wdwarfdate: A Python Package to Derive Bayesian Ages of White Dwarfs

The calculation of the total age using wdwarfdate relies on a precise measurement of T_eff and log g. The estimation of these two parameters is done from observational data, and it is described in Section 2.2. From T_eff and log g, the process to estimate the total age of a white dwarf is divided into four stages, summarized in Figure 1:

• 1.  
  From the T_eff and log g input values combined with cooling evolutionary tracks for the white dwarf we obtain its mass (final mass, m_f) and cooling age (t_cool).
• 2.  
  From the m_f we estimate the mass of the progenitor star (initial mass, m_i) using an IFMR, which are, in most cases, empirically calibrated.
• 3.  
  With the m_i combined with stellar evolution tracks we estimate its evolutionary lifetime, meaning the time since the star formed until it became a white dwarf, which we will call main-sequence age or lifetime in this work (t_ms).
• 4.  
  Finally, adding t_cool and t_ms, we obtain the total age of the object (t_tot).

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These steps depend on models which assume single-star evolution. If a white dwarf interacted with another object at any point, then the evolutionary models will not reproduce its parameters correctly (e.g., Marsh 1995; Brown et al. 2011). In Section 4 we discuss different scenarios where the single-star evolution assumption might not be valid.

The accuracy and precision of the estimated fundamental white dwarf parameters using wdwarfdate depend strongly on how the two input observables were determined, the obtained values, and uncertainties. In addition, to initialize wdwarfdate the user needs to choose between cooling models for hydrogen-atmosphere (DA) and helium-atmosphere (non-DA) white dwarfs, an IFMR, and a stellar evolution model for the progenitor star (metallicity and rotation). In this section, we describe the methods generally used to estimate T_eff and log g in the literature and how they affect predictions by wdwarfdate. We also describe the models included in wdwarfdate for each of the choices described above, as summarized in Table 1.

The determination of temperature and surface gravity generally rely on either a photometric or a spectroscopic approach. Photometric methods rely on accurate trigonometric parallaxes and high-precision photometry (e.g., Koester et al. 1979) to reconstruct the spectral energy distribution of a white dwarf, and require an accurate knowledge of the extinction caused by interstellar dust. Spectroscopic methods, however, rely on atmosphere models to reproduce the detailed profiles of hydrogen or helium lines, and depend on their theoretical line profiles (e.g., Bergeron et al. 1992; Beauchamp et al. 1999). These line profiles in most cases approximate the white dwarf atmosphere as a one-dimensional object, which can lead to biases in some extreme cases of white dwarf properties, and in most cases ignore the impact of magnetic fields which can be significant for the most massive white dwarfs. Recent studies using data from Gaia DR2 and the spectroscopic method show that the results from these two techniques generally agree within the uncertainties (Tremblay et al. 2019), except for T_eff ≲ 6500 K where the parallax-based photometric estimations of log g were found to be more accurate than spectroscopic estimations (Napiwotzki et al. 2020; Kawka & Vennes 2012).

wdwarfdate relies on the theoretical evolutionary sequences of the Montreal White Dwarf Group (Bédard et al.2020). These models assume that the white dwarf has a core composed of a uniform carbon and oxygen mixture (X_C = X_O = 0.5), surrounded by a He layer (with a fractional mass q_He = M_He/M_* = 10⁻²) surrounded by an outermost H layer. The evolutionary tracks assume that this outermost H layer is "thick" (q_H ≡ M_H/M_* = 10⁻⁴) for DA white dwarf and "thin" (q_H = 10⁻¹⁰) for non-DA (Bédard et al. 2020). The publicly available Montreal cooling tracks were calculated for masses in the range 0.2−1.3 M_⊙ in steps of 0.05 M_⊙. These tracks are shown in Figure 2 in black, together with the limits within which wdwarfdate was designed to function in terms of T_eff and log g, shown in a blue dashed line. These limits are based on the following limitations: (1) the lowest temperature covered by the cooling tracks (1500 K); (2) the full range of white dwarf masses covered by the cooling tracks (0.2−1.3 M_⊙); and (3) the 0.3 Myr isochrone, given that cooling ages below this value are unreliable. In summary, the approximate limits within which wdwarfdate will rely on the cooling tracks are: 1500≤ T_eff ≲ 100,000 K and 7.0 ≤ log g ≲ 9.3. Section 4 details additional limitations of wdwarfdate.

