The K2 M67 Study: Establishing the Limits of Stellar Rotation Period Measurements in M67 with K2 Campaign 5 Data

M67 was observed during K2 Campaign 5, which commenced on 2015 April 27 and ended on July 10. The pointing was centered on R.A. = 130^h09^m2753 and decl. = 16°49'4661.¹⁰ Individual K2 targets for Campaign 5 were proposed for observation by members of the community. Our proposal (PI. R. Matthieu) included 12,935 candidate members, with 12,521 drawn from the EPIC database, 266 from the 2MASS calibration field (see, e.g., Sarajedini et al. 2009), and the last 148 targets filling in those missing from the EPIC database around K_p ∼ 19 (R. L. Gilliland 2018, private communication). Membership for stars brighter than K_p = 15 is based on radial velocity and proper motion, as explained in G15, taking those with membership probabilities of 20% or greater to make a more complete catalog. For the remainder, we accepted those stars whose photometric membership made them likely members, unless contradicted by known radial velocities or proper motions. These stars formed our master target list. M67 was located on channels 1 and 2 of CCD module 6, near the edge of the FOV. Figure 2 shows the entire Campaign 5 FOV, with M67 targets represented by the teal circles on CCD module 6, along with a scattering of orange diamonds that represent the sample of hot stars used in this analysis, which will be discussed later.

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The stars located in the less-crowded, outer parts of the cluster were observed in the standard manner, by downloading a small window of pixels centered on each star of interest. A total of ∼25,000 individual star postage stamps were collected during Campaign 5; of these, 1877 fell on output channels 6.1 and 6.2. For the crowded inner regions of the cluster, a "superstamp" was used: a large, contiguous region created by juxtaposing many postage stamps. Approximately 2210 of the proposed M67 targets were located within the superstamp (the number of stars in the superstamp for which light curves are actually extracted depends on the method used). Figure 3 shows the locations of the M67 SAP sample (in blue) and the superstamp stars (in red) where they fall on channels 6.1 and 6.2 using two images of M67 taken from the ESO Online Digitized Sky Survey, centered on the channels' respective pointing.

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Most postage stamps were read out in "long-cadence" mode, or one observation every 30 minutes, with a total of 3663 cadences during the campaign. After basic reduction (correction of pixel-level instrumental effects), each series of postage stamps was combined into a Target Pixel File (TPF), and the TPFs were made publicly available at the Mikulski Archive for Space Telescopes (MAST) a few months after the end of the campaign.

In the rest of this section, we describe the two parallel light curve preparation procedures, or "pipelines," from two different institutions (Oxford and Harvard-CfA in collaboration with NASA Ames), which we tested against each other in this study. The K2 M67 study team includes core members of the teams behind the K2SC and VJ pipelines, which we implement here. While other K2 pipelines have been successful in various contexts, we did not include them in this study for a variety of reasons. The EVEREST pipeline, for example, does not perform optimally with a crowded field (Luger et al. 2016, 2017). Furthermore, the K2 light curves from K2VARCAT (Armstrong et al. 2014, 2016) only encompass Campaigns 0–4. Alternatively, Cody et al. (2018) have recently released Campaign 5 M67 light curves, using an approach to superstamp data similar to that of Aigrain et al. (2015a) (which we employ here). Nardiello et al. (2016) have also produced Campaign 5 M67 light curves using the PSF-fitting approach of Libralato et al. (2016) in addition to aperture and optimal-mask photometry. However, the light curves from Cody et al. (2018) were unavailable when we started this study, and we did not realize those of Nardiello et al. (2016) were publicly available until after we completed it. A future comparison with both sets of light curves would be interesting, especially since the root mean square error of the latter using aperture photometry at the bright end is similar to that of the pipelines we use (see Section 2.4).

