# An analytical formulation for $\phi^{4}$ field-potential dynamics

Arash Ghahraman
<EMAIL_ADDRESS>
Kurosh Javidan
<EMAIL_ADDRESS>
Department of physics, Ferdowsi university of Mashhad
91775-1436 Mashhad Iran

###### Abstract

An analytical model for adding a space dependent potential to the $\phi^{4}$
field is presented, by constructing a collective coordinate for solitary
solution of this model. Interaction of the $\phi^{4}$ field with a delta
function potential barrier and also delta function potential well is
investigated. Most of the characters of the interaction are derived
analytically while they are calculated by other models numerically. We will
find that the behaviour of the solitary solution is like a point particle
’living’ under the influence of a complicated potential which is a function of
the field initial conditions and the potential parameters.

##  1 Introduction

Dynamical evolution of a field in the presence of an external potential, in
which case the parameters of the model are functions of space is an important
phenomenon from the mathematical point of view and also because of its
applications. An external potential can be added to the equation of motion as
perturbative terms [1, 2]. These effects also can be taken into account by
making some parameters of the equation of motion to be function of space or
time [3, 4]. The external potential also can be added to the field through the
metric of background space-time [5, 6]. This method is mainly suitable for
nonlinear field theories contain solitonic solutions.

As is well known, when waves scatter on a potential, they can be partly
reflected and partly transmitted. For the fields with solitonic solutions, the
situation is more complicated as solitons cannot split and thus must either
bounce, pass through or become trapped inside the potential. This behaviour is
very sensitive to the value of all the parameters of the model as well as to
the initial conditions for the scattering. Most of the researches are in base
of numerical studies because such these systems are generally non-integrable.
So it is clear that we need suitable models with analytic solutions to test
the validity of such phenomenon and predict their behaviour.

In this paper an analytical model for the interaction of $\phi^{4}$ field with
an external potential is presented. This method can be used for other field
theories and from this viewpointin, we will present an example for explaining
the method. However the results are perfectly valid for the $\phi^{4}$ field
theory. So a model for the $\phi^{4}$ field in an space dependent potential is
presented in section 2. The analytic model is introduced and will be solved in
section 3. The results for the field-potential barrier system are presented in
section 4. In section 5 field-potential well system is discussed. Some
conclusion and remarks will be presented in section 6.

##  2 $\phi^{4}$ field and a space dependent potential

The general form of the action in an arbitrary metric is:

$I=\int{{\cal L}(\phi,\partial_{\mu}\phi)\sqrt{-g}d^{n}xdt}$ (1)

where ”g” is the determinant of the metric $g^{\mu\nu}(x)$. Energy density of
the ”field + potential” can be found by varying ”both” the field and the
metric [7]. For the lagrangian of the form

${\cal L}=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-U(\phi)$ (2)

the equation of motion becomes [7, 8]

$\frac{1}{\sqrt{-g}}\left(\sqrt{-g}\partial_{\mu}\phi\partial^{\mu}\phi+\partial_{\mu}\phi\partial^{\mu}\sqrt{-g}\right)+\frac{\partial
U(\phi)}{\partial\phi}=0$ (3)

One can add a space dependent potential to the lagrangian of the system by
introducing a suitable nontrivial metric for the back ground space-time,
without missing the topological boundary conditions [6, 7]. In other words,
the metric carries the introduced potential. The suitable metric in the
presence of a weak potential $v(x)$ is [5, 6, 7]:

$g_{\mu\nu}(x)\cong\left(\begin{array}[]{clrr}1+V(x)&0\\\
0&-1\end{array}\right)$ (4)

The equation of motion (3) (describes by Lagrangian (2)) in the background
space-time (4) is

$\left(1+V(x)\right)\frac{\partial^{2}\phi}{\partial
t^{2}}-\frac{\partial^{2}\phi}{\partial
x^{2}}-\frac{1}{2\left|1+V(x)\right|}\frac{\partial V(x)}{\partial
x}\frac{\partial\phi}{\partial x}+\frac{\partial U(\phi)}{\partial\phi}=0$ (5)

For the $\phi^{4}$ model, we have $U(\phi)=\left(\phi^{2}-1\right)^{2}$. The
$\phi^{4}$ model has solitary wave solution

$\phi(x,t)=tanh\left(\sqrt{2}\frac{x-X(t)}{\sqrt{1-v^{2}}}\right)$ (6)

where $X(t)=x_{0}-vt$ and $v$ is the velocity of the solitary wave. By
inserting the solution (6) in the lagrangian (2) and using the metric (4),
with adiabatic approximation [1, 2] we have

${\cal
L}=\sqrt{1+v(x)}\left(\left(1+v(x)\right)\dot{X}^{2}-2\right)sech^{4}\left(\sqrt{2}\left(x-X(t)\right)\right)$
(7)

For the weak potential $v(x)$ (7) becomes

${\cal
L}\approx\left(\left(1+\frac{3}{2}v(x)\right)\dot{X}^{2}-2\left(1+\frac{1}{2}v(x)\right)\right)sech^{4}\left(\sqrt{2}\left(x-X(t)\right)\right)$
(8)

##  3 Collective coordinate variable

The center of a soliton can be considered as a particle, if we look at this
variable as a collective coordinate. The collective coordinate could be
related to the potential by using the above model. This model is able to give
us an analytic solution for most of the features of the soliton-potential
system. For the lagrangian (8) X(t) remains as a collective coordinate if we
integrate (8) over variable x

$L=\int{{\cal
L}dx}=\frac{2\sqrt{2}}{3}\dot{X}^{2}-\frac{4\sqrt{2}}{3}+\left(\frac{3}{2}\dot{X}^{2}-1\right)\int{sech^{4}\left(\sqrt{2}\left(x-X(t)\right)\right)v(x)dx}$
(9)

If we take the potential $v(x)=\epsilon\delta(x)$ then (9) becomes

$L=\frac{2\sqrt{2}}{3}\dot{X}^{2}-\frac{4\sqrt{2}}{3}+\left(\frac{3}{2}\dot{X}^{2}-1\right)\epsilon
sech^{4}\left(\sqrt{2}X(t)\right)$ (10)

The equation of motion for the variable X(t) is derived from (10)

$\left(\frac{2\sqrt{2}}{3}+\frac{3\epsilon}{2}sech^{4}\left(\sqrt{2}X\right)\right)\ddot{X}-\sqrt{2}\epsilon
tanh(\sqrt{2}X)sech^{4}\left(\sqrt{2}X\right)\left(3\dot{X}^{2}+2\right)=0$
(11)

The above equation shows that the peak of the soliton moves under the
influence of a complicated force which is a function of its position and its
velocity. If $\epsilon>0$ we have a barrier and $\epsilon<0$ creates a
potential well. Fortunately equation (11) has an exact solution as follows

$\frac{3\dot{X}^{2}+2}{3\dot{X}_{0}^{2}+2}=\frac{\frac{2\sqrt{2}}{3}+\frac{3\epsilon}{2}sech^{4}\left(\sqrt{2}\left(X_{0}\right)\right)}{\frac{2\sqrt{2}}{3}+\frac{3\epsilon}{2}sech^{4}\left(\sqrt{2}\left(X\right)\right)}$
(12)

where $X_{0}$ and $\dot{X}_{0}$ are initial position and initial velocity
respectively. The energy of the soliton in the presence of the potential
$v(x)=\epsilon\delta(x)$ becomes

$E=\left(\frac{2\sqrt{2}}{3}+\frac{3\epsilon}{2}sech^{4}\left(\sqrt{2}X\right)\right)\dot{X}^{2}+\frac{4\sqrt{2}}{3}+\epsilon
sech^{4}\left(\sqrt{2}X\right)$ (13)

Subtituting $\dot{X}$ in (13) from (12) shows that the energy is a function of
initial conditions $X_{0}$ and $\dot{X}_{0}$ only. Therefore the energy of the
sytem is conserved. Some features of the soliton-potential dynamics can be
investigated using equations (12) and (13) analytically which are discussed in
the following.

