1. Introduction NASA's Kepler mission (W. J. Borucki et al. 2010) transformed our understanding of stellar rotation, with light curves that allowed measurement of rotation periods for tens of thousands of stars, extending up to ;100 days (e.g., A. McQuillan et al. 2014; R. Angus et al. 2015; A. R. G. Santos et al. 2021). Building on Kepler's success, the repurposed K2 mission (J. E. Van Cleve et al. 2016) targeted young stellar populations along the ecliptic plane, enabling the development of gyrochronological sequences for stellar associations with ages from around 10 Myr to over 2 Gyr (e.g., L. M. Rebull et al. 2018; S. T. Douglas et al. 2019; J. L. Curtis et al. 2020).

NASA's Transiting Exoplanet Survey Satellite (TESS; G. R. Ricker et al. 2014) now offers an opportunity to expand stellar rotation studies with its near-complete sky coverage and <1% photometric precision down to T ; 15 (G. R. Ricker et al. 2015). TESS provides variability metrics for a diverse set of stars, including those of scientific interest such as stars with spectroscopic monitoring (e.g., S. Hojjatpanah et al. 2020), wide binaries (e.g., N. R. Deacon et al. 2016), eclipsing systems (e.g., A. Prša et al. 2022), and M dwarfs (e.g., E. K. Pass et al. 2022).

TESS has already significantly advanced our knowledge of stellar rotation. Rotation periods derived from TESS light curves have been instrumental in confirming diffuse coeval populations (e.g., B. M. Tofflemire et al. 2021; M. L. Wood et al. 2023), estimating ages of exoplanet-hosting stars (e.g., G. Zhou et al. 2021), and investigating properties of nearby brown dwarfs (D. Apai et al. 2021). As TESS continues observing, covering new sky regions and extending time baselines, the scope of rotation-based research in stellar and planetary sciences will continue to grow.

A major challenge with TESS data is the 27.4 day observing window, which limits detection of rotation periods beyond ;13 days within a single sector. While multiple sectors (particularly in TESS's continuous viewing zones) theoretically provide longer baselines for period measurement, efforts to use these longer baselines have generally struggled to recover reliable long-period rotations (e.g., J. T. VanderPlas 2018; B. L. Canto Martins et al. 2020). Some progress has been made using machine learning techniques (e.g., Y. Lu et al. 2020; Z. R. Claytor et al. 2024) or combining TESS data with ground-based observations (W. S. Howard et al. 2021).

Additional challenges in detecting rotation signals from TESS data include scattered-light contamination (C. Hedges et al. 2020), TESS's relatively small aperture for faint stars, TESS's large pixel scale (21″) and contamination from nearby stars, and data gaps during each sector's data downlink period. These factors can hinder period detection, and in some cases, as with scattered-light interference, may even introduce erroneous periodic signals (e.g., the ;13.7 day lunar-synchronous signal; S. Hattori et al. 2022).

Given the ubiquity of TESS light curves and the potential limitations in deriving accurate rotation periods, it is essential to answer two key questions: How reliable are TESS-derived rotation periods, and, when periods are undetectable, what can be inferred about the star's rotation? For the former, we assess reliability-the accuracy of detected rotation periods. For the latter, we evaluate completeness-the thresholds below which rotation signals cannot be reliably detected.

Both reliability and completeness vary with observable factors such as rotation period and stellar brightness. With a robust map of both as a function of common observables, we can more accurately compare theoretical rotation period distributions, such as those from a single-aged P rot -color sequence, with observed distributions. Similar methods are used in exoplanet research, where knowledge of both falsepositive rates and pipeline completeness is crucial for deriving occurrence rates (S. E. Thompson et al. 2018). While injectionrecovery tests are common in exoplanet studies, simulating rotation signals is more challenging as there is no analytical model that describes stellar rotation comparable to that of transits. The best alternative is to compare rotation periods measured from TESS to that of a more reliable/complete set, like those from Kepler or K2.

This study aims to chart the completeness and reliability of rotation periods derived from TESS light curves using widely accessible, commonly adopted methods. In the long term, we aim to translate TESS-based rotation measurements and nondetections into meaningful indicators of stellar age or membership in young associations. We therefore seek an extraction method that preserves variability signals even for faint stars in nearby young clusters, as well as a straightforward, interpretable algorithm for periodic signal measurement that can be efficiently applied across large stellar samples.

The rest of the paper is organized in the following fashion. Section 2 describes the sample of benchmark rotation periods used as to quantify the precision, reliability, and completeness of the TESS rotation pipeline. Section 3 describes the method for extracting light curves and measuring P rot . Section 4.1 quantifies the empirical uncertainties of TESS P rot measurements, while Sections 4.2 and 4.3 explore the reliability and completeness of single-sector P rot measurements, respectively. In Section 4.4, we discuss the reliability and completeness of P rot measurements from stitched consecutive-sector light curves. We show an application of our results in Section 5 by applying our method to a set of nearby stellar clusters. We discuss some of the implications of our work in Section 6, and list the key takeaways for our study alongside future improvements in Section 7.

To detect the limits of TESS P rot measurements, we required a large set of stars in the TESS field of view with reliable rotation periods (the benchmark set). To this end, it was important to include both stars with longer rotation periods and those that are fainter than the limits we expect for targets observed by TESS. That is, we want to detect the limits (in P rot and T) where TESS fails to recover rotation periods, which requires targets landing beyond those limits.

