# Transient Corotating Clumps Around Adolescent Low-Mass Stars From Four Years
of TESS

Luke G. Bouma 51 Pegasi b Fellow  Department of Astronomy, MC 249-17,
California Institute of Technology, Pasadena, CA 91125, USA  Rahul Jayaraman
MIT Kavli Institute and Department of Physics, 77 Massachusetts Avenue,
Cambridge, MA 02139  Saul Rappaport  MIT Kavli Institute and Department of
Physics, 77 Massachusetts Avenue, Cambridge, MA 02139  Luisa M. Rebull
Department of Astronomy, MC 249-17, California Institute of Technology,
Pasadena, CA 91125, USA  Lynne A. Hillenbrand  Department of Astronomy, MC
249-17, California Institute of Technology, Pasadena, CA 91125, USA  Joshua N.
Winn Department of Astrophysical Sciences, Princeton University, 4 Ivy Lane,
Princeton, NJ 08540, USA  Alexandre David-Uraz Department of Physics and
Astronomy, Howard University, Washington DC, 20059  Center for Research and
Exploration in Space Science and Technology, and X-ray Astrophysics
Laboratory, NASA/GSFC, Greenbelt, MD 20771, USA  Gáspár Á. Bakos Department of
Astrophysical Sciences, Princeton University, 4 Ivy Lane, Princeton, NJ 08540,
USA

(Received 2023 September 8; Revised —; Accepted —)

###### Abstract

Complex periodic variables (CPVs) are stars that exhibit highly structured and
periodic optical light curves. Previous studies have indicated that these
stars are typically disk-free pre-main-sequence M dwarfs with rotation periods
ranging from 0.2 to 2 days. To advance our understanding of these enigmatic
objects, we conducted a blind search using TESS 2-minute data of 65,760 K and
M dwarfs with $T$$<$16 and $d$$<$150 pc. We found 50 high-quality CPVs, and
subsequently determined that most are members of stellar associations. Among
the new discoveries are the brightest ($T$$\approx$9.5), closest
($d$$\approx$20 pc), and oldest ($\approx$200 Myr) CPVs known. One exceptional
object, LP 12-502, exhibited up to eight flux dips per cycle. Some of these
dips coexisted with slightly different periods, and the shortest-duration dips
precisely matched the expected timescale for transiting small bodies at the
corotation radius. Broadly, our search confirms that CPVs are mostly young
($\lesssim$150 Myr) and low-mass ($\lesssim$0.4 $M_{\odot}$). The flux dips
characteristic of the class have lifetimes of $\approx$100 cycles, although
stellar flares seem to induce sudden dip collapse once every few months. The
most plausible explanation for these phenomena remains corotating
concentrations of gas or dust. The gas or dust is probably entrained by the
star’s magnetic field, and the sharp features could result from a multipolar
field topology, a hypothesis supported by correspondences between the light
curves of CPVs and of rapidly rotating B stars known to have multipolar
magnetic fields.

Weak-line T Tauri stars (1795), Periodic variable stars (1213), Circumstellar
matter (241), Star clusters (1567), Stellar magnetic fields (1610), Stellar
rotation (1629)

††software:  astrobase (Bhatti et al., 2021), astropy (Astropy Collaboration
et al., 2013, 2018, 2022), lightkurve (Lightkurve Collaboration et al., 2018),
numpy (Harris et al., 2020), pyGAM (Servén & Brummitt, 2018), scipy (Virtanen
et al., 2020), TESS-point (Burke et al., 2020), wotan (Hippke et al., 2019).
††facilities:  Astrometry: Gaia (Gaia Collaboration et al., 2018, 2023).
Imaging: Second Generation Digitized Sky Survey. Spectroscopy: Keck:I (HIRES;
Vogt et al. 1994). Photometry: TESS (Ricker et al., 2015), Broadband
photometry: 2MASS (Skrutskie et al., 2006), APASS (Henden et al., 2016), Gaia
(Gaia Collaboration et al., 2018, 2023), SDSS (York et al., 2000), WISE
(Wright et al., 2010).

##  1 Introduction

(a)  (b)

Figure 1:  Complex periodic variables (CPVs): Top: Phase-folded TESS light
curves for three CPVs. Each panel shows the average of the data accumulated
over one month, relative to the mean stellar brightness. Gray circles are raw
2-minute data; black circles are binned to 300 points per cycle. The period in
hours is printed in the bottom right corner. Left-to-right, the objects are LP
12-502 (TIC 402980664; Sector 19), TIC 94088626 (Sector 10), and TIC 425933644
(Sector 28). Bottom: Cartoon configurations for magnetically-entrained
corotating material. The dust clump scenario (left) and gas prominence
scenario (right) propose different opacity sources, and different occultation
geometries.

All young stars vary in optical brightness, and the origin of such variability
is, in most cases, understood. Well-explored sources of optical variability
include inhomogeneities on stellar surfaces such as starspots and faculae
(e.g. Basri, 2021), occultations by circumstellar disks (e.g. Bodman et al.,
2017), and, in geometrically favorable circumstances, eclipses by stars and
planets (e.g. Rizzuto et al., 2020). More exotic sources of optical
variability that are potentially relevant to this work include transiting
exocomets (e.g. $\beta$ Pic; Zieba et al. 2019), disintegrating rocky bodies
(e.g. KOI-2700; Rappaport et al. 2014), and occultations by circumstellar
plasma clumps (e.g. $\sigma$ Ori E; Townsend et al. 2005; Townsend & Owocki
2005).

Data from K2 (Howell et al., 2014) and TESS (Ricker et al., 2015) have
revealed a new class of variable star for which the root cause of variability
is only beginning to become clear: complex periodic variables (CPVs). These
objects are identified from their optical light curves, which show nearly
periodic troughs that are either sharp or broad; these troughs are often
superposed on quasi-sinusoidal spot-like modulation (Stauffer et al., 2017,
2018b; Zhan et al., 2019). Some CPVs show up to eight dips per cycle. Most
CPVs are pre-main-sequence M dwarfs with ages of $\approx$5-150 million years
(Myr), and rotation periods of 0.2–2 days. They are observed to comprise
$\approx$1-3% of M dwarfs younger than 100 Myr (Rebull et al., 2016; Günther
et al., 2022). They generally do not show near-infrared excesses indicative of
dusty disks, but the wavelength-dependent dip amplitudes of some CPVs is
consistent with reddening by dust (Onitsuka et al., 2017; Bouma et al., 2020;
Günther et al., 2022; Koen, 2023). The dip amplitudes and phases usually
evolve gradually over tens to hundreds of cycles, although they have
occasionally been observed to change abruptly within one cycle (e.g. Stauffer
et al., 2017; Palumbo et al., 2022; Popinchalk et al., 2023).

The sharp features of CPV light curves can have durations as short as 5% of
the rotation period ($P_{\rm rot}$), which is too short to be caused by
starspots rotating into and out of view. Starspots produce flux variations
with characteristic timescales of $P_{\rm rot}$ and $0.5\,P_{\rm rot}$. With
finely-tuned viewing geometries, starspots can produce dip durations as short
as $\approx$$0.2\,P_{\rm rot}$, but in such cases, limb darkening causes the
dip amplitudes to be smaller than the observed amplitudes of $\sim$1% (see
Stauffer et al. 2017, Figures 37-41). Thus, a “starspot-only” scenario can be
ruled out for many CPVs (Stauffer et al., 2017; Zhan et al., 2019; Koen,
2021). Given that many CPVs cannot be explained by starspots alone, and
working under the assumption that all CPVs share the same basic physical
scenario, we discard the “starspot-only” model. Instead, the correct
explanation probably involves spatially concentrated circumstellar material
(e.g. Stauffer et al., 2017; Günther et al., 2022).

Figure 1 illustrates two proposed configurations for the extrinsic material.
The first scenario invokes opaque dust “clumps” that orbit near the Keplerian
corotation radius [$R_{\rm c}=(GM/\Omega^{2})^{1/3}$, where
$\Omega=2\pi/P_{\rm rot}$] and periodically transit the star (e.g. Stauffer et
al., 2017; Farihi et al., 2017; Sanderson et al., 2023). The second scenario
invokes “prominences”, long-lived condensations of cool, dense, marginally-
ionized gas that are embedded within the hotter corona and that corotate with
the star (Collier Cameron & Robinson, 1989; Jardine & Collier Cameron, 2019;
Waugh & Jardine, 2022). These hypothetical prominences are analogous to
quiescent prominences and filaments seen in the solar corona (see e.g. Vial &
Engvold, 2015), though rather than existing at a fraction of the stellar
radius as in the solar case, they would exist at distances of a few stellar
radii. A final possibility is that an optically-thick ring obscures a narrow
band of the stellar photosphere (Zhan et al., 2019); hot spots passing behind
such a ring could produce sudden dips. We disfavor this scenario for reasons
described in Appendix A.

While the dust clump and gas prominence hypotheses both invoke magnetically-
entrained material, the two pictures differ in the origin and the composition
of the occulting material. “Dust clumps” invoke opacity from dust, which would
need to be collisionally charged (Sanderson et al., 2023), and which might be
sourced from a low-mass debris disk. “Gas prominences” invoke opacity from
partially-ionized gas, perhaps bound-free transitions in hydrogen or a
molecular opacity. The gas might be sourced from a stellar wind. Unambiguous
evidence in support of either scenario has yet to be acquired. Such evidence
might include a spectroscopic detection of silicate 10 $\mu$m dust absorption
during a dip, or perhaps detection of transient Balmer-line excesses as a
function of cycle phase, similar to observations made in systems such as AB
Dor (see Collier Cameron, 1999) or PTFO 8-8695 (Johns-Krull et al., 2016).

In both models, the corotation radius is the location at which matter
concentrates. The empirical basis for this is that the sharp CPV features are
superposed over smooth, quasi-sinusoidal starspot profiles. The theoretical
importance of the corotation radius has been noted in previous studies of
magnetic rotators (e.g. Lamb et al., 1973; Nakajima, 1985; Königl, 1991; Long
et al., 2005). In regions where the magnetic field dominates the flow (i.e.
$B^{2}/8\pi>\rho v^{2}/2$), matter is dragged along with the field lines.
Within such regions, charged gas or dust can become trapped at corotation
because of the four relevant forces – gravity, Lorentz, inertial Coriolis, and
inertial centrifugal – the Lorentz and Coriolis only act perpendicular to
field lines, while gravity and the centrifugal force are in balance at $R_{\rm
c}$ (e.g. Townsend & Owocki, 2005, their Section 2). Another way to phrase
this statement is that, in the corotating frame, the effective potential
experienced by charged particles tends to have local minima at $R_{\rm c}$;
given a flow from either the star or from a tenuous accretion disk, this local
potential minimum enables material to build up (Townsend & Owocki, 2005).

While theoretical heritage for understanding magnetic rotators exists, CPVs
have remained mysterious because they have been both hard to discover and hard
to characterize. They have been hard to discover because they are rare: CPVs
comprise $\approx$1% of the youngest $\approx$1% of M dwarfs (Rebull et al.,
2018). Out of the millions of stars monitored by K2 and TESS, about 70 CPVs
have been reported to date (Rebull et al., 2016; Stauffer et al., 2017, 2018b;
Zhan et al., 2019; Bouma et al., 2020; Stauffer et al., 2021; Günther et al.,
2022; Popinchalk et al., 2023). They have been hard to characterize because
many of the known CPVs are faint; the initial K2 discoveries (Rebull et al.,
2016; Stauffer et al., 2017) were M2-M6 dwarfs at distances $\gtrsim$100 pc,
with optical brightnesses of $V$$\approx$15.5 to $V$$>$20\. At such
magnitudes, high-resolution time-series spectroscopy is out of reach with
current facilities, despite the potential utility of such observations.

In this work, we aim to find bright and nearby CPVs, since these objects will
be the most amenable to detailed photometric and spectroscopic analyses. To do
this, we use 2-minute cadence data acquired by TESS between 2018 July and 2022
September (Sectors 1-55; Cycles 1-4). We present our search methods in Section
2, and the resulting CPV catalog in Section 3. The observed evolution of many
CPVs over a two-year baseline is described in Section 4, including a deep-dive
into the behavior of an especially interesting object, LP 12-502. We discuss a
few implications in Section 5, and conclude in Section 6.

Some comments on nomenclature are needed. What we are calling “complex
periodic variables” (Koen, 2023) have also been called “complex rotators”
(Zhan et al., 2019; Günther et al., 2022; Popinchalk et al., 2023), “transient
flux dips”, “persistent flux dips”, and “scallop shells” (Stauffer et al.,
2017). The CPVs should not be conflated with “dippers”, which are classical T
Tauri stars with infrared excesses, and which show large-amplitude variability
linked to obscuring inner disk structures and accretion hot spots (Cody et
al., 2014; Robinson et al., 2021). The phenomenology and stellar properties of
CPVs and dippers are quite different (though see Sections 3.3 and 5.9). The
defining phenomenological features of the CPVs are that their light curves are
complex, relative to quasi-sinusoidal starspots, and the complex features are
periodic, meaning they typically repeat for at least tens of days. While
rotation likely does play a central role in explaining their physical
behavior, the acronym for “complex rotator” is already used in the
astrophysical literature for cosmic rays. Given these considerations, we refer
to the stars as complex periodic variables (CPVs); our preferred explanation
for their behavior is that transient clumps of gas or dust orbit at their
corotation radii.

##  2 Methods

###  2.1 Stellar selection function

We searched for CPVs by analyzing the short-cadence data acquired by TESS
between 2018 July 25 and 2022 September 1 (Sectors 1-55). Specifically, we
used the 2-minute cadence light curves produced by the Science Processing and
Operations Center at the NASA Ames Research Center (Jenkins et al., 2016).
While the TESS data products from these sectors also included full frame
images with cadences of 10 and 30 minutes for a larger number of sources, we
restricted our attention to the 2-minute data for the sake of uniformity and
simplicity in data handling. In exchange, we sacrificed both completeness and
homogeneity of the selection function. While TESS cumulatively observed
$\approx$90% of the sky for at least one lunar month between 2018 July and
2022 September, the 2-minute cadence data were collected for only a subset of
observable stars that were preferentially nearby and bright (see Fausnaugh et
al., 2021). The total 2-minute data volume from Sectors 1-55 included
1,087,475 short-cadence light curves, which were available for 428,121 unique
stars.

