Transient Corotating Clumps around Adolescent Low-mass Stars from Four Years of TESS

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 "short-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 ≈90% of the sky for at least 1 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 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:

$$\begin{eqnarray}&&T\lt 16\quad (\mathrm{Amenable}\ \mathrm{with}\ \mathrm{TESS}),\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&{G}_{\mathrm{BP}}-{G}_{\mathrm{RP}}\gt 1.5\quad (\mathrm{Redstars}\ \mathrm{only}),\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&{M}_{{\rm{G}}}\gt 4\quad (\mathrm{Dwarf}\ \mathrm{stars}\ \mathrm{only}),\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&d\lt 150\,\mathrm{pc}\quad (\mathrm{Close}\ \mathrm{stars}\ \mathrm{only}).\end{eqnarray} \tag{ 4 }$$

Here, ${M}_{{\rm{G}}}=G+5\mathrm{log}({\varpi }_{\mathrm{as}})+5$ is the Gaia G-band absolute magnitude, ϖ_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 Data Release 2 (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 the conditions from Equations (1)–(4). The results are shown in Figure 2. TESS 2 minute data exist for ≈50% of T<16 M and late-K dwarfs at ≈50 pc. Within 20 pc, ≳80% of the T < 16 M and late-K dwarfs have at least one sector of short-cadence data. Beyond 100 pc, ≲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 ≈M5.5V, corresponding to a main-sequence mass of ≈0.12 M_⊙.

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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 Canum Venaticorum variable (RS CVn) stars).

For our dip-counting pipeline, we began with the PDC_SAP light curves for each sector (Smith et al. 2017), removed nonzero quality flags, and normalized the light curve by dividing out its median value. We then flattened the light curve using a 5 days 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, although for most stars, fewer knots were preferred based on cross-validation using an ℓ²-norm penalty. An example fit is shown in panel (e) of Figure 3.

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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 68^th percentile of the distribution of the residuals from the median value of δ f_i ≡ 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 ϕ ∈ [0, 1]. We instead performed the analysis described above using a phase-folded light curve spanning ϕ ∈ [ − 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 reidentify 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 2 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

2.2.3. Manual Vetting

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 Σ (Gagné et al. 2018); ⁷ this algorithm calculates the probability that a given star belongs either to 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 3D position and 3D velocity space. We used the Gaia DR2 sky positions, proper motions, and distances to calculate the membership probabilities. BANYAN Σ 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 Σ yielded ambiguous results, we consulted the meta-catalog of young, age-dated, and age-dateable stars assembled by Bouma et al. (2022), and also searched the local volume around each star for comoving companions; ⁸ see also Tofflemire et al. (2021). A few important sources in the former meta-catalog included the Theia groups from Kounkel & Covey (2019), Kounkel et al. (2020), and the SPYGLASS stars from Table 1 of Kerr et al. (2021). Finally, to provide a base for comparison, we also ran the BANYAN Σ membership analysis on our entire 65,760 target star sample.

Table 1. Bona Fide, Candidate, and Debunked Complex Periodic Variables from the TESS 2 Minute Data

