Photometric White Dwarf Rotation

The TESS telescope offers long-time light curves and high-frequency resolution. However, because of its large plate scale of 21″ × 21″ per pixel, it exhibits low spatial resolution. Consequently, signals from nearby sources contaminate many TESS light curves. We overcome this problem by using the TESS_localize software (Higgins & Bell 2023) to determine whether the detected period arises from the WD or a nearby star. For each star on our target list, we run TESS_localize for all sectors in which the star was observed, setting the “Principal Components” parameter to 3 (see Higgins & Bell 2023).

There are two crucial TESS_localize output parameters that need to be checked to ensure the quality of the fit: “Height” and “p-value.” When TESS_localize fits the position of the signals, the strength of the signals is represented as the best-fit height parameter. Similar to the challenge of identifying stars in an image, the signal needs to stand out significantly from the background noise. Higgins & Bell (2023) recommend that the height parameter exceed 5 times its uncertainty for the localization to be considered significantly above the noise floor. Another critical parameter is the p-value; this number supports the hypothesis that the location of the fit of the signal is consistent with the position of the source. Because we do not reject this hypothesis, the p-value should be reasonably high. We adopt the same threshold as Pedersen & Bell (2023): p-value ≥ 0.05. Applying these criteria, we select only those sectors for which TESS_localize achieved a significant localization of the signal of interest that is consistent with the location of a known Gaia source brighter than G = 18 mag. Therefore, since we excluded noisy sectors, we do not need to be concerned about the variation in noise from sector to sector.

The final constraint step involves an examination of the “Relative Likelihood,” also an output parameter of TESS_localize. Essentially, it ranks the proximity of Gaia sources to the localized position of the signal within the pixels. For the same target, TESS_localize can fit distinct values of relative likelihood for each available sector of data. So, for analysis purposes, we denote the mean relative likelihood over all selected sectors as “Like.” To be consistent, we only consider that the signal originates from the WD if TESS_localize indicates the WD as the most probable source for all selected sectors.

The two-sample Kolmogorov–Smirnov (K-S) test is a robust statistical method used to assess the likelihood of two distinct samples originating from the same population. To ensure that we are not including false-positive data in our sample, we performed the two-sample K-S test comparing samples with different thresholds on the selection parameter like. The K-S tests indicated that we would be discarding valuable data if we embraced a like limit higher than 0.75. Given that, and the fact that the software estimates like ≃ 0.75 even for confirmed periods from known variables (see Appendix A), we selected all the stars with like ≥0.75.

The TESS_localize analysis with strict conditions reduced by 56% our initial sample of 854 candidate stars. However, the sample of 854 WDs is already a consequence of a preliminary selection using TESS_localize that resulted in a reduction of 61%. Therefore, in total, TESS_localize reduced our sample of 2193 WDs, with nonpulsating periods detected in 83%. It is important to point out that in this second selection, 89% of the stars discarded by TESS_localize were rejected because the software was unable to perform a high-quality fit, presenting a low p-value and height. Only 11% were rejected because the software indicated a low probability of the signal having the WD as the source.

Our “complete sample“ consists of 318 WDs, including 76 in binary systems and 30 with detected magnetic fields, counting one WD in a binary system that exhibits a magnetic field. Table 1 lists all stars that compose our samples, providing information about their Gaia ID, R.A., decl., variability period, relative and absolute amplitude, like, mean height parameter normalized by uncertainty, temperature, mass, and additional information. Table 1 also reports the fraction of sectors whose TESS_localize fit achieved our quality criteria as “Q/S.” WDs confirmed to be in binary systems or with a detected magnetic field can be identified in Table 1 by the column “Info.” The character “+” indicates binarity (e.g., DA+M and WD+pair), while the characters “H” and “P” indicate magnetic field (e.g., DAH and DAP). WDs in binary systems with the designation “WD+pair” were identified through proper motion and parallax, while the ones with the designation “eclipse” were identified by the shape of the light-curve variability.

