High-order Gravity-mode Period Spacing Patterns of Intermediate-mass ($1.5 \, M_\odot < M < 3 \, M_{\odot}$) Main-sequence Stars I. Perturbative Analysis

Abstract

Theoretical study of high-order gravity-mode period spacing ($ΔP_g$) pattern is relevant for the better understanding of internal properties of intermediate-mass ($1.5 \, M_\odot < M < 8 \, M_{\odot}$) main-sequence g-mode pulsators. In this paper, we carry out the first-order perturbative analysis to evaluate effects of a sharp, though not discontinuous, transition in the Brunt-Väisälä (BV) frequency on the $ΔP_g$ pattern. Such a finite-width transition in the BV frequency, whose scale height can be comparable to the local wavelength of gravity waves, is expected to develop in relatively low-mass ($1.5 \, M_\odot < M < 3 \, M_{\odot}$) main-sequence stars, causing a bump in the second derivative of the BV frequency. Inspired by Unno et al.'s formulation, we treat the bump in the second derivative of the BV frequency as a small perturbation, which allows us to derive an analytical expression of the $ΔP_g$ pattern. The analytical expression shows that the amplitude of the oscillatory $ΔP_g$ pattern is determined by a weighted average of the bump in the second derivative of the BV frequency where the weighting function is given by the g-mode eigenfunction. Tests with low-mass ($\sim 2 \, M_\odot$) main-sequence stellar models show that the analytical expression can reproduce the $ΔP_g$ patterns numerically computed reasonably well. The results of our perturbative analysis will be useful for, e.g., improving semi-analytical expressions of the $ΔP_g$ pattern, which would enable us to investigate $ΔP_g$ patterns of SPB stars and $γ$ Dor stars for inferring chemical composition profile and rotation rates.

Figures (7)

