Asteroseismic Study of KUV03442+0719 with Parallax Constraints

Hydrogen atmosphere white dwarf KUV03442+0719 was first reported as a pulsator by Gianninas et al. in 2006. Follow-up campaigns by Su et al. revealed more periods. Some spectroscopic results suggest that KUV03442+0719 has a slightly below-average mass and an effective temperature of 11,000 K. However, Gaia data (parallax and magnitude) suggest that it may be a low-mass white dwarf. Such an object would have a helium core. We perform asteroseismic fitting of KUV03442+0719, modeling it both as a carbon/oxygen normal mass white dwarf, and a helium-core low-mass white dwarf. To perform the study, we perform a grid search with WDEC models, refined by simplex minimization of the best fits. Both analyses result in best-fit models that are comparable in terms of quality of fit. More pulsation data would be required to allow us to distinguish between the two scenarios. We present and contrast our results with expectations from stellar evolution. We also provide analytic formulae for a temperature-dependent mass–radius relationship for helium-core white dwarfs.


Introduction
The focus of this paper is the asteroseismic fitting of pulsating white dwarf KUV03442+0719 ns, a hydrogenatmosphere pulsating white dwarf (ZZ Ceti star or DAV).With the recent addition of DAVs discovered with the Transiting Exoplanet Survey Satellite mission (Romero et al. 2022), there are 500 objects in this class.White dwarfs are the end result of the evolution of lower-mass stars (∼98% of all stars) and as such, hold in their interiors the fossil records of the evolution of their progenitor.White dwarf asteroseismology allows us to determine the interior structure of pulsating white dwarfs.The resulting chemical profiles help constrain physical processes such as nuclear fusion, core overshooting, mass loss, and diffusion.
Pulsations observed in white dwarfs are driven in the convection zone and are g modes.Because of geometric cancellations, we do not expect to observe modes past ℓ = 2 (Dziembowski 1977).Modes are also described by their radial overtone.In this paper, we shall use k to denote that number, even though it is often called n in the literature.Most observed pulsation spectra consist of fewer than a dozen modes.Sometimes, rotationally split triplets allow us to identify modes as most likely ℓ = 1, while quintuplets positively point to ℓ = 2 modes.While it is often not possible to do, positive identification of the modes before attempting the asteroseismic fitting is desirable, as best-fit solutions can change according to which mode identification one adopts.This is why having constraints beyond the pulsation spectrum is key.The first pulsations in KUV03442+0719 were reported in Gianninas et al. (2006).Su et al. (2014) performed a 3 yr observing campaign that yielded more periods.
For KUV03442+0719, there are a number of spectroscopic studies (Section 2.2).Two of these studies point to an averagemass hydrogen-atmosphere white dwarf.However, the third, based on Gaia data, suggests that KUV03442+0719 may be a low-mass white dwarf (0.29 M e ; Gentile Fusillo et al. 2021).The Gaia parallax allows the determination of the radius if one assumes that the source is a single object.Based on that radius, KUV03442+0719 is larger than the average white dwarf and therefore has a lower mass.White dwarfs are considered low mass (LMWDs) below 0.45 M e .A subclass of LMWDs are the extremely low-mass white dwarfs (ELMS), with masses between 0.18 and 0.20 M e .KUV03442+0719 does not have a known companion.Because the main-sequence lifetime of the low-mass progenitors of low-mass white dwarfs is longer than the age of our galaxy, it is commonly hypothesized that such stars are the result of binary system evolution, where the white dwarf progenitor loses mass to its companion during the red giant phase (Althaus et al. 2013, and references therein).Surveys have indeed found ELMs by looking at binary systems (Brown et al. 2022).However, work on both the observational front (Kilic et al. 2007) and the theoretical front (Justham et al. 2010) have suggested that a significant fraction of low-mass white dwarfs could be found outside of binary systems.Formation mechanisms for such single objects include massive episodes of mass loss during the red giant phase due to high metallicities, or formation in wider binaries, where the white dwarf subsequently gets separated from its companion.Regardless of the formation mechanism, LMWDs are expected to have cores made up of helium instead of carbon and oxygen (Althaus et al. 2017).
Another hypothesis is that KUV03442+0719 is a member of an unresolved binary system (or more generally, that there is a line-of-sight object caught in the aperture).In that case, we may be looking at a normal-mass carbon-oxygen core white dwarf.While one can argue that this second scenario is more likely, the first scenario cannot be discarded based on the available scientific evidence.As it is not possible to determine which of these hypotheses is the right one, we proceed with studying KUV03442+0719 under each hypothesis.One where we treat it as a helium-core white dwarf and use constraints from Gentile Fusillo et al. (2021), and one where we treat it as a carbon/oxygen core white dwarf and use constraints from Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.Gianninas et al. (2011) and Koester et al. (2009).In the latter case, we are unable to use distance constraints from Gaia, as the Gaia magnitude is that of the combined object.We begin this paper by revisiting the period analysis of Su et al. (2014) to produce a list of periods to use in the asteroseismic fitting.We then introduce and process the nonpulsational observational constraints we have for the object.In Section 3 we introduce our fitting methods.We then proceed with the asteroseismic fitting of KUV03442+0719 under the assumption of different core makeups.We conclude in Section 5.