To simplify our search, we defined our target sample as stars with 2-minute
cadence TESS light curves satisfying the following four conditions:

$\displaystyle T$ $\displaystyle<16\quad$ $\displaystyle(\mathrm{Amenable\
with\ TESS})$ (1) $\displaystyle G_{\rm BP}-G_{\rm RP}$
$\displaystyle>1.5\quad$ $\displaystyle(\mathrm{Red\ stars\ only})$ (2)
$\displaystyle M_{\rm G}$ $\displaystyle>4\quad$ $\displaystyle(\mathrm{Dwarf\
stars\ only})$ (3) $\displaystyle d$ $\displaystyle<150\,{\rm pc}\quad$
$\displaystyle(\mathrm{Close\ stars\ only}).$ (4)

Here, $M_{\rm G}=G+5\log(\varpi_{\rm as})+5$ is the Gaia $G$-band absolute
magnitude, $\varpi_{\rm as}$ is the parallax in units of arcseconds, and $d$
is a geometric distance defined by inverting the parallax and ignoring any
zero-point correction. We performed this selection by cross-matching TIC8.2
(Stassun et al., 2019; Paegert et al., 2021) against the Gaia DR2 point-source
catalog (Gaia Collaboration et al., 2018). We opted for Gaia DR2 rather than
DR3 because the base catalog for TIC8 was Gaia DR2, which facilitated a one-
to-one crossmatch using the Gaia source identifiers. The target sample
ultimately included 65,760 M dwarfs and late-K dwarfs, down to $T$$<$16 and
out to $d$$<$150 pc. For stars with multiple sectors of TESS data available,
we searched for CPV signals independently. In total, our 65,760 star target
list included 180,017 month-long light curves.

We assessed the completeness of our selection function by comparing the number
of stars with TESS Sector 1-55 short-cadence data against the number of Gaia
DR2 point sources. We required all stars to meet conditions 1–4. The results
are shown in Figure 2. TESS 2-minute data exist for $\approx$50% of $T$$<$16 M
and late-K dwarfs at $\approx$50 pc. Within 20 pc, $\gtrsim$80% of the
$T$$<$16 M and late-K dwarfs have at least one sector of short-cadence data.
Beyond 100 pc, $\lesssim$10% of such stars have any short-cadence data
available. This can be translated into our sensitivity for the lowest mass
stars by considering that the spectral type of a $T$=16 star at $d$=50 pc is
$\approx$M5.5V, corresponding to a main-sequence mass of $\approx$0.12
$M_{\odot}$.

Figure 2:  Completeness of the TESS 2-minute data for late-K and early M
dwarfs near the Sun, from Sectors 1-55. The orange dotted curve shows the
number of stars in successive radial shells, each with a width of 10 pc. To be
part of our selection function, these stars must meet the following
conditions: they must be red dwarf stars ($G_{\rm BP}-G_{\rm RP}$¿$1.5$;
$M_{\rm G}$¿4) amenable for TESS observations ($T$¡16). The blue solid curve
shows the fraction of such stars with at least one sector of TESS 2-minute
cadence data acquired between Sectors 1-55.

###  2.2 CPV discovery

Prior to this study, most CPVs have been found by visually examining all the
light curves of stars in young clusters (Rebull et al., 2016; Stauffer et al.,
2017; Popinchalk et al., 2023), or by flagging light curves with short periods
and strong Fourier harmonics for visual inspection (Zhan et al., 2019). In
this work, we implemented a new search approach based on counting the number
of sharp local minima in phase-folded light curves, while also using the
Fourier approach. We applied these two search techniques independently to our
65,760 targets.

####  2.2.1 Counting dips

The dip counting technique aims to count sharp local minima in phase-folded
light curves. The most remarkable CPVs often show three or more dips per
cycle, which distinguishes them from other types of variables such as
synchronized and spotted binaries (RS CVn stars).

For our dip-counting pipeline, we began with the PDC_SAP light curves for each
sector (Smith et al., 2017), removed non-zero quality flags, and normalized
the light curve by dividing out its median value. We then flattened the light
curve using a 5-day sliding median filter, as implemented in wotan (Hippke et
al., 2019). We computed a periodogram of the resulting cleaned and flattened
light curve, opting for the Stellingwerf (1978) phase dispersion minimization
(PDM) algorithm implemented in astrobase (Bhatti et al., 2021) due to its
shape agnosticism. If a period $P$ below 2 days was identified, we reran the
periodogram at a finer grid to improve the accuracy of the period
determination.

Once a star’s period was identified, we binned the phased light curve to 100
points per cycle. To separate sharp local minima from smooth spot-induced
variability, we then iteratively fit robust penalized splines to the wrapped
phase-folded light curve, excluding points more than two standard deviations
away from the local continuum (Hippke et al., 2019). The wrapping procedure is
discussed below. In this fitting framework, the maximum number of equidistant
spline knots per cycle is the parameter that controlled the meaning of “sharp”
— we allowed at most 10 such knots per cycle, though for most stars fewer
knots were preferred based on cross-validation using an $\ell^{2}$-norm
penalty. An example fit is shown in panel (e) of Figure 3.

We then identified local minima in the resulting residual light curve using
the SciPy find_peaks utility (Virtanen et al., 2020), which is based on
comparing adjacent values in an array. For a peak to be flagged as
significant, we required it to have a width of at least $0.02\,P$, and a
height of at least twice the noise level. The noise level was defined as the
68th percentile of the distribution of the residuals from the median value of
$\delta f_{i}\equiv f_{i}-f_{i+1}$, where $f$ is the flux and $i$ is an index
over time. In panel (e) of Figure 3, automatically-identified local minima are
shown with the gray triangles.

Wrapping is necessary to eliminate edge effects when fitting the light curve
and when identifying local minima in the residuals. A phased light curve would
usually cover phases $\phi\in[0,1]$. We instead performed the analysis
described above using a phase-folded light curve spanning $\phi\in[-1,2]$,
which was created by duplicating and concatenating the ordinary phase-folded
light curve. The free parameters we adopted throughout the analysis – for
instance the maximum number of spline knots per cycle, and the height and
depth criteria for dips – were chosen during testing based on the desire to
correctly re-identify a large fraction ($>$90%) of previously known CPVs,
while also being able to consistently reject common false positives such as
spot-induced variability and eclipsing binaries.

In short, CPV candidates were identified by requiring a peak PDM period below
two days and the presence of at least three sharp local minima, based on at
least one sector of the TESS 2-minute data. Candidates were then inspected
visually as described in Section 2.2.3.

####  2.2.2 Fourier analysis

We performed an independent search using a Fourier-based approach, following
Zhan et al. (2019) and Pribulla et al. (2023, their Section 1.3). Starting
with the PDC_SAP light curves, we normalized each light curve, and then re-
binned it into equal width 2-minute bins to account for the uneven spacing in
the TESS data, as well as the data gap caused by satellite downlink during
each sector. We then padded the data to ensure that the light curve had a
length that was a power of two, as described by Zhan et al. After taking the
Fourier transform of the padded light curve using numpy.fft, we searched for
peaks with a significance exceeding 12-$\sigma$ within a set of 500 frequency
bins.

Peaks of significance were found for $\approx$10% of the searched stars. For
all such cases, we generated an interim “summary sheet” with information about
the star, its full and folded light curves, Fourier transform, potential
contaminating stars, and information about these contaminating stars. We then
reviewed each summary sheet, and tentatively classified each light curve based
on visual inspection of its morphology (with common categories including
eclipsing binary, CPV, RS CVn, and cataclysmic variable).

####  2.2.3 Manual vetting

Figure 3:  Validation plots used to classify CPVs. The complete figure set
contains one image per sector for each of the 66 objects in Table Transient
Corotating Clumps Around Adolescent Low-Mass Stars From Four Years of TESS,
and is accessible both through the online journal and via
https://zenodo.org/record/8327508. Panels are as follows. a): Phase-folded
light curve; gray points are raw 2-minute data and black points are binned to
200 points per cycle. The adopted period is given in the lower-right corner.
b): Phase-dispersion minimization (PDM) periodogram. Dotted lines show up to
the 10th harmonic and subharmonic. c): DSS finder chart, with 21′′ and 42′′
radius circles for scale. One TESS pixel has a full side length of 21′′. d):
Cleaned light curve, binned to 20-minute cadence, in Barycentric TESS Julian
Date (BTJD). e): Phase-folded light curve, binned to 100 points per cycle. The
gray line denotes the spline-fit to the wrapped phase-folded light curve, and
small gray triangles denote automatically identified local minima. f): Phase-
folded light curve at twice the peak period. g): Phase-folded light curve at
half the peak period. h): Phase-folded time-series within the “background”
aperture defined in the SPOC light curves. i): Phase-folded flux-weighted
centroid in the column direction. j): Phase-folded flux-weighted centroid in
the row direction. k): Gaia DR2 color–absolute magnitude diagram. The gray
background denotes stars within 100 pc from Gaia Collaboration et al. (2021).
l): Information from Gaia DR2, TIC8, and the automated dip-counting search
pipeline. “Neighbors”, abbreviated “nbhr”, are listed within apparent
distances of 2 TESS pixels if $\Delta T$$<$2.5. m): BANYAN $\Sigma$ v1.2
association probabilities, calculated using positions, proper motions, and the
parallax.

We homogeneously assessed whether the objects identified using the dip-
counting (Section 2.2.1) and Fourier (Section 2.2.2) approaches were
consistent with expectations for CPVs by assembling the data shown in Figure
3. We labeled a star as a “good” CPV if it met all of the following criteria
for at least one TESS sector:

  * •

$P<2$ days.

  * •

At least three dips per cycle, or else otherwise oddly-shaped dips relative to
expectations for quasi-sinusoidal starspot-induced modulation.

  * •

Persistent dips over multiple consecutive rotation cycles.

We also noted ”possible” CPVs for which the classification was more ambiguous,
and “false positives” that are definitely not CPVs. The most common false
positives for both the Fourier and dip-counting techniques were eclipsing
binaries, ordinary spotted rapid rotators, and light curves that were complex
due to multiple stars contributing to the photometric aperture. Our
specialized dip-counting pipeline flagged 368 unique stars for visual
inspection; about 20% were subsequently labeled either good or possible CPVs.
From the more general Fourier pipeline, $\approx$0.5% of stars that passed the
12-$\sigma$ peak threshold were eventually classified as CPVs.

###  2.3 Stellar properties

####  2.3.1 Ages

While most of our target stars were field stars, color–absolute magnitude
diagrams suggested that most CPVs tended to be on the pre-main-sequence (e.g.
panel (k) of Figure 3). We therefore estimated stellar ages by checking for
probabilistic spatial and kinematic associations between the CPVs and known
clusters in the solar neighborhood. For most stars in our sample, we did this
using BANYAN $\Sigma$ (Gagné et al.,
2018).111https://github.com/jgagneastro/banyan_sigma, git commit 394b486 This
algorithm calculates the probability that a given star belongs to either the
field, or to any of 27 young clusters (“associations”) within 150 pc of the
Sun. This is achieved by modeling the field and cluster populations as
multivariate Gaussian distributions in 3-D position and 3-D velocity space. We
used the Gaia DR2 sky positions, proper motions, and distances to calculate
the membership probabilities. BANYAN $\Sigma$ in turn analytically
marginalizes over the radial velocity dimension. The probabilities returned by
this procedure are qualitatively helpful, but should be interpreted with
caution because the assumption of Gaussian distributions is questionable for
most groups within the solar neighborhood (see e.g. Kerr et al., 2021, Figure
10).

For a few cases where BANYAN $\Sigma$ yielded ambiguous results, we consulted
the meta-catalog of young, age-dated, and age-dateable stars provided by Bouma
et al. (2022), and also searched the local volume around each star for co-
moving companions.222https://github.com/adamkraus/Comove, git commit 278b372
Finally, to provide a base for comparison, we also ran the BANYAN $\Sigma$
membership analysis on our entire 65,760 target star sample.

####  2.3.2 Effective temperatures, radii, and masses

We determined the stellar effective temperature and radii for the CPVs by
fitting the broadband spectral energy distributions (SEDs); we then estimated
the masses by interpolating against the sizes, temperatures, and ages of the
PARSEC v1.2S models (Bressan et al., 2012; Chen et al., 2014).

For the SED fitting, we used astroARIADNE (Vines & Jenkins, 2022). We adopted
the BT-Settl stellar atmosphere models (Allard et al., 2012) assuming the
Asplund et al. (2009) solar abundances, and the Barber et al. (2006) water
line lists. The broadband magnitudes we considered included $GG_{\rm BP}G_{\rm
RP}$ from Gaia DR2, $Vgri$ from APASS, $JHK_{\rm S}$ from 2MASS, SDSS $riz$,
and the WISE $W1$ and $W2$ passbands. We omitted UV flux measurements from our
SED fit to avoid any possible bias induced by chromospheric UV excess. We
omitted WISE bands $W3$ and $W4$ due to reliability concerns. astroARIADNE
compares the measured broadband flux measurements against pre-computed model
grids, and by default fits for six parameters: $\\{T_{\rm
eff},R_{\star},A_{\rm V},\log g,[{\rm Fe/H}],d\\}$. The distance prior is
drawn from Bailer-Jones et al. (2021). The surface gravity and metallicity are
generally unconstrained. Given our selection criteria for the stars, we
assumed the following priors for the temperature, stellar size, and
extinction:

$\displaystyle T_{\rm eff}/{\rm K}$ $\displaystyle\sim\mathcal{N}(3000,1000),$
(5) $\displaystyle R_{\star}/R_{\odot}$ $\displaystyle\sim\mathcal{T}_{\rm
N}(0.5,0.3,0.1,1.5),$ (6) $\displaystyle A_{\rm V}/{\rm mag}$
$\displaystyle\sim\mathcal{U}(0,0.2),$ (7)

for $\mathcal{N}$ the Gaussian and $\mathcal{U}$ the uniform distributions,
and $\mathcal{T}_{\rm N}(\mu,\sigma,a,b)$ a truncated normal distribution with
mean $\mu$, standard deviation $\sigma$, and lower and upper bounds $a$ and
$b$. We validated our chosen upper bound on $A_{\rm V}$ using a 2MASS color-
color diagram. Finally, using Dynesty (Speagle, 2020), we sampled the
posterior probability assuming the default Gaussian likelihood, and set a
stopping threshold of ${\rm d}\log\mathcal{Z}<0.01$, where $\mathcal{Z}$
denotes the evidence.