In addition to the sample of 318 WDs, we found that for only three of the 14 stars in our seismological sample the WD is also the source of the nonpulsating periods we detected. These three stars are discussed separately in Section 4.4. Table 2 presents the details of the nonpulsating periods we identified, including amplitude, and quality parameters from the TESS_localize fit.

Stars in binary systems have a completely different evolution from single stars if their companions are close enough to interact, affecting their structure. Consequently, variability periods from close binary systems represent a distinct population from single WD rotational periods. Given that, we divided our sample of 318 WDs between those that probably evolved as single stars and those that might have evolved by binary interaction.

Our complete sample comprises 76 WDs in binary systems: 22 identified through parallax and proper motion, 16 identified by the light-curve shape, and 38 identified through spectral analysis from the literature. The WDs with pairs we identified through parallax and proper motion are all dozens of astronomical units away from their companions; consequently, they probably evolved single stars. The binary systems in our sample identified through spectral observations comprise a WD and a main-sequence star of spectral type M, L, or K. About 75% of the stars in this type of binary system are far enough from their pair to evolve as single stars (Willems & Kolb 2004). However, we cannot distinguish the specific evolutionary status of each system; thus, we will also assume that the 38 WDs with pairs identified by spectral analysis had their evolution affected by their pairs. Finally, the 16 WDs that show eclipses on their light curves are likely to be close to their pairs, since the periodicities of the eclipses are shorter than 13 days for all cases. Therefore, we select a sample labeled “WDs with close pair” encompassing the 54 WDs in our sample with pairs identified by spectra and eclipses.

It is important to recognize that our sample may include more WDs originally in binary systems than those that we detect. WDs with a mass lower than 0.45 M_⊙ are formed by binary interaction (e.g., Marsh et al. 1995; Kilic et al. 2007), since it would take longer than the age of the Universe for them to evolve as single stars. This does not necessarily imply that these stars are currently in binary systems, but they certainly originate from such a population. Therefore, we classified the stars in our sample with a mass less than 0.45 M_⊙ as “potential WDs with pair.” We also included stars with an effective temperature higher than 40,000 K, based on the fact that “WDs with close pairs” might exhibit overestimated temperatures due to the reflection effect caused by a brighter companion (e.g., Schaffenroth et al. 2023; Steen et al. 2024). Once high-temperature WDs are uncommon, we assumed that WDs with temperatures higher than 40,000 K, an arbitrary choice, are more likely to have a pair than to be a single high-temperature WD. Furthermore, we assumed that the remaining stars—those neither confirmed to have companions nor meeting the criteria for potential pairs—are "likely single WDs.”

We performed a K-S test comparing the sample of potential WDs with pair with the sample of WDs with close pairs; the test indicated that these samples may originate from the same population. That suggested that the mass and temperature parameters that we used as a criterion for dividing our sample of WDs with no confirmed pairs into likely single WDs and potential WDs with pair were well chosen.

Our complete sample consists of 318 WDs, including 54 WDs with close pairs, 149 potential WDs with pairs, and 115 likely single WDs. Further details of these samples are available in Table 3, which summarizes information about the quantities of magnetic WDs in each sample, their median periods , and the period median absolute deviation (MAD).

Table 3. Compilation of Our Samples’ Information: Their Star Number, Number of Magnetic WDs, Median Period (), and MAD

Sample	Stars	Mag.	MAD
			(hr)
Complete sample	318	30	6.8 ± 5.2
WDs with close pairs	54	1	8.1 ± 5.4
Potential WDs with pairs	149	4	7.6 ± 5.2
Likely single WDs	115	25	3.9 ± 3.5

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Figures 4 (5) show the dependence of the detected period—on a logarithmic scale—on the effective temperature (stellar mass) in the left panel and the effective temperature (mass) distribution in the right panel. The red points and bins denote magnetic WDs, whereas the blue points and bins correspond to WDs with close pairs. It is important to point out that the blue and red bins are independent distributions; their overlap does not correspond to a sample of WDs with both close pairs and magnetic fields detected.