Therefore, for each star, we applied our two pipelines to extract the light curves, correct for pointing-related systematics, and correct for residual common-mode systematics. In an ideal world, one would try every possible combination of each of these steps, resulting in eight versions of each light curve from the three separate data sets, each of which would have been injected with 42 simulated signals. Unfortunately, this would have led to an unmanageably large set of tests (almost 1,000,000 in total), and would have taken far too long to complete, especially when applying Pre-search Data Conditioning (PDC)–maximum a posteriori (MAP) for common-mode correction (see Section 2.3.3). While mixing the pipelines may have produced slightly more optimal results, we treat each pipeline as a unit, and produced only two versions of each light curve.

The "Oxford" pipeline is comprised of the following.

• 1.  
  For light curve extraction: the Simple Aperture Photometry (SAP) component of the standard Kepler pipeline, or the procedure described in Aigrain et al. (2015a) for cases where the SAP light curves are not available.
• 2.  
  For pointing systematics correction: the K2SC pipeline Aigrain et al. (2016).
• 3.  
  For the correction of residual common-mode systematics: a Principal Component Analysis (PCA) step.

2.2.1. Light Curve Extraction

2.2.2. Star-by-star Systematics Correction

2.2.3. Common-mode Systematics Correction

Visual inspection of the K2SC-processed light curves reveals significant, residual, long-term trends common to many light curves, which appear to be present whatever the extraction and detrending method used. Barnes et al. (2016), who used the MAST PDC versions of the light curves, already noticed them; they used a PCA approach to remove them. These particular trends are likely due to stellar drift from differential velocity aberration (DVA), which causes photometric aperture losses or variations from sensitivity differences within a pixel.

We also use a PCA approach to remove residual systematics. We normalize each light curve by dividing by its median. We then construct a matrix where each row is the logarithm of the normalized light curve. We also experimented with the more standard approach of re-scaling each light curve to have zero mean and unit variance, but found that this gave poorer results. We then use singular value decomposition to compute: the eigenvectors of the matrix, which are the principal components (PCs); the eigenvalues, which tell us what fraction of the total variance of the data set is explained by each trend; and a matrix of coefficients relating each light curve to each PC. The first few PCs represent the dominant trends in the data, and these can be removed by subtracting each of them, multiplied by the corresponding coefficient from each light curve.

Because M67 falls on two different CCD channels, the two subsets might be expected to show different systematics. However, we found no significant difference between the sets of PCs extracted from the entire set or from each output channel in turn. We therefore processed all the SAP stars on the two channels, both cluster members and non-members, together. On the other hand, there were much more significant differences between the PCs extracted from the SAP and SS sets, as illustrated in Figure 4, so these were processed separately. The SS PCs appear noisier than the SAP PCs, as a result of the relative dominance of the common-mode systematics in each set. The first, most dominant PC of the SAP set accounts for roughly 84% of the total variance of the light curves, while the first PC of the SS set only accounts for about 43%. The amplitude of the SS PCs are therefore much smaller, relative to the noise, than the SAP PCs.

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If PCA is applied blindly, large-amplitude features in individual light curves can dominate the PCs, skewing the correction and introducing these features in the corrected light curves of other stars. This can be diagnosed easily, however, as the PCs are linear combinations of the light curves, and the linear combination becomes dominated by a single star. Two problematic stars in the SAP set were identified in this way, and excluded from the PC estimation: EPIC 211391083 (an eclipsing binary) and EPIC 211327533 (whose light curve contains a large discontinuity). We also excluded the first 85 cadences (∼1.8 days) from the light curves before evaluating the PCs, as the systematics were particularly pronounced in that early phase of the campaign, and the PCA correction was of significantly lower quality when they were included (perhaps because the assumption of linearity, which is intrinsic in PCA, was violated more radically in that period).

Usually, a threshold in the fraction of explained variance (i.e., the eigenvalue) associated with each PC is used to decide how many PCs to include in the correction. We experimented with different values for this threshold, and settled on thresholds of 2.5% and 7.5% for the SAP and SS sets, respectively, which correspond to two PCs in both cases; see Figure 4. We tested a range of thresholds for each data set and found that overly high thresholds only produced one PC, which was not sufficient to explain all of the dominant, common modes seen in the light curves when examined by eye. Going too low, on the other hand, produced three or more PCs, in which case the extras were superfluous or introduced unwanted features in the data. A range of thresholds can produce the same two PCs, but the ones we chose were the on the lower end of those ranges for both data sets.