With the effective temperatures and stellar radii from the SED fit, we
estimated the stellar masses by interpolating against the PARSEC isochrones
(v1.2S Chen et al., 2014). The need for models that incorporate some form of
correction for young, active M dwarfs is well-documented (e.g. Stassun et al.,
2012; David & Hillenbrand, 2015; Feiden, 2016; Somers et al., 2020). Plausible
explanations for the disagreement between observed and theoretical M dwarf
colors and sizes include starspot coverage (e.g. Gully-Santiago et al., 2017)
and potentially incomplete line lists (e.g. Rajpurohit et al., 2013). In the
PARSEC models, Chen et al. (2014) performed an empirical correction to the
temperature–opacity relation drawn from the BT-Settl model atmospheres, in
order to match observed masses and radii of young eclipsing binaries. This is
sufficient for our goal of estimating stellar masses. Given our estimates of
$\\{\tilde{T}_{\rm eff},\tilde{M}_{\star},\tilde{t}\\}$, and approximating
their uncertainties as Gaussian $\sigma_{\tilde{T}_{\rm eff}}$,
$\sigma_{\tilde{M}_{\star}}$ and $\sigma_{\tilde{t}}$, we define a distance
metric $\Delta$ to each model PARSEC grid-point $\\{T_{\rm
eff},M_{\star},t\\}$ via

$\Delta^{2}=\left(\frac{\tilde{T}_{\rm eff}-T_{\rm
eff}}{\sigma_{\tilde{T}_{\rm
eff}}}\right)^{2}+\left(\frac{\tilde{M}_{\star}-M_{\star}}{\sigma_{\tilde{M}_{\star}}}\right)^{2}+\left(\frac{\tilde{t}-t}{\sigma_{\tilde{t}}}\right)^{2},$
(8)

where the division by the uncertainties helps to assign equal importance to
each dimension. The mass reported in Table Transient Corotating Clumps Around
Adolescent Low-Mass Stars From Four Years of TESS is the model mass that
minimizes the distance. The reported uncertainties in the masses are based on
propagating the statistical uncertainties in the radii, temperatures, and
ages.

####  2.3.3 Binarity

The main types of binaries of interest in this work are those that are
unresolved, because they can lead to misinterpretations of the data. For
instance, unresolved binaries might produce multiple photometric signals and
hinder our ability to correctly identify the star hosting the CPV signal.
Unresolved binaries could also bias photometric magnitude and color
measurements, which would affect our stellar parameter estimates. To attempt
to identify binaries, we considered the following lines of information.

Radial velocity scatter—We examined diagrams of the Gaia DR3 “radial velocity
error” as a function of stellar color for all 63 CPVs and candidate CPVs.
Since this quantity represents the standard deviation of the non-published
Gaia RV time series, outliers can suggest single-lined spectroscopic binarity.
These plots showed two clusters of stars, at $\lesssim$10 km s-1and 20-25 km
s-1. We therefore adopted a threshold of $20$ km s-1 to flag possible single-
lined spectroscopic binaries, which selected three stars: TIC 405910546, TIC
224283342, and TIC 280945693.

RUWE—We examined plots of Gaia DR3 RUWE as a function of color.333For an
explanation of the renormalized unit weight error (RUWE), see the GAIA DPAC
technical note http://www.rssd.esa.int/doc_fetch.php?id=3757412. Elevated
RUWEs imply excess astrometric noise relative to a single-source model. This
can be caused by marginally resolved binaries, intrinsic photometric
variability, or intrinsic astrometric motion. Based on this exercise, we
adopted a threshold of RUWE${}_{\rm DR3}>2$ to flag sources with excess
astrometric noise. This threshold was met by 16/50 high-quality CPVs and by
0/13 of the ambiguous CPVs. The choice of the threshold RUWE is somewhat
subjective, since the RUWE distribution has an extended tail (e.g. Penoyre et
al., 2022). If we had instead required RUWE${}_{\rm DR3}>1.4$, 21/50 high-
quality CPVs and 2/13 of the ambiguous sample would have been flagged.

Gaia DR3 non-single stars—Gaia DR3 included a non_single_star column that
flagged eclipsing, astrometric, and spectroscopic binaries. None of the stars
in our CPV sample were identified as binaries in this column.

Multiple periodic TESS signals—During our visual analysis of the TESS light
curves and PDM periodograms, we flagged sources with beating light curves, and
with PDM periodograms that showed multiple periods. For such cases, we then
subtracted the mean CPV signal over each sector, and repeated the phase-
dispersion minimization analysis. The resulting secondary periods, $P_{\rm
sec}$, are listed in Table Transient Corotating Clumps Around Adolescent Low-
Mass Stars From Four Years of TESS; we required these to be at least 5%
different from the primary period. The majority of secondary signals showed
morphologies corresponding to starspot modulation. This process yielded 22/50
high-quality CPVs with secondary periods; 3/13 of the ambiguous sample met the
same criterion. Of the 16 good CPVs with ${\rm RUWE}_{\rm DR3}>2$, 15 also
showed secondary periods in the TESS light curves. Considering the weaker
threshold of ${\rm RUWE}_{\rm DR3}>1.4$, 18/21 such CPVs showed secondary TESS
periods. The latter results strongly suggest that the secondary periods are
associated with bound binary companions.

Table Transient Corotating Clumps Around Adolescent Low-Mass Stars From Four
Years of TESS summarizes each of the sources of binarity information into a
single bitwise column. We describe detailed results concerning binarity in
Section 3.4, and summarize those results Section 5.2.

Figure 4:  CPVs found in the TESS 2-minute data. Phased TESS light curves over
one month are shown for 50 CPVs in the high quality sample. Gray are raw
2-minute data; black bins to 300 points per cycle. Objects are ordered such
that sources with the most TESS data available are on top (see Section 3.1).
Zero phase is chosen to correspond to minimum light. Each panel is labeled by
the TIC identifier, the TESS sector number, the period in hours, and the
three-bit binarity flag from Table Transient Corotating Clumps Around
Adolescent Low-Mass Stars From Four Years of TESS, which denotes Gaia DR3
radial_velocity_error outliers (bit 1), Gaia DR3 ruwe outliers (bit 2), and
stars with secondary TESS periods (bit 3).

##  3 Results

###  3.1 CPV catalog

Table Transient Corotating Clumps Around Adolescent Low-Mass Stars From Four
Years of TESS lists the 66 objects identified by our search. The 50 stars in
the “good” sample demonstrated what we deemed to be the key characteristics of
the CPV phenomenon in at least one TESS sector. The classification of 13 CPV
candidates was ambiguous, and the 3 remaining objects were notable false
positives that we discuss below. The quality column in the table divides the
three classes; additional data from TESS or other instruments could help
resolve the classification of the ambiguous cases. Of the 63 CPVs and
candidate CPVs, 32 were found using both the dip-counting and Fourier
techniques, 23 were found using only the dip-counting technique, and 8 were
found using only the Fourier technique. In the following, we will focus our
discussion on the good sample, irrespective of discovery method. We will often
refer to stars by their TIC identifiers; these can be referenced against the
figures in most digital document readers using a “find” (Ctrl+F) utility.

Figure 4 is a mosaic of phased light curves for the 50 CPVs. The objects are
sorted first in order of the number of TESS 2-minute cadence sectors in which
they clearly demonstrated the CPV phenomenon, and secondarily by descending
brightness. The top five objects by this metric are TIC 300651846 (12
sectors); TIC 402980664 (7 sectors); TIC 89463560 (5 sectors); TIC 363963079
(5 sectors); and TIC 294328887 (4 sectors). The brightest five CPVs span
9.3$<$$T$$<$11.1; the faintest five span 14.5$<$$T$$<$15.0. The fastest five
have periods spanning 3.6 hr$<$$P$$<$6.2 hr, and the slowest five span 27
hr$<$$P$$<$38 hr.

The light curves show between two and eight local minima per cycle. Some stars
show ordinary sinusoidal modulation during one portion of the phased light
curve, and highly structured modulation in the remainder of the cycle (e.g.
TIC 206544316, TIC 224283342, TIC 402980664). Others show structured
modulation over the entire span of a cycle (e.g. TIC 2234692, TIC 425933644,
TIC 142173958). Others show some mix between these two modes.

A small number of objects at first glance seem reminiscent of eclipsing
binaries, such as TIC 193831684, TIC 59836633, or TIC 5714469. We believe
these cases are unlikely to be eclipsing binaries due to the additional
coherent peaks and troughs in the light curves, which are distinct from any
binary phenomena of which we are aware.

###  3.2 Ages of CPVs

Of our 63 confirmed and candidate CPVs, 61 were associated with a nearby
moving group or open cluster, primarily using BANYAN $\Sigma$ as described in
Section 2.3.444Two of the 61 memberships were made with low confidence and are
flagged in Table Transient Corotating Clumps Around Adolescent Low-Mass Stars
From Four Years of TESS. The assignment of TIC 397791443 to IC 2602 was based
not on BANYAN $\Sigma$ but instead on a literature search (e.g. Cantat-Gaudin
& Anders, 2020). The relevant groups are listed in Table Transient Corotating
Clumps Around Adolescent Low-Mass Stars From Four Years of TESS; their ages
span $\approx$5-200 Myr. For comparison, BANYAN $\Sigma$ assigned high-
probability ($>$95%) field membership to 59,361 out of the 65,760 target
stars. Most stars in our target sample are old; the CPVs returned by our blind
search are young.

The groups that contain the largest number of CPVs in our catalog are Sco-Cen,
Tuc-Hor, and Columba. Six CPVs were also identified in the Argus association
(Zuckerman, 2019), which serves as an indirect line of evidence supporting the
reality and youth of that group. The large contribution from Sco-Cen is not
surprising since Sco-Cen contains the majority of pre-main-sequence stars in
the solar neighborhood, and many of its stars were selected for TESS 2-minute
cadence observations by guest investigators. Given the $\lesssim$$10\%$
completeness of our data beyond 100 pc (Figure 2), there may be many more CPVs
in Sco-Cen that remain to be discovered.

There were two stars for which neither BANYAN $\Sigma$ nor a literature search
led to a confident association with any young group. Both stars display CPV
signals over multiple TESS sectors. Both are photometrically elevated relative
to the main sequence, an indication of youth. Both were also noted by Kerr et
al. (2021) as being in the “diffuse” population of $<$50 Myr stars near the
Sun.

Our search confirms that the CPV phenomenon persists for at least $\approx$150
Myr. Table Transient Corotating Clumps Around Adolescent Low-Mass Stars From
Four Years of TESS includes three $\approx$150 Myr CPVs in AB Dor (Bell et
al., 2015), a $\approx$112 Myr old Pleiades CPV (Dahm, 2015), and a similarly-
aged Psc-Eri member (Ratzenböck et al., 2020). To our knowledge, TIC 332517282
in AB Dor ($t$$=$$149^{+51}_{-19}$ Myr; Bell et al. 2015) was the previous
record-holder for the oldest-known CPV (Zhan et al., 2019; Günther et al.,
2022); at least one unambiguous CPV (EPIC 211070495) and a few other
candidates were also previously known in the Pleiades (Rebull et al., 2016).

The maximum age of CPVs might even exceed 200 Myr, based on the candidate
membership of TIC 294328887 in the Carina Near moving group (Zuckerman et al.,
2006). The estimated age of this group, $200\pm 50$ Myr, is based on the
lithium sequence of its G-dwarfs (Zuckerman et al., 2006), which shows a
coeval population of stars older than the Pleiades and younger than the 400
Myr Ursa Major moving group. However, the formal BANYAN $\Sigma$ membership
probability is somewhat low (only 6%), perhaps due to the missing radial
velocity. This lack of information could be rectified by acquiring even a
medium-resolution spectrum. An independent assessment of the group’s
kinematics using Gaia data, and its rotation sequence using TESS, could also
bear on the question of whether TIC 294328887 is a member.

###  3.3 Infrared excesses of CPVs

Most CPVs in our catalog did not show infrared excesses in the $W1$-$W4$
bands, which is typical for this class of object (Stauffer et al., 2017).
Visually inspecting the SEDs of our 66 star sample and the WISE images
available through IRSA, we labeled two objects as having reliable infrared
excesses (both $W3$ and $W4$ fluxes are more than 3$\sigma$ above the
photospheric prediction): TIC 193136669 (TWA 34) and TIC 57830249 (TWA 33).
However, neither is considered a “good” CPV for the reasons that follow.

Both of the stars with IR excesses are in the TW Hydrae association
($\approx$10 Myr). They have periods of 38 hr and 44 hr, respectively. In our
initial labeling, we labeled both as “ambiguous” CPVs because the dips in
their Sector 36 light curves seemed to stochastically evolve over only one or
a few cycles, which is atypical for CPVs; their periods were also long in
comparison with most of the other CPVs. Inspection of additional sectors
clarified that both sources are dippers, not CPVs (see the online plots in
Figure 3). For TIC 57830249, the Sector 10 light curve shows completely
different behavior from Sector 36, with variability amplitudes of $\pm 50\%$
and no obvious periodicity. TIC 57830249 also shows continuum emission at 1.3
mm (Rodriguez et al., 2015), which suggests that cold dust grains are present.

The dipper classification of TIC 193136669 is less obvious; the main
indication that it is a dipper is that Sectors 62 and 63 show its dips
appearing and disappearing within the span of one cycle. None of the CPVs in
our sample exhibit this property. Independently, TIC 193136669 is known to
have a cold disk of dust and molecular gas, based on 1.3 mm continuum emission
and resolved ${}^{12}{\rm CO}(2-1)$ emission (Rodriguez et al., 2015). It was
labeled a dipper by Capistrant et al. (2022); we agree with their designation,
and label it an “impostor” CPV in Table Transient Corotating Clumps Around
Adolescent Low-Mass Stars From Four Years of TESS. Section 5.9 highlights
plausible evolutionary connections between CPVs and dippers in light of these
“misclassifications”.