TESS Cycles 4 and 5 cover the ecliptic plane, offering a large sample of stars with both TESS and K2 data (in addition to earlier TESS cycles with some K2 overlap). K2-TESS overlap studies have performed extensive searches of the K2 data for rotation periods (e.g., S. T. Douglas et al. 2016Douglas et al. , 2017;;L. M. Rebull et al. 2018). Importantly, K2's campaigns were significantly longer than a TESS sector (>70 days versus ;27 days), and K2 could reach fainter sources than TESS (K2's aperture is 95 cm versus 10 cm for TESS's). Comparisons with ground-based data and across multiple K2 campaigns have shown K2 periods to be reliable (R. Rampalli et al. 2021). Thus, rotation periods derived from K2 data can serve as a reasonable benchmark to test the reliability of TESS P rot measurements.

We used the K2 rotation periods from T. Reinhold & S. Hekker (2020, hereafter RH20). RH20 performed a comprehensive search for P rot between 1 and 44 days across all available K2 light curves, reporting an empirically estimated error e Prot and a signal strength metric, H peak , for each P rot . To select stars from this catalog, we adopted RH20's quality criterion of H peak > 0.3 and limited our analysis to P rot < 40 days. Extending our sample in period yielded only a negligible increase in the number of stars. RH20 suggests greater confidence for stars with H peak > 0.5, but our results were consistent across the extended range 0.3 < H peak < 0.5. Additionally, variations in spot coverage over time and the wavelength differences between K2 and TESS make H peak only a rough proxy for expected signals in TESS data.

To identify stars from the RH20 catalog within the TESS field of view, we matched EPIC catalog coordinates for each star to the TESS-point software (C. J. Burke et al. 2020), determining which stars from RH20 have been observed by TESS. This process provided a sample of approximately 23,000 stars observed by both TESS and K2 as of 2024 January.

As we show in Figure 1, the resulting RH20-TESS overlap sample is diverse, spanning a wide range of color (T eff ), rotation period, and number of sectors of TESS data. There is an excess of stars with ;20-25 day periods compared to periods outside this range; this is seen in Kepler data as well (T. Reinhold et al. 2013).

The default TESS light curves, produced by the TESS Science Processing Operations Center (SPOC), are optimized for exoplanet detection rather than preserving stellar variability (J. M. Jenkins et al. 2016). To better capture stellar rotation signals, we generated custom light curves using the unpopular package (S. Hattori et al. 2022).foot_0 Unpopular was specifically designed to retain stellar variability (S. Hattori et al. 2022) and has been increasingly popular for rotation period studies with TESS data (e.g., N. Vowell et al. 2023;M. L. Wood et al. 2023).

For estimating rotation periods, we used a Lomb-Scargle (LS) periodogram. While machine learning (A. R. G. Santos et al. 2021) and autocorrelation (A. McQuillan et al. 2014) are popular methods to estimate rotation periods from Kepler data, we opted to use LS because it has an easily accessible and commonly used implementation within astropy (Astropy Collaboration et al. 2022), it handles unevenly sampled and nonsinusoidal data, and it has higher usage with space-based photometry generally. More generally, LS periodograms have been a standard tool for measuring rotation periods in data sets from Kepler, K2, and TESS (e.g., T. Reinhold et al. 2013;L. M. Rebull et al. 2016;A. W. Mann et al. 2016A. W. Mann et al. , 2017;;R. Rampalli et al. 2021).

Comprehensive details on causal pixel model (CPM) extraction are available in S. Hattori et al. (2022), with the broader method described in D. Wang et al. (2016) and D. Wang et al. (2017). Briefly, unpopular constructs light curves by modeling the pixel response within the target aperture using pixels outside the aperture. This approach assumes that nonaperture and aperture pixels are linked only by nonastrophysical signals (e.g., scattered-light and instrumental systematics). The final CPM light curve is produced by subtracting the modeled systematics from the raw aperture light curve, removing instrumental noise without impacting the stellar signal.

For each target (and sector of observations for that target), we downloaded 50 × 50 TESS Full Frame Image cutouts from the Barbara A. Mikulski Archive for Space Telescopes using the TESScut interface (C. E. Brasseur et al. 2019;STScI 2022). We searched for targets by their TESS Input Catalog (TIC) IDs and performed background subtraction on the raw data using the median flux from the 300 dimmest pixels in each cutout. To extract the light curves with unpopular, we applied a 1 pixel aperture centered on the target to minimize contamination from background stars, given TESS's large 21″ pixels. We selected 100 predictive pixels, excluding a 5 × 5 pixel region around the target, and used the "Similar Brightness" method in unpopular.

Figure 1. The RH20-TESS overlap: characteristics of the 23,000-star RH20-TESS overlap sample used in this study. On top, we show the sky position of targets colored by the number of TESS sectors available as of 2024 January. The bottom-left panel shows the rotation period distribution (as measured by K2) for the subset of stars that comprise our final sample used for analysis (see Section 3.2). Effective temperatures are calculated with the empirical color-effective temperature relation from J. L. Curtis et al. (2020) with Gaia Data Release 2 G BP -G RP colors dereddened using STILISM dust maps (R. Lallement et al. 2019). The middle-right panel shows the target's position on a Gaia color-magnitude diagram, with members of the three major clusters overplotted. The bottom-right panel shows a (stacked) histogram of the RH20 rotation periods colored by number of consecutive TESS sectors. The sample covers a wide range in stellar type, rotation period, brightness, and data coverage-ideal for testing completeness and reliability with TESS.

For the CPM model, we split each light curve into 100 sections for training and testing, applying an L2 regularization value of 0.1. In this configuration, each section was detrended using a model trained on the remaining 99 sections, with the regularization value setting the strength of the model's regularization. Example K2, CPM, and SPOC light curves are displayed in Figure 2.

This configuration for unpopular has been successfully employed in previous analyses (e.g., M. G. Barber et al. 2022; M. L. Wood et al. 2023). Our conclusions are also robust against modest variations in this setup (e.g., in the L2 regularization value or aperture size; S. Hattori et al. 2022).

