Analysis of Damped Wave Frequencies

Asteroseismology is the study of stellar oscillation modes, which can help determine interior properties that are hard to determine from the surface temperature and brightness of a star. The structure and interior of massive stars must be understood to accurately predict the end of their lives (producing neutron stars or black holes), as well as their impact on galaxy evolution (Cantiello et al. 2021). Recent observations of massive stars show low-frequency photometric variability (Bowman et al. 2019). This variability is thought to be caused by internal gravity waves. These waves are generated in the cores of massive stars by convection and can propagate to the surface. One way to compare theories of wave generation and propagation to observations is to consider the characteristic frequency of the observed variability. The observed characteristic frequency is much lower than the convective frequency; furthermore, the observed frequency decreases with stellar age over the main sequence, whereas the convective frequency increases (Cantiello et al. 2021). To improve theoretical estimates of the wave frequency, we include the effects of radiative diffusion on the internal gravity waves. Radiative diffusion damps the waves as they move from the core to the surface. Because diffusion acts preferentially on low-frequency waves, this acts to increase the typical frequency of waves above the convective frequency, possibly alleviating the discrepancy described in (Cantiello et al. 2021).

We evaluate three important frequencies using stellar models of main sequence stars computed using Modules for Experiments in Stellar Astrophysics (MESA). These models have masses ranging from 5 solar masses to 120 solar masses. The first is the Brunt–Väisälä frequency (N), the maximum possible frequency an internal gravity wave can have. To calculate a typical value, we average (N) from the top of the core convection zone to the bottom of the iron convection zone. Cantiello et al. (2021) compared the convective frequency to the typical frequency of observed variability. The convective frequency is the lowest frequency at which waves are significantly excited by convection.

Finally, we include the effects of radiative damping on the waves. Following (Bowman et al. 2019), we assume convection excites waves with an energy flux spectrum and calculate the frequency with maximum wave flux.

Here l is spherical harmonic degree, which we assume to be unity, K is the thermal diffusivity, r is the radius of the star, and omega is the wave frequency. The integral in the above equation is taken from the top of the core convection zone to the bottom of the iron convection zone (or the surface for low-luminosity stars with no iron convection zone). As expected, we find the typical frequency of waves including damping is between the convective frequency and the Brunt–Väisälä frequency. In the figure below, we plot this wave frequency on the spectroscopic HR-diagram, along with the observed frequencies. We find that the typical frequency of waves including damping decreases as a star ages, matching the trend of observed variability.

Conclusion & Future Steps: By calculating the typical frequency of damped waves for main sequence stars of different masses and ages, we were able to see that the typical frequency of waves including damping decreases as a star ages. This matches the trend of the observed variability of massive stars. This adds evidence in favor of the hypothesis that the variability is due to internal gravity waves generated by core convection. Future work should include the effects of rotation on wave propagation.

This material is based upon work supported by the National Science Foundation under grant No. AST-1757792.