11institutetext: Departamento de Física, Universidad Técnica Federico Santa
María, Av. España 1680, Valparaíso, Chile
11email<EMAIL_ADDRESS>22institutetext: Millenium Nucleus for
Planet Formation, Valparaíso, Chile 33institutetext: Department of Physics &
Astronomy, University of Wyoming, Laramie, WY 82071 44institutetext:
Department of Astronomy, California Institute of Technology, 1200 E.
California Blvd., Pasadena, CA 91125, USA 55institutetext: Department of
Astronomy and Theoretical Astrophysics Center, University of California,
Berkeley, CA 94720, USA

# Evidence for saturated and disrupted magnetic braking from samples of
detached close binaries with M and K dwarfs

Diogo Belloni 11 Matthias R. Schreiber 1122 Maxwell Moe 33 Kareem El-Badry 44
Ken J. Shen 55

(Received…; accepted …)

###### Abstract

Context.  Recent observations of close detached eclipsing M and K dwarf
binaries have provided substantial support for magnetic saturation when stars
rotate sufficiently fast, leading to a magnetic braking (MB) torque
proportional to the spin of the star.

Aims.  We investigated here how strong MB torques need to be to reproduce the
observationally-inferred relative numbers of white dwarf plus M dwarf post-
common-envelope binaries under the assumption of magnetic saturation.

Methods.  We carried out binary population simulations with the BSE code
adopting empirically-derived inter-correlated main-sequence binary
distributions as initial binary populations and compared the simulation
outcomes with observations.

Results.  We found that the dearth of extreme mass ratio binaries in the
inter-correlated initial distributions is key to reproduce the large fraction
of post-common-envelope binaries hosting low-mass M dwarfs (${\sim 0.1-0.2}$
M⊙). In addition, orbital angular momentum loss rates due to MB should be high
for M dwarfs with radiative cores and orders of magnitude smaller for fully
convective stars to explain the observed dramatic change of the fraction of
short-period binaries at the fully convective boundary.

Conclusions.  We conclude that saturated but disrupted, that is, dropping
drastically at the fully convective boundary, MB can explain the observations
of both close main-sequence binaries containing M and K dwarfs and post-
common-envelope binaries. Whether a similar prescription can explain the spin
down rates of single stars and of binaries containing more massive stars needs
to be tested.

###### Key Words.:

binaries: close – methods: numerical – stars: evolution – white dwarfs

##  1 Introduction

Understanding how a magnetized wind extracts angular momentum from a star, so
called magnetic braking, is a key ingredient to understand the evolution of
close binaries as important as cataclysmic variables (CVs), low-mass X-ray
binaries, ultra-compact X-ray binaries, or double white dwarfs. Despite this
importance, the strength and main dependencies of magnetic braking, in
particular on the mass and rotation period of the star, remain puzzling.

In early studies magnetic braking was calibrated using the spin down rates of
solar-type stars (Skumanich 1972) but recently it has become clear that, in
particular for lower mass main-sequence stars, the situation is more
complicated (e.g. Barnes 2003; Newton et al. 2016) which most likely hints
towards different and mass dependent magnetic braking laws. One frequently
discussed attempt to describe magnetic braking is based on the observation
that chromospheric activity, coronal X-ray emission, flare activity, and
magnetic field strengths in low-mass main-sequence stars are correlated and
increase with rotation up to a mass dependent critical rotation rate above
which the relation between activity and rotation saturates. The assumption
that these observables also relate with magnetic braking led to postulating
saturated magnetic braking prescriptions in which the dependence of the
magnetic braking torque on the spin period becomes shallower above a given
rotation rate (e.g. Chaboyer et al. 1995; Sills et al. 2003; Andronov et al.
2003).

In binaries with orbital periods shorter than $\sim 5-10$ d (e.g. Fleming et
al. 2019), the spin period is synchronised with the orbital period and
magnetic braking therefore leads to orbital angular momentum loss. Changes in
the orbital period, or distributions of representative samples of close binary
stars, which are in principle easier to measure than rotation rates of single
stars, can therefore be used to constrain the dependencies and strength of
magnetic braking.

In semi-detached binary stars with a white dwarf accreting from a main-
sequence star companion, the previously mentioned CVs, magnetic braking is
often assumed to be absent (or very weak) in case the donor star is fully
convective but very efficient in case it still has a radiative core. The
prescription used for binary population models of CVs, called disrupted
magnetic braking, assumes efficient magnetic braking from Rappaport et al.
(1983), which is based on the Skumanich (1972) prescription, combined with the
assumption that magnetic braking becomes inefficient as soon as the main-
sequence star becomes fully convective. These assumptions reasonably well
explain the increased radii of the main-sequence stars at long orbital periods
($\sim 3-5$ hr), a dearth of systems in the orbital period range of $\approx
2-3$ hr, and most of the mass transfer rates derived from observations below
the period gap (e.g. Knigge et al. 2011; McAllister et al. 2019; Pala et al.
2017, 2022).

However, CVs are far from ideal systems to constrain magnetic braking because
of two main reasons. First, mass transfer may be increased by so-called
consequential angular momentum loss (Schreiber et al. 2016) which is not well
understood and contaminates the relation between mass transfer rates and
angular momentum loss due to magnetic braking. Second, mass transfer rates are
difficult to measure and the few measurements that are available show quite a
large scatter at a given orbital period. It might therefore not be too
surprising that the current model for CV evolution fails to explain some
important observables (e.g. Belloni et al. 2020; Pala et al. 2020; Fuentes-
Morales et al. 2021; Pala et al. 2022; Belloni & Schreiber 2023a).

Instead of using single stars or CVs we here combine observational constraints
from the two cleanest and most suitable types of system towards a better
understanding of magnetic braking. The first has been provided by Schreiber et
al. (2010). They observed a large number of detached binaries consisting of a
white dwarf with an M dwarf companion and find a strong dependence of the
relative number of short orbital period systems, which are post-common-
envelope binaries (PCEBs), on the mass of the main-sequence star. This
measurement is very clean because the orbital period evolution of these
systems is not affected by mass transfer, the masses of the stellar components
can relatively easily be estimated, and we know that the orbital period
distribution of the PCEBs peaks at a few hours and that there are very few
systems with orbital periods exceeding one day (Nebot Gómez-Morán et al.
2011). Using these systems to constrain angular momentum loss through magnetic
braking has been suggested more than a decade ago (e.g. Politano & Weiler
2006; Zorotovic et al. 2010) but no dedicated simulations have ever been
performed.

The second clean observational constraint that we take into account here comes
from eclipsing close main-sequence binaries (El-Badry et al. 2022). The
observed orbital period distributions of these systems provide evidence for a
magnetic braking torque that has a shallower dependence on the star spin than
assumed by Rappaport et al. (1983) and can be reasonably well understood
assuming saturated magnetic braking laws. While the eclipsing main-sequence
binary star sample is very useful to constrain the dependence of magnetic
braking on the spin period, it is less sensitive to its strength or possible
dependencies on the stellar mass. In this work we investigate if a
prescription of (saturated) magnetic braking exists that can explain these two
critical observational constraints from close detached binary stars using
binary population synthesis.

