# Rotation at the Fully Convective Boundary: Insights from Wide WD + MS Binary
Systems

Federica Chiti Institute for Astronomy, University of Hawai‘i at Mānoa, 2680
Woodlawn Dr., Honolulu, HI 96822, USA  Jennifer L. van Saders Institute for
Astronomy, University of Hawai‘i at Mānoa, 2680 Woodlawn Dr., Honolulu, HI
96822, USA  Tyler M. Heintz Department of Astronomy & Institute for
Astrophysical Research, Boston University, 725 Commonwealth Ave, Boston, MA,
02215, USA  J. J. Hermes Department of Astronomy & Institute for Astrophysical
Research, Boston University, 725 Commonwealth Ave, Boston, MA, 02215, USA  J.
M. Joel Ong (王加冕)  Institute for Astronomy, University of Hawai‘i at Mānoa,
2680 Woodlawn Dr., Honolulu, HI 96822, USA  NASA Hubble Fellow  Daniel R. Hey
Institute for Astronomy, University of Hawai‘i at Mānoa, 2680 Woodlawn Dr.,
Honolulu, HI 96822, USA  Michele M. Ramirez-Weinhouse  Lāna’i High and
Elementary School, 555 Fraser Ave, Lanai City, HI 96763  Alison Dugas
Institute for Astronomy, University of Hawai‘i at Mānoa, 2680 Woodlawn Dr.,
Honolulu, HI 96822, USA

###### Abstract

Gyrochronology, a valuable tool for determining ages of low-mass stars where
other techniques fail, relies on accurate calibration. We present a sample of
327 wide ($>$$100$ au) white dwarf + main sequence (WD + MS) binary systems.
Total ages of WDs are computed using all-sky survey photometry, Gaia
parallaxes, and current hydrogen atmosphere WD models. Using a magnetic
braking law calibrated against open clusters, along with assumptions about
initial conditions and angular momentum transport, we construct gyrochrones to
predict the rotation periods of the MS stars. Both data and models show that,
near the fully convective boundary, MS stars with WD ages up to 7.5 Gyr
experience a rotation period increase by up to a factor of $\approx$$3$ within
a $<50\,\mathrm{K}$ effective temperature range. We suggest that rapid braking
at this boundary is driven by a sharp rise in the convective overturn
timescale ($\tau_{\mathrm{cz}}$) caused by structural changes between
partially and fully convective stars and the ${}^{3}\textrm{He}$ instability
occurring at this boundary. While the specific location in mass (or
temperature) of this feature varies with model physics, we argue that its
existence remains consistent. Stars along this feature exhibit rotation
periods that can be mapped, within 1$\sigma$, to a range of gyrochrones
spanning $\approx 6$ Gyr. Due to current temperature errors
($\simeq$$50\,\mathrm{K}$), this implies that a measured rotation period
cannot be uniquely associated to a single gyrochrone, implying that
gyrochronology may not be feasible for M dwarfs very close to the fully
convective boundary.

stellar ages, stellar rotation, stellar evolution, white dwarfs

##  1 Introduction

Ages of stars are critical to our understanding of the evolution of
astrophysical systems and yet are one of the most difficult stellar properties
to measure. The only star for which we have a precise and accurate age is the
Sun; for any other star, age can only be estimated or inferred. There are many
techniques to estimate stellar ages, but there is no single method that is
applicable to all spectral types (Soderblom, 2010).

K and M dwarfs are the most numerous stars in the Galaxy and have lifetimes
longer than the age of the Milky Way disk, meaning that they preserve a record
of the history of star formation and chemical evolution of the Galaxy. They
are, however, resistant to standard age-dating techniques. Isochrone fitting
fails to provide constraints on stellar ages when used on low-mass stars, due
to their slow nuclear evolution (Takeda et al., 2017). Similarly,
asteroseismology, which provides precise ages for Sun-like stars, cannot be
used to date low-mass stars like K and M dwarfs due to their low oscillation
amplitudes (Chaplin et al., 2011).

A promising tool in this low mass regime is gyrochronology (Barnes, 2007),
which derives ages for cool MS stars by exploiting the fact that they spin
down with time (Skumanich, 1972) due to magnetic braking. Magnetic braking is
the mechanism by which a star loses angular momentum to magnetized stellar
winds over time as the result of the interaction between mass loss and dynamo-
driven stellar magnetic fields. Epstein & Pinsonneault (2014) showed that
under Skumanich-type spin down, rotation-based age dating is potentially among
the most precise methods available.

Calibration of period-age relations for low mass stars requires a large sample
of old, well-dated low mass stars. However, only a handful of stars below 0.8
$M_{\odot}$ are currently available, and most of them are in young clusters (4
Gyr at the oldest, Dungee et al., 2022). Furthermore, observations of these
clusters have shown that standard braking models fail to reproduce the
observed rotational sequences, suggesting that stellar spin-down may not be as
simple as it once appeared.

Recent measurements of the rotation period ($P_{\mathrm{rot}}$) of stars in
the benchmark open clusters Praesepe ($\approx$700 Myr, Douglas et al., 2019)
and NGC 6811 (1Gyr, Meibom et al., 2011; Janes et al., 2013) show that a
simple power law with a braking index $n$ ($n=0.5$ in the Skumanich law) fails
at predicting the observed rotational sequences in these clusters (Curtis et
al., 2020). While solar-type stars in NGC 6811 have longer periods compared to
their counterparts in the younger cluster Praesepe, the two sequences merge at
$M<0.8$ $M_{\odot}$ (K0 to M0-type stars, Curtis et al., 2019; Douglas et al.,
2019). In other words, the spin-down appears to “stall” (or reduce) for low-
mass stars in NGC 6811. Spada & Lanzafame (2020) demonstrated that this
phenomenon can be explained by relaxing the assumption of solid-body rotation
and allowing for angular momentum (AM) transport between the core and the
envelope (i.e. radial differential rotation). The spin-down stalling observed
in K and early M-type stars in NGC 6811 is putatively an epoch when AM
transport from the core to the envelope balances the AM loss at the surface
due to winds. The lack of a radiative core in fully convective stars that can
support such core-envelope interaction may be responsible for the closure of
the intermediate period gap discovered with Kepler by McQuillan et al. (2013)
at the fully convective boundary (Lu et al., 2021).

Near the fully convective boundary, another relevant feature is the
underdensity of stars observed near $M_{G}=10.2$ in the main sequence on the
Hertzsprung-Russell diagram found by Jao et al. (2018) using Gaia DR2
measurements. It has been proposed that this Gaia M-dwarf gap is a
manifestation of the location where stars transition from partially to fully
convective, which is predicted to occur at a mass of $\sim 0.35$ $M_{\odot}$.
Earlier theoretical work by van Saders & Pinsonneault (2012a) demonstrated
that stars slightly above this threshold undergo a structural instability due
to non-equilibrium 3He burning during the first few billion years on the main
sequence. This results in the development of a convective core, separated from
a deep convective envelope by a thin radiative layer. The continuous
accumulation of central 3He causes the radiative zone separating them to thin
even further, initiating fully convective episodes. Using stellar models and
stellar population synthesis, Feiden et al. (2021) confirmed that the 3He
instability is responsible for the appearance of the M-dwarf gap.

With the distinct structural changes across the fully convective boundary,
there have been attempts to comprehend whether there are any consequences for
the magnetic properties, activity levels, and rotation rates of stars. For
instance, Donati et al. (2008) and Reiners & Basri (2009) demonstrated that
$0.34-0.36$ $M_{\odot}$ stars are prone to undergo sudden alterations in their
large-scale magnetic topologies, which could lead to observable surface
activity signatures. More recently, Jao et al. (2023) conducted a high-
resolution spectroscopic H$\alpha$ emission survey of M dwarfs spanning the
Gaia M-dwarf gap and argued that stars above the top gap edge exhibit
H$\alpha$ emission while stars within the gap or below do not display any
emission. Thus, stars near the fully convective boundary provide a powerful
laboratory for testing the physics of M-dwarf stars, including those affected
by the 3He instability. Moreover, having a reliable spin-down model that can
predict the rotational evolution of these stars is crucial for determining a
precise period-age relation for old, low-mass stars.

A primary limitation is the lack of empirical anchors of known age for old K
and M-type stars. Open clusters have been the major contributors of
calibrators to date; however, due to their short dissipation timescales
($\sim$200 Myr; Wielen, 1971), old clusters are rare and tend to be more
distant and challenging to observe. The standard gyrochronology calibrators
and the recent observations of late K- and early M- dwarfs in M67 (Dungee et
al., 2022) do not extend beyond 4 Gyr for stars below 0.8 $M_{\odot}$.
Likewise, the asteroseismic calibrator sample that has been important for
understanding braking in solar-mass stars (van Saders et al., 2016; Hall et
al., 2021) does not extend beyond $\approx 0.8$ $M_{\odot}$.

Wide binaries that contain a white dwarf (WD) companion provide a distinctive
opportunity to determine the ages of field stars. WDs are the end product of
stars with initial masses less than $8-10$ $M_{\odot}$ and, as they no longer
undergo nuclear fusion in their core, they gradually cool with time becoming
dimmer and colder. Because their effective temperature and mass uniquely
correspond to a single cooling age (given a composition), WDs have been
utilized as stellar clocks for decades (Fontaine et al., 2001). The
advancement of robust cooling models (Bergeron et al., 1995) allows WDs to
serve as precise and dependable age indicators. However, to determine the
complete age of a WD, one needs to consider the time from its zero-age main
sequence (ZAMS) to its present state as a WD. This involves using a semi-
empirical initial-final mass relation (IFMR) in conjunction with stellar
evolution model grids to ascertain the progenitor lifetimes from the ZAMS to
the WD phase. Using photometry only, Heintz et al. (2022) found that WD ages
are precise at the 25% level for WDs with masses $>0.63\,M_{\odot}$.

Wide, coeval binaries are sufficiently distant ($>100$ au) that the two stars
can be expected to evolve as single stars without any interaction between them
(White & Ghez, 2001). Therefore, the WD companion offers an independent age
estimate of the entire system, making it possible to extend period-age
relationships to the age of the galactic disk for the most common stars in our
galaxy.

The number of wide coeval binary systems has significantly increased since the
launch of the Gaia spacecraft. From Gaia DR2, El-Badry & Rix (2018)
constructed a catalog of over $\sim$$53{,}000$ binaries, $\sim$$3000$ of which
contained a WD and a main sequence star, which represented a tenfold increase
in the number of known coeval binaries (Holberg et al., 2013). With the
release of Gaia eDR3, El-Badry et al. (2021) published an extensive catalog of
1.3 million spatially resolved binary stars within $\approx$$1$ kpc of the
Sun, including more than $16{,}000$ WD + MS binaries, which increased the
sample by another order of magnitude. This increase presents an opportunity to
infer precision ages for cool, old stars, which are the focus of this work.

In Section 2 we describe the physics of the rotational evolution models, their
calibration and the technique used to determine WD ages. In Section 3, we
present the sample selection of WD + MS systems. The ages of these systems as
revealed by our models and WDs are discussed in Section 4, where we also
compare our sample to other datasets. Finally, the conclusions of our results
are presented in Section 5.

##  2 Methods

###  2.1 Gyrochronology Models

We use the rotation code rotevol (van Saders & Pinsonneault, 2013; Somers et
al., 2017) to model the AM evolution of a star. We use as input non-rotating
tracks with stellar masses between 0.18 and 1.15 M⊙, generated using the Yale
Rotating Evolution Code (YREC, see van Saders & Pinsonneault, 2012b;
Pinsonneault et al., 1989; Bahcall & Pinsonneault, 1992). The models include
helium and heavy element diffusion following Thoul et al. (1994), but with the
diffusion coefficients multiplied by a factor of 0.753 to match the
helioseismically determined helium abundance (Basu & Antia, 1995) in the Sun
at solar age. We adopt boundary conditions using the Allard et al. (1997)
atmospheric tables, OPAL opacities (Iglesias & Rogers, 1996) with low-
temperature opacities from Ferguson et al. (2005) for a Grevesse & Sauval
(1998) solar mixture. We adopt nuclear reaction rates from Adelberger et al.
(2011) with weak screening (Salpeter, 1954) and the equation of state from the
OPAL project (Rogers et al., 1996; Rogers & Nayfonov, 2002). We assume no
overshooting, and a mixing length theory of convection (Cox & Giuli, 1968;
Vitense, 1953). Our solar-calibrated model at 4.57 Gyr (Bahcall et al., 1995)
has $Z=0.01709$, $X=0.71642$ and mixing length parameter
$\alpha_{ml}=1.94243$. We run models at a range of surface spot covering
fractions in YREC (0%, 25% and 50%) following the prescription of Somers &
Pinsonneault (2015) and Somers et al. (2020) with $x_{spot}=0.8$ but retain
otherwise identical physical ingredients in the spotted models.

We run YREC in the non-rotating configuration and compute the rotational
evolution post hoc using the tracer code rotevol (van Saders & Pinsonneault,
2013; Somers et al., 2017). The benefit of this approach is that we can
rapidly search parameter space when fitting the magnetic braking law to the
observations; the downside is that rotation cannot influence the structure.
While this is a reasonable assumption for all but the most rapid rotators,
ideally one would actively couple the starspot filling fraction to the
rotation rate (e.g. Cao & Pinsonneault, 2022; Cao et al., 2023). We leave this
exercise to future work, and instead examine the behavior of a range of fixed
spot covering fractions.

To model the rotation period evolution of a star as a function of time, one
must choose appropriate initial starting conditions as well as to specify
prescriptions for three processes that drive the angular momentum (AM)
evolution, early disk interactions, AM loss at the stellar surface through
magnetized winds (“braking law”), and internal AM transport. In this section,
we describe the ingredients and assumptions of the stellar evolutionary
models.

####  2.1.1 Initial conditions

The rotation period $(P_{\mathrm{rot}})$ of a star approaching the MS is
determined largely by two factors: the initial rotation period
$(P_{\mathrm{init}})$ and the protostellar disk lifetime
$(\tau_{\mathrm{disk}})$. Observations of stars in young clusters indicate
both that there is a strong mass-rotation correlation in the
$P_{\mathrm{init}}$ distribution for low-mass stars and that stars at a fixed
mass can have a wide range of initial rotation rates (e.g. Somers et al.,
2017). This variation in initial rotation conditions results in a range of
rotation periods (approximately $\approx$ 0.1 to 10 days) when stars reach the
zero age main-sequence (ZAMS). Although initial conditions become less
significant for older stars (van Saders & Pinsonneault, 2013; Epstein &
Pinsonneault, 2014; Bouvier, 2013), there are certain combinations of stellar
ages and masses, namely the K and M dwarfs of interest, where this initial
variability is important (Bouvier, 2013; Gallet & Bouvier, 2015).

The models start on the pre-MS, where the distribution of initial rotation
conditions is set up by the young cluster Upper Sco at 10 Myr. By this age
massive accretion disks are nearly absent (Williams & Cieza, 2011) and
significant AM loss has not occurred yet (Somers et al., 2017; Rebull et al.,
2018). Initial conditions are chosen based on the observed rotation period
distribution in Upper Sco at $\sim$$10$ Myr, as shown in Fig. 1. We adopt a
1-D smoothing spline fit to the $10^{\mathrm{th}}$ and $50^{\mathrm{th}}$
percentile rotation periods in mass bins spanning the $10^{\mathrm{th}}$ to
the $90^{\mathrm{th}}$ percentile masses, with a 10% increment. This approach
enables us to encompass the entire spectrum of initial rotation periods, while
also mitigating the influence of rare tidally locked binaries and background
sources on our fitting process.