Contamination ratio. To minimize the risk of contamination, we excluded stars with a TIC contamination ratio greater than 1. This ratio quantifies the degree of flux blending from nearby sources, with higher values indicating greater contamination (K. G. Stassun et al. 2018). Contamination risk increases with T due to TESS's large pixel size and the higher stellar density at fainter magnitudes.

Among the 22,986 stars in our sample, 13,506 had contamination ratios below 1 in the TIC. Most of the difference was from stars with no contamination ratio in the TIC. For example, none of the 545 stars in our sample with T > 16 had contamination ratios listed in the TIC. To address this, we recalculated the contamination ratio for each star using the tic_contam.py script from M. Paegert et al. (2021), increasing the number of stars with contamination ratios below 1 from 13,506 to 22,403 (97% of the sample), including 486 stars with T > 16.

Removing binaries. Unresolved binary companions can introduce periodic signals into the target star's light curve, potentially leading to the recovery of the binary companion's period instead of the target star's, resulting in discrepancies between the RH20 period and our measured period. To mitigate this, we excluded stars with a Gaia RUWE >1.2 and non_single_star == 1, both of which are indicative of an unresolved binary (e.g., C. Ziegler et al. 2020) or extreme youth (S. Fitton et al. 2022). We note that this cut will not remove most binaries (M. L. Wood et al. 2021). Among the 22,986 stars in our sample, 5363 had a RUWE > 1.2, and 548 were flagged with non_single_star == 1.

After all cuts, our sample was reduced to 16,752 stars. This is the set of stars used for all subsequent analyses.

To estimate rotation periods from the CPM light curves, we used the LS (N. R. Lomb 1976, J. D. Scargle 1982) algorithm with the nifty-ls method in astropy. 3We searched for rotation periods across a grid from 0.2 to 20 days, using 100,000 evenly spaced steps. Other parameters for the LS periodogram were kept at their default values. An evenly spaced grid was chosen because the default astropy grid is sparsely sampled at rotation periods 10 days, which can lead to incorrect period determinations for longer-period rotators. The cutoff at 0.2 days was applied because stars with rotation periods below 0.2 day are exceptionally rare (E. R. Newton et al. 2016;M. N. Günther et al. 2022) and challenging to detect in the longest-cadence (30 m) TESS data.

To identify the peak in the periodogram, we located the highest local maximum in period space and adopted this as the rotation period (P rot,CPM ) for further analysis. For stars observed in more than one sector, we independently measured the rotation period in each sector and selected the period with the highest LS power as the final rotation period for that star. Sector-to-sector variations are discussed in Section 4.1.

We repeated this analysis for all stars with data from at least two consecutive TESS sectors. For stars with multiple consecutive sectors, we analyzed only the first two sectors by stitching them into a single light curve and applying the LS periodogram with the same parameters as described above. We then extended this analysis by stitching all available sectors for each star to create a composite light curve, on which we again applied the LS periodogram. Results are presented in Section 4.

To begin, we will only consider rotation periods from individual sectors. We will address the results from stitched sectors in Sections 4.4 and 4.5.

To estimate true period uncertainties, we used an empirical approach, comparing rotation measurements P i obtained from multiple data sets (i.e., multiple TESS sectors and K2 campaigns) with a chosen benchmark period P true . The relative differences, (P i -P true )/P true , should be normally distributed around zero, with the standard deviation representing the uncertainty on P rot (assuming the values are the same across stars within a bin). Assuming P true is the true period and has negligible uncertainty, this method will capture both datarelated uncertainties and astrophysical effects, as the light curves are separated by timescales longer than those of spot evolution and other relevant astrophysical changes.

We applied this comparison to our P rot,CPM estimates, treating each star's P rot,RH20 as P true and each available singlesector P rot,CPM as P i . To analyze the results, we grouped data into 2 day bins in P rot,RH20 , increasing to 4 days in the final bin due to lower counts. We show the distribution of each bin in Figure 3.

Within each bin, we performed a least-squares fit assuming a Gaussian plus a constant to the binned distribution of (P i -P true )/P true . We only considered (P i -P true )/P true values in the [-0.45, 0.45] region. The constant and the restricted window are meant to focus on the empirical uncertainties in cases where the period is the correct one, not the effects of aliasing or erroneous measurements (a topic for Section 4.2).

As we show in Figure 4, the fractional uncertainties scale linearly with P rot,CPM . Period uncertainties are below 3% for stars with P rot < 5 days, roughly doubling by 12 days. We fit this with a line, which yielded a period uncertainty relation of ( ) ( ) / P % 0.005577 days 0.001768. 1

This equation is valid for periods <12 days. While the analysis includes measurements out to 14 days, a small fraction of these are 12-14 days and those that are have low reliability (Section 4.2). Equation (1) provides a fractional uncertainty. To get the absolute uncertainty on the rotation period, take the output from Equation (1) and multiply it by the measured rotation period. For example, for a rotation period of 5 days, Equation (1) predicts a fractional uncertainty of 0.030, or 3%. Multiplied by 5 days, this gives an uncertainty of 0.15 days.

There is a noticeable bias in the longest P rot,CPM periods, such that P rot,CPM is, on average, smaller than the equivalent P rot,RH20 . The effect can be seen in the bottom 8-10 day and 10-14 day bins of Figure 3 as well as the unbinned comparison (Figure 5). This bias is small for periods shorter than 10 days; in the 8-10 day bin, the offset is 1.5%, below the uncertainty (4.8%), which is the width of the Gaussian fit to the data (the standard deviation). The offset being less than the uncertainty in this bin indicates the bias is not the dominant contributor to the error budget in this bin. This effect can be significant over the sample but not for any individual star.