##  2 Initial binary populations

In previous population models of white dwarf plus main-sequence star binaries
(PCEBs and CVs), the initial binary population (IBP) has been based on
uncorrelated distributions of initial parameters for the main-sequence
binaries (e.g. de Kool 1992; de Kool & Ritter 1993; Willems & Kolb 2004;
Politano & Weiler 2006; Davis et al. 2010; Toonen & Nelemans 2013; Zorotovic
et al. 2014; Camacho et al. 2014; Schreiber et al. 2016; Cojocaru et al. 2017;
Belloni et al. 2018, 2020). However, assumptions for uncorrelated
distributions are not consistent with observational surveys performed in the
past decades. In addition, the measurement we aim to compare with predictions
in this work is the fraction of PCEBs among white dwarf plus M dwarf binaries
identified by the Sloan Digital Sky Survey (Schreiber et al. 2010), that is,
the observable we want to explain is the number of close binaries that evolved
through common-envelope evolution among all spectroscopically identified white
dwarf plus M dwarf binaries, which includes wide (but still unresolved)
systems. Therefore, the number of wide systems, that is, systems in which the
two stars are so far apart that they never interacted, is as important as the
number of PCEBs which implies that we need to consider a large range of
initial orbital periods.

Moe & Di Stefano (2017, hereafter MD17) and Offner et al. (2023) investigated
dozens of surveys related to main-sequence binaries and, after combining the
samples from such surveys and correcting for their respective selection
effects, concluded that the distributions of periods, masses, and mass ratios
are not independent at a statistically significant level and fitted joint
probability density functions ${f(M_{1},q,P_{\rm orb},e)\neq
f(M_{1})f(q)f(P_{\rm orb})f(e)}$ to the corrected distributions, where $M_{1}$
is the primary mass, $q=M_{2}/M_{1}$ is the mass ratio and $M_{2}$ is the
secondary mass, $P_{\rm orb}$ is the orbital period and $e$ is the
eccentricity. These fitted correlated distributions are the most realistic
ones currently available and should be incorporated into binary population
models. In addition, we expect them to be crucial if a large range of initial
parameters play an important role as in the case of the present
investigations. We incorporated the distributions published by MD17 and
compared the resulting predictions with those of the standard
“flat”uncorrelated distributions and the observations.

###  2.1 Standard “flat” uncorrelated distributions

The IBPs commonly adopted in population synthesis related to PCEBs and CVs are
very simplistic, as the focus is often on testing parameters of binary
evolution, rather than inspecting how the assumed initial distributions might
affect the results. Apart from the initial mass function, distributions for
the mass ratio, separation and eccentricity are usually assumed to be flat,
being the separation flat in log-scale and the eccentricity flat in squared-
scale (i.e. thermal eccentricity distribution). As we intend to test the
impact of correlated distributions derived by MD17 (Sect. 2.2), we also
perform binary population synthesis with these commonly adopted flat initial
distributions.

Our “flat” IBP corresponds to ${\approx 6.15\times 10^{5}}$ zero-age main-
sequence binaries characterized by: (i) $M_{1}$ is obtained from the canonical
Kroupa (2001) initial mass function, in the range $[1,8]$ M⊙, as most white
dwarf progenitors belong to this mass range; (ii) $M_{2}$ is obtained assuming
a uniform mass ratio distribution, where $M_{2}\leq M_{1}$, and requesting
that $M_{2}\geq 0.069$, in order to avoid sub-stellar secondaries; (iii) the
semi-major axis $(a)$ follows a log-uniform distribution in the range between
$1.1$ times the sum of the radii of the two stars and $10^{5.5}$ R⊙; and (iv)
the eccentricity is uniform in $e^{2}$, which means that $e$ follows a thermal
distribution in the range $[0,1]$.

###  2.2 Inter-correlated distributions of Moe & Di Stefano (2017)

In our second IBP, we also selected ${\approx 6.15\times 10^{5}}$ zero-age
main-sequence binaries and picked $M_{1}$ from the canonical Kroupa (2001)
initial mass function in the range between $1$ and $8$ M⊙. However, we adopted
the correlated distributions derived by MD17, in which the $P_{\rm orb}$
distribution depend critically on $M_{1}$ and the binary fraction and $e$ and
$q$ distributions depend on both $P_{\rm orb}$ and $M_{1}$.

We incorporate the frequency $f_{{\rm log}P_{\rm orb};q>0.1}(M_{1},P_{\rm
orb})$ of companions with mass ratio ${q>0.1}$ per decade of $P_{\rm orb}$
from MD17, but made three slight modifications for the current study. First,
we convolve their analytic relations for $f_{{\rm log}P_{\rm orb};q>0.1}$ with
a Gaussian 2D kernel of width ${\delta\log({P_{\rm orb}/{\rm d})}=0.4}$ and
${\delta\log({M_{1}}/{\rm M}_{\odot})=0.1}$ to ensure smooth transitions
across the parameter space. Second, we interpolate between their value of
$f_{{\rm log}P_{\rm orb};q>0.1}$ at ${P_{\rm orb}=2}$ d and zero at the
minimum possible orbital period, which corresponds to Roche-lobe filling on
the zero-age main sequence. Finally, we extend their distributions toward
slightly longer periods ${\log(P_{\rm orb}/{\rm d})=9}$.

For ${M_{1}=1}$ M⊙ primaries, the binaries follow a log-normal period
distribution with peak of ${f_{{\rm log}P_{\rm orb};q>0.1}=0.12}$ at
${\log(P_{\rm orb}/{\rm d})=5}$ (${\approx 50}$ au) and a value of ${f_{{\rm
log}P_{\rm orb};q>0.1}=0.03}$ at ${\log(P_{\rm orb}/{\rm d})=1}$. By
integrating ${f_{{\rm log}P_{\rm orb};q>0.1}}$ across all orbital periods, the
mean multiplicity frequency of companions per ${M_{1}=1}$ M⊙ primary is
${f_{\rm mult}=0.58}$. This orbital period distribution and multiplicity
frequency are consistent with previous parameterizations of solar-type
binaries (Duquennoy & Mayor 1991; Raghavan et al. 2010).

Meanwhile, the frequency of companions to more massive primaries is larger,
especially at short orbital periods. For ${M_{1}=8}$ M⊙, we find a peak of
${f_{{\rm log}P_{\rm orb};q>0.1}=0.28}$ at ${\log(P_{\rm orb}/{\rm d})=4}$
(${\approx 10}$ au), a value of ${f_{{\rm log}P_{\rm orb};q>0.1}=0.16}$ at
${\log(P_{\rm orb}/{\rm d})=1}$, and a multiplicity frequency of ${f_{\rm
mult}=1.6}$. These statistics are consistent with the observed properties of
B-type main-sequence binaries (Abt et al. 1990; Rizzuto et al. 2013). A
significant fraction of PCEBs evolve from main-sequence binaries with
intermediate orbital periods ${\log{\rm(P_{\rm orb}/d)}=2-4}$. Across this
period interval, the frequency of companions increases by nearly a factor of
four between ${M_{1}=1}$ M⊙ and ${M_{1}=8}$ M⊙ primaries.

Figure 1: Mass-ratio distribution of zero-age main-sequence binaries with
orbital period ${P_{\rm orb}=10^{2}-10^{4}}$ d, separated according to the
primary mass $M_{1}$.

