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Contact: F.X.Timmes
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On Trapped Modes In Variable White Dwarfs As Probes Of The 12C(α,γ)16O Reaction Rate (2022)

In this article, we seek signatures of the current experimental $^{12}$C$(\alpha,\gamma)^{16}$O reaction rate probability distribution function in the pulsation periods of carbon-oxygen white dwarf models. We find that adiabatic g-modes trapped by the interior carbon-rich layer offer potentially useful signatures of this reaction rate probability distribution function. Probing the carbon-rich region is relevant because it forms during the evolution of low-mass stars under radiative helium burning conditions, mitigating the impact of convective mixing processes. We make direct quantitative connections between the pulsation periods of the identified trapped g-modes in variable white dwarf models and the current experimental $^{12}$C$(\alpha,\gamma)^{16}$O reaction rate probability distribution function. We find an average spread in relative period shifts of $\Delta P/P \simeq \pm$ 2% for the identified trapped g-modes over the $\pm$3$\sigma$ uncertainty in the $^{12}$C$(\alpha,\gamma)^{16}$O reaction rate probability distribution function — across the effective temperature range of observed DAV and DBV white dwarfs and for different white dwarf masses, helium shell masses, and hydrogen shell masses. The g-mode pulsation periods of observed white dwarfs are typically given to 6-7 significant figures of precision. This suggests that an astrophysical constraint on the $^{12}$C$(\alpha,\gamma)^{16}$O reaction rate could, in principle, be extractable from the period spectrum of observed variable white dwarfs.

Left: $^{12}$C($\alpha,\gamma$) reaction rate ratios, $\sigma_i/\sigma_0$, as a function of temperature. For our models, $\sigma_i$ spans -3.0 to 3.0 in 0.5 step increments, with $\sigma_0$ being the current nominal rate. Negative $\sigma_i$ are gray curves and positive $\sigma_i$ are green curves. The $\pm$1,2,3 $\sigma_i$ curves are labeled. The blue band show the range of temperatures encountered during core and shell He burning. Right: Mass fraction profiles of the evolutionary DAV models resulting from the $^{12}$C($\alpha,\gamma$) reaction rate uncertainties $\sigma_i$ after each model has cooled to T$_{\rm eff}$ = 11,500 K. The nominal $\sigma$=0 reaction rate is the black curve, negative $\sigma_i$ are gray curves and positive $\sigma_i$ are green curves. Solid curves are for $^{12}$C and $^{16}$O, dashed curves are for $^{1}$H and $^{4}$He. The trace isotopes $^{14}$N, $^{20}$Ne, $^{22}$Ne, and $^{56}$Fe are labeled. Key regions and transitions are also labeled.

Integer multiples $q$ of the radial wavelength $\lambda_r$ profiles versus radius for the $g_5$, and $g_{10}$ modes, for $\sigma_0$. The trapped $g_5$ mode is shown by the dark blue solid line, and the dotted curves depict the trapped $g_{10}$ mode. Solid black segments depict the width of the regions R1, R2, R3, and R4, as defined by distance between labeled chemical transitions.

Top panels: Relative period differences from $\sigma$=0 for the $g_5, g_6$, and $g_{10}$ modes as the evolutionary DAVs cool. The range of T$_{\rm eff}$ in observed DAV WDs is marked, with the vertical dashed black line selecting the T$_{\rm eff}$ = 11,500 K midpoint. Bottom panel: Relative period differences for $g_5$ versus the $^{12}$C($\alpha$,$\gamma$)$^{16}$O reaction rate uncertainties $\sigma_i$ at T$_{\rm eff}$ =11,500 K. Scatter points are the raw data values and the curve is a polynomial fit.

Relative period differences for model sequences of varying WD masses, shell masses, and classes. The wd_builder DAVs were measured at T$_{\rm eff}$ = 11,500K and the wd_builder DBVs were measured at T$_{\rm eff}$ = 25,000 K. Labeled are the $\pm$3$\sigma$ uncertainties on the experimental $^{12}$C$(\alpha,\gamma)^{16}$O reaction rate in steps of 0.5$\sigma$, the trapped g-mode that most distinctly probes the radiatively-formed, carbon-rich region R2, and the mean relative period difference for each sequence.


