# Measurement of the angular momenta of pre–main-sequence stars: early
evolution of slow and fast rotators and empirical constraints on spin-down
torque mechanisms

Marina Kounkel Department of Physics and Astronomy, Vanderbilt University, VU
Station 1807, Nashville, TN 37235, USA<EMAIL_ADDRESS>Keivan
G. Stassun Department of Physics and Astronomy, Vanderbilt University, VU
Station 1807, Nashville, TN 37235, USA  Lynne A. Hillenbrand  Department of
Astronomy, California Institute of Technology, Pasadena, CA 91125, USA  Jesús
Hernández Universidad Nacional Autónoma de México, Instituto de Astronomía, AP
106, Ensenada 22800, BC, México  Javier Serna Universidad Nacional Autónoma de
México, Instituto de Astronomía, AP 106, Ensenada 22800, BC, México  Jason Lee
Curtis Department of Astronomy, Columbia University, 550 West 120th Street,
New York, NY 10027, USA

###### Abstract

We use TESS full-frame imaging data to investigate the angular momentum
evolution of young stars in Orion Complex. We confirm recent findings that
stars with rotation periods faster than 2 d are overwhelmingly binaries, with
typical separations of tens of AU; such binaries quickly clear their disks,
leading to a tendency for rapid rotators to be diskless. Among (nominally
single) stars with rotation periods slower than 2 d, we observe the familiar,
gyrochronological horseshoe-shaped relationship of rotation period versus
$T_{\rm eff}$, indicating that the processes which govern the universal
evolution of stellar rotation on Gyr timescales are already in place within
the first few Myr. Using spectroscopic $v\sin i$ we determine the distribution
of $\sin i$, revealing that the youngest stars are biased toward more pole-on
orientations, which may be responsible for the systematics between stellar
mass and age observed in star-forming regions. We are also able for the first
time to make empirical, quantitative measurements of angular momenta and their
time derivative as functions of stellar mass and age, finding these
relationships to be much simpler and monotonic as compared to the complex
relationships involving rotation period alone; evidently, the relationship
between rotation period and $T_{\rm eff}$ is largely a reflection of mass-
dependent stellar structure and not of angular momentum per se. Our
measurements show that the stars experience spin-down torques in the range
$\sim 10^{37}$ erg at $\sim$1 Myr to $\sim 10^{35}$ erg at $\sim$10 Myr, which
provide a crucial empirical touchstone for theoretical mechanisms of angular
momentum loss in young stars.

††software: TOPCAT (Taylor, 2005), Plotly (Inc., 2015)

##  1 Introduction

Young stars are born from large clouds of gas, and even at the earliest
stages, these pre-stellar objects exhibit rotation (Covey et al., 2005). As
the natal core transfers material through the protoplanetary disk onto the
protostar sitting at the center, and the protostar itself contracts, through
the conservation of angular momentum ($L$), the rotational speed would have
increased until reaching break-up velocity. However, there are several
mechanisms in place to regulate its rotation. A protostar is tidally locked to
the inner part of the protoplanetary disk, matching its Keplerian rotation at
the inner wall, removing the excess angular momentum through winds and jets
that form bipolar outflows (Bouvier et al., 1997; Matt et al., 2012; Gallet et
al., 2019).

After the outer envelope is consumed, and a young star eventually depletes its
disk, and, in the absence of another reservoir that would enable mass to flow,
the angular momentum would not increase further. This young star will continue
to contract, reducing its size by a factor of few until it settles onto the
main sequence. Despite this, the vast majority of stars will not rotate more
rapidly over this time as a result of the contraction, as their angular
momentum is decreasing due to magnetic braking (Skumanich, 1972). In large
part, this spin down of the angular momentum is very regular as a function of
mass, enabling to estimate the age of a star through the measurement of its
rotational period via gyrochronology (e.g., Barnes, 2003).

There have been many subsequent efforts over the past 20 years to constrain
gyrochronology relations, both theoretically (e.g., Gossage et al., 2021), and
empirically (e.g., Angus et al., 2020; Curtis et al., 2020), but they mainly
focused on main sequence stars, with only a limited consideration for pre-main
sequence stars. Recently, Kounkel et al. (2022, hereafter, Paper I) have used
TESS full frame images (FFI) to measure rotational periods of $\sim$100,000
stars with known ages, producing the largest such catalog to-date that enabled
the analysis angular momentum evolution of stars between ages of $\sim$10 Myr
to $\sim$1 Gyr. Stars younger than that present a particular complexity due to
their rapid evolution, requiring a more careful analysis of their ages, as
such they were largely ignored in the analysis.

In this paper, we seek to fill in this gap in the current understanding of the
rotational evolution of young stars. In particular, we examine rotation of
stars within the Orion Complex, which is one of the closest massive star
forming regions, spanning the area on the sky between
$76^{\circ}<\alpha<88^{\circ}$ and $-10^{\circ}<\delta<15^{\circ}$ containing
multiple sub-populations with ages from 1 to 10 Myr (Kounkel et al., 2018). It
covers the crucial age that is often missed by previous gyrochronology
studies. These data enable a precise measurement of a number of parameters
beneficial for characterization of stellar rotation in this region.

Stellar rotation in Orion has been examined repeatedly in the past (e.g.,
Stassun et al., 1999; Rebull, 2001; Herbst et al., 2002), although the
completeness of the census and the level of precision of the underlying data
cannot be compared to the modern surveys. More recently, Serna et al. (2021)
have also revisited rotation in this region. They have measured rotational
periods for 517 member of the Orion Complex using TESS FFI, of which 352 stars
also had available rotational velocity $v\sin i$ measurements. As expected,
they find a significant anti-correlation between rotation period and $v\sin
i$. They also note that in the stars younger than $<5$ Myr, $v\sin i$ appears
to systematically decrease, through some type of angular momentum extraction
mechanism (such as through disk locking), but that at the ages older than
that, $v\sin i$ increases due to conservation of angular momentum.
Additionally, accreting/disk-bearing young stars (Classical T Tauri stars -
CTTSs / Class II) appear to rotate slower than non-accreting/diskless stars
(Weak-lined T Tauri stars - WTTSs / Class III) during the first 1.5 Myr, but
they appear to have a more comparable distribution at later ages.

While Serna et al. (2021) have primarily examined the distribution of $v\sin
i$, in this work we aim to examine the rotational period evolution of these
young stars, significantly expanding the sample. We also leverage empirically
determined stellar radii from Kounkel et al. (2020) and precise individual age
estimates newly determined by McBride et al. (2021) to quantify the stellar
angular momenta and its evolution as a function of stellar mass and age.