Pulsation Spectrum
While Su et al. (2014) extract 31 independent modes in their work, based upon observations collected in 2010, 2011, and 2012, we do not think that the Fourier transforms shown in the paper support such an extensive list of periods.The signal-tonoise ratio and spectral windows indicate that we can trust the two highest-amplitude modes detected each year.We also check for beat frequencies and find none.The six frequencies resulting from this process are listed in Table 1.For ease of cross-referencing with Su et al. (2014), we use the same frequency labels.Frequencies with labels that start with "0" are present in the 2010 light curve, while frequencies with labels that start with "1" and "2" are present in the 2011 and 2012 light curves, respectively.The fact that the modes are sorted by frequencies is a coincidence.We also include the period found in the discovery paper (Gianninas et al. 2006).
It is an established fact that higher-period modes in white dwarfs interact with the base of the convection zone.This would be particularly true of KUV03442+0719 ns, as it is a red-edge pulsator.Montgomery et al. (2020) have quantified the effect on the frequencies.Higher k modes can vary in frequency by as much as 5 μHz.In Table 1, mode numbers 116 and 117 are less than 5 μHz apart.The variations observed by Montgomery et al. (2020) were measured over a time period of 75 days, while the 2011 observing run that gave rise to frequencies 116 and 117 was 7 nights long.However, given that the convective response timescales of white dwarfs are on the order of 100 s (Provencal et al. 2012) it is possible that 116 and 117 correspond to the same mode.Out of the pair, we include only mode 117 in our fitting, because of its much smaller error.