Figure 4 shows that our sample of WDs is distributed around approximately 12,000 K, a representative value of the DA population, which comprises about 80% of the WDs spectroscopically identified (e.g., Kepler et al. 2021). The left panel of Figure 4 shows that there does not appear to be a dependence of the detected periods on the effective temperature. Moreover, the close binaries exhibit a mean temperature of 26,300 K, similar to the complete sample, whose mean temperature is 27,700 K. On the other hand, magnetic WDs present a trend toward low temperatures, with a mean effective temperature of 17,100 K.

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The left panel of Figure 5 shows that there is no strong dependence of the detected periods on the stellar mass for the complete sample. However, this plot reveals a negative correlation for the likely single WDs, suggesting a trend to shorter rotational periods for more massive WDs. Moreover, we can also observe a positive correlation between period and mass for the WDs with close pairs.

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The left panel of Figure 5 also shows that most magnetic WDs in our sample exhibit periods shorter than 10 hr and that they are among the targets with the shortest periods. It is essential to recognize that our sample has a selection effect. According to Brinkworth et al. (2013), the rotational period of magnetic WDs exhibits a bimodal distribution, with one group showing periods on the order of minutes to hours and the other displaying periods much longer than their 4 yr of observations. However, their total sample consists of only 26 stars, which is smaller than our sample of 30 magnetic WDs. Each observation cycle (sector) of the TESS telescope lasts ≈27 days, with an interruption of data transfer in the middle, limiting the detection of periods to a few hundred hours. In this way, we detect only the fastest portion of magnetic WDs. Furthermore, massive WDs are fainter and tend to rotate more rapidly (e.g., Hermes et al. 2017b; Caiazzo et al. 2021; Williams et al. 2022). Although TESS is a small telescope, in fact, most of our magnetic sample consists of massive WDs rotating with periods shorter than 10 hr.

The histogram in Figure 5 shows that the mass distribution exhibits two peaks—one sharp peak centered on 0.2 M_⊙ and a broader distribution around approximately 0.55 M_⊙. The distribution around 0.55 M_⊙ is consistent with the values expected for the mass distribution of WDs (e.g., O’Brien et al. 2023). On the other hand, the distribution around 0.2 M_⊙, which contains 16% of the sample, cannot be explained by the evolution of single stars.

We included in this work all WDs that the TESS telescope was able to observe with sufficient signal-to-noise ratio (S/N)—mostly WDs with apparent magnitudes lower than approximately 17.5. Therefore, our sample is limited by magnitude rather than volume. As the mass of WDs is inversely proportional to their radius and the brightness of a star is proportional to its radius, the brighter the WD, the less massive it is. This leads to magnitude-limited samples of WDs being biased toward low-mass stars.

To overcome this bias, we corrected the mass distribution of our sample using the method of Schmidt (1968; see, e.g., Liebert et al. 2005; Kepler et al. 2007). In this method, the contribution of each star is weighted by , where V_max is the volume defined by the maximum distance at which the star would still be visible. To correct for the nonuniform distribution of stars in our Galaxy, we assumed a Galactic disk scale height of 250 pc, following Fleming et al. (1986) and Liebert et al. (2005). We applied this method using the parallax, apparent magnitude, and galactic latitude from Gaia DR3 (Gaia Collaboration et al. 2018). A Gaia parallax was not available for only one star, which was not included in Figure 6.

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Figure 6 exhibits the mass distribution of our sample corrected by the volume. It shows that the distribution of low-mass WDs around 0.2 M_⊙ has disappeared, confirming that the excess of brighter low-mass WDs in our sample is an observational bias. Furthermore, the distribution also indicates a lack of WDs with approximately 0.6 M_⊙, which was supposed to be the most common mass for WD stars. Instead, there is an excess of massive WDs in our sample. Furthermore, it shows that the density of our complete sample is dominated by the likely single WDs sample and that the magnetic WDs of our sample dominate the density of the massive WDs. These results could indicate that our likely single WDs sample is dominated by magnetic WDs resulting from binary interactions (e.g., Toonen et al. 2017; Kilic et al. 2023).

It is relevant to point out that our initial sample of 854 WDs does not present this lack of 0.6 M_⊙ stars (see Appendix B). Instead, it exhibits a mass distribution corrected by , as expected for WDs: centered around 0.6. M_⊙ (e.g., Kepler et al. 2021; O’Brien et al. 2023). Therefore, this gap only appears after the high-quality TESS_localize selection.