On The Impact Of 22Ne On The Pulsation Periods Of Carbon-Oxygen White Dwarfs With Helium Dominated Atmospheres (2021)

In this article, we explore changes in the low-order g-mode pulsation periods of 0.526, 0.560, and 0.729 M$_{\odot}$ carbon-oxygen white dwarf models with helium-dominated envelopes due to the presence, absence, and enhancement of 22Ne in the interior. The observed g-mode pulsation periods of such white dwarfs are typically given to 6-7 significant figures of precision. Usually white dwarf models without 22Ne are fit to the observed periods and other properties. The root-mean-square residuals to the ≈ 150-400 s low-order g-mode periods are typically in the range of $\sigma_{\rm rms}$ $\lesssim$ 0.3 s, for a fit precision of $\sigma_{\rm rms}/ P$ $\lesssim$ 0.3 %. We find average relative period shifts of $\Delta P/P$ ≈ $\pm$ 0.5 % for the low-order dipole and quadrupole g-mode pulsations within the observed effective temperature window, with the range of $\Delta P/P$ depending on the specific g-mode, abundance of 22Ne, effective temperature, and mass of the white dwarf model. This finding suggests a systematic offset may be present in the fitting process of specific white dwarfs when 22Ne is absent. As part of the fitting processes involves adjusting the composition profiles of a white dwarf model, our study on the impact of 22Ne can provide new inferences on the derived interior mass fraction profiles. We encourage routinely including 22Ne mass fraction profiles, informed by stellar evolution models, to future generations of white dwarf model fitting processes.

DB WD from a 2.1 M$_{\odot}$, Z=0.02, ZAMS model
element diffusion in the 0.560 M$_{\odot}$ WD
propagation diagram
approximation to Brunt-Väisälä frequency
mass-radius with markers of key transitions
period evolution
period changes in the 0.560 M$_{\odot}$ WD with zero or supersolar 22Ne
what causes the period changes
period changes in the 0.526 and 0.729 M$_{\odot}$ WD with zero and supersolar 22Ne

The Impact of White Dwarf Luminosity Profiles on Oscillation Frequencies (2018)

KIC 08626021 is a pulsating DB white dwarf of considerable recent interest, and first of its class to be extensively monitored by Kepler for its pulsation properties. Fitting the observed oscillation frequencies of KIC 08626021 to a model can yield insights into its otherwise-hidden internal structure. Template-based white dwarf models choose a luminosity profile where the luminosity is proportional to the enclosed mass, $L_r \propto M_r$, independent of the effective temperature $T_{\rm eff}$. Evolutionary models of young white dwarfs with $T_{\rm eff} \gtrsim$ 25,000 K suggest neutrino emission gives rise to luminosity profiles with $L_r$ $\not\propto$ $M_r$.

In this article we explore this contrast by comparing the oscillation frequencies between two nearly identical white dwarf models: one with an enforced $L_r \propto M_r$ luminosity profile and the other with a luminosity profile determined by the star's previous evolution history. We find the low order g-mode frequencies differ by up to $\simeq$ 70 $\mu$Hz over the range of Kepler observations for KIC 08626021.

This suggests that by neglecting the proper thermal structure of the star (e.g., accounting for the effect of plasmon neutrino losses), the model frequencies calculated by using an $L_r \propto M_r$ profile may have uncorrected, effectively-random errors at the level of tens of $\mu$Hz. A mean frequency difference of 30 $\mu$Hz, based on linearly extrapolating published results, suggests a template model uncertainty in the fit precision of $\simeq$ 12% in white dwarf mass, $\simeq$ 9% in the radius, and $\simeq$ 3% in the central oxygen mass fraction.


white dwarf structure
propagation diagram
mode frequency differences
weight function shifts
white dwarf cooling