Figure 5:  Properties of CPVs identified by our search. CPVs are mostly pre-
main-sequence M dwarfs, younger than $\approx$150 Myr, with rotation periods
faster than $\approx$1 day. The 50 bona fide CPVs in Table Transient
Corotating Clumps Around Adolescent Low-Mass Stars From Four Years of TESS are
the dark blue dots; 13 ambiguous CPV candidates are light blue dots.
Unresolved binaries (red rings) are objects for which the Gaia DR3 radial
velocity scatter exceeded 20 km s-1, or if Gaia ${\rm RUWE}_{\rm DR3}$$>$2 and
multiple photometric signals were present in the TESS light curve. The top
panels show the 65,760 target stars with 2-minute cadence TESS data as the
shaded gray background; darker regions correspond to a larger relative number
of searched stars. The lower-left panel compares the rotation–color
distribution of CPVs against the rotation periods of K and M dwarfs in the
Pleiades from Rebull et al. (2016). The lower-middle panel plots the derived
corotation radii $R_{\rm c}=(GM/\Omega^{2})^{1/3}$ in units of stellar radii
against the measured CPV periods, in units of hours. Ages in the final panel
are known from cluster membership.

###  3.4 Binarity of CPVs

####  3.4.1 Binary statistics

A significant fraction of the CPVs show indications of unresolved binarity.
Excess noise above the Gaia single-source astrometric model is common (16/50
high-quality CPVs have RUWEDR3$>$2), as is the presence of multiple periods in
the TESS light curves (22/50). Elevated astrometric noise almost always
implies multiple detectable TESS periods (15/16 high-quality CPVs). The latter
observation corroborates the claim that most sources with RUWEDR3$>$2 are
binaries with projected apparent separations below 1′′, and projected physical
separations $\lesssim$50 AU. These observations are also in agreement with
previous analyses of multi-periodic low-mass objects discovered by K2, which
found that such systems are almost always binaries (Tokovinin & Briceño, 2018;
Stauffer et al., 2018a).

####  3.4.2 Do K dwarf CPVs exist?

To date, the only stars reported to show the CPV phenomenon are M dwarfs, with
typical stellar masses $\lesssim$0.3 $M_{\odot}$ (Stauffer et al., 2017;
Günther et al., 2022). However the two most massive CPVs in our sample, TIC
405754448 and TIC 405910546, were assigned masses of $\approx$0.82 $M_{\odot}$
and $\approx$0.60 $M_{\odot}$ respectively. The next-highest mass in our
sample belongs to TIC 59836633 ($\approx$0.48 $M_{\odot}$), with all remaining
CPVs having masses $\lesssim$0.40 $M_{\odot}$.

The locations of TIC 405754448 and TIC 405910546 in color–absolute magnitude
diagrams, combined with their probable membership in Lower Centaurus Crux,
support the conclusion that these stars have relatively high masses. However
in detail, both objects are subject to ambiguities in interpretation. The TIC
405910546 light curve has a unique shape, suggestive of an eclipsing binary.
Independently, TIC 405910546 was one of only three CPVs flagged with a Gaia
DR3 radial velocity scatter exceeding 20 km s-1. Combined, these factors
suggest that TIC 405910546 could be a pre-main-sequence eclipsing binary; it
should be studied further to clarify this classification.

For the other object, TIC 405754448, the evidence for binarity is stronger.
The RUWEDR3 statistic is 6.8, and the raw light curves in Sectors 11, 37, and
38 show both the CPV signal with period 12.9 hr and amplitude $\approx$1% and
an additional sinusoidal signal with a period $\approx$6.5 days and amplitude
$\approx$0.3%, likely from a second star. If TIC 405754448 is a K+M binary,
then the flux ratio between the primary and secondary would be expected to be
$\approx$10:1. Thus, if the K star were the source of the CPV signal, its
intrinsic variability amplitude would be $\approx$1%, while if the M star were
responsible its intrinsic variability amplitude would be $\approx$10%.

In short, these two objects suggest that the CPV phenomenon may extend up in
mass to pre-main-sequence K dwarfs, but more data are needed to substantiate
this claim.

####  3.4.3 An astrophysical CPV false positive: TIC 435903839

We originally classified TIC 435903839, with RUWEDR3=17.7, as an “ambiguous”
CPV with a 10.8 hr period, because this period minimized the dispersion in the
phase-folded light curve. More careful inspection revealed an impostor: this
source is a photometric blend of two ordinary rotating stars with $P_{0}$=3.60
hr, and $P_{1}$=5.41 hr, giving a beat period $(P_{0}^{-1}-P_{1}^{-1})^{-1}$
of 10.8 hr. This is a novel false positive scenario for CPVs: two rapid
rotators near the 3:2 period commensurability. The beat between the two
rotation signals produces the apparent CPV signal. Such false positives can be
excluded through careful accounting of all peaks in a periodogram. For
instance, TIC 435903839 shows a peak at 16.27 hr, which is not an integer
multiple of the dispersion-minimizing 10.82 hr period.

####  3.4.4 Multiple CPVs in the same system: TIC 425937691 and TIC 142173958

TIC 142173958 and TIC 425937691 both show evidence for two separate CPV
signals in their TESS light curves. For TIC 142173958, the signals have
periods of 11.76 hr and 12.84 hr. For TIC 425937691, the two periods are 4.82
hr and 3.22 hr, near the 3:2 period commensurability. Given that both sources
have two photometric signals and elevated RUWEs, each source is probably an
unresolved binary consisting of two CPVs. To our knowledge, these are the
third and fourth such systems known: EPIC 204060981 has two CPVs with periods
of 9.59 hr and 9.12 hr (Stauffer et al., 2018b), and TIC 242407571 has two
CPVs with periods of 11.33 hr and 13.63 hr, near the 6:5 period
commensurability (Stauffer et al., 2021).

##  4 Evolution of CPV Behavior

Figure 6:  Evolution of CPV light curves over two years. Out of the 50 CPVs in
Figure 4, 32 had 2-minute cadence TESS data available for a baseline of at
least two years; the 27 brightest are shown here due to space constraints.
Each panel shows one sector of TESS data, and is phased to its deepest minimum
in flux. Each panel’s title shows the TIC identifier and period in hours. Text
insets show the TESS sector numbers, which generally span two years, or at
least 1,000 cycles. The vertical scale is fixed across sectors to clarify
shape changes. Gray circles are raw 2-minute data; colored circles bin to 300
points per cycle.  Figure 7:  LP 12-502 (TIC 402980664) light curve, where
each time segment represents one TESS orbit. Data were acquired in Sectors
18-19, 25-26, 53, and 57-58. Flares are drawn in gray. The light curve is
binned to 15-minute intervals so that there are 96 points per day, and each
point is connected by a line. Data gaps have nothing plotted. The red vertical
lines highlight apparently instantaneous state changes in the shape of the dip
pattern.

###  4.1 Evolution over two year baseline

Figure 6 shows “before” and “after” views of 27 CPVs for which TESS 2-minute
cadence observations were available at least two years apart. Such a baseline
was available for 32 of the 50 confirmed CPVs in our catalog; for plotting
purposes we show the brightest 27. We have defined $\phi=0$ for each sector to
be the time of minimum light observed in that sector, rather than using a
consistent phase definition across multiple sectors. This is because for most
of the sources we do not know the period at the precision necessary to be able
to accurately propagate an ephemeris over two years. The achievable period
precision, $\sigma_{P}$, can be estimated as

$\sigma_{P}=\frac{\sigma_{\phi}P}{N_{\rm baseline}},$ (9)

for $N_{\rm baseline}$ the number of cycles in the observed baseline and
$\sigma_{\phi}$ the phase precision with which any one feature (e.g. a dip, or
the overall shape of the sinusoidal envelope) can be tracked. Assuming
$\sigma_{\phi}$$\approx$$0.02$ and a 20-day baseline over a single TESS sector
yields $\sigma_{P}$$\approx$$0.25^{+0.38}_{-0.14}$ minutes for the population
shown in Figure 6; propagated forward 1,000 cycles yields a typical ephemeris
uncertainty range of 2 to 11 hours. Measuring the period independently for
each sector did not reveal evidence for significant ($>$3$\sigma$) changes in
period, implying a period stability of $\lesssim$0.1% over two years.

A few objects in Figure 6 show the CPV phenomenon in one sector, and only
marginal signs or no sign of CPV behavior in the other sector. In our
subjective assessment, cases for which at least one sector would be flagged as
“ambiguous” include TIC 368129164 (Sector 23 might be labeled an EB), TIC
177309964 (Sector 38 would be simply a rotating star), TIC 404144841 (Sector
38 looks like a rotating star), TIC 201898222 (Sector 3 looks like a rotating
star), TIC 144486786 (Sector 32 might be an RS CVn), and TIC 38820496 (Sector
28 might be an RS CVn). TIC 193831684, assessed on a single-sector basis,
would probably be labeled an eclipsing binary—in fact, Justesen & Albrecht
(2021) already gave this source such a label. However, based on the shape
evolution between Sectors 13 and 39, it is a CPV.

Based on the fraction of sources that “turned off”, the observed shape
evolution implies that CPVs have an on-off duty cycle of $\approx$75%.
Correcting for the duty cycle might be important in population-level estimates
of the intrinsic frequency of the CPV phenomenon (e.g. Günther et al., 2022).

Figure 8:  Evolution of LP 12-502 ($P$=18.5611 h) at fixed period and epoch
over three years. Each panel shows one (averaged) TESS orbit; small text
denotes relative cycle number. There are 200 binned black points per cycle.
The TESS pointing law dictates the large time gaps between cycles 64-248,
315-1233, and 1264-1410; larger gaps tend to yield larger shape changes. The
dips usually evolve over tens to hundreds of cycles. However cycles 1233-1264
show a dip that switched from a depth and duration of 3% and 3 hr to 0.3% and
1 hr over less than one cycle (cf. Figure 7).

(a)

(b)

Figure 9:  River plots of the LP 12-502 light curve, showing (clockwise from
top-left) Sectors 18-19, 25-26, 53, and 58-59. A two-harmonic sinusoid has
been subtracted to highlight the sharp dips. A fixed period and phase are
adopted for all sectors; the dips across all observations are bounded by
$\phi\in[-0.35,0.35]$. In Sectors 25-26 (cycles 248-315), periods are visible
at the fundamental period of 18.5611 hr, as well as at faster
($\phi$$\approx$0-0.07) and slower ($\phi$$\approx$0.25-0.27) relative periods
based on the presence of blue dips with distinct slopes. Multiple simultaneous
periods are also visible in Sector 53 (cycles 1234-1263) and Sectors 58-59
(cycles 1411-1479). White chunks denote missing data. The state changes noted
with red markers in Figure 7 occur in cycles 261, 309, and 1241.

###  4.2 Evolution over consecutive sectors, & LP 12-502

A few of our complex periodic variables were near the TESS continuous viewing
zones (Figure 5, top right). Out of this already small sample, LP 12-502 (TIC
402980664; $d$=21 pc, $J$=9.4, $T$=11.1) stood out due to the quality and
content of its data. We discuss another interesting source, TIC 300651846, in
Appendix B. In this section, we describe the LP 12-502 observations and the
possible implications.

####  4.2.1 LP 12-502 observations

Whenever LP 12-502 was located within a TESS sector, it was observed at
2-minute cadence. Figure 7 shows all the available data, from Sectors 18, 19,
25, 26, 53, 58, and 59. Vertical offsets were applied to separate the data
from different spacecraft orbit numbers; there are always two orbits per
sector. We binned the light curve to 15-minute intervals to facilitate visual
inspection. Points more than 2.5$\sigma$ above the median are drawn in gray,
to prevent outliers from seizing attention. Data gaps are not connected by
lines (a common source of confusion in light curve visualization). Figure 8
shows the same data after phase-folding each TESS spacecraft orbit, assuming
$P$=18.5611 hr and a fixed reference epoch of BTJD=1791.5367. Finally, Figure
9 shows “river plots” of the same data, split into similar intervals: the
Sector 18-19 data, 25-26 data, 53 data, and 58-59 data. The river plots are
subject to one additional processing step: we fitted and subtracted a maximum-
likelihood two-harmonic sinusoid independently from the Sector 18-19 data,
25-26 data, and 53, 58, and 59 data in order to accentuate changes in the dip
timing and structure.

The average period, determined by measuring the PDM peak period over each
sector independently, was $\langle P\rangle=18.5560$ hr. The range between the
maximum and minimum sector-specific periods was measured to be about one
minute. However, a period shift of $\pm$1 minute leads to large phase drifts
over the entire timespan of observations. One minute is $\approx$1/1000th of a
period, and we have observed 1500 cycles. By folding with a fine grid of trial
periods, we found that the choice $P=18.5611\pm 0.0001$ hr causes more of the
features in the LP 12-502 light curve to maintain constant phases over the
entire dataset.

We now attempt to describe the complex morphology of the light curve and its
evolution. For the first 64 cycles, the star shows four obvious local minima.
We dub these dips $\\{1,2,3,4\\}$ at phases $\\{-0.28,-0.08,0,0.25\\}$,
respectively. Dips 2 and 3 are part of the same “global” minimum, which
otherwise resembles a long eclipse. Over cycles 0-64, the depth of dips 1 and
3 remain roughly fixed. Dip 4 decreases in depth by about 2%, and dip 2
increases in depth by about the same amount (see Figure 8). A subtle fifth dip
may also be present at phase +0.08, at the end of the global minimum that
includes dips 2 and 3.

There is then a 6-month (184-cycle) gap to Cycles 248-315, which show two
highly structured dip complexes, plus a small leading dip. The leading dip has
the same phase (relative to minimum light) as in cycles 0-64, and therefore
seems likely to be due to the same structure. Along a similar line of logic,
it seems plausible that the first “dip complex” during cycles 248-264
represents an evolution and reduction in amplitude of dips 2 and 3 that were
seen during cycles 0-64. During cycles 266-310, an additional local minimum
develops between the two complexes; this feature is best visualized on the
river plots (Figure 9), where it is seen to have a shorter period than the
other dips (as described below).