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To estimate the progenitor mass from the white dwarf mass, or vice versa, the user can choose among the following IFMRs: Cummings et al. (2018; MIST- and PARSEC-based); Marigo et al. (2020); Salaris et al. (2009); and Williams et al. (2009), shown in Figure 3 with semi-empirical data from Cummings et al. (2018). Below we describe each of these IFMRs. Cummings et al. (2018) used white dwarf members of open clusters with known ages to derive the expected progenitor mass and constrain the IFMR of relatively massive (and young) white dwarfs. Their analysis is based on a spectroscopic determination of the white dwarfs T_eff and log g, which was then used to estimate the cooling age and mass of the white dwarf using the Fontaine et al. (2001) cooling tracks. These models are valid for T_eff ≲ 30,000 K and were updated by the more recent Bédard et al. (2020) cooling tracks. By subtracting the cooling age from the age of the host open cluster, they obtained the main-sequence age of the progenitor star, which they then used to estimate the progenitor mass using the PAdova and TRieste Stellar Evolution Code (PARSEC; Bressan et al. 2012) and the models computed using the Modules for Experiments in Stellar Astrophysics (MESA; Paxton et al. 2011, 2013, 2015, 2018) Isochrones & Stellar Tracks (MIST; Choi et al. 2016; Dotter 2016). These two families of stellar evolutionary models resulted in two distinct IFMRs, both of which are included in wdwarfdate. The difference between the initial masses calculated for the two IFMR are generally within 5%. By default, wdwarfdate uses the MIST-based IFMR as recommended by Cummings et al. (2018), due to the nonrotating MIST isochrones estimating more precise ages for young stars (∼100 Myr), as tested with the Pleiades cluster. However, the PARSEC-based IFMR should be considered especially for high-mass progenitor stars (>5 M_⊙), given that the MIST models tend to underestimate masses in this range.

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The semi-empirical IFMR from Marigo et al. (2020) was also obtained from white dwarfs from known clusters. In particular, they included seven new white dwarf members of the old open clusters NGC 752 (1.55 Gyr) and Ruprecht 147 (2.5 Gyr), which allowed them to study the low-mass end (around 0.5 M_⊙) of the IFMR in detail. In addition, Marigo et al. (2020) used a new analysis technique, which combines photometric and spectroscopic data to estimate precise T_eff, log g, and m_f, and to confirm cluster membership for the white dwarfs. As shown in Figure 3, they found a kink at low masses, which they associated with the formation of carbon stars in the Galaxy. The IFMR from Salaris et al. (2009) also used white dwarfs from known clusters to calibrate the relation, but put special emphasis in the models used, and performed a detailed analysis of the sources of uncertainties and scatter in the relation. Finally, the IFMR from Williams et al. (2009) studied white dwarfs from the intermediate-age open cluster M35 (NGC 2168), which provided information to constrain the higher-mass end of the IFMR.

The IFMRs included in wdwarfdate depend on cluster objects that are younger and therefore more massive than average field white dwarfs, which could cause a bias in the relation. In addition, as some of the IFMRs were done prior to the Gaia mission, and many of these objects were assumed to be cluster members based on their kinematics and colors, without parallaxes, some of these objects might not be true members, biasing the relation. The IFMR will also place an extra boundary in the limits of T_eff and log g for which wdwarfdate can estimate a total age. For example, Cummings et al. (2018) MIST-based IFMR was calibrated for white dwarf masses between 0.5−1.3 M_⊙, which will place a tight constraint specially in the lowest value possible of log g. We will discuss these limitations further in Section 4.

The IFMRs from Cummings et al. (2018) and Marigo et al. (2020) are calibrated for m_i > 0.83 M_⊙, and we included them in the code extending this limit to 0.45 M_⊙, assuming a constant m_f between 0.45 < m_i < 0.83 (gray line in Figure 3). This extension allows wdwarfdate to estimate parameters for white dwarfs with masses m_f ≲ 0.6 M_⊙, which is the peak of the white dwarf mass distribution (e.g., Genest-Beaulieu & Bergeron 2019).

To estimate the main-sequence age from the initial mass, or progenitor mass, we adopted in wdwarfdate the MIST isochrones. These tracks cover a mass range between 0.1−300 M_⊙ with a step of 0.05 M_⊙, and an age range from 0.1 Myr to 13.8 Gyr. To select the optimal stellar evolution track, the user has to choose the metallicity and rotation of the progenitor star among the following options: [Fe/H] = {−4,−1,0,0.5} (dex), [α/Fe] = 0, and v/vcrit = {0.0,0.4}. We will discuss the effects of using different tracks on the total age in Section 4.