TIC	T	d	GBP − GRP	RUWE	P	Assoc	Age	Teff	R⋆	M⋆	Rc	${P}_{\sec }$	Quality	Bin	Nsector
—	mag	pc	mag	⋯	hr	⋯	Myr	K	R⊙	M⊙	R⋆	hr	⋯	⋯	⋯
368129164	9.29	18.3	2.89	6.95	6.44	ABDMG	149	3127	0.72	0.4	1.79	2.60	1	011	3
405754448	9.63	92.6	1.75	6.81	12.92	LCC	15	4278	1.5	0.82	1.74	134.4	1	011	5
167664935	10.31	62.5	2.52	5.05	14.05	UCL	16	3316	1.43	0.38	1.49	10.71	1	011	3
311092148	11.03	26.8	3.03	1.5	7.86	COL	42	3071	0.48	0.28	2.71	⋯	1	000	1
402980664	11.11	21.3	3.04	1.48	18.56	COL	42	3090	0.37	0.22	5.78	⋯	1	000	10
50745567	11.28	38.0	3.22	3.95	6.34	BPMG	24	2982	0.67	0.28	1.67	28.55	1	011	2
59836633	11.38	61.9	2.71	1.21	14.96	BPMG	24	3243	0.84	0.45	2.79	⋯	1	000	3
425933644	11.4	43.2	2.82	10.29	11.67	THA	45	3136	0.62	0.41	3.12	⋯	1	010	6
142173958	11.61	70.6	3.09	2.9	11.76	TWA	10	3019	1.14	0.26	1.47	12.84	1	011	3
146539195	11.62	48.2	3.37	2.86	6.73	BPMG	24	2882	0.81	0.24	1.37	7.29	1	011	2
206544316	11.63	42.8	2.89	1.26	7.73	THA	45	3116	0.57	0.35	2.43	⋯	1	000	6
335598085	11.9	105.5	2.85	2.79	15.85	LCC	15	3163	1.3	0.28	1.61	17.94	1	011	3
405910546	12.11	111.9	2.36	1.09	37.99	LCC	15	3463	0.93	0.6	5.19	⋯	1	100	4
272248916	12.15	80.5	2.83	5.5	8.9	UCL	16	3188	0.81	0.4	1.97	50.7	1	011	3
178155030	12.17	46.8	2.91	1.29	11.67	THA	45	3094	0.49	0.3	3.53	⋯	1	000	4
224283342	12.29	38.0	3.04	1.27	21.3	COL	42	3068	0.39	0.23	6.06	⋯	1	100	3
89026133	12.31	131.6	2.82	4.0	11.2	UCL	16	3176	1.32	0.3	1.28	27.83	1	011	3
234295610	12.51	48.1	3.04	1.13	18.29	THA	45	3047	0.45	0.26	5.0	⋯	1	000	3
118449916	12.54	97.1	3.09	25.18	12.31	TAU	2	3022	1.05	0.28	1.68	6.71	1	011	4
67897871	12.55	148.2	3.01	2.55	6.23	USCO	10	3096	1.5	0.18	0.65	6.72	1	011	2
353730181	12.65	106.6	2.75	1.23	13.51	TAU	2	3221	0.81	0.4	2.6	⋯	1	000	4
201898222	12.68	42.2	3.21	1.29	10.7	THA	45	2971	0.39	0.2	3.64	13.62	1	001	5
264767454	12.73	123.3	2.93	12.3	10.01	COL(?)	42	3096	1.11	0.38	1.52	20.62	1	011	13
442571495	12.75	80.8	3.03	1.64	9.59	UCL	16	3066	0.67	0.3	2.27	13.82	1	001	3
2234692	12.8	53.7	3.0	1.2	6.52	COL	42	3077	0.44	0.25	2.53	59.8	1	001	7
94088626	12.88	57.6	3.07	1.12	6.6	ARG	45	3064	0.46	0.27	2.51	⋯	1	000	2
264599508	12.88	79.7	3.01	1.89	7.9	COL	42	3097	0.6	0.35	2.34	8.99	1	001	7
363963079	12.92	83.1	3.09	8.0	7.82	ARG	45	3041	0.67	0.35	2.11	7.41	1	011	7
193831684	13.03	51.6	3.23	1.16	31.02	BPMG	24	2979	0.42	0.2	6.87	⋯	1	000	3
177309964	13.1	91.0	2.94	1.15	10.88	CAR	45	3128	0.61	0.38	2.94	⋯	1	000	34
425937691	13.18	43.1	3.77	2.86	4.82	THA	45	2780	0.41	0.16	1.92	3.22	1	011	5
141146667	13.28	57.6	3.28	1.23	3.93	FIELD	NaN	2972	0.42	NaN	NaN	⋯	1	000	6
332517282	13.29	39.0	3.27	1.05	9.67	ABDMG	149	2966	0.28	0.21	4.84	⋯	1	000	3
144486786	13.3	77.4	3.05	15.05	6.82	COL	42	3084	0.49	0.29	2.43	11.49	1	011	4
38820496	13.3	44.1	3.37	1.08	15.73	THA	45	2916	0.34	0.16	5.15	⋯	1	000	5
289840926	13.31	40.2	3.75	1.16	4.8	BPMG	24	2785	0.36	0.14	2.08	15.64	1	001	3
404144841	13.33	77.1	3.19	1.11	10.74	TWA	10	3008	0.53	0.24	2.91	⋯	1	000	4
89463560	13.45	123.9	2.97	1.31	9.43	ARG	45	3073	0.75	0.4	2.21	7.76	1	001	10
300651846	13.49	109.2	2.86	1.16	8.26	CAR	45	3159	0.61	0.4	2.51	⋯	1	000	31
267953787	13.49	130.5	3.59	1.2	17.46	TAU	2	2833	1.07	0.12	1.57	⋯	1	000	4
68812630	13.6	123.8	3.22	1.63	9.04	TAU	2	3000	0.76	0.28	1.89	5.28	1	001	3
141306513	13.65	50.2	3.4	1.08	13.36	THA	45	2930	0.32	0.16	4.79	⋯	1	000	2
201789285	14.03	45.4	3.82	1.19	3.64	THA	45	2754	0.3	0.12	1.99	⋯	1	000	5
294328887	14.23	97.1	3.22	1.05	8.51	CARN	200	2998	0.45	0.35	3.32	⋯	1	000	35
312410638	14.3	136.9	3.12	1.09	28.06	UCL	16	3049	0.58	0.25	5.11	⋯	1	000	3
38539720	14.52	129.4	3.37	1.2	9.16	PERI	120	2952	0.56	0.27	2.57	⋯	1	000	1
359892714	14.53	95.5	4.04	1.07	11.33	EPSC	3	2669	0.56	0.13	2.34	⋯	1	000	6
118769116	14.58	119.0	3.6	1.13	8.56	TAU	2	2845	0.55	0.18	2.18	⋯	1	000	4
440725886	14.69	135.1	2.96	1.06	3.92	PLE	112	3117	0.45	0.35	1.98	⋯	1	000	5
397791443	15.01	151.1	3.1	1.06	6.95	IC2602	46	3023	0.48	0.26	2.45	⋯	1	000	6
160329609	9.65	8.7	3.4	1.18	24.31	ARG	45	2913	0.35	0.16	6.62	⋯	0	000	3
148646689	12.14	140.4	2.44	1.7	10.63	UCL	16	3441	1.27	0.55	1.58	13.37	0	001	3
280945693	12.27	98.2	2.97	1.16	15.27	LCC	15	3074	1.09	0.31	1.94	⋯	0	100	5
165184400	12.37	43.2	3.02	1.24	15.91	THA	45	3094	0.42	0.25	4.81	⋯	0	000	4
245834739	12.55	115.4	2.85	1.49	10.47	TAU	2	3131	1.01	0.36	1.71	9.86	0	001	6
125843782	13.01	127.7	2.86	1.21	44.17	TAU	2	3128	0.91	0.35	4.9	⋯	0	000	4
244161191	13.17	44.7	3.54	1.28	7.17	COL	42	2867	0.37	0.16	2.76	8.39	0	001	3
231058925	13.17	51.2	3.25	1.35	8.87	THA	45	2966	0.39	0.2	3.28	⋯	0	000	5
301676454	13.4	70.7	3.07	1.24	9.18	ARG	45	3057	0.45	0.27	3.19	⋯	0	000	1
58084670	13.58	140.2	2.82	1.06	11.16	FIELD	NaN	3142	0.77	NaN	NaN	⋯	0	000	6
67745212	13.63	27.8	3.8	1.11	5.12	COL	42	2804	0.21	0.09	3.2	⋯	0	000	2
5714469	13.73	78.3	3.65	1.12	10.35	UCL	16	2825	0.54	0.18	2.53	⋯	0	000	3
259586708	13.82	95.6	2.93	1.17	22.52	COL	42	3128	0.46	0.29	5.81	⋯	0	000	7
435903839	11.95	80.7	2.49	17.7	10.82	ABDMG(?)	149	3412	0.74	0.51	2.66	⋯	−1	010	6
57830249	11.96	48.8	3.2	1.34	43.82	TWA	10	2956	0.69	0.25	5.69	⋯	−1	000	3
193136669	13.06	61.1	3.49	1.22	37.64	TWA	10	2853	0.58	0.2	5.77	⋯	−1	000	4