Spectroscopy, Photometry, and Parallax
We list in Table 2 the effective temperature and surface gravity determinations that have been published for the star.Three results were obtained using solely spectroscopic modeling.They point to a white dwarf of average mass (logg ∼8).The fourth surface gravity determination combines the spectroscopic measurement of the effective temperature, the Gaia (G) magnitude of the object, and its parallax.It is brighter than an average-mass white dwarf would be at that distance, indicating a larger-than-normal radius.This in turn leads to a lower surface gravity.
This points to two possible scenarios.In the first scenario, there is a line-of-sight companion to KUV03442+0719 that augments its brightness.In performing the spectroscopic study of the object, we assume that the hydrogen lines used in the spectroscopic analysis belong to the pulsating white dwarf.Given the way the high surface gravities of white dwarfs shape their hydrogen lines, this does not seem completely Note.All are assumed to be m = 0 modes.For the Bayes information criterion (BIC) parameter, n obs = 6.For the C/O models, n par = 5, while for the He model, n par = 6 (based on the refined fitting, where M env was fixed).unreasonable, but we have to recognize that we are making an assumption here when leveraging the logg and effective temperature to constrain our asteroseismic fitting.
In the second (less likely) scenario, we are looking at a single object, and it must be a low-mass, helium-core white dwarf.If we assume that the G magnitude is entirely due to the white dwarf, then the photometry and the Gaia parallax allow us to place constraints on the mass and effective temperature of our best fit(s).This requires the use of a mass-radius relationship.We derive one from our own grid of helium-core models (see Section 3), allowing us to obtain constraints that are selfconsistent.In the method below, we take as input solely the photometry and the parallax for KUV03442+0719 ns, with error bars.Those data are listed in the top three rows of Table 2.
From the Stefan-Boltzmann law, we have where In Equation (2), the distance to the white dwarf, d WD can be obtained from Gaia data, and the ratio of the fluxes where G corr is the Gaia magnitude corrected for reddening (Casagrande & VandenBerg 2018): A polynomial fit to WDEC models for helium-core DA's yields the following mass-radius relationship: where the radius R is in centimeters.Parameters a, b, c, and d are themselves cubic functions of the effective temperature.We discuss this mass relationship further in the Appendix and supply values for the parameters in Equation (5).In the Appendix, we discuss the dependence of the mass-radius relation on the internal structure and effective temperature of the models.We find a strong dependence on the hydrogen envelope mass M env2 (see Figure 1).
Once we have the radius, we use the mass-radius relationship to find the corresponding mass.We thus obtain a relationship between the mass and the effective temperature.That relationship is plotted in Figure 2 as the two bold solid diagonal lines.We obtain two lines because we propagate the uncertainties on the distance and G magnitude.The dependence of the mass-radius relation on M env2 propagates to these lines, and we obtain different swaths for different values of M env2 (holding the other chemical structure parameters to that of the fiducial model).

Mode Identification
As we do not have any clear triplets or quintuplets, we begin with the assumption that all observed modes are m = 0 modes.All of the periods are greater than 1100 s.For g-mode pulsations, this means that the period spectrum is close to the   4).The color scale is in tenths of seconds.The red box indicates the boundaries in effective temperature and mass based on the spectroscopy of Gentile Fusillo et al. (2021).The diagonal lines with positive slopes are constructed by combining the parallax data with the mass-radius relationship described in Section 2.2.The dashed lines correspond to a model with an envelope mass of M env2 = 10 −2.2 , the solid lines to M env2 = 10 −5.0 , and the dotted-dashed lines to an envelope mass of 10 −11.4 .There is a pair of lines for each, a result of propagating the uncertainties in the distance and G magnitude.The closed circle marks the location of the best-fit model (Section 4.2).We also plotted lines of asymptotic period spacings for l = 1 modes calculated for our WDEC, helium-core white dwarf models (labeled every 5 s).
asymptotic limit and we expect little mode trapping (i.e., we expect the periods to be evenly spaced).This allows us to make use of the asymptotic period spacing to identify which modes are ℓ = 1 modes and which modes are ℓ = 2.
In the theory of nonradial stellar oscillations, if we assume long periods (and so low frequencies), we find that the periods of the modes are given by Unno et al. (1989): where N is the Brunt-Väisäila frequency integrated between the turning points of the modes, and k is the radial overtone of the mode.For a given star, the integral is a constant and we see that the periods are proportional to the factor ) .This means that there is a constant period spacing associated with each given ℓ.We compute such average ℓ = 1 period spacings for a fiducial helium-core model and show lines of constant period spacing in Figure 2.For carbon and oxygen core models of average mass (∼0.60 M e ) and on the cool end of the ZZ Ceti instability strip, we find that the ℓ = 1 period spacing is ∼50 s, while for ℓ = 2 it is ∼30 s.With the period spacings as a guide, and given that we have only 6 periods to fit, it is not difficult to try every possible combination of ℓ = 1 and ℓ = 2 identifications and select the one that yields the best fits.We repeated the exercise for the helium-core fitting.For the latter, Figure 2 provides an idea of the expected ℓ = 1 period spacings.The ℓ = 2 spacing is a factor of 3 smaller.