In this section, we describe the Bayesian method implemented in wdwarfdate. Our code derives Bayesian ages of white dwarfs, with a posterior probability given by

$$\begin{eqnarray}&&p({m}_{{\rm{i}}},{\mathrm{log}}_{10}{t}_{\mathrm{cool}},{{\rm{\Delta }}}_{{\rm{m}}}| {T}_{\mathrm{eff}},\mathrm{log}g),\end{eqnarray} \tag{ 1 }$$

where m_i is the initial mass, or the mass of the progenitor star, and t_cool is the cooling age of the white dwarf. Δ_m is an extra parameter which takes into account the scatter in the IFMR of white dwarfs, and it is described below. T_eff and log g are the inputs for the effective temperature and surface gravity, respectively. To simplify notation we did not include uncertainties in Equation (1), although these are also required in the code. Using the Bayes theorem, the posterior probability in Equation (1) can be written as

$$\begin{eqnarray}&&\begin{array}{l}p({m}_{{\rm{i}}},{\mathrm{log}}_{10}{t}_{\mathrm{cool}},{{\rm{\Delta }}}_{{\rm{m}}}| {T}_{\mathrm{eff}},\mathrm{log}g)\propto \\ p({T}_{\mathrm{eff}},\mathrm{log}g| {m}_{{\rm{i}}},{\mathrm{log}}_{10}{t}_{\mathrm{cool}},{{\rm{\Delta }}}_{{\rm{m}}})\\ p({\mathrm{log}}_{10}{t}_{\mathrm{cool}})p({m}_{{\rm{i}}})p({{\rm{\Delta }}}_{{\rm{m}}}),\end{array}\end{eqnarray} \tag{ 2 }$$

where p(log ₁₀ t_cool), p(m_i), and p(Δ_m) are the priors on the cooling age, the initial mass, and Δ_m respectively, which are described below, and

$$\begin{eqnarray}&&\begin{array}{l}p({T}_{\mathrm{eff}},\mathrm{log}g| {m}_{{\rm{i}}},{\mathrm{log}}_{10}{t}_{\mathrm{cool}},{{\rm{\Delta }}}_{{\rm{m}}})\\ \propto \exp \left[-\displaystyle \frac{{\left(\overline{{T}_{\mathrm{eff}}}-{T}_{\mathrm{eff}}\right)}^{2}}{2{\sigma }_{{T}_{\mathrm{eff}}}^{2}}\right]\\ \exp \left[-\displaystyle \frac{{\left(\overline{\mathrm{log}g}-\mathrm{log}g\right)}^{2}}{2{\sigma }_{\mathrm{log}g}^{2}}\right]\end{array}\end{eqnarray} \tag{ 3 }$$

is the likelihood. $\overline{{T}_{\mathrm{eff}}}$ and $\overline{\mathrm{log}g}$ in Equation (3) are the modeled effective temperature and surface gravity of the white dwarf. These modeled quantities are calculated from the three parameters that are being sampled (m_i, t_cool and Δ_m) as shown by the probabilistic graphical model in Figure 4, which represents the posterior probability of the problem. In summary: 1) From the initial mass (m_i) we estimate the main-sequence age (t_ms); 2) by adding t_ms to the cooling age (t_cool) we obtain the total age (t_tot) of the white dwarf; 3) from m_i and the delta parameter (Δ_m) we estimate a final mass (m_f, described below); 4) from m_f and t_cool we estimate the modeled effective temperature and surface gravity ($\overline{{T}_{\mathrm{eff}}}$ and $\overline{\mathrm{log}g}$), which are compared to the input parameters (T_eff and log g) to estimate the posterior probability. The models used in each step are described in Section 2.2.

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The Δ_m parameter in Equations (1), (2), and (3) was included to model the scatter in the IFMR. This scatter has different causes including contamination from noncluster members, measurement errors, metallicity variations, inconsistencies with the models used, and age determination of the clusters (e.g., Weidemann 2000; Williams et al. 2009; Casewell et al. 2009). Parameter Δ_m is drawn from a normal distribution centered at zero with standard deviation σ_m, so that

$$\begin{eqnarray}&&p({{\rm{\Delta }}}_{{\rm{m}}})={ \mathcal N }(0,{\sigma }_{{\rm{m}}}).\end{eqnarray} \tag{ 4 }$$

We estimated parameter σ_m from the Cummings et al. (2018) data and adopted σ_m = 0.03 M_⊙. The inclusion of the scatter in the posterior is indicated in Figure 4 as ${\hat{m}}_{{\rm{f}}}={m}_{{\rm{f}}}+{{\rm{\Delta }}}_{{\rm{m}}}$, where m_f is the true mass of the white dwarf, and ${\hat{m}}_{{\rm{f}}}$ is the scattered mass, because it was calculated using the IFMR.