The power-law slope $\alpha$ of the eccentricity $e$ distribution
${f_{e}\propto e^{\alpha}}$ also depends on the primary mass and orbital
period. We adopt the parameterization for ${\alpha(M_{1},P_{\rm orb}}$) as
given in MD17. We also limit the distribution to the maximum eccentricity
$e_{\rm max}$ without Roche-lobe filling at periastron. Specifically, we adopt
the power-law slope $\alpha$ across the interval ${0<e<0.7e_{\rm max}}$, and
then model $f_{e}$ as a linearly declining function until it reaches zero at
${e=e_{\rm max}}$.

We adopt the three-component mass-ratio distribution $f_{q}$ as described in
MD17: a power-law slope across small mass ratios ${q=0.1-0.3}$, a power-law
slope across large mass ratios ${q=0.3-1.0}$, and an excess fraction of twins.
All three components of the mass-ratio distribution depend on $M_{1}$ and
$P_{\rm orb}$. When modelling PCEBs, we are particularly concerned with the
mass-ratio distribution across intermediate orbital periods ${P_{\rm
orb}=10^{2}-10^{4}}$ d, which we display in Fig. 1. In the current study, we
model the twin component as a function that quickly rises from ${q=0.85}$ and
peaks at ${q=1.0}$ (Tokovinin 2000; Halbwachs et al. 2003; Moe & Di Stefano
2013). The mass-ratio distribution of solar-type binaries with intermediate
orbital periods is close to flat, possibly with a small excess at ${q=0.3}$
and ${q>0.85}$ (Duquennoy & Mayor 1991; Halbwachs et al. 2003; Raghavan et al.
2010). Meanwhile, the mass-ratio distribution of A-type and B-type main-
sequence binaries with intermediate orbital periods is considerably skewed
towards ${q=0.3}$ with a flattening across ${q=0.1-0.3}$ (Rizzuto et al. 2013;
Gullikson et al. 2016; Murphy et al. 2018). Combining all the statistics,
there are ${\approx 8}$ times more companions with $q=0.2$ and ${\log{(P_{\rm
orb}/{\rm d})}=2-3}$ around ${M_{1}=8}$ M⊙ primaries compared to their
counterparts around ${M_{1}=1}$ M⊙ primaries.

We incorporate the frequency $f_{{\rm log}P_{\rm orb};q>0.1}(M_{1},P_{\rm
orb})$ of companions with mass ratio ${q>0.1}$ per decade of $P_{\rm orb}$ as
explained above. For ${q<0.1}$, the statistics are unfortunately poorly
constrained from observations. At separations ${\lesssim 1}$ au, there is a
brown dwarf desert, that is, an extreme deficit of ${q<0.1}$ companions to
both solar-type primaries (Grether & Lineweaver 2006; Ma & Ge 2014) and
A/early-F primaries (Murphy et al. 2018). Across separations ${\sim 10-100}$
au, the companion mass distribution is relatively uniform across
${q=0.01-0.2}$, that is, there are about as many ${q=0.01-0.1}$ companions as
there are ${q=0.1-0.2}$ companions (Wagner et al. 2019; Nielsen et al. 2019).
There are no constraints on ${q<0.1}$ companions to B stars except for beyond
separations ${>100}$ au. Finally, there are no good constraints on ${q<0.1}$
companions of any spectral type across separations ${\sim 1-10}$ au, which is
precisely the range from which most PCEBs should originate.

Binary formation models of disk fragmentation, accretion, and migration can
reproduce many of the observed features of the initial binary population,
including the brown dwarf desert at separations ${<1}$ au and the sizeable
population of brown dwarf companions beyond ${>10}$ au (Tokovinin & Moe 2020).
However, the transition between these two regimes is substantially model
dependent. Given the incompleteness and selection effects of the observed
samples and the systematic errors in the physical modelling, the occurrence
rate of brown dwarf companions to B–G primaries across separations of ${\sim
1-10}$ au is uncertain by a factor of ${\sim 3}$.

We assume here an absence of companions with ${q<0.1}$ (e.g. Duchêne et al.
2023), but bear in mind that some systems must have come from extreme mass
ratio binaries. In particular, there are several observed PCEBs hosting brown
dwarfs (Zorotovic & Schreiber 2022), which could only be explained if they
originated from zero-age main-sequence binaries with ${q<0.1}$. Similarly,
PCEBs with massive WDs (${\gtrsim 1}$ M⊙) having M dwarf companions should
come from zero-age main-sequence binaries with ${q<0.1}$ as the WD progenitors
in these cases are initially more massive than ${\sim 5}$ M⊙. Despite that,
the contribution of systems with ${q<0.1}$ to the total population of PCEBs is
most likely negligible as the fraction of PCEBs with either high-mass WDs or
brown dwarfs makes up only a few per cent (Zorotovic et al. 2010; Nebot Gómez-
Morán et al. 2011; Brown et al. 2023).

When compared with the “flat” uncorrelated IBP described in Sect. 2.1, the
inter-correlated distributions of Moe & Di Stefano (2017) contain two
improvements, which are crucial for the binary star populations we investigate
in this paper. The first one is the dearth of binaries with ${q<0.1}$ we have
just described above. The other one is the dependence of the binary properties
on the primary mass. In particular, with increasing primary mass, the binary
fraction increases across separations of ${1-10}$ au and their mass-ratio
distribution becomes skewed toward ${q\sim 0.3}$, with the aforementioned
turnover below ${q<0.1}$. This implies that the fraction of binaries that
evolve via common-envelope evolution with M-dwarf companions is substantially
higher than in the other IBP. The remaining features of the IBP from Moe & Di
Stefano (2017) have little impact on the binary populations considered in this
work.

##  3 Binary population models

For both IBPs (Sect. 2) we carried out binary population synthesis using the
BSE code assuming solar metallicity (i.e. ${Z=0.02}$) and a constant star
formation rate (e.g. Weidner et al. 2004; Kroupa et al. 2013; Recchi & Kroupa
2015; Schulz et al. 2015) over the age of the Galactic disc (${\approx 10}$
Gyr, Kilic et al. 2017). For the common-envelope evolution, we adopted an
efficiency of $0.25$, that is, we assumed that $25$% of the change in orbital
energy during the spiral-in is used to unbind the common envelope, with no
contributions from other energy sources, which is consistent with the
increasing evidence that PCEB progenitors experience strong orbital shrinkage
during common-envelope evolution (e.g. Zorotovic et al. 2010; Toonen &
Nelemans 2013; Camacho et al. 2014; Cojocaru et al. 2017; Belloni et al. 2019;
Hernandez et al. 2022; Zorotovic & Schreiber 2022; Scherbak & Fuller 2023).
The binding energy parameter was calculated according to fitting scheme
provided by Claeys et al. (2014, their Appendix A), which is based on the
detailed numerical stellar evolution calculations by Dewi & Tauris (2000) and
takes into account the structure and the evolutionary stage of the red giant
donor. All other stellar and binary evolution parameters not clearly mentioned
are set as the standard in BSE (e.g. Hurley et al. 2002; Belloni et al. 2018;
Banerjee et al. 2020). Finally, in this work, we only considered PCEBs that
are pre-CVs, that is, we do not consider detached CVs crossing the period gap.