In Section 2 we summarize the sources of photometric and spectroscopic data we
use to assemble our study sample of $\sim$9000 stars with rotation periods,
precise individual ages between 1–10 Myr, disk classifications, and empirical
radius determinations. Section 3 presents the principal results of our
investigation, and in Section 4 we discuss those results in the context of
empirical constraints on theoretical spin-down torque mechanisms,
understanding the nature of the rapid rotators, the relationship of rotation
to stellar disks, and an intriguing bias in the distribution of stellar
inclination angles as a function of age. We conclude with a summary of our
findings in Section 5.

##  2 Data

Figure 1: Distribution of rotational periods in the sample as a function of
$T_{\rm eff}$. Left panel: all sources in the Orion Complex from (Paper I),
using $T_{\rm eff}$ estimate from TIC (Stassun et al., 2019). Right: a subset
of sources observed by APOGEE, with spectroscopically determined $T_{\rm
eff}$.

The latest census of kinematically selected members of the Orion Complex
consists of over 10,000 optically bright stars (Kounkel et al., 2020). Paper I
have analyzed TESS FFI data for all of these sources, and have observed
periodic signatures in the light curves for $\sim$5700 stars of this sample.
As these stars are young, they tend to have strong magnetic fields resulting
in large spots, resulting in obvious rotational signatures. Furthermore, with
typical rotational periods $<$10 days for the stars at these ages, full period
data can easily be recovered even in a single TESS sector. The distribution of
rotational periods for the Orion Complex (Paper I) is shown in Figure 1.

APOGEE spectra are available for several thousand stars across the Complex (Da
Rio et al., 2016; Cottle et al., 2018; Kounkel et al., 2018). APOGEE is a high
resolution ($R\sim$22,500) spectrograph covering H-band, operating as part of
Sloan Digital Sky Survey (Wilson et al., 2019). These data have been
extensively analyzed, improving the fidelity of spectral parameters for these
young stars, in comparison to the primary pipeline used in public releases
(Abdurro’uf et al., 2022). APOGEE Net (Olney et al., 2020; Sprague et al.,
2022) provides $T_{\rm eff}$ and $\log g$ estimates that have been calibrated
relative to the pre-main sequence isochrones. Kounkel et al. (2019) have
estimated rotational velocity $v\sin i$ of the stars. They also characterized
multiplicity within this sample, including identification of both single-lined
(SB1) and double-lined (SB2) spectroscopic binaries. In total, APOGEE spectra
of 1831 periodic variables from Paper I in the Orion Complex are available. We
adopt these 1831 stars as the primary sample throughout the paper, as only
they have the full set of necessary measurements described in this section.

In addition to the directly measured spectroscopic parameters, supplementary
data for these stars have been computed. In particular, Kounkel et al. (2018)
have estimated angular diameters for these stars through the fitting of the
spectral energy distribution (SED), using an atmospheric model corresponding
to $T_{\rm eff}$ of each star. Coupled with trigonometric parallaxes, this
results in a direct estimate of stellar radii. This is an independent radius
measurement to those also provided as part of Gaia DR3 (Fouesneau et al.,
2022) or in the TESS Input Catalog (Stassun et al., 2019). While all three
have comparable estimates, the radii in Kounkel et al. (2018) can be
considered more robust for this sample, as a) they are obtained through
spectroscopically derived $T_{\rm eff}$ to identify the best model (in field
stars, this can produce $<$2% precision; Stassun et al., 2017), b) the SED fit
spans all of the available fluxes across the full electromagnetic spectrum, c)
a special consideration is given to the stars with IR excess, which would
affect the SED fit.

However, $T_{\rm eff}$ used for the SED fitting in Kounkel et al. (2018) were
derived using the IN-SYNC pipeline (Cottaar et al., 2014), which had a number
of known systematics. $T_{\rm eff}$ from the APOGEE Net pipeline are more
reliable for these young stars, they offer a better calibration relative to
not only the isochrones but also to the more evolved field star. APOGEE Net
$T_{\rm eff}$ are on average $\sim$300 K cooler than the IN-SYNC $T_{\rm
eff}$. As such, we renormalize the previously reported radii using updated
$T_{\rm eff}$through Stefan–Boltzmann law, resulting in $R$ estimate that are
on average 20% larger than previously reported. We note that the effect of
spots is not considered; we discuss the implications from them in Section 3.3.

We identify all of the known disk-bearing and/or accreting stars in the sample
using a combination of different catalogs. They include classification based
on H$\alpha$ emission (Suárez et al., 2017; Briceño et al., 2019),
classification of NIR excess from Spitzer and other high sensitivity surveys
of various regions (Hernández et al., 2007, 2010; Megeath et al., 2012;
Großschedl et al., 2019), as well as a classification of NIR excess based on
WISE data across the entire sky (Marton et al., 2016). While accretion
signature and disk classification are not necessarily interchangeable, they
are expected to be equivalent in a vast majority of cases. Together, these
correspond to 580 out of 1831 stars in our sample, which we consider to be
Class II/CTTS. The remaining sources are considered to be Class III/WTTS.

We estimate masses for this sample using MassAge (Hernandez J. et al. in
prep), through comparing $T_{\rm eff}$ and bolometric luminosity to MIST
isochrones (Choi et al., 2016). Finally, we estimate ages of stars using
neural net Sagitta (McBride et al., 2021). This neural net is utilizing Gaia &
2MASS photometry (G, BP, RP, J, H, K), Gaia parallaxes, and the typical
extinction along the line of sight to predict the likelihood of a star being
pre-main sequence, as well as the age of a young star, up to $\sim$80 Myr. It
was trained on the census of pre-main sequence stars found in moving groups
from Kounkel (2020), with the age of the stars typically inherited from their
parent population (subdividing the moving groups into smaller meaningful
populations with a narrow age spread), supplemented with the sources ages of
which have been determined independently in literature. Appendix A motivates
the selection and compares this age model to other techniques. As masses and
ages are derived using different models, this may introduce some systematics.
But, given that these stars are typically moving vertically down the Hayashi
tracks without a significant change in $T_{\rm eff}$, there should not be a
significant variance with mass as a function of age.

The distribution of these ages for the stars in Orion is shown in Figure 2.
The uncertainties are generated through propagating the typical uncertainties
in the input astrometry and photometry through the model, they typically have
a range of 0.05-07 dex. We do note that there is some overlap between the
stars that have originally been used for training Sagitta and the stars in
this work; we do not expect this to significantly affect the quality of the
derived ages.