Figure 1: The Upper Sco rotation distribution at 10 Myr (Somers et al., 2017)
from which we adopt the initial rotation periods for the model tracks. The
solid blue and yellow lines show a 1-D smoothing spline fit to the 10th and
50th period percentiles in each mass bin, respectively.

####  2.1.2 Magnetic braking

We adopt the van Saders & Pinsonneault (2013) formulation of the classic wind
braking law proposed by Kawaler (1988)

$\frac{dJ}{dt}=\left\\{\begin{array}[]{ll}f_{K}K_{M}\omega\left(\frac{\omega_{\mathrm{sat}}}{\omega_{\odot}}\right)^{2}&\omega_{\mathrm{sat}}\leqslant\omega\frac{\tau_{cz}}{\tau_{cz,\odot}}\\\
f_{K}K_{M}\omega\left(\frac{\omega\tau_{cz}}{\omega_{\odot}\tau_{cz,\odot}}\right)^{2}&\omega_{\mathrm{sat}}>\omega\frac{\tau_{cz}}{\tau_{cz,\odot}}\end{array},\right.$
(1)

where $f_{K}$ is a normalization constant tuned to reproduce the observed
rotation at known age; $\omega$ is the rotation rate; $\omega_{\odot}$ is the
rotation rate of the Sun ($2.86\times 10^{-6}\,\mathrm{rad\,s}^{-1}$);
$\omega_{\mathrm{sat}}$ is the saturation threshold; $\tau_{cz}$ is the
convective overturn timescale; $K_{M}$ is the product

$\frac{K_{M}}{K_{M,\odot}}=c(\omega)\left(\frac{R}{R_{\odot}}\right)^{3.1}\left(\frac{M}{M_{\odot}}\right)^{-0.22}\left(\frac{L}{L_{\odot}}\right)^{0.56}\left(\frac{P_{\mathrm{ph}}}{P_{\mathrm{ph},\odot}}\right)^{0.44}$

(2)

with luminosity $L$, mass $M$, radius $R$, photospheric pressure
$P_{\mathrm{ph}}$; $c(\omega)$ is the centrifugal correction from Matt et al.
(2012). As in van Saders & Pinsonneault (2013), we assume that the magnetic
field scales as

$\frac{\dot{B}}{\dot{B_{\odot}}}=\left(\frac{P_{\mathrm{phot}}}{{P_{\mathrm{phot},\odot}}}\right)^{0.5}\left\\{\begin{array}[]{ll}\frac{\omega_{sat}}{\omega_{\odot}}&\omega_{\mathrm{sat}}\leqslant\omega\frac{\tau_{cz}}{\tau_{cz,\odot}}\\\
\frac{\omega\tau_{cz}}{\omega_{\odot}\tau_{cz,\odot}}&\omega_{\mathrm{sat}}>\omega\frac{\tau_{cz}}{\tau_{cz,\odot}}\end{array},\right.$
(3)

and tie the mass loss rate to the empirical scaling of $\dot{M}$ with X-ray
luminosity from Wood et al. (2005), with X-ray luminosity given by the
Pizzolato et al. (2003) scaling with Rossby number
($\mathrm{Ro}=P_{\mathrm{rot}}/\tau_{cz}$)

$\frac{\dot{M}}{\dot{M_{\odot}}}=\left(\frac{L_{bol}}{L_{bol,\odot}}\right)\left\\{\begin{array}[]{ll}\left(\frac{\omega_{sat}}{\omega_{\odot}}\right)^{2}&\omega_{\mathrm{sat}}\leqslant\omega\frac{\tau_{cz}}{\tau_{cz,\odot}}\\\
\left(\frac{\omega\tau_{cz}}{\omega_{\odot}\tau_{cz,\odot}}\right)^{2}&\omega_{\mathrm{sat}}>\omega\frac{\tau_{cz}}{\tau_{cz,\odot}}\end{array},\right.$
(4)

We do not include weakened magnetic braking (van Saders et al., 2016) as the
low-mass stars in our sample are not expected to reach the relevant Ro within
the age of the galactic disk (van Saders et al., 2019).

###  2.2 AM Redistribution

For the internal AM transport, we adopt the prescription for core-envelope
coupling as described in Denissenkov et al. (2010). The basic assumption of
this model, which was originally proposed by MacGregor & Brenner (1991) as the
double-zone model, is that the core and the envelope rotate rigidly but not
necessarily at the same rate. This assumption is roughly consistent with the
current rotational state of the solar interior (Denissenkov et al., 2010). The
rate at which the two zones are allowed to exchange angular momentum is
defined by the core-envelope coupling timescale $\tau_{c}$, which is assumed
to be constant along the evolution and a function of stellar mass as in Spada
& Lanzafame (2020)

$\tau_{c}=\tau_{c,\odot}\left(\frac{M_{*}}{M_{\odot}}\right)^{-\alpha}$ (5)

where $\tau_{c,\odot}$ is the solar rotational coupling timescale ($\approx
22$ Myr, Spada & Lanzafame 2020) and $\alpha$ is a power-law exponent of this
mass-dependent timescale for transport. This scaling was found to remain
consistent regardless of the choice of wind braking law and in good agreement
with the separate analysis of core-envelope re-coupling by Somers &
Pinsonneault (2016). Lanzafame & Spada (2015) found $\alpha=7.28$ and Spada &
Lanzafame (2020) refined this estimate to $\alpha=5.6$ using new data of the
clusters Praesepe and NGC 6811 that extended to lower mass stars. More
recently, Cao et al. (2023) found $\alpha=11.8$ by using spotted models to fit
the rotational sequences in the Pleiades and Praesepe. We leave it as a free
parameter in our calibration fits.

Thus, our model has six parameters, of which three are set as follows:
$P_{\mathrm{init}}$ is given by the Upper Sco rotational distribution shown in
Fig. 1; the disk-locking timescale $\tau_{\mathrm{disk}}$ is set to the age of
Upper Sco, 10 Myr; the solar rotational coupling timescale is fixed at 22 Myr.
We fit for the remaining three parameters: the normalization constant $f_{K}$
and the saturation threshold $\omega_{\mathrm{sat}}$ in the braking law, and
the exponent $\alpha$ in the core-envelope coupling timescale mass dependence.

To constrain $f_{K}$, $\omega_{\mathrm{sat}}$ and $\alpha$, we calibrate the
models such that they can reproduce the rotational distributions in the
Pleiades, Praesepe, M67 and the rotation period of the Sun at solar age.

####  2.2.1 Model Calibration

We split the cluster data into temperature bins with an equal number of
points; for each temperature bin, we divide the observed rotation periods into
two percentile groups, the 0-20th and the 20-80th groups. We initialize non-
spotted ($f_{\mathrm{spot}}=0\%$) tracks with masses between 0.18 $M_{\odot}$
and 0.6 $M_{\odot}$ with a 0.01 $M_{\odot}$ step and between 0.6 $M_{\odot}$
and 1.15 $M_{\odot}$ with a 0.5 $M_{\odot}$ step. We match the tracks
metallicity to that of the clusters: solar metallicity tracks are used for the
Pleiades and M67; $[\mathrm{Fe/H}]=+0.2$ tracks are used for Praesepe.

We evolve each track using values of $P_{\mathrm{init}}$ drawn from a 1-D
smoothing spline fit to the 10th and 50th percentiles in Upper Sco (Fig. 1) at
the stellar mass of the track and assume that the fast rotators behave as
solid body rotators (Denissenkov et al., 2010; Somers & Pinsonneault, 2016),
while median rotators are allowed to undergo core-envelope decoupling.
Furthermore, we treat fully convective stars as rigid rotators, since their
core is no longer distinct. For each $P_{\mathrm{init}}$ percentile, we
separately interpolate $T_{\mathrm{eff}}$ and $P_{\mathrm{rot}}$ as a function
of age and evaluate them at the age of the clusters. We obtain a gyrochrone by
further interpolating $P_{\mathrm{rot}}$ and $T_{\mathrm{eff}}$ across the
full stellar mass range.

We quantify agreement between the model and observed $P_{\mathrm{rot}}$ by
computing the
$\chi^{2}=\Sigma(P_{\mathrm{obs}}-P_{\mathrm{mod}})^{2}/\sigma^{2}_{P}$, where
$\sigma_{P}$ is the error on $P_{\mathrm{obs}}$ and is assumed to be 10% of
$P_{\mathrm{obs}}$ (Epstein & Pinsonneault, 2014). Since the models are also
calibrated to reproduce the solar rotation period ($P_{\odot}=25.4$ days,
Denissenkov et al. 2010) at solar age ($t_{\odot}=4.57$ Gyr, Bahcall et al.
1995), we normalize the $\chi^{2}$ computed for each temperature-period
percentile group in a cluster by the number of data points in the group such
that each cluster’s $\chi^{2}$ has an equal contribution to the total
$\chi_{\mathrm{tot}}^{2}=\chi_{\mathrm{Pleiades}}^{2}+\chi_{\mathrm{Praesepe}}^{2}+\chi_{\mathrm{M67}}^{2}+\chi_{\odot}^{2}$.

Thus, the fit includes 405 data points and three free parameters,
$f_{k},\,\omega_{\mathrm{sat}},\,\alpha$. The best-fitting values of the
parameters ($f_{K}=6.81$, $\omega_{\mathrm{sat}}=3.90\times 10^{-5}$ and
$\alpha=10.97$; $\chi^{2}_{\mathrm{tot}}=158$) are obtained by minimizing
$\chi^{2}_{\mathrm{tot}}$ through the differential evolution (DE) function
from the Python library yabox (Mier, 2017). The calibrated gyrochrones
evaluated at the clusters ages are shown in Fig. 2.

Figure 2: Two sets of gyrochrones launched from the Upper Sco rotation
distribution, corresponding to the 10th and 50th percentiles (colored blue and
orange, respectively) are shown in each plot against observed cluster member
rotational sequences. These gyrochrones are fitted to the cluster’s observed
periods according to matching bins (0-20th percentiles in blue, 20-80th
percentiles in orange). Shaded regions around each gyrochrone account for the
uncertainty in the cluster’s age at which the models are being evaluated. The
bottom right panel shows the rotational evolution of the Sun up to its current
age (indicated as a black marker) as predicted by our model and the Spada &
Lanzafame (2020) model.

###  2.3 White Dwarf Cosmochronology

White dwarfs are the final evolutionary stage of stars with initial masses of
less than roughly 8-10 $M_{\odot}$. Because they no longer undergo nuclear
fusion in their cores, their evolution consists of a cooling phase dominated
by the leaking of residual thermal heat from the non-degenerate ions in the
electron-degenerate core. The key idea behind using WDs as cosmochronometers
is that their effective temperature and mass map uniquely onto a single
cooling age. The effective temperature and surface gravity of the WD, which
yields its mass, can be derived from either spectroscopy or photometry coupled
with model atmospheres. Once the cooling age has been determined using the WD
atmospheric parameters, the next step is to estimate its progenitor MS and
post-MS lifetimes. This is done by using IFMRs (e.g., Cummings et al. 2018) to
correlate the final WD mass to initial ZAMS masses, from which the progenitor
lifetimes are estimated. The total age of the WD is given by the sum of the
cooling age and progenitor MS and post-MS lifetimes.

Due to the lack of spectroscopic observations for all the WDs in the sample,
we use spectral energy distribution (SED) fitting of the mean fluxes in
different bands from all-sky surveys including Gaia, the Sloan Digital Sky
Survey (SDSS), the Panoramic Survey Telescope and Rapid Response System
(PanSTARRS) and the SkyMapper Southern Survey. We compute total ages of the
WDs following the methods outlined in Heintz et al. (2022), which we summarize
below for completeness.

####  2.3.1 Fitting Routine

We convert model DA white dwarf spectra, spanning effective temperatures of
3000 K to $40{,}000$ K and surface gravities of 6.25 to 9.5 dex, from Koester
(2010) to synthetic fluxes by using the sensitivity of each band-pass and the
appropriate AB magnitude zeropoints. The assumption that all WDs are DAs can
introduce systematic mass errors of $10-15$% (Giammichele et al., 2012)
however, due to the lack of spectral information for the majority of the WDs,
this is the simplest assumption that we can adopt. The observed magnitudes are
also converted to absolute fluxes at 10 pc using AB zeropoints and the
weighted mean parallax of the binary from Gaia. For SDSS $u$ and $z$, the
magnitudes are shifted $0.04$ and $0.02$ mag, respectively, to account for the
shift relative to the AB mag system (Eisenstein et al., 2006). The weighted
mean parallax of the binary system is dominated by the brighter MS star and is
on average six times more precise than the individual WD parallax which in
turn allows for a more precise age determination. The observed fluxes are de-
reddened using extinction values from Gentile Fusillo et al. (2021), which are
obtained using the 3D extinction maps from Lallement et al. (2022). These
observed de-reddened fluxes are related to the model synthetic fluxes through
the radius of the WD through the following relation

$F_{X}=\frac{R^{2}}{(3.08568\times
10^{19}\mathrm{cm})^{2}}F_{X,\mathrm{mod}}(\mathrm{T_{eff},log\ g})$ (6)

where $F_{X}$ is the observed flux at 10 pc in bandpass $X$, $R$ is the radius
of the WD in cm, and $F_{X,mod}$ is the synthetic flux in bandpass $X$ that is
a function of effective temperature and surface gravity.

We use a Markov Chain Monte Carlo approach and make use of the python package
emcee (Foreman-Mackey et al., 2013) to get best-fit temperatures and radii,
which are represented by the 50th percentiles of the MCMC posterior
distributions. These are converted to surface gravities, masses, and cooling
ages using the cooling models from Bédard et al. (2020), which assume that the
WD inherits a “thick” hydrogen layer from its progenitor and thus retains its
DA spectral type throughout its life. We use flat priors for the temperatures
and surface gravities that cover the full range of the models. A lower limit
on the magnitude uncertainties of 0.03 mag is set to account for systematics
in the conversion of magnitudes to average fluxes. We also impose a lower
limit on the uncertainty on the surface gravities of 0.03 dex and a lower
uncertainty of 1.2% on the effective temperatures to account for any unknown
systematics in the models (e.g., Liebert et al. 2005).

####  2.3.2 Photometric Cleaning

There is often a large luminosity contrast between the WD and the MS star in
the binary, therefore an added measure of cleaning of the photometry is needed
to obtain reliable parameters. We first remove photometry that is flagged for
several issues in SDSS, PanSTARRS, and SkyMapper. We remove photometry from
SDSS with EDGE, PEAKCENTER, SATUR, and NOTCHECKED flags. We only use
photometry from PanSTARRS with rank detections of 0 or 1. We also remove any
photometry from SkyMapper that has any raised flags.

Going beyond our method in Heintz et al. (2022), we also systematically remove
photometry that is not consistent with the Gaia fluxes. To do this, we run the
fitting routine described in Section 2.3.1. Then, we fit a line to the
residuals of the resulting SED fit to account for an incorrect temperature
estimate. We compare the residuals of this linear fit to the largest absolute
percent deviation between Gaia $\mathrm{G,G_{BP},\ and\ G_{RP}}$. Any
photometric bands that are more than $3\sigma$ away from this absolute percent
deviation are removed, where $\sigma$ is the uncertainty for the individual
photometric band. A new linear fit to the SED residuals is performed with
these bands removed. Then, a new comparison to the largest absolute percent
deviation of the Gaia bands is performed, including the previously removed
bands. This iterative process continues until a consistent list of photometric
bands are removed, and a final SED fit is performed. SDSS $u$-band is not
subjected to this stage of photometric cleaning since it can be a strong
indicator of whether the WD is a DA or non-DA due to the presence of the
Balmer jump in DA WDs. The larger residual of SDSS $u$ can be indicative of a
non-DA and not because the photometry is suspect (e.g. Bergeron et al. 2019).
We find that $~{}30$ WDs show anomalously discrepant SDSS $u$-band photometry
that suggests there may be some non-DA in the sample. However, since $>65$% of
WDs in the Gaia magnitude-limited samples are DAs (Kleinman et al., 2013),
adopting a DA model is a reasonable assumption.