After a PCEB is formed, it evolves towards shorter periods through orbital
angular momentum loss. In addition to magnetic braking, we also included
emission of gravitational waves as mechanism to remove orbital angular
momentum as described in Hurley et al. (2002, section 2.4, equation 48).
Regarding magnetic braking, we adopted the following prescription with
magnetic saturation, which was first proposed by Chaboyer et al. (1995),

$\dot{J}_{\rm MB,SAT}\,=\,-\beta\left(\frac{R_{2}}{{\rm
R}_{\odot}}\,\frac{{\rm
M}_{\odot}}{M_{2}}\right)^{1/2}\left\\{\begin{array}[]{ll}\Omega_{2}^{3},&{\rm
if}\;\Omega_{2}\leq\Omega_{\rm crit},\\\ \Omega_{2}\,\Omega_{\rm
crit}^{2},&{\rm if}\;\Omega_{2}>\Omega_{\rm crit},\end{array}\right.$ (1)

where ${\beta=2.7\times 10^{47}}$ erg s-1 (Andronov et al. 2003), and $M_{2}$,
$R_{2}$, and $\Omega_{2}$ are the mass, radius, and spin frequency (in s-1) of
the main-sequence star, respectively. $\Omega_{\rm crit}$ is the threshold
angular velocity beyond which saturation occurs and is assumed to be (El-Badry
et al. 2022)

$\Omega_{\rm crit}\ =\ 10\,\Omega_{\odot}\left(\frac{\tau_{\odot}}{\tau_{\rm
2}}\,\right),$ (2)

where ${\Omega_{\odot}=3\times 10^{-6}\leavevmode\nobreak\ {\rm s}^{-1}}$ and
$\tau_{\rm 2}$ is the convective turnover time-scale of the main-sequence star
given by (Wright et al. 2011)

$\log_{10}\left(\frac{\tau_{\rm 2}}{\rm
d}\right)=1.16-1.49\log_{10}\left(\frac{M_{2}}{{\rm
M}_{\odot}}\right)-0.54\log_{10}^{2}\left(\frac{M_{2}}{{\rm
M}_{\odot}}\right).$ (3)

According to Eqs. 3 and 2, the lower the mass of the main-sequence star, the
longer the convective turnover time-scale and the longer the critical spin
period below which magnetic braking is saturated. In particular, the
saturation spin period is as long as ${\sim 21.6}$ d, for a $0.1$ M⊙ star, and
as short as ${\sim 2.85}$ d, for a $0.9$ M⊙ star.

To test different strength of magnetic braking and different levels of
disrupted magnetic braking, we introduced two multiplicative factors. First,
we add a factor $K$ with which we can scale the strength of magnetic braking.
Second, for fully convective stars, that is, those less massive than ${\sim
0.35}$ M⊙, we added an additional parameter $\eta$ to the expression such that
magnetic braking is reduced by a factor of $\eta$ for these stars.

$\dot{J}_{\rm MB}\,=\,\left\\{\begin{array}[]{ll}K\,\dot{J}_{\rm SAT}\,,&{\rm
if}\;M_{2}>0.35\,{\rm M}_{\odot},\\\ &\\\ \left(K\,\dot{J}_{\rm
SAT}\right)/\eta\,,&{\rm if}\;M_{2}\leq 0.35\,{\rm
M}_{\odot}\leavevmode\nobreak\ ({\rm fully\leavevmode\nobreak\
convective}).\end{array}\right.$ (4)

We here focus on this prescription and test whether it can also explain the
fraction of PCEBs amongst the entire population of white dwarf plus M dwarf
binaries, and for which combination of $K$ and $\eta$. Naturally, Eq. 4
reduces to Eq. 1 when ${K=1}$ and ${\eta=1}$. In addition, magnetic braking
becomes entirely disrupted for fully convective M dwarfs when
${\eta\rightarrow\infty}$ is assumed.

##  4 Results

Figure 2: Comparison between the observed fractions of PCEBs amongst white
dwarf plus M dwarf binaries across the M dwarf mass (Schreiber et al. 2010)
and the predicted with Eq. 4 for several combinations of model parameters.
Both predicted and observed fractions were normalized at their highest values.
Each panel corresponds to a different choice for $K$, while the line colour
and type indicate the assumed value of $\eta$. We can see that any combination
of the parameters such that ${K\gtrsim 50}$ and ${\eta\gtrsim 50}$ is able to
reasonably well explain the high fraction of systems for M dwarf masses
${\lesssim 0.3}$ M⊙ as well as the huge reduction of systems at $\sim 0.5$ M⊙.
Figure 3: Similar to 2, but assuming the inter-correlated IBP derived by MD17.
Analogous to what we have found while adopting an IBP draw from flat
distributions, any combination of the parameters such that ${K\gtrsim 50}$ and
${\eta\gtrsim 50}$ is able to reasonably well explain the huge reduction of
systems at ${\sim 0.5}$ M⊙. However, with a more realistic IBP, we can also
reproduce the high fraction of systems for M dwarf masses ${\lesssim 0.3}$ M⊙.
Figure 4: Comparison between the observed orbital period distributions for
different mass bins (El-Badry et al. 2022) and the predicted assuming ${K=50}$
and several values of $\eta$. The line types and colours indicate the assumed
value of $\eta$, while each panel corresponds to a different primary mass bin,
being ${0.10-0.30}$ M⊙ (top left panel), ${0.30-0.40}$ M⊙ (top right panel),
${0.40-0.65}$ M⊙ (bottom left panel), and ${0.65-0.90}$ M⊙ (bottom right
panel). It is clear from the figure, especially when the binaries host only
fully convective stars (top left panel), that the strength of magnetic braking
does not strongly contribute to shape the distributions. On the other hand,
the orbital period distribution of main-sequence binaries is strongly affected
by how the magnetic braking torque depends on the star spins as shown by El-
Badry et al. (2022). Figure 5: Angular momentum loss rates obtained with the
often adopted RVJ prescription (right panel) and the saturated and disrupted
prescription (Eq. 4), assuming ${K=\eta=50}$ (left panel), for different main-
sequence star masses and fixed white dwarf mass of $0.6$ M⊙. The torques
provided by the RVJ prescription strongly depend on the orbital period, unlike
the saturated prescription. For main-sequence stars with radiative cores that
are near Roche lobe filling, both prescriptions lead to comparable angular
momentum loss rates. For fully convective main-sequence stars, magnetic
braking according to the saturated and disrupted prescription is order of
magnitude weaker in comparison with the non-fully convective counterparts.

We show in Figs. 2 and 3 the fractions of PCEBs amongst white dwarf plus M
dwarf binaries as function of the M dwarf mass for several combinations of the
model parameters $K$ and $\eta$ and for the two IBPs we adopted and compare
with the observationally-inferred fractions (Schreiber et al. 2010). The
observational and all predicted distributions are normalized at their highest
fractions since our goal is to reproduce the qualitative shape of the observed
distribution as well as the relative changes in the fractions across the M
dwarf mass. From this comparison we can derive clear constraints on the
assumed IBPs as well as the strengths and mass dependencies of magnetic
braking.