Figure 2: Left: Distribution of the adopted ages for the stars in Orion
Complex. Right: Typical uncertainties in the sample.

##  3 Results

In this section, we present the results of the distributions of stellar
rotation periods for our study sample of Orion Complex stars, typically having
ages $\lesssim$10 Myr. The sample includes stars that are rapid rotators and
slow rotators, stars that have disks (CTTSs) and those that lack disks
(WTTSs), and stars that are single or in binaries. We consider these subsets
and their relationships in turn.

However, to begin, we note that the overall distribution of stellar rotation
periods among the nominally single stars in the sample (Figure 1) already at
these very young ages manifest the familiar “horseshoe-shaped” pattern in the
$P_{\rm rot}$ vs. $T_{\rm eff}$ diagram that has become the basis for
gyrochronology at later ages (e.g., Barnes, 2003). This suggests that, despite
significant scatter at any given age, the dominant physical mechanisms that
govern the evolution of angular momentum are already in place and very quickly
sculpting the distribution of angular momenta into the universal patterns with
stellar mass and age that make gyrochronology possible.

###  3.1 Rapid rotators

Examining the period distribution of CTTS and WTTS objects in Figure 3, there
is a clear difference between the two samples. Very few CTTS have rotational
periods faster than 2 days. On the other hand, $\sim$1/3 of WTTSs have
rotational periods $<$2 days. We use this 2 day period to delineate between
“slow” and “fast” rotators.

Examining their distribution as a function of age (Figure 4), we find that the
overall fraction of rapid rotators remains more or less constant, however
there are important differences between the WTTSs and CTTSs with regards to
this trend. Among WTTSs, the overall fraction of fast rotators decreases from
$\sim$45% at ages $<$1 Myr to $\sim$30% at ages $>3$ Myr. The reason for this
trend is likely to be increasing dilution of the diskless stars over time by
more slow rotators, as all stars irrespective of their rotation eventually
deplete their disks, until the age where almost all stars are diskless
($\sim$3 Myr; see, e.g., Ribas et al., 2014). Among CTTSs, there is an
opposite trend: there are almost no rapid rotators with disks at an age $<$1
Myr, but their fraction increases moderately to $\sim$10% at ages $>3$ Myr. We
return to discuss the implications of this trend in Section 4.2.

At ages $<$1 Myr, rapid rotators tend to have rotational periods between 1 and
2 days. As they age, however, the fraction of systems with periods $<1$ day
increase relative to the total number of stars. This process appears to
continue at older ages, as in populations older than $\sim$20–30 Myr, most
rapid rotators appear to favor periods $<$0.5 days (Paper I). Most likely,
without an effective method of reducing a sufficient fraction of their angular
momentum, the stars continue to spin up as they shrink towards their main
sequence radius, and their spin-up increases until the magnetic braking can
overcome this speed increase, or until they stop contracting (e.g., Rebull et
al., 2022).

Figure 3: Period distribution between CTTS and WTTS objects.

Figure 4: Fraction of rapid rotators among CTTSs and WTTSs with $T_{\rm
eff}$$<$6700 K as a function of age. Top panel shows all rapid rotators with
periods $<2$ days, bottom panel is restricted to the subset with periods $<1$
day.

###  3.2 Slow rotators

Figure 5: Dependence of radius, rotational period, angular momentum, and
specific angular momentum on temperature and age of stars in Orion with $P>2$
days. The typical uncertainties in the data are 30 K in $T_{\rm eff}$, 0.1 R⊙
in radius, 0.01 day in period, 0.05 dex in L, and 0.04 dex in H. In young
disk-bearing stars typical uncertainties in radius are larger due to the
infrared excess affecting the quality of the SED fitting.  Figure 6: A fit of
angular momentum L. Yellow dots correspond to the data within each age bin.
The red line shows the running median within the data. Thick blue line shows
the best fit for L for a given age. Thin blue lines are offset in age by
$\pm$0.3 dex, and dotted lines are offset in age by $\pm$0.6 dex. The typical
scatter between the data and the resulting fit of L is shown in each panel.
Open circles represent CTTSs, filed circles represent WTTSs.

Figure 7: Rate of loss of angular momentum $L$ (left) and specific angular
momentum $H$ (right), as a function of temperature and age, assuming the
empirical relation in Figure 6 and Table 1.

In the analysis of slow rotators, we limit the sample only to the sources with
rotational periods $>$2 days. We examine the trends in this sample as a
function of age (Figure 5).

There is no appreciable difference in the distribution of periods relative to
the ages of the stars. However, over the first 10 Myr, the radii of the stars
rapidly contract. Because of this, there is a significant evolution in the
angular momentum. We estimate $L$ assuming a solid body rotation via
$L=2/5MR^{2}\Omega$, where $\Omega=2\pi/P$. $L$ is not conserved during this
contraction, as such the stars are losing angular momentum, most likely due to
magnetic braking, despite the lack of significant evolution in rotational
periods.

One complexity that can skew this approximation of $L$ is the differential
rotation in stars, with rotation at the poles being slower than near the
equator. At the moment we cannot determine the precise latitude of the spots.
However, pre-main sequence stars moving on Hayashi tracks are expected to have
significantly smaller differential rotation than, e.g., the Sun, with the
shear of only 3.9%, as it decreases with increasing convective zone depth
(Küker & Stix, 2001; Marsden et al., 2011). As such, a solid body
approximation is reasonable for the young stars in this sample.

The resulting distribution of L appears to be mostly consistent between slowly
rotating CTTSs and WTTSs, suggesting that the overall angular momentum content
of the stars evolves largely irrespective of their disks. And, indeed, this
spin down will continue throughout the lifetime of low mass stars, long past
their shedding of the disks or entering the main sequence (Paper I).

We bin the data using a kernel of $\pm$0.1 dex in age, $\pm$200 K in $T_{\rm
eff}$, and find the running median in $L$ in each bin. Similarly to Paper I,
using the resulting average $L$ in the 2D plane, we perform an empirical fit
using a formalism of

$\begin{split}\log L=a_{0}+a_{1}\log T_{\rm eff}+a_{2}(\log T_{\rm
eff})^{2}+a_{3}(\log T_{\rm eff})^{3}+\\\ b_{0}\log t+b_{1}\log t\log T_{\rm
eff}+b_{2}\log t(\log T_{\rm eff})^{2}\end{split}$ (1)

where $\log T_{eff}$ is the $\log_{10}$ of $T_{\rm eff}$ in K, and $\log t$ is
$\log_{10}$ of age in years. The best fit is shown in Figure 6, and the
coefficients are included in Table 1. Independently, we also fit specific
angular momentum per unit mass ($H$). We note that this functional form is
arbitrary, and a comparable fit can be achieved with a different power
dependence on both $T_{\rm eff}$ and $\log t$. In Paper I this form was chosen
as the lowest power polynomial that could fit the age range from 10 Myr to 1
Gyr and be easily reversible with respect to age. In the Orion sample exending
only up to 10 Myr, a maximum power of 2 instead of 3 can also result in a good
fit, but for the sake of consistency we retain the same formalism.