As already explained in Section 1, some inhomogeneity at the stellar surface of single WDs must be present for us to be able to detect their rotational rates through photometry. The most common way of generating inhomogeneities is by magnetic fields (e.g., Mathys et al. 2024). The origin of the WD magnetic fields is still an open question; however, it is well known that interactions can result in WDs with strong magnetic fields (Tout et al. 2008). One possibility is that the TESS_localize high-quality parameters may have selected targets with higher amplitude variability and, consequently, those with higher magnetic fields. One piece of evidence is that the TESS_localize criteria reduced our initial sample by 56%; however, it reduced only by 11.8% our initial sample of 34 known magnetic WDs.

Figure 7 shows the histograms of periods up to 100 hr (right) and up to 30 hr (left) for each sample listed in Table 3. Each line of this figure displays the histograms of one sample, indicated by their legends. The last line of Figure 7 shows the same histograms but for the rotational periods determined by asteroseismology, which are listed in Table 4. The red bins represent the magnetic WDs in each sample.

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In the first line of Figure 7 we observe the distribution of our complete sample with its population of 30 magnetic WDs indicated by red bins. The right panel shows that the variability periods are mostly concentrated in the shortest bins. Even considering only periods shorter than 30 hr (left panel) the two shortest bins encompass 36% of our sample. Moreover, this plot shows that our sample of magnetic WDs is concentrated in the shortest periods, presenting a median period of 3.4 hr.

The second line of Figure 7 shows the period histograms for the sample of WDs with close pairs. This sample is composed of 54 stars, one of them with a detected magnetic field. The periods detected for this sample have a median of 8.1 hr. The third line of Section 4.4 illustrates the period distributions for the sample of potential WDs with pairs. This sample is composed of 149 WDs, including four with magnetic fields detected. Their median period is 7.6 hr, close to the median period of the WDs with close pairs.

Finally, the fourth line of Section 4.4 shows the period distribution of the likely single WDs sample. This sample is composed of 115 stars, 25 of them with magnetic fields detected. The right panel shows that the photometric rotation periods are very concentrated in the shortest-period bin, with 69% of periods being shorter than 10 hr. The likely single WDs exhibit a median period of 3.9 hr, very close to the median period exhibited by our complete magnetic sample (3.4 hr). The 25 magnetic WDs included in the likely single WDs sample present an even shorter median period of 1.9 hr.

The last line of Figure 4.4 displays the distribution of the rotational periods estimated through asteroseismology, compiled in Table 4. Despite the difference in sample sizes, this figure shows significant disparities between the rotational distributions derived from photometry and seismology. The seismological sample has a median rotational period of 24.2 hr, which is over 6 times longer than the photometric rotational median period. Additionally, the distribution of rotational periods derived from seismology is broader compared to photometry, with the seismological and the photometric samples exhibiting a MAD of 13.9 hr and 3.5 hr, respectively.

It is important to acknowledge that the rotation period calculated through asteroseismological models is an estimate of the mean internal rotational period of the star; while the photometric rotational period represents the rotation of the stellar surface. Moreover, the seismological sample and the likely single WDs sample are representative of different populations. The seismological sample is representative of the pulsating WDs, whose instability strips are very narrow in temperature range. In contrast, our likely single WDs sample is representative of the WDs with surface inhomogeneities, probably mostly caused by magnetic fields.

As explained in Section 2, we excluded pulsation periods. However, to eliminate any doubts about the short periods in our likely single WDs sample, Figure 8 shows the histogram of periods longer than 0.3 hr. It is important to note that previous work has already detected periods shorter than 0.3 hr for massive WDs (e.g., Barstow et al. 1995; Caiazzo et al. 2021). The inset plot confirms that even with a conservative approach of excluding periods shorter than 0.3 hr, the majority of our sample still exhibit rotation periods shorter than 10 hr.