For the 10-14 day bin, the effect is ;10%, a nontrivial effect even considering the larger uncertainty (6%-7%).

This bias is likely due to the shorter TESS observing window (∼30 days) compared to K2 (∼70 days). For stars with true periods of 10-14 days, a single TESS sector covers only two or three rotations and the LS periodogram will have fewer samples at increasing periods. This would also explain why the effect is not seen when we compare periods from TESS data to other periods from TESS data (discussed below).

We repeated our experiment by comparing P rot,CPM against the single-sector P rot,CPM measurement with the highest LS Figure 3. Distribution of the difference between the measured period from TESS data and the benchmark period, taken from K2 data (RH20; yellow) or the best (highest LS power) sector of data from TESS (red). The histograms show the fractional difference between the rotation periods, broken into six period bins. The black lines show fits assuming a Gaussian distribution with a constant offset to account for nonmatches (which have random difference).

power. The results are shown as purple bins in Figure 3 and the purple points and line in Figure 4.

If the P rot,RH20 uncertainties are much smaller than those from TESS, the TESS-TESS comparison should yield larger uncertainties at all periods. Interestingly, it gave smaller uncertainties for the shortest periods (<4 days). This could be due to the 6 hr signal in K2 light curves (from the roll and thruster fire; J. E. Van Cleve et al. 2016;A. Vanderburg & J. A. Johnson 2014). This would manifest as higher uncertainties in P rot,RH20 for the fastest rotators.

The TESS-TESS comparison yielded much larger uncertainties in the 6-8 day and 10-14 day ranges. The 10-14 day bin contains the scattered-light signal (∼13.7 days), while the 6-8 day bin contains its half-alias. This impacts both periods used in the comparison, resulting in much higher uncertainties. This also manifests as an asymmetry in the profiles, particularly in the 6-8 day bin due to a bias toward 6.85 days (half the scattered-light signal).

In order to quantify reliability (and completeness in the next section), we needed to quantify if our measured P rot,CPM matched P rot,RH20 . We considered a period a match if P rot,CPM and P rot,RH20 were within 3σ of the uncertainty predicted by the linear fit in Equation (1), and a similar (proportional) criteria for an alias. Formally,

( ) M P P P A P NR Recovery , 3 , 3 , o t h e r w i s e , 2 P P P P rot,CPM rot,RH20 rot,RH20 rot,CPM 2 2 rot,CPM rot,RH20 rot,RH20 rot,CPM s s

with M representing a match, A an alias, and NR representing a nonrecovery. We used fractional period uncertainties to match our fractional uncertainties from Equation (1), and we opted for three standard deviations as this range will include the appropriate spread of recoveries at any P rot,CPM length.

A common source of failures were aliases, i.e., a recovered period that is an integer ratio of the true one. Due to the short observation window of TESS, these are overwhelmingly measurements at half the true period. This effect has been seen in prior studies using TESS data, as well (e.g., M. Kounkel et al. 2022). The number of periods double the true period (and other integer ratios) was consistent with the background of mismatches (see Figure 5), so we defined aliases to only consider the half-period case (Equation 2). As with the matches, we see a bias for aliases of P rot,CPM to yield shorter periods than P rot,RH20 at P rot,RH20  8 days.

We measured reliability as a function of three parameters: LS power, T magnitude, and the signal-to-noise ratio (SNR). For the SNR, we are interested in the ratio of the variability amplitude to that of the random noise and scaled by the number of rotational cycles in the data. We approximate this as ( )

where A 90-10 is the 90th percentile of the amplitude of the light curve minus the 10th percentile, P2P RMS is the point-to-point variation in the light curve, and N cycles is the number of rotation cycles present in the light curve. Since our light curves have cadences ranging from 20 s to 30 minutes, we binned each light curve to a 1 hr cadence before calculating the point-to-point variation (so this is a 1 hr SNR). The resulting equation is not a true SNR, which would need to account for other factors like morphology in the light curve. Instead, this is a useful proxy for SNR.

For LS power, we defined reliability as

where M is the number of period matches, and N the total number of stars that satisfy the LS constraint. This represents the fraction of stars with a matching P rot,CPM and LS power in The points are derived from the bins shown in Figure 3. The lines show the best-fit linear relation assuming the true period is the RH20 period (yellow), while the red points show the same assuming the true period is the P rot,CPM measurement with the highest LS power. The blue line shows the same analysis repeated using periods derived from light curves made from consecutive sectors instead of single-sector light curves and adopting RH20 as the true period.

some range divided by all stars in that range. We repeated these calculations using aliases instead of matches, as well as replacing LS power with T magnitude and SNR (Equation 3) to compute the effect of all parameters on the period reliability.

Figure 6 shows our measurement of reliability as a function of rotation period for varying bins in LS power, TESS magnitude, and SNR. Here, we report the SNRs as percentiles to allow direct comparison between single-sector and consecutive-sector completeness/reliability measurements. Due to the N cycles term in Equation (3), the consecutive-sector light curves will generally have a higher SNR than the single-sector light curves. As we will see later, while a SNR of 10 might manifest as a strong rotation signal in a single-sector light curve, the same might not be true for a consecutive-sector light curve.

Reliability is high (>80% even with no quality cuts) and relatively flat with increasing period out to ;10 days. Reliability drops rapidly around 12 days (half of a TESS sector). There is a corresponding effect seen in the alias fraction, of a quick rise between 10 and 15 days. The X-axis is defined as P rot,RH20 , meaning ;30% of the stars with true periods ;14 days are retrieved as 7 day rotators in TESS data (for example).