In analogy with the Oxford pipeline, the "CfA" pipeline is comprised of the following three segments:

• 1.  
  Light curve extraction from calibrated K2 TPFs, as described by Vanderburg & Johnson ( 2014) and Vanderburg et al. (2016).
• 2.  
  Correction for the 6 hr roll systematics, again as described by Vanderburg & Johnson (2014) and Vanderburg et al. (2016).
• 3.  
  Correction of non-roll systematics using the Kepler team's PDC technique (Smith et al. 2012; Stumpe et al. 2014).

2.3.1. Light Curve Extraction

2.3.2. Star-by-star Systematics Correction

2.3.3. Common-mode Systematics Correction

We here compute the scatter of the M67 SAP and SS stars as a function of Kepler magnitude at each step of the Oxford and CfA pipelines, in order to conduct a preliminary assessment of pipeline performance prior to the injection tests. We use a variation of the median of absolute deviations (MAD) (Hoaglin et al. 1983) scatter, estimated in the following manner:

$$\begin{eqnarray}&&{\rm{MAD}}=1.48{\rm{median}}(| {f}_{i}-{m}_{f}| ),\end{eqnarray} \tag{ 1 }$$

where f_i is the flux at each cadence i, and m_f is the median flux. We use the median as opposed to the mean because it is less influenced by outliers. Technically, the median term itself is the MAD scatter; we include a factor of 1.48 as a scaling factor that, under the assumption of a Gaussian distribution of f, makes the scatter equivalent to the standard deviation.

The scatter is given in Figure 5. The Oxford pipeline results are shown in black and the CfA, using single-scale PDC–MAP, in red. From top to bottom, the figure shows the scatter for the raw light curves, after star-by-star systematics correction, and after common-mode systematics removal, with the SAP sample on the left and the SS sample on the right. In the SAP sample, the raw scatters produced by the two pipelines are fairly similar, though the CfA pipeline performs slightly better. For both pipelines, the star-by-star systematic correction reduces the scatter somewhat over the whole magnitude range and forms tighter relationships, but the final common-mode systematic correction has the most drastic effect, leading to scatter values of a few tens of ppm at the bright end. The final scatter obtained by the CfA pipeline is slightly lower than that obtained by the Oxford pipeline over the entire magnitude range. The final median scatters at K_p = 12 for the Oxford and CfA SAP samples are 544 ppm and 321 ppm, respectively.

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By contrast, the raw outputs of the two pipelines are very different for the SS sample. The extraction step used by the Oxford pipeline, which involves moving apertures, already lessens the pointing-related systematics in the raw light curves of fainter stars. The effect of the star-by-star systematic correction on the scatter is then relatively minor for both pipelines, but the common-mode systematic correction is particularly effective over the whole magnitude range for the CfA pipeline, and for bright stars in the Oxford pipeline. The final results of the two pipelines are fairly similar, though the CfA pipeline performs slightly better for brighter stars (albeit not to the extent of the SAP sample). Among the fainter stars, there are several instances where the CfA pipeline leads to final scatters on the order of 1%, whereas the corresponding scatter for the Oxford pipeline is much lower. In these cases, the Oxford pipeline generally tends to correct the systematics better. The final median SS scatter at K_p = 12 is 203 ppm for the Oxford pipeline and 275 ppm for the CfA pipeline.

The final scatter distributions are very similar for the SAP and SS samples, as well as for the two pipelines. It is worth noting, however, that the scatter between the CfA SAP and SS data sets are closer to each other than their counterparts in the Oxford pipeline. This is most likely due to the fact that the CfA light curves are all extracted using the same technique, while the two Oxford data sets are extracted via different methods, so they have different starting points with respect to noise levels. Regardless, these numbers should give us increased confidence that the light curves we are using, from both pipelines, are relatively robust, and there is no obvious effect in either sample that is not handled reasonably well by either pipeline.