The priors indicated in the posterior probability of Equation (2) are given by the star formation history (SFH) for t_cool and the initial mass function (IMF) for m_i. We assume a constant SFH (e.g., Madau & Dickinson 2014) and applied it as a flat prior on t_tot, where we required t_tot < 15 Gyr to avoid a sharp cut at the age of the universe in the age probability distributions. As t_tot = t_cool + t_ms, the Jacobian of the coordinate transformation is 1, and this prior could be used without extra modifications. In addition, we assumed the IMF to be m_i ^−2.3 (Kroupa 2001; Chabrier 2003), which describes the range of stellar masses expected to form a white dwarf.

wdwarfdate obtains the probability distributions by explicitly integrating over the likelihood and priors to perform the normalization. We set the code to evaluate the posterior in a grid made of discrete arrays for each parameter: m_i, log ₁₀ t_cool, and Δ_m. The code first performs a wide grid evaluation with the parameters ranging between 0.3 Myr < t_cool < 15 Gyr, 0.1 M_⊙ < m_i < 10 M_⊙ and −0.1 M_⊙ < Δ_m < 0.1 M_⊙. Then it obtains the probability distribution for each parameter by setting the minimum and maximum evaluation limits according to where the probability is higher. These limits can also be adjusted by the user. We use these results combined with the models described in Section 2.2 to derive probability distributions for the rest of the parameters (total age, final mass, and main-sequence age).

One of the limitations of wdwarfdate is given by the IFMR: the total age of a white dwarf can be estimated only in the cases where its mass is between the allowed limits of the IFMR. For the cases where a total age cannot be estimated but the T_eff and log g are within the limits of the cooling tracks, we included in wdwarfdate a fast-test method to estimate the final mass and cooling age only. For this method, wdwarfdate generates two normal distributions for the effective temperature and surface gravity, using the input values (T_eff and log g) as means, and the uncertainties (σ_Teff and σ_logg ) as the standard deviations of the distributions such as

$$\begin{eqnarray}&&\begin{array}{l}{X}_{{T}_{\mathrm{eff}}}\sim { \mathcal N }({T}_{\mathrm{eff}},{\sigma }_{{T}_{\ \mathrm{eff}}})\\ {X}_{\mathrm{log}g}\sim { \mathcal N }(\mathrm{log}g,{\sigma }_{\mathrm{log}g}).\end{array}\end{eqnarray} \tag{ 5 }$$

The code does a Monte Carlo uncertainty propagation by running the two distributions through the process described in Section 2.1 and in Figure 1, and obtains a distribution for each of the parameters of interest: final mass, initial mass, cooling age, main-sequence age, and total age. wdwarfdate will automatically run this method when the final mass of the white dwarf is outside of the allowed ranges by the IFMR. The main difference between the fast-test and the Bayesian methods are the priors: the fast-test method does not include the IMF and SFH as priors on the initial mass and the total age, respectively, but it does constrain the total age to be smaller than 15 Gyr.

To test that the final mass and cooling age estimated with the fast-test and the Bayesian methods are comparable, we run a grid of white dwarfs with T_eff = 1500−100,000 K and log g = 7−9.3 and uncertainties of 10% and 1%, respectively, using the two methods. For both methods we used the DA white dwarf cooling sequences, the Cummings et al. (2018) MIST-based IFMR, the MIST tracks with [Fe/H] = 0 and v/vcrit = 0.0, and the uncertainties were calculated using the 16th and 84th percentile of the distributions. The results show that the final mass estimations agree within the uncertainties between both methods (right panel of Figure 5), and the cooling age estimations agree in most cases (left panel Figure 5). We color-coded in purple the points for which the cooling age differs by more than 1σ, and we found that these points agree in the final mass within the uncertainty, although the cooling age values seem to be more discrepant as the cooling age decreases. This discrepancy does not depend on the cooling age because there are some white dwarfs with small cooling ages that have the same result in the fast-test and Bayesian method. The purple points on Figure 5 have some of the highest temperatures and surface gravities, as shown by the (T_eff−log g) inset diagram in the left panel of Figure 5. Moreover, the purple points follow the area where the cooling tracks are closer to each other in Figure 2. This increases the uncertainty in the cooling age, making the likelihood probability distribution function (PDF) of these objects wider. Therefore, the prior, in particular the IMF, has a bigger effect on the estimated values, making the cooling ages estimated with the Bayesian method larger, given that it favors smaller initial masses.

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The final masses between 0.65 and 0.8 M_⊙ in the right panel of Figure 5 are slightly lower with the Bayesian method than with the fast-test method. This effect is also caused by the prior, in a similar way as described above. By inspecting the cooling tracks in Figure 2, we noticed that the cooling tracks in the mass range 0.65−0.8 M_⊙ are closer to each other, making the likelihood less constrained. The differences for the final mass and cooling age described above show the advantage of applying the Bayesian method, which adds extra information using the priors (IMF and SFH), unlike the fast-test method, which provides the highest likelihood value. In summary, when the uncertainties in T_eff and log g are small (∼10% and ∼1%, respectively) the fast-test and the Bayesian methods are comparable in the estimation of final masses and cooling ages, except for white dwarfs with both T_eff > 40,000 K and log g > 8.4, which coincide with dense track areas. However, these exceptions represent only 2% of our sample, and some have extremely high temperatures; therefore these cases are not common.