The match fraction appears to be 30%-60% past 15 days. However, this is an overestimate driven by definition (Equation ( 2)). For periods below 8 days, 3σ corresponds to 1 day, and hence encompasses a small fraction of the total period space. The odds of a chance match are small (5%). By 15 days, 3σ is about ±4 days, which would capture 20%-40% of periods in the parent sample by chance. Similarly, at the longest periods, the number of recovered aliases is in line with the number of aliases that would be recovered by random guessing. We do not consider this to be statistically significant for the same reasons as above: The uncertainty budget is so large by P rot = 25 days, the number of random matches covers a huge swath of parameter space.

As expected, LS power, T, and SNR all impact on the reliability of measured periods. The impact of LS power is particularly strong: Curves with the highest LS power have reliability 90% even out to 10 days. In other words, this analysis indicates that if you have a collection of measurements that all have LS power greater than 0.1, more than 80% of those measurements will be the "correct" rotation period until P rot ≈ 8 days, dropping to 80% at 10 days, and hitting the sharp drop to less than 50% by 13 days.

Target brightness has a modest effect for stars T < 15, after which reliability drops quickly. Periods for the faintest stars (T > 16) are generally unreliable. SNR appears to have a strong effect: Targets with SNR > 10 are significantly more reliable at periods below 10 days. A lot of this is a consequence of the way SNR is calculated. SNR scales with N cycles , and fast rotators tend to have larger amplitudes (M. G. Barber & A. W. Mann 2023). As a result, fast rotators are overwhelmingly in the high-SNR bin and the few that are not tend to be around faint stars.

One piece that is not captured in Figure 6 is why some periods do not match, especially when the LS power is high and the period is short. Of the stars with LS power > 0.2, P rot, RH20 < 10 days, and T < 15, less than 5% are not matches or a half-alias. Examination of these stars suggests most of them are periods that landed just outside the requirements set in Equation (2). Those might be due to larger than typical astrophysical variation (e.g., from differential rotation and spot evolution). Other mismatches in this category are binaries that show two distinct periods in the periodogram, and where analysis of the K2 data favored a different period than the TESS data. Often such systems have two reported periods in the literature (e.g., L. M. Rebull et al. 2018; S. T. Douglas et al. 2019). We also reanalyzed the K2 data for a random subset of these mismatches, and recovered a period consistent with our TESS estimate, suggesting the P rot,RH20 was incorrect for 1% of the overlap sample. We treat all these cases as "unreliable" based on our strict definition, but the reliability in this bin is 3%-5% points higher if we account for these effects.

Separate from the questions of precision and reliability is the question of completeness-a measure of the fraction of a sample where you expect to be able to measure P rot . This relates to reliability, but completeness considers stars that were removed because of a given sample cut (in LS power, T, or SNR). In general, completeness drops as reliability increases. Ideally, we want cuts on quality that will improve reliability for small costs in completeness. In practice, one selects cuts based on the demands of reliability and completeness set by the science.

The most robust measure of completeness would be injection/ recovery, as is done with exoplanet occurrence calculations (e.g., Z. Wahhaj et al. 2013;E. A. Petigura et al. 2013;A. C. Rizzuto et al. 2017). Unlike for exoplanets, however, we do not have an analytic model of stellar rotation. Instead, we estimate how completeness changes as a function of LS power, T, and SNR empirically using our cross-match K2-TESS sample. A downside of this approach is that it includes any biases already present in the K2-TESS overlap sample, but it is still a good proxy for how completeness changes with P rot and data quality cuts. We discuss this issue further in Section 7.

Figure 6. Single-sector reliability results. The left column shows the match fractions for LS power, T, and SNR, while the right column shows the same analysis for aliases. The blue line shows the reliability fraction without cuts, and the gray line represents the chance that a randomly chosen rotation period will match the RH20 period. Bins with fewer than three total points have been excluded from these plots.

We define completeness for LS power as

As with Equation (4), M represents the number of matches satisfying a given criteria. Unlike Equation (4), N is the total number of stars in the K2-TESS cross-match sample. Thus, completeness reflects the overall success rate for the full sample (and the number of stars lost in any given cut), while reliability measures the success rate only among stars that meet the detection criteria (i.e., how accurate are the measured rotation periods for stars above the detection threshold).

Figure 7 shows how completeness changes with P rot,RH20 and each of the three tested parameters. As with reliability, completeness drops rapidly at 10-15 days, a reflection of the low match rate. Below 10 days, completeness is no higher than ;80%. That is, we cannot recover about 20% of the periods for stars with <10 day rotation periods. The great majority of these 20% are stars that land in low-reliability regimes (e.g., low LS power and faint).

SNR cuts have a small effect on the fastest rotators, as almost all of the fast rotators have a high SNR. However, SNR cuts dramatically reduce the number of ;10 day rotators. LS power cuts have a more smooth effect, reducing the sample across all periods. The highest LS cut we considered (LS > 0.2) yielded Figure 7. Single-sector completeness results. The left column shows the match fractions for LS power, T, and SNR, while the right column shows the same analysis for aliases. The gray line represents the chance that a randomly chosen rotation period will match the RH20 period. The maximum completeness is ∼80% because a large number of TESS rotation measurements have LS power <0.2, many of which do not match the true RH20 period. See Section 4.3 for a more detailed description of how to interpret completeness results.

highly reliable periods below 10 days (90%-100%), but at a heavy cost. This cut removed more than half the sample.

Reliability was only weakly impacted by magnitude cuts out to T ∼ 16, and cut out a large fraction of the sample. On the faint end, T < 16 and T < 17 are almost identical. This is because there are few stars T < 16, and few of those are matches (Figure 6). The implication is that magnitude cuts below T = 16 come at a significant cost in completeness for marginal gains in reliability-nearly all of which are covered by an LS power cut.