Table 1: Fitted coefficients for gyrochronology relations  Coefficient | Value ($L$) | Value ($H$)
---|---|---
$a_{0}$ |  $-$1926.1822 | 1554.6763
$a_{1}$ | 1352.6072 |  $-$1590.9434
$a_{2}$ |  $-$299.8859 | 524.9803
$a_{3}$ | 21.2112 |  $-$56.0318
$b_{0}$ | 139.5928 | 186.4053
$b_{1}$ |  $-$76.1140 |  $-$102.8078
$b_{2}$ | 10.3335 | 14.1278

Through taking a partial derivative with respect to time of the resulting
function, we observe the evolution in the rate of spin down (Figure 7). The
loss of angular momentum is most effective at the youngest ages, with
d$L$/d$t$ decreasing by more than an order of magnitude over the first 10 Myr.
The primary source of angular momentum loss is likely to be magnetic breaking:
young stars may have other factors that would affect their rotation (including
accretion and outflows), but the combined net d$L$/d$t$ from these processes
appears to be significantly smaller than the one we observe (e.g., Gallet et
al., 2019).

The spin down is more effective among the higher mass stars, this appears to
be the case both with d$L$/d$t$ as well as d$H$/d$t$ (See also Section 4.1 for
discussion). The rate of loss of specific angular momentum appears to be
somewhat proportional to $H$ itself; i.e., at a given age, a 5000 K star
appears to have (on average) a specific angular momentum $\sim$3.6 times
higher than a 3500 K star, and, similarly the ratio of d$H$/d$t$ for the same
stars is also $\sim$3.6.

While the underlying trend of decreasing $L$ and $H$ at older ages is
significant, at all ages there is significant scatter in their distribution,
to a degree that is unlikely to improve with more precise measurements of
mass, radius, or age. As such, while after a few hundred Myr a population of a
given age would be able to form a relatively narrow gyrochrone (e.g., Douglas
et al., 2019), early on there is a wide range of the initial conditions
pertaining to the angular momentum in the system that are yet to be erased.

Nonetheless, it is striking that the angular momenta of the stars evidently
follow much simpler, more monotonic relationships with stellar mass and age
than rotation periods alone (i.e., compare various panels in Figure 5 to the
upper right panel in Figure 5).

###  3.3 Inclination

The stellar radii measurements derived from the SED fitting are available from
Kounkel et al. (2018). An alternative way to estimate the radius is through a
combination of the rotational period and rotational velocity, however as the
latter is only available as $v\sin i$, the resulting radius estimate ${\rm
R}\sin i$ would also have inclination dependence. As such, comparing these two
estimates enables a constraint on $\sin i$ (e.g., Jeffries, 2007; Jackson &
Jeffries, 2010).

Mathematically, the upper limit of $\sin i$ is 1. In practice the ratio
between ${\rm R}\sin i$/$R$ does occasionally exceed 1 by more than the formal
errors suggest, pointing to a systematic calibration issue in some of the
measurements. Here we analyze some of the consideration that can lead to
systematic issues that a) result in unreasonably large $\sin i$ and b) create
a distribution of orientations of stars that, due to observational biases, do
not appear to be entirely random.

####  3.3.1 $T_{\rm eff}$

Young stars tend to be strongly magnetically active, with a very high spot
fraction coverage. Because of this, it may be difficult to describe these
stars with a single temperature. Depending on the wavelength that is used to
fit $T_{\rm eff}$, it is possible to have a systematic difference of a few 100
K depending on the wavelength of the spectrum (e.g., Lodieu et al., 2018), as
some regions of the spectrum are more sensitive to the temperature of the
photosphere ($T_{\rm phot}$) or the spot ($T_{\rm spot}$), as opposed to the
true $T_{\rm eff}$. The estimate of radius from the SED fitting is directly
dependent on $T_{\rm eff}$, obtained via $R\propto\sqrt{F/T^{4}_{\rm eff}}$,
where $F$ is the total flux.

We note that as the radii reported in Kounkel et al. (2018) relied on
temperatures that was systematically hotter than what is adopted here. Without
the renormalization with updated $T_{\rm eff}$, this systematic bias
propagated from $T_{\rm eff}$ to radii would further exaggerate the number of
sources with $\sin i$$>$1\. As such, this demonstrates the improvement in
temperature calibration in APOGEE Net compared to earlier estimates,
significantly reducing the bias in $\sin i$ from $T_{\rm eff}$. However,
without performing an explicit two temperature fit, spectroscopically measured
temperature will be lower than the temperature of the unspotted photosphere,
but higher than the actual $T_{\rm eff}$. Due to the resulting luminosity
differences of templates, during the SED fitting $R$ can becomes
underestimated.

For example, a star with $T_{\rm phot}=4000$ K and $T_{\rm spot}=$3500 K with
50% spot coverage would have $T_{\rm eff}$ of 3775 K. However, in the H band,
the resulting spectrum would be best matched by a 3900 K template, resulting
in $R$ difference of 10%.

####  3.3.2 Differential rotation

As previously mentioned, differential rotation in stars with large convective
zones is expected to be relatively small (Küker & Stix, 2001; Marsden et al.,
2011), affecting young stars only on a few % level. However, if a spot is
found at higher latitudes, the rotation period at the equator would be
somewhat faster than the one that is observed. This will result in
overestimating ${\rm R}\sin i$, which will also overestimate $\sin i$.

####  3.3.3 Binaries

The excess in $\sin i$$>1$ distribution is most pronounced for sources that
have previously been identified as SB2, reaching values up to 10. In such
sources, $v\sin i$ tend to be overestimated, as the line broadening in the
spectra is affected by the presence of both stars in the system, which the
spectroscopic fit of $v\sin i$ did not take into the account. Similarly, there
may be additional issues, such as mismatch in the source of origin between
$v\sin i$ and the rotational period, as well as the excess flux from the
companion in the SED fitting.