##  4 Potential barrier

Suppose that a potential barrier of the height $\epsilon$ is located at the
origin. When the soliton is far from the center of the potential
($X\rightarrow\infty$) (13) reduces to
$E=\frac{2\sqrt{2}}{3}\dot{X}_{0}^{2}+\frac{4\sqrt{2}}{3}$. It is the energy
of a particle with a mass of $\frac{4\sqrt{2}}{3}$ and a velocity of
$\dot{X}_{0}$. A soliton with a low velocity reflects back from the barrier
and a high energy soliton climbs over the barrier and passes over it. So we
have a critical value for the velocity of the soliton which separates these
two situations. The energy of a soliton in the origin (X=0) comes from
(13)$E(X=0)=\left(\frac{2\sqrt{2}}{3}+\frac{3\epsilon}{2}\right)\dot{X}_{0}^{2}+\frac{4\sqrt{2}}{3}+\epsilon$.
The minimum energy for a soliton in this position is
$E=\frac{4\sqrt{2}}{3}+\epsilon$. On the other hand, a soliton which comes
from the infinity with initial velocity $v_{c}$ has the energy of
$E\left(X=\infty\right)=\frac{2\sqrt{2}}{3}v_{c}^{2}+\frac{4\sqrt{2}}{3}$.
Therefore it can pass through the barrier if
$v_{c}>\sqrt{\frac{3\sqrt{2}}{4}\epsilon}$. The same result is derived by
substituting $\dot{X}=0$, $\dot{X}_{0}=v_{c}$, $X_{0}=\infty$ and $X=0$ in
(12).

If the soliton is located at some position like $X_{0}$ (which is not
necessary infinity) the critical velocity will not be
$\sqrt{\frac{3\sqrt{2}}{4}\epsilon}$. The soliton can pass over the barrier if
the soliton energy is greater than the energy of a static soliton at the top
of the barrier. So a soliton in the initial position $X_{0}$ with initial
velocity of $\dot{X}_{0}$ has the critical initial velocity if its velocity
becomes zero at the top of the barrier $X=0$. Consider a soliton with initial
conditions of $X_{0}$ and $\dot{X}_{0}$. If we set $X=0$ and $\dot{X}=0$ in
equation (12) then $v_{c}=\dot{X}_{0}$. Therefore we have

$v_{c}=\sqrt{\frac{\epsilon\left(1-sech^{4}\left(\sqrt{2}X_{0}\right)\right)}{\frac{2\sqrt{2}}{3}+\frac{3\epsilon}{2}sech^{4}\left(\sqrt{2}x_{0}\right)}}$
(14)

When the initial velocity is less than the $v_{c}$ then there exists a return
point in which the velocity of the soliton is zero. For this situation we have

$\left(\frac{3}{2}\dot{X}_{0}^{2}+1\right)\left(\frac{2\sqrt{2}}{3}+\frac{3\epsilon}{2}sech^{4}\left(\sqrt{2}X_{0}\right)\right)-\frac{2\sqrt{2}}{3}=\frac{3\epsilon}{2}sech^{4}\left(\sqrt{2}X_{stop}\right)$
(15)

Therefore this model predicts that $sech^{4}\left(\sqrt{2}X_{stop}\right)$ is
proportional to $sech^{4}\left(\sqrt{2}X_{0}\right)$. Also (15) shows that
$sech^{4}\left(\sqrt{2}X_{stop}\right)$ is proportional to
$\frac{1}{\epsilon}$.

A soliton with initial conditions $X_{0}$ and $\dot{X}_{0}$ will go to the
infinity after the interaction with a potential barrier. The final velocity of
the soliton after the interaction is

$\dot{X}=\sqrt{\dot{X}_{0}^{2}+\frac{3\epsilon}{2\sqrt{2}}\left(\frac{3\dot{X}_{0}^{2}}{2}+1\right)sech^{4}\left(\sqrt{2}X_{0}\right)}$
(16)

which is greater than the initial velocity $\dot{X}_{0}$.

Equations (12) and (13) show that the soliton finds its initial velocity after
the interaction when it reaches its initial position. This means that the
interaction is completely elastic. Numerical simulation are in agreement with
above analytical results [7, 9, 10].

##  5 Soliton-well system

The soliton-well system is very interesting problem. Suppose a particle moves
toward a frictionless potential well. It falls in the well with increasing
velocity and reaches the bottom of the well with its maximum speed. After
that, it will climb the well with decreasing velocity and finally pass through
the well. Its final velocity after the interaction is equal to its initial
speed.

Changing $\epsilon$ to $-\epsilon$ in (12) changes potential barrier to
potential well. The solution for the system is

$\frac{3\dot{X}^{2}+2}{3\dot{X}_{0}^{2}+2}=\frac{\frac{2\sqrt{2}}{3}-\frac{3\epsilon}{2}sech^{4}\left(\sqrt{2}\left(X_{0}\right)\right)}{\frac{2\sqrt{2}}{3}-\frac{3\epsilon}{2}sech^{4}\left(\sqrt{2}\left(X\right)\right)}$
(17)

There is not a critical velocity for a soliton-well system, but an escape
velocity can be defined. A soliton with initial position $X_{0}$ reaches the
infinity with a zero final velocity if its initial velocity is

$\dot{X}_{escape}=\sqrt{\frac{\epsilon
sech^{4}\left(\sqrt{2}X_{0}\right)}{\frac{2\sqrt{2}}{3}-\frac{3\epsilon}{2}sech^{4}\left(\sqrt{2}X_{0}\right)}}$
(18)

In other words, a soliton which is located in the initial position $X_{0}$ can
escape to infinity if its initial velocity $\dot{X}_{0}$ is greater than the
escape velocity $\dot{X}_{escape}$.

Consider a potential well with the depth of $\epsilon$ and a soliton at the
initial position $X_{0}$ which moves toward the well with the initial velocity
$\dot{X_{0}}$ smaller than the $\dot{X}_{escape}$. The soliton interacts with
the potential and reaches a maximum distance $X_{max}$ from the center
potential with a zero velocity and then come back toward the well. The soliton
oscillates around the well with the amplitude $X_{max}$. The required initial
velocity to reach $X_{max}$ is found from (17) as

$\dot{X}_{0}=\sqrt{\frac{\epsilon\left(sech^{4}\left(\sqrt{2}X_{max}\right)-sech^{4}\left(\sqrt{2}X_{0}\right)\right)}{\frac{2\sqrt{2}}{3}-\frac{3\epsilon}{2}sech^{4}\left(\sqrt{2}X_{max}\right)}}$
(19)

If the initial velocity is lower than the escape velocity the soliton
oscillates around the well. The period of oscillation can be calculated
numerically using equation (17).

##  6 Conclusion and Remarks

A model for the $\phi^{4}$ field-potential interaction has been presented. The
model predicts a critical velocity for the soliton-barrier interaction as a
function of initial conditions of the field and the potential characters. The
model predicts specific relations between some functions of initial conditions
and other functions of final state of the field during the interaction. An
escape velocity has been derived for the field-well system. This model is able
to explain most of the features of the system analytically.

This model can be used for prediction the results of other field theories
beside the $\phi^{4}$ model.

## References

  * [1] Kivshar Y S, Fei Z and Vasquez L 1991 Phys. Rev. Lett 67 1177
  * [2] Fei Z, Kivshar Y S and Vasquez L 1992 Phys. Rev.A46 5214
  * [3] Piette B, Zakrzewski W J and Brand J 2005 J. Phys. A: Math. Gen.38 10403-10412
  * [4] Piette B. Zakrzewski W.J.2007 J. Phys. A: Math. Theor.40 No 2, 329-346
  * [5] Kalbermann G 1999 Phys. Lett A252 37-42
  * [6] Javidan K and Sarbishaei M 2001 Indian J. Physics B75 (5) 413-418
  * [7] Javidan K. 2006 J. Phys. A: Math. Gen. 39 No 33 10565-10574
  * [8] Felsager B. (1998) Geomrtry, Particles, and fields (New York: Springer-Verlag) pp 434-437
  * [9] Al-Alawi Jassem H. , Zakrzewski W.J. (2007) J. Phys. A 40, 11319-1131
  * [10] Javidan K arXiv:0807.1618


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

Rocio Kiman Kavli Institute for Theoretical Physics, University of California,
Santa Barbara, CA 93106, USA  Department of Physics, Graduate Center, City
University of New York, 365 5th Ave, New York, NY 10016, USA  Department of
Astrophysics, American Museum of Natural History, Central Park West at 79th
St, New York, NY 10024, USA  Hunter College, City University of New York, 695
Park Ave, New York, NY 10065, USA Rocio Kiman<EMAIL_ADDRESS>Siyi Xu
(许bsmi偲gbsn艺) Gemini Observatory/NSF’s NOIRLab, 670 N. Aóhoku Place, Hilo, HI
96720, USA  Jacqueline K. Faherty Department of Astrophysics, American Museum
of Natural History, Central Park West at 79th St, New York, NY 10024, USA
Jonathan Gagné Planétarium Rio Tinto Alcan, Espace pour la Vie, 4801 av.
Pierre-de Coubertin, Montréal, Québec, Canada  Institute for Research on
Exoplanets, Université de Montréal, Département de Physique, C.P. 6128 Succ.
Centre-ville, Montréal, QC H3C 3J7, Canada  Ruth Angus Department of
Astrophysics, American Museum of Natural History, Central Park West at 79th
St, New York, NY 10024, USA  Center for Computational Astrophysics, Flatiron
Institute, 162 5th Avenue, New York, NY 10010 USA  Department of Astronomy,
Columbia University, 116th St & Broadway, New York, NY 10027, USA  Timothy D.
Brandt Department of Physics, University of California, Santa Barbara, Santa
Barbara, CA 93106, USA  Sarah L. Casewell School of Physics and Astronomy,
University of Leicester, University Road, Leicester, LE1 7RH, UK  Kelle L.
Cruz Department of Physics, Graduate Center, City University of New York, 365
5th Ave, New York, NY 10016, USA  Department of Astrophysics, American Museum
of Natural History, Central Park West at 79th St, New York, NY 10024, USA
Hunter College, City University of New York, 695 Park Ave, New York, NY 10065,
USA  Center for Computational Astrophysics, Flatiron Institute, 162 5th
Avenue, New York, NY 10010 USA