Our completeness analysis is designed to answer the question: Given a sample cut on LS power, T, or SNR, how many rotation periods are removed from your sample? For example, if you choose to only consider stars with LS power >0.2, your measurements will be highly reliable (see Figure 6) but will have very low completeness (see Figure 7), because very few stars have a strong enough rotation signal to have LS power >0.2. In fact, of the 5723 stars in our sample where our TESS period matched the RH20 period, only 1532 (27%) had LS power >0.2.

The above analysis considers only single TESS sectors. The systematics treatment using CPM (as done with unpopular) should preserve astrophysical signals, even over multiple sectors. However, this requires testing; time-dependent signal degradation and long-term systematics may dominate even with careful correction. Here, we repeat our analysis after stitching together sectors, exploring the impact on completeness and reliability.

To start, we focus on targets with two consecutive sectors of data. The sectors are combined as described in Section 3.3, and our rotation period estimates are done in the exact same way as for our single-sector analysis.

As in Section 4.1, we estimate empirical uncertainties by comparing the resulting periods to those from RH20. The resulting comparison is shown in Figure 4. The period uncertainties are lower, particularly at long periods. The gain is about a factor of 2 by 10 days, but negligible at periods below 4 days. At short periods, precision is likely dominated by other systematics (including from the K2 periods).

We show consecutive-sector reliability results in Figure 8. Results below 10 days are comparable to single-sector results. There are gains in reliability for stars with periods 10-20 days, but these are mostly targets with the highest LS power (although the reliability is still low). For example, in the LS power >0.2 bin, the reliability at 15 days is 20% for single sectors, and 40% using consecutive sectors. Other changes are small or consistent with noise.

We show the completeness results in Figure 9. Similarly to the single-sector analysis in Section 4.2, a LS power cut comes at a significant cost to completeness, with even a modest cut of LS power = 0.05 decreasing completeness by 20% for P rot < 10 days. The same trend is seen for T and SNR.

Our final analysis explores whether the stitching of all available sectors of TESS data (consecutive or not) improves the results. Unfortunately, very few stars had more than two consecutive sectors, so this often meant merging light curves with large gaps between the data set. The net result was only marginal improvement. The period uncertainties were within 1% of the results from the consecutive-sector analysis. Completeness and reliability metrics also showed little change from the consecutive-sector results; most were a decrease in reliability or a change consistent with random variations.

To demonstrate one application of our study, we compiled membership lists for three benchmark associations from the literature: α Persei (α Per; A. W. Boyle & L. G. Bouma 2023), Pisces-Eridanus (J. L. Curtis et al. 2019), and Group X (S. Messina et al. 2022). These lists were refined following the criteria in L. G. Bouma et al. (2023), retaining only stars with flag_benchmark_period == True. We then used unpopular to generate TESS light curves for each star and measured rotation periods using a LS periodogram, employing the same setup as described in Section 3.

We then assigned reliability estimates using our TESS-RH20 sample above as a function of the three main parameters: LS power, T, and SNR. Reliability was calculated using LS power for α Per, T for Pisces-Eridanus, and SNR for Group X (one parameter was used for a given group). We then select a set of "nearby" stars within the relevant parameter space. This is always 1 day in period, then ±0.05 around the LS power, ±0.5 mag around the T magnitude, or ±2.5 for the SNR. Any combination of parameters can be used (in addition to period), although for this test we are exploring just one parameter at a time.

Figure 10 presents these results, with each star color-coded by its recovery (top row), match (middle row), and alias (bottom row) probabilities. The recovery probability represents the likelihood that the measured rotation period is either a match or an alias of the true period. Rotation periods shorter than 1 day are excluded from reliability calculations due to RH20's search limits, and stars with such periods (below 1 day for matches, 0.5 day for aliases) are shown in gray without probabilities.

Our analysis indicates that the overwhelming majority of stars have high (80%-100%) reliability. Longer-period rotators (P rot > 10 days) in Pisces-Eridanus exhibit lower reliability (40%-60%) when assessed using T; these are also the most discrepant points from the sequence. Overall, we estimate that 11, 7, and 11 rotation measurements are unreliable across α Per, Pisces-Eridanus, and Group X, respectively. This analysis is somewhat misleading, because our reliability estimates depend on the true period, not the measured one. However, reliability is quite high out to 10 days, which encompasses most of the stars considered here, although it likely underestimates the number of half-aliases (with real periods 10-20 days) in the sample. One could correct for this by repeating the analysis using the measured period, although this still requires some knowledge of the relative occurrence of stars in a given period bin (i.e., P(P rot )).

We can perform an analogous analysis with completeness. Stars with measured periods likely have higher LS power than their nonmeasured counterparts (although the effect of the other parameters is less clear). However, assuming measured periods are typical, the number of missing detections (M) is expected to be

where C i are the individual completeness estimates for each of N stars with successful measurements. The summed value is effectively the number of "missed" period measurements for each successful one. In this manner, we estimate there are 17, 5, and 14 members for which we would have failed to measure their rotation period in α Per, Psc-Eri, and Group X, respectively.

Completeness or Reliability? In principle, longer time baselines can help resolve low-amplitude modulations and better sample slower rotations. Indeed, combined sectors yield lower period uncertainties (Figure 4). One might expect consecutive-sector observations to outperform single-sector data in terms of completeness and reliability. Figure 11 illustrates the results by showing the fraction difference between consecutive-sector and single-sector results. Positive values mean consecutive-sector measurements outperform single-sector measurements, whereas negative values indicate they underperform.