Figure 3: Top: Contribution of the cooling age to the total age of the WD as a
function of mass. Bottom: WD total age as a function of mass; the
uncertainties on the WD ages have been inflated by empirically determined
factors (Heintz et al., 2022) to compensate for systematics on the total ages.

Moreover, Heintz et al. (2022) found that, when assuming DA spectral types for
all the WD in their sample, the ages are good to 25% and provided error
inflation factors to account for inaccuracies in the WD ages, including the
assumption of an incorrect spectral type.

####  2.3.3 Progenitor Lifetimes

To get the progenitor lifetimes of the WDs, we use an IFMR from Heintz et al.
(2022) which uses a theoretically motivated shape to the IFMR from Fields et
al. (2016), fit to WDs in solar metallicity clusters (Cummings et al., 2015,
2016), in conjunction with the stellar evolutionary tracks from Modules for
Experiments in Stellar Astrophysics (MESA, Paxton et al. 2011, Paxton et al.
2013, Paxton et al. 2015). The errors on these values are determined by using
the $1\sigma$ uncertainties on the WD mass to determine an upper and lower MS
mass. The difference between the central value and the upper and lower MS mass
are quoted as upper and lower errors, respectively. The same process is
carried out for the progenitor lifetimes as well.

Figure 4: MS stars in the full wide WD+MS sample in a
T${}_{\mathrm{eff}}-$Prot space. 1985 rotation periods are retrieved from ZTF;
384 periods are from ATLAS; 102 periods are from Gaia DR3; K2 provides 82
periods; 78 periods are from ASAS-SN; 23 periods are retrieved from Kepler; 15
periods are from Santos et al. (2021); 20 periods are from Santos et al.
(2020); 9 periods are from MEarth North and 3 periods are from MEarth South.
The most opaque markers represent the MS stars in the WD + MS binaries that
made it into the final selection (327 systems) described in Sec. 3. The
McQuillan Kepler field (McQuillan et al., 2014a) of 34,030 MS stars below
$6500\,\mathrm{K}$ is shown in the background, together with the additional
detection of $15{,}640$ M and K-type Kepler stars by Santos et al. (2019).

####  2.3.4 Precision of Total Ages

The total age of the WDs in the sample is primarily determined by their mass,
and therefore uncertainties in the WD mass have a significant impact on the
accuracy of the estimated total ages. Heintz et al. (2022) found that the
total ages derived from WDs with $M<0.63$ $M_{\odot}$ become very noisy.
Moreover, accurately determining the ages of low-mass WDs poses a challenge
due to the lack of well-defined constraints at the lower end of the IFMR. WDs
with masses below $0.575$ $M_{\odot}$ may not provide reliable age estimates
since they might not have formed through the evolution of a single star. While
their cooling ages can offer a minimum estimate of their total age, it is
essential to consider the possibility that their low mass is a result of
contamination in the photometry, especially if they are formed through binary
interactions.

Thus, we adopt the following: we ignore total ages obtained from WDs with
$M<0.575$ $M_{\odot}$; for WDs with $M>0.575$ $M_{\odot}$, we adopt the total
ages computed as the sum of the cooling age and progenitor lifetimes. To avoid
contamination from the MS companion, we filter sources with a Gaia BP-RP
corrected excess factor $<0.1$ (Riello et al., 2021). Because formal age
uncertainties are often underestimated for higher-mass WDs, we inflate the age
uncertainties by a factor computed following the comparison to wide WD+WD
described in Heintz et al. (2022).

Our final sample predominantly comprises of massive WDs (with masses greater
than $0.67$ $M_{\odot}$), in which the total ages are primarily influenced by
cooling rather than the IFMR, as illustrated in Fig. 3. This results in an
average age uncertainty of 10% prior to inflation and 20% post-inflation.

##  3 Sample Selection

We construct the wide binary sample using the El-Badry et al. (2021) catalog,
which contains 1.3 (1.1) million binaries with a $>$ 90% ($>$ 99%) probability
of being bound. Stars are classified as MS or WD based on their location on
the Gaia color-absolute magnitude diagram (CMD). The absolute magnitude is
defined as $M_{G}=G+5\,\mathrm{log}(\omega)-10$, where $G$ is the $G$-band
mean magnitude and $\omega$ is the parallax in mas; stars with
$M_{G}>3.25(G_{\mathrm{BP}}-G_{\mathrm{RP}})+9.625$ are classified as WDs; all
other stars with measured $G_{\mathrm{BP}}-G_{\mathrm{RP}}$ are classified as
MS stars (El-Badry & Rix, 2018). We only select systems containing a WD and an
MS star and find $22{,}563$ such systems.

Figure 5: Properties of the 327 binaries with precise WD ages in the sample.
We distinguish fully convective (FC) MS stars from partially convective (PC)
MS stars based on their absolute Gaia magnitude, $M_{G}$, and Gaia BP-RP color
measurements using Jao’s gap (Jao et al., 2018). Figure 6: Gaia color-
magnitude diagram showing the full data sample (2701 systems, grey markers) of
WD + MS binaries for which we estimated ages from the WD companions. We
highlight the 327 binaries with the most precise WD total ages (average
uncertainty $<10\%$). The Jao et al. (2018) gap is shown as a solid black
line.

We search for rotation periods of these MS stars in several rotation surveys,
including the Asteroid Terrestrial-impact Last Alert System (ATLAS) variable
stars database (Heinze et al., 2018), the All-Sky Automated Survey for
SuperNovae (ASAS-SN) variable star database (Shappee et al., 2014; Jayasinghe
et al., 2018), the Calar Alto high-Resolution search for M dwarfs with
Exoearths with Near-infrared and optical Echelle Spectrographs (CARMENES)
catalog (Díez Alonso et al., 2019), the Gaia third Data Release (DR3, Gaia
Collaboration (2022)), the Hungarian-made Automated Telescope Network (HATNet)
Exoplanet survey (Hartman et al., 2010), the Kilodegree Extremely Little
Telescope (KELT) database (Oelkers et al., 2018), the Kepler (McQuillan et
al., 2014b; Santos et al., 2020, 2021) space mission, the K2 space mission
(Reinhold & Hekker, 2020), the MEarth Observatory (Newton et al., 2016, 2018)
and the Zwicky Transient Facility (ZTF, Chen et al., 2020; Lu et al., 2022).
We find 5005 binaries that feature an MS star with a measured rotation period,
with approximately 85% of the rotation periods sourced from the ZTF catalog.
Lu et al. (2022) found that nearly 50% of stars with ZTF periods $<$ 10 days
are likely to be incorrect, therefore we exclude all binaries with such ZTF
fast rotators. This reduces the number of WD + MS with measured rotation
periods to 2701.

We cross-match these MS stars with various spectroscopic catalogs, including
the Galactic Archaeology with HERMES (GALAH) survey (Buder et al., 2018), the
Large Sky Area Multi-Object Fibre Spectroscopic Telescope (LAMOST) (Zhong et
al., 2020), the Apache Point Observatory Galactic Evolution Experiment
(APOGEE) survey (Abdurro’uf et al., 2022), and Gaia DR3 (Recio-Blanco et al.,
2022). We retrieve spectroscopic properties for 430 MS stars. We use
spectroscopic data when available, but do not require spectroscopy to be
included in the sample.

We obtain WD ages for these 2701 binaries and select 1065 binaries with WD
mass $>0.575$ $M_{\odot}$ and Gaia BP-RP excess factor $<0.1$ (see Sec. 2.3.4
for more details). Finally, we require WD ages to have the average of the
uninflated low and upper uncertainties less than 20%. The final sample
contains 327 binaries with precise WD ages, which we report in Table
LABEL:samp in the Appendix. The distribution of rotation periods across the
different catalogs used to create the sample is presented in Fig. 4. Effective
temperatures of the MS stars were computed from GBP-GRP measurements using a
polynomial fit taken from Curtis et al. (2020), assuming no extinction. Their
color-temperature relation was constructed using nearby benchmark stars,
including a sample of low mass stars with 3056 K $<$ $T_{\mathrm{eff}}$ $<$
4131 K and $-0.54$ dex $<~{}\mathrm{[Fe/H]}~{}<$ +0.53 dex from Mann et al.
(2015). The basic properties and CMD of the sample are shown in Fig. 5 and 6,
respectively.

##  4 Results and Discussion

###  4.1 Model Assessment

In general, our gyrochrones reasonably match the rotation sequences observed
in the clusters, with the exception of K-dwarfs in Praesepe. Specifically, for
K-dwarfs with effective temperatures between $4250\,\mathrm{K}$ and
$5000\,\mathrm{K}$, our models predict rotation periods that are a few days
shorter than those observed. We note that there are no super-solar metallicity
atmospheric tables available from Allard et al. (1997), therefore we have
approximated the metal rich case with the solar atmosphere. However, we do not
get very different best-fit parameters if we use solar metallicity tracks for
all clusters, including Praesepe. At $\sim 670$ Myr, these K-dwarfs experience
core-envelope decoupling. The fitted coupling timescale in this work has a
strong mass dependence, therefore, these low-mass stars take an extended
period before resuming their spin-down. Here we adopt a constant, rotation-
rate independent coupling timescale, although we expect this timescale to
change with time. We suspect that this coupling timescale prescription is
contributing to the morphology mismatch between the cluster and the model,
which appears to be a more significant concern than differences in atmosphere
or metallicity. Exploring more nuanced prescriptions that depend on
evolutionary state and rotation rate are well-motivated, but beyond the scope
of this work.

The bottom right panel in Fig. 2 shows that our models predict the rotation of
the Sun at solar age within $\sim 1$ day. Our models reproduce the 4 Gyr M67
cluster sequence in the cool stars Dungee et al. (2022) (Fig 2, lower left
panel).

Another assessment of our models is presented in Fig. 7, where we show a set
of gyrochrones against the ZTF rotation period catalog from Lu et al. (2022).
The ZTF distribution shows an overdensity of fully convective stars rotating
slowly ($P_{\mathrm{rot}}>40$ days) past the closing of the intermediate
period gap, a period dearth in the
T${}_{\mathrm{eff}}-\mathrm{P}_{\mathrm{rot}}$ space for low-mass stars that
was first detected with Kepler by McQuillan et al. (2013). This increase in
rotation period for such stars is also predicted by gyrochrones older than 2
Gyr. We note that while the apparent agreement with the ZTF is good, the
earlier McQuillan et al. (2014a) sample contains lower amplitude, more slowly
rotating stars at these temperatures that are not fit by our gyrochrones.
These stars are presumably older than 4 Gyr, and suggest that our core-
envelope coupling prescription may be overly simplistic. Even if this is the
case, it does not fundamentally alter the conclusions of this paper.

The best-fit parameters obtained from the clusters fit are in agreement with
those from Cao et al. (2023) ($f_{k}=9.79\pm 0.37$,
$\omega_{\mathrm{sat}}=(3.466\pm 0.085)\times 10^{5}\,\mathrm{rad\,s^{-1}}$
and $\alpha=11.8\pm 1.0$), who calibrated spotted models (Somers et al., 2020)
to the Pleiades and Praesepe. We find a higher value of $\alpha$ compared to
the value reported in Spada & Lanzafame (2020) ($\alpha\approx 5.6$), although
their models did not include stellar spots (which alter the mass-temperature
relation and therefore apparent mass-dependece of $\alpha$) and assumed a
fixed initial rotation period of 8 days for all stellar masses rather than a
distribution of $P_{init}$. Moreover, their two-zone model has been calibrated
for stellar masses down to $\approx 0.4M_{\odot}$.

Figure 7: ZTF stars from Lu et al. (2022) are plotted as grey circles. Our
gyrochrones are shown as solid lines and color-coded by their age. The
gyrochrones are constructed using a non-spotted solar-metallicity grid and
launched from the median $P_{\mathrm{init}}$ percentile in Upper Sco.

###  4.2 A spike in $P_{\mathrm{rot}}$ at the fully convective boundary

Using the best-fit parameters, we construct gyrochrones to predict the
rotation period of the stars in our sample at the age inferred from their WD
companions. To make a direct comparison between model and observed rotation
periods, we create a grid of $f_{\mathrm{spot}}=0\%$, solar-metallicity tracks
for stellar masses between 0.18 and 1.15 $M_{\odot}$. For each stellar mass,
we launch a track with $P_{\mathrm{init}}$ values between the
$5^{\mathrm{th}}$ and the $95^{\mathrm{th}}$ percentiles of the Upper Sco
period distribution (Fig. 1). For each star in our data sample, the model
rotation period $\overline{P_{w}}$ is computed as the likelihood weighted
average of the rotation periods in the grid

$\overline{P_{w}}=\frac{\sum_{i=1}^{n}\Delta t\Delta
m\mathcal{L}_{i}P_{i}}{\sum_{i=1}^{n}\Delta t\Delta m\mathcal{L}_{i}},$ (7)

where $\Delta t$ and $\Delta m$ are the time and mass increments between each
point on our non-uniformly sampled model grid, respectively; $P_{i}$ is the
$i^{\mathrm{th}}$ model rotation period in the grid; $\mathcal{L}_{i}$ is its
corresponding likelihood. The likelihood function accounts for the uncertainty
on the WD age, taken as the average of the inflated (as per Heintz et al.
2022) lower and upper uncertainties, and the effective temperature of the MS
star, computed as the root sum of the squares of the typical temperature
precision ($\pm 50$ K) and the uncertainty obtained from propagation of the
Gaia $G_{\mathrm{BP}}-G_{\mathrm{RP}}$ uncertainties involved in the
color-$T_{\mathrm{eff}}$ relation (Curtis et al., 2020). The discrepancy
between the observed and model rotation periods is quantified as the
difference between $P_{\mathrm{rot}}$ and $\overline{P_{w}}$, i.e.
$\lvert\Delta P_{\mathrm{rot}}\rvert$, divided by the likelihood weighted
standard deviation $\sigma_{w}$ of the model periods, defined as

$\sigma_{w}=\sqrt{\frac{\sum_{\mathrm{i}=1}^{\mathrm{n}}\Delta t\Delta
m\mathcal{L}_{\mathrm{i}}\left(\mathrm{P}_{\mathrm{rot}}-\overline{P_{w}}\right)^{2}}{\sum_{\mathrm{i}=1}^{\mathrm{n}}\Delta
t\Delta m\mathcal{L}_{\mathrm{i}}}}$ (8)

The lower the ratio $\lvert\Delta P_{\mathrm{rot}}/\sigma_{w}\rvert$, the
smaller the discrepancy is. The results are presented in Fig. 8. 65% of the
rotation periods predicted by non-spotted, solar-metallicity models are within
$3\sigma_{w}$ from the observed rotation periods and fall in the region
bounded by the gyrochrones computed at the lower and upper WD age bounds in
each bin.

At the fully convective boundary, our sample shows a rapid increase in the
rotation period of MS stars with WD ages up to 7.5 Gyr. The same trend is
confirmed by the gyrochrones ([Fe/H]$=+0.0$, $f_{\mathrm{spot}}=0\%$), which
span periods between 30 and 80 days across a narrow temperature range ($\sim
50$ K) for ages up to 2.5 Gyr. Similarly, between 2.5 and 5 Gyr, the models
show a sharp rise in rotation period from 50 to 170 days and up to 200 days
between 5 and 7.5 Gyr. Thus, both the models and data suggest that, at the
fully convective boundary, stars with relatively long rotation periods are not
necessarily old, in contrast to the standard picture of stellar spin-down. In
addition, at this boundary, a measured rotation period cannot be uniquely
associated (within reasonable observed errors in $\mathrm{T_{eff}}$ of
$\simeq$50 K) to a single gyrochrone – rather, gyrochrones spanning several
billions of years all provide reasonable matches to the observed
($P_{\mathrm{rot}},\,T_{\mathrm{eff}}$) combination. This significantly
inflates the age uncertainties on rotation-based ages in this
$T_{\mathrm{eff}}$ range as the rotation period of a star along this vertical
incline is predicted, within 1$\sigma$, by gyrochrones between 2 Gyr and 8
Gyr.