The second dip complex during cycles 248-315 shows the most substructure.
During e.g. cycles 283-298, this single complex shows six local minima. The
first and deepest dip is sharp: it shows a flux excursion of 3.5% over about
22 minutes (0.02 $P$), which is the steepest slope exhibited anywhere in the
LP 12-502 dataset. After the sharp dip, there is a roughly exponential return
to the baseline flux spanning about a quarter of a period, punctuated by
coherent local minima and maxima that the river plot (Figure 9) reveals to
have slightly longer periods than the sharp dip. The sharp leading dip remains
roughly constant in amplitude until a sudden “state change” at BTJD 2030.7
(cycle 309) that occurred at the same time as a flare, and left the trailing
dips seemingly unaffected. This apparent state change, and two others, are
marked with red lines in Figure 7.

The behavior during Sectors 53–58 (cycles 1233-1481) is comparatively tame;
the light curve shows only four to six dips per cycle. Some dips remain stable
in depth and duration over this five-month interval. Other dips grow, like the
one at $\phi=+0.06$ between cycles 1458 and 1481. Other dips, such as the one
at $\phi=+0.12$ in cycles 1233-1264, disappear entirely. The most dramatic
state change occurs during cycle 1241, when a large dip switches from a depth
of 3% and a duration of 3 hours to a depth of 0.3% and a duration of 1 hour.

####  4.2.2 Lessons from LP 12-502

State-changes reveal dip independence—The state-changes seen in cycles 261,
309, and 1241 confirm that dips can disappear in less than one cycle. While
such behavior was also noted by Stauffer et al. (2017), the data presented
here show further that the dips can be independent and additive. For example,
throughout cycles 1233-1264, there are three sharp dips between phases of 0
and 0.3 with different amplitudes but similar slopes. During the transition,
the leading dip nearly disappeared while the other two dips hardly changed;
compare the centermost two panels of Figure 8. Evidently, the material or
process responsible for one dip can vary independently of the materials or
processes responsible for other dips. The state changes during cycles 261 and
309 support the same conclusion, while also hinting that the leading dip of a
complex is most prone to disappearing, leaving the trailing dips unchanged in
its wake.

Slow growth; rapid death—LP 12-502 shows at least three instances in which
dips switch off over less than one cycle; we did not see any such instances of
dips switching on. Dip growth seems to happen more slowly. For instance, the
dip at phase 0-0.1 between cycles 258-290 begins to become detectable during
cycle 258, and growths in depth by about 2% over the next eight cycles. The
evolution of this particular dip is most clear in the river plots. The
evolution of the dip group at phases 0.1-0.3 during cycles 1410-1481 is
another example of this slow mode of dip growth.

Dip durations—The shortest dip duration for any of the individual LP 12-502
dips seems to be $\approx$0.06 $P$ $\approx$ 1.08 hr. This is very similar to
the characteristic timescale of a transiting small body at the corotation
radius,

$T_{\rm dur}\equiv R_{\star}P_{\rm rot}/(\pi a)=1.02\pm 0.10\,{\rm hr},$ (10)

where we have inserted the stellar radius and mass derived in Section 2.3.
Thus, the shortest-duration dips are likely produced by transits of bodies or
distributions of material that are smaller than the star. The corotation
radius corresponds to $a/R_{\star}\approx 5.8$, i.e., the transit of a body at
the corotation radius has a duration about six times shorter than a feature on
the stellar photosphere that is carried across the visible hemisphere by
rotation. On the other hand, some dip durations are sufficiently long that an
explanation involving transits would require structures that are larger than
the star along the direction of orbital motion.

Dip periods—Most of the LP 12-502 dips repeat with a period of $P=18.5611\pm
0.0001$ hr. However the river plots (Figure 9) reveal that a few dips have
detectably distinct periods. For instance, in sectors 25-26, the dip that
develops around cycle 262 has a period shorter than the mean period by
$\approx$0.1%, and some of the trailing local minima in the main dip complex
have periods slower than the mean period by $\approx$0.04%. In addition to the
fundamental period, we were able to identify at least four distinct periods
shown by specific dips over the full Sectors 18-59 dataset: 18.5683, 18.5672,
18.5473, and 18.5145 hr, with a measurement uncertainty of $\approx$0.0002 hr.
Possibly, the different periods belong to clumps of dust or prominences of gas
at slightly different orbital distances surrounding the corotation radius.

##  5 Discussion

###  5.1 Typical and extreme CPVs

Referring back to Figure 5, typical CPV masses span 0.1-0.4 $M_{\odot}$,
typical ages span 2-150 Myr, and relative to the Pleiades, the CPVs are among
the more rapidly rotating half of M dwarfs. The CPV mass and age range
includes both fully convective stars and stars with a combination of radiative
cores and convective envelopes; the dividing line for these ages is at around
$M_{\star}=0.25\,M_{\odot}$ (Baraffe & Chabrier, 2018). We found no obvious
differences in light curve morphology for CPVs above and below this fully-
convective pre-main-sequence boundary.

The closest CPV in our catalog is DG CVn (TIC 368129164), a member of AB Dor
at $d$$=$18 pc. The three brightest CPVs are DG CVn ($T$=9.3), TIC 405754448
($T$=9.6), and TIC 167664935 ($T$=10.3). The shortest period, 3.64 hr, belongs
to TIC 201789285. The longest period, 37.9 hr, belongs to TIC 405910546. Based
on the GaiaDR3 RV scatter, the latter source may turn out to be an eclipsing
binary; if so, the longest-period CPV in our catalog would be TIC 193831684
(31.0 hr). By definition, we required the periods to be below 48 hr.

The lowest mass ($\approx$$0.12$ $M_{\odot}$) belongs to TIC 267953787. The
catalog contains a few other stars with similar mass. We cannot rule out the
possibility that CPVs exist with even lower masses, given the small number of
such low-mass stars in our target sample. Perhaps even brown dwarfs can be
CPVs, although it might be difficult to distinguish the type of variability we
associate with CPVs from the usual variability of brown dwarfs caused by
clouds and latitudinal bands (e.g. Apai et al., 2021; Vos et al., 2022).

###  5.2 Is binarity important for CPVs?

For CPVs, binarity seems to provide either nuisances or curiosities. The
nuisances include astrophysical false positives with two beating rapidly
rotating stars, as well as uncertainty about which star produces the CPV
signal in binary systems. Our two candidate K dwarf CPVs suffer from this
latter concern (see Section 3.4).

Curiosities include the four binary systems that are now known to each host
two separate CPVs. CPVs are sufficiently rare that such systems may have
physical import. Recent work has shown that the orbits of binaries closer than
$\lesssim$700 AU tend to be aligned with their planetary systems (e.g.
Christian et al., 2022). If we assume that observing CPV variability requires
high line-of-sight inclinations, and that the inclinations in binaries are
correlated, then we would expect the detection of one CPV in a binary system
to raise the probability that the other star is a CPV. The limitations of the
current catalog prevent further exploration of this issue, but it might be
interesting for future study.

A separate possible relation between binarity and CPVs is that the presence of
a binary companion could cause early disk dispersal, freeing the star to
contract. The CPV phenomenon seems to require rapid rotation; thus if binary
systems tend to produce more rapidly rotating stars, we would expect a sample
of CPVs to have a larger binary fraction than the field. However, we see
minimal evidence for such an effect (Section 3.4). The multiplicity rate of M
dwarfs near the Sun is 26.8$\pm$1.4% (Winters et al., 2019). The $\approx$32%
multiplicity fraction in our CPV sample (from RUWEDR3; see Section 3.4) seems
approximately consistent with this field fraction.

###  5.3 Transience of CPV dips

While CPV periods appear to remain fixed over thousands of cycles, the light
curve shapes evolve over typical timescales of 10 to 1,000 cycles (e.g.
Figures 6 and 9). Although we refer to them as “periodic”, the CPVs are
therefore actually quasiperiodic, with coherence timescales of $\approx$100
cycles. This marks a qualitative departure from the “persistent” vs.
“transient” flux dip distinction previously described by Stauffer et al.
(2017), which was based on $\lesssim$100 cycle K2 baselines. The observation
that CPVs have a population-averaged on-off duty cycle of $\approx$75% (Figure
6) is also new. Appendix B for instance shows $\approx$1,000 cycles of a
source, TIC 300651846, with between zero and five sharp local minima per
cycle. During the “zero” epochs (cycles $\approx$503-542), the source would
likely be labeled an ordinary rotating star.

###  5.4 Special phases of CPV dips

An independent peculiarity of CPV evolution is that the dips do not explore
all phase angles with equal weight. LP 12-502, and other CPVs, exhibit
preferred phases lasting for at least two years. For LP 12-502, all of the
dips happen over phases corresponding to only two thirds of the period
(Figures 8 and 9). The remaining third seems to be “out of limits” for dips
over the timespan of observations. This could be evidence that the stellar
magnetic field is not azimuthally symmetric. Alternatively, the source of the
material (e.g. a planetesimal swarm) might be distributed over an arc rather
than occurring randomly around the entire orbit.

###  5.5 Dip asymmetries?

The asymmetry of a dip around the time of minimum light might be caused by the
variation in optical depth of the occulting material as a function of orbital
phase angle. Sharp leading edges with trailing exponential egresses, for
instance, have been previously seen for transiting exocomets and
disintegrating rocky bodies (e.g. Rappaport et al., 2012; Brogi et al., 2012;
Vanderburg et al., 2015; Zieba et al., 2019).

Examining Figure 4, it is clear that CPV dips can be asymmetric but it is not
obvious whether there is a preference for sharper ingresses or sharper
egresses. In some cases (e.g. TIC 425933644), the flux variations do not
resemble isolated dips, making the meaning of “ingress” and “egress” unclear.
In other cases, such as Sector 36 of TIC 89463560, there is a sharp drop with
an exponential return to the baseline flux, resembling the signatures of
exocomets (e.g. Rappaport et al., 2018; Zieba et al., 2019), and the
outflowing exospheres of some transiting planets (e.g. McCann et al., 2019;
MacLeod & Oklopčić, 2022).

###  5.6 What causes the CPV phenomenon: dust vs. gas

Both the dust clump and the gas prominence scenarios (Figure 1 and Section 1)
invoke clumps of material at the corotation radius; one property that
distinguishes the two ideas is the composition of the material.

####  5.6.1 What is a prominence?

The prominence idea is based on a loose analogy with quiescent
prominences/filaments in the solar corona that last as long as a few weeks
(see Vial & Engvold, 2015). In the context of the Sun, a prominence is a clump
of cold, partially ionized hydrogen viewed in emission against the dark
backdrop of space. A filament is the same clump of plasma, but viewed in
absorption against the solar disk. In an extrasolar context, spectroscopic
detections of transient Balmer- and resonance-line absorption seen for stars
such as AB Dor and Speedy Mic (e.g. Collier Cameron & Robinson, 1989;
Jeffries, 1993; Dunstone et al., 2006; Leitzinger et al., 2016) have been
interpreted as prominences that scatter a star’s chromospheric emission (see
Collier Cameron & Robinson, 1989). The short-term mechanical stability of such
gas configurations is theoretically plausible for rapid rotators (Ferreira,
2000; Waugh & Jardine, 2022). To our best knowledge, this class of
spectroscopic observation also has no viable alternative explanations.

We performed a simple visual examination of the TESS light curves for five
prominence-hosting systems studied by Jardine & Collier Cameron (2019)–AB Dor,
Speedy Mic, LQ Lup, HK Aqr, and V374 Peg–and detected no CPV behavior. While
individual prominences may only last one to tens of rotation cycles, the
prominence system itself is thought to always be “on”, due to the repeatable
detectability of spectroscopic transients (e.g. Collier Cameron et al., 1990,
and references therein). Assuming that spectroscopically observable prominence
systems indeed do not turn off, this would imply that they are not always
accompanied by photometric CPV-like dips: a link between the spectroscopic
prominences that may exist around rapidly rotating low-mass stars and the CPV
phenomenon has yet to be made.

####  5.6.2 What is the microphysical source of opacity?

CPVs show broadband flux variations that can be 1-2$\times$ deeper in the blue
than in the red (Onitsuka et al., 2017; Bouma et al., 2020; Günther et al.,
2022; Koen, 2023). Dust can naturally explain this chromaticity, since it has
a larger absorption cross-section in the blue than the red (e.g. Cardelli et
al., 1989). Gas might also explain the observed chromaticities (Gray, 1992).
While bound-bound absorption can be excluded, since it provides opacity only
at narrow resonant lines, the hydrogen opacity due to bound-free absorption is
“jagged” (see Gray, 1992, Figure 8.5 and Eq. 8.8), such that at temperatures
of $\approx$3,000 K to $\approx$10,000 K the opacity can be larger at blue
wavelengths than at red wavelengths. Bound-free absorption of ${\rm H}^{-}$ is
often important at such temperatures, but this opacity source is stronger in
the red than the blue, the opposite of what is required to produce deeper dips
in the blue than in the red. Likewise, Thomson scattering is too gray to be
the dominant opacity source. From hydrogen alone, bound-free absorption
therefore seems like the most plausible opacity source. However it remains to
be demonstrated whether a sufficient population of excited states could be
maintained, particularly given the short ($\approx$microsecond) radiative
decay timescales.

An instructive point of comparison is the rapidly rotating magnetic B star,
$\sigma$ Ori E, which shows dips that are deeper in the blue than in the red
(Hesser et al., 1977). Photometric and spectroscopic observations of this star
have been understood in terms of a warped torus of corotating circumstellar
material (Landstreet & Borra, 1978; Nakajima, 1985; Townsend et al., 2005).
The circumstellar material is unlikely to be dust, which would sublimate
quickly at the distance of the torus from the star.555Zhan et al. (2019)
explored the sublimation timescales for a canonical CPV with
$M_{\star}=0.2\,M_{\odot}$, $R_{\star}=0.3\,R_{\odot}$, and $T_{\rm
eff}=3200\,{\rm K}$. They found that non-shielded, generic silicate dust
mixture (Draine, 1985) with a single size of 0.1 $\mu$m reached the
$\approx$1500 K sublimation temperature at $\approx$3 $R_{\star}$. This
suggests that dust sublimation could be an important effect even for CPVs.
The opacity source for $\sigma$ Ori E and its analogs is instead thought to be
bound-free absorption by neutral hydrogen (Nakajima, 1985), although to our
best knowledge direct evidence for this conclusion has yet to be acquired.
Separate and smaller-amplitude continuum flux brightenings in $\sigma$ Ori E
may also come from electrons scattering photospheric light toward the observer
when the clouds are not transiting (Berry et al., 2022).