###  4.1 Importance of the dearth of extreme mass ratio binaries

The observed fraction of PCEBs remains roughly constant for M dwarfs masses
smaller than ${\sim 0.3}$ M⊙, that is, for fully convective M dwarfs. From
this point on, the larger the M dwarf mass the lower the fraction of PCEBs.

Assuming uncorrelated flat IBP (Fig. 2), we can clearly see that the high
fraction of PCEBs with M dwarf of masses ${\sim 0.1-0.2}$ M⊙ cannot be
reproduced, irrespective of the assumed strength of magnetic braking. While
the observed fraction remains very high for the lowest mass main-sequence
stars, the simulations predict a significant decrease of the fractions of
PCEBs towards lower main-sequence star masses.

This decrease is easy to understand. Assuming a flat IBP, for M dwarf masses
of ${\sim 0.1-0.2}$ M⊙, most white dwarf plus M dwarf binaries have progenitor
mass ratios smaller than ${\sim 0.1}$. The initially very wide systems of
these extreme mass ratio systems, that is, those in which both stars do not
interact, become wide white dwarf plus main-sequence binaries regardless of
the main-sequence star mass. In contrast, the fraction of PCEB progenitors
with such extreme mass ratios that survive common-envelope decreases with
decreasing mass of the main-sequence star. This lowers the number of PCEBs
towards smaller masses and in turn the fraction of PCEBs with respect to all
white dwarf plus main-sequence binaries. Given that this feature of the
predicted distributions occurs independent on the assumed magnetic braking
parameters, it indicates that the assumed flat IBP is probably wrong.

Indeed, when comparing the results of both IBPs, we can see that there is no
problem at all to reproduce the high fraction of PCEBs with M dwarf companions
of masses ${\sim 0.1-0.2}$ M⊙ in the case of the MD17 IBP (Fig. 3). This is a
consequence of the previously mentioned lower limit in the mass ratio
distribution at ${q=0.1}$ in this IBP, which is motivated by observations
(e.g. Grether & Lineweaver 2006; Moe & Di Stefano 2015; Murphy et al. 2018).
In other words, the PCEB fraction among white dwarf plus main-sequence
binaries with low-mass M dwarfs can be explained if the dearth of extreme mass
ratio systems, for solar type stars often called the brown dwarf desert, is
taken into account.

One may think that the lack of binaries with ${q<0.1}$ in the IBP should
reduce both the number of PCEBs with low-mass M dwarfs and the number of wide
systems with low-mass M dwarfs, and for this reason there should be no
differences at all between both IBPs adopted here. However, by removing
${q<0.1}$ binaries from the IBP, those binaries that are progenitors of PCEBs
with low-mass M dwarfs can more easily survive CE evolution since these low-
mass M dwarfs are on average more massive. This removal of ${q<0.1}$ binaries
results in a reduction in the number of PCEBs with low-mass M dwarfs (a factor
of ${\lesssim 4}$) that is smaller than the reduction in the number of wide
systems (a factor of ${\sim 10}$). This leads in turn to the above mentioned
agreement between prediction and observations. We confirmed that by running
the same models for the flat IBP but enforcing that ${q>0.1}$ always, and we
obtained fractions similar to those shown in Fig. 3.

###  4.2 Evidence for disrupted magnetic braking

Towards main-sequence star masses larger than $0.2$ M⊙, the observed fraction
of PCEBs continuously decreases. At an M dwarf mass of ${\sim 0.5}$ M⊙, the
observed fraction of PCEBs has dropped by a factor of ${5-10}$ compared to
systems with low-mass M dwarfs.

The observed fraction of PCEBs with M dwarfs of masses ${\gtrsim 0.3}$ M⊙ can
be fairly well reproduced regardless of the adopted IBP, as long as the
magnetic braking torque is sufficiently strong, which occurs when ${K\gtrsim
50}$, and drops at the fully convective boundary by at least a factor of
${\eta\sim 50}$. This result holds for both IBPs because, in this mass range,
the common-envelope survival rate is not dramatically affected by the number
of binaries with extreme mass ratios.

Therefore, the drop of the fraction of PCEBs with respect to white dwarf plus
main-sequence binaries is consistent with the disrupted magnetic braking
scenario, i.e. because the orbital angular momentum loss due to magnetic
braking drastically decreases at the fully convective boundary. Efficient
magnetic braking causes zero-age PCEBs with main-sequence stars that have a
radiative core to evolve into CVs on a short time-scale compared to PCEBs with
fully convective main-sequence stars. We will discuss in more detail in what
follows how $K$ and $\eta$ shape the distribution of the fraction of PCEBs
across the M dwarf mass using the outcomes when the MD17 IBP is adopted.

###  4.3 Evidence for strong and weak magnetic braking

Comparison with the observations does not only provide evidence for disrupted
magnetic braking for fully convective main-sequence stars but also an
increased strength of magnetic braking for main-sequence stars with a
radiative core. Assuming ${K=1}$ or ${K=10}$ (top panels of Fig. 3), the high
fractions of PCEBs hosting fully convective M dwarfs can be nicely reproduced
for all values of $\eta$, except ${\eta=1}$. However, even no magnetic braking
(${\eta\rightarrow\infty}$) does not provide evolutionary time scales
different enough to reproduce the decrease at of the fraction of PCEBs at the
fully convective boundary.

Only if the strength of magnetic braking is increased by a factor of at least
${K\sim 50}$ and disrupted (i.e. ${\eta\gtrsim 50}$), the relative numbers of
PCEBs with fully convective M dwarfs and the number of those with more massive
M dwarfs agree with the observations. In other words, magnetic saturation
alone cannot account for the relatively larger number of PCEBs with fully
convective M dwarfs in comparison with PCEBs with more massive M dwarfs.
Magnetic braking needs to be stronger (${K\gtrsim 50}$) for main-sequence
stars with radiative cores than provided by the standard saturated magnetic
braking prescription (i.e. Eq. 1).

###  4.4 Main-sequence binaries

In the previous subsections we have provided clear evidence for disrupted
magnetic braking from observations of close binary stars. At this point one
might be wondering whether the constraints for the detached eclipsing low-mass
main-sequence binaries provided by El-Badry et al. (2022) would be violated or
not by arbitrarily changing the strength of magnetic braking.

We carried out main-sequence binary population synthesis with the assumptions
described in Sect. 3. We picked the primary mass from the canonical Kroupa
(2001) initial mass function and the secondary from a uniform mass ratio
distribution, assuming a minimum mass of $0.1$ M⊙. The orbital period was also
picked from a uniform distribution assuming a maximum of $5$ d and a minimum
corresponding to a separation equal to $1.1$ times the sum of the primary and
secondary radii. The orbit was assumed to be circular, which is consistent
with the strong tidal interaction expected to take place in such close
binaries. The age of each binary was chosen from a uniform distribution
extending up to $10$ Gyr. To better compare with El-Badry et al. (2022), we
split the IBP into four sub-IBPs, each of which having $2\times 10^{5}$ main-
sequence binaries, according to the primary mass, namely (i) ${0.10-0.30}$ M⊙,
(ii) ${0.30-0.40}$ M⊙, (iii) ${0.40-0.65}$ M⊙, and (iv) ${0.65-0.90}$ M⊙.