Notes. The non-truncated machine-readable versions are accessible both through the online journal, and at https://zenodo.org/record/8327508. 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 ≈ ± 10%, but can be asymmetric. The median statistical uncertainties on the temperature, radius, and mass are ±50​​​​​​ K, ±4%, and ±9% respectively. N_sector denotes the number of TESS sectors for which any data are expected to be acquired between approximately 2018 July and 2024 October. 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). AB Doradus moving group (ABDMG; Bell et al. 2015). Argus (ARG; Zuckerman 2019). β Pic moving group (BPMG; Bell et al. 2015). Carina near moving group (CARN; Zuckerman et al. 2006). Columba (COL; Bell et al. 2015). ​​​​​ Chamaeleontis (EPSC; Murphy et al. 2013). Lower Centaurus Crux (LCC; Pecaut & Mamajek 2016). Pisces–Eridani (PERI; Curtis et al. 2019). Pleiades (PLE; Dahm 2015). Taurus (TAU; Kenyon & Hartmann 1995). Tucana-Horologium association (THA; Bell et al. 2015). TW Hydrae association (TWA; Bell et al. 2015). Upper Centaurus Lupus (UCL; Pecaut & Mamajek 2016). Upper Scorpius (USCO; Pecaut & Mamajek 2016). The "(?)" string denotes low-confidence membership.