Given these complexities, it seems important for a future theoretical study to
be conducted to determine to what degree the observed chromaticities in CPVs
match, or do not match, expectations from radiative transfer. This issue has a
key ability to resolve the question of whether the CPVs are explained by dust
or by gas, which has bearing on whether the material producing the dips is
coming from the star, or whether it is a byproduct of the protoplanetary disk.

####  5.6.3 The lifetime constraint

The observed lifetime of the CPV phenomenon could provide another way to
discern between the gas and dust clump scenarios. Based on the available
statistics from Rebull et al. (2022) and references therein, it seems
plausible that CPV occurrence decreases with stellar age from $\approx$3% at
10 Myr (Sco-Cen), to $\approx$1% at 100 Myr (Pleiades), down to 0% by the
$\approx$700 Myr age of Praesepe. This is odd in the context of the prominence
scenario, because pre-main-sequence M dwarfs spin up over the first 100 Myr;
prominences might therefore be expected to be more common at 100 Myr than at
10 Myr, under the assumption that the production of prominences depends only
on the stellar rotation rate. The dust clump scenario would hold a natural
explanation: the lower occurrence of CPVs around older stars would simply
reflect a finite supply of dust. One possible complication however is that the
magnetic field topology of rapidly rotating M dwarfs may depend on factors
other than the rotation rate (e.g. the age), which might alter the production
of prominences.

###  5.7 Planets or planetesimals near corotation?

Close-in planets are common around M dwarfs; studies from Kepler have shown
that early M dwarfs have $\approx$0.1 planets per star with sizes between 1-4
$R_{\oplus}$ and orbital periods within 3 days (Dressing & Charbonneau, 2015).
The frequency of planets per star increases to $\approx$0.7 when considering
periods as long as 10 days. Extrapolating to all small (0.1-4 $R_{\oplus}$)
planets within 10 days, it is reasonable to expect nearly all M dwarfs to have
at least one planet.

In the context of disk-driven planet migration, the stopping location for the
innermost planet is set by the protoplanetary disk’s truncation radius (e.g.
Izidoro & Raymond, 2018, and references therein). The truncation radius is
often calculated by equating the magnetic pressure from the stellar
magnetosphere with the ram pressure of the inflowing gas. As it happens, the
truncation radius is close to the corotation radius for low accretion rates
(e.g. Romanova & Owocki, 2015; Li et al., 2022). These considerations invite
us to imagine one or more planets migrating inward due to gas drag, and
arriving at $\approx$5-10 stellar radii before the disk is depleted.

With this picture in mind, it is tempting to attribute features of the CPV
light curves to transits of material ejected by planets or planetesimals.
Young rocky bodies are expected to be hot, and they might expel either gas or
dust. The Jupiter-Io system (e.g. Saur et al., 2004) is analogous, in that a
small rocky body feeds the construction of a plasma torus. We emphasize that
although this type of configuration seems a priori plausible, no direct
evidence currently supports it.

The main logical function of the planetesimals would be to serve as a source
for the occulting gas or dust; they would not necessarily need to explain the
observed phases of the observed dips. The azimuthal angle of the eventual
entrainment could be entirely dictated by the stellar magnetic field. In this
scenario, the obscuring material would inspiral from one or more rocky bodies
well beyond the corotation radius. The planetesimals themselves would not
necessarily need to transit. However if they did, they would need to be
$\lesssim 1$ $R_{\oplus}$ based on their non-detections in the TESS data.
Possibly analogous systems include K2-22 (Sanchis-Ojeda et al., 2015) and
KOI-2700 (Rappaport et al., 2014), though the obscuring material in the CPVs
would need to be observed much further from the emitting planet than for those
two examples.

A more restrictive variant of the planetesimal scenario would be to posit that
the obscuring material remains close to the launching body, similar to comets,
or to the aforementioned K2-22 and KOI-2700 systems. If so, then the
planetesimals would need to be at the corotation radius. One prediction would
therefore be that certain orbital phases would produce recurrent dips when
observed over sufficiently long baselines, because the launching planetesimal
would be massive enough to remain in orbit, while stochastically ejecting
material. For most CPVs (Figure 6), the data seem to be in tension with this
expectation because the relative spacing between dips is almost never
conserved. With that said, certain sources do seem to exhibit “special
phases”, including LP 12-502 (TIC 402980664), DG CVn (TIC 368129164), TIC
193831684, and TIC 146539195. One possible explanation for this might be if
obscuring material is remaining close to its launching body, or bodies. An
alternative explanation could be that the stellar magnetic field
configurations responsible for confining said material are stable over the
existing two-year baseline.

###  5.8 Mass flux estimate

Assuming for the moment that the obscuring material is dust, we can estimate
the mass of a transiting clump. First, we convert the transit depth into an
effective cloud radius, $R_{\rm cloud}$. For most CPVs in Figure 4, this
yields $\approx$2-20 $R_{\oplus}$. A minimum constraint on the number density
of dust particles is obtained by requiring the cloud to be optically thick.
For cases like LP 12-502, this is reasonable because the transit duration of
the shortest dips implies $R_{\rm cloud}\ll R_{\star}$. Carrying out the
relevant calculation assuming the dust grains are 1 $\mu$m in size, Sanderson
et al. (2023) reported minimum cloud masses of order $10^{12}$ kg (their Eq.
23), which scale linearly with both the optical depth and dust grain radius.
This is comparable to a small asteroid; the asteroid belt itself has a mass of
order $\approx$1021 kg (Park et al., 2019). A similar calculation that assumed
occulting clumps of hydrogen, rather than dust, derived gas prominence masses
of at least $10^{14}$ kg (Collier Cameron et al., 1990), about 100$\times$
larger than the lower limit on the dust mass.

If the disappearance of a dip represents the permanent loss of the obscuring
material – for example, if it is the result of a dust clump being accreted or
ejected – then we can also estimate the rate at which mass is flowing through
the structures that lead to dips. For instance, LP 12-502 showed three “state-
switch” events over the six months of available TESS observations, during
cycles 261, 309, and 1241 (Figure 9). The other source for which we performed
a comparable analysis, TIC 300651846 (Appendix B), showed two state-switches
over 11 months. In all such cases, at least one dip turned off. For purposes
of estimation, we will take LP 12-502 as our prototype. Assuming the occulting
material is dust, the corresponding $\dot{M}\equiv M\cdot{\rm d}N/{\rm d}t$
time-averaged over six months is $\approx$$1\times 10^{-12}M_{\oplus}\,{\rm
yr}^{-1}$. Considered cumulatively over the $\approx$$10^{8}$ years for which
the CPV phenomenon is observed, this yields a cumulative moved dust mass of
$10^{-4}\,M_{\oplus}$, of order the Solar System’s asteroid belt. If the
occulting material is gas, the lower mass bounds would be of order 100 times
larger. For cases in which we observe the growth of dips, such as the Sector
29 data for TIC 224283342, or Sector 5 of TIC 294328885, the dip depths
typically increase by of order a few percent over ten to twenty days. This
growth rate yields a mass flux one order of magnitude larger than the earlier
estimate.

###  5.9 From dippers to debris disks

About one in three young stars with infrared-detected inner dusty disks show
quasiperiodic or stochastic dimming over timescales of roughly one day (e.g.
Alencar et al., 2010; Cody & Hillenbrand, 2010). The dimming amplitudes can
reach a few tenths of the stellar brightness, and dips with identical depths
and phases rarely recur. These “dipper” stars are probably explained by
occulting circumstellar dust in the inner disk (e.g. Cody et al., 2014;
Ansdell et al., 2016; Robinson et al., 2021; Capistrant et al., 2022). While
the phenomenon can persist beyond $\approx$10 Myr (Gaidos et al., 2019, 2022),
in all such cases it seems to be associated with the presence of infrared
excesses. Phenomenologically, dippers are different from CPVs in that their
dips are usually deeper, less periodic, and more variable in depth over
timescales of only one or a few cycles. Dipper stars also tend to be younger,
since they tend to be classical T Tauri stars with infrared excesses.

In identifying the two candidate CPVs with outlying SEDs (TICs 193136669 and
TIC 57830249; Section 3.3), we were prompted to reconsider our light curve-
based labeling, and ultimately concluded that these sources are dippers. This
episode suggests that there could be overlap between CPVs and dippers. Taking
TIC 57830249 as one example, the Sector 36 TESS data are suggestive of a CPV,
with relatively periodic, sharp dips with depths of a few percent. The Sector
10 data are completely different, varying in apparent flux by a factor of two,
with no discernible periodicity at all. Perhaps this source becomes a “dipper”
when an inflow of dust reaches the inner disk wall, and is otherwise a “CPV”
when the inner disk is starved of dust.

Although TIC 57830249 is an intriguing outlier, the general picture is that
stars without infrared excesses have more stable optical light curves than
those with infrared excesses. While some dippers may evolve into CPVs after
the disk is mostly gone, this would be generically expected based on
population statistics: young objects become old. There may be no other causal
connection between the two evolutionary stages. With that said, a common
mystery between the CPVs and dippers is how exactly the narrowness of their
flux dimmings is produced. A similar mechanism may operate for both types of
object, tied perhaps to a shared magnetic topology, or perhaps to a preference
for dust to inspiral to the star in clumped structures.

(a)  (b)

(c)

Figure 10:  The magnetic B star connection. $\sigma$ Ori E and HD 345439 (top
row) are magnetic B stars with predominantly dipolar magnetic fields known to
host circumstellar plasma tori. HD 37776 and HD 64740 (middle row) are
analogous magnetic B stars with field topologies potentially dominated by high
order multipoles. The bottom row compares the latter systems against the
“best-matching” CPV light curves, selected by eye from Figure 4. CPVs have
light curves that are visually similar to the topologically complex magnetic B
stars. Stellar masses rounded to one significant figure are given in the lower
right of each panel; the star, TESS sector, and period are listed in each
subtitle.

###  5.10 Strengthening the magnetic B star connection

Stauffer et al. (2017) previously noted a possible connection between the CPVs
and rapidly rotating magnetic B stars such as $\sigma$ Ori E, which can have
circumstellar gas clouds trapped in corotation (Townsend et al., 2005). The
$\sigma$ Ori class is distinct from Be-star decretion disks, which are found
to systematically not host detectable magnetic fields (Rivinius et al., 2013;
Wade et al., 2016).

An argument against the connection between CPVs and the $\sigma$ Ori E analogs
is that the light curve of $\sigma$ Ori E is simpler than those in Figure 4,
with only two broad local minima, and one “hump” (Figure 10; see also
Jayaraman et al. 2022). Within the model proposed by Townsend et al., the
simplicity of the light curve is the result of a simple dipolar magnetic
field, which is typical of magnetic B stars (Aurière et al., 2007; Donati &
Landstreet, 2009). The magnetic axis needs to be tilted relative to the
stellar spin axis in order to match the qualitative behavior of both the
broadband light curves, and the line-profile variations seen in hydrogen,
helium, and carbon (Oksala et al., 2012).

Two interesting and possibly telling exceptions to the rule that magnetic B
stars have simple light curves are HD 37776 and HD 64740. HD 37776 is known
from spectropolarimetry to have an extreme field geometry dominated by high
order multipoles (Kochukhov et al., 2011). The field geometry of HD 64740,
while potentially less extreme, also shows evidence for a non-dipolar
contribution (Shultz et al., 2018). Recent TESS light curves of these two B
stars appear surprisingly similar to the CPV light curves (Mikulášek et al.,
2020). The middle row of Figure 10 shows the phased TESS light curves for
these two stars, with by-eye best-matching CPVs shown underneath for
comparison. The number of dips per cycle, the shapes of the dips, and the dip
depths relative to the sinusoidal envelope are all similar. This connection
suggests that the highly structured light curves of both the M dwarfs and the
B stars are associated with (and perhaps caused by) strong non-dipolar
magnetic fields. Non-dipolar fields for M dwarfs are plausible, given that
Zeeman Doppler Imaging has revealed non-axisymmetric magnetic field patterns
for the few M dwarfs for which this technique is technically feasible (see
Kochukhov, 2021, and references therein).

The physical similarity between the B stars and the M dwarfs could have its
origin in the existence of a “centrifugal magnetosphere” (see Petit et al.,
2013). In other words, both classes of objects might satisfy the condition
$R_{\rm m}>R_{\rm c}$, for $R_{\rm m}$ the magnetosphere radius (sometimes
called the Alfvén radius). Provided that charged particles are confined to
move along magnetic field lines, material can then build up at the corotation
radius (e.g. Romanova & Owocki, 2015, Sec. 4, and references therein). In the
converse “dynamical” case, when $R_{\rm m}<R_{\rm c}$, material interior to
the magnetospheric radius returns to the stellar surface over the free-fall
timescale. A simple estimate assuming a dipole field with $B_{0}\approx
1\,{\rm kG}$ at the star’s surface, a local plasma number density $n\approx
10^{9}\,{\rm cm^{-3}}$, and a plasma temperature $10^{6}\,{\rm K}$ gives
magnetospheric radii of order a few times the corotation radii, $R_{\rm c}$.
This suggests that the existence of a centrifugal magnetosphere is plausible
for young, rapidly rotating M dwarfs.

##  6 Conclusions

In this work, we searched 2-minute cadence TESS data collected from 2018 July
to 2022 September for complex periodic variables (CPVs). The target stars were
65,760 late-K and early-to-mid M dwarfs within 150 pc and with TESS magnitudes
$T$¡16. The selection function included $>$$80\%$ of such stars within 30 pc,
and $<$$10\%$ of such stars at distances exceeding 100 pc (Figure 2).