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Moss et al. (2023) obtained time-resolved spectroscopy for two magnetic WDs in our sample at the Gemini Observatory. Based on the shifting positions of the Zeeman-split H_α components, they inferred a rotation period of 0.648 hr for the star TIC 392797216 (LHS 2273). This star is included in our sample of likely single WDs, and the period reported is very close to the 0.68 hr period we found in the TESS data. Using the same method, they found that the star TIC 262548040 (LHS 1243) has a rotation period of 0.216 hr. Due to its low mass of 0.41 M_⊙, we classified this star as a possible WD with pair. Figure 9 presents the FT of the data obtained by TESS for this star, showing that only one peak at 2617 μHz (0.106 hr) is above the detection limit (horizontal dashed red line). The vertical dotted lines indicate the peak of 0.106 hr and its double (0.212 hr), showing that there is no significant peak at 0.212 hr in the TESS data. Therefore, this shows that, in this case, the signal we detect in the TESS data is the harmonic of the real rotation period. Despite the harmonic issue, Moss et al. (2023) confirm the rotation rates for two of the fastest rotators in our sample.

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From the 61 stars listed in Table 4, 45 were observed by TESS. We analyzed all of them, searching for nonpulsating periods. Most pulsating WDs do not show any frequency with amplitude above the detection limit, other than pulsation periods, or show only low amplitude at frequencies smaller than 4.6 μHz (10 times the frequency resolution of a sector) in the FT. Some pulsating WDs exhibit dozens of pulsation frequencies in their FTs, leading to linear combinations at low frequencies. We only identified reliable variability periods outside the pulsation range (periods longer than 1500 s) for three of these stars: NGC 1501, TIC 7675859, and G226-29. Their TESS_localize output parameters are compiled in Table 2 and are described in the following sections: 4.4.1, 4.4.2, and 4.4.3.

4.4.1. NGC 1501

The pulsating pre-WD NGC 1501 was discovered to pulsate with periods from 5235 s to 1154 s, by Bond et al. (1996). They estimate a rotation period of 1.17 days from a mean splitting of 9.9 μHz. This star was observed by TESS during Sector 19 for 25 days with 120 s exposures. We analyze its FT and detect three variability frequencies outside its pulsating frequency range. TESS_localize confirmed that the source of these signals is NGC 1501. Figure 10 shows this FT up to 200 μHz, where the three nonpulsating frequencies that we have detected are indicated by the vertical dotted lines and correspond to 4.11, 39, and 138 hr. None of these periods is consistent with the rotation period estimated by Bond et al. (1996). Moreover, NGC 1501 is a planetary nebula covered by an ionized gas disk. Therefore, considering the complexity of this system, we are unable to interpret these periods.

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4.4.2. TIC 7675859

Romero et al. (2024) determined the asteroseismological rotation period of 7.15 hr for the DAV TIC 7675859 based on data from Sectors 25, 26, 40, 52, 53, and 54. We observed the three components of a high-amplitude triplet in the data of Sectors 52–54 (see Figure 11). The triplet at 1738 μHz is perfectly symmetric, with a frequency spacing Δν = 22.47 μHz. Moreover, we also observed a significant peak at 22.44 μHz (corresponding to 12.3 hr), which is a very close value to the mean frequency spacing of this triplet. The peak at 22.44 μHz is higher than all components of the triplet at 1738 μHz, but we cannot eliminate the possibility that it is a linear combination (Kurtz et al. 2015). This peak of 12.3 hr can also be a subharmonic of the real rotation rate or even the real rotation rate of the stellar surface itself, indicating differential rotation.

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4.4.3. G226-29

G226-29 is the brightest known DAV, with a known pulsation triplet around 109 s (Kepler et al. 1983). When studying this triplet, Kepler et al. (1995) estimated the rotation period of the WD as 8.9 hr. G226-29 was observed by TESS with 120 s exposures from Sector 14 to 26, and with 20 s exposures from Sectors 40 to 41 and 47 to 60. In total, it was observed along 29 sectors. Data from all sectors show an S/N ≥4.1 wandering peak from 0.93 hr (297.3 μHz) to 3.25 hr (85.5 μHz). As an example, Figure 12 plots the running FT of Sector 51 data. This figure shows the known stable triplet around ∼9200 μHz and a peak that wanders from ∼145 μHz to ∼168 μHz. Figure 13 shows the complete FT of all 20 s data combined, where the wandering peak appears as a broad band at low frequencies (see the inset plot).