Cummings et al. (2018) used a sample of 79 white dwarfs from 13 different clusters of known ages with T_eff and log g measurements from the literature to calibrate the IFMR. In addition, the authors remeasured T_eff and log g for 8 of the white dwarfs. Cummings et al. (2018) is one of the most comprehensive studies to date of white dwarfs from clusters. They used the cooling tracks from the Montreal White Dwarf Group to estimate final masses and cooling ages from T_eff and log g, and the MIST isochrones to estimate initial masses from the cooling and total ages of the objects (see Section 2.2 for a detailed description of their work). Given that they estimated all the parameters which can be obtained with our code using a similar procedure, Cummings et al. (2018) is an ideal study to compare the performance of wdwarfdate.

We run wdwarfdate on all the stars in Cummings et al. (2018) using the T_eff and log g and uncertainties published in their work. We compared the final mass, cooling age, initial mass, and main-sequence age in Figure 6. Most of the values agree within 1σ with their results. The uncertainties calculated with wdwarfdate are comparable to the estimated by Cummings et al. (2018) for the cooling age and final mass. For the main-sequence age and initial mass, the uncertainties are smaller in the Cummings et al. (2018) results because they were propagated from the age of the cluster that has a small uncertainty, and not the IFMR as in wdwarfdate.

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We color-coded the white dwarfs for which any of the parameters in Figure 6 did not agree within 1σ to study them in detail. The black dot is the white dwarf GD 50, and although the parameters agree within the uncertainties it is color-coded because it is discussed below. The white dwarfs color-coded in green and purple are outliers in the IFMR (Figure 3), which explains why the initial mass and main-sequence age do not agree. We refer to Cummings et al. (2018) for the analysis of the outliers. The final mass and cooling age for the green objects agree within uncertainties, which is expected because we are using the same T_eff and log g to estimate them. In the case of the three white dwarfs color-coded in purple, wdwarfdate estimated a higher cooling age than expected according to the parameters in Cummings et al. (2018). However, these objects have T_eff > 50,000 K, and the cooling tracks used in Cummings et al. (2018) and Fontaine et al. (2001) are not suitable for temperatures >30,000 K. We confirmed our calculations of the cooling ages with the Montreal White Dwarfs Data Base (MWDD; Dufour et al. 2017) ¹⁴ , which contains the updated models by Bédard et al. (2020), which we implemented in wdwarfdate.

To compare the total age estimated with wdwarfdate with the cluster age, we excluded the outliers (points color-coded in green and purple in Figure 6) in Figure 7. The total estimated age agrees within 1σ with the cluster age for 65% of the white dwarfs, and within 2σ for 94%. We also calculated a median relative difference of 12% between the median age for each white dwarf that was not an outlier and the age according to the cluster membership, which shows wdwarfdate is doing a good job estimating these total ages.

The white dwarf color-coded in black is GD 50, which is a young and ultramassive white dwarf of 1.28 ± 0.08 M_⊙. For this object we obtained a main-sequence age of 46.21 ^+4.76 _−3.74 Myr, a cooling age of 82.62^+11.19 _−14.08 Myr, a total age of ${128.71}_{-11.92}^{+10.54}$ Myr, a final mass of 1.27 ± 0.02 M_⊙, and an initial mass of 7.5 ± 0.3 M_⊙. Our progenitor mass calculation differs slightly from the value in Cummings et al. (2018), but agrees with the results of Gagné et al. (2018), who calculated a mass of 7.8 ± 0.6 M_⊙. They found GD 50 is likely member of the AB Doradus moving group and performed a detailed estimation of its parameters using the Montreal white dwarf evolutionary models and the MIST models, accounting for all possible C/O/Ne core compositions, and finding a total age of 117 ± 26 Myr. Our result is between this value and the one estimated by Cummings et al. (2018) that assumed GD 50 belongs to the the Pleiades (based on Dobbie et al. 2006) and assigned it an age of 135 ± 35 Myr, which is slightly larger than the current measured age of this cluster (112 ± 5 Myr; Dahm 2015). We also used wdwarfdate on GD 50 using the PARSEC-based IFMR from Cummings et al. (2018), as was suggested in Section 2.2 for progenitor masses >5 M_⊙, and we obtained a total age of 114.61^+12.72 _−11.68 Myr, significantly closer to the results of Gagné et al. (2018).