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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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_BP G_RP from Gaia DR2, Vgri from APASS, JHK_S from Two Micron All Sky Survey (2MASS), SDSS riz, and the Wide-field Infrared Survey Explorer (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 precomputed model grids, and by default fits for six parameters: $\{{T}_{\mathrm{eff}},{R}_{\star },{A}_{{\rm{V}}},\mathrm{log}g,[\mathrm{Fe}/{\rm{H}}],d\}$. 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:

$$\begin{eqnarray}&&{T}_{\mathrm{eff}}/{\rm{K}}\sim { \mathcal U }(2000,8000),\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&{R}_{\star }/{R}_{\odot }\sim { \mathcal U }(0.1,1.5),\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&{A}_{{\rm{V}}}/\mathrm{mag}\sim { \mathcal U }(0,0.2),\end{eqnarray} \tag{ 7 }$$

for ${ \mathcal U }$ the uniform distribution. We validated our chosen upper bound on A_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 $d\mathrm{log}{ \mathcal Z }\lt 0.01$, 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 forM dwarfs is well-documented (e.g., Boyajian et al. 2012; Stassun et al. 2012; David & Hillenbrand 2015; Feiden 2016; Kesseli et al. 2018; Morrell & Naylor 2019; 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 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 $\{{\tilde{T}}_{\mathrm{eff}},{\tilde{R}}_{\star },\tilde{t}\}$, and approximating their uncertainties as Gaussian ${\sigma }_{{\tilde{T}}_{\mathrm{eff}}}$, ${\sigma }_{{\tilde{R}}_{\star }}$, and ${\sigma }_{\tilde{t}}$, we define a distance metric Δ to each model PARSEC grid-point {T_eff, R_⋆, t} via

$$\begin{eqnarray}&&{{\rm{\Delta }}}^{2}={\left(\displaystyle \frac{{\tilde{T}}_{\mathrm{eff}}-{T}_{\mathrm{eff}}}{{\sigma }_{{\tilde{T}}_{\mathrm{eff}}}}\right)}^{2}+{\left(\displaystyle \frac{{\tilde{R}}_{\star }-{R}_{\star }}{{\sigma }_{{\tilde{R}}_{\star }}}\right)}^{2}+{\left(\displaystyle \frac{\tilde{t}-t}{{\sigma }_{\tilde{t}}}\right)}^{2},\end{eqnarray} \tag{ 8 }$$

where the division by the uncertainties helps to assign equal importance to each dimension. The mass reported in Table 1 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

Table 1 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.

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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.

Of our 63 confirmed and candidate CPVs, 61 were associated with a nearby moving group or open cluster, primarily using BANYAN Σ as described in Section 2.3. ¹⁰ The relevant groups are listed in Table 1; their ages span ≈5–200 Myr and are plotted in Figure 5. For comparison, BANYAN Σ 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 ≲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 Σ 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 ≈150 Myr. Table 1 includes three ≈150 Myr CPVs in AB Dor (Bell et al. 2015), a ≈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 = ${149}_{-19}^{+51}$ 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±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 Σ 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.