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The methodology described in Section 2.1 assumes that the variability signals are stable. Therefore, to verify whether this wandering peak comes from G226-29, we had to apply the TESS_localize test differently. For each sector, we obtain the highest peak exhibited in such an FT range and run the TESS_localize code for this period. As detailed in Table 2, TESS_localize confirmed that this wandering peak indeed comes from G226-29; however, this variability remains a mystery.

The rotational period from seismology is indicated by a vertical black dashed line in Figure 13, showing the signal we are detecting is not consistent with the rotation of the stellar interior. On the other hand, this signal is very long to be consistent with pulsations. This leads us to suspect differential rotation; that is, the wandering peak could be a manifestation of the rotation of the stellar surface, which is rotating faster than the interior. Although the star has an upper limit on the magnetic field around 10 kG (Valyavin et al. 2006), subsurface magnetic fields, like those in Section 5.2, are a possibility. Radial differential rotation alone does not explain why this peak varies so much, but an angular differential rotation, like the Sun (see, e.g., Snodgrass & Ulrich 1990; Beck 2000), could.

Using ground-based photometric observation time series from main-sequence stars spanning late-F to early-M spectral types, Fritzewski et al. (2021) determined 279 rotation periods. They found that the rotation periods range from 0.5 days to 32 days, with trends of slow-rotating FGK stars and fast-rotating K dwarfs. In another recent investigation, Labadie-Bartz et al. (2023) explored the rotational characteristics of chemically peculiar (CP) stars using TESS data. Among their findings, they found that the rotation period distribution of CP stars is centered around 3 days. CP stars are observed within the spectral types from early B to early F. Since all stars with spectral types roughly from A to K will become WDs, both works provide evidence that the rotational period of WD progenitors is of the order of days. We have shown in this work that the median photometric rotation period of WDs is 3.9 hr; therefore, these findings suggest that a robust mechanism of internal angular momentum transfer operates during the evolution of these stars.

The asteroseismology of low- and intermediate-mass stars enabled the measurement of the internal rotation rate of WD progenitors at different evolutionary phases (e.g., Deheuvels et al. 2012, 2014, 2015, 2020; Mosser et al. 2012; Di Mauro et al. 2016, 2018; Gehan et al. 2018; Tayar et al. 2019). These inferred rotation rates have been used to constrain models of angular momentum transfer processes. Eggenberger et al. (2019b) and Moyano et al. (2022) found that the transport of angular momentum must be more efficient in more massive stars during the subgiant and red giant phases, even if the efficiency of the internal transport of angular momentum decreases with the evolution of the star. They suggest that the physical nature of the additional mechanism may be different in subgiant and red giant stars. Eggenberger et al. (2022) find that the core rotation rates of the red giant branch models are nearly insensitive to the initial rotation velocity.

On theoretical grounds, Fuller et al. (2019) deduced a new expression for the Tayler instability that leads to larger magnetic field amplitudes, more efficient angular momentum transport, and smaller shear than predicted by the original Tayler–Spruit dynamo. Eggenberger et al. (2019a) applied the new prescription for subgiant and red giant models and found that it leads to low core rotation rates after the main sequence that are in better global agreement with asteroseismic measurements than those predicted by the original Tayler–Spruit dynamo. However, it fails simultaneously to reproduce the asteroseismic measurements available for subgiant and red giant stars.

den Hartogh et al. (2019) found that the rotational periods of the models with additional viscosity are too large in the WD phase. However, the rotational periods match WD rotation periods of the order of 1 day when they exclude the additional effect during the core-helium-burning phase. Therefore, they proposed that the efficiency of the unknown angular momentum transfer process must decrease during the core-helium-burning phase. Applying the Fuller et al. (2019) formalism, den Hartogh et al. (2020) found that when the dynamo effect is turned off at the end of the core-helium-burning phase, the WD rotation rates decrease to the order of days. They concluded that the Fuller et al. (2019) dynamo formalism cannot be the sole solution to the missing process of angular momentum transport in intermediate-mass stars.

