Unified $g$-mode and $r$-mode Analysis of Accreting White Dwarf Stars

Abstract

Dwarf novae are a subset of cataclysmic variables that accrete material intermittently in short-duration outbursts with sometimes long quiescent intervals in between. During the quiescent state, the white dwarf (WD) photosphere may be observable. Some of these systems show periodic variability consistent with a non-radial oscillation. Asteroseismology has become a unique tool for the measurement of internal structure of the WDs, such as their masses, radii, temperatures and rotation profiles. A few stable periodicities have been observed for accreting WDs, but the lack of complete and accurate theoretical models has hindered the real diagnosis of the observed pulsations. Though the associated pulsations in accreting WDs are thought to be $g$-modes, some work in the literature suggests that these pulsations could be Rossby modes ($r$-modes). Here, to elucidate this, we present a first simultaneous analysis of $g$- and $r$-mode pulsations in accreting white dwarfs including a full computation of visibility accounting for the distribution of variation over the WD surface. We show that, up to the second lowest degree ($\ell =2$), neither $g-$ nor $r$-modes have a clear advantage in visibility. Although a few retrograde $r$-mode orders exhibit a larger visibility, the low-order $g$ modes possess higher frequency in the star's frame, making them more likely to be driven within the convective driving scenario commonly applied to isolated WDs. Therefore, we favor a $g$-mode origin for the observed periods in accreting WDs, though $r$-modes will be important for stars with more observed modes.

Figures (9)

• Figure 1: HR diagram (top) and effective temperature evolution (bottom) of a 0.78 $M_\odot$ WD through its evolutionary stages from pre-main-sequence (PMS) to cooling. Tracks highlight accretion (on/off) phases with element diffusion and dwarf nova (DN) cycles. The sharp drops in $T_{\rm eff}$ in the second and third bottom panels (black arrows) correspond to changes in boundary conditions, specifically the adopted optical depth ($\tau$) at the outermost zone. We perform our seismological analysis on the final model, marked with a '+'.
• Figure 2: Displayed are the various elemental abundance profiles in the top two panels in linear and logarithmic scale plotted against the pressure for the 0.78 $M_{\odot}$ model, three months after the dwarf nova outburst (short-term accretion). Brunt-Väisälä and Lamb frequency are shown in the bottom panel. Mass fractions are shown on the log scale in the middle panel to highlight the other species produced, which are difficult to note in the linear scale.
• Figure 3: Eigenvalue ($\lambda$) of Laplace's Tidal equation (\ref{['eqn:LTE']}) plotted against the spin parameter ($q = \frac{2\Omega}{\omega}$). $\Omega$ is the angular rotation frequency of the star, and $\omega$ is the mode frequency in the star's frame. Eigenvalues of both $r$ (Rossby)- and $g$ (gravity)-modes are displayed in this plot. The gray dashed vertical axis is the zero rotation ($\Omega = 0$) line. Modes with $m < 0$ ($m > 0$) propagate in the prograde (retrograde) direction in the co-rotating frame, respectively, and are marked on the right (left) side of the panel. Solid lines correspond to $r$ modes, and dashed (dot-dashed) lines indicate retrograde (prograde) $g$ modes, respectively. At zero rotation, $\lambda$ approaches $\ell(\ell + 1)$, which is the solution of LTE in the non-rotating limit indicated as red dots and marked with green text. The prograde and retrograde mode lines don't connect at zero rotation because the values shown are for actual modes in a finite rotation frequency star, thus having a maximum $\omega$, and corresponding minimum $q$.
• Figure 4: Absolute radial and horizontal displacement perturbations of the sixth radial order of, $m = 1$, $k = -1$, $r$ mode are shown in the top panel. The WD core is to the left. The dashed red line indicates that 8.4% of the WD core is crystallized. The corresponding unnormalized angular eigenfunctions (also known as Hough functions), which are the solutions of LTE, with spin parameter $q = 2\Omega$/$\omega$ = 2.56, $\Theta, \hat{\Theta}$, and $\Tilde{\Theta}$, are shown in the middle panel. The vertical dashed line indicates the boundary of the equatorial waveguide, the critical angle ($\mu_c$), $\mu = 1/q$. The bottom panel exhibits the dependence of the full displacement eigenfunction, $\xi_r(r)\Theta(\theta)$, on both $r/R$ and $\theta$ in the meridional plane.
• Figure 5: Unnormalized but scaled (explained in the text) Hough eigenfunctions $\Theta$, $\hat{\Theta}$, and $\Tilde{\Theta}$ are plotted as functions of $\mu =\mathrm{cos \theta}$ for the lowest order retrograde $g$ mode (top panel) with $|m| =1, k=0$, where solid lines are for $q = 3.02$ and dashed lines are for $q = 1.33$. The bottom panel shows the even order $r$ mode with $|m|=2, k=-2$ for $q = 10.01$. The vertical dashed line denotes the location of the critical angle $1/q$ ($\mu_{c1} = 1/1.33$ and $\mu_{c2} = 1/3.02$ on the top panel). Amplitudes of the Hough functions are set similarly to Lee_Saio_1997.