To test the accuracy of wdwarfdate at lower masses, we estimated the parameters of the white dwarfs from Canton et al. (2021). The authors measured T_eff and log g from high-resolution spectra for 22 white dwarfs from the M67 cluster, which is 3.5 Gyr and has solar metallicity. They used the cooling tracks from Fontaine et al. (2001) to estimate the final masses from these parameters, and the PARSEC isochrones to estimate initial masses from the age of the cluster. They estimated initial masses only for a smaller vetted sample, and discarded the rest because, for example, they were potential He-core white dwarfs or potential blue straggler remnants. Using their measured T_eff and log g, and uncertainties with wdwarfdate, we found good agreement for the final mass and cooling age between our results and Canton et al. (2021)—as shown in Figure 8—for all but two white dwarfs. However, our results for these two white dwarfs agree with the MWDD; therefore the differences are likely due to Canton et al. (2021) using an older version of the Montreal cooling tracks (Fontaine et al. 2001). In addition, we compared our estimation of the total age and the initial mass for the vetted sample from Canton et al. (2021), and found that our results for both parameters agree within 1σ, as shown by the bottom panels in Figure 8. The uncertainties on the total age of the white dwarfs are large (±5 Gyr), which is expected given that these are low-mass white dwarfs and the largest contribution to the total age comes from the main-sequence age, which tends to have higher uncertainties. Furthermore, the uncertainties on the initial mass calculated by Canton et al. (2021) are smaller because they used the known total age to estimate these masses, which is well constrained. To further check our code, we run wdwarfdate using the IFMR from Marigo et al. (2020) which differs from Cummings et al. (2018) for low masses, and found the same results for the median value of the parameters, although the shape of the posterior was different in some cases, due to the change in the shape of the IFMR.

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We used wdwarfdate to estimate ages of 30 white dwarfs identified to be candidate wide comovers with M dwarfs by Kiman et al. (2021). These M dwarfs constitute a new set of age calibrators, and we refer the reader to Kiman et al. (2021) for information about the main-sequence comoving star. Kiman et al. (2021) estimated the white dwarf ages using wdwarfdate, and the T_eff and log g calculated by Gentile Fusillo et al. (2019) with photometry from Gaia DR2. For our calculations we used the updated version of that work (Gentile Fusillo et al. 2021), which took advantage of the precise photometry and parallaxes from Gaia EDR3 and estimated T_eff and log g for 359,073 high-confidence white dwarf candidates. However, we did not find improvement between Gaia DR2 and EDR3 in the relative uncertainty of T_eff and log g for the 30 white dwarfs, and we found a relative difference for the values of 8.4% and 2.2%, respectively.

All of the 30 objects have a white dwarf probability >95% assigned by Gentile Fusillo et al. (2021). However, one of the white dwarfs does not have an estimation for T_eff and log g. We discarded that source along with the white dwarfs with masses <0.5 M_⊙, or >1.1 M_⊙, according to their position in the color–magnitude diagram (Figure 9) and white dwarfs with G_BP−G_RP > 0.9. In short, the reasons for these cuts are that high- and low-mass white dwarfs are thought to be the result of binary evolution, and therefore do not follow the single-star evolution assumption made by the cooling tracks included in wdwarfdate (Bédard et al. 2020). In addition, the color cut avoids contamination from nonwhite dwarfs such as unresolved binaries. For a more detailed discussion of the limitations of wdwarfdate see Section 4.

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We used wdwarfdate to estimate the ages of the remaining 18 white dwarfs. Five of these objects were identified as DA white dwarfs in the MWDD, two as DB, and we assumed DA for the rest, and used the corresponding T_eff and log g from Gentile Fusillo et al. (2021). Given that most white dwarfs are DAs (∼75%), this is a good assumption (Fontaine et al. 2001; Gentile Fusillo et al. 2015). To run our code we used the cooling sequence with a thick H outermost layer for DA white dwarfs and with a thin layer for non-DA, the stellar evolution models for [Fe/H] = 0 and v/vcrit = 0.0, and the Cummings et al. (2018) MIST-based IFMR. We also calculated these white dwarf ages using the Marigo et al. (2020) IFMR and found similar results, which are not displayed here. The Gaia source id of the white dwarfs, the input T_eff and log g, and the parameters estimated with wdwarfdate are in Table 2. The reported values in this table are the median and 16th and 84th percentiles as uncertainties. We were able to estimate the total age for 13 of the 18 white dwarfs; the rest have masses outside of the allowed range by the IFMR. For these 13 white dwarfs we used both the Bayesian and fast-test methods included in wdwarfdate, and both results are shown in the table, while for the remaining 5 we only used the fast-test method to estimate the final mass and cooling age.