Fitting
Armed with the list of periods listed in Table 1, we proceed with the asteroseismic fitting.We calculated grids of models with the White Dwarf Evolution Code (WDEC).For details about the code see Bischoff-Kim & Montgomery (2018).The code is open source and may be obtained from GitHub. 1 A key feature of WDEC is the ability to vary the interior chemical profiles.Instead of using time-dependent diffusion to calculate core chemical profiles based on some starting chemical composition, the code instead accepts the profiles as an input and then calculates a model that satisfies the equations of stellar structure and calculates the associated nonradial oscillation modes.The code allows us to vary a maximum of 15 parameters: mass, effective temperature, convection, and 12 chemical profile parameters, as described in Bischoff-Kim & Montgomery (2018).The details of the parameterization of the oxygen profile used in this work may be found in Bischoff-Kim (2018).For the goodness of fit, we use the quantity where n obs is the number of periods present in the pulsation spectrum and the weights w i are the inverse square of the errors listed for each period in Table 1.For the period given in the discovery paper (Gianninas et al. 2006), we had to estimate a weight.We assigned the same weight as for modes 006 and 007, based on the similarities between the spectral windows and the fact that all are higher amplitude modes.
In order to help place the goodness of fit that we find for KUV03442+0719 in context, it is useful to also provide a statistic called the Bayes information criterion (BIC).BIC takes the number of parameters versus the number of constraints (here periods) into account and gives a measure of quality of fit that considers the fact that fewer periods fit with more parameters will lead to a smaller σ RMS .Two BICs have been used in the white dwarf asteroseismology literature, that of Koen & Laney (2000) and Liddle (2007), Equations ( 9) and (10) respectively.The two quantities differ by a constant factor.

Carbon-Oxygen Core White Dwarf Fit
We begin with an assessment of the sensitivity of each available parameter to the period spectrum.This is a concern for KUV03442+0719 ns, as we only have six modes and because they all have higher periods, we expect a weak sensitivity to the core structure.However, it is worth checking,  as it has been shown that there were exceptions to such rules (Bischoff-Kim 2017; Charpinet et al. 2017).

Parameter Selection
In order to quantify the influence each parameter has on the fits, we selected a fiducial model in an area of parameter space that is close to where we expect KUV03442+0719 to land.For this guess, we used the spectroscopy of Gianninas et al. with 3D corrections (see Table 2), and stellar evolution models by Althaus et al. (2010).Most notably, we adopted thick helium and hydrogen envelopes.At this stage, it is not crucial to have the best-fit model for the star, but we do need to be in the right ballpark.The 15 parameters of the fiducial model are listed in Table 3.
For that fiducial model, we varied one parameter at a time and observed the effect of varying that parameter on the quality of the fit.We tried his exercise for every possible ℓ identification of the modes (this yielded 16 sets of periods to fit).The results for one representative set are shown in Figure 3.We find consistent results regardless of ℓ identifications.A perfect fit has a σ RMS of 0 s.Parameters M env through alph2 dictate the shape of the helium and hydrogen composition profiles (they are envelope parameters).M env is defined such that M env =10 −2 separates the outer 0.01 (1%) of the model where the helium and hydrogen reside from the inner 99%, where only carbon and oxygen are present.M he and M H are defined the same way, with the former marking the location of the base of the pure helium layer and the latter the location of the base of the hydrogen layer.alph1 and alph2 set how gradual the transitions are.The transition from pure helium to pure hydrogen is not parameterized but instead calculated according to diffusive equilibrium.The parameter called "alpha" sets the strength of mixing-length theory (MLT) convection, while the remaining parameters dictate the shape of the oxygen composition profile (Bischoff-Kim 2018).
In Figure 3, we highlighted with color the parameters we ultimately decided to vary in the fits.While stellar mass has a significant effect on the quality of the fits, there is no monotonous descent to any given minimum so we did not think we would learn much by varying that parameter.While M env shows a clear minimum, its value is constrained by the thickness of the hydrogen layer.As M H falls in category 3 (unruly), we decided to fix that parameter to a canonical M H =10 −4 .M env cannot be any smaller than 10 −2 if M H =10 −4 .We fixed it to what stellar evolution calculations predict, around 10 −1.6 .The last envelope parameter, M he , does not affect the fits to a significant level.On the flip side, we find that even though we only have long-period modes in the pulsation spectrum, they do affect parameters that set chemical profiles deeper in the interior (Xhebar, h 1 and h 2 ).WDEC can either treat convection as a free parameter or state-of-the-art models to calculate it.We opted for the latter and did not treat that as a free parameter.We detail the parameters of the grid for CO core white dwarfs in Table 3.