Beyond the fully convective boundary, our models return to a reasonable
behavior without distinct features. This suggests the feasibility of applying
gyrochronology to the coolest fully convective stars, at least from a model
standpoint.

Figure 8: MS stars binned by their companion WDs ages. The grey shaded regions
show the range of rotation periods spanned by $f_{\mathrm{spot}}=0$, solar-
metallicity gyrochrones launched from the median percentile of Upper Sco at
the WD age bounds, which are plotted as black solid line. Gyrochrones with
$f_{\mathrm{spot}}=25\%$ and $f_{\mathrm{spot}}=50\%$ launched with the same
$P_{\mathrm{init}}$ at the same ages are plotted as black dashed and dotted
lines, respectively, and the corresponding region of allowed rotation periods
is shown in light pink and green. Stars are color coded by $|\Delta
P_{\mathrm{rot}}/\sigma_{w}|$ (see Sec. 4.2). Blue points show good agreement,
while the redder points show worse agreement between the observed and the
predicted rotation periods.

###  4.3 Model description of the spike

Stellar interior theory predicts that as the stellar mass decreases, the
convection zone (CZ) deepens in the interior of the star until the star
becomes fully convective at $\sim 0.35\,\mathrm{M}_{\odot}$. The convective
overturn timescale refers to the characteristic timescale of convective
motions. In this work, we compute the characteristic convective overturn
timescale as the local $\tau_{cz}=H_{P}/v$, where $H_{P}$ is the pressure
scale height at the base of the convective zone and $v$ is the convective
velocity (from a mixing length theory of convection) one pressure scale height
above the convective zone boundary. As we approach the fully convective
boundary, the convective envelope gets deeper, occupying a larger part of the
total stellar mass, the pressure scale height increases and the convective
velocity decreases, as predicted by the mixing-length theory (Böhm-Vitense,
1958). This leads to an increase in $\tau_{cz}$.

However, it has been shown that the behavior of models near the transition to
the fully convective regime is not smooth. The van Saders & Pinsonneault
(2012a) instability predicts that low-mass stars at the boundary undergo non-
equilibrium 3He burning, which gives rise to a small convective core separated
from the convective envelope by a thin radiative zone. As the amount of
central 3He increases, the convective regions grow in mass and the convective
envelope deepens, until they merge, leading to a fully convective episode.
This process repeats until the total 3He concentration is high enough that the
star remains fully convective.

Figure 9: The sizes of the convective zone relative to the total size of the
star in mass and radius coordinates as a function of time are plotted in the
top and middle panels, respectively. The bottom panel shows the convective
overturn timescale normalized by the solar value as a function of time. The
black arrow shows the jump between partially convective (PC) and fully
convective (FC) stars. In all panels, the saw-toothed curves represent stars
undergoing fully convective episodes driven by non-equilibrium 3He burning
(van Saders & Pinsonneault, 2012a). All tracks have solar-metallicity and a
25% spot covering fraction.

In the models used in this work, the van Saders & Pinsonneault (2012a)
instability occurs for masses between 0.30 $M_{\odot}$ and 0.37 $M_{\odot}$,
depending on the metallicity and spot covering fraction. For instance, in a
solar-metallicity, $f_{\mathrm{spot}}=25\%$ model grid, the van Saders &
Pinsonneault (2012a) instability affects models in the $0.31-0.34$ $M_{\odot}$
range and the fully convective boundary is at $0.31$ $M_{\odot}$. This is
shown in Fig. 9. In the top panel, we see that as we move from a
$0.4\,M_{\odot}$ partially convective star to $0.31\,M_{\odot}$, the
contribution of the mass of the convective zone to the total stellar mass
increases, until the star is fully convective and $M_{cz}/M_{tot}=1$.
Similarly, the middle panel shows that the base of the convective zone is
eating downward in mass and deepening in the interior as we approach the fully
convective boundary, which makes $\tau_{\mathrm{cz}}$ longer. Furthermore, due
to fully convective episodes initiated by non-equilibrium 3He burning, the
convective zone base of stars in the range $0.34-0.31$ $M_{\odot}$ suddenly
moves from a fractional depth $R_{cz}/R_{tot}=0.4-0.3$ to the center of the
star $R_{cz}=0$, which results in discontinuous jumps in $\tau_{\mathrm{cz}}$,
as seen in the bottom panel. The convective overturn timescale maxima are in
phase with drops in $R_{cz}/R_{tot}$ and peaks in $M_{cz}/M_{tot}$ and
correspond to fully convective episodes.

We suggest that it is the rise in $\tau_{\mathrm{cz}}$ due to differences in
the structure of partially and fully convective stars that causes the vertical
feature in $P_{\mathrm{rot}}$ for low-mass stars older than 1 Gyr, and that
its sharpness is caused by the fact that the CZ boundary does not smoothly
move towards the core (as a function of mass) until the star is fully
convective, but rather jumps from a partially convective configuration in
stars undergoing the van Saders & Pinsonneault (2012a) instability.

To show how $\tau_{\mathrm{cz}}$ affects stellar spin down at the fully
convective boundary, we consider the relative contribution of all terms
involved in the braking law. We evaluate the weight of the factors affecting
$dJ/dt$ as a function of stellar mass at the median age of the data sample
($\sim 2$ Gyr). Since the majority of the stars in the sample are M dwarfs
that slowly evolve on the main sequence and the structure is very stable after
the fully convective episodes, the contribution of the structural terms in the
braking law do not significantly depend on the age at which they are
evaluated. The results are shown in Fig. 10: while the stellar mass, radius,
luminosity and pressure factors vary smoothly for stars with masses between
$1.15M_{\odot}$ and $0.2M_{\odot}$, the convective overturn timescale factor
changes abruptly at the fully convective boundary, between $0.33M_{\odot}$ and
$0.31M_{\odot}$. In this narrow mass range, $\tau_{\mathrm{cz}}$ rapidly
increases as we approach $0.33\,M_{\odot}$, reaches a peak at
$0.31\,M_{\odot}$, and then drops modestly. No other stellar property exhibits
such a distinct feature at this boundary.

Figure 10: Contributions of the stellar mass $M$, radius $R$, luminosity $L$,
photospheric pressure $P_{\mathrm{phot}}$ and convective overturn timescale
$\tau_{cz}$ factors to the total $dJ/dt$ using solar-metallicity models with
25% spot covering fraction. The fully convective boundary is at
$M=0.31M_{\odot}$.

The peak in the $\tau_{\mathrm{cz}}$ curve in the bottom panel of Fig. 10 is
reached by the model with the largest mass — and thus largest $H_{P}$ — that
becomes fully convective, which is the $0.31\,M_{\odot}$ model in the solar-
metallicity, $f_{\mathrm{spot}}=25\%$ model grid. Below this point, stars are
fully convective since they never have a radiative core. However, they are
also smaller in radius and mass of the CZ, therefore $\tau_{\mathrm{cz}}$
drops.

####  4.3.1 Calibrations for $\tau_{cz}$ across the fully convective boundary

Literature sources do not agree on a single method for computing the
convective overturn timescale from a stellar model, but we argue here that the
sharp increase in the convective overturn timescale that drives rapid braking
is a ubiquitous feature across common prescriptions for inferring the
timescale.

Figure 11: Comparisons of $\tau_{cz}$ values as a function of mass (left
panel) and effective temperature (right panel). We show our fiducial, solar
metallicity unspotted stellar models as a solid black curve. Models with
identical physics but a 25% spot covering fraction are shown in gray. The
Wright et al. (2018) empirical calibration is shown in purple, and the Barnes
& Kim (2010) “global” model-based $\tau_{cz}$ are shown in turquoise. Metal-
rich ($\textrm{[Fe/H]}=+0.2$) models are shown as the dotted curve.

When using models to compute convective overturn timescales, there are two
primary approaches: the “local” prescription (used here) and a “global”
prescription, where one instead computes some suitable average $\tau_{cz}$
over the entire convection zone. Both approaches yield fundamentally the same
behavior modulo a scale factor, since the deep portions of the CZ probed by
the local approach are also the most heavily weighted in the global average
(Kim & Demarque, 1996). We show in Fig. 11 that our fiducial model, which uses
a local approach, displays fundamentally the same behavior as that in Barnes &
Kim (2010), which utilizes a global approach. Once normalized by their
respective solar convective overturn timescales, both methods show the same
behavior as a function of mass and a rapid increase in $\tau_{cz}$ near the
fully convective boundary as the $\tau_{cz}$ computation begins to probe the
structure of the near core.

While the location in mass (or temperature) of the sharp rise in $\tau_{cz}$
depends on properties like the metallicity and spot covering fraction, both
produce only modest shifts in the precise location of the rise in $\tau_{cz}$,
also shown in Fig. 11. In the case of metallicity, stars are more convective
at higher metallicity but fixed mass, shifting the rise in $\tau_{cz}$ and
onset of full convection to slightly higher masses in metal-rich stars,
although this vertical feature does not significantly move in temperature, as
shown in Fig. 12.

Figure 12: Gyrochrones constructed using a metal-poor (left), a solar
metallicity (center) and a metal-rich (right) non-spotted model grid. We show
gyrochrones for ages between 0.1 and 10 Gyr, with a step of 0.5 Gyr. While the
location of the steep rise in $P_{\mathrm{rot}}$ is shifted to slightly higher
masses in metal-rich stars, the feature does not vary significantly in
temperature and occurs at $\approx$$3500$ K across all model grids.

Adding spots to the surface of the model — which may be an important component
in modeling young, low-mass stars (Cao et al., 2023) — decreases the observed
effective temperature, with only modest impacts on the structure of the deep
interior (see Somers & Pinsonneault, 2015), which shifts the onset of deep
convection and large $\tau_{cz}$ values to lower effective temperatures but
not significantly lower stellar masses. We find that the maximum stellar mass
undergoing fully convective episodes decreases with increasing spot covering
fraction: 0.34 $M_{\odot}$ for 0% spot covering fraction; 0.33 $M_{\odot}$ for
25% spot covering fraction; 0.32 $M_{\odot}$ for 50% spot covering fraction.
Furthermore, a higher spot covering fraction leads to a lower peak rotation
period, which is a direct consequence of the onset of the 3He instability
shifting to lower masses and therefore lower convective overturn timescales,
as shown in Fig. 13. While the precise location of the steep rise in
$\tau_{cz}$ depends on the model physics, the existence of a steep rise does
not.

Figure 13: Solar-metallicity gyrochrones up to 10 Gyr constructed using three
model grids with different spot covering fractions $f_{\mathrm{spot}}$. The
sharp rise in rotation period occurs for the same 0.30-0.34 $M_{\odot}$ mass
range across the three model grids, however the location of this feature
varies across a $\sim$$350$ K temperature window between the three model grids
with different spot-covering fractions.

Finally, attempts to develop purely empirical calibrations of $\tau_{cz}$ also
predict an increase in overturn timescale across the fully convective
boundary. Wright et al. (2018) made the assumption that fully convective M
dwarfs obeyed the same Ro-activity relation as partially convective stars, and
then found the values of $\tau_{cz}$ as a function of color (mass) that
minimized scatter in the Ro$-$X-ray luminosity relation. Although the
resulting relation does not trace the model predictions exactly (Fig. 11), it
does indicate a reasonably steeply rising $\tau_{cz}$ across the fully
convective boundary.

###  4.4 A Few Complications

We find that 46% of the MS stars shown in Fig. 8 have $\Delta
P_{\mathrm{rot}}/\sigma_{w}>3\sigma_{w}$, i.e. their rotation rate as
predicted by our spin-down model is at odds with the apparent system age given
by the WD. This percentage goes down to 26% if we exclude MS stars with an
effective temperature within 200 K from the fully convective boundary. We
argue that these discrepancies may have multiple potential sources.

####  4.4.1 Stellar Spots

The model rotation periods of M dwarfs in our sample shown in Fig. 8 were
obtained using a model grid with solar metallicity and
$f_{\mathrm{spot}}=0\%$. However, by adopting a model grid with a non-zero
spot covering fraction, we can extend the region probed by our models to
cooler temperatures, as shown in Fig. 13. For instance, Fig. 8 shows that MS
stars cooler than 3200 K that fall outside of the grey region bounded by the
$f_{\mathrm{spot}}=0\%$ gyrochrones, are found within the region bounded by
the $f_{\mathrm{spot}}=25\%$ and $f_{\mathrm{spot}}=50\%$ gyrochrones.
Therefore, knowing the $f_{\mathrm{spot}}$ of these stars would be helpful to
choose the most appropriate tracks to model their spin-down and improve the
comparison between the predicted and observed rotation periods.

####  4.4.2 Metallicity

Metallicity may also be responsible for some of the discrepancies observed in
our data. In the absence of spectroscopic data, we have assumed a solar-
metallicity for our sample. Modern braking laws, including the van Saders &
Pinsonneault (2013) prescription used in this work, suggest that metallicity
can have a strong impact on the rotational evolution of low-mass stars
(Claytor et al., 2020; Amard & Matt, 2020). The convective overturn timescale
is a direct consequence of the stellar structure and therefore is affected by
the chemical composition of the star. Stars with a high abundance of elements
heavier than He have a higher opacity, which steepens radiative temperature
gradients, leading to deeper convective envelopes (van Saders & Pinsonneault,
2012a; Amard et al., 2019), higher pressure scale height and, therefore, a
longer $\tau_{\mathrm{cz}}$ and more efficient braking. We find that 94% of
the stars with $\lvert\Delta P_{\mathrm{rot}}\rvert/\sigma_{w}>3$ have
observed rotation periods that are longer than the model periods (i.e. a
positive $\Delta P_{\mathrm{rot}}$). This percentage does not significantly
change when not accounting for stars within 200 K from the fully convective
boundary (91%).

Since higher metallicity leads to stronger braking, by adopting a metal-rich
model grid we expect to recover longer model rotation period that may provide
a better match to the measured rotation periods. Fig. 14 shows the difference
between computing $\Delta\mathrm{P}_{\mathrm{rot}}/\sigma_{w}$ using a solar-
metallicity model grid, like the one used in Fig. 8, versus a higher-
metallicity model grid. By adopting an [Fe/H] = +0.2 grid, we obtain an
improvement in the predictions of rotation periods for only a handful of stars
with $\Delta P_{\mathrm{rot}}/\sigma_{w}>3$ when computed with a solar-
metallicity grid, as shown in Fig. 14. Nevertheless, marginalizing over
metallicity when constructing a model grid may improve the predictions of the
rotation periods for systems with known metallicities.

Figure 14: Comparison between $\Delta\mathrm{P}_{\mathrm{rot}}/\sigma_{w}$
computed for the MS stars in our sample using a [Fe/H] = +0.0 and a [Fe/H] =
+0.2 model grids. Stars are color-coded by their observed rotation period. A
1:1 line is plotted as a black dashed line. The hatched region in the bottom
right corner highlights stars for which
$\Delta\mathrm{P}_{\mathrm{rot}}/\sigma_{w}$ decreases to $3\sigma_{w}$ or
lower if a higher metallicity model grid is adopted instead of a solar
metallicity one. Only a handful stars show improved values of
$\Delta\mathrm{P}_{\mathrm{rot}}/\sigma_{w}$ when using an [Fe/H] = +0.2 model
grid.

