Relationship Between Major Stellar Physical Parameters and Normal Mode Frequencies in Accreting White Dwarf Stars

TL;DR

This work addresses how major stellar parameters of accreting white dwarfs in cataclysmic variables shape observable $g$-mode frequencies. It employs forward modeling with MESA to build a grid of WD structures across five masses, varied core temperatures $T_c$, and accreted-layer masses $M_{\rm acc}$, including diffusion and a rotating pulsation analysis via GYRE under the Traditional Approximation of Rotation. A key contribution is the comprehensive mapping of $g$-mode frequencies as functions of $M_{\rm WD}$, $M_{\rm acc}$, and $T_c$, complemented by a novel mode-identification diagnostic based on the time evolution of pulsation periods after accretion events, i.e., the $|\ abla P|$ versus $P_{\rm avg}$ behavior. The results show systematic increases in mode frequencies with higher $T_c$ and $M_{\rm acc}$, reveal how crystallization and envelope structure modulate the inner and outer cavities, and suggest retrograde modes as the most observable in CVs, offering a practical path to probe accreted layers and WD interiors through asteroseismology.

Abstract

White dwarfs (WDs) are the final fate of about 97\% of the stars in our galaxy, making them vital tracers of stellar history. A fraction of WDs exist in cataclysmic variable (CV) systems, accreting matter from a nearby companion star. A subset of CVs undergo episodic rapid mass transfer, termed dwarf novae (DNe) outbursts. Some accreting WDs exhibit near sinusoidal photometric variations, interpreted as $g$ mode pulsations. However, identifying pulsation modes in accreting WDs remains challenging due to the paucity of available observed modes. In this work, we present a comprehensive computation of the observable $g$ mode frequencies across a range of WD parameters, varying the WD mass, size of the newly accreted layer and core temperature. We also introduce a novel method for mode identification based on the time evolution of pulsation periods following an accretion episode. Our mode identification method does not rely on the direct detection of the consecutive radial mode orders, frequently required in isolated WDs. Moreover, this work improves upon our previous WD modeling efforts. We use a more realistic core temperature in addition to thermohaline mixing and element diffusion enabled during the accretion phase.

Figures (14)

• Figure 1: Temperature profiles against the fractional depth ($\log (1-m/M)$) at the beginning of the long-term accretion phase (top panel, solid) and the hydrogen luminosity $\log(L_{\rm H}/L_\odot)=2$ (top panel, dashed)and 5 (bottom panel, dot-dashed), for three different core temperatures with $T_{\rm c} = 5,7$, and $9\times 10^6$ K of a 0.78 $M_\odot$ WD model.
• Figure 2: WD compositions and frequency profiles after a long-term accretion phase with $M_\text{acc} = 1.5\times 10^{-4}~M_\odot$ at core temperatures $T_{\rm c} = 5$ and 7 $\times 10^{6}$ K of a 0.78 $M_{\odot}$ WD model at $T_{\rm eff} = 14000$ K. 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 significant species produced that are difficult to note on the linear scale (top panel). The fraction of solid core is denoted by the vertical dashed line and is about 34% in radius for $T_{\rm c} = 5\times 10^6$ K. The WD is entirely liquid at $T_{\rm c} = 7\times 10^6$ K.
• Figure 3: Evolution of the WD surface temperature during the long-term accretion phase is shown for a 0.78 $M_{\odot}$ model. The figure presents 12 different models with $\langle\dot{M}\rangle$ (right to left): $8,8.2, 8.4, 8.6, 8.8, 9, 10, 10.5, 11, 11.5, 12$, and $13 \times 10^{-11}~ M_\odot~{\rm yr}^{-1}$. Each model is characterized by a $T_{\rm c} = 5\times 10^{6}$ K and a final effective temperature $T_{\rm eff} = 14000$ K. The minimum and maximum $\langle\dot{M}\rangle$ values end at $M_{\rm acc}$ of 3.467 and $0.837 \times 10^{-4}~M_\odot$, respectively. The highest $\langle\dot{M}\rangle$ results in the WD reaching $T_{\rm eff} = 14000$ K in the shortest timescale, within in approximately one million years.
• Figure 4: Gravity mode frequencies in the inertial frame against time (in months) since the accretion outburst for 0.78 $M_\odot$ mass model. The left panel is for $T_{\rm c} = 5\times 10^6$ K (crystallized core), and the right panel is for $T_{\rm c} = 7\times 10^6$ K, both for a rotation period of 209 s. Top to bottom indicates $n_{\rm g}=2–16$. Three different colors and line styles indicate the three values of the azimuthal eigenfunction index $m$ that make up the dipole triplets.
• Figure 5: First 14 $g$-mode frequencies in the inertial frame are shown as a function of $T_{\rm c}$ calculated for the dipole mode for a 0.78 $M_\odot$ WD model following the long-term accretion phase. The models have an accreted mass of $M_{\rm acc} = 1.5\times 10^{-4}~M_\odot$. Solid purple lines are the retrograde modes, dashed blue lines are the prograde modes, and dotted orange lines are the zonal modes ($m=0$).