For the purposes of the injection tests, we define the following three stellar samples.

• 1.  
  The SAP sample consists of the 1877 stars observed with individual postage stamps in channels 6.1 and 6.2, 1319 of which match to our master catalog and 236 of which are confirmed members from G15.
• 2.  
  The SS sample consists of the 976 stars included in the superstamp with matches in the master catalog, 359 of which are confirmed members from G15. Seventy-five of the SS overlap with the SAP.
• 3.  
  In addition, we define a hot star sample consisting of 89 main sequence stars with spectral types A through F located on CCD modules 6, 11, and 16, with effective temperatures ranging from 6300 to 10715 K (the majority falling between 6300 and 7300 K), B − V values of −1.0 to 0.45, and log g values greater than 4.0. This information was all acquired from MAST. These stars were chosen because they lack an outer convection zone, or have only a very thin one. This means they are not able to support a large-scale magnetic field, and so do not spin down quickly (Schatzman 1962). Thus, if they do display periodic variability, it is expected to occur on timescales of ∼2 days or less, whether it is due to pulsations (many of these stars lie in the classical instability strip) or rotation (as these hot stars are rapid rotators) (Nielsen et al. 2013). Therefore, their light curves are expected to contain only noise and systematics, at least on timescales longer than ∼2–7 days, which makes them ideal targets for injection tests. Twenty-two of the hot stars overlap with the SAP set.

We inject sinusoidal signals by directly multiplying the following into the raw light curves of each test sample:

$$\begin{eqnarray}&&{f}_{\mathrm{inj}}=\displaystyle \frac{{a}_{\mathrm{inj}}}{2}\sin \left(\displaystyle \frac{2\pi }{{P}_{\mathrm{inj}}}t+{\phi }_{\mathrm{inj}}\right)+1,\end{eqnarray} \tag{ 2 }$$

where f_inj is the injected signal, a_inj is the injected amplitude, P_inj is the injected period, ϕ_inj is the injected phase, and t is time in days. We vary P_inj from five to 35 days, in intervals of five days, leading to seven period values. We also vary a_inj, as follows: 0.05%, 0.10%, 0.30%, 0.50%, 1.00%, and 3.00%. Each light curve, then, is injected with 42 different combinations of periods and amplitudes, leading to over 120,000 injections in total for each pipeline to process. The phase, ϕ_inj, is selected at random for each injection, but we keep track of its value. Where we have extracted the SS light curves using multiple aperture sizes from the Oxford pipeline, we inject the signals into the version that minimizes the scatter for the corresponding, non-injected light curve.

Because these injection tests involve only sinusoidal signals, we use a modified version of the widely used Lomb–Scargle (LS) periodogram Scargle (1982) to recover them. The LS periodogram is essentially equivalent to least-squares fitting of sinusoidal signals (Irwin et al. 2006), and is thus, in principle, optimal to recover stable sinusoidal signals in the presence of white Gaussian noise of known variance.

Real stellar rotation signals are not strictly sinusoidal; they contain significant power at harmonics of the true period due to the scattered distribution of the active regions on the stellar surface, evolve over time with the active region evolution, and may contain signals at a range of periods due to differential rotation. For this reason, rotation period searches in Kepler and K2 data often use other period search methods that do not depend on the assumption of sinusoidality or strict periodicity (McQuillan et al. 2013a; Angus et al. 2018). These methods have been shown to outperform approaches based on least-squares sine-fitting in real and simulated data sets (Aigrain et al. 2015b). However, if we know the signal of interest is sinusoidal, then a sine-fitting approach is likely to do at least as well as these alternative, more flexible methods.