Unresolved binaries that are widely separated enough to have their orbital
motion being negligible in $v\sin i$, will also have issue in their $R$
estimate from the SED fitting. As $F$ will be contaminated by the second star
in the system, $R$ will be artificially inflated, not representative of the
radius of either the primary or the secondary. Since $R$ is too large, but
${\rm R}\sin i$ in these wide binaries be more similar to systems with just a
single star, the resulting $\sin i$ will be underestimated. When we compare
the distribution of slow and fast rotators, fast rotators appear to favor
significantly smaller $\sin i$ than the slow rotators.

Paper I have found that fast rotators generally occupy the binary sequence in
a given population. The difference in the $\sin i$ distribution between slow
and rapid rotators is consistent with originating from unresolved binaries. As
such, to provide a more equitable distribution of $\sin i$ that is dependent
on other stellar properties, in evaluating other biases we focus solely on
slow rotators.

Figure 8: Distribution of ${\rm R}\sin i$/R ratios, as a proxy of $\sin i$.
Top: a comparison between slow and fast rotators. Middle: a comparison of the
sources with strong ($>$5% of the total flux) and weak ($<1$%) variability
(limited just to the slow rotators with P$>$ 2 days), as well as double lined
spectroscopic binaries from Kounkel et al. (2019). Bottom: a comparison of
CTTSs and WTTSs, also limited just to slow rotators.

####  3.3.4 Detectability of periods and $v\sin i$

It is important to consider the biases of the underlying catalog that
influence the resulting distribution. In order to be able to measure rotation
period, a star (of any age) needs to be spotted, and these spots need to
rotate in and out of view. The strongest signal is going to be produced by the
edge-on systems. In the face-on systems, rotation cannot be observed. In the
intermediate orientations, the systems with intrinsically larger spot sizes
would be favored, since otherwise an induced variability from a weaker spot
that may not fully rotate out of the field of view may be too shallow to be
detected. We find that sources with a large amplitude of variability appear to
favor edge-on inclinations, in comparison to other sources (Figure 8). If
initially sources with strong and weak variability have spots of the same size
(given that their ages and masses are comparable), weakly variable stars will
have more intermediate inclinations, (resulting in a portion of the spot to
never rotate out of the field of view or, alternatively, never rotate in the
field of view), and strongly variable stars will be edge-on.

Similarly $v\sin i$ would also be biased against face-on systems, due to the
necessary resolution in the spectra to confidently measure very weak
broadening. As such, the expected distribution of inclination angles is not
expected to be random, but rather favor $\sin i$$\longrightarrow$1.

This does have implications for the stars in older populations in which
magnetic activity is weaker, leading to smaller spots. At more advanced ages,
if a star is detected as variable at more advanced ages, it is more likely to
be edge on, as inclined systems might have the amplitude of variability less
than the instrumental sensitivity.

####  3.3.5 Extinction

Young stars have additional biases in their detection. For example, $\sin i$
of CTTSs tends to be slightly smaller than $\sin i$ of WTTS stars. Applying a
two-sided KS test shows that the two populations are different at 3$\sigma$
level. This difference is driven solely by the deficit of the edge-on CTTSs;
restricting both samples to $\sin i$$<$0.9, these inclined CTTSs and WTTSs are
consistent to originating from the same distribution at $<$1$\sigma$ level.

CTTSs still have dusty disks, and an edge-on disk orientation would produce a
highly reddened source, most likely rendering it to be too faint to be
observed in a magnitude-limited survey like APOGEE. As such, while WTTSs would
favor an edge-on orientation, CTTSs in the sample would tend to the sources
where the photosphere is not fully obscured by the disk, but still as close to
$\sin i$$\longrightarrow$1 as possible within this constraint. We further
discuss the implications of this in Section 4.4.

##  4 Discussion

###  4.1 Empirical constraints on models of angular momentum loss

The fact that young, low-mass stars must somehow deplete their angular
momentum by at least an order of magnitude prior to the main sequence has been
understood as a challenge since the earliest measurements of their rotational
characteristics (e.g., Cohen & Kuhi, 1979; Herbst et al., 1982; Bouvier et
al., 1986; Basri, 1987). Whereas the rotational evolution of low-mass stars on
the main sequence has long been well understood as the result of magnetized
stellar winds acting on Gyr timescales (e.g., Skumanich, 1972; Kawaler, 1988;
Barnes et al., 2001), it has also been recognized that standard stellar winds
cannot achieve the necessary angular momentum losses for PMS stars on the
short timescale of $\sim$107 yr. Consequently, a number of mechanisms have
been proposed that are unique to the PMS phase, most of which involve
harnessing the inertia of circumstellar disks and/or the power of accretion
from those disks during the disk lifetime of the first few Myr.

For example, early attempts to model the exchange of angular momentum from
star to disk in CTTSs were based on the model of Ghosh & Lamb (1979), in which
the star transfers angular momentum to the inner part of the disk where the
stellar magnetosphere intersects (and truncates) it. Shu et al. (1994)
developed a modified form of the Ghosh & Lamb (1979) model—the so-called
“X-wind” model—that invokes the pinching of field lines in the immediate
vicinity of the co-rotation radius in the disk to regulate the star’s
rotation, keeping the star’s rotation period constant even as the star
contracts (i.e., “disk-locking”), and the excess stellar angular momentum
ejected in a disk wind outflow (e.g., Koenigl, 1991).

Alternatives to the X-wind model include those that still utilize the star-
disk interaction but remove the angular momentum from the star more directly.
For example, the magnetospheric ejection model (e.g., Zanni & Ferreira, 2013)
is based on MHD simulations of the sporadic opening and reconnection of
magnetospheric field lines, which eruptively launch the magnetically confined
mass; this is conceptually similar to stellar coronal mass ejections but with
the long lever arm of the star-disk interaction region. Another example is the
so-called accretion powered stellar wind model (e.g., Matt et al., 2012), in
which disk accretion itself acts to open stellar field lines and launch a
powerful stellar wind, thereby extracting angular momentum directly from the
stellar surface.

All of these mechanisms make predictions for the torque that can be exerted on
the star. With our empirically determined angular momentum loss rates for
TTSs, anchored in directly measured rotation rates, distances, and empirically
determined radii for a very large number of stars with ages 106–107 yr (i.e.,
Figure 7), we are in a position to quantitatively confront the model
predictions beyond previous comparisons to stellar rotation period
distributions (e.g., Stassun et al., 1999; Rebull et al., 2004; Irwin &
Bouvier, 2009).