Helium-core White Dwarf Fit
A representative chemical composition profile for the helium-core grid is shown in Figure 1.Five parameters describe the hydrogen profile.The location of the transition from pure helium to the He/H mix region is M env2 .The sharpness of that transition is described by the two diffusion parameters α 1 and α 2 , with a higher value denoting a sharper transition.α 1 cannot be much below 10; otherwise, composition profiles show a discontinuity at the edge of the pure helium core for thicker hydrogen layers.The thickness of the pure hydrogen layer is given by parameter M H .In Figure 1, lines show the value of M env2 and M H for the fiducial model.M H cannot be any smaller than 10 −2.6 ; otherwise, the transition zone is not smooth.There is also the constraint M H > M env2 .The hydrogen abundance in the mixed He/H region is denoted by X H .To that we add the mass, effective temperature, and efficiency of MLT convection (Bohm & Cassinelli 1971).Because we do not have an oxygen profile for helium-core white dwarfs, six parameters become irrelevant.We also no longer need the location of the base of the helium layer.This reduces the number of possible parameters to vary to seven, a computationally manageable number.Constraints in the masseffective temperature plane from distances and magnitudes, described in Section 2.2, help in the determination of a unique best fit.The parameters, the symbols used, and the range and step sizes used for the grids are listed in Table 4.

Carbon and Oxygen Core
Among all possible ℓ identification of modes, we find two clear best-fit models.The parameters of these models (model 1 and model 2) are listed in Table 3. Model 1 is marginally better.In Figure 4 three contour maps show the quality of the fits for that model.In choosing among all the possible ℓ identification we selected the combinations that yielded the best fits that fell between 10,400 and 11,100 K, to be consistent with the constraints from spectroscopy (Table 2).The maps for C/O model 2 are very similar.The two best-fit models differ by the ℓ identification of one of the higher error modes and land on identical or adjacent grid points for all parameters varied.We refined the best fit by performing a simplex search, using the ℓ identification of model 1.We graph the chemical profiles of C/ O model 1 in the top panel of Figure 5, along with chemical profiles for a 0.525 M e , 10858 K model from Althaus et al. (2010), their lowest mass model.Model 1 has nearly the same effective temperature (10900 K), but a lower mass (0.465 M e ).
Even if we cannot exactly compare the two models because of the mass discrepancy, a central oxygen abundance below 50% is far below what is expected from stellar evolution calculations (Córsico et al. 2019), especially for lower-mass white dwarfs.That model also features a pure carbon layer between the C/O core and the He/H envelope, a feature that is very difficult to reproduce in stellar evolution calculations.If we constrain the central oxygen abundance to be greater than 50%, we find a third good fit, labeled as "model 3" in Table 3.We also refined that best fit with a simplex search (constrained central oxygen abundance best fit) in that same table.In Table 1, we list the periods of the two best-fit models (the lower central oxygen abundance, global best fit, and the best fit constrained to a higher central oxygen abundance), with their ℓ and k identifications.We also include BIC measures of quality of fit for each (Equations ( 9) and (10)).We graph the chemical profiles of the higher central oxygen abundance best-fit model in Figure 5 2).The third graph was produced using only the models that had effective temperatures in the spectroscopic range.

Helium Core
Because of the greater number of parameters, we started with a lower resolution grid, refined with a finer grid more narrowly focused on promising regions of parameter space.We started by fitting different period sets discussed in Section 2.3 on the coarse grid.In order to choose the optimal period identification, we looked for best-fit models that landed in or near the spectroscopic box shown in Figure 2.This gave us a constraint on the envelope mass (M env2 ∼10 −5.0 ) and helped us clearly identify the mode identification shown in Table 1.We refined the grid and finished the fitting with a simplex search to hone in on the best-fit parameters listed in Table 4.The corresponding chemical are shown in Figure 1.