Generally, consecutive-sector analyses yield 5%-10% points lower reliability and completeness. There are some gains in reliability at lower LS power (0.05-0.10) and SNR (25-50 percentiles) and 5-10 day periods, suggesting that the longer baseline does provide advantages in specific cases. Using all available sectors was worse, yielding no significant gains in precision, and additional losses in reliability and completeness.

Inspection of the results suggests that merging sectors increases both the signal of interest and any instrumental systematics. This also explains why the reliability drop is largest at 12-17 days and for fainter stars, where scattered light is expected to dominate. Merging sectors enhances these systematics because many are common between observations. Changes in spot morphology can happen on timescales of months (R. Rampalli et al. 2021), so the contribution from the star often grows more slowly with increasing number of sectors than other persistent signals.

Although CPM systematic correction is meant to remove instrumental signals while preserving stellar ones, the results suggest this process was not entirely successful. Instead, consecutive-sector measurements should only be used in cases where one is already confident the period is reliable and wants the gain in precision. A better option is to analyze each sector separately and combine the results; simply using the sector of data that yielded the highest LS power provided the highest reliability. An alternative would be to use external ground-based photometry (e.g., W. S. Howard et al. 2021) to separate out systematics. Absent that, improvement will require new methods to merge light curves from multiple sectors and techniques to suppress nonastrophysical signals. These results align with other studies that have struggled to measure long-period rotations from consecutive sectors of TESS data (M. Kounkel et al. 2022). However, more advanced techniques, such as machine learning techniques (Z. R. Claytor et al. 2024), appear to be making progress toward reliably extracting long-period rotation measurements from TESS data.

One goal of our analysis is to identify the quality cuts that maximally improve reliability, minimize spurious detections, and/or maintain high completeness. The three parameters explored here (LS power, T magnitude, and SNR) are similar or identical to the most frequently applied parameters (e.g., L. G. Bouma et al. 2021;T. Fetherolf et al. 2023).

Our results show the reliability/completeness trade-off. The highest LS power cut (>0.2), for example, yields >90% reliability for periods below 10 days. However, even among fast rotators, most light curves do not yield such high LS power. So this cut would catch only 40%-50% of the sample. Similar effects are seen from cuts on T. The pattern with SNR is more complex, in part because there are few fast-rotating low-SNR targets. The codependence of quality cuts is more complicated, as each of the parameters we explored are correlated to each other.

To facilitate setting cuts specific to a given science case, we provide a code that calculates the completeness and reliability for any given subset of the sample. 4 In this way, one can design their cuts around the science requirements or determine period reliability on a per-star basis over the survey.

We evaluated the precision, completeness, and reliability of stellar rotation measurements derived from TESS data. To this end, we used a sample of 23,000 stars with TESS light curves and rotation period measurements from K2 data. By assuming the K2-based periods are highly reliable, we could assess the performance of periods derived from TESS light curves as a function of LS power, T magnitude, SNR, and rotation period. . Application to young associations. The recovery, match, and alias probability of rotation periods measured by TESS for three benchmark associations: α Per (t ∼ 80 Myr), Pisces-Eridanus (t ∼ 130 Myr), and Group X (t ∼ 300 Myr). Effective temperatures were estimates from B P -R P color using the same methodology as described in Figure 1. Match and alias probability were calculated using using the results of our reliability analysis in Section 4.2, with the reliability for α Per calculated using LS power, Pisces-Eridanus using T, and Group X using SNR. The recovery probability is the chance that either a match or an alias is found. Stars are colored by their probability, with gray points falling in regions where our analysis does not apply.

In terms of precision, we find as follows:

1. Periods derived from a simple LS of TESS data are good to 1%-2% for periods below 5 days, increasing roughly linearly to 6% by 12 days. 2. The precision at periods from 6 to 8 days is significantly improved by merging sectors, although the gains are similar to computing each sector separately and combining the resulting measurements. 3. TESS-based periods are systematically biased by ;10% (too short) for stars with periods of 10-14 days. This effect is weak in the 8-10 day bin (<3%) and not measurable at shorter periods.

Because we compared periods estimated from different instruments and separated by years (over which the stellar signal chances) these estimates cover any instrumental and astrophysical effects.

In terms of reliability, we find as follows:

1. Rotation periods derived using a single sector of TESS data are 70%-80% reliable out to periods of 10 days, even without any quality cuts. Modest cuts can improve this to more than 90%. 2. Beyond 10 days, reliability drops below 40%, and by 15 days assigned periods are only marginally better than random assignment. 3. LS power is a strong predictor of reliability. The highest LS cut (>0.2) gives periods that are reliable 90%-95% of the time out to 10 days. Brightness has a more modest impact on reliability, mostly in the faint end (T > 15). 4. Alias detection is minimal (∼10%) for periods shorter than 5 days, but increases to ∼30% for periods of 10 days or more.

5. Stitching consecutive or multiple sectors does not enhance reliability; single-sector measurements are generally as effective.

For completeness, our findings indicate as follows:

1. Maximum completeness is ∼80%, though completeness drops significantly with stricter LS power, T, or SNR cuts. 2. Stitching multiple sectors does not improve completeness.

One of the goals of developing a probabilistic assessment of each rotation period measurement was to include rotation data into higher-level analyses. Specifically, we are interested in both identifying new members of known associations and using rotation to find entirely new associations.

A Bayesian selection of association members follows the form

( ) P M P M P M P , 7 q q q = where ( | ) P M q is the probability that a star is a member given the set of measurement parameters q. In most cases, θ is a set of kinematic and spatial coordinates (e.g., UVWXYZ) that can be compared to a model of the known group(s) and of the Galactic field (A. C. Rizzuto et al. 2011;L. Malo et al. 2014).