Magnetic fields are present in low-mass stars and, when strong enough, can change the rotation by exerting a torque capable of reducing the differential rotation or even imposing uniform rotation throughout the star. Moyano et al. (2023) showed that the transport of angular momentum in radiative zones during the main sequence of low-mass stars must be efficient. They suggested that the internal magnetic fields are a strong candidate for the missing physical ingredient in stellar interiors. In addition, they point out that at least one efficient process able to neutralize the development of differential rotation in stellar interiors must act during the whole evolution of these stars.

We computed evolutionary sequences using the Modules for Experiments in Stellar Astrophysics (MESA) code (Paxton et al. 2011, 2013, 2015, 2018, 2019; Jermyn et al. 2023), release 22.11.1. Our models have an initial mass of M_i = 1.5 M_⊙ (following den Hartogh et al. 2019) and an initial metallicity of Z = 0.02, 0.002, and 0.001. The initial mass was chosen to be above the lower limit of 1.3 M_⊙ for the convective core in the main sequence (e.g., Aerts & Tkachenko 2023), leading to a WD with a mass close to the mean mass of most WD stars (e.g., Kepler et al. 2007; O’Brien et al. 2023). The models are computed from the zero-age main sequence (ZAMS) until a WD cooling effective temperature of 8500 K. Our models start with equatorial surface rotation velocities of 10 and 50 km s⁻¹, typical of main-sequence stars with 1 and 2 solar masses. These values are assumed to be the solid-body rotations at the ZAMS, which is a reasonable and common assumption for this type of star (see, e.g., Bouvier et al. 1997; Granada & Haemmerlé 2014; Amard et al. 2019; Deal et al. 2020; Nguyen et al. 2022; Douglas et al. 2024). For further information on the models, see Appendix C.

The mixing of elements and the transport of angular momentum due to rotation are implemented in MESA closely following Heger et al. (2000, 2005). Two efficiency factors must be set to calibrate the diffusion coefficients: the contribution of rotationally induced instabilities to the diffusion coefficient is reduced by the factor f_c = 1/30, and the sensitivity of rotationally induced mixing is f_μ = 0.05.

We consider the following angular momentum transport mechanisms: dynamical shear instability, Solberg–Hoiland, secular shear instability, Eddington–Sweet circulation, Goldreich–Schubert–Fricke and the Tayler–Spruit dynamo.

The final masses of our models are 0.546 M_⊙ for Z = 0.02, 0.577 M_⊙ for Z = 0.002, and 0.585 M_⊙ for Z = 0.001. This aligns with the expectations that stars with higher metallicity should form less massive CO cores and lose more mass in the AGB phase than stars with the same initial mass but lower metallicity (e.g., Dominguez et al. 1999; Catalán et al. 2008; Romero et al. 2015; Choi et al. 2016). Similar models with initial masses of 2.5 M_⊙ reach the WD phase with masses around 0.6 M_⊙.

In Figure 14, we show the evolution of the surface rotational period (P) versus the effective temperature (T_eff) during the cooling track for a set of model sequences with different metallicity and initial rotation velocity. The surface rotational period is computed via P = 2π r/v, where r is the radius of the model and v is the rotational velocity at the equator. Sequences that consider the six mechanisms of angular momentum transport are shown as dotted lines (AMT = ON), and solid lines represent sequences with all mechanisms turned off (AMT = OFF).

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For v_i = 10 km s⁻¹ and Z = 0.02 (the purple line in Figure 14), we find that the WD models with AMT = OFF decrease the average equatorial rotation period on the surface from P = 300 hr to ∼ 30 hr when cooling from ∼ 17 kK to 8500 K. The sequence with AMT = ON does not even reach 300 hr before cooling down to the same temperature (and therefore does not appear in the plot).