(Received April 21 2022; Revised June 6 2022; Accepted June 8 2022)

###### Abstract

White dwarfs have been successfully used as cosmochronometers in the
literature, however their reach has been limited in comparison to their
potential. We present wdwarfdate, a publicly available Python package to
derive the Bayesian age of a white dwarf, based on its effective temperature
($T_{\rm eff}$) and surface gravity ($\log g$). We make this software easy to
use with the goal of transforming the usage of white dwarfs as
cosmochronometers into an accessible tool. The code estimates the mass and
cooling age of the white dwarf, as well as the mass and main-sequence age of
the progenitor star, allowing for a determination of the total age of the
object. We test the reliability of the method by estimating the parameters of
white dwarfs from previous studies, and find agreement with the literature
within measurement errors. By analysing the limitation of the code we find a
typical uncertainty of $10\%$ on the total age when both input parameters have
uncertainties of $1\%$, and an uncertainty of $25\%$ on the total age when
$T_{\rm eff}$ has an uncertainty of $10\%$ and $\log g$ of $1\%$. Furthermore,
wdwarfdate assumes single star evolution, and can be applied to calculate the
total age of a white dwarf with parameters in the range $1,500\lesssim T_{\rm
eff}\lesssim 90,000$ K and $7.9\lesssim\log g\lesssim 9.3$. Finally, the code
assumes a uniform mixture of C/O in the core and single star evolution, which
is reliable in the range of white dwarf masses $0.45-1.1\,\mbox{${\rm
M}_{\sun}$}$ ($7.73\lesssim\log g\lesssim 8.8$).

White dwarf stars (1799), Fundamental parameters of stars (555), Stellar ages
(1581), Bayesian statistics (1900), Open source software (1866)

††journal: The Astronomical Journal††software: scipy (Virtanen et al., 2020);
numpy (Oliphant, 2006; Van Der Walt et al., 2011); matplotlib (Hunter, 2007);
astropy (Astropy Collaboration et al., 2013, 2018)

##  1 Introduction

White dwarfs are the perfect candidates to use as cosmochronometers. These
degenerate objects are formed with high temperatures from the death of stars
with masses $<8\,\mbox{${\rm M}_{\sun}$}$, and cool over time. The cooling
process is understood to the point that by deriving the effective temperature
($T_{\rm eff}$ [K]) and surface gravity ($\log g$ [cm ${\rm s^{-2}}$], units
for $\log g$ will be implicit from now on to simplify notation) from
observational data we can estimate how long it has been cooling with a
relatively high accuracy (e.g., Fontaine et al., 2001; Bédard et al., 2020).
These cooling models, combined with stellar evolutionary models (e.g., Choi et
al., 2016; Dotter, 2016) and the initial-to-final mass relation (IFMR, e.g.,
Cummings et al., 2018; El-Badry et al., 2018; Marigo et al., 2020), provide a
reliable method to estimate the total age of a white dwarf. In addition, with
the release of Gaia DR2 and EDR3 (Gaia Collaboration et al., 2016, 2018; Brown
et al., 2021), a large number of new white dwarfs have been discovered –about
$10$ times more than previously known– that have precise proper motions and
parallaxes measured by Gaia (e.g., Gentile Fusillo et al., 2019, 2021;
McCleery et al., 2020).

The Gaia mission also allowed for the identification of binaries with
unprecedented precision (e.g., El-Badry & Rix, 2018; El-Badry et al., 2021).
Given that it can be assumed that the components of a binary system were born
at approximately the same time, identifying white dwarfs with wide co-moving
companions allows us to calculate the age of the system under the assumption
that each object has likely evolved as a single star, without phases of mass
transfer or common envelope evolution (Bodenheimer, 2011). Stellar age is one
of the most difficult fundamental properties to measure directly; although
some methods such as gyrochronology (e.g., Skumanich, 1972; Barnes, 2003; Van
Saders et al., 2016), asteroseismology (e.g., Chaplin et al., 2014; Aguirre et
al., 2017) and isochrone fitting (e.g., Baraffe et al., 2015; Agüeros et al.,
2018; Berger et al., 2020) can yield relatively precise ages for single stars,
they are limited to specific ranges of masses, or stages of main-sequence
evolution, in which they can be applied (e.g., Chabrier & Baraffe, 1997;
Baraffe et al., 2015; Rodríguez et al., 2016). For example, none of these
methods can be applied to low-mass stars ($<0.6\,\mbox{${\rm M}_{\sun}$}$).
Binary stars provide a powerful tool to estimate stellar ages “independently
of mass”: the age of a star of any mass can be estimated if it is co-moving
with another object which can be age-dated.

White dwarf cosmochronology is the subject of extensive literature. Some
examples include Kalirai (2012), who calculated the ages of white dwarfs in
the Milky Way halo to study its formation; Gagné et al. (2018), who estimated
the age of the ultra-massive white dwarf GD 50 to refine the age of the AB Dor
moving group which GD 50 belongs to; Anguiano et al. (2017), who estimated
white dwarf parameters, including their ages, to study the formation of the
Galactic disk; Catalán (2015) who used white dwarf-white dwarf binaries to
calibrate the low-mass end of the IFMR; and Kilic et al. (2019), who used
white dwarf ages to study the Milky way inner halo. In addition to these,
other studies (e.g., Garcés et al., 2011 and Fouesneau et al., 2019) have
estimated the age of main-sequence stars by calculating the age of co-moving
white dwarf companions. Furthermore, previous studies have presented software
libraries that can be used to estimate white dwarf ages: WD_models111Available
at https://github.com/SihaoCheng/WD_models (S. Cheng, priv. comm.) is a Python
library to estimate white dwarf parameters from Gaia photometry, although it
does not provide measurement uncertainties for these parameters; and
BASE-9222Available at https://base-9.readthedocs.io/en/latest/ (von Hippel et
al., 2006, 2014) is a C++  library that estimates star cluster and stellar
parameters from photometry, and in particular can be used to estimate white
dwarf and progenitor masses and ages. BASE-9 uses a Bayesian technique,
combined with a Markov chain Monte Carlo and numerical integration techniques
to estimate the posterior probability distributions of the different
parameters, including age. As a consequence, BASE-9 results complicated to
implement, in particular when only the parameters of the white dwarf are
needed. This shows the incredible potential of estimating total ages of white
dwarfs, and the lack of tools to make it accessible.

In this paper, we present wdwarfdate, a new Python library to estimate the
total age of a white dwarf with the associated uncertainty, from $T_{\rm eff}$
(K), $\log g$, and atmospheric composition. The goal of this software is to
make the usage of white dwarf as cosmochronometers accessible. Therefore, this
code is designed to be easily used, with warnings for the corresponding
limitations of the theoretical models applied in the process. The estimation
of the white dwarf age is done using Bayesian statistics under the assumptions
of a white dwarf core with uniform mixture of C/O and single star evolution.
wdwarfdate also calculates the mass and cooling age of the white dwarf and the
mass and main-sequence age of its progenitor star, along with their respective
uncertainties (and their full probability densities) which include scatter in
the IFMR and propagate the measurement errors of the white dwarf parameters
used as an input.

This paper is divided into the following sections: in Section 2, we describe
the general method used to estimate the total age of a white dwarf (Section
2.1), the models that were included for the age determination (i.e., cooling
tracks, IFMRs and stellar evolutionary models; Section 2.2), the Bayesian
framework on which wdwarfdate relies (Section 2.3), and the fast-test mode, an
alternative method to estimate the parameters of the white dwarf included in
the code (Section 2.4). In Section 3, we apply wdwarfdate to calculate the
total ages of a reference sample of age-calibrated white dwarfs to validate
our method. We also estimate the ages of $18$ white dwarfs in wide co-moving
pair candidates with M dwarfs (Section 3.3) to provide a sample of age-
calibrated M dwarfs for future studies. In Section 4, we detail the
constraints and assumptions that are required in using wdwarfdate. Our results
are summarized in Section 5.