Of the full sample, only 28 MS stars have measured metallicities. We find no
evident trend with metallicity in those stars where measurements are
available.

####  4.4.3 WD Age Resets in Triple Systems

We have neglected the possibility of triple systems where the inner WD binary
merges and resets the apparent system age. Modeling the evolution of single
star and binary populations has shown that the age of a merger remnant can be
underestimated by a factor of three to five if single star evolution is
assumed for a WD (Temmink et al., 2020). The same study found that WDs from
binary mergers make up about 10-30% of all observable single WDs and 30-50% of
massive ($>$0.9 $M_{\odot}$) WDs. Similarly, Heintz et al. (2022) estimated
that $21-36$% of WD+WD pairs likely started as a triple system. These values
are consistent with the fraction of binaries in our sample ($\approx$30%) for
which the models are unable to predict the rotation periods of the MS
companions.

Figure 15: In the background, a 2D histogram shows the distribution of $\Delta
P_{\mathrm{rot}}/\sigma_{w}$ relative to the ages of WDs in a sample drawn
from a synthetic population. Each bin is color-coded based on the fraction of
triples (i.e. the number of WDs in the bin that have had their age reset),
while the transparency reflects the total count of WDs in the bin. Warmer
colors signify a higher proportion of WDs merger products, and greater opacity
indicates a larger overall WD count in the bin. The black circles represent
the distribution of the observed sample. At young WDs ages, the observed
sample exhibits an increased occurrence of systems with $\Delta
P_{\mathrm{rot}}/\sigma_{w}>2$ in a region prone to triple contamination, as
indicated by the synthetic population.

To test whether the discrepant systems in our sample may be reasonably
accounted for by binary mergers, we randomly draw values of age and mass from
a uniform distribution of ages up to 10 Gyr and a uniform distribution of
masses between 0.18 and 1.15 $M_{\odot}$, respectively. We find the point in
our model grid that is closest to each age-mass combination and construct a
synthetic population of 500 MS stars with an age, mass, rotation period and
effective temperature. We reset the age of 30% of the stars in this synthetic
population to a number randomly drawn from a uniform distribution between 0
and the true model age to simulate an apparent age that has been reset by a
merger event. From this synthetic population, we draw a sample of 400 stars
that matches the observed distribution of WD ages and compute $\Delta
P_{\mathrm{rot}}/\sigma_{w}$ for each star in the sample.

We find that 42% of the MS stars in our data sample have a positive $\Delta
P_{\mathrm{rot}}/\sigma_{w}>3$, i.e the age that we would infer from the
rotation period of the MS star is older than what we estimate from their WD
companions. The synthetic population reveals that 20% of the stars have a
positive $\Delta P_{\mathrm{rot}}/\sigma_{w}>3$. The 2D histogram in the
background of Fig. 15 shows the distribution of $\Delta
P_{\mathrm{rot}}/\sigma_{w}$ as a function of the WD age after reset in the
synthetic population and highlights a tail of discrepant $\Delta
P_{\mathrm{rot}}/\sigma_{w}$ values at $t<$ 3 Gyr in a region with a high
fraction of triples per bin. Such a distribution is well matched by the
distribution of $\Delta P_{\mathrm{rot}}/\sigma_{w}$ of our sample, as shown
by the higher concentration of stars with $\Delta
P_{\mathrm{rot}}/\sigma_{w}>3$ at $t<3$ Gyr and the decrease of the number of
stars with $\Delta P_{\mathrm{rot}}/\sigma_{w}>3$ with age.

While we cannot identify with certainty which WDs in our sample may be the
products of binary mergers, our findings suggest that a fraction of the
systems showing $\Delta P_{\mathrm{rot}}/\sigma_{w}>3$ may have been triple
systems that experienced merger events. Consequently, the ages we estimate for
the WDs in such systems may underestimate the true system age, leading our
models to predict shorter rotation periods for the companion MS stars than
what is observed. We suspect, in particular, that the top panel of Fig. 8 is
subject to this bias, and that some significant portion of the long period
outliers may be these former triple systems.

###  4.5 Comparison to other datasets

####  4.5.1 Gyro-kinematic ages of Kepler stars

We compare the WD ages with the empirical gyro-kinematic ages from Lu et al.
(2021). Gyro-kinematic ages leverage on the idea that the velocity dispersion
of a stellar population at a given age increases with time due to
gravitational interactions between the star and gas clouds (Spitzer &
Schwarzschild, 1951). By making this assumption, Lu et al. (2021) used the
rotation periods of around $30{,}000$ Kepler stars to determine their coeval
nature (i.e. they assigned the same age to stars showing similar rotation
periods and temperatures) and applied age-velocity-dispersion relations to
estimate average stellar ages for groups of coeval stars.

We restricted our analysis to gyro-kinematic ages of stars with
$3500\,\mathrm{K}<T_{\mathrm{eff}}<5000\,\mathrm{K}$, where the gyro-kinematic
ages should not be impacted by weakened braking (van Saders et al., 2019). To
compare this sample to our WD + MS sample, for each MS star in our sample, we
created a bin centered at its $P_{\mathrm{rot}}$ and $T_{\mathrm{eff}}$ and
selected gyro-kinematic stars with a $P_{\mathrm{rot}}$ and a
$T_{\mathrm{eff}}$ within 5 days and 100 K from the $P_{\mathrm{rot}}$ and
$T_{\mathrm{eff}}$ of the MS star, respectively. If the bin did not contain a
minimum number of data points, we increased the size of the bin in the
$P_{\mathrm{rot}}$ and $T_{\mathrm{eff}}$ directions by 10%; we repeated this
process up to three times and until the bin contained a sufficient number of
data points to obtain a median age representative of the gyro-kinematic age of
the bin. We computed the gyro-kinematic age associated to the MS star in our
sample as the median of the gyro-kinematic ages of the stars within the bin.

Fig. 16 shows that there is a general disagreement between WD and gyro-
kinematic ages. In particular, we identify two bands in the plot: a lower
band, where the age predicted by the WD companion varies between 0.1 and 4 Gyr
while the gyro-kinematic age is roughly constant at 1.5 Gyr, and an upper
band, where the age inferred from the WD companions is younger than the gyro-
kinematic age. Our hypothesis is that the elongation in the lower band may be
due to core-envelope decoupling, which would cause these stars to have a
similar rotation period but different ages. WDs more accurately track the true
system age, while the gyro-kinematic age is confused by groups of stars with
different ages having similar rotation periods. The discrepancies in the stars
populating the upper band are likely caused by a combination of two factors:
1) some fraction of the WDs in our sample are WD merger products, therefore
the age inferred from the WD age is an underestimate of the true age of the
system; 2) at the fully convective boundary, the gyrochrones are compressed,
therefore stars at the same rotation period along this boundary are not
necessarily coeval.

We find that 50% of the stars located in this upper band show a $\Delta
P_{\mathrm{rot}}/\sigma_{w}>3\sigma_{w}$ and a positive $\Delta
P_{\mathrm{rot}}$, which supports the WD merger hypothesis for these systems,
as discussed in Section 4.4.3. Furthermore, the MS stars populating the upper
band of Fig. 16 are distributed along the sharp rise in rotation period at the
fully convective boundary and the disagreement between the WD and gyro-
kinematic ages increases as we move toward longer periods, as shown in Fig.
17. Therefore, gyro-kinematic ages, which assume that stars at similar periods
and temperatures have the same ages, are likely not reliable for stars at the
fully convective boundary.

Other factors that affect the precision of WD total ages are their mass and
the IFMR. Precise mass measurements and well-constrained IFMRs are required to
obtain precise ages of low-mass WDs ($M<0.63\,M_{\odot}$) since their
progenitor lifetimes represent a major part of their total age. However, the
WD companions of the MS stars in the upper band of Fig. 17 are all high-mass
WDs ($M>0.67\,M_{\odot}$), which have short ZAMS progenitor lifetimes, thus
their age precision is not significantly affected by their mass and choice of
IFMR. Furthermore, because formal age uncertainties of higher-mass WDs are
often underestimated, we have applied inflation factors (Heintz et al., 2022).
Lastly, the distribution of WD ages in our sample (grey histogram along the
$x$-axis in Fig. 16) more closely resembles that of the Kepler-APOGEE Cool
Dwarfs sample (Claytor et al., 2020) and Kepler field stars (Silva Aguirre et
al., 2018). These age distributions peak at around 1-2 Gyr and do not exhibit
the double peak observed in the gyro-kinematic age distribution (grey
histogram along the $y$-axis in Fig. 16).

Figure 16: Comparison between the WD ages from this work and the median gyro-
kinematic ages (Lu et al., 2021) computed for bins of Kepler stars around the
MS stars in the sample. The data points are color-coded by the effective
temperature of the MS stars. The uncertainties represent the upper and lower
uncertainties on the WD ages (inflated as prescribed by Heintz et al. 2022).
The histograms show the distributions of WD and gyro-kinematic ages. Figure
17: Kepler stars are color-coded by their gyro-kinematic ages (Lu et al.,
2021) in greys. The MS stars in our sample are plotted as larger circles and
color-coded by the difference between the median of the gyro-kinematic ages of
the neighboring binned Kepler stars and the age inferred from the WD
companions. The dashed vertical line shows the $T_{\mathrm{eff}}$ of the
coolest star in the Kepler sample. Beyond this line, the cooler MS stars in
our sample are plotted as white circles since there are not enough Kepler
stars in the surroundings to make an age comparison. The typical uncertainty
in $T_{\mathrm{eff}}$ is shown at the top.

The sharp increase of rotation periods at the fully convective boundary also
challenges the hypothesis that the closing of the intermediate period gap
detected in the Kepler distribution (McQuillan et al., 2013) is caused by the
disappearance of the convective core, as proposed by Lu et al. (2022). We
argue that perhaps it is not a lack of core-envelope decoupling, but rather
the scatter of ages along the steep incline responsible for the period gap
closure. Since we see a scatter of ages at the fully convective boundary, any
feature that is a function of time, such as the intermediate period gap, gets
“spread” across the boundary. The fact that we can have a range of ages at the
fully convective boundary implies that we cannot interpret the closing of the
intermediate period gap at this location as strong evidence for core-envelope
decoupling, although the feature is still fundamentally tied to the loss of a
radiative core.

####  4.5.2 Gyro-kinematic ages from Lu et al. (2023b)

The shearing flows of the solar tachocline, the transition region between the
convective zone and the underlying radiative core (Schou et al., 1998), are
considered to play a key role in the process of magnetic field generation.
Fully convective stars do not have a tachocline and therefore are expected to
have a different dynamo mechanism. Recent observations of X-ray emissions from
fully convective stars reveal that these stars host a dynamo with a rotation-
activity relationship that closely resembles that of solar-like stars (Wright
et al., 2018), implying that the presence of the tachocline may not be a
critical factor in the creation of the stellar magnetic field. However, a
recent work by Lu et al. (2023b) suggested that the dynamos of partially and
fully convective stars may be fundamentally different.

Using gyro-kinematic ages of a dataset that combines the Kepler stars from Lu
et al. (2021) and stars with ZTF rotation periods from Lu et al. (2022) and Lu
et al. (2023a), they found that fully convective stars exhibit a $\sim
2.25\times$ higher AM loss rate than partially convective stars. To account
for this, they suggest that fully convective stars necessitate a dipole field
strength approximately 2.5 times greater, or an approximately fourfold
increase in the rate of mass loss, or a blend of both factors.

We use solar-metallicity tracks with $f_{\mathrm{spot}}=0\%$ to compute the
ratio of AM loss rate of a 0.28 $M_{\odot}$ star and a 0.40 $M_{\odot}$ star
at the same rotation period. We chose these masses to represent fully and
partially convective stars, respectively, while also avoiding stars along the
vertical feature in rotation period that we find at the fully convective
boundary. For a range of rotation periods between 10 and 90 days, we find that
the fully convective star always shows a higher AM loss rate than the
partially convective star by at least a factor of 2, except when its rotation
period is shorter than 16 days. Therefore, we suggest that invoking a
modification to the stellar dynamo mechanism is not necessary to explain the
stronger magnetic braking at the fully convective boundary. By scaling the
torque with Rossby number, our models naturally reproduce the sharp rise in
rotation period at the fully convective boundary.

###  4.6 Activity signatures

Because we are invoking changes in rotation period, Rossby number, and
convective overturn timescales to explain the observed behavior, it is natural
to ask whether there are observable activity signatures of such a physical
transition.

Although the rotation periods increase across the fully-convective boundary,
the convective overturn timescales also increase, meaning that we expect very
modest values of the Rossby number. Stars near the “spike” in the gyrochrones
achieve Rossby numbers less than solar ($\sim 2$) but greater than saturation
($\sim 0.1$) despite the extremes they represent in both rotation period and
convective overturn timescale. Rotation-activity correlations have emerged
from a variety of activity signatures such as X-rays (Wright et al., 2011,
2018), H$\alpha$ emission (Newton et al., 2017), UV (France et al., 2018), and
show that the activity level decreases with increasing rotation period and
increases with decreasing Rossby number. If magnetic activity levels truly do
track Rossby number, we expect these stars to be active, but neither unusually
active nor unusually quiet compared to field stars of mixed ages at slightly
hotter or slightly cooler temperatures.

Stars “on the spike” do have Rossby numbers that are $\sim 0.2-0.7$ lower than
stars immediately hotter or cooler than the feature, depending on the age. If
we assume, for example, that magnetic activity scales as $\textrm{Ro}^{-2}$,
then this corresponds to a factor of $\sim 2-20$ enhancement in activity for
stars in a narrow mass range at the fully convective boundary. Observational
$L_{X}/L_{bol}-\textrm{Ro}$ relations display an order of magnitude of spread
in the X-ray luminosities at fixed Rossby number, making the predicted
activity signature at the fully convective boundary relatively subtle in
comparison. Precision activity measurements in controlled environments like
open clusters may represent the best hope of detecting an activity feature at
the fully convective boundary.

There have been observational efforts to examine the activity of stars in the
vicinity of the fully convective boundary. Jao et al. (2023) claims that stars
above the observed M dwarf luminosity gap (higher mass) are more active. In
contrast, our models do not predict a lower Rossby number and higher activity
rates just above the gap; instead, they predict that fully convective stars
coolward (less massive) than the gap have lower Rossby numbers, and would thus
appear more active (their long rotation periods are balanced by a larger
$\tau_{cz}$). Because existing field samples are relatively small and
challenging to control for binarity, it is not yet obvious if this apparent
tension is robust.

Boudreaux et al. (2024) similarly studied the activity of gap stars, finding
that there was a larger scatter in both the observed rotation rates and
activity levels on the cool (lower mass) side of the gap. We create a simple
stellar population where ages are drawn from a Gaussian centered on 3 Gyr with
a width of 2 Gyr, truncated at 0 Gyr and 14 Gyr or the main sequence turnoff
age, whichever is younger for each mass in our model grid. In this toy model
the dispersion in rotation periods does indeed increase across the fully
convective boundary (by about a factor of 3), as does the dispersion in
predicted activity levels (again a factor $\sim 3$, assuming a
$\textrm{Ro}^{-2}$ scaling for activity proxies).

##  5 Conclusions

In this work, we constructed a sample of 327 wide, coeval WD + MS binaries
with a measured rotation period for the MS companions, which are mostly K and
M dwarfs. We infer effective temperatures, surface gravities, and masses of
the WD companions by using hydrogen-dominated atmosphere models and fitting
photometry from a variety of all-sky surveys. Using these atmospheric
parameters, we computed the total age of each WD using WD cooling models, a
theoretically motivated and observationally calibrated IFMR, and stellar
evolution model grids. Our sample is dominated by massive
($M>0.67\,M_{\odot}$) WDs for which the total age is primarily governed by
cooling processes. This allowed us to achieve an average uncertainty of 10% on
the WD total age.