Summary and Conclusions
We performed the asteroseismic fitting of KUV03442 +0719 , modeling it both as a carbon-oxygen core, 0.465 M e white dwarf, and as a low-mass, helium-core white dwarf.For the former, given Gaia parallax and magnitude data, we have to assume that there is a line-of-sight object that adds to the brightness of the white dwarf, while not leaving a signature in the spectrum.In the latter, the Gaia data is consistent with the lower mass (and therefore larger radius) of the white dwarf and we can assume that we are simply looking at the white dwarf.This allows us to use constraints from Gaia and from the spectroscopic temperature determination to limit our search in the mass-effective temperature plane.One useful product of this study is a mass-radius relationship for helium-core white dwarfs, based on WDEC models.
We contrast the interior chemical profiles of the best-fit models we find for both the carbon-oxygen core white dwarfs and helium-core white dwarfs with those that result from stellar evolution in Figures 5 and 1 respectively.While we do not have entirely equivalent models, it is worth noting that for the helium-core fit, we find a best-fit model that has the expected hydrogen layer mass, but a thinner envelope mass.The carbonoxygen model fits better than the helium-core model, when we take into consideration the fact that we had more parameters varied in the latter, but only marginally so.To perform our study, we used periods published by Su et al. (2014) but only selected the highest-amplitude modes.Improved pulsation data might help better distinguish between the two scenarios in the future.
The other parameters have a negligible effect on the mass-radius relation.
Gaia absolute magnitude of the white dwarf G and the absolute magnitude of the Sun (M = 4.83):

Figure 1 .
Figure 1.Core profiles for the fiducial model that serves as a basis for the mass-radius relationship (dashed lines, Section 2), best-fit model (solid lines), and interior profile based on the 0.2724 M e , 16481 K model of Calcaferro et al. (2017; dashed-dotted line).The center is on the left.The core is composed of helium, while the envelope of hydrogen.The fiducial model, listed in Table4, was chosen to have a thick helium/hydrogen envelope, but a thin pure hydrogen layer.This allows maximum freedom in analyzing the dependence of the mass-radius relation on the different parameters.The best-fit model is the result of the fitting procedure described in Section 3. The labeled vertical lines illustrate how the values of the envelope parameters M env2 and M H are defined.They are pictured for the fiducial model.

Figure 2 .
Figure2.Constraints for the asteroseismic fitting, along with a contour plot of the quality of fit of models comprising the master grid for helium-core white dwarfs (see Table4).The color scale is in tenths of seconds.The red box indicates the boundaries in effective temperature and mass based on the spectroscopy of Gentile Fusillo et al.(2021).The diagonal lines with positive slopes are constructed by combining the parallax data with the mass-radius relationship described in Section 2.2.The dashed lines correspond to a model with an envelope mass of M env2 = 10 −2.2 , the solid lines to M env2 = 10 −5.0 , and the dotted-dashed lines to an envelope mass of 10 −11.4 .There is a pair of lines for each, a result of propagating the uncertainties in the distance and G magnitude.The closed circle marks the location of the best-fit model (Section 4.2).We also plotted lines of asymptotic period spacings for l = 1 modes calculated for our WDEC, helium-core white dwarf models (labeled every 5 s).

Figure 3 .
Figure 3. Parameter sensitivity for KUV03442+0719 's pulsation spectrum.For a description of the parameters, see text and Bischoff-Kim (2018).The lightly colored boxes highlight parameters that we varied in the grid search.
with a 0.525 M e , 11,359 K model from Althaus et al. (2010).

Figure 4 .
Figure 4. Location of the best fits in three cuts in parameter space for C/O model 1.All five parameters varied in the C/O core fitting are featured.C/O model 2 presents very similar contour plots.The vertical lines indicate the range of effective temperatures that correspond to the spectroscopy (Table2).The third graph was produced using only the models that had effective temperatures in the spectroscopic range.

Figure 6 .
Figure 6.Mass-radius relationships for different effective temperatures or structure parameters.The base model is the fiducial model listed in Table4.The other parameters have a negligible effect on the mass-radius relation.

Table 3
Parameters Used in the Fits for the Grid of CO Core Models

Table 5
Parameters for the Mass-Radius Relation (Equation (5) and Subsequent Text)