• Figure 1: Comparison between $[k_r^2 - \epsilon]$ (the greenish curves, computed with Equation (\ref{['eq:rev2']})) and $k_r^2$ (the black dotted curves, computed with Equation (\ref{['eq:2']})) as a function of the fractional radius $r/R_\star$, where $R_\star$ is the stellar radius, in the case of $2 \, M_\odot$ main-sequence stellar models with different evolutionary stages, i.e., $X_\mathrm{c} = 0.4$ (left) and $0.2$ (right), where $X_\mathrm{c}$ is the hydrogen mass content at the center (details in the model computations can be found in Section \ref{['sec:2-2']} of the main text). The eigenfrequency is chosen to be $1$ cycle per day, which is typical for the main-sequence g-mode pulsators 2016MNRAS.455L..67M. It is apparent that $[k_r^2 - \epsilon]$ is well approximated by $k_r^2$ in almost all the region (see the top panels), although we find a few differences in the expanded looks (see the bottom panels). While the locations of the lower turning point (yellow) are almost the same irrespective of whether $[k_r^2 - \epsilon]$ or $k_r^2$ is used, the locations of the upper turning point are slightly different (orange in the case of $[k_r^2 - \epsilon]$, and red in the case of $k_r^2$). We also see the divergence of $[k_r^2 - \epsilon]$ in the outer envelope (see, e.g., $r/R_\star \sim 0.93$ in the left top panel), which is due to the fact that $P$ becomes zero at the point (see Equations (\ref{['eq:rev3']}) and (\ref{['eq:rev4']})). Despite the differences above, it is expected that high-order g modes of intermediate-mass ($\sim 2 \, M_\odot$) main-sequence stars can be described by the approximated $k_r^2$ because we see little differences between $[k_r^2 - \epsilon]$ and $k_r^2$ in the deep region ($r /R_\star \sim 0.1$) where the radial wavenumber is so large that g-mode properties are mostly determined by the structure there.
• Figure 2: Internal structures around the convective core boundary of the four stellar models with different masses ($M = 2 \, M_\odot$ or $4 \, M_\odot$, greenish or reddish, respectively) and different evolutionary stages ($X_\mathrm{c} \sim 0.4$ or $0.2$, darker or lighter, respectively). The top row shows the BV frequency profiles $N$ in units of $\mathrm{d}^{-1}$ as a function of the stellar fractional radius $r/R_\star$, where $R_\star$ is the stellar radius. The convective boundary is indicated by the yellow dashed line below which there is no value for $N$ because $N^2<0$ in the convective core. Just above the convective core boundary is the hump in $N$, which originates from the chemical composition gradients left behind the receding the nuclear burning core 2008MNRAS.386.1487M. The region where the first derivative of the BV frequency transitions sharply is highlighted by the grey shaded area. The bottom row shows the $f$-terms (solid curves) of the corresponding stellar models in the same ranges of the fractional radius as are shown in the top row. Because numerical differentiation of $N$ of the stellar models leads to the shaky $f$-terms especially inside the convective core, we lighten the color of the $f$-term in the convective core for visual aid. We also draw analytical $f$-terms (black dotted) in the convective core that is computed with Equation (\ref{['eq:5']}) assuming $k_r^2 \sim -L^2 / r^2$. The blue dashed curve represents $\zeta$ that is defined as Equation (\ref{['eq:7']}). It is seen that, in the radiative zone where $N$ is finite, $\zeta$ is much larger than the $f$-terms except for the grey shaded area where the first derivative of $N$ transitions sharply. In contrast, inside the convective core, the $f$-term (black dotted) is comparable to or larger than (the absolute value of) $\zeta$ (blue dashed).
• Figure 3: Comparisons between the toy models (colored curves) and the stellar models (grey dashed curves) with different masses ($M = 1.6 \, M_\odot$ or $2 \, M_\odot$, blueish or greenish, respectively) and different evolutionary stages ($X_\mathrm{c} \sim 0.4$ or $0.2$, darker or lighter, respectively). In the same manner as in Figure \ref{['fig:1']}, the BV frequencies $N$ (in units of $\mathrm{d}^{-1}$) and the $f$-terms are shown in the top and bottom rows, respectively. We see good agreement between the toy models and stellar models, which is especially the case for the $f$-terms. Note that $N$ and the $f$-term are shown only for the radiative zone because the convective core is not relevant in the first-order perturbative approach we demonstrated in this study (see discussions in Section \ref{['sec:2-3']}).
• Figure 4: Comparison among the three kinds of eigenfunctions: the eigenfunction obtained by numerically solving Equation (\ref{['eq:1']}) where the toy BV frequency models are used (black), that proportional to $\sqrt{3} \mathrm{Ai} + \mathrm{Bi}$ (red), and that proportional to $\mathrm{Ai}$ (yellow), where $\mathrm{Ai}$ and $\mathrm{Bi}$ are the first and second kind of the Airy function. The eigenfunctions are shown as a function of the stelar fractional radius $r/R_\star$, and the convective boundary is represented by the green dashed line. The spherical degree and the radial order of the eigenfunctions are $n = -30$ and $\ell = 1$. Because our first-order perturbative approach focuses only on the radiative zone, the analytical eigenfunctions (red and yellow) are not drawn in the convective core. It is seen that the eigenfunction proportional to $\sqrt{3} \mathrm{Ai} + \mathrm{Bi}$ (red) can reproduce the spatial phase of the numerically computed eigenfunction (black) reasonably well not only around the convective boundary (the left panel) but also in a broader region of the radiative envelope (the right panel). This is not the case between the numerical eigenfunction and that proportional to $\mathrm{Ai}$; the locations at which the eigenfunction is zero are different. Note that the difference in the amplitude of the eigenfunctions are attributed to the fact that the numerical eigenfunction (black) is $v$ while the analytical eigenfunctions (red and yellow) are $W$ (remember the relation $v = (|\mathrm{d}r/\mathrm{d}\zeta|)^{1/2} W$). Accordingly, we multiply the eigenfunctions with some suitable constants for visual aid. The values of the constants are different for the left and right panels. It should nevertheless be emphasized that the locations where the eigenfunction is zero do not change even if we consider the factor $(|\mathrm{d}r/\mathrm{d}\zeta|)^{1/2}$. Thus, the phase match between the numerical eigenfunction and that proportional to $\sqrt{3} \mathrm{Ai} + \mathrm{Bi}$ is well founded.
• Figure 5: Comparison of the "numerical" (the grey diamonds) and the "perturbative" (the colored curves) $\Delta P_g$ patterns in the case of the toy BV frequencies that are constructed based on the four stellar models with different masses ($M = 1.6 \, M_\odot$ or $2 \, M_\odot$, blueish or greenish, respectively) and different evolutionary stages ($X_\mathrm{c} \sim 0.4$ or $0.2$, darker or lighter, respectively). The g-mode period and $\Delta P_g$ are expressed in units of $\mathrm{d}$. We see excellent agreements between the two types of $\Delta P_g$ patterns in terms of the both amplitude and phase of the oscillatory component in the $\Delta P_g$ patterns.