To model the rotational evolution of the MS stars, we adopted an angular
momentum loss prescription for magnetized winds from van Saders & Pinsonneault
(2013) and modelled the internal angular momentum transport as in the standard
two-zone model from Denissenkov et al. (2010). We calibrated gyrochronology
models to reproduce the rotational sequences of the open clusters Pleiades,
Praesepe, M67 and the rotation period of the Sun at solar age. We used the
calibrated gyrochrones to predict the rotation periods of the MS stars in the
sample given their effective temperature and the age from their WD companions.

We find that the rotation period steeply increases across a narrow temperature
range for stars near the fully convective boundary and up to $\sim 8$ Gyr.
This sharp rise in rotation period is evident in both the models and the data
and suggests that stars rotating slowly at the fully convective boundary are
not necessarily old.

We propose that the rise in rotation period at this boundary is driven by an
increase in convective overturn timescale due to structural differences
between partially and fully convective stars. As the convective envelope
extends deeper into the star, encompassing a larger fraction of the overall
stellar mass, it results in a rise in the pressure scale height and a
reduction in convective velocity, leading to an increase in $\tau_{cz}$.
Furthermore, we argue that the sharpness of such rise in $\tau_{cz}$ is
induced by non-equilibrium 3He burning occurring for stars just short of the
fully convective boundary (van Saders & Pinsonneault, 2012a). Although the
exact location of this vertical feature in $\tau_{cz}$, and consequently
$P_{\mathrm{rot}}$, depends on properties like metallicity, spot covering
fraction and $\tau_{cz}$ prescriptions, the existence of this feature does
not.

Due to the current uncertainties in temperature measurements, the rotation
periods of stars situated along this distinct feature can be associated to a
broad spectrum of gyrochrones, spanning a range of $\sim 6$ Gyr. Consequently,
despite gyrochronology being regarded as a promising approach for determining
the ages of low-mass stars, our findings suggest that age estimation via this
method might pose greater challenges when applied to stars located at the
fully convective boundary.

Future work is planned to obtain spot covering fractions for the MS stars in
our sample to allow for a better comparison between the observed and the model
rotation periods. Furthermore, as discussed in Sec. 4, metallicity has a non-
negligible impact on stellar spin-down, therefore more robust predictions of
rotation period will be achieved by taking into account the inherent
variability and uncertainty associated with this parameter in our models.
Having metallicity values for more stars in our sample would also allow for
more accurate comparison between the observed and model rotation periods as
well as better WD age estimates, since the IMFR is likely to be sensitive to
the metallicity of the progenitor stars (Cummings et al., 2019). Knowing the
metallicity of some of the WDs in the sample will be useful to refine the
IFMRs and obtain even more precise WD ages (Raddi et al., 2022). Thus, these
systems represent optimal targets for wide field spectroscopic surveys. For
example, the Milky Way Mapper of the Sloan Digital Sky Survey V (Kollmeier et
al., 2017) is collecting APOGEE infrared spectra for 6 million stars across
the entire Milky Way and will provide metallicities for all these stars.

Having kinematic ages that remain unaffected by the rotation of main sequence
stars would be advantageous. Such ages could serve as a supplementary
assessment for the rapid increase in rotation period observed at the fully
convective boundary. Furthermore, they have the potential to offer insights
into the underlying causes of the discordance observed between the gyro-
kinematic ages detailed in Lu et al. (2021) and the ages deduced from our WD
sample.

Finally, our sample represents a subset drawn from a pool of 5005 WD + MS
binary systems with measured rotation periods. It is worth noting that there
exists a total of $22{,}563$ such systems (El-Badry et al., 2021). Access to a
greater number of rotation periods would prove invaluable in expanding our
sample size, probing even older age ranges, and enhancing our understanding of
the rotational evolution of low-mass stars. While the count of TESS-derived
rotation periods for cool, MS stars through machine learning techniques is on
the rise (Claytor et al., 2023), there is also promise in forthcoming space
missions like the Nancy Grace Roman Space Telescope (Spergel et al., 2015)
which will enable many new rotation period measurements.

We would like to thank Dhvanil Desai, Zachary Claytor, Lyra Cao and Nicholas
Saunders for helpful discussions. We acknowledge support from the National
Science Foundation under Grants No. AST-1908119 and AST-1908723. JMJO
acknowledges support from NASA through the NASA Hubble Fellowship grant HST-
HF2-51517.001, awarded by STScI, which is operated by the Association of
Universities for Research in Astronomy, Incorporated, under NASA contract
NAS5-26555.