The recent work of Gallet et al. (2019) provides an especially useful basis
for comparison of the model predictions to our measurements, as those authors
constructed a family of models that attempt to self-consistently incorporate
the action of all of the disk/accretion-based mechanisms described above (cf.,
their Figure 6). The models suggest that, for accretion rates that are not too
high (i.e., $\dot{M}\lesssim 10^{-8}$ M⊙/yr; otherwise, the net torque on the
star is positive), spin-down torques as strong as $10^{36}$ erg are possible
during the first $\sim$1 Myr and declining below $10^{35}$ erg after 1–2 Myr.
By comparison, our measurements (Figure 7) imply that the stars in fact
experience torques as strong as $10^{37}$ erg during the first $\sim$1 Myr,
declining to $\sim$1036 erg after $\sim$2 Myr, and finally below $\sim$1035
erg at $\sim$10 Myr. Thus, it appears that the disk/accretion-based mechanisms
may not supply the full torque budget that we have measured, at least not on
their own; an additional source of torque may be required that does not rely
on disk inertia or accretion power.

One possibility is extreme coronal mass ejections (CMEs). Aarnio et al. (2012)
quantitatively explored this possibility by translating the empirical solar
relationship between X-ray energy and CME ejected mass rates (see Aarnio et
al., 2011) to the observation by the Chandra Orion Ultradeep Project of
extremely large flaring loops on TTSs in Orion (Favata et al., 2005). Aarnio
et al. (2012) found that the stars can eject very massive CMEs with a high
frequency during the first $\sim$10 Myr, and that the high observed CME losses
are independent of whether the stars have retained their disks or are actively
accreting (see also Aarnio et al., 2013). These studies estimated the torque
from extreme CMEs as ranging from $10^{35}$ erg to $10^{37}$ erg (cf., Figure
4 in Aarnio et al., 2012), comparable to the angular momentum losses we have
determined empirically in this work (Figure 7).

###  4.2 Nature of rapid rotators

Previous observations have suggested that rapid rotators appear to be almost
ubiquitously associated with unresolved binaries (Stauffer et al., 2016, 2018;
Simonian et al., 2019; Gillen et al., 2020; Bouma et al., 2021; Kounkel et
al., 2022). In populations with a clearly defined cluster sequence on the HR
diagram, they tend to occupy the binary sequence Given the resolution of the
data used to make this assessment, here “unresolved” binaries includes all of
the separations up to several tens of au.

In younger regions ($<$10 Myr) such as Orion, an age spread of just a few Myr
is apparent in the HR diagram due to a rapid stellar evolution at these ages.
As such, the binary sequence can be difficult to observe directly due to
confusion with younger stars. However, as has been discussed in Section 3.3.3,
the $\sin i$ distribution of slow and fast rotators in Figure 8 is consistent
with the assumption of rapid rotators in Orion also being associated with wide
but still unresolved multiples, similarly to what has been previously found
older clusters.

We note that rapid rotators appear less likely to form around shorter period
binaries with separation $<$ a few au. We examine their occurrence rate around
SB1s (which favors systems with periods $<$1000 days Kounkel et al., 2019).
There are 4 times fewer rapid rotators among SB1s than in the full sample (6%
vs 27%). We discuss this in the following subsection. Thus, the range of
separations of binaries hosting rapid rotators is expected to be a few to a
few tens of au.

Given this range of separations, it may be possible to resolve these systems
with high resolution imaging. A few such studies have been conducted in the
Orion Nebula (Duchêne et al., 2018; De Furio et al., 2019, 2022) reaching
separations down to 10 au, but, due to a limited sample size and the magnitude
limit so far, there are no sources we identify as rapid rotators in common in
these studies. In future, high resolution imaging of the sources identified as
rapid rotators may be more fruitful in finding resolved binaries in comparison
to a blind search.

However, more in-depth studies have been conducted in other star forming
regions, such as Taurus molecular clouds. Due to its proximity, resolved
binaries have been detected down to separations of 3 au (Kraus et al., 2011).
Cross-matching this census with the rotational periods from Paper I yields a
sample of 18 stars, of which 5 have been resolved as binaries. Four of these
binaries are rapid rotators, with projected separations ranging from 4 to 20
au consistent with our interpretation (Figure 9). The sole non-rapidly
rotating resolved binary has a separation of 150 au, which suggests that the
systems with separations larger than few tens of au are less likely to result
in spin-up.

Figure 9: Top: Rotational period as a function of color for the sample from
Kraus et al. (2011) in Taurus, with the periods from Paper I. Blue dots
highlight binaries that have been resolved through high resolution imaging,
red diamonds — the sources in which no resolved companion has been identified
to date. Black line shows the period of 2 days, delineating slow and rapid
rotators (see also Figure 4 in Stauffer et al., 2018). Bottom: rotational
period as a function of projected separation in resolved binaries

###  4.3 Disk evolution in rapid rotators

We observed in Figure 3 a clear connection between stellar rotation period and
the presence of a disk, in the sense that the rapid rotators are
overwhelmingly found among the WTTSs (i.e., diskless stars). This association
of rapid rotation with the lack of a disk may be a direct consequence of the
association between rapid rotation and binarity (see Section 4.2), and the
fact that binaries with separations of a few au to tens of au tend to disperse
their disks on faster timescales than single stars (e.g., Kraus et al., 2012).

While it is well understood that very close, tidally locked binaries (with
orbital periods of only a few days) will naturally be forced to rotate rapidly
(e.g., Melo et al., 2001), it is not immediately intuitive why binary systems
with separations up to several dozen au should imbue their stars with a
greater amount of angular momentum. One possibility suggested by recent
simulations of binary star formation (Kuruwita et al., 2017) is that binaries
are less effective at removing angular momentum from the system: for example,
in those simulations, a binary with separation of 45 au has an efficiency of
only 42% in transporting angular momentum away from the system via outflows,
relative to a single star. In that case, our observation that the fastest
rotators tend to be in binaries that are wider than the reach of tidal
interaction would be a vestige of the binary formation process.

Kuruwita et al. (2017) also find that close binary systems with separations of
2.5 au have an efficiency of 87% in transporting angular momentum away from
the system in comparison to the single stars. That is to say, compact binaries
are significantly less likely to have angular momentum excess in comparison to
the wider systems with separations of a few tens of au. This is consistent
with our observations of close binaries being less likely to develop rapid
rotation. Most likely, this is due to the compact systems resembling single
stars in their interaction with the protoplanetary disk, with the outflows
being magnetocentrifugally driven, compared to the outflows being driven by
magnetic pressure gradient in wider binaries (Kuruwita et al., 2017).