##  2 Method

In this section, we describe the sequential steps required to calculate the
mass and cooling age of the white dwarf, as well as the mass and the main-
sequence age of the progenitor star, and finally the total age of the object.
wdwarfdate will derive ages using Bayesian statistics –Bayesian age– by
default, but the code also includes a fast-test mode, an alternative method to
estimate the parameters of a white dwarf, and we explain both in this section.
We also describe the models in which wdwarfdate relies for the parameter
estimation. For an example on how to use wdwarfdate see Appendix A.

###  2.1 Outline of the algorithm

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

  1. 1.

From the $T_{\rm eff}$ and $\log g$ input values combined with cooling
evolutionary tracks for the white dwarf we obtain its mass (final mass,
$m_{\rm f}$) and cooling age ($t_{\rm cool}$).

  2. 2.

From the $m_{\rm f}$ we estimate the mass of the progenitor star (initial
mass, $m_{\rm i}$) using an IFMR, which are, in most cases, empirically
calibrated.

  3. 3.

With the $m_{\rm i}$ 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
($t_{\rm ms}$).

  4. 4.

Finally, adding $t_{\rm cool}$ and $t_{\rm ms}$, we obtain the total age of
the object ($t_{\rm tot}$).

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.

Figure 1: Steps required to estimate a white dwarf total age from an effective
temperature and surface gravity ($T_{\rm eff}$ and $\log g$). In this diagram,
$m_{\rm f}$ and $m_{\rm i}$ are the final mass (mass of the white dwarf) and
initial mass (mass of the progenitor star), and $t_{\rm cool}$, $t_{\rm ms}$
and $t_{\rm tot}$ are the cooling age of the white dwarf, the main sequence
age of the progenitor star and the total age of the object, respectively.
$\sigma_{\rm x}$ indicates the uncertainty of each parameter x.

###  2.2 Models incorporated in wdwarfdate

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 DA white dwarf
and non-DA, 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 $T_{\rm eff}$ 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 requires 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 1-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 $T_{\rm eff}\lesssim 6500$ 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 ($X_{\rm
C}=X_{\rm O}=0.5$), surrounded by a He layer (with a fractional mass $q_{\rm
He}=M_{\rm He}/M_{*}=10^{-2}$) surrounded by an outermost H layer. The
evolutionary tracks assume that this outermost H layer is “thick” ($q_{\rm
H}\equiv M_{\rm H}/M_{*}=10^{-4}$) for hydrogen-atmosphere white dwarfs (DA
white dwarf), and “thin” ($q_{\rm H}=10^{-10}$) for helium-atmosphere white
dwarfs (non-DA white dwarf; Bédard et al., 2020). The publicly available
Montreal cooling tracks were calculated for masses in the range
$0.2-1.3\,\mbox{${\rm M}_{\sun}$}$ in steps of $0.05\,\mbox{${\rm
M}_{\sun}$}$. These tracks are shown in Figure 2 in black, together with the
limits within which wdwarfdate was designed to function in terms of $T_{\rm
eff}$ 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 ($0.2-1.3\,\mbox{${\rm M}_{\sun}$}$); 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: $1,500\leqslant T_{\rm eff}\lesssim 100,000$ K and
$7.0\leqslant\log g\lesssim 9.3$. Section 4 details additional limitations of
wdwarfdate.

Figure 2: White dwarf cooling tracks from Bédard et al. (2020) for thick
($q_{\rm H}\equiv M_{\rm H}/M_{*}=10^{-4}$) and thin ($q_{\rm H}=10^{-10}$) H
layers (left and right panels, respectively). The cooling tracks cover white
dwarf masses in the range $0.2-1.3\,\mbox{${\rm M}_{\sun}$}$ in steps of
$0.05\,\mbox{${\rm M}_{\sun}$}$. The limits of region in the ($\log g$,
$T_{\rm eff}$) plane where wdwarfdate relies on cooling tracks are shown with
blue dashed lines. We also included isochrones between $0.3$ Myr$-3$ Gyr in
gray for comparison and show where the density of isochrones increases.

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 $T_{\rm eff}$ 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 $T_{\rm eff}\lesssim 30,000$ 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): the MESA 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 non-rotating MIST isochrones estimating
more precise ages for young stars ($\sim 100$ Myr), as tested with the
Pleiades cluster. However, the PARSEC-based IFMR should be considered
especially for high-mass progenitor stars ($>5\,\mbox{${\rm M}_{\sun}$}$),
given that the MIST models tend to underestimate masses in this range.

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
$0.5\,\mbox{${\rm M}_{\sun}$}$) 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 $T_{\rm eff}$, $\log g$ and $m_{\rm
f}$, 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 colours, 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 $T_{\rm eff}$ 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 $0.5-1.3\,\mbox{${\rm
M}_{\sun}$}$, 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 $m_{\rm i}>0.83\,\mbox{${\rm M}_{\sun}$}$, and we included them in the
code extending this limit to $0.45\,\mbox{${\rm M}_{\sun}$}$, assuming a
constant $m_{\rm f}$ between $0.45<m_{\rm i}<0.83$ (gray line in Figure 3).
This extension allows wdwarfdate to estimate parameters for white dwarfs with
masses $m_{\rm f}\lesssim 0.6\,\mbox{${\rm M}_{\sun}$}$, which is the peak of
the white dwarf mass distribution (e.g., Genest-Beaulieu & Bergeron, 2019).

Figure 3: Comparison of the initial-to-final mass relations included in
wdwarfdate: Cummings et al. (2018) MIST-based (solid blue line) and PARSEC-
based (dashed-dotted light blue line), Marigo et al. (2020, dotted orange
line), Salaris et al. (2009, solid thin red line) and Williams et al. (2009,
dashed green line). We include the data from Cummings et al. (2018) as gray
points with uncertainties for comparison. As a gray line we show the short
extension we included in wdwarfdate of the IFMR from Cummings et al. (2018)
and Marigo et al. (2020) at the low-mass end.

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 $0.1-300\,\mbox{${\rm M}_{\sun}$}$ with a step of $0.05\,\mbox{${\rm
M}_{\sun}$}$, 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:
${\rm[Fe/H]}=\\{-4,-1,0,0.5\\}$ (dex), $[{\rm\alpha/Fe}]=0$, and ${\rm
v/vcrit}=\\{0.0,0.4\\}$. We will discuss the effects of using different tracks
on the total age in Section 4.

Table 1: Models included in wdwarfdate.  Models | Model included
---|---
Cooling tracks |  Bédard et al. (2020)aaAvailable online http://www.astro.umontreal.ca/~bergeron/CoolingModels/:
| DA and non-DA white dwarfs
IFMR | Marigo et al. (2020)
|  Cummings et al. (2018) MIST
|  Cummings et al. (2018) PARSEC
| Salaris et al. (2009)
| Williams et al. (2009)
Stellar evolution tracks | MIST isochronesbbAvailable online http://waps.cfa.harvard.edu/MIST/
|  (Choi et al., 2016; Dotter, 2016):
| $[{\rm Fe/H}]=\\{-4,-1,0,0.5\\}$
| $[{\rm\alpha/Fe}]=0$
| ${\rm v/vcrit}=\\{0.0,0.4\\}$

###  2.3 Bayesian framework

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

$p(m_{\rm i},\log_{10}t_{\rm cool},\Delta_{\rm m}|T_{\rm eff},\log g)$ (1)

where $m_{\rm i}$ is the initial mass, or the mass of the progenitor star, and
$t_{\rm cool}$ is the cooling age of the white dwarf. $\Delta_{\rm m}$ is an
extra parameter which takes into account the scatter in the IFMR of white
dwarfs, and it is described below. $T_{\rm eff}$ 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 Bayes theorem, the posterior probability in
Equation 1 can be written as

$\begin{split}p(m_{\rm i},\log_{10}t_{\rm cool},&\Delta_{\rm m}|T_{\rm
eff},\log g)\propto\\\ &p(T_{\rm eff},\log g|m_{\rm i},\log_{10}t_{\rm
cool},\Delta_{\rm m})\\\ &p(\log_{10}t_{\rm cool})p(m_{\rm i})p(\Delta_{\rm
m}),\end{split}$ (2)

where $p(\log_{10}t_{\rm cool})$, $p(m_{\rm i})$ and $p(\Delta_{\rm m})$ are
the priors on the cooling age, the initial mass and $\Delta_{\rm m}$
respectively, which are described below, and

$\begin{split}p(&T_{\rm eff},\log g|m_{\rm i},\log_{10}t_{\rm
cool},\Delta_{\rm m})\propto\\\ &\exp\left[-\frac{(\overline{T_{\rm
eff}}-T_{\rm eff})^{2}}{2\sigma_{T_{\rm
eff}}^{2}}\right]\exp\left[-\frac{(\overline{\log g}-\log
g)^{2}}{2\sigma_{\log g}^{2}}\right]\end{split}$ (3)

is the likelihood. $\overline{T_{\rm eff}}$ and $\overline{\log g}$ 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 ($m_{\rm i}$, $t_{\rm cool}$ and $\Delta_{\rm m}$) 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
($m_{\rm i}$) we estimate a main sequence age ($t_{\rm ms}$); 2) by adding
$t_{\rm ms}$ to the cooling age ($t_{\rm cool}$) we obtain the total age
($t_{\rm tot}$) of the white dwarf; 3) from $m_{\rm i}$ and the delta
parameter ($\Delta_{\rm m}$) we estimate a final mass ($m_{\rm f}$, described
below); 4) from $m_{\rm f}$ and $t_{\rm cool}$ we estimate the modeled
effective temperature and surface gravity ($\overline{T_{\rm eff}}$ and
$\overline{\log g}$) which are compared to the input parameters ($T_{\rm eff}$
and $\log g$) to estimate the posterior probability. The models used in each
step are described in Section 2.2.