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##  Appendix A Data Sample

Table 1:  Sample of 327 MS + WD wide binaries. The reported values of $\mathrm{T_{eff}}$ and $\mathrm{P_{rot}}$ refer to the MS stars in the sample. $\sigma_{L}$ and $\sigma_{U}$ are the computed lower and upper uncertainties on the WD total age, respectively. MS Gaia DR3 ID | WD Gaia DR3 ID |  $\mathrm{T_{eff}}$ (K) |  $\mathrm{P_{rot}}$ (days) | WD Total Age (Gyr) |  $\sigma_{L}$ (Gyr) |  $\sigma_{U}$ (Gyr)
---|---|---|---|---|---|---
1016412914066913792 | 1016427924977024768 | 3532 | 37.2 | 2.09 | 0.10 | 0.18
1018436324699616000 | 1018436320403748480 | 3551 | 22.4 | 1.83 | 0.05 | 0.06
1028188821038873344 | 1028188816743640064 | 3702 | 26.9 | 1.87 | 0.08 | 0.13
1033502588937866368 | 1033495266017587712 | 4165 | 24.7 | 0.32 | 0.02 | 0.02
1036217489305134848 | 1036217489305134976 | 3286 | 15.9 | 1.63 | 0.08 | 0.10
1053864376052875648 | 1053864371756927232 | 3400 | 11.8 | 2.76 | 0.17 | 0.17
1056274986577005312 | 1056274608619883392 | 3669 | 35.0 | 0.55 | 0.05 | 0.08
1059690653808481152 | 1059690649512906368 | 3551 | 21.6 | 1.50 | 0.04 | 0.05
1068143974080812160 | 1068340988524781952 | 3541 | 11.7 | 2.76 | 0.03 | 0.04
1089162891193023488 | 1089159966319435904 | 3461 | 24.4 | 1.05 | 0.05 | 0.09
1102405649756361472 | 1101648597344318464 | 3511 | 21.4 | 1.35 | 0.05 | 0.09
115344202888681216 | 115344095513403520 | 3550 | 64.2 | 4.49 | 0.10 | 0.11
1175151533777264256 | 1175151538072196992 | 3416 | 65.8 | 2.04 | 0.07 | 0.10
1202830540713288064 | 1202830540713287936 | 3372 | 41.8 | 1.09 | 0.04 | 0.05
1215379026400365568 | 1215379026400366080 | 3318 | 85.0 | 1.99 | 0.06 | 0.06
1233273646861764480 | 1233273646861782656 | 3352 | 77.8 | 1.90 | 0.11 | 0.13
1237798377728024320 | 1237600053318254848 | 3339 | 24.7 | 2.49 | 0.05 | 0.06
1246229948286292096 | 1246230047071154944 | 3534 | 16.7 | 2.10 | 0.08 | 0.07
1246239294135248512 | 1246239294135248384 | 3659 | 35.8 | 0.96 | 0.03 | 0.03
1252651160616571136 | 1252618213922867584 | 3066 | 63.9 | 4.12 | 0.12 | 0.13
1291263917336163456 | 1291263913040087936 | 3401 | 18.7 | 2.52 | 0.10 | 0.20
1322796261553157760 | 1322796261553165824 | 3787 | 13.0 | 1.06 | 0.05 | 0.11
1340289109302083584 | 1340289113593902080 | 3245 | 36.2 | 1.62 | 0.04 | 0.04
1341558083155952512 | 1341557984372129536 | 3253 | 55.8 | 2.77 | 0.12 | 0.11
1342071937339096192 | 1342071933043493120 | 3260 | 89.3 | 3.92 | 0.12 | 0.16
2187901837176956800 | 2187901832880159488 | 3480 | 31.3 | 2.15 | 0.17 | 0.13
2187942828344053888 | 2187942789690550272 | 3376 | 76.4 | 1.63 | 0.17 | 0.12
1369864498912398208 | 1369861200377513088 | 3066 | 128.5 | 1.01 | 0.05 | 0.05
2217771234476476672 | 2217771135693888256 | 3420 | 30.6 | 1.23 | 0.15 | 0.16
2221182095642372480 | 2221184019786046336 | 3413 | 55.5 | 2.61 | 0.14 | 0.18
2185748271854641024 | 2233787511119888000 | 3978 | 119.0 | 1.46 | 0.18 | 0.19
1464104190284155392 | 1464104190283536000 | 3609 | 20.8 | 0.94 | 0.03 | 0.03
2254761455631944448 | 2254761593070911360 | 3858 | 10.4 | 0.46 | 0.07 | 0.06
1481817597487494144 | 1481817593190278272 | 3422 | 116.8 | 0.97 | 0.05 | 0.06
1495234113246413440 | 1495233387395677952 | 3173 | 26.3 | 1.24 | 0.04 | 0.05
2275852459474001664 | 2275852455177465088 | 3366 | 28.8 | 2.77 | 0.11 | 0.13
2299189284536000256 | 2299190006090503808 | 3494 | 14.2 | 0.49 | 0.02 | 0.02
1557284811442449408 | 1557284807145814912 | 3243 | 82.9 | 1.09 | 0.05 | 0.08
2508478574102156544 | 2508478569806892928 | 3322 | 13.4 | 2.79 | 0.13 | 0.16
2512689428758531456 | 2512689424463488256 | 3658 | 29.3 | 2.69 | 0.06 | 0.10
1681875731024076160 | 1681875726730147968 | 3424 | 102.0 | 1.63 | 0.05 | 0.06
2539792321663876736 | 2539792317371533440 | 3310 | 13.3 | 1.46 | 0.10 | 0.20
2559242888517624320 | 2559255150648826112 | 3489 | 48.4 | 1.25 | 0.12 | 0.09
1718710573146454400 | 1718710568850372864 | 3385 | 84.8 | 1.19 | 0.07 | 0.11
2576762025758278656 | 2576762266276447104 | 4076 | 14.0 | 0.69 | 0.03 | 0.03
1766826194114207360 | 1766826194114406656 | 4612 | 21.3 | 0.77 | 0.03 | 0.03
265001616763644928 | 265001612467640448 | 3709 | 44.7 | 1.47 | 0.04 | 0.05
2656915701868610944 | 2656915697573351680 | 3458 | 43.0 | 1.52 | 0.04 | 0.04
266049206532575232 | 266049206532585472 | 3473 | 19.8 | 1.09 | 0.05 | 0.06
267776887192982784 | 267777604451603328 | 3421 | 29.2 | 3.95 | 0.05 | 0.07
268123027195914368 | 268123332136902016 | 3119 | 200.0 | 1.53 | 0.04 | 0.04
2719749527220933120 | 2719749527220932992 | 3469 | 102.0 | 0.61 | 0.04 | 0.04
2724335723363858944 | 2724335723364297344 | 3268 | 12.4 | 1.55 | 0.05 | 0.07
2736527898566892672 | 2736128947644568832 | 3078 | 12.4 | 5.14 | 0.08 | 0.09
2739560214200407168 | 2739560218493488512 | 3638 | 33.6 | 0.93 | 0.09 | 0.15
2742785532774608896 | 2742785322319919232 | 3387 | 88.1 | 0.96 | 0.03 | 0.05
1857669979411064576 | 1857670009471539968 | 3737 | 12.2 | 0.50 | 0.06 | 0.10
1874965739686844032 | 1874965499168713600 | 3858 | 14.5 | 3.25 | 0.08 | 0.10
2774575643747747968 | 2774583920149511040 | 3936 | 15.7 | 0.91 | 0.05 | 0.07
2781085405419253504 | 2781085401124115328 | 3448 | 159.7 | 3.38 | 0.12 | 0.14
2805156498090147968 | 2805486793959872512 | 3559 | 45.0 | 0.53 | 0.02 | 0.02
2824510273561942272 | 2824510239202203392 | 3364 | 33.0 | 2.20 | 0.08 | 0.11
1911106416309693952 | 1911106416309735552 | 3087 | 14.3 | 1.02 | 0.05 | 0.04
2828957042181061376 | 2828959305628252672 | 4283 | 170.1 | 0.88 | 0.05 | 0.05
2837635727977879168 | 2837615520155969792 | 3330 | 57.9 | 3.30 | 0.12 | 0.15
2840176317097371008 | 2840368456753528704 | 3546 | 30.7 | 1.42 | 0.07 | 0.08
2857544378863193472 | 2857547329505058176 | 3297 | 159.0 | 3.94 | 0.15 | 0.17
286958004978753536 | 287038990881256448 | 3303 | 170.1 | 0.60 | 0.06 | 0.06
292660381515098880 | 292656528929179008 | 3640 | 28.6 | 2.00 | 0.06 | 0.09
296479462139576576 | 296425242472999808 | 3417 | 65.4 | 1.64 | 0.09 | 0.18
2065535813733429888 | 2065535916813680256 | 3498 | 41.9 | 1.30 | 0.03 | 0.03
3063799218201750272 | 3063800661311469312 | 4761 | 10.7 | 2.15 | 0.15 | 0.10
208556331928147840 | 208541312424592896 | 3516 | 57.6 | 0.67 | 0.08 | 0.07
3094667942729601408 | 3094662853191563392 | 3718 | 89.0 | 0.35 | 0.06 | 0.03
3096007732007670144 | 3096007762070816128 | 3843 | 14.1 | 0.43 | 0.05 | 0.06
3137340435682881920 | 3137340435679753728 | 3752 | 18.3 | 0.52 | 0.13 | 0.08
3143591601666495360 | 3143596966081381120 | 3386 | 10.6 | 0.60 | 0.11 | 0.09
3145894292546764544 | 3145894288249489152 | 3375 | 23.2 | 1.92 | 0.10 | 0.19
3160143344767586176 | 3160143344767585536 | 3204 | 22.6 | 1.92 | 0.13 | 0.10
3163784102643971328 | 3163784441944402176 | 4003 | 13.5 | 0.31 | 0.05 | 0.05
3165392310198209664 | 3165392305902629504 | 3407 | 106.7 | 2.62 | 0.03 | 0.04
3266435259830726016 | 3266393448324042752 | 3295 | 24.7 | 4.56 | 0.09 | 0.11
3276414466021403008 | 3276414466021403264 | 3186 | 111.1 | 3.06 | 0.05 | 0.05
3283182338046329472 | 3283182265031074560 | 3206 | 17.4 | 1.50 | 0.14 | 0.10
3290811230756139008 | 3290811226460959360 | 3522 | 46.5 | 4.33 | 0.17 | 0.19
2238010155466716544 | 2238010185526881792 | 3372 | 27.4 | 2.73 | 0.16 | 0.15
3317635947223274752 | 3317636703137518208 | 3315 | 125.4 | 3.99 | 0.10 | 0.08
3330006857484987008 | 3330030565704492800 | 3069 | 138.5 | 0.64 | 0.08 | 0.07
3361149463489086720 | 3361149527909420928 | 3350 | 41.0 | 2.13 | 0.05 | 0.06
3369278874508583424 | 3369278767129521792 | 3667 | 16.5 | 4.40 | 0.14 | 0.14
3380358618818995328 | 3380357205774532480 | 3754 | 29.2 | 4.07 | 0.11 | 0.14
3387417517828617088 | 3387417513531897984 | 3377 | 51.6 | 1.34 | 0.09 | 0.17
340111386361933056 | 340111420721666048 | 3554 | 55.4 | 1.06 | 0.09 | 0.20
3423222461152705536 | 3423222559935120000 | 3990 | 17.7 | 1.91 | 0.02 | 0.02
3457932320957284992 | 3457949260309952512 | 3508 | 23.2 | 1.14 | 0.07 | 0.14
348345663302119936 | 348345659004467840 | 3213 | 144.9 | 3.27 | 0.16 | 0.17
3604099511263969408 | 3508014250968880256 | 3651 | 17.5 | 3.74 | 0.09 | 0.10
3610422562117192192 | 3610422798340344704 | 3434 | 34.2 | 3.03 | 0.09 | 0.11
3630646463602449664 | 3630648387747801088 | 3396 | 21.9 | 1.02 | 0.01 | 0.02
3724362340763265536 | 3724362336470991488 | 3067 | 24.9 | 1.91 | 0.05 | 0.05
3726944921778222080 | 3726944951843168768 | 3457 | 37.1 | 1.25 | 0.07 | 0.13
3727701042180797952 | 3727701007821059072 | 3237 | 40.0 | 1.64 | 0.05 | 0.06
3733304978070296320 | 3733305179932909312 | 3094 | 173.7 | 4.73 | 0.11 | 0.11
2434159727027561728 | 2434159727027564032 | 4666 | 20.2 | 1.64 | 0.08 | 0.13
374254624016996352 | 374254619721557504 | 4233 | 12.8 | 1.12 | 0.03 | 0.05
2519281859960487424 | 2519281855665451008 | 3288 | 75.3 | 3.16 | 0.11 | 0.14
3831257784633494016 | 3831257823288140544 | 3302 | 37.9 | 1.42 | 0.03 | 0.03
2567879174396642944 | 2567879174396643072 | 3437 | 29.2 | 0.58 | 0.04 | 0.05
2618616409043165824 | 2618616409043298688 | 5004 | 21.0 | 1.24 | 0.03 | 0.03
3870805121940366720 | 3870804537824812800 | 3458 | 129.3 | 1.22 | 0.09 | 0.17
3897698321657960832 | 3897698317362400384 | 3568 | 24.0 | 1.00 | 0.03 | 0.05
390689402277884544 | 390689397980291072 | 3520 | 89.1 | 2.94 | 0.13 | 0.13
3907936385524737792 | 3907943330487532928 | 3126 | 26.2 | 3.96 | 0.05 | 0.06
3912311376651443968 | 3912311376651444224 | 3221 | 77.7 | 1.30 | 0.04 | 0.03
3931756484602030976 | 3931756484602030848 | 3290 | 100.1 | 2.40 | 0.05 | 0.06
3941959093514800128 | 3941964964734084864 | 3855 | 14.6 | 1.70 | 0.03 | 0.04
3944167939359417088 | 3944167939360964480 | 3582 | 13.0 | 0.86 | 0.07 | 0.08
395377200164554368 | 395377513698597888 | 3130 | 14.5 | 5.09 | 0.07 | 0.08
3956995877097644416 | 3956995666643307008 | 4235 | 43.5 | 1.20 | 0.07 | 0.04
2743171942392459008 | 2743171938097183616 | 3470 | 13.3 | 0.45 | 0.07 | 0.06
3995728682445851648 | 3995728953028752768 | 3233 | 108.7 | 5.03 | 0.13 | 0.13
4016142868162417792 | 4016142868162408576 | 3333 | 87.3 | 2.65 | 0.07 | 0.07
411948871922982528 | 411951822555835648 | 3570 | 15.6 | 2.07 | 0.17 | 0.13
2806310641702070016 | 2806310603047483904 | 3589 | 84.5 | 0.83 | 0.05 | 0.07
2816359731303136384 | 2816359727009816960 | 3141 | 29.8 | 7.46 | 0.10 | 0.15
2828203292601777536 | 2828203288307357952 | 3333 | 157.9 | 1.25 | 0.02 | 0.03
424458153152543616 | 424459076562158464 | 3804 | 19.1 | 3.02 | 0.11 | 0.13
427169926782173056 | 427169583175190144 | 3740 | 28.4 | 1.69 | 0.12 | 0.11
2836648190441772160 | 2836648190441773056 | 3542 | 90.6 | 1.16 | 0.02 | 0.03
4283721804641828480 | 4283721800317089920 | 3319 | 138.5 | 3.75 | 0.07 | 0.09
4310991582806231296 | 4310991685885480064 | 3329 | 126.7 | 1.57 | 0.08 | 0.13
2858833212649253504 | 2858833208354052096 | 3486 | 51.0 | 4.52 | 0.15 | 0.18
434845926690890496 | 434857952593321088 | 4276 | 22.3 | 1.20 | 0.03 | 0.03
2867316941290457856 | 2867316975650195968 | 4023 | 19.7 | 1.10 | 0.05 | 0.09
2874159648826704896 | 2874159644531724288 | 3457 | 11.4 | 0.53 | 0.04 | 0.04
3074742111972735488 | 3074742107676979584 | 3498 | 30.9 | 0.56 | 0.04 | 0.05
4419026537287781632 | 4418831438398101376 | 3788 | 193.7 | 0.54 | 0.03 | 0.04
4424520144057157248 | 4424527561464289664 | 3962 | 18.3 | 4.71 | 0.12 | 0.13
4447660805076826880 | 4447663760014329344 | 3351 | 54.5 | 1.43 | 0.07 | 0.09
3129688040911585152 | 3129692434664991104 | 3277 | 39.1 | 2.37 | 0.05 | 0.06
4532268744537179904 | 4532250327717123968 | 3317 | 73.8 | 1.62 | 0.10 | 0.18
4563562151123622656 | 4563562426001533696 | 3446 | 54.6 | 2.35 | 0.09 | 0.11
4565593739374133504 | 4565593735079515904 | 3215 | 178.0 | 4.06 | 0.15 | 0.16
4590962805523660672 | 4590962766864961024 | 3373 | 190.7 | 1.92 | 0.06 | 0.06
4603527229834516096 | 4603528020108454528 | 3308 | 12.9 | 1.09 | 0.06 | 0.08
470826482637310080 | 470826482637310848 | 3233 | 87.1 | 2.63 | 0.03 | 0.03
508627314444951680 | 508627344501773952 | 3595 | 14.8 | 1.08 | 0.04 | 0.04
544555471783803136 | 544555467486553984 | 4044 | 10.5 | 2.21 | 0.17 | 0.16
564321564114510080 | 564509271364217344 | 3417 | 93.2 | 1.85 | 0.09 | 0.08
584616487177143424 | 584616482882438400 | 4112 | 134.5 | 0.53 | 0.03 | 0.05
594439627139102720 | 594439519764767232 | 4713 | 13.0 | 0.44 | 0.04 | 0.04
615513794709603200 | 621567804516385792 | 3590 | 32.3 | 1.01 | 0.09 | 0.16
6222883046075139968 | 6222882251504782720 | 3262 | 274.2 | 0.61 | 0.02 | 0.01
3839256319408219008 | 3839256319408293248 | 5527 | 12.5 | 1.94 | 0.10 | 0.18
634350314384284032 | 634367395469443840 | 3237 | 23.6 | 3.44 | 0.05 | 0.06
642718770397533568 | 642715055250964608 | 3332 | 64.9 | 0.35 | 0.04 | 0.05
642837139695905024 | 642837135401004672 | 3434 | 46.7 | 2.58 | 0.10 | 0.18
666745023612674816 | 666640707447423616 | 3553 | 35.2 | 5.03 | 0.06 | 0.05
3907622513610306432 | 3907622479250567808 | 3401 | 57.0 | 2.66 | 0.12 | 0.10
3919287679146000640 | 3919287674850641792 | 5204 | 14.6 | 1.12 | 0.03 | 0.06
678153006506212480 | 678153006506212608 | 3374 | 79.4 | 0.60 | 0.04 | 0.06
3941164971241835776 | 3941164966946431360 | 3631 | 43.8 | 1.43 | 0.07 | 0.10
70802299655353344 | 70802226637418496 | 3329 | 13.8 | 1.24 | 0.07 | 0.09
714403252037918464 | 714403247741604992 | 3853 | 11.3 | 2.40 | 0.11 | 0.10
4024591309351630848 | 4024591305056699136 | 3241 | 54.5 | 3.02 | 0.16 | 0.16
719440149164429824 | 719439423313644672 | 3375 | 11.6 | 2.09 | 0.08 | 0.06
720018767158643072 | 720018767158642944 | 3757 | 43.3 | 0.81 | 0.05 | 0.08
739321307963291136 | 739321312258441216 | 3300 | 16.8 | 1.35 | 0.08 | 0.05
748521441443836288 | 748521544523051136 | 3229 | 61.9 | 0.95 | 0.08 | 0.16
750669371768362752 | 750669367473333760 | 3412 | 109.1 | 3.00 | 0.16 | 0.15
779970154057928320 | 779970154057955840 | 3539 | 110.5 | 2.44 | 0.11 | 0.15
793941132918315392 | 793917660919464960 | 3801 | 26.0 | 1.68 | 0.06 | 0.06
794265282689661056 | 794265278392449664 | 3500 | 11.0 | 2.03 | 0.05 | 0.06
821363846267866368 | 821363807610959104 | 3347 | 52.7 | 3.03 | 0.12 | 0.15
847233808680375424 | 847233804386021248 | 3473 | 12.6 | 1.54 | 0.06 | 0.09
85649623637225856 | 85649413183595392 | 3112 | 41.4 | 3.70 | 0.14 | 0.19
4413351854700039168 | 4413351919121477632 | 3859 | 26.4 | 0.66 | 0.06 | 0.08
4434749725365419648 | 4434749725365419520 | 3440 | 88.2 | 1.03 | 0.02 | 0.02
1345042955196670208 | 1345041748308955648 | 3250 | 17.1 | 4.14 | 0.16 | 0.18
1355692412505713152 | 1355692408211800704 | 3634 | 18.5 | 0.44 | 0.03 | 0.04
1356340054918762240 | 1356303083840138368 | 3671 | 11.4 | 2.56 | 0.16 | 0.16
1357831271859374080 | 1357826525919678208 | 3381 | 13.0 | 1.56 | 0.04 | 0.04
1363780969795550336 | 1363761827125936128 | 4184 | 33.2 | 3.96 | 0.15 | 0.18
1365015515194773632 | 1365015545259769600 | 3157 | 93.6 | 3.99 | 0.12 | 0.14
4460569316402173184 | 4460569312107772672 | 3412 | 55.0 | 1.30 | 0.06 | 0.04
1397652112683641984 | 1397649806285796352 | 3750 | 10.3 | 0.40 | 0.04 | 0.06
1399620238497200896 | 1399620268561690240 | 4089 | 10.8 | 1.61 | 0.09 | 0.10
1410448641324559744 | 1410448259071414528 | 3201 | 130.4 | 1.79 | 0.05 | 0.03
1412158789927363328 | 1412158789927364224 | 3460 | 10.3 | 0.57 | 0.06 | 0.09
1428806289324715392 | 1428804262100101632 | 3473 | 16.9 | 0.40 | 0.08 | 0.05
4466814817325615360 | 4466814881747732480 | 3327 | 58.0 | 0.53 | 0.04 | 0.04
143817499599533440 | 143817499597885824 | 3288 | 129.2 | 1.84 | 0.08 | 0.13
1444622768385439232 | 1444622768385439104 | 3315 | 81.3 | 1.67 | 0.04 | 0.03
1519088739564567936 | 1519088735269074688 | 3460 | 54.1 | 1.79 | 0.06 | 0.07
1526498829461895936 | 1526498863822158592 | 3339 | 12.7 | 2.37 | 0.07 | 0.07
1541323338581798144 | 1541323334283923200 | 3163 | 103.6 | 3.12 | 0.13 | 0.13
1541905598706074880 | 1541905560050370816 | 3485 | 16.9 | 0.45 | 0.06 | 0.05
1549715047545373056 | 1549715047547083136 | 3244 | 88.8 | 0.79 | 0.03 | 0.04
4611559819405628672 | 4611559815111559552 | 3517 | 33.7 | 1.64 | 0.09 | 0.14
1588270938897733888 | 1588270934604149632 | 3964 | 30.8 | 1.75 | 0.09 | 0.08
1625475663602558208 | 1625661648570402304 | 3492 | 10.0 | 1.40 | 0.09 | 0.05
545940409758573824 | 545940164944375296 | 3502 | 40.4 | 2.44 | 0.11 | 0.11
1640003991777199872 | 1640200215946705024 | 3346 | 70.8 | 1.67 | 0.06 | 0.07
546388323308593280 | 546388319012071040 | 3253 | 19.5 | 3.85 | 0.18 | 0.20
1656963645101470336 | 1656967459032459520 | 3448 | 29.9 | 1.61 | 0.17 | 0.15
1664401914278345472 | 1664401914277704192 | 3372 | 68.5 | 2.99 | 0.09 | 0.14
166587938734739456 | 166587938734739328 | 3212 | 29.6 | 1.52 | 0.05 | 0.05
1684035206221069824 | 1684011837303097856 | 3803 | 19.2 | 1.05 | 0.04 | 0.05
865042323638265472 | 865042319342035328 | 3405 | 157.6 | 2.22 | 0.07 | 0.08
873994719110827136 | 873994719110827264 | 3322 | 18.6 | 2.36 | 0.06 | 0.06
907150801482530944 | 907156397824275328 | 3499 | 10.5 | 1.04 | 0.03 | 0.04
909162529804018688 | 909162525508848000 | 3250 | 17.2 | 1.26 | 0.02 | 0.02
936641902362276736 | 936641833642799744 | 3702 | 176.8 | 1.50 | 0.03 | 0.03
9500918630942208 | 9500914335389952 | 3425 | 131.0 | 1.10 | 0.05 | 0.06
642134100794299776 | 642134135154038528 | 3074 | 18.0 | 1.22 | 0.03 | 0.02
6527346680622426880 | 6527346680622426752 | 3392 | 22.5 | 0.48 | 0.05 | 0.06
657819944131129472 | 657820012850606080 | 3480 | 38.1 | 1.47 | 0.03 | 0.04
990883830321907456 | 990883830322597376 | 3403 | 120.0 | 1.21 | 0.02 | 0.03
991811169597579520 | 991811165304213760 | 3327 | 73.7 | 1.73 | 0.10 | 0.17
995300431092665856 | 995312903678079232 | 3865 | 144.7 | 2.54 | 0.03 | 0.04
1737075642850176000 | 1737075642850176128 | 3480 | 47.8 | 1.74 | 0.04 | 0.04
1759287667517352960 | 1759334156243365248 | 3381 | 20.3 | 1.54 | 0.06 | 0.08
1765702257007857024 | 1765700912683411456 | 3417 | 29.0 | 2.55 | 0.16 | 0.14
1791515487198244736 | 1791505454155264128 | 3469 | 11.6 | 1.10 | 0.16 | 0.13
1791860218454403968 | 1791857435315484032 | 4009 | 13.9 | 0.38 | 0.04 | 0.06
1812554680855696896 | 1812554676560349568 | 3465 | 292.4 | 0.56 | 0.03 | 0.03
1819001289330307200 | 1819001289323819520 | 3484 | 32.4 | 1.42 | 0.08 | 0.11
1822194099312577024 | 1822193996219126784 | 3302 | 59.7 | 3.26 | 0.15 | 0.16
1830600484179388416 | 1830600484161582464 | 3654 | 17.9 | 0.44 | 0.05 | 0.07
184735996626896000 | 184735992329821312 | 3409 | 45.7 | 0.82 | 0.06 | 0.06
18493721155296640 | 18493721155296768 | 3312 | 64.7 | 1.09 | 0.12 | 0.18
719562332394139776 | 719562332394140032 | 3343 | 14.0 | 0.54 | 0.03 | 0.05
741037684269171072 | 741037679973569408 | 3504 | 26.2 | 1.06 | 0.02 | 0.03
844085460212939392 | 844085490277157376 | 3713 | 34.8 | 0.43 | 0.05 | 0.06
1937074166540526976 | 1937827194563435648 | 3502 | 51.9 | 1.45 | 0.04 | 0.05
855594185859622272 | 855594181563866880 | 3508 | 34.0 | 0.63 | 0.03 | 0.03
856837802230613504 | 856837729215097984 | 3991 | 42.7 | 0.86 | 0.07 | 0.14
1969985710668480256 | 1969985710668477184 | 3252 | 143.5 | 1.96 | 0.05 | 0.06
197048465274295936 | 197048460976343168 | 3068 | 29.6 | 4.27 | 0.12 | 0.12
2010645375776770304 | 2010692306878145280 | 3406 | 14.4 | 0.32 | 0.04 | 0.02
2018812170226865664 | 2018818045723591680 | 3830 | 12.7 | 1.19 | 0.09 | 0.07
860411043221913984 | 860411038927355264 | 3614 | 20.8 | 1.48 | 0.07 | 0.10
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2048599211506044160 | 2048599314570668544 | 3816 | 36.0 | 2.74 | 0.09 | 0.12
2053240524958351360 | 2053240524960871680 | 3407 | 41.0 | 1.43 | 0.08 | 0.11
2053585565450552960 | 2053584770878226304 | 5293 | 13.3 | 0.68 | 0.04 | 0.03
2063157913677426304 | 2063157260843901952 | 3444 | 24.8 | 1.70 | 0.07 | 0.09
2069622492291282176 | 2069622487994113408 | 3563 | 17.1 | 0.43 | 0.02 | 0.02
896972484905743872 | 898473932456801024 | 3298 | 40.1 | 1.39 | 0.03 | 0.02
2097547922975807488 | 2097543451912361984 | 3545 | 46.5 | 1.51 | 0.05 | 0.07
2099643935736477696 | 2099656953779008000 | 4852 | 20.0 | 0.54 | 0.06 | 0.04
2111700802286462848 | 2111335107294795520 | 3457 | 33.4 | 1.11 | 0.05 | 0.09
2137699338842695552 | 2137699712500128256 | 3466 | 37.1 | 0.98 | 0.09 | 0.15
2145132896436922496 | 2145132892139909632 | 3321 | 22.6 | 1.76 | 0.06 | 0.06
2157583383856242304 | 2157582872757579136 | 3490 | 14.2 | 0.65 | 0.10 | 0.08
2163324969169731072 | 2163324969157050112 | 3687 | 86.5 | 0.50 | 0.09 | 0.10
2167838052078071552 | 2167861618555210368 | 3425 | 153.5 | 3.65 | 0.15 | 0.16
1055800238072103296 | 1055800165056700416 | 3229 | 100.5 | 4.00 | 0.21 | 0.27
1214994506568260608 | 1214994502274335360 | 3483 | 46.6 | 6.71 | 0.26 | 0.24
1222565537480885376 | 1222565533185185024 | 3343 | 177.6 | 4.06 | 0.22 | 0.35
1299293685113944448 | 1299293680818146176 | 3284 | 76.8 | 3.60 | 0.18 | 0.22
1332138536976820096 | 1332138532682327552 | 3436 | 21.2 | 2.29 | 0.13 | 0.25
2272112333234818944 | 2224073910005362688 | 3184 | 97.9 | 2.01 | 0.11 | 0.21
2255996069750790528 | 2255996069748606976 | 3534 | 21.3 | 4.32 | 0.19 | 0.25
2579247712310922240 | 2579247708016296448 | 3420 | 18.3 | 2.57 | 0.26 | 0.40
2700089675200729216 | 2700089675200402048 | 3198 | 108.4 | 4.83 | 0.21 | 0.25
2709850830194329856 | 2709858217538113920 | 3656 | 11.7 | 4.44 | 0.17 | 0.23
271639471546035584 | 271639432884899712 | 3380 | 55.8 | 3.47 | 0.19 | 0.27
2745741638504666112 | 2745741634209187072 | 3775 | 37.6 | 2.67 | 0.23 | 0.27
286792799059592192 | 286739236524151040 | 3430 | 31.9 | 2.04 | 0.17 | 0.24
196589900204859392 | 196589861548739584 | 4034 | 20.8 | 3.88 | 0.14 | 0.20
293717390146554624 | 293717454571113984 | 4001 | 169.2 | 2.06 | 0.14 | 0.24
3047076750159238016 | 3047076750150944512 | 3591 | 270.4 | 5.05 | 0.20 | 0.33
3235085949939746304 | 3235085949939746176 | 3075 | 92.1 | 4.69 | 0.27 | 0.60
3238194445407460864 | 3238194372390557312 | 4268 | 58.1 | 2.92 | 0.22 | 0.28
3278241888706974336 | 3278241888707395584 | 3108 | 117.5 | 3.33 | 0.28 | 0.39
3309178916095700224 | 3309188669965291264 | 3727 | 11.7 | 1.78 | 0.15 | 0.20
3323096740440611712 | 3323096740438265728 | 3291 | 92.9 | 3.36 | 0.26 | 0.54
3389371380055187968 | 3389371384349749376 | 3095 | 29.3 | 3.03 | 0.22 | 0.50
3413008990266533376 | 3413009093345760768 | 3367 | 70.4 | 3.10 | 0.25 | 0.41
3435597013551204736 | 3435597013551206784 | 3273 | 10.7 | 2.73 | 0.25 | 0.28
3593826155650669568 | 3785887605544154624 | 3720 | 20.3 | 3.20 | 0.34 | 0.41
395932320399086464 | 395932247377032320 | 4225 | 12.0 | 1.03 | 0.10 | 0.20
4461916252508018176 | 4461908070590901376 | 4114 | 13.1 | 2.07 | 0.12 | 0.28
44769884397015296 | 44767955954797312 | 3392 | 19.0 | 2.31 | 0.15 | 0.26
4517946196857223808 | 4517946192529261824 | 4649 | 17.9 | 2.29 | 0.23 | 0.27
498275652982958848 | 498275652982959104 | 3351 | 12.8 | 1.84 | 0.10 | 0.21
506944889856368256 | 506944889847883008 | 3637 | 40.8 | 2.80 | 0.28 | 0.45
52639226555782656 | 52639153540042496 | 3877 | 31.9 | 3.73 | 0.20 | 0.28
549835601498785536 | 549788563014393728 | 3514 | 39.3 | 3.73 | 0.20 | 0.24
569795207875481344 | 569795203578976128 | 3377 | 72.9 | 5.45 | 0.18 | 0.31
583948877461123584 | 583949251122623232 | 3462 | 66.4 | 2.23 | 0.10 | 0.25
594439627139102720 | 594439519764767232 | 4713 | 13.0 | 0.44 | 0.04 | 0.04
6033508695270898432 | 6033508690936569344 | 3472 | 239.8 | 0.40 | 0.02 | 0.02
6236729088627705216 | 6236729329145889792 | 5412 | 26.0 | 6.78 | 0.15 | 0.22
3961323073827903360 | 3961323069532233984 | 3844 | 12.9 | 3.75 | 0.28 | 0.34
692919207148481920 | 692919202851887104 | 3307 | 130.4 | 4.18 | 0.31 | 0.55
748247452595214336 | 748247456890075136 | 3086 | 25.8 | 3.38 | 0.17 | 0.21
765964300065414144 | 765965051684223360 | 3392 | 60.0 | 3.85 | 0.20 | 0.30
77783305135054720 | 77782544925602432 | 3239 | 68.2 | 3.49 | 0.20 | 0.25
790463549441746944 | 790463545141273984 | 3600 | 112.4 | 2.80 | 0.14 | 0.31
4371782034475387776 | 4371782030179914624 | 3263 | 26.3 | 3.67 | 0.22 | 0.32
823582763810855168 | 823582759515662592 | 3544 | 24.6 | 3.33 | 0.26 | 0.30
851088662087300992 | 851088662087301120 | 3606 | 56.8 | 5.37 | 0.35 | 0.56
855190699452263040 | 855190695156439680 | 3204 | 19.3 | 4.28 | 0.18 | 0.23
861184515292492288 | 861184515291473536 | 3234 | 26.7 | 6.31 | 0.23 | 0.47
861951493372439424 | 861951489076842880 | 3337 | 20.9 | 2.52 | 0.16 | 0.45
4465117102652542336 | 4465117098356736000 | 3657 | 39.1 | 3.14 | 0.18 | 0.27
1604198258179547264 | 1604200899583092992 | 3400 | 90.4 | 4.75 | 0.24 | 0.42
909192835093328896 | 909192830801295104 | 3320 | 15.8 | 3.36 | 0.26 | 0.40
1748817160020646784 | 1748816983925915776 | 5340 | 40.0 | 4.06 | 0.20 | 0.27
1787683727830825856 | 1787683723535019776 | 3466 | 20.8 | 3.17 | 0.16 | 0.25
181306276262296192 | 181306168886213120 | 3675 | 10.2 | 2.63 | 0.31 | 0.36
1855910211052222592 | 1855910275463661440 | 3522 | 27.7 | 1.54 | 0.10 | 0.27
1868011126788924672 | 1867262393723489536 | 3262 | 146.7 | 5.64 | 0.20 | 0.36
192725009459729152 | 192701576117862400 | 3159 | 10.9 | 4.19 | 0.31 | 0.37
19693185261940096 | 19693180966870656 | 3469 | 128.4 | 4.00 | 0.28 | 0.47
1988383701288404992 | 1985376365185555200 | 3794 | 12.2 | 4.03 | 0.22 | 0.25
1996725077535282944 | 1996725077535283200 | 3654 | 104.1 | 5.27 | 0.27 | 0.36
2053585565450552960 | 2053584770878226304 | 5293 | 13.5 | 0.68 | 0.05 | 0.03
889041879334169344 | 889041668879549696 | 3643 | 18.1 | 3.81 | 0.18 | 0.24
2067604574919660416 | 2067604574919660544 | 3288 | 22.3 | 1.18 | 0.08 | 0.16
2080422720138489344 | 2080422720133121152 | 3779 | 163.1 | 2.60 | 0.23 | 0.28
2099643935736477696 | 2099656953779008000 | 4852 | 19.7 | 0.54 | 0.06 | 0.04
2101241457413709568 | 2101241281314863872 | 5438 | 25.1 | 2.09 | 0.11 | 0.23
2142368311889141248 | 2142368346248880896 | 3418 | 10.8 | 2.66 | 0.22 | 0.22