It is interesting in this context that we have observed a small fraction of
rapid rotators among CTTSs, and that this prevalence of rapid rotators among
the disked stars appears to increase with time (see Figure 4). This has also
been observed by Serna et al. (2021). We may again interpret these rapid
rotators as likely being binaries as well, but whose orbital configurations
permitted the disk to survive for some time (thus remaining as CTTSs for this
duration). In that case, the increasing prevalence of rapid rotators among the
CTTSs with time may suggest that the increased angular momentum deposited into
the inner system by the binary formation process (see above) was partially
stored in the disk, and then gradually accreted onto the stars over the disk
lifetime, thus allowing the stars to later emerge as rapid rotators just like
their diskless binary cousins.

###  4.4 Age-dependent inclination bias

Figure 10: Kernel density estimate distribution of ${\rm R}\sin i$/$R$ ratios
of CTTSs and WTTSs as a function of age. Note that the last two age bins are
different between the two panels, due to a decreased number of CTTSs at older
ages. The distribution is often bimodal, the sources that appear to be
inclined are likely to be unresolved binaries.

As can be seen in Figure 8, the average difference in $\sin i$ between CTTSs
and WTTSs is $\sim$10–15%. If we do assume that we see WTTSs across all
inclinations between 0 and 90∘, then the maximum typical inclination of CTTSs
in the sample would range between 0∘ and 55–65∘, with other inclinations being
too extinct. This translates to the dust being found at the disk scale height
of $\sim$0.5–0.6. This is in excess of what is typically found in
protoplanetary disks (scale height of $\sim$0.2 has been reported in some
systems, but this is also considered to be quite high Natta et al., 2001;
Olofsson et al., 2013; Montesinos et al., 2021). Some dust can be present at
$>3$ times the scale height, and some extinction can be expected even from the
modestly inclined disks (D’Alessio et al., 2006).

However, the difference in the observed inclination may not necessarily be
tied to the presence of the protoplanetary disk itself, but rather, the
relative difference in age, owning to CTTSs being on average younger than
WTTSs. If we compare $\sin i$ distribution for CTTSs and WTTSs separately
across different age bins (Figure 10), then the younger sources have a
distribution of $\sin i$ that favors smaller values than those that are more
evolved. This appears to hold true regardless of if the star has a significant
IR excess from disk/has active accretion, or not. CTTSs still favor a slightly
smaller $\sin i$ than WTTSs at a given age bin, but the difference is less
drastic than in the overall distribution.

As such the source of opacity that reddens these stars likely comes from the
outer debris disk/envelope that would not necessarily show significant IR
excess in all but the longest of wavelengths, which may have been missed by
the selection of disk-bearing stars. It is likely to settle or dissipate in a
few Myr, but a puffy debris disk appears to be relatively common at ages $<$2
Myr, even if a star has already lost its inner disk. Regardless of the
dominant source of opacity, be it the disk or the envelope, it is the dust
along the line of sight that results in a deficit of young edge-on systems in
a magnitude-limited sample. The younger a star is, the less likely is the dust
to have been settled. No other age-dependent properties are likely to be
responsible for this bias.

The difference in the distribution of inclinations between the stars of
different ages may be an important challenge in being able to determine
stellar ages. If the spots are not distributed randomly on the photosphere,
but, e.g., spots are preferentially located along the pole instead of the
equator, as has been demonstrated in several young systems (Donati et al.,
2013, 2014, 2015) or more evolved stars with strong magnetic fields (e.g.,
Roettenbacher et al., 2016), then this may impose a bias on the stellar age.
The systems where the spots are prominently visible would appear to be less
luminous than their unspotted counterparts. Coupled with the biases on the
inclination, as well as the relative spot size and the spot contrast at
different $T_{\rm eff}$ this may be responsible for the trends in age as a
function of mass in a given population in Figure 11.

When separated into different age bins, $\sin i$ distribution in Figure 10
often appears as bimodal. It is possible that the more inclined peak is
dominated by the unresolved binaries. Although the rapid rotators are excluded
from the sample presented in that figure, other binaries are still expected to
be present in the sample, and, just like rapid rotators, they would have
systematically underestimated $\sin i$.

##  5 Conclusions

Using TESS full-frame imaging data, APOGEE spectroscopic data, broadband
spectral energy distributions, and Gaia parallaxes, we assemble the largest
collection to date of stars younger than 10 Myr with rotation periods from
Paper I, spectroscopic $T_{\rm eff}$ and $v\sin i$, disk classifications,
empirically determined radii from Kounkel et al. (2020), and precise
individual age estimates from McBride et al. (2021). With the resulting sample
of $\sim$9000 stars in the Orion Complex, we examine the angular momentum
content of young stars and the evolution of angular momentum with stellar age.

Similarly to previous studies, we find that stars with rotation periods faster
than 2 d are predominantly binaries, with typical separations of tens of AU.
Such binaries are known to rapidly clear their disks, and consequently we
observe an association between rapid rotation and disklessness. It is not
clear how such binaries, which are much wider than the reach of tidal effects,
come to possess greater amounts of total angular momentum, however it is
consistent with recent simulations which find that outflows from binaries are
less efficient at removing angular momentum from the inner system as compared
to single stars.

Among the (nominally single) stars with rotation periods slower than 2 d, we
find that the angular momentum loss is most effective at the youngest ages,
slowly decelerating its effect over time. Additionally, higher mass stars have
higher total angular momentum (as well as specific angular momentum per unit
mass) in comparison to their lower mass counterparts, but they experience more
efficient angular momentum depletion, even at ages $<$10 Myr.

More generally, we observe the familiar, gyrochronological horseshoe-shaped
relation between rotation period and $T_{\rm eff}$, implying that the
processes responsible for the universal evolution of stellar rotation on Gyr
timescales are already in place within the first few Myr. We also find that
our empirically quantified stellar angular momenta exhibit much simpler and
monotonic relationships with stellar mass and age as compared to the
relationships involving rotation period alone. We conclude that the
relationship between rotation period and $T_{\rm eff}$ is largely a
manifestation of the mass dependence of stellar structure and not of angular
momentum per se.