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). 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 3σ 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 (≈10 Myr). They have periods of 38 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 ±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 ¹²CO(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 1. Section 5.10 highlights plausible evolutionary connections between CPVs and dippers in light of these misclassifications.

3.4.1. Binary Statistics

3.4.2. Do K Dwarf CPVs Exist?

3.4.3. An Astrophysical CPV False Positive: TIC 435903839

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 and 12.84 hr. For TIC 425937691, the two periods are 4.82 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 and 9.12 hr (Stauffer et al. 2018b), and TIC 242407571 has two CPVs with periods of 11.33 and 13.63 hr, near the 6:5 period commensurability (Stauffer et al. 2021).

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Figure 6 shows before and after views of 27 CPVs for which TESS 2 minute cadence observations were available at least 2 yr 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 ϕ = 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 2 yr. The achievable period precision, σ_P , can be estimated as

$$\begin{eqnarray}&&{\sigma }_{P}=\displaystyle \frac{{\sigma }_{\phi }P}{{N}_{\mathrm{baseline}}},\end{eqnarray} \tag{ 9 }$$

for N_baseline is the number of cycles in the observed baseline, and σ_ϕ the phase precision with which any one feature (e.g., a dip, or the overall shape of the sinusoidal envelope) can be tracked. Assuming σ_ϕ ≈0.02 and a 20 day baseline over a single TESS sector yields σ_P ≈${0.25}_{-0.14}^{+0.38}$ minutes for the population shown in Figure 6; propagating forward 1000 cycles yields a typical ephemeris uncertainty range of 2–11 hr. Measuring the period independently for each sector did not reveal evidence for significant (>3σ) changes in period, implying a period stability of ≲0.1% over 2 yr.

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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 eclipsing binary), 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 ≈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).

A few of our CPVs 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 2.5σ 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.

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The average period, determined by measuring the PDM peak period over each sector independently, was 〈P〉 = 18.5560 hr. The range between the maximum and minimum sector-specific periods was measured to be about 1 minute. However, a period shift of ±1 minute leads to large phase drifts over the entire timespan of observations. 1 minute is ≈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 ± 0.0001 hr causes more of the features in the LP 12-502 light curve to maintain constant phases over the entire data set.

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 remains 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 data set. 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, is 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 5 month interval. Other dips grow, like the one at ϕ = + 0.06 between cycles 1458 and 1481. Other dips, such as the one at ϕ = + 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 hr to a depth of 0.3% and a duration of 1 hr.

4.2.2. Lessons from LP 12-502

S tate-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 grows 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.

D ip durations. The shortest dip duration for any of the individual LP 12–502 dips seems to be ≈0.06 P ≈1.08 hr. This is very similar to the characteristic timescale of a transiting small body at the corotation radius,

$$\begin{eqnarray}&&{T}_{{dur}}\equiv {R}_{\star }{P}_{\mathrm{rot}}/(\pi a)=1.02\pm 0.10\,\mathrm{hr},\end{eqnarray} \tag{ 10 }$$

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_⋆ ≈ 5.8, i.e., the transit of a body at the corotation radius has a duration about 6 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 ± 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 ≈0.1%, and some of the trailing local minima in the main dip complex have periods slower than the mean period by ≈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 data set: 18.5683, 18.5672, 18.5473, and 18.5145 hr, with a measurement uncertainty of ≈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.

Referring back to Figure 5, typical CPV masses span 0.1–0.4 M_⊙, 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_⋆ = 0.25 M_⊙ (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 Gaia _DR3 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 (≈0.12 M_⊙) 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).

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 3.4.2).

Curiosities include the four binary systems that are now known to each host two separate CPVs (see Section 3.4.4). CPVs are sufficiently rare that such systems may have a physical import. Recent work has shown that the orbits of binaries closer than ≲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.

While CPV periods appear to remain fixed over thousands of cycles, the light-curve shapes evolve over typical timescales of 10–1000 cycles (e.g., Figures 6 and 9). Although we refer to them as periodic, the CPVs are therefore actually quasiperiodic, with coherence timescales of ≈100 cycles. This marks a qualitative departure from the "persistent" versus "transient" flux dip distinction previously described by Stauffer et al. (2017), which was based on ≲100 cycle K2 baselines. The observation that CPVs have a population-averaged on–off duty cycle of ≈75% (Figure 6) is also new. Appendix B for instance shows ≈1000 cycles of a source, TIC 300651846, with between zero and five sharp local minima per cycle. During the zero epochs (cycles ≈503–542), the source would likely be labeled an ordinary rotating star.