To properly consider rotation as a measurement parameter requires P(P rot |M), the probability if measuring P rot given that a star is a member. For this, we can use gyrochronology relations (e.g., L. G. Bouma et al. 2021) to build a model of P rot for a given star including astrophysical variation. But turning the model into a probability of measuring that value requires both an estimate of the fraction of unreliable periods and the probability that a period was missed even if it is present (i.e., the reliability and completeness). The remaining missing element is P(P rot ), although this can be derived from a larger parent population, like Kepler, or from a model of rotational spin-down (S. P. Matt et al. 2012).

This framework allows membership algorithms to capture the full uncertainty in the data; if a star's rotation period is not well measured (e.g., the reliability is low), the algorithm can degrade its membership probability accordingly. Over many stars, this has the advantage of preventing systematically incorrect rotation periods from skewing cluster properties such as age estimates (through gyrochronology), rotation dispersion, or spin-down rates. Lastly, this method facilitates the selection of more stars further from the group core and helps remove nonmembers near a group by chance.

A separate application is identifying new associations. Many searches for stellar associations have favored the hierarchical clustering algorithm HDBSCAN (L. McInnes et al. 2017). HDBSCAN enables clustering in an arbitrary number of dimensions, meaning one can add P rot as a parameter in addition to the spatial and kinematic parameters used by similar searches (M. Kounkel & K. Covey 2019; R. M. P. Kerr et al. 2021).

When clustering with rotation as an input parameter, precision and reliability can be used to weight certain points in the clustering metric. As a result, the final cluster assignments will reflect the degree of confidence in each rotation measurement, yielding a cleaner separation of rotational sequences and fewer misclassifications.

A drawback of these conclusions is that we are limited to the set of rotation periods inside the K2 sample. That is, completeness is only measured with respect to stars with periods below 44 days, and the comparison set is likely missing many targets at the high end of that period distribution. It also exudes rapidly rotating stars with weak spot signatures (e.g., some high-temperature stars), although such cases are rare. This has minimal impact on period precision estimates or our measure of reliability, but it means we are overestimating our completeness compared to the true parent population of rotators.

Equation ( 5) is therefore more useful as a relative completeness (how it changes with various cuts). Absolute completeness is significantly lower, as there are many stars with periods longer than 40 days. This could be computed from our measurements using Bayes' theorem (P(P rot |LS,T,SNR)) given the underlying distribution of rotation periods (P(P rot )). A partial fix for this is to use a benchmark sample with longerperiod rotators, like Kepler or ground-based surveys with long baselines (e.g., E. R. Newton et al. 2016).

Our analysis used a LS periodogram as it is the most common method for estimating rotation periods from light curves. Over such large samples, machine learning methods may yield better results (Z. R. Claytor et al. 2024). The results here can still be used as a benchmark for testing performance (i.e., to test if the new method outperforms a simple LS).

A major area for improvement would be methods to stitch the sectors. Right now, combining sectors worsens the output, which is hard to overcome absent a method to absolutely calibrate the fluxes in each sector. One could fit for a flux offset between each sector, and adopt the combination that yields the strongest period. However, this is not computationally practical when running over >10 4 stars. Such an approach would also need to be combined with a method to suppress scattered light and other signals not connected to the star. statistical noise. These results verify that the trends we recovered in our earlier analysis with the K2-TESS overlap are real.

As an additional tool for evaluating period reliability, we constructed receiver operating characteristic (ROC) curves for our three parameters (LS power, T, and SNR). These curves allow us to assess each parameter's ability to distinguish between rotation period measurements that agree with the benchmark K2 periods and those that do not. To compute ROC curves, we first classify each measurement as a true positive, false positive, true negative, or false negative. These categories are defined as follows:

1. True positive (TP). A rotation period from a star that passes a given threshold (e.g., LS power >0.2) and for which the K2 and TESS periods agree within 3σ. 2. False positive (FP). A star that passes the threshold but for which the TESS and K2 periods do not agree within 3σ. 3. True negative (TN). A star that falls below the threshold and for which the periods do not agree. 4. False negative (FN). A star that falls below the threshold and for which the K2 and TESS periods agree. The true positive rate (TPR) and false positive rate (FPR) are defined as ( ) TPR TP TP FN ; FPR FP FP TN . B 1 = + = +

A ROC curve is then made by repeatedly varying a parameter threshold and calculating the TPR and FPR at each instance. A perfect model would follow a path that rises vertically to TPR = 1 before turning horizontally to the upperright corner (FPR = 1). A diagonal line represents random guessing. The optimal parameter value typically corresponds to the point on the curve closest to the top-left corner.

Figure 13 shows the resulting ROC curves for LS power, T, and SNR. For each, we mark the point closest to the ideal upper-left corner and annotate the corresponding threshold. We find as follows:

1. LS power performs best at differentiating TPs and FPs, with the ROC curve significantly above the diagonal and an optimal threshold near LS ≈0.1. 2. SNR also performs well, with an optimal threshold near SNR ≈10. 3. T performs poorly; its curve lies below the diagonal, indicating that it is worse than random at separating TP and FP periods in our sample.

These results are consistent with our earlier reliability analysis, and confirm that LS power and SNR are the most informative metrics for screening reliable rotation periods. T is a poor discriminator when used in isolation-likely because it is only indirectly tied to signal quality, whereas LS power and SNR are more direct measures of the strength of a rotation signal.

We caution that these ROC curves are specific to the K2-TESS overlap sample. The optimal thresholds may differ for other stellar populations (e.g., young clusters with faster rotators or higher amplitudes). Nonetheless, this analysis offers a quantitative framework for choosing quality cuts that balance reliability and completeness.