The model we are considering is of the form:

$$\begin{eqnarray}&&m(t)={m}_{\mathrm{dc}}+\alpha \sin (\omega t+\phi ),\end{eqnarray} \tag{ 3 }$$

where m_dc is the light curve's vertical offset, α is the amplitude, ϕ is the phase, and ω is the angular frequency. For each value of ω (or period, P = 2π/ω), this model can be expressed as a linear basis model with three basis functions: a constant, a sine term, and a cosine term, each with period P. For each trial period, we can solve for the values of m_dc, α, and ϕ that minimize the sum of the squared residuals, or χ² (in space data, the relative measurement uncertainties can be treated as approximately constant for a given star, so the two are equivalent). We evaluate the relative reduction in χ² with respect to a constant model, $S=({\chi }^{2}(0)-{\chi }^{2}(P))/{\chi }^{2}(0)$, as a function of period P, to construct a periodogram. For this study, we search for 490 periods ranging from 2 to 100 days, using an evenly spaced grid in frequency space.

3.3.1. Periodogram Normalization

If the light curves consisted simply of a stable sinusoidal signal plus white Gaussian noise, locating the peak from the Lomb–Scargle periodogram would give us the best-fit period, and evaluating the significance of the detection would be relatively straightforward (see, e.g., Horne & Baliunas 1986). However, this is not the case, even after our light curves are subjected to the systematics-correction steps of our two pipelines. The residual systematics, which remain in the light curves despite our best efforts, often lead to broad peaks in the range of 30–60 days, which can overwhelm the peak due to the injected signal. To address this, we perform a collective normalization of the periodograms before searching for the period.

To normalize the periodograms, we compute the median periodogram as a function of trial period for stars we expect to share the same systematic noise properties (i.e., separately for the SAP and SS sets, and for the Oxford and CfA pipelines). Each star's periodogram is then divided by the median periodogram, thereby suppressing the peaks, or excess power, seen in many light curves. The normalized periodogram is thus given by $S^{\prime} (P)=S(P)/\hat{S}(P)$, where $\hat{S}(P)$ is the median periodogram value for period P. The median periodograms are evaluated from the non-injected versions of the light curves, to avoid the induced power at fixed periods from their injected counterparts; they can be seen in Figure 6, with the Oxford SS and SAP median periodograms on the left and the CfA on the right. The SAP median periodograms for both data sets include the hot stars, as they were processed in both cases using the same techniques. However, Figure 6 also shows the hot star median periodograms separate from the rest of the SAP set for both pipelines, given by red dashed lines. This emphasizes that the broad peaks in both pipelines at long periods are most likely not a result of true astrophysical signals, as we would not expect periods much longer than about five days in the hot star samples. We also note that the median periodogram for the SAP set from the Oxford pipeline is higher overall and peaks at longer periods than the SS sample, hinting that the former likely contains systematics with larger amplitudes on greater timescales. This is consistent with the relative amplitudes of the PCA trends compared to noise for the two samples, which are smaller for the SS (see Figure 4). This also means that long-period signals will be more heavily suppressed by normalization in the SAP than in the SS sample.

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Figure 7 illustrates the effect of the periodogram normalization for an individual star from the SAP set. The top panel shows the light curve of EPIC 211355490 injected with a period of 20 days and amplitude of 0.1%, fully processed with both the Oxford (in gray) and CfA (in blue) pipelines. The middle panel depicts the original periodograms prior to normalization for both pipelines, while the bottom panel gives the normalized versions. The respective periods are indicated by vertical lines. There is a relatively strong, long-period signal in the Oxford pipeline that is not present in the CfA (unsurprising, given the differences in Figure 6), and this is what the Lomb–Scargle finds in the former, settling on a period of 62 days. On the other hand, the normalized Oxford periodogram has a peak at 19.2 days, which closely matches the injected period. The normalization of the periodogram thus allowed us to find the "true" period of Oxford version of this light curve when otherwise it would have been lost by the systematic, long-term trend. The CfA, however, appears to have done a better job of cleaning up systematics in the light curve; it found peaks of 20.0 days in both the original and normalized periodograms.