In addition, using the combination of rotational periods, rotational $v\sin
i$, as well as stellar radii measured from the SED fitting, we estimate the
$\sin i$ of the stars, revealing a variety of biases in their distribution. We
find that most variable systems tend to prefer an edge-on orientation, as this
is most favorable configuration to observe star spots rotating in and out of
view. We also find that in a magnitude-limited sample, the younger systems
tend to have on average lower $\sin i$ than the systems that are more evolved.
This applies both to the stars with protoplanetary disks, as well as those
that have already managed to disperse it. Most likely this difference in the
distribution is driven by the extinction from the dust in the outer envelopes
/ debris disks, taking a few Myr to fully dissipate, even if the inner disk is
fully gone. If there is a preferential distribution of spots along the
photosphere, this difference in inclination distribution as a function of age
may introduce a bias in the age estimate of some stars.

Most importantly, our directly measured rotation periods, together with
precise Gaia distances and our empirically determined stellar radii for very
large sample of stars with ages 1–10 Myr, permit us to construct empirical
determinations of stellar angular momentum distributions and thus empirical
determinations of angular momentum loss rates. Our measurements show
quantitatively that the stars experience spin-down torques in the range $\sim
10^{37}$ erg at $\sim$1 Myr to $\sim 10^{35}$ erg at $\sim$10 Myr. Recent
modeling efforts that utilize the inertia of circumstellar disks and/or the
power of disk accretion to extract stellar angular momentum during the disk
lifetime predict torques in the range $\sim 10^{36}$ erg at $\sim$1 Myr to
$\lesssim 10^{35}$ erg by 2–3 Myr (see, e.g., Gallet et al., 2019). Other
mechanisms that do not depend solely on disk inertia or accretion power may
therefore be required. For example, empirical observations of the extreme
coronal mass ejections (CMEs) associated with X-ray super-flares observed in
stars with ages 1–10 Myr appear capable of exerting torques ranging from $\sim
10^{35}$ erg to $\sim 10^{37}$ erg (see, e.g., Aarnio et al., 2012, 2013). Our
empirical measurements of the spin-down torques experienced by young stars
should serve as a touchstone for these and other theoretical mechanisms of
angular momentum loss in young stars.

##  Appendix A Age consistency

Figure 11: Average systematic trends in the distribution of ages as a function
of $T_{\rm eff}$ across Orion, in different age ranges. Included models are
MIST (Choi et al., 2016), PARSEC (Marigo et al., 2017), Baraffe et al. (2015),
SPOTS with 0%, 17%, 34%, and 51% spot coverage (Somers et al., 2020), and
Sagitta (McBride et al., 2021). The black line shows the constant age across
all $T_{\rm eff}$.

We estimate ages for stars through a variety of different means. Similarly to
Serna et al. (2021), we have inferred the ages of these stars by using $T_{\rm
eff}$, as well as Gaia and 2MASS photometry relative to various isochrones
(Baraffe et al., 2015; Choi et al., 2016; Marigo et al., 2017; Somers et al.,
2020) using MassAge code (Hernandez J. in prep). We also use Sagitta (McBride
et al., 2021), which is a neural net trained on the photometry of young stars
and the typical ages of star forming regions in which they are found. Finally,
we also use spectroscopically derived $\log g$ values, which strongly
correlates with stellar age. All of these estimates are model-dependent.
Although they all show similar progression of evolution, ordering sources from
oldest to youngest in a similar manner, the relative zero point calibration,
particularly as a function of mass, is systematically different between all of
the approaches.

Previously, Kounkel et al. (2018) deconvolved the Orion Complex into 190
kinematically coherent subgroups. Collectively, these groups can be used to
examine the spatial and temporal structure of the region. Independently,
however, they are relatively compact and stars within a given group are
expected to be coeval; i.e., they have self-consistent age distribution
regardless of their mass.

Table 2: Systematic offset in the mean age between different models  Model |  $\log t_{\rm Sagitta}-\log t_{\rm Model}$ (dex)
---|---
MIST | +0.14
PARSEC | +0.17
Baraffe |  $-$0.05
SPOTS-0 | +0.10
SPOTS-17 | +0.01
SPOTS-34 |  $-$0.10
SPOTS-51 |  $-$0.19

Assuming this coevality, we can test the performance of various models in
determining stellar ages, regardless of the absolute zero point calibration
between them. For a given group, we estimate a median age that is reported by
a given model, considering only the stars with $T_{\rm eff}$ between 3800 and
4500 K. Then, we estimate an age in a small bin within 150 K of a given
temperature, for all of the bins containing at least 4 stars, within each
individual group. The difference between the two estimates shows the age
gradient imposed by the underlying model as a function of mass. Afterwards,
all of the groups of a given age range are averaged together to characterize
the systematic trends.

We find that for $T_{\rm eff}$ of 3800–5000 K, all of the models appear to be
relatively stable, producing a coeval distribution of stars. However, this
does not hold for cooler stars. In the populations younger than $<4$ Myr,
isochrone models tend to overestimate the ages of stars by $\sim$0.1 dex, or
$\sim$25%, with the excess being more extreme for the youngest groups. PARSEC
isochrones (Marigo et al., 2017) appear to have the worst performance,
producing an excess of up to $\sim$0.3 dex, overestimating the ages of cool
stars by a factor of $\sim$2\. For groups older than 4 Myr, the trend is
reversed, and the ages can be underestimated by as much as $\sim$0.2 dex.

In all cases, strongly magnetic models with large spot coverage (Somers et
al., 2020) tend to be offset from the same models without spots by $\sim$0.2
dex. But, because of the underlying trends in the data, in groups $<4$ Myr,
non-magnetic tracks show greater coevality, whereas in groups $>$4 Myr,
strongly magnetic tracks appear to be coeval. We note that magnetic tracks
infer significantly older ages for a given group as a whole, with the
difference of 0.3 dex between tracks with spot sizes of 51% and those without
spots (Table 2).

Across all ages, Sagitta (McBride et al., 2021) tends to have the greatest
consistency in ages as a function of stellar mass, with only a modest
overestimation in ages in groups $<$2 Myr (we discuss this trend in Section
4), but otherwise appears stable across other age ranges. In part, this is by
construction, as the underlying model is empirical, trained on the average
ages of stars in a given population across the solar neighborhood. The other
models are theoretical, and may be lacking some of the physics for a precise
representation of very cool stars.

For this reason, to ensure coevality between low mass and high mass stars in a
given region, in subsequent sections we adopt ages from Sagitta. It’s absolute
calibration is most similar to the models from Baraffe et al. (2015), as well
as the magnetic tracks with $\sim$17% spot coverage. But, as the offset in age
between the models is systematic, using a different set of models does not
significantly influence the results presented in this work.

This work has made use of data from the European Space Agency (ESA) mission
Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing
and Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement.

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