The $\Delta_{\rm m}$ parameter in Equations 1, 2 and 3 was included to model
the scatter in the IFMR. This scatter has different causes including
contamination from non-cluster 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). The parameter $\Delta_{\rm m}$ is drawn from a normal distribution
centered at zero with standard deviation $\sigma_{\rm m}$, so that

$p(\Delta_{\rm m})=\mathcal{N}(0,\sigma_{\rm m}).$ (4)

We estimated the parameter $\sigma_{\rm m}$ from the Cummings et al. (2018)
data, and adopted $\sigma_{\rm m}=0.03\,\mbox{${\rm M}_{\sun}$}$. The
inclusion of the scatter in the posterior is indicated in Figure 4 as
$\hat{m}_{\rm f}=m_{\rm f}+\Delta_{\rm m}$, where $m_{\rm f}$ 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 $t_{\rm cool}$ and the Initial Mass
Function (IMF) for $m_{\rm i}$. We assume a constant SFH (e.g., Madau &
Dickinson, 2014) and applied it as a flat prior on $t_{\rm tot}$, where we
required $t_{\rm tot}<15$ Gyr to avoid a sharp cut at the age of the Universe
in the age probability distributions. As $t_{\rm tot}=t_{\rm cool}+t_{\rm
ms}$, 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 $m_{\rm i}^{-2.3}$ (Kroupa, 2001; Chabrier, 2003), which describes the
range of stellar masses expected to form a white dwarf.

Figure 4: Probabilistic Graphical Model for the Bayesian method of wdwarfdate.
In short, $T_{\rm eff}$ and $\log g$ are calculated from the parameters being
sampled ($m_{\rm i}$, $t_{\rm cool}$ and $\Delta_{\rm m}$), and these values
are compared to the input parameters. In the figure, $\hat{m_{\rm f}}$ is the
scattered final mass (mass of the white dwarf) calculated from the initial
mass using an IFMR, and $\Delta_{\rm m}$ is an extra parameter to take into
account the scatter in this relation. Therefore, $m_{\rm f}$ is the true final
mass, calculated by adding $\Delta_{\rm m}$ to $\hat{m_{\rm f}}$.

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: $m_{\rm i}$, $\log_{10}t_{\rm cool}$ and $\Delta_{\rm m}$. The code
first performs a wide grid evaluation with the parameters ranging between
$0.3\,{\rm Myr}<t_{\rm cool}<15\,{\rm Gyr}$, $0.1\,\mbox{${\rm
M}_{\sun}$}<m_{\rm i}<10\,\mbox{${\rm M}_{\sun}$}$ and $-0.1\,\mbox{${\rm
M}_{\sun}$}<\Delta_{\rm m}<0.1\,\mbox{${\rm M}_{\sun}$}$. 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).

###  2.4 Fast-test method

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 $T_{\rm eff}$ 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 ($T_{\rm eff}$ and $\log g$) as means, and the uncertainties
($\sigma_{T_{\rm eff}}$ and $\sigma_{\log g}$) as the standard deviations of
the distributions such as

$\begin{split}X_{T_{\rm eff}}&\sim\mathcal{N}(T_{\rm eff},\sigma_{T_{\rm
eff}})\\\ X_{\log g}&\sim\mathcal{N}(\log g,\sigma_{\log g}).\end{split}$ (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
$T_{\rm eff}=1,500-100,000$ 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 ${\rm[Fe/H]}=0$ and ${\rm v/vcrit}=0.0$, and the
uncertainties were calculated using the $16^{\rm th}$ and $84^{\rm th}$
percentile of the distributions. The results show that the final mass
estimations agree within the uncertainties between both methods (right panel
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 $1\sigma$, 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
which 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 $(T_{\rm eff}-\log g)$ 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.

The final masses between $0.65-0.8\,\mbox{${\rm M}_{\sun}$}$ 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 $0.65-0.8\,\mbox{${\rm M}_{\sun}$}$ 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 $T_{\rm eff}$ and $\log g$ are small
($\sim 10\%$ and $\sim 1\%$, respectively) the fast-test and the Bayesian
method are comparable in the estimation of final masses and cooling ages,
except for white dwarfs with both $T_{\rm eff}>40,000$ 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.

Figure 5: Comparison of the fast-test and Bayesian methods included in
wdwarfdate for the cooling age and final mass (left and right panels,
respectively). We included the one-to-one relation in a black solid line. We
run a grid of white dwarfs with $T_{\rm eff}=1,500-100,000$ K and $\log
g=7-9.3$ and uncertainties of $10\%$ and $1\%$, respectively. This comparison
shows that the estimation of final mass and cooling age using both methods is
similar. The purple points in both panels represent calculations of cooling
age that differ more than $1\sigma$ between the two methods. The small panel
of $T_{\rm eff}$ and $\log g$ shows that these points are situated in a dense
area of isochrones (Figure 2), which makes the likelihood less constrained. As
a consequence, the prior has more influence and makes the Bayesian cooling
ages larger.

##  3 Results of applying wdwarfdate to literature and new white dwarfs

In this section we compare the results from wdwarfdate with those from
previous studies. The goal of this section is to test the validity of
wdwarfdate results. We also calculate the ages of white dwarfs in candidate
binaries with M dwarfs which make a new set of age calibrators. Unless
indicated otherwise, the estimations were made using the Bayesian method and
the Cummings et al. (2018) MIST-based IFMR, with the DA white dwarf cooling
sequences, and ${\rm[Fe/H]}=0$ and ${\rm v/vcrit}=0.0$ stellar evolution
models. The uncertainties were calculated using the $16^{\rm th}$ and $84^{\rm
th}$ percentile of the distributions.

###  3.1 Comparison to white dwarfs from clusters from Cummings et al. (2018)

Cummings et al. (2018) used a sample of $79$ white dwarfs from $13$ different
clusters of known ages with $T_{\rm eff}$ and $\log g$ measurements from the
literature to calibrate the IFMR. In addition, the authors re-measured $T_{\rm
eff}$ 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 $T_{\rm eff}$ 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 $T_{\rm
eff}$ and $\log g$ and uncertainties published in their work. We compared
final mass, cooling age, initial mass and main sequence age in Figure 6. Most
of the values agree within $1\sigma$ 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 which has a
small uncertainty, and not the IFMR as in wdwarfdate.

We color-coded the white dwarfs for which any of the parameters in Figure 6
did not agree within $1\sigma$ 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 $T_{\rm eff}$ 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 $T_{\rm eff}>50,000$ K, and the cooling
tracks used in Cummings et al. (2018) (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)333https://www.montrealwhitedwarfdatabase.org/, which contains the
updated models by Bédard et al. (2020), which we implemented in wdwarfdate.

Figure 6: We run the white dwarfs in Cummings et al. (2018) using wdwarfdate
to compare the results. Most of the parameters are within $1\sigma$ and we
color-coded the rest. Both the green and purple points are outliers in the
IFMR. In addition, the three purple white dwarfs with higher cooling age have
high effective temperatures, which are better modeled with the new cooling
tracks implemented in our code (Bédard et al., 2020). The black dot is the
massive white dwarf GD 50.

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.
The total estimated age agrees within $1\sigma$ with the cluster age for
$65\%$ of the white dwarfs, and within $2\sigma$ 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.

Figure 7: We compare the total age posterior PDF obtained with wdwarfdate for
the white dwarfs in Cummings et al. (2018), with the age of the cluster they
belong to. Also, we divided the white dwarfs according to their membership and
added in a black vertical line the age of the group. We found good agreement
between the age of the cluster and the age derived with wdwarfdate.