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3.3.2. Detection Threshold

Once we have identified the peak in each normalized periodogram, we must decide whether it is significant. After testing a number of different possible schemes, we opted for the following detection threshold definition:

$$\begin{eqnarray}&&{S}_{{\rm{T}}}^{{\prime} }=\mathrm{MAX}({S}_{90}^{{\prime} }\times C,{S}_{\min }^{{\prime} }),\end{eqnarray} \tag{ 4 }$$

where ${S}_{T}^{{\prime} }$ is the threshold value and ${S}_{90}^{{\prime} }$ is the 90th percentile of the normalized periodogram for a given light curve. Here, C and S_min are tuning parameters that we can vary to alter the relative and absolute components of the threshold, respectively. For example, if ${S}_{90}^{{\prime} }=4$, and the value for C is set at 3, then the maximum power of the normalized periodogram must be at 12 or greater for the associated period measurement to be considered a detection. S_min comes into play when ${S}_{90}^{{\prime} }\times C$ is so low that most peaks would qualify as a detection; it ensures that there is a minimum value for the threshold. Both C and S_min can then be varied to test their effect on the completeness and reliability of the period search, which we discuss in Section 3.4.4.

If the highest peak in the normalized periodogram passes the threshold defined in Equation (4), then we have a detection and record the corresponding period, as well as the best-fit amplitude and phase. Otherwise, a result of "no detection" is recorded.

To evaluate the results of the injection tests, we need to quantify how sensitive the period search is, i.e., what fraction of the injected signals are successfully recovered—but also how trustworthy the detections are, i.e., what fraction of the detections are "valid," meaning that the measured period and phase are within some tolerance of the injected values.

3.4.1. Valid Detections, Completeness, and Reliability

3.4.2. Calculating Uncertainties

To estimate the uncertainties, we assume a Poisson distribution with respect to both completeness and reliability, and define the uncertainty as:

$$\begin{eqnarray}&&\sigma =\pm \displaystyle \frac{1}{\sqrt{N}},\end{eqnarray} \tag{ 5 }$$

where N is the number of injections per injected period and amplitude bin (i.e., the number of injected amplitudes multiplied by the number of injected periods multiplied by the number of stars in each test sample) in the case of completeness. For reliability, N is the total number of detections in each detected period and measured amplitude range. Therefore, for each test set within each pipeline, the individual completeness values will have the same uncertainty while the reliability will vary from bin to bin. We recognize this is a little simplistic, but the more sophisticated approach would require repeating the injections many times, which would be far too time-consuming. For our purposes, this is a good first approximation.

3.4.3. Intrinsically Variable Stars

3.4.4. Varying the Threshold

The detection threshold obviously has an effect on the results of this study. Lower thresholds increase completeness, but also lead to a drop in reliability in the same bins. If measured rotation periods are to be compared with theoretical models, it is preferable to prioritize higher reliability over slight gains in completeness—so long as the completeness itself is well-measured, to account for missed detections. We therefore recorded the normalized periodogram peak $S^{\prime} ;$ the corresponding period, phase, and amplitude; and the 90th percentile of the normalized periodogram, ${S}_{90}^{{\prime} }$, for each injected signal in each light curve. It was then trivial to vary the value of C and S_min in (4) and examine the impact that this had on completeness, reliability, and the "variables" excluded from the statistics.

We experimented with C = 0 to C = 6 while holding S_min at 5, and varying S_min from 0 to 12 while holding C at 4, in order to test the effect that each parameter has on completeness and reliability. We used the Oxford pipeline to perform these two tests. A sample of the results of these tests can be seen in Figures 8 and 9. Figure 8 shows the effect on completeness for changing both tuning parameters at an injected period of 25 days across all injected amplitudes. The top row of each figure shows the effect of changing C, while the bottom row shows the effect of changing S_min. In both panels, we print the number of stars in each injected period and injected amplitude bin. We can see that, at C = 0 and C = 1, the results are dominated by the minimum threshold value, but they have relatively few stars per bin. Completeness then peaks at C = 2 before starting to decline again. There is also an increase of ∼500 stars per bin at C = 2, followed by another 500 stars at C = 3. Beyond C = 3, the number of stars per bin still grows, but less rapidly. For S_min, the completeness essentially stays the same until about S_min = 8. By S_min = 12, the completeness drops to about half the values at S_min = 0 at amplitudes of 0.5% and less. Not visible in Figure 8 is the fact that all sensitivity at periods greater than 25 days, at any amplitude, is lost. The number of stars per bin increases with S_min, but only very slightly. Thus, as expected, C has a much greater effect on the number of stars per bin—and more importantly, on completeness—than S_min, as long as S_min is not too high (in the case of completeness).