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 2 yr. 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.

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 the 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., Brogi et al. 2012; Rappaport 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).

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?

5.6.2. What Is the Microphysical Source of Opacity?

5.6.3. The Lifetime Constraint

The CPV dips are usually superposed on nearly sinusoidal modulation. The sinusoidal modulation is probably induced by brightness inhomogeneities on the stellar surface. If the dips are explained by material orbiting the star, then the proximity of the spot and dip periods is surprising. For instance, the dust clumps modeled by Sanderson et al. (2023) tend to accumulate near—but not exactly at—corotation. To explore the proximity of the dip and spot periods, we generated synthetic light curves that superposed a sinusoid and a single eclipse. We imposed periods, amplitudes, sampling, and noise properties similar to typical CPVs in Figure 4. We then tuned the period of the eclipse signal to differ slightly from the sinusoidal period, and processed the resulting synthetic signals through the same period-finding routine to which we subjected the real data. Provided the dip and sinusoid periods agreed to within ≈0.1%, we found that phase-folding on the dominant period (that of the larger-amplitude starspots) yielded dip signals analogous to those in Figure 4. However, increasing the period difference beyond ≳0.3%, the dip signal quickly becomes smeared out and unidentifiable when phase-folding. This exercise highlights that our search was sensitive only to stars with dip and spot periods that agreed to within a few parts per thousand. Some CPVs may exceed this threshold; we encourage future work aimed at determining whether such systems exist.

Close-in planets are common around M dwarfs; studies from Kepler have shown that early M dwarfs have ≈0.1 planets per star with sizes between 1 and 4 R_⊕ and orbital periods within 3 days (Dressing & Charbonneau 2015). The frequency of planets per star increases to ≈0.7 when considering periods as long as 10 days. Extrapolating to all small (0.1–4 R_⊕) 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 ≈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 ≲1 R_⊕ based on their nondetections 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 farther 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 2 yr baseline.

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_cloud. For most CPVs in Figure 4, this yields ≈2-20 R_⊕. 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_cloud ≪ R_⋆. Carrying out the relevant calculation assuming the dust grains are 1 μm in size, Sanderson et al. (2023) reported minimum cloud masses of order 10¹² kg (their Equation (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 ≈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 10¹⁴ kg (Collier Cameron et al. 1990), about 100× 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 6 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 $\dot{M}\equiv M\cdot {dN}/{dt}$ time-averaged over 6 months is ≈1 × 10⁻¹² M_⊕ yr⁻¹. Considered cumulatively over the ≈10⁸ yr for which the CPV phenomenon is observed, this yields a cumulative moved dust mass of 10⁻⁴ M_⊕, of the order of 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 an order of a few percent over 10 to 20 days. This growth rate yields a mass flux 1 order of magnitude larger than the earlier estimate.

About one in three young stars with infrared-detected inner dusty disks shows quasiperiodic or stochastic dimming over timescales of roughly 1 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 ≈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 an 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 2, 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.

Stauffer et al. (2017) previously noted a possible connection between the CPVs and rapidly rotating magnetic B stars such as σ Ori E, which can have circumstellar gas clouds trapped in corotation (Townsend et al. 2005). The σ 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 σ Ori E analogs is that the light curve of σ 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. (2005), 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).

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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 nondipolar 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 nondipolar magnetic fields. Nondipolar fields for M dwarfs are plausible, given that Zeeman Doppler Imaging has revealed nonaxisymmetric 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_m > R_c, for R_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, their Section 4, and references therein). In the converse dynamical case, when R_m < R_c, the material interior to the magnetospheric radius returns to the stellar surface over the freefall timescale. A simple estimate assuming a dipole field with B₀ ≈ 1 kG at the star's surface, a local plasma number density n ≈ 10⁹ cm⁻³, and a plasma temperature 10⁶ K gives magnetospheric radii of order a few times the corotation radii, R_c. This suggests that the existence of a centrifugal magnetosphere is plausible for young, rapidly rotating M dwarfs.