Slowly Pulsating B (SPB) stars cannot explain the amplitudes attained by the pulsations studied. In this poster we discuss our preliminary results on resonant non-linear interactions among these pulsations, and discuss our progress in understanding how such interactions can saturate the pulsation amplitudes.

On this accompanying Zenodo page, we provide additional information on the linear and non-linear pulsation models, and provide additional theoretical background on (weak) non-linear interactions among gravito-inertial pulsations (for a more thorough review, see e.g. Van Beeck et al. 2022). We specifically compute the coupling coefficients or interaction strengths \(\eta\) for interactions among three gravito-inertial modes to predict (resonant) mode amplitudes in KIC5941844, an SPB star selected as a high-priority target for non-linear asteroseismic modeling by Van Beeck et al. (2021).

Linear pulsation modeling: GYRE models

Linear pulsation models serve as input for computing the strength of the non-linear interactions among the gravito-inertial modes in SPB stars (e.g. Lee, 2012 or Van Beeck et al. 2022). We generated linear adiabatic and non-adiabatic pulsation models using the stellar pulsation code GYRE (Townsend & Teitler 2013, Townsend et al. 2018, Goldstein and Townsend, 2020), where we used a MESA stellar evolution model (Paxton et al. 2011, 2013, 2015, 2018, 2019) of KIC5941844 by Pedersen et al. (2021) as input. Rotation is taken into account within the 'Traditional approximation for rotation' (TAR, see e.g. Longuet-Higgins, 1968 or Lee & Saio, 1997).

The GYRE inlist used for these computations, named 'gyre_KIC5941844.in', can be downloaded from this page.

(Weak) Non-linear pulsation modeling

The time evolution of the amplitudes of (weak) non-linearly coupled pulsation modes are governed by the so-called 'amplitude equations' (see e.g. Van Hoolst, 1994a,b for an example on the derivation of such set of equations). In Van Beeck et al. (2022) we derive the amplitude equations for resonant interactions of the kind \(\omega_p \simeq \omega_{d1} + \omega_{d2}\) among three gravito-inertial modes in SPB stars (i.e. non-linear interactions on the quadratic level, where \(\omega_p\)is the parent mode angular frequency in the co-rotating frame and \(\omega_{di}\) is the angular frequency of the daughter mode \(i\) in that frame), taking rotation into account using the TAR. The amplitude equation for the parent mode amplitude \(A_p\) can be written down as

\(\dfrac{\partial A_p}{\partial t} = \gamma_p\, A_p + 2\, A_{d1} \,A_{d2}\,\omega_p\,\eta\,\sin\Gamma\)

where \(\gamma_p\) is the parent mode linear driving rate, \(t\) denotes time, \(\Gamma\) is a relative phase factor, and where \(A_{d 1}\) and \(A_{d2}\)represent the amplitudes of the daughter modes. Similar equations exist that describe the time evolution of the amplitudes of the daughter modes. We furthermore have derived equations for the time evolution of the individual mode phases and \(\Gamma\) (see Van Beeck et al. 2022).

Stationary/time-independent solutions of this set of amplitude equations can be derived (we refer to e.g. Lee, 2012¹ and Van Beeck et al. 2022 for their derivation) and are relevant for our purposes because the (observed) amplitudes derived in Van Beeck et al. (2021) are stationary/time-independent. Specifically, we look at the stable stationary solutions, that is, those solutions that are stable to small disturbances away from the stationary state (i.e. these disturbances are damped over time).

To create Figure 1 on the poster we numerically integrated this set of amplitude equations for two different initial conditions using the SciPy initial value problem solving method for ODEs (we evolved for 100 damping timescales of the parent mode using the Radau integration method, 10 000 integration points, with a relative tolerance of 1e-10). Both initial conditions represent situations where we are investigating a parametric resonant coupling \((\gamma_{p} > 0,\, \gamma_{d1} < 0, \,\gamma_{d2} < 0)\), yet differ in their linear daughter (angular) pulsation frequencies \(\omega_{d1}\)and \(\omega_{d2}\). The linear (cyclic) frequency detunings defined as \(\delta \nu = \nu_p - \nu_{d1} - \nu_{d2}\) for both numerical integrations are therefore different, and the stable stationary solution is attained for the set of more detuned frequencies (i.e. with larger \(\delta \nu\)). Even though the stationary solutions are not stable in the limit cycle case, they oscillate around their stable stationary solutions.

Figure 2 is the result of a preliminary non-linear asteroseismic modeling approach applied to the linear GYRE pulsation models generated for KIC5941844 using the MESA model of Pedersen et al. (2021). It shows that the theoretically predicted surface luminosity fluctuations of the coupled modes (which are dependent on the theoretical mode amplitudes \(A_i\)) produced due to weak non-linear three-mode coupling are an order of magnitude larger than the observed amplitudes derived in Van Beeck et al. 2021. We note that these theoretically predicted values are upper bounds that still need to be multiplied with a factor of unity (or less) due to observer position, adding uncertainty to the comparison with observational quantities.

It may therefore be necessary in future non-linear asteroseismic modeling to take into account higher-order non-linear interactions among modes (e.g. cubic interactions as in Van Hoolst, 1994a,b) or so-called multi-mode coupling, in which multiple quadratic-order non-linear mode interactions are considered in a linked network (see e.g. O'Leary & Burkart, 2014).

Please let the corresponding author know if you have any further questions.

References

• Goldstein, J. & Townsend, R. H. D. 2020, ApJ, 899, 116
• Lee, U. 2012, MNRAS, 420, 2387
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• Van Beeck, J., Bowman, D. M., Pedersen, M. G., et al. 2021, A&A, 655, A59
• Van Beeck, J., Van Hoolst, T., Fuller, J., Aerts, C. 2022, Unpublished Manuscript/Work In Progress
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¹: We note that Lee (2012) uses a different definition of the azimuthal wave number \(m\) than Van Beeck et al. (2022). This quantum number is for example used to denote prograde and retrograde directions for modes in rotating stars, and therefore has an impact on the coupling results. Van Beeck et al. (2022) furthermore introduces a slightly different definition of the coupling coefficient \(\eta\) than Lee (2012), introducing additional ambiguity in the direct comparison of our results (Van Beeck et al. 2022) with those of Lee (2012).