We found 50 objects that showed complex quasiperiodic behavior over at least
one TESS sector. These 50 bona fide CPVs are listed in Table Transient
Corotating Clumps Around Adolescent Low-Mass Stars From Four Years of TESS.
This table also includes 13 ambiguous CPVs, whose designation is less certain,
and 3 impostors. We inferred ages for all but two of the 66 objects based on
memberships in young stellar associations; we also derived temperatures and
radii using SED fitting, and inferred stellar masses by interpolating against
stellar evolutionary models. We caution that our sample is far from being
volume-limited and is not even magnitude-limited: the TESS 2-minute stellar
sample had a heterogeneous selection function which may have been biased in
favor of young stars over field stars. Previous work however has shown that
$\approx$1-3% of M dwarfs younger than $\approx$100 Myr show the CPV
phenomenon (Rebull et al., 2016; Günther et al., 2022; Rebull et al., 2022).

Analyzing the TESS light curves and stellar properties of our CPVs, we draw
the following conclusions.

  1. 1.

The sharpest CPV dips have durations of $\approx$0.05 $P$ and depths of
$\approx$1-3% (Figures 4 and 6). Explaining dips this sharp requires material
extrinsic to the stellar surface (see Section 1).

  2. 2.

The shortest CPV dips, also with durations of $\approx$0.05 $P$, match the
expected transit duration for a small body at the corotation radius, $T_{\rm
dur}\equiv R_{\star}P_{\rm rot}/(\pi a)$ (see Section 4.2.2). Such dips are
therefore likely produced by transits of bodies or distributions of optically-
thick material that are smaller than the star.

  3. 3.

Many CPV dips have durations a few times longer than $T_{\rm dur}$ (Figure 4).
The dips are often superposed on a quasi-sinusoidal signal that presumably
originates from starspots and faculae on the stellar surface. The only viable
explanation currently known for sharp dips being superposed on the starspot
signals is that concentrations (“clumps”) of circumstellar material corotate
with the star. Assuming that the longer dips have the same physical origin as
the shortest dips, the corotating clumps must also be capable of having sizes
comparable to the star.

  4. 4.

The mean periods of CPVs remain fixed to within a relative precision
$\lesssim$0.1% over the two-year ($\approx$1,000 cycle) baseline of available
observations. The light curve shapes always evolve over this timescale (Figure
6).

  5. 5.

The dips in CPV light curves can have slightly different periods. LP 12-502,
for instance, showed dips with four distinct periods within $\pm 0.3\%$ of its
fundamental period, sometimes simultaneously, and each lasting for up to 50
cycles (Figure 9).

  6. 6.

The CPV peaks and dips evolve over timescales that are both secular
($\approx$100 cycles) and impulsive ($<$1 cycle). Dip growth seems to happen
over durations of at least ten cycles, and slow dip decay can also occur.
“State-switches” correspond to dips collapsing instantaneously; they occur
once every few months for both LP 12-502 and TIC 300651846. State-switches are
almost always linked with observed optical flares. Such switches are
suggestive of magnetic reconnection opening the “magnetic cage” that traps the
dust.

  7. 7.

The detailed morphology changes exhibited by LP 12-502 during its state-
switches (e.g. Figure 8, cycles 1233-1264) imply that the flux dips are
additive and independent.

  8. 8.

The on-off duty cycle for CPVs is $\approx$$75$%, based on the fraction of
bona fide CPVs that either turned on or turned off during TESS re-
observations, two years after the initial observation (Figure 6).

  9. 9.

The CPV phenomenon persists for $\gtrsim$150 Myr, based on the existence of
multiple CPVs in AB Dor, the Pleiades, and Psc-Eri (Section 2.3). It may even
extend to 200 Myr, based on the one CPV we found in the Carina Near moving
group (TIC 294328887; $\approx$200 Myr). The lack of detected CPVs in the
Hyades and Praesepe suggests that the lifetime of the phenomenon is limited to
the first few hundred million years.

  10. 10.

Most CPVs are M dwarfs with masses 0.1-0.4 $M_{\odot}$. Two sources, TIC
405754448 and TIC 405910546, have masses that appear to exceed 0.5
$M_{\odot}$. Both are potentially binaries, and this may confuse our ability
to accurate identify the source of the CPV signal (Section 3.4.2). We
encourage additional scrutiny of these objects in future work.

  11. 11.

The closest CPVs to the Sun are at distances of 15-20 pc; the brightest have
$V$$\approx$12 ($J$$\approx$7.5). We have found most of the close exemplars in
this work, since our CPV sample was $\gtrsim$80% complete within 30 pc. The
lack of CPVs in the volume-complete $<$15 pc sample of 0.1-0.3 $M_{\odot}$
stars analyzed by Winters et al. (2021) is consistent with this estimate.
Expanding our analysis of the TESS data to the full frame images would yield a
truly volume-limited selection function, and would expand the CPV census by
about a factor of two within 50 pc, and by a factor of ten within 100 pc.

  12. 12.

Surprising analogs to CPVs exist in two magnetic B stars, one of which is
known to have an extreme multipolar field topology (Section 5.10). Since most
magnetic B stars have dipolar magnetic fields, this suggests that the CPV dips
and warps are similarly being sculpted by the stellar magnetic fields, and
that the magnetic fields themselves are potentially also multipolar.

  13. 13.

The rate of dip evolution can be used to place a model-dependent lower bound
on how much material is either being accreted or ejected during the state
changes (Section 5.8). Order of magnitude estimates require at least an
asteroid belt’s worth of dust ($10^{-4}$ $M_{\oplus}$) over $10^{8}$ years, or
at least $\approx$$10^{-2}$ $M_{\oplus}$ if the occulting material is gas.

While many questions remain, two in particular will be important for
clarifying what these objects might teach us in a broader astrophysical
context: 1) Is the eclipsing material responsible for the phenomenon gas or
dust? 2) What sets the characteristic clumping size for the circumstellar
material?

The distinction between gas or dust is important because it could clarify
whether the CPV phenomenon is intrinsic, so that material comes from the star,
or extrinsic, so that it is sourced through some generic evolutionary phase of
debris disks. This knowledge would in turn propagate to our understanding of
whether the phenomenon is primarily teaching us about dust production and
processing in gas-poor disks, or whether it is teaching us about the ability
of cold gas to remain stable in hot stellar coronae for long durations.
Observationally, acquisition of medium- or high-resolution time-series spectra
holds a good chance at resolving the gas vs. dust question. Given our observed
$\approx$75% on-off duty cycles, such data must be acquired simultaneously
with photometric time-series observations (e.g. during TESS re-observation) in
order for detections and non-detections to be interpretable.

In both the gas and dust scenarios, CPVs are preferentially viewed edge-on.
This implies that after correcting for the line-of-sight inclination, roughly
one third of low mass stars (those that rotate rapidly enough; Günther et al.,
2022) could trap circumstellar material in the same way. It also suggests that
CPVs may preferentially show transiting planets at larger distances than the
corotating material, though this conclusion would be dependent on whether the
magnetic and stellar spin axes tend to be aligned. Given these points,
observational follow-up work should include searching for outer transiting
planets, and measuring equatorial velocities in order to test whether the
stellar inclination angles are indeed preferentially edge-on. Any source of
empirical information on the stellar magnetic field, whether from the Zeeman
effect (e.g. Kochukhov, 2021) or perhaps radio emission (e.g. Hallinan et al.,
2015), could also help clarify the strength of the magnetospheres for these
objects.

On the theoretical front, building a physical understanding of what sets the
characteristic size scale of the clumping material would help clarify why the
light curves have the bizarre shapes that are observed. The relevant puzzles
in plasma physics and radiative transfer could perhaps be connected to our
understanding of the close-in rocky planets that are expected to be present
around most of these stars.

LGB is grateful for support from the Heising-Simons 51 Pegasi b Fellowship,
and for helpful conversations with J. Spake, A. Mann, G. Laughlin, and B.
Draine. We are also grateful for the assistance of S. Yee, L. Weiss, H.
Isaacson, and A. Howard in acquiring and reducing the HIRES spectra. This
paper relied primarily on data collected by the TESS mission; the specific
2-minute cadence observations can be accessed via DOI
https://doi.org/10.17909/t9-nmc8-f686 (catalog 10.17909/t9-nmc8-f686). Funding
for the TESS mission is provided by NASA’s Science Mission Directorate. We
thank the TESS Architects (G. Ricker, R. Vanderspek, D. Latham, S. Seager, and
J. Jenkins) and the many TESS team members for their efforts to make the
mission a continued success. LP 12-502 in particular was observed at 2-minute
cadence thanks to the TESS Guest Investigator programs G022252 (PI: J.
Schlieder; Sectors 18, 19, 25, 26) and G04168 (PI: R. Jayaraman; Sector 53).
ADU acknowledges support by NASA under award number 80GSFC21M0002, as well as
ROSES award 22-ADAP22-0070. LGB conceived the project, performed the dip-
counting search, light curve classification, cluster membership, SED,
variability, and secondary-period analyses, and wrote the manuscript. RJ and
SR performed the Fourier-based analysis and contributed to light curve
classification. LR cross-examined the light curve classification, and
contributed an independent SED analysis. ADU identified the magnetic B star
connection. LAH contributed to project design. JNW, SR, and LAH significantly
improved the clarity of the manuscript. GÁB acquired and maintained the
servers used to run the dip-finding pipeline.

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Table 1: Bona fide, candidate, and debunked complex periodic variables from
the TESS 2-minute data. The non-truncated machine readable versions are
accessible both through the online journal, and at
https://zenodo.org/record/8327508.