For sequences with Z = 0.002 (the dark green lines in Figure 14), we find that if AMT = OFF (solid line), the models brake from P = 300 hr to 40 hr only at the very end of the cooling track (T_eff < 10 kK), while the AMT = ON sequence (dotted line) has already spun down to 300 hr when T_eff < 17 kK, although it reaches only P ∼ 60 hr at T_eff = 8500 K.

The internal redistribution of the angular momentum is very efficient early in the evolution of low-mass stars (e.g., Mosser et al. 2012). This suggests that when AMT = ON, a large amount of mass lost by winds during the AGB phase carries away most of the angular momentum, and thus the surfaces of the models rotate slower in the WD phase.

The difference in turning the angular momentum transport mechanisms on and off is much more pronounced for low metallicity (Z = 0.001; the light blue line in Figure 14). While AMT = OFF (solid line) follows the same trend as for higher metallicities (that is, it only reaches low P at the end of the cooling track), the sequence with AMT = ON (dotted line) cools from T_eff = 40 kK down to 8500 K, with an approximately constant rotation period of P = 25 hr, closer to most of the observed period distributions.

In Figure 14, we also show two sequences of models with v_i = 10 km s⁻¹, where the Tayler–Spruit mechanism was turned off and all the other five mechanisms were kept on throughout the evolution (ST = OFF; dotted–dashed lines). For Z = 0.001 (the light green line), the result is similar to that for the same metallicity, but with all six angular momentum transport mechanisms turned off. This suggests that the Tayler–Spruit mechanism dominates over the other mechanisms from the point of view of rotation analysis during the WD phase.

Finally, we present two sequences with v_i = 50 km s⁻¹ and Z = 0.02. These models show that observational data with a shorter rotation period (P < 50 hr) are better adjusted by models with AMT = OFF, while models with AMT = ON present a rotation period shorter than 300 hr only at the end of WD cooling, when T_eff < 10 kK. This suggests that if we disregard the Tayler–Spruit mechanism throughout the evolution, models with a higher initial velocity (50 km s⁻¹) fit the WD rotation data better if the initial mass is 1.5 M_⊙.

The dominant factors for the spin-up of the external layers during the WD cooling phases are the He convection zones that arise at T_eff ≃ 32,000 K, followed by the H convection zone, if the star has an H envelope, around T_eff ≃ 14,000 K. The movement of mass in the convection zones efficiently redistributes the angular momentum.

Also, the models with AMT = ON generate internal magnetic fields due to differential rotation, as shown in Figure 15 for a model sequence with M_i = 1.5 M_⊙, Z = 0.001, and v_i = 10 km s⁻¹ (corresponding to the dotted light blue line in Figure 14). According to Spruit (2002), the Maxwell stresses transport angular momentum in the radial direction at a rate proportional to the product B_r B_ϕ . In Figure 15, the x-axis represents the mass coordinate of the model, with the core on the left and the surface on the right, and the y-axis shows the intensity of the magnetic field components B_r and B_ϕ in the internal part of the models at two different temperatures during the cooling track. The chosen temperatures, 25,000 K and 11,000 K, are representative of the DAV and DBV variable classes.

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Figure 15 shows that during cooling, the azimuthal component of the magnetic field is always greater than the radial component. In some regions, the magnetic field reaches an intensity of ∼10⁴ G. We also found that both components tend to be less intense as the models cool, and the outer ∼5% (by mass) of the models shows no significant magnetic fields.

As a general behavior, the models suggest the following. During cooling, the Tayler–Spruit dynamo mechanism acts smoothly in the core, flattening the rotation profile up to the He layer. On the other hand, a convective zone appears at the base of the He envelope around 32,000 K and increases in size inward as the models cool, causing angular momentum to be transferred from the core and the He zone.

Around 14,000 K, a convective zone develops at the base of the H envelope, and then the rotational speed on the surface is as high as that of the fastest part of the core and the He zone. The exact temperature for this to occur depends on the metallicity and, especially, on the history of mass loss at the end of the AGB phase. This is because the last thermal pulses directly interfere with the amount of H remaining in the model that enters the cooling track, modifying the convection in this zone.

Thus, the internal rotation profile of the models during cooling depends, to some extent, on angular momentum transfer mechanisms, but also strongly depends on convection in the He and H zones.