##  Appendix B Rotational Isochrones

Table 2:  Rotational isochrones constructed using a solar-metallicity, $f_{\mathrm{spots}}=0\%$ model grid. Surface rotation periods are reported in days. Effective temperature are given as computed from a 2.0 Gyr gyrochrone. $M(M_{\odot})$ |  $T_{\mathrm{eff}}$ (K) | Ages (Gyr)
---|---|---
|  | 0.5 | 1.0 | 2.0 | 3.0 | 4.0 | 5.0 | 6.0 | 7.0 | 8.0 | 9.0 | 10.0
0.18 | 3200 | 0.48 | 1.13 | 6.36 | 20.15 | 39.02 | 51.46 | 61.47 | 70.09 | 77.79 | 84.83 | 91.35
0.19 | 3221 | 0.54 | 1.29 | 7.48 | 24.15 | 42.19 | 54.6 | 64.71 | 73.48 | 81.34 | 88.54 | 95.23
0.20 | 3241 | 0.6 | 1.46 | 8.77 | 27.93 | 45.38 | 57.83 | 68.08 | 77.02 | 85.06 | 92.45 | 99.31
0.21 | 3259 | 0.67 | 1.65 | 10.25 | 31.49 | 48.61 | 61.16 | 71.6 | 80.74 | 88.98 | 96.56 | 103.6
0.22 | 3276 | 0.74 | 1.86 | 11.95 | 34.96 | 51.95 | 64.67 | 75.32 | 84.68 | 93.13 | 100.91 | 108.14
0.23 | 3292 | 0.82 | 2.09 | 13.83 | 38.37 | 55.39 | 68.34 | 79.25 | 88.85 | 97.54 | 105.52 | 112.95
0.24 | 3307 | 0.9 | 2.34 | 16.0 | 41.89 | 59.07 | 72.31 | 83.5 | 93.37 | 102.29 | 110.49 | 118.12
0.25 | 3321 | 0.99 | 2.62 | 18.45 | 45.54 | 63.01 | 76.6 | 88.11 | 98.27 | 107.45 | 115.89 | 123.73
0.26 | 3335 | 1.09 | 2.92 | 21.2 | 49.43 | 67.33 | 81.33 | 93.2 | 103.68 | 113.14 | 121.84 | 129.91
0.27 | 3348 | 1.2 | 3.24 | 24.27 | 53.66 | 72.13 | 86.62 | 98.91 | 109.74 | 119.52 | 128.49 | 136.82
0.28 | 3360 | 1.3 | 3.59 | 27.67 | 58.38 | 77.61 | 92.7 | 105.46 | 116.7 | 126.83 | 136.12 | 144.73
0.29 | 3372 | 1.42 | 3.96 | 31.48 | 63.92 | 84.13 | 99.93 | 113.26 | 124.96 | 135.49 | 145.14 | 154.07
0.30 | 3383 | 1.54 | 4.35 | 35.71 | 70.69 | 92.19 | 108.86 | 122.86 | 135.11 | 146.1 | 156.15 | 165.45
0.31 | 3394 | 1.66 | 4.78 | 40.4 | 79.62 | 102.9 | 120.68 | 135.48 | 148.37 | 159.91 | 170.43 | 180.15
0.32 | 3405 | 1.8 | 5.24 | 45.64 | 93.25 | 119.18 | 138.4 | 154.18 | 167.82 | 179.97 | 191.02 | 201.21
0.33 | 3416 | 1.95 | 5.82 | 37.21 | 109.37 | 143.24 | 165.55 | 183.09 | 197.93 | 210.99 | 222.78 | 233.61
0.34 | 3426 | 2.16 | 6.53 | 31.0 | 43.21 | 69.29 | 140.69 | 172.77 | 195.51 | 213.87 | 229.62 | 243.6
0.35 | 3437 | 2.33 | 7.61 | 31.78 | 43.93 | 51.94 | 59.44 | 66.68 | 73.45 | 79.84 | 91.44 | 153.94
0.36 | 3446 | 1.92 | 4.52 | 19.78 | 32.97 | 43.15 | 51.13 | 57.94 | 64.0 | 68.91 | 73.57 | 78.31
0.37 | 3458 | 2.13 | 5.28 | 21.2 | 32.87 | 41.58 | 48.64 | 52.94 | 58.46 | 63.63 | 68.31 | 72.61
0.38 | 3470 | 2.36 | 6.18 | 22.09 | 32.71 | 40.01 | 44.45 | 50.4 | 55.81 | 60.57 | 64.94 | 68.85
0.39 | 3484 | 2.62 | 7.3 | 22.78 | 31.99 | 37.19 | 43.13 | 48.98 | 53.96 | 58.42 | 62.35 | 65.97
0.40 | 3497 | 2.92 | 8.7 | 23.29 | 30.87 | 36.24 | 42.56 | 47.97 | 52.64 | 56.72 | 60.26 | 63.48
0.41 | 3512 | 3.25 | 10.55 | 23.75 | 30.19 | 36.27 | 42.32 | 47.35 | 51.64 | 55.34 | 58.58 | 61.35
0.42 | 3526 | 3.65 | 12.1 | 23.81 | 29.86 | 36.12 | 41.82 | 46.46 | 50.34 | 53.63 | 56.45 | 58.79