TIC | $T$ | $d$ | $G_{\rm BP}-G_{\rm RP}$ | RUWE | $P$ | Assoc | Age | $T_{\rm eff}$ | $R_{\star}$ | $M_{\star}$ |  $R$c | $P_{\rm sec}$ | Quality | Bin | $N_{\rm sector}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
– | mag | pc | mag | – | hr | – | Myr | K | $R_{\odot}$ | $M_{\odot}$ | $R_{\star}$ | hr | – | – | –
368129164 | 9.29 | 18.3 | 2.89 | 6.95 | 6.44 | ABDMG | 149 | 3140 | 0.71 | 0.4 | 1.81 | 2.60 | 1 | 011 | 3
405754448 | 9.63 | 92.6 | 1.75 | 6.81 | 12.92 | LCC | 15 | 4273 | 1.5 | 0.82 | 1.74 | 134.4 | 1 | 011 | 5
167664935 | 10.31 | 62.5 | 2.52 | 5.05 | 14.05 | UCL | 16 | 3325 | 1.42 | 0.38 | 1.5 | 10.71 | 1 | 011 | 3
311092148 | 11.03 | 26.8 | 3.03 | 1.5 | 7.86 | COL | 42 | 3035 | 0.5 | 0.27 | 2.6 | - | 1 | 000 | 1
402980664 | 11.11 | 21.3 | 3.04 | 1.48 | 18.56 | COL | 42 | 3080 | 0.37 | 0.22 | 5.76 | - | 1 | 000 | 10
50745567 | 11.28 | 38.0 | 3.22 | 3.95 | 6.34 | BPMG | 24 | 3014 | 0.67 | 0.28 | 1.68 | 28.55 | 1 | 011 | 2
59836633 | 11.38 | 61.9 | 2.71 | 1.21 | 14.96 | BPMG | 24 | 3282 | 0.82 | 0.48 | 2.91 | - | 1 | 000 | 3
425933644 | 11.4 | 43.2 | 2.82 | 10.29 | 11.67 | THA | 45 | 3151 | 0.63 | 0.4 | 3.06 | - | 1 | 010 | 6
142173958 | 11.61 | 70.6 | 3.09 | 2.9 | 11.76 | TWA | 10 | 3028 | 1.15 | 0.26 | 1.45 | 12.84 | 1 | 011 | 3
146539195 | 11.62 | 48.2 | 3.37 | 2.86 | 6.73 | BPMG | 24 | 2898 | 0.8 | 0.24 | 1.38 | 7.29 | 1 | 011 | 2
206544316 | 11.63 | 42.8 | 2.89 | 1.26 | 7.73 | THA | 45 | 3114 | 0.57 | 0.35 | 2.44 | - | 1 | 000 | 6
335598085 | 11.9 | 105.5 | 2.85 | 2.79 | 15.85 | LCC | 15 | 3119 | 1.34 | 0.28 | 1.56 | 17.94 | 1 | 011 | 3
405910546 | 12.11 | 111.9 | 2.36 | 1.09 | 37.99 | LCC | 15 | 3455 | 0.92 | 0.6 | 5.26 | - | 1 | 100 | 4
272248916 | 12.15 | 80.5 | 2.83 | 5.5 | 8.9 | UCL | 16 | 3193 | 0.81 | 0.4 | 1.97 | 50.7 | 1 | 011 | 3
178155030 | 12.17 | 46.8 | 2.91 | 1.29 | 11.67 | THA | 45 | 3097 | 0.49 | 0.3 | 3.53 | - | 1 | 000 | 4
224283342 | 12.29 | 38.0 | 3.04 | 1.27 | 21.3 | COL | 42 | 3050 | 0.39 | 0.22 | 6.08 | - | 1 | 100 | 3
89026133 | 12.31 | 131.6 | 2.82 | 4.0 | 11.2 | UCL | 16 | 3188 | 1.33 | 0.31 | 1.29 | 27.83 | 1 | 011 | 3
234295610 | 12.51 | 48.1 | 3.04 | 1.13 | 18.29 | THA | 45 | 3074 | 0.44 | 0.27 | 5.09 | - | 1 | 000 | 3
118449916 | 12.54 | 97.1 | 3.09 | 25.18 | 12.31 | TAU | 2 | 3025 | 1.04 | 0.28 | 1.7 | 6.71 | 1 | 011 | 4
67897871 | 12.55 | 148.2 | 3.01 | 2.55 | 6.23 | USCO | 10 | 3082 | 1.5 | 0.18 | 0.65 | 6.72 | 1 | 011 | 2
353730181 | 12.65 | 106.6 | 2.75 | 1.23 | 13.51 | TAU | 2 | 3253 | 0.8 | 0.41 | 2.65 | - | 1 | 000 | 4
201898222 | 12.68 | 42.2 | 3.21 | 1.29 | 10.7 | THA | 45 | 2996 | 0.39 | 0.2 | 3.69 | 13.62 | 1 | 001 | 5
264767454 | 12.73 | 123.3 | 2.93 | 12.3 | 10.01 | COL(?) | 42 | 3150 | 1.0 | 0.42 | 1.76 | 20.62 | 1 | 011 | 13
442571495 | 12.75 | 80.8 | 3.03 | 1.64 | 9.59 | UCL | 16 | 3099 | 0.65 | 0.3 | 2.35 | 13.82 | 1 | 001 | 3
2234692 | 12.8 | 53.7 | 3.0 | 1.2 | 6.52 | COL | 42 | 3098 | 0.44 | 0.26 | 2.56 | 59.8 | 1 | 001 | 7
94088626 | 12.88 | 57.6 | 3.07 | 1.12 | 6.6 | ARG | 45 | 3090 | 0.46 | 0.27 | 2.53 | - | 1 | 000 | 2
264599508 | 12.88 | 79.7 | 3.01 | 1.89 | 7.9 | COL | 42 | 3098 | 0.62 | 0.4 | 2.4 | 8.99 | 1 | 001 | 7
363963079 | 12.92 | 83.1 | 3.09 | 8.0 | 7.82 | ARG | 45 | 3040 | 0.67 | 0.4 | 2.21 | 7.41 | 1 | 011 | 7
193831684 | 13.03 | 51.6 | 3.23 | 1.16 | 31.02 | BPMG | 24 | 2971 | 0.42 | 0.2 | 6.87 | - | 1 | 000 | 3
177309964 | 13.1 | 91.0 | 2.94 | 1.15 | 10.88 | CAR | 45 | 3125 | 0.62 | 0.4 | 2.95 | - | 1 | 000 | 34
425937691 | 13.18 | 43.1 | 3.77 | 2.86 | 4.82 | THA | 45 | 2782 | 0.41 | 0.16 | 1.91 | 3.22 | 1 | 011 | 5
141146667 | 13.28 | 57.6 | 3.28 | 1.23 | 3.93 | FIELD | NaN | 2968 | 0.42 | NaN | NaN | - | 1 | 000 | 6
332517282 | 13.29 | 39.0 | 3.27 | 1.05 | 9.67 | ABDMG | 149 | 2975 | 0.28 | 0.2 | 4.87 | - | 1 | 000 | 3
144486786 | 13.3 | 77.4 | 3.05 | 15.05 | 6.82 | COL | 42 | 3074 | 0.51 | 0.3 | 2.38 | 11.49 | 1 | 011 | 4
38820496 | 13.3 | 44.1 | 3.37 | 1.08 | 15.73 | THA | 45 | 2903 | 0.34 | 0.16 | 5.13 | - | 1 | 000 | 5
289840926 | 13.31 | 40.2 | 3.75 | 1.16 | 4.8 | BPMG | 24 | 2807 | 0.36 | 0.14 | 2.05 | 15.64 | 1 | 001 | 3
404144841 | 13.33 | 77.1 | 3.19 | 1.11 | 10.74 | TWA | 10 | 3008 | 0.52 | 0.22 | 2.86 | - | 1 | 000 | 4
89463560 | 13.45 | 123.9 | 2.97 | 1.31 | 9.43 | ARG | 45 | 3055 | 0.75 | 0.37 | 2.15 | 7.76 | 1 | 001 | 10
300651846 | 13.49 | 109.2 | 2.86 | 1.16 | 8.26 | CAR | 45 | 3136 | 0.62 | 0.4 | 2.44 | - | 1 | 000 | 31
267953787 | 13.49 | 130.5 | 3.59 | 1.2 | 17.46 | TAU | 2 | 2826 | 1.06 | 0.12 | 1.59 | - | 1 | 000 | 4
68812630 | 13.6 | 123.8 | 3.22 | 1.63 | 9.04 | TAU | 2 | 2996 | 0.76 | 0.27 | 1.86 | 5.28 | 1 | 001 | 3
141306513 | 13.65 | 50.2 | 3.4 | 1.08 | 13.36 | THA | 45 | 2964 | 0.32 | 0.16 | 4.85 | - | 1 | 000 | 2
201789285 | 14.03 | 45.4 | 3.82 | 1.19 | 3.64 | THA | 45 | 2757 | 0.3 | 0.12 | 2.02 | - | 1 | 000 | 5
294328887 | 14.23 | 97.1 | 3.22 | 1.05 | 8.51 | CARN | 200 | 2994 | 0.45 | 0.35 | 3.33 | - | 1 | 000 | 35
312410638 | 14.3 | 136.9 | 3.12 | 1.09 | 28.06 | UCL | 16 | 3030 | 0.58 | 0.25 | 5.11 | - | 1 | 000 | 3
38539720 | 14.52 | 129.4 | 3.37 | 1.2 | 9.16 | PERI | 120 | 2924 | 0.57 | 0.25 | 2.43 | - | 1 | 000 | 1
359892714 | 14.53 | 95.5 | 4.04 | 1.07 | 11.33 | EPSC | 3 | 2675 | 0.55 | 0.13 | 2.36 | - | 1 | 000 | 6
118769116 | 14.58 | 119.0 | 3.6 | 1.13 | 8.56 | TAU | 2 | 2852 | 0.56 | 0.2 | 2.18 | - | 1 | 000 | 4
440725886 | 14.69 | 135.1 | 2.96 | 1.06 | 3.92 | PLE | 112 | 3109 | 0.45 | 0.35 | 1.99 | - | 1 | 000 | 5
397791443 | 15.01 | 151.1 | 3.1 | 1.06 | 6.95 | IC2602 | 46 | 3031 | 0.48 | 0.26 | 2.46 | - | 1 | 000 | 6
160329609 | 9.65 | 8.7 | 3.4 | 1.18 | 24.31 | ARG | 45 | 2912 | 0.35 | 0.16 | 6.6 | - | 0 | 000 | 3
148646689 | 12.14 | 140.4 | 2.44 | 1.7 | 10.63 | UCL | 16 | 3466 | 1.25 | 0.55 | 1.61 | 13.37 | 0 | 001 | 3
280945693 | 12.27 | 98.2 | 2.97 | 1.16 | 15.27 | LCC | 15 | 3103 | 1.09 | 0.31 | 1.94 | - | 0 | 100 | 5
165184400 | 12.37 | 43.2 | 3.02 | 1.24 | 15.91 | THA | 45 | 3076 | 0.42 | 0.25 | 4.74 | - | 0 | 000 | 4
245834739 | 12.55 | 115.4 | 2.85 | 1.49 | 10.47 | TAU | 2 | 3112 | 1.02 | 0.36 | 1.68 | 9.86 | 0 | 001 | 6
125843782 | 13.01 | 127.7 | 2.86 | 1.21 | 44.17 | TAU | 2 | 3135 | 0.9 | 0.35 | 4.97 | - | 0 | 000 | 4
244161191 | 13.17 | 44.7 | 3.54 | 1.28 | 7.17 | COL | 42 | 2860 | 0.38 | 0.18 | 2.8 | 8.39 | 0 | 001 | 3
231058925 | 13.17 | 51.2 | 3.25 | 1.35 | 8.87 | THA | 45 | 2978 | 0.38 | 0.2 | 3.33 | - | 0 | 000 | 5
301676454 | 13.4 | 70.7 | 3.07 | 1.24 | 9.18 | ARG | 45 | 3009 | 0.47 | 0.25 | 2.96 | - | 0 | 000 | 1
58084670 | 13.58 | 140.2 | 2.82 | 1.06 | 11.16 | FIELD | NaN | 3138 | 0.77 | NaN | NaN | - | 0 | 000 | 6
67745212 | 13.63 | 27.8 | 3.8 | 1.11 | 5.12 | COL | 42 | 2781 | 0.21 | 0.09 | 3.17 | - | 0 | 000 | 2
5714469 | 13.73 | 78.3 | 3.65 | 1.12 | 10.35 | UCL | 16 | 2828 | 0.54 | 0.18 | 2.53 | - | 0 | 000 | 3
259586708 | 13.82 | 95.6 | 2.93 | 1.17 | 22.52 | COL | 42 | 3133 | 0.46 | 0.29 | 5.84 | - | 0 | 000 | 7
435903839 | 11.95 | 80.7 | 2.49 | 17.7 | 10.82 | ABDMG(?) | 149 | 3458 | 0.76 | 0.54 | 2.66 | - | -1 | 010 | 6
57830249 | 11.96 | 48.8 | 3.2 | 1.34 | 43.82 | TWA | 10 | 2948 | 0.7 | 0.25 | 5.63 | - | -1 | 000 | 3
193136669 | 13.06 | 61.1 | 3.49 | 1.22 | 37.64 | TWA | 10 | 2855 | 0.58 | 0.19 | 5.62 | - | -1 | 000 | 4

Note. — This table includes 50 good CPVs (Quality flag 1), 13 ambiguous CPVs
(Quality flag 0), and 3 impostors (Quality flag -1). The three-bit binarity
flag “Bin” is for Gaia DR3 radial_velocity_error outliers (bit 1), Gaia DR3
ruwe outliers (bit 2), and stars with multiple TESS periods (bit 3). The
machine-readable version, available online, includes additional columns for
the Gaia DR2 and DR3 source identifiers, as well as the stellar parameter
uncertainties. The age uncertainties are typically $\approx\pm 10\%$, but can
be asymmetric. The median statistical uncertainties on the temperature,
radius, and mass are $\pm$50 K, $\pm$4% and $\pm$9% respectively. $N_{\rm
sector}$ denotes the number of TESS sectors for which any data are expected to
be acquired between 2018 July and 2024 Oct. This number is generally greater
than the number of sectors for which 2-minute cadence data exist. Association
names and provenance follow conventions adopted by Gagné et al. (2018): ABDMG:
AB Doradus moving group (Bell et al., 2015). ARG: Argus (Zuckerman, 2019).
BPMG: $\beta$ Pic moving group (Bell et al., 2015). CARN: Carina Near moving
group (Zuckerman et al., 2006). COL: Columba (Bell et al., 2015). EPSC:
$\epsilon$ Chamaeleontis (Murphy et al., 2013). LCC: Lower Centaurus Crux
(Pecaut & Mamajek, 2016). PERI: Pisces-Eridani (Curtis et al., 2019). PLE:
Pleiades (Dahm, 2015). TAU: Taurus (Kenyon & Hartmann, 1995). THA: Tucana-
Horologium assocation (Bell et al., 2015). TWA: TW Hydrae assocation (Bell et
al., 2015). UCL: Upper Centaurus Lupus (Pecaut & Mamajek, 2016). USCO: Upper
Scorpius (Pecaut & Mamajek, 2016). The “(?)” string denotes low-confidence
membership.

##  Appendix A The ring hypothesis

One hypothesis for the CPVs, presented by Zhan et al. (2019), is that the star
might be “orbited by one or more rings composed of dust-size or somewhat
larger particles…The ring particles would move in Keplerian orbits at
relatively large distances from the star, and therefore the sublimation
lifetime would not be an issue even if the particles are dust-like in size.” A
sketch of this scenario was presented by Zhan et al. (2019), in their Figure
11. An example set of proposed parameters involved a ring inclined with
respect to the stellar spin axis by a few degrees, and with inner and outer
radii of 10 and 15 stellar radii.

One concern with the ring hypothesis is that if a cool spot were to transit
behind the ring, it would produce a brightening, not a dimming. Most CPVs show
dimmings. The ring scenario would therefore imply that large hot spots are
common in the photospheres of pre-main-sequence M dwarfs. Empirical evidence
however suggests that cool spots dominate the optical variability of disk-free
pre-main-sequence stars. This evidence includes flux excursions caused by
spot-crossings during planetary transits (e.g. Rizzuto et al., 2020; Gilbert
et al., 2022), correlations between simultaneous photometric and chromospheric
time-series (Reinhold et al., 2019), and stellar spectra that show molecules
that only form at cool temperatures (e.g. Gully-Santiago et al., 2017; Pérez
Paolino et al., 2023).

An independent concern with the ring hypothesis is that it is fine-tuned. The
model requires specific locations for the inner edge and the outer edge of the
ring, an inclination that yields a band with a specific apparent size, and
material in the ring that must be optically thick while also being homogeneous
enough to not induce any apparent photometric variability. It is challenging
to ascribe specific probabilities to any one of these factors. However the
requirement that they all be simultaneously met seems sufficiently severe to
disfavor this scenario.

(a)  Figure 11:  Light curve evolution of TIC 300651846. All available
2-minute cadence data as of 2023 Aug 11 are shown. Cycles 0 to 622 span TESS
Sectors 32-39 (Nov 2020–June 2021); cycles 2296-2676 span Sectors 61-65
(Jan–June 2023). We assumed a 8.254 hr period and a fixed reference epoch
(BTJD 2174.127) for both panels. Light curve segments are split based on the
presence of gaps longer than three hours. Cycle numbers are listed in the
lower-right of each light curve segment.

(a)  Figure 12:  River plots of TIC 300651846. This is an alternative
visualization of the data in Figure 11. All available 2-minute cadence data as
of 2023 Aug 11 are shown. Cycles 0 to 622 span TESS Sectors 32-39 (Nov
2020–June 2021); cycles 2296-2676 span Sectors 61-65 (Jan–June 2023). We
assumed $P$$=$8.254 hr and $t_{0}$=2174.127 [BTJD]. Note that the two panels
have slightly different color scales.

##  Appendix B TIC 300651846

Figures 11 and 12 show 2-minute cadence data for TIC 300651846, a CPV in the
TESS continuous viewing zone. If it were not for the existence of LP 12-502,
this source would have received greater attention. With the exception of a few
sectors, TESS data will exist for TIC 300651846 for at least Sectors 1-12,
27-39, and 61-69. While most of the available data exist in the full frame
images, Figures 11 and 12 focus only on the currently available 2-minute
cadence data.

During Sectors 32-39, the source shows between one and four local minima per
cycle. During the early portions of Sectors 61-65, it is more complex, with at
least five clear local minima per cycle. As the source evolves, its shape
becomes simpler, and the sharpness of one global minimum appears to increase.

State-switches analogous to those observed in LP 12-502 occur at cycles 498
and 2554. During the cycle 498 switch, two narrow dips at