Table 2. Candidate White Dwarf Comovers with M Dwarfs

Name	SPT a	Teff [K] b	$\mathrm{log}g$ b	Met. c	tms [Gyr]	tcool [Gyr]	ttot [Gyr]	mi [M⊙]	mf [M⊙]
Gaia EDR3 2643218398328129664	⋯	4661 ± 251	7.90 ± 0.23	FT	⋯	${5.28}_{-1.89}^{+2.79}$	⋯	⋯	${0.53}_{-0.13}^{+0.14}$
SDSS J123304.34+030245.6	⋯	4735 ± 263	7.83 ± 0.27	FT	⋯	${4.33}_{-1.53}^{+3.13}$	⋯	⋯	${0.48}_{-0.13}^{+0.16}$
Gaia EDR3 1323951092358466304	⋯	5652 ± 842	8.49 ± 0.58	B	${1.67}_{-1.31}^{+4.05}$	${5.42}_{-2.35}^{+2.75}$	${8.20}_{-2.76}^{+3.32}$	${1.81}_{-0.58}^{+1.49}$	${0.64}_{-0.06}^{+0.16}$
				FT	${0.14}_{-0.09}^{+0.42}$	${6.09}_{-2.00}^{+2.68}$	${6.51}_{-1.81}^{+2.72}$	${4.58}_{-1.78}^{+2.19}$	${0.96}_{-0.25}^{+0.23}$
Gaia EDR3 1393854635743747200	⋯	6469 ± 617	8.09 ± 0.32	B	${2.04}_{-1.46}^{+5.28}$	${2.90}_{-1.01}^{+2.07}$	${5.96}_{-2.07}^{+4.30}$	${1.68}_{-0.54}^{+1.11}$	${0.63}_{-0.05}^{+0.09}$
				FT	${0.47}_{-0.28}^{+1.70}$	${3.40}_{-1.38}^{+1.92}$	${4.56}_{-1.24}^{+1.84}$	${2.97}_{-1.32}^{+1.13}$	${0.74}_{-0.12}^{+0.17}$
SDSS J155516.95+315307.3	DA	6740 ± 108	8.02 ± 0.05	B	${4.43}_{-2.59}^{+4.86}$	${1.78}_{-0.13}^{+0.15}$	${6.17}_{-2.47}^{+4.80}$	${1.32}_{-0.26}^{+0.42}$	${0.60}_{-0.02}^{+0.03}$
				FT	${3.31}_{-1.59}^{+4.09}$	${1.85}_{-0.14}^{+0.16}$	${5.11}_{-1.38}^{+4.11}$	${1.44}_{-0.31}^{+0.35}$	${0.60}_{-0.02}^{+0.03}$
SDSS J114850.50+325406.5	DA	6981 ± 977	7.71 ± 0.57	FT	⋯	${1.31}_{-0.62}^{+2.24}$	⋯	⋯	${0.47}_{-0.19}^{+0.32}$
Gaia EDR3 4467448853280423296	⋯	7151 ± 935	8.27 ± 0.45	B	${1.53}_{-1.22}^{+5.19}$	${2.68}_{-1.16}^{+2.49}$	${5.36}_{-2.08}^{+4.41}$	${1.87}_{-0.70}^{+1.57}$	${0.65}_{-0.06}^{+0.18}$
				FT	${0.27}_{-0.19}^{+0.92}$	${3.41}_{-1.71}^{+1.59}$	${3.99}_{-1.30}^{+1.65}$	${3.59}_{-1.44}^{+2.06}$	${0.86}_{-0.19}^{+0.22}$
Gaia EDR3 860411038927355264	⋯	8557 ± 1021	8.06 ± 0.41	B	${2.06}_{-1.66}^{+6.18}$	${1.29}_{-0.42}^{+1.06}$	${3.90}_{-1.60}^{+5.51}$	${1.67}_{-0.58}^{+1.48}$	${0.63}_{-0.06}^{+0.15}$
				FT	${0.43}_{-0.30}^{+1.72}$	${1.52}_{-0.61}^{+1.49}$	${2.60}_{-0.85}^{+1.79}$	${3.07}_{-1.42}^{+1.61}$	${0.76}_{-0.14}^{+0.21}$
Gaia EDR3 3221072914762838144	⋯	9249 ± 492	7.72 ± 0.19	FT	⋯	${0.56}_{-0.13}^{+0.17}$	⋯	⋯	${0.45}_{-0.09}^{+0.10}$
SDSS J003000.03-002738.9	DA	9359 ± 1061	7.94 ± 0.45	FT	${2.40}_{-1.96}^{+6.41}$	${0.97}_{-0.30}^{+0.67}$	${3.59}_{-1.61}^{+5.91}$	${1.59}_{-0.51}^{+1.46}$	${0.62}_{-0.05}^{+0.14}$
Gaia EDR3 793351038769083776	⋯	10286 ± 1114	7.93 ± 0.41	FT	${2.67}_{-2.13}^{+6.33}$	${0.73}_{-0.22}^{+0.40}$	${3.45}_{-1.74}^{+6.10}$	${1.54}_{-0.47}^{+1.31}$	${0.62}_{-0.05}^{+0.11}$
Gaia EDR3 636417842920590208	⋯	12338 ± 2743	8.45 ± 0.44	B	${1.13}_{-0.97}^{+5.10}$	${1.18}_{-0.73}^{+2.65}$	${3.36}_{-1.91}^{+5.59}$	${2.19}_{-1.00}^{+2.19}$	${0.67}_{-0.08}^{+0.27}$
				FT	${0.17}_{-0.10}^{+0.47}$	${0.86}_{-0.48}^{+1.27}$	${1.36}_{-0.58}^{+1.36}$	${4.30}_{-1.62}^{+1.94}$	${0.93}_{-0.23}^{+0.21}$