We compare in Fig. 4 predicted and observed distributions fixing ${K=50}$. Our
simulations fit the observations as well as the one based on saturated
magnetic braking by El-Badry et al. (2022). This is not surprising because the
normalized period distribution of main-sequence binaries only constrains the
dependence of magnetic braking on the spin period which is identical in our
prescription and any other saturated magnetic braking prescription.

##  5 Discussion

We have combined two very clean constraints on orbital angular momentum loss
through magnetic braking (observations of detached eclipsing main-sequence
binaries and detached white dwarf plus M dwarf binaries) and found a purely
empirical prescription for magnetic braking that can reproduce both
observations. To explain the observed PCEB distribution, the strength of
magnetic braking needs to significantly change at the fully convective
boundary. We need a ${\gtrsim 50}$ times stronger magnetic braking for stars
that still contain a radiative core compared to fully convective stars to
explain the dramatically increased fraction of PCEBs among white dwarf plus M
dwarf binaries with fully convective main-sequence stars (see also Schreiber
et al. 2010).

To also reproduce the flat period distributions observed of main-sequence
binaries for all type of main-sequence M dwarf binary stars (El-Badry et al.
2022), both magnetic braking prescriptions need to weakly depend on the
orbital period as predicted by the saturated magnetic braking prescriptions
(e.g. Sills et al. 2003). Combining both these constraints leads to a
prescription that can simultaneously explain both observational facts. In what
follows we briefly discuss to which degree alternative explanations of the
observations might exist and the implications of our results for other types
of close binary stars.

###  5.1 Do reasonable alternative explanations exist?

If the significant decrease of the fraction of PCEBs among white dwarf plus M
dwarf binaries at the fully convective boundary was not caused by a dramatic
change in the efficiency of orbital angular momentum loss through magnetic
braking, the observations would need to be explained by previous evolutionary
effects that make the formation of PCEBs with M dwarfs with a radiative core
unlikely. This would imply that for some reason, common-envelope evolution
leads to the merger or very short post-common-envelope orbital periods of
white dwarf plus early M dwarf companions while fully convective stars are
more likely to emerge at longer periods.

This possibility, however, appears to be very unlikely. First, for a given
primary mass and orbital period, the available orbital energy is larger for
more massive secondary stars which makes it actually easier to survive the
common-envelope phase. Second, we do not see evidence for a relation between
common-envelope efficiency and main-sequence star mass in the observed samples
of PCEBs (Zorotovic et al. 2014; Zorotovic & Schreiber 2022), and last but not
least we find a significant number of descendants from PCEBs with early low-
mass main-sequence stars in observed samples of CVs (Pala et al. 2020, 2022).

Concerning the main-sequence binary orbital period distribution as measured by
El-Badry et al. (2022), we do not see any reasonable alternative explanation
than a magnetic braking prescription that depends weakly on the spin period.
One could in principle think of a flat birth distribution combined with
extremely weak magnetic braking but this would not only disagree strongly with
the fraction of PCEBs but also with observations of the spin down rates of
single stars (e.g. Newton et al. 2016). It therefore appears to us that the
two samples we analysed in this work provide solid evidence for a disrupted
and saturated magnetic braking prescription for binaries hosting main-sequence
stars with masses between $\sim 0.1-0.9$ M⊙.

###  5.2 Implications for single stars and other binary stars

The prescription of a disrupted and saturated magnetic braking we derived is
purely empirical and based on observations of binary stars with main-sequence
stars less massive than $\sim 0.5$ M⊙ and $\sim 0.9$ M⊙ in the case of the
PCEBs and main-sequence binary stars samples, respectively. In CVs, a white
dwarf accretes hydrogen rich material from a low-mass main-sequence star and
our empirical prescription should therefore properly describe angular momentum
loss through magnetic braking in CVs. We think that this is most likely the
case.

The decrease of at least a factor of ${\sim 50}$ at the fully convective
boundary will clearly generate a period gap, that is, a dearth of semi-
detached systems with orbital periods between two and three hours. This gap
occurs because of the following. Non-fully convective main-sequence stars are
driven out of thermal equilibrium due to strong magnetic braking and the
correspondingly large mass transfer rate and evolve towards shorter orbital
periods. As soon as the main-sequence star becomes fully convective, at an
orbital period of about three hours, magnetic braking becomes much less
efficient, the main-sequence star has enough time to relax to its equilibrium
radius and consequently does not continue to fill its Roche lobe, and the
binary evolves as a detached binary until mass transfer resumes at an orbital
period of about two hours. The increased main-sequence radius above the gap
(Knigge et al. 2011) as well as detached CVs crossing the gap have been
observed (Zorotovic et al. 2016). The period gap itself is evident in large
magnitude limited samples (Knigge et al. 2011) but hardly visible in volume
limited samples (Pala et al. 2020), which is likely caused by the small sample
size.

Our model should also be able to reasonably well produce mass transfer rates
similar to the observed ones above and below the period gap as magnetic
braking for main-sequence stars with radiative cores is much stronger than
that for fully convective main-sequence stars (Fig. 5). However, in general,
we advocate caution when using mass transfer rates derived from observations
of CVs to constrain magnetic braking prescriptions. This is because secular
mass transfer rates are rather difficult to measure. The radii of the main-
sequence stars (Knigge et al. 2011; McAllister et al. 2019) and the white
dwarf effective temperatures (Pala et al. 2017, 2022) have been used but for
systems with non-fully convective main-sequence stars only a handful of
reliable measurements exist. We finally note that weak and continuous
saturated magnetic braking as suggested previously will neither produce mass
transfer rates similar to the observed ones nor lead to the prediction of an
orbital period gap unless both features are produced mainly by consequential
angular momentum loss. This though seems very unlikely as consequential
angular momentum loss is regulated by the mass transfer process itself and for
this reason cannot drive CV evolution.

Other types of close binaries whose evolution is largely influenced by angular
momentum loss through magnetic braking are progenitors of AM CVn binaries,
low-mass X-ray binaries, ultra-compact X-ray binaries, and progenitors of
close binaries hosting millisecond pulsars paired with helium white dwarfs.
The characteristics of the observed samples of these binaries seem to be
explained best assuming a convection and rotation boosted (CARB) magnetic
braking (Van et al. 2019; Van & Ivanova 2019; Chen et al. 2021; Soethe &
Kepler 2021; Deng et al. 2021; Belloni & Schreiber 2023b) for stars that are
nuclear evolved and more massive than ${\sim 1}$ M⊙. While our prescription
predicts relatively strong magnetic braking for stars with a radiative core
and a convective envelope, one would need to run dedicated simulations to find
out whether it can explain the observations as well as results achieved with
the CARB prescription. Alternatively, our prescription could only be valid for
lower mass stars (${\lesssim 0.9}$ M⊙) while the CARB model applies for
nuclear evolved and more massive stars ($\gtrsim 1$ M⊙). In case of the
latter, one would need to understand the physical reasons behind this apparent
switch in the strength and dependencies of magnetic braking.