The white dwarf color-coded in black is GD $50$, which is a young and
ultramassive white dwarf of $1.28\pm 0.08\,\mbox{${\rm M}_{\sun}$}$. 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
$128.71^{+10.54}_{-11.92}$ Myr, a final mass of $1.27\pm 0.02\,\mbox{${\rm
M}_{\sun}$}$, and an initial mass of $7.5\pm 0.3\,\mbox{${\rm M}_{\sun}$}$.
Our progenitor mass calculation differs slightly from the value in Cummings et
al. (2018), but agrees with the results of Gagné et al. (2018), that
calculated a mass of $7.8\pm 0.6\,\mbox{${\rm M}_{\sun}$}$. 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 $117\pm 26$ 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 $135\pm 35$ Myr, which is slightly larger than the
current measured age of this cluster ($112\pm 5$ 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 $>5\,\mbox{${\rm
M}_{\sun}$}$, and we obtained a total age of $114.61^{+12.72}_{-11.68}$ Myr,
significantly closer to the results of Gagné et al. (2018).

###  3.2 Comparison to white dwarfs from M$67$ from Canton et al. (2021)

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
$T_{\rm eff}$ and $\log g$ from high resolution spectra for $22$ white dwarfs
from the M$67$ cluster which is $3.5$ Gyr and solar metallicity. They used the
cooling tracks from Fontaine et al. (2001) to estimate 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
$T_{\rm eff}$ and $\log g$, and uncertainties with wdwarfdate, we found good
agreement for 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 $1\sigma$, as shown by the bottom panels in Figure 8. The
uncertainties on the total age of the white dwarfs are large ($\pm 5$ 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.

Figure 8: Comparison of the performance of wdwarfdate at lower-masses, with
the results from Canton et al. (2021). These white dwarfs belong to the M$67$
cluster ($3.5$ Gyr). For the final mass, cooling age and initial mass we show
the one-to-one line in black, and for the total age we show the age
distributions with the age of the cluster in a black line. For the total age
and initial mass we only show the vetted white dwarfs from Canton et al.
(2021). We found good agreement between our results and their study. The
uncertainties in the initial mass calculated by Canton et al. (2021) are
smaller than the point in some cases because they used a well constrained
total age to estimate these masses.

###  3.3 Wide, co-moving white dwarf companions

We used wdwarfdate to estimate ages of $30$ white dwarfs identified to be
candidate wide co-movers 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 co-moving star. Kiman et
al. (2021) estimated the white dwarf ages using wdwarfdate, and the $T_{\rm
eff}$ 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), that took advantage of the precise photometry
and parallaxes from Gaia EDR3 and estimated $T_{\rm eff}$ 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 $T_{\rm
eff}$ 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 $T_{\rm eff}$ and $\log g$. We discarded that source along
with the white dwarfs with masses $<0.5\,\mbox{${\rm M}_{\sun}$}$, or
$>1.1\,\mbox{${\rm M}_{\sun}$}$, according to their position in the color-
magnitude diagram (Figure 9) and white dwarfs with $G_{\rm BP}-G_{\rm
RP}>0.9$. 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 non-white dwarfs such as unresolved binaries. For a
more detailed discussion of the limitations of wdwarfdate see Section 4.

Figure 9: Color-magnitude diagram for the $30$ white dwarf which are candidate
wide co-movers with M dwarfs (Kiman et al., 2021). We show as red empty
circles the white dwarfs which were discarded because they are outside the
ranges which are likely to have followed single star evolution:
$0.5\,\mbox{${\rm M}_{\sun}$}<m_{\rm f}<1.1\,\mbox{${\rm M}_{\sun}$}$, and
also white dwarfs with $G_{\rm BP}-G_{\rm RP}<0.9$ to avoid contamination from
non-single white dwarfs. The only object within these limits which is also
discarded did not have an estimation of $T_{\rm eff}$ and $\log g$ in Gentile
Fusillo et al. (2021). We also show the models of constant mass
$0.5\,\mbox{${\rm M}_{\sun}$}$, $0.7\,\mbox{${\rm M}_{\sun}$}$,
$0.9\,\mbox{${\rm M}_{\sun}$}$ and $1\,\mbox{${\rm M}_{\sun}$}$ for DA white
dwarfs (Bergeron et al., 1995; Fontaine et al., 2001; Holberg & Bergeron,
2006; Kowalski & Saumon, 2006; Bergeron et al., 2011; Tremblay et al., 2011;
Blouin et al., 2018).

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 $T_{\rm eff}$
and $\log g$ from Gentile Fusillo et al. (2021). Given that most white dwarfs
are DAs ($\sim$ $75\%$), this is a good assumption (Fontaine et al., 2001;
Gentile Fusillo et al., 2015). To run our code we used the cooling sequence
with thick H outermost layer for DA white dwarfs and with thin layer for non-
DA, the stellar evolution models for ${\rm[Fe/H]}=0$ and ${\rm v/vcrit}=0.0$,
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 $T_{\rm eff}$ and $\log g$, and the parameters estimated with wdwarfdate
are in Table 2. The reported values in this table are the median, $16^{\rm
th}$, and $84^{\rm th}$ percentile 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.

The white dwarfs from Table 2 exemplify the impact of the $T_{\rm eff}$ 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 $T_{\rm eff}$ 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
($0.64^{+0.16}_{-0.06}\,\mbox{${\rm M}_{\sun}$}$ using the Bayesian method and
$0.97^{+0.23}_{-0.26}\,\mbox{${\rm M}_{\sun}$}$ 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.

Table 2: Candidate white dwarf co-movers with M dwarfs.  Name | SPTaaClassification in the Montreal White Dwarf Data Base. |  $T_{\rm eff}$ [K]bbParameters from Gentile Fusillo et al. (2019), estimated from photometry. |  $\log g$bbParameters from Gentile Fusillo et al. (2019), estimated from photometry. | Met.ccMethod used in wdwarfdate: B = Bayesian, F = Fast-test. |  $t_{\rm ms}$ [Gyr] |  $t_{\rm cool}$ [Gyr] |  $t_{\rm tot}$ [Gyr] |  $m_{\rm i}$ [M$\odot$] |  $m_{\rm f}$ [M$\odot$]
---|---|---|---|---|---|---|---|---|---
Gaia EDR3 2643218398328129664 |  $\cdots$ | $4661\pm 251$ | $7.90\pm 0.23$ | FT |  $\cdots$ | $5.28_{-1.89}^{+2.79}$ |  $\cdots$ |  $\cdots$ | $0.53_{-0.13}^{+0.14}$
SDSS J123304.34+030245.6 |  $\cdots$ | $4735\pm 263$ | $7.83\pm 0.27$ | FT |  $\cdots$ | $4.33_{-1.53}^{+3.13}$ |  $\cdots$ |  $\cdots$ | $0.48_{-0.13}^{+0.16}$
Gaia EDR3 1323951092358466304 |  $\cdots$ | $5652\pm 842$ | $8.49\pm 0.58$ | 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 |  $\cdots$ | $6469\pm 617$ | $8.09\pm 0.32$ | 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 | $6740\pm 108$ | $8.02\pm 0.05$ | 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 | $6981\pm 977$ | $7.71\pm 0.57$ | FT |  $\cdots$ | $1.31_{-0.62}^{+2.24}$ |  $\cdots$ |  $\cdots$ | $0.47_{-0.19}^{+0.32}$
Gaia EDR3 4467448853280423296 |  $\cdots$ | $7151\pm 935$ | $8.27\pm 0.45$ | 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 |  $\cdots$ | $8557\pm 1021$ | $8.06\pm 0.41$ | 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 |  $\cdots$ | $9249\pm 492$ | $7.72\pm 0.19$ | FT |  $\cdots$ | $0.56_{-0.13}^{+0.17}$ |  $\cdots$ |  $\cdots$ | $0.45_{-0.09}^{+0.10}$
SDSS J003000.03-002738.9 | DA | $9359\pm 1061$ | $7.94\pm 0.45$ | 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 |  $\cdots$ | $10286\pm 1114$ | $7.93\pm 0.41$ | 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 |  $\cdots$ | $12338\pm 2743$ | $8.45\pm 0.44$ | 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 |  $\cdots$ | $12371\pm 2799$ | $8.07\pm 0.50$ | 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 | $12859\pm 1751$ | $8.18\pm 0.22$ | 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 |  $\cdots$ | $14390\pm 2095$ | $7.83\pm 0.28$ | FT |  $\cdots$ | $0.18_{-0.09}^{+0.16}$ |  $\cdots$ |  $\cdots$ | $0.51_{-0.12}^{+0.15}$
RMB 103 | DB | $14096\pm 518$ | $7.93\pm 0.08$ | 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 | $15137\pm 682$ | $7.97\pm 0.08$ | 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 | $15602\pm 2052$ | $8.11\pm 0.30$ | 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 | $0.49_{-0.26}^{+1.74}$ | $0.29_{-0.12}^{+0.21}$ | $0.89_{-0.26}^{+1.57}$ | $2.92_{-1.29}^{+0.94}$ | $0.73_{-0.11}^{+0.15}$

##  4 Limitations and assumptions

White dwarf cosmochronology is a powerful tool to estimate stellar ages.
However, it is based on a number of models and assumptions that need to be
taken into account. In this section, we discuss the constraints and model
choices when using wdwarfdate.