Gaia EDR3 2536705752006690304	⋯	12371 ± 2799	8.07 ± 0.50	B	${2.05}_{-1.69}^{+5.70}$	${1.00}_{-0.61}^{+2.71}$	${4.40}_{-2.68}^{+5.81}$	${1.68}_{-0.56}^{+1.61}$	${0.63}_{-0.06}^{+0.17}$
				FT	${0.35}_{-0.25}^{+1.20}$	${0.64}_{-0.35}^{+0.97}$	${1.32}_{-0.54}^{+1.61}$	${3.29}_{-1.44}^{+1.97}$	${0.80}_{-0.16}^{+0.23}$
SDSS J082233.92+213047.3	DA	12859 ± 1751	8.18 ± 0.22	B	${1.82}_{-1.37}^{+5.43}$	${0.46}_{-0.17}^{+0.32}$	${2.38}_{-1.24}^{+5.51}$	${1.75}_{-0.61}^{+1.27}$	${0.64}_{-0.06}^{+0.11}$
				FT	${0.47}_{-0.22}^{+1.20}$	${0.46}_{-0.17}^{+0.28}$	${1.09}_{-0.32}^{+1.06}$	${2.97}_{-1.17}^{+0.77}$	${0.74}_{-0.11}^{+0.13}$
Gaia EDR3 703753485491279488	⋯	14390 ± 2095	7.83 ± 0.28	FT	⋯	${0.18}_{-0.09}^{+0.16}$	⋯	⋯	${0.51}_{-0.12}^{+0.15}$
RMB 103	DB	14096 ± 518	7.93 ± 0.08	B	${6.23}_{-3.61}^{+4.83}$	${0.25}_{-0.03}^{+0.04}$	${6.50}_{-3.62}^{+4.77}$	${1.19}_{-0.18}^{+0.36}$	${0.58}_{-0.03}^{+0.03}$
				FT	${5.38}_{-3.28}^{+5.40}$	${0.25}_{-0.03}^{+0.04}$	${5.62}_{-3.23}^{+5.39}$	${1.24}_{-0.23}^{+0.42}$	${0.59}_{-0.02}^{+0.03}$
LAMOST J095620.88+272729.8	DA	15137 ± 682	7.97 ± 0.08	B	${4.52}_{-2.82}^{+5.23}$	${0.20}_{-0.03}^{+0.04}$	${4.72}_{-2.81}^{+5.21}$	${1.31}_{-0.26}^{+0.48}$	${0.60}_{-0.03}^{+0.03}$
				FT	${2.91}_{-1.54}^{+4.82}$	${0.20}_{-0.03}^{+0.04}$	${3.10}_{-1.51}^{+4.79}$	${1.50}_{-0.38}^{+0.50}$	${0.61}_{-0.03}^{+0.04}$
PB 6042	DB	15602 ± 2052	8.11 ± 0.30	B	${2.49}_{-1.89}^{+6.10}$	${0.26}_{-0.10}^{+0.19}$	${2.80}_{-1.79}^{+6.02}$	${1.57}_{-0.49}^{+1.16}$	${0.62}_{-0.05}^{+0.10}$
				FT	${0.49}_{-0.26}^{+1.74}$	${0.29}_{-0.12}^{+0.21}$	${0.89}_{-0.26}^{+1.57}$	${2.92}_{-1.29}^{+0.94}$	${0.73}_{-0.11}^{+0.15}$

Notes.

^a Classification in the Montreal White Dwarf Data Base. ^b Parameters from Gentile Fusillo et al. (2019), estimated from photometry. ^c Method used in wdwarfdate: B=Bayesian, F=Fast-test.

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The white dwarfs from Table 2 exemplify the impact of the T_eff and log g uncertainties in the estimation of the parameters, and show the advantage of using the Bayesian method. For example, Gaia EDR3 1323951092358466304 has a 15% and 7% uncertainty on T_eff and log g, respectively. Although the parameters estimated for this white dwarf are within the uncertainties between the fast-test and the Bayesian method, there is a significant difference between, for example, the final masses (0.64^+0.16 _−0.06 M_⊙ using the Bayesian method and ${0.97}_{-0.26}^{+0.23}\,{\text{}}{M}_{\odot }$ using the fast-test method). In this case where the likelihood is not well constrained due to the uncertainties, the prior in the Bayesian method (in particular the IMF) is adding extra information preferring lower-mass white dwarfs, while the fast-test method selects the highest likelihood value. This was discussed in detailed in Section 2.4 and will be discussed further in Section 4.