The situation is equally complicated for single stars. Braking prescriptions
derived from rotation rates of single M dwarfs in star clusters have been
claimed to not show a discontinuity at the fully convective boundary (see
Godoy-Rivera et al. 2021, their section 6.2) which, if true, contradicts our
finding. Taking a detailed look at the K2 data for the Praesepe cluster,
Douglas et al. (2017) found indications for an increased braking efficiency
for early M dwarfs relative to solar-type stars and fully convective stars
which agrees with our prescription. Samples of older field M stars indicate
that fast rotators are far more frequently found among fully convective stars
(Newton et al. 2016) which could agree with our prediction of less efficient
braking in these stars. Recently, Pass et al. (2022) analysed the rotation of
M dwarfs that are members of wide binary stars which allowed an age
determination and found that fully convective stars remain fast rotators up to
an age of ${2-3}$ Gyr but spin down at later stages. This has been interpreted
as a time dependence in the strength of angular momentum loss in young single
stars (Brown 2014). However, old M dwarfs experiencing spin–orbit interactions
as members of unresolved binaries are forced to rapid rotation and activity
persist (Pass et al. 2022). This might indicates that the results from the
spin-down of single stars might not be applicable to close binary stars.

We conclude that the empirical braking laws we derive from two samples of
detached binary stars are a suitable magnetic braking prescription for close
binaries with stars less massive than ${\lesssim 0.9}$ M⊙. To what degree the
same braking can be applied to more massive potentially nuclear evolved stars
in binaries or to single stars of the same mass remains highly uncertain.

##  6 Conclusions

We have performed population synthesis for white dwarf plus M dwarf binary
stars and found that:

i)

a magnetic braking prescription that is disrupted, i.e drops by a factor of
${\gtrsim 50}$ at the fully convective boundary, can explain the observed drop
of the fraction of post-common-envelope binaries at the fully convective
boundary;

ii)

magnetic braking needs to be stronger than assumed in standard prescriptions
of saturated magnetic braking for M dwarfs with a radiative core and weaker in
the case of fully convective main-sequence stars;

iii)

the observed large fraction of post-common-envelope binaries with low-mass
main-sequence stars (${\sim 0.1-0.2}$ M⊙) is naturally reproduced if the
dearth of extreme mass ratio binaries among main-sequence binaries is taken
into account in the initial binary distributions, that is, the brown dwarf
desert observed at separations ${<1}$ au must extend to wider separations
${\sim 1-10}$ au, albeit not as dry as observed at very close separations.

As we assumed that in addition to being disrupted, magnetic braking is
saturated, i.e depends weakly on the orbital period, the predictions of our
prescription is in reasonable agreement with the period distribution of main-
sequence binary stars. We discussed possible alternative explanations for the
observations but did not find a convincing one. We therefore conclude that
saturated and disrupted magnetic braking represents an adequate magnetic
braking law for ${\sim 0.1-0.9}$ M⊙ main-sequence stars that are members of
close binary systems. However, a physically rather than empirically motivated
saturated magnetic braking law is required to eventually understand magnetic
braking.

###### Acknowledgements.

We would like to thank an anonymous referee for the comments and suggestions
that helped to improve this manuscript. We thank the Kavli Institute for
Theoretical Physics (KITP) for hosting the program “White Dwarfs as Probes of
the Evolution of Planets, Stars, the Milky Way and the Expanding Universe” and
the Munich Institute for Astro-, Particle and BioPhysics (MIAPbP) for hosting
the program “Stellar Magnetic Fields from Protostars to Supernovae”. This
research was supported in part by the National Science Foundation under Grant
No. NSF PHY-1748958 and by the Munich Institute for Astro-, Particle and
BioPhysics (MIAPbP) which is funded by the Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation) under Germany’s Excellence Strategy –
EXC-2094 – 390783311. We thank Jim Fuller for pleasant and helpful discussions
during the KITP program. DB acknowledges financial support from FONDECYT grant
number 3220167. MRS was supported from FONDECYT grant number 1221059 and ANID,
– Millennium Science Initiative Program – NCN19_171. KJS was supported by NASA
through the Astrophysics Theory Program (80NSSC20K0544).