We run a grid of $100$ white dwarfs with $1,500\,{\rm K}<T_{\rm eff}<100,000$
K and $7<\log g<9.3$ with different relative uncertainties to test the
resulting uncertainty of the Bayesian total ages derived with wdwarfdate. In
this study we used the Cummings et al. (2018) MIST-based IFMR, with the DA
white dwarf cooling sequences and ${\rm[Fe/H]}=0$ and ${\rm v/vcrit}=0$
stellar evolution models. We found that for uncertainties of $1\%$ in both
$T_{\rm eff}$ and $\log g$, the median relative uncertainty in total age is
$\sim 11\%$, as shown in Figure 10. If the uncertainty of $T_{\rm eff}$ is
increased to $10\%$, the total age uncertainty increases to $\sim 25\%$. On
the other hand, using an uncertainty of $1\%$ for $T_{\rm eff}$ and $5\%$ for
$\log g$, the total age uncertainty increases to $\sim 52\%$, showing the
importance of precise input measurements, in particular of $\log g$. For a
study of the total age relative uncertainty as a function of $T_{\rm eff}$ and
$\log g$ see Appendix B. We repeated this analysis for non-DA cooling models
and found no significant difference in the results.

Figure 10: Median relative uncertainty in the total age calculated using the
Bayesian method in wdwarfdate. The median is calculated over all $T_{\rm eff}$
and $\log g$ results with the same input uncertainties, and the uncertainty in
the median is calculated as the standard deviation. In the left panel, the
relative uncertainty in $T_{\rm eff}$ is fixed at $1\%$ and $10\%$. In the
right panel, the relative uncertainty in $\log g$ is fixed at $1\%$ and $5\%$.

In addition to the limits on $T_{\rm eff}$ and $\log g$ set by the cooling
tracks discussed in Section 2.2, the IFMR imposes extra limits to the input
values for which wdwarfdate will be able to estimate a total age, main
sequence age and initial mass. To quantify this limit, we run wdwarfdate on a
grid of $T_{\rm eff}$ and $\log g$ and recorded which parameters were
estimated. In this exercise, we adopted the MIST-based Cummings et al. (2018)
IFMR, the cooling sequences for DA white dwarfs, and the stellar evolution
models for ${\rm[Fe/H]}=0$ and ${\rm v/vcrit}=0.0$. The values of $T_{\rm
eff}$ and $\log g$ for which wdwarfdate can estimate a total age are color-
coded by final mass in Figure 11. In summary, wdwarfdate will derive Bayesian
total ages for $1,500\,{\rm K}\lesssim T_{\rm eff}\lesssim 90,000$ K and
$7.9\lesssim\log g\lesssim 9.3$, and only final mass and cooling age using the
fast-test method for $7.0\lesssim\log g\lesssim 7.9$ (green-blue five point
stars in Figure 11).

Figure 11: Limits of $T_{\rm eff}$ and $\log g$ for wdwarfdate: In brown are
the values for which the code does not estimate any parameters because they
are outside of the allowed range by the cooling tracks. In green-blue five-
point stars are the values for which only a final mass and cooling age are
estimated using the fast-test method, but not the rest of the parameters
because of the limits where the MIST-based Cummings et al. (2018) IFMR (the
chosen relation for this example) is calibrated. We color-coded by final mass
the values for which wdwarfdate estimates all the parameters using the
Bayesian method: final mass, cooling age, initial mass, main sequence age and
total age.

The masses for C/O core white dwarfs ranges from $0.45-1.1\,\mbox{${\rm
M}_{\sun}$}$. For lower masses, the core is composed of He and for higher
masses of O and/or Ne (Fontaine et al., 2001; Garcia-berro et al., 1997). As
we implemented cooling tracks that assume a C/O core in wdwarfdate, parameter
estimation using our code outside this range should be taken with caution. We
discuss further the lower and upper mass limits below.

Within a Hubble time, it is not possible to have a white dwarf with a mass
smaller than $0.45\,\mbox{${\rm M}_{\sun}$}$ that was formed by single stellar
evolution. Low-mass white dwarfs ($<0.45\,\mbox{${\rm M}_{\sun}$}$) cannot
ignite He, therefore they have a He core, and are thought to be the result of
binary evolution (e.g., Marsh, 1995; Brown et al., 2011). There are studies
which propose that these objects could also be formed from a red giant branch
star which had a high mass loss rate (e.g., Kilic et al., 2007). In addition,
binary stars could also appear to be white dwarfs of masses
$<0.45\,\mbox{${\rm M}_{\sun}$}$, for example unresolved white dwarf binaries
such as double degenerate stars, look like one single over-luminous star. With
the same $T_{\rm eff}$, this means bigger radius, which means smaller mass
according to the mass-radius relation of degenerate objects (Bergeron et al.,
2001; Bédard et al., 2017; Kilic et al., 2020). This phenomenon could mimic a
white dwarf of mass $<0.45\,\mbox{${\rm M}_{\sun}$}$.

Higher-mass white dwarfs ($>1.1\,\mbox{${\rm M}_{\sun}$}$) should also be
viewed with caution. These stars are potentially magnetic white dwarfs and may
have come from a merger (Ferrario et al., 2015), which would make the single
star evolution assumption fail, and therefore the models would not reproduce
the true evolution of the object.

The typical structure of a white dwarf consists of a C/O core with a thin
shell of He, surrounded by a thin layer of H. Both the core and the surface
layers have an important role in how the white dwarfs cool: the thermal energy
is stored in the core while the heat outflow is regulated by the opaque
envelope (Hansen et al., 2004). For example, DA white dwarfs with thick H
layers cool slower than non-DA objects because the H envelope is a better
insulator (Fontaine et al., 2001). In addition, the chemical composition of
the core will affect the heat capacity of the object, and therefore the
cooling rate (Hansen et al., 2004). For instance, the estimated cooling age of
a white dwarf assuming a pure C core is an upper limit, while assuming a pure
O core is a lower limit, with differences of a few Gyr for the oldest objects
(Fontaine et al., 2001). Although the core composition is still a source of
uncertainty for white dwarf cosmochronology (Simon et al., 2015), it is
necessary to assume one to estimate the cooling age and mass of a white dwarf
with evolutionary models. wdwardate uses cooling tracks from Bédard et al.
(2020) which assume a C/O core with the same proportions of C and O with the
option of choosing between a thin or thick outermost H layer.

The parameter estimation using wdwarfdate depends also on the relation between
the progenitor mass and the white dwarf mass, meaning the IFMR. This is an
empirical relation, which is improving as more precise measurements of the
parameters of white dwarfs and their progenitors become available. The IFMRs
included in wdwarfdate are similar to each other (See Figure 3), and we have
seen no significant difference in the median values of the ages and masses
estimated in this study when changing the IFMR. However, the IFMR should be
chosen carefully when using wdwarfdate because the shape of the posterior PDF
can be affected by it. Our recommendation among the available IFMRs is to use
the MIST-based Cummings et al. (2018) relation, because this is the most
complete and simplest relation to use with our method. Although Marigo et al.
(2020) is more recent than Cummings et al. (2018), the difference is small
between the relations, and the kink in the former complicates the sampling of
the posterior without significant difference in the median mass or age of the
white dwarf.

The main sequence age of the progenitor is obtained from a stellar evolution
model, based on its mass, abundances and rotation, which can modify the
estimation of the total age (Moss et al., 2022). We tested the parameter
estimation for $50$ randomly selected white dwarfs with $1500\,{\rm K}<T_{\rm
eff}<100,000\,{\rm K}$ and $7<\log g<9.3$, with uncertainties of $10\%$ and
$1\%$ respectively, using all the possible combinations of
${\rm[Fe/H]=\\{-4,-1,0.5\\}}$ (dex) and ${\rm v/vcrit=\\{0.0,0.4\\}}$. For all
the runs we used the MIST-based Cummings et al. (2018) IFMR, and the cooling
sequences for DA white dwarfs. We compared the results to the parameters
estimated using ${\rm[Fe/H]}=0$, and ${\rm v/vcrit}=0.0$ to estimate the
effect of using different stellar evolution models. We found no significant
change in the estimated values of the final mass or cooling age, which is
expected because we did not modify the parameters of the white dwarf. For the
initial mass we found a small difference of up to $0.16\,\mbox{${\rm
M}_{\sun}$}$ when ${\rm[Fe/H]}=-4$ for both cases of ${\rm v/vcrit}$. For the
main sequence age and total age we found differences of up to $0.7$ Gyr when
${\rm[Fe/H]}=-4$, again not changing significantly with ${\rm v/vcrit}$. This
shows that the parameters of the progenitor star need to be chosen carefully
given that they can affect the results. In case there is no information of the
parameters of the progenitor star, it is reasonable to choose solar
metallicity (${\rm[Fe/H]}=0$), taking into account the possible variations
described above.