## References

  * Abt et al. (1990) Abt, H. A., Gomez, A. E., & Levy, S. G. 1990, ApJS, 74, 551
  * Andronov et al. (2003) Andronov, N., Pinsonneault, M., & Sills, A. 2003, ApJ, 582, 358
  * Banerjee et al. (2020) Banerjee, S., Belczynski, K., Fryer, C. L., et al. 2020, A&A, 639, A41
  * Barnes (2003) Barnes, S. A. 2003, ApJ, 586, 464
  * Belloni et al. (2019) Belloni, D., Giersz, M., Rivera Sandoval, L. E., Askar, A., & Ciecielag, P. 2019, MNRAS, 483, 315
  * Belloni & Schreiber (2023a) Belloni, D. & Schreiber, M. R. 2023a, in Handbook of X-ray and Gamma-ray Astrophysics. Edited by Cosimo Bambi and Andrea Santangelo, Springer, 129
  * Belloni & Schreiber (2023b) Belloni, D. & Schreiber, M. R. 2023b, A&A, 678, A34
  * Belloni et al. (2020) Belloni, D., Schreiber, M. R., Pala, A. F., et al. 2020, MNRAS, 491, 5717
  * Belloni et al. (2018) Belloni, D., Schreiber, M. R., Zorotovic, M., et al. 2018, MNRAS, 478, 5639
  * Brown et al. (2023) Brown, A. J., Parsons, S. G., van Roestel, J., et al. 2023, MNRAS, 521, 1880
  * Brown (2014) Brown, T. M. 2014, ApJ, 789, 101
  * Camacho et al. (2014) Camacho, J., Torres, S., García-Berro, E., et al. 2014, A&A, 566, A86
  * Chaboyer et al. (1995) Chaboyer, B., Demarque, P., & Pinsonneault, M. H. 1995, ApJ, 441, 865
  * Chen et al. (2021) Chen, H.-L., Tauris, T. M., Han, Z., & Chen, X. 2021, MNRAS, 503, 3540
  * Claeys et al. (2014) Claeys, J. S. W., Pols, O. R., Izzard, R. G., Vink, J., & Verbunt, F. W. M. 2014, A&A, 563, A83
  * Cojocaru et al. (2017) Cojocaru, R., Rebassa-Mansergas, A., Torres, S., & García-Berro, E. 2017, MNRAS, 470, 1442
  * Davis et al. (2010) Davis, P. J., Kolb, U., & Willems, B. 2010, MNRAS, 403, 179
  * de Kool (1992) de Kool, M. 1992, A&A, 261, 188
  * de Kool & Ritter (1993) de Kool, M. & Ritter, H. 1993, A&A, 267, 397
  * Deng et al. (2021) Deng, Z.-L., Li, X.-D., Gao, Z.-F., & Shao, Y. 2021, ApJ, 909, 174
  * Dewi & Tauris (2000) Dewi, J. D. M. & Tauris, T. M. 2000, A&A, 360, 1043
  * Douglas et al. (2017) Douglas, S. T., Agüeros, M. A., Covey, K. R., & Kraus, A. 2017, ApJ, 842, 83
  * Duchêne et al. (2023) Duchêne, G., Oon, J. T., De Rosa, R. J., et al. 2023, MNRAS, 519, 778
  * Duquennoy & Mayor (1991) Duquennoy, A. & Mayor, M. 1991, A&A, 248, 485
  * El-Badry et al. (2022) El-Badry, K., Conroy, C., Fuller, J., et al. 2022, MNRAS, 517, 4916
  * Fleming et al. (2019) Fleming, D. P., Barnes, R., Davenport, J. R. A., & Luger, R. 2019, ApJ, 881, 88
  * Fuentes-Morales et al. (2021) Fuentes-Morales, I., Tappert, C., Zorotovic, M., et al. 2021, MNRAS, 501, 6083
  * Godoy-Rivera et al. (2021) Godoy-Rivera, D., Pinsonneault, M. H., & Rebull, L. M. 2021, ApJS, 257, 46
  * Grether & Lineweaver (2006) Grether, D. & Lineweaver, C. H. 2006, ApJ, 640, 1051
  * Gullikson et al. (2016) Gullikson, K., Kraus, A., & Dodson-Robinson, S. 2016, AJ, 152, 40
  * Halbwachs et al. (2003) Halbwachs, J. L., Mayor, M., Udry, S., & Arenou, F. 2003, A&A, 397, 159
  * Hernandez et al. (2022) Hernandez, M. S., Schreiber, M. R., Parsons, S. G., et al. 2022, MNRAS, 517, 2867
  * Hurley et al. (2002) Hurley, J. R., Tout, C. A., & Pols, O. R. 2002, MNRAS, 329, 897
  * Kilic et al. (2017) Kilic, M., Munn, J. A., Harris, H. C., et al. 2017, ApJ, 837, 162
  * Knigge et al. (2011) Knigge, C., Baraffe, I., & Patterson, J. 2011, ApJS, 194, 28
  * Kroupa (2001) Kroupa, P. 2001, MNRAS, 322, 231
  * Kroupa et al. (2013) Kroupa, P., Weidner, C., Pflamm-Altenburg, J., et al. 2013, in Oswalt T. D., Gilmore G (, eds, Planets, Stars and Stellar Systems, Vol. 5. Springer, Berlin), 115
  * Ma & Ge (2014) Ma, B. & Ge, J. 2014, MNRAS, 439, 2781
  * McAllister et al. (2019) McAllister, M., Littlefair, S. P., Parsons, S. G., et al. 2019, MNRAS, 486, 5535
  * Moe & Di Stefano (2013) Moe, M. & Di Stefano, R. 2013, ApJ, 778, 95
  * Moe & Di Stefano (2015) Moe, M. & Di Stefano, R. 2015, ApJ, 801, 113
  * Moe & Di Stefano (2017) Moe, M. & Di Stefano, R. 2017, ApJS, 230, 15
  * Murphy et al. (2018) Murphy, S. J., Moe, M., Kurtz, D. W., et al. 2018, MNRAS, 474, 4322
  * Nebot Gómez-Morán et al. (2011) Nebot Gómez-Morán, A., Gänsicke, B. T., Schreiber, M. R., et al. 2011, A&A, 536, A43
  * Newton et al. (2016) Newton, E. R., Irwin, J., Charbonneau, D., et al. 2016, ApJ, 821, 93
  * Nielsen et al. (2019) Nielsen, E. L., De Rosa, R. J., Macintosh, B., et al. 2019, AJ, 158, 13
  * Offner et al. (2023) Offner, S. S. R., Moe, M., Kratter, K. M., et al. 2023, in Astronomical Society of the Pacific Conference Series, Vol. 534, Astronomical Society of the Pacific Conference Series, ed. S. Inutsuka, Y. Aikawa, T. Muto, K. Tomida, & M. Tamura, 275
  * Pala et al. (2022) Pala, A. F., Gänsicke, B. T., Belloni, D., et al. 2022, MNRAS, 510, 6110
  * Pala et al. (2020) Pala, A. F., Gänsicke, B. T., Breedt, E., et al. 2020, MNRAS, 494, 3799
  * Pala et al. (2017) Pala, A. F., Gänsicke, B. T., Townsley, D., et al. 2017, MNRAS, 466, 2855
  * Pass et al. (2022) Pass, E. K., Charbonneau, D., Irwin, J. M., & Winters, J. G. 2022, ApJ, 936, 109
  * Politano & Weiler (2006) Politano, M. & Weiler, K. P. 2006, ApJ, 641, L137
  * Raghavan et al. (2010) Raghavan, D., McAlister, H. A., Henry, T. J., et al. 2010, ApJS, 190, 1
  * Rappaport et al. (1983) Rappaport, S., Verbunt, F., & Joss, P. C. 1983, ApJ, 275, 713
  * Recchi & Kroupa (2015) Recchi, S. & Kroupa, P. 2015, MNRAS, 446, 4168
  * Rizzuto et al. (2013) Rizzuto, A. C., Ireland, M. J., Robertson, J. G., et al. 2013, MNRAS, 436, 1694
  * Scherbak & Fuller (2023) Scherbak, P. & Fuller, J. 2023, MNRAS, 518, 3966
  * Schreiber et al. (2010) Schreiber, M. R., Gänsicke, B. T., Rebassa-Mansergas, A., et al. 2010, A&A, 513, L7
  * Schreiber et al. (2016) Schreiber, M. R., Zorotovic, M., & Wijnen, T. P. G. 2016, MNRAS, 455, L16
  * Schulz et al. (2015) Schulz, C., Pflamm-Altenburg, J., & Kroupa, P. 2015, A&A, 582, A93
  * Sills et al. (2003) Sills, A., Deiters, S., Eggleton, P., et al. 2003, New Astronomy, 8, 605
  * Skumanich (1972) Skumanich, A. 1972, ApJ, 171, 565
  * Soethe & Kepler (2021) Soethe, L. T. T. & Kepler, S. O. 2021, MNRAS, 506, 3266
  * Tokovinin & Moe (2020) Tokovinin, A. & Moe, M. 2020, MNRAS, 491, 5158
  * Tokovinin (2000) Tokovinin, A. A. 2000, A&A, 360, 997
  * Toonen & Nelemans (2013) Toonen, S. & Nelemans, G. 2013, A&A, 557, A87
  * Van & Ivanova (2019) Van, K. X. & Ivanova, N. 2019, ApJ, 886, L31
  * Van et al. (2019) Van, K. X., Ivanova, N., & Heinke, C. O. 2019, MNRAS, 483, 5595
  * Wagner et al. (2019) Wagner, K., Apai, D., & Kratter, K. M. 2019, ApJ, 877, 46
  * Weidner et al. (2004) Weidner, C., Kroupa, P., & Larsen, S. S. 2004, MNRAS, 350, 1503
  * Willems & Kolb (2004) Willems, B. & Kolb, U. 2004, A&A, 419, 1057
  * Wright et al. (2011) Wright, N. J., Drake, J. J., Mamajek, E. E., & Henry, G. W. 2011, ApJ, 743, 48
  * Zorotovic & Schreiber (2022) Zorotovic, M. & Schreiber, M. 2022, MNRAS, 513, 3587
  * Zorotovic et al. (2010) Zorotovic, M., Schreiber, M. R., Gänsicke, B. T., & Nebot Gómez-Morán, A. 2010, A&A, 520, A86
  * Zorotovic et al. (2014) Zorotovic, M., Schreiber, M. R., & Parsons, S. G. 2014, A&A, 568, L9
  * Zorotovic et al. (2016) Zorotovic, M., Schreiber, M. R., Parsons, S. G., et al. 2016, MNRAS, 457, 3867