The last aspect to take into account when running wdwarfdate is the choice of
priors. Currently the code assumes a constant SFH and the IMF to be $m_{\rm
i}^{-2.3}$ when using the Bayesian mode (see Section 2.3), and it does not
allow changes in these priors. As discussed in Sections 2.4 and 3.3, when the
uncertainty in the input parameters is large, the likelihood will be less
constrained and the priors will have a bigger effect, making the initial and
the final masses smaller, and therefore affecting the estimated ages as well.
In this case, the user could consider running the fast-test method instead of
the Bayesian method. In this alternative method, wdwarfdate will perform a
Monte Carlo uncertainty propagation to estimate the white dwarf parameters (as
it was described in Section 2.4) and will not be affected by the priors.

##  5 Summary

In this work we presented wdwarfdate, a publicly available Python package
which derives Bayesian total ages of white dwarfs from an effective
temperature ($T_{\rm eff}$) and a surface gravity ($\log g$), assuming single
star evolution. wdwarfdate obtains the probability distributions by explicitly
integrating over the likelihood and priors to perform the normalization for
the following parameters: mass of the progenitor star, cooling age of the
white dwarf and $\Delta_{\rm m}$, which models the scatter in the IFMR. From
these parameters, the code obtains probability distributions for the mass of
the white dwarf, the main sequence age of the progenitor and the total age.
The derived ages depend on the input parameters and on the models chosen when
initializing wdwarfdate, which are summarize in Table 1 (See Appendix A for an
example on how to use wdwarfdate).

We tested wdwarfdate on white dwarfs cluster members from Cummings et al.
(2018) and Canton et al. (2021). In general we found good agreement between
our results and theirs for the mass and age of the white dwarf as well as the
properties of the progenitor star. In addition, we used wdwarfdate to estimate
the ages of a set of $18$ white dwarfs which are candidate co-movers with M
dwarfs (Kiman et al., 2021), and therefore conform a group of low-mass star
age-calibrators.

We also described a detailed analysis of the constraints on the code which
need to be taken into account before using. In summary:

  * •

We found a typical uncertainty of $10\%$ on the Bayesian total age for $T_{\rm
eff}$ and $\log g$ with uncertainties of $1\%$, and of $25\%$ with input
uncertainties of $10\%$ and $1\%$ for $T_{\rm eff}$ and $\log g$,
respectively.

  * •

Combining the limitation of the cooling tracks with the restrictions of the
IFMR, the values for which wdwarfdate can derive Bayesian total ages are
$1,500\,{\rm K}\lesssim T_{\rm eff}\lesssim 90,000$ K and $7.9\lesssim\log
g\lesssim 9.3$ (cm ${\rm s^{-2}}$).

  * •

Given that the cooling tracks assume single star evolution and C/O core for
the white dwarfs, the best range of final masses to use wdwarfdate is
$0.45-1.1\,\mbox{${\rm M}_{\sun}$}$, because objects outside this range are
not likely to have evolved as a single star.

  * •

The IFMRs included in our code are similar to each other and the results are
not modified significantly from using one over the other. However we recommend
using the MIST-based Cummings et al. (2018) IFMR, given it is one of the most
complete studies of white dwarfs in clusters currently available, and has a
relatively simple functional form.

  * •

The choice of ${\rm[Fe/H]}$ can modify the main sequence age and total age by
up to $0.7$ Gyr, but if there is no information about the progenitor star
available, solar metallicity (${\rm[Fe/H]}=0$) is a reasonable assumption.

  * •

When the uncertainty of the input parameters is large, the prior in the
Bayesian method will affect the estimation of the age given that the
likelihood is not well constrained. The user can consider using the fast-test
method in these cases.

Although Gaia has increased the number of known white dwarfs by a factor of
$10$, we still need more white dwarfs which belong to known clusters with
spectra to improve the calibration of the IFMR. This relation is one of the
main limitations when estimating the total age of a white dwarf because the
complete functional form of the IFMR is still not known with great accuracy,
especially in the low-mass regime.

In future work we plan to include the estimation of $T_{\rm eff}$ and $\log g$
from photometry in wdwarfdate, and extra priors in case the final mass or
total age of the white dwarf are known, to improve the estimation of the other
parameters.

To learn about how to use wdwarfdate, check out the source code (Kiman,
2022)444https://github.com/rkiman/wdwarfdate.git and the online
documentation555https://wdwarfdate.readthedocs.io.

##  6 Acknowledgements

The authors would like to thank Jeff Andrews, Lars Bildsten, Javier Roulet and
Antoine Bédard for useful discussions.

Support for this project was provided by a PSC-CUNY Award, jointly funded by
The Professional Staff Congress and The City University of New York.

This material is based upon work supported by the National Science Foundation
under Grant No. 1614527.

This work has been supported by NASA K2 Guest Observer program under award
80NSSC19K0106.

This work was supported by the SDSS Faculty and Student Team (FAST)
initiative.

Support for this work was provided by the William E Macaulay Honors College of
The City University of New York

S.X. is supported by the international Gemini Observatory, a program of NSF’s
NOIRLab, which is managed by the Association of Universities for Research in
Astronomy (AURA) under a cooperative agreement with the National Science
Foundation, on behalf of the Gemini partnership of Argentina, Brazil, Canada,
Chile, the Republic of Korea, and the United States of America.

J.F. and S.X. acknowledge support from the Heising-Simons Foundation.

S.L.C. acknowledges the support of an STFC Ernest Rutherford Research
Fellowship.

##  Appendix A Running wdwarfdate

Below we include an example of how to run wdwarfdate, where we selected a
white dwarf from Cummings et al. (2018), and we are using the cooling tracks
for a DA white dwarf, the stellar evolution track corresponding to [Fe/H] = 0
and no rotation, and the MIST-based IFMR from Cummings et al. (2018). By
default wdwarfdate will run the Bayesian method, but the fast-test method can
be selected by indicating method=’fast_test’ in the WhiteDwarf class. The
object results is an astropy
Table666https://docs.astropy.org/en/stable/table/index.html with the total age
of the object, main sequence age and mass of the progenitor and cooling age
and mass of the white dwarf, and their uncertainties. This example takes $21$
seconds to run in a personal computer. For more information on how to run
wdwarfdate see the documentation of the code.

    import wdwarfdate

    teff = 19250
    teff_err = 500
    logg = 8.16
    logg_err = 0.084

    WD = wdwarfdate.WhiteDwarf(teff,teff_err,logg,logg_err,
                               model_wd=’DA’,
                               feh=’p0.00’,
                               vvcrit=’0.0’,
                               model_ifmr = ’Cummings_2018_MIST’)
    WD.calc_wd_age()

    results = WD.results

##  Appendix B Relative uncertainty in total age.

To study the relative uncertainty in the Bayesian total age estimated with
wdwarfdate we run a grid of $100$ white dwarfs with $1,500\,{\rm K}<T_{\rm
eff}<100,000$ K and $7<\log g<9.3$, with different uncertainties. In this
study we used the Cummings et al. (2018) MIST-based IFMR, with the DA white
dwarf cooling sequences and ${\rm[Fe/H]}=0$ and ${\rm v/vcrit}=0$ stellar
evolution models. In Figure 12 we show the relative uncertainty in the total
age ($t_{\rm tot}$) for four cases of input uncertainties: $\Delta T_{\rm
eff}/T_{\rm eff}=1\%$ and $\Delta\log g/\log g=1\%$, $\Delta T_{\rm
eff}/T_{\rm eff}=10\%$ and $\Delta\log g/\log g=1\%$, $\Delta T_{\rm
eff}/T_{\rm eff}=1\%$ and $\Delta\log g/\log g=5\%$, and $\Delta T_{\rm
eff}/T_{\rm eff}=10\%$ and $\Delta\log g/\log g=5\%$. As discussed in Section
4, the biggest effect on the uncertainty of $t_{\rm tot}$ comes from
increasing the uncertainty of $\log g$. In addition, we show that higher
$T_{\rm eff}$ are the most affected by the increase in the uncertainty of the
input parameters. We repeated this analysis for non-DA cooling models and
found no significant difference.

Figure 12: Relative uncertainty in the Bayesian total age estimated using
wdwarfdate. We include four cases of different input uncertainties to show the
variation as a function of $T_{\rm eff}$ and $\log g$.

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