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Likewise, in Figure 9, we can see an illustration of how C and S_min affect reliability across the ranges of measured periods in this study. Here, we have fixed the measured amplitude to the range of 0.25–0.45%, just above the solar range. Again, the top row shows the effect of changing C. As expected, reliability generally increases as C increases, noticeably so until about C = 4. The effect of changing S_min, however, is largely negligible except at the long period ranges. There, it decreases until S_min = 8. Here, the reliability is zero, but with very few detections, and beyond, the number of detections drops to zero for periods longer than about ∼30 days.

Figures 8 and 9 demonstrate the clear trade-off between completeness and reliability. Using large values for C and S_min, i.e., a stringent detection threshold, leads to low completeness and large threshold error, but excellent reliability. A high detection threshold also means that fewer stars are marked as intrinsically variable, so that more injected light curves are used to compute the final statistics. As the threshold is gradually lowered, completeness increases and threshold error decreases, at the cost of reduced reliability. The number of variable stars also grows, and almost all stars are excluded by very low detection thresholds, leaving relatively few in the calculation of the results. Therefore, we settled on C = 4 and S_min = 4 as the best compromise. When using these values, 238 of the 1877 stars in the Oxford SAP set were marked as intrinsically variable, as were 120 of the 976 stars in the SS set and 23 of the 89 hot stars, leaving 1639, 856, and 66 stars in each set, respectively. Using the same tuning parameters for the CfA pipeline, we are left with 1587 SAP stars, 848 SS stars, and 70 hot stars.

In Figures 10 through 15, for the hot star, SAP, and SS samples from the Oxford and CfA pipelines, each figure is made up of six panels, corresponding to the injected amplitudes of 3.0%, 1.0%, and 0.50% (top row), and 0.3%, 0.1%, and 0.05% (bottom row). Within each panel, the colored circles represent detections, while the cases that did not pass the detection threshold are shown as smaller black dots. The gray-shaded area in each plot shows the period validity range, though recall that a ±20% phase match is also required. Phase considerations aside, a black point in the shaded area is a missed valid detection, whereas a colored circle outside the shaded area is an invalid detection. The stars marked as intrinsically variable were excluded from these figures.

The completeness results for the Oxford and CfA pipelines are shown in the top and bottom rows of Figure 16, respectively. We have identified in blue the completeness for the solar case—where the injected amplitude is 0.1% and the injected period is 25 days—for each data set. Tables 1 through 6 in Appendix A record the completeness values and uncertainties for each injected period–amplitude bin for all test samples from each pipeline.

Table 1.  Oxford Hot Star Completeness (%)

${a}_{\mathrm{inj}}\setminus {P}_{\mathrm{inj}}$	5 days	10 days	15 days	20 days	25 days	30 days	35 days
3.00%	98	95	95	94	94	92	88
1.00%	95	94	88	88	83	73	59
0.50%	92	89	80	70	65	52	35
0.30%	86	80	71	50	48	26	23
0.10%	65	47	27	17	17	11	3
0.05%	41	24	17	12	9	3	2

Note. Total number of injections per period–amplitude bin: 66; σ = ±12.3%.

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Table 2.  CfA Hot Star Completeness (%)

${a}_{\mathrm{inj}}\setminus {P}_{\mathrm{inj}}$	5 days	10 days	15 days	20 days	25 days	30 days	35 days
3.00%	100	99	94	91	87	83	80
1.00%	97	93	91	84	81	77	76
0.50%	93	90	86	83	77	73	60
0.30%	91	87	84	80	69	49	34
0.10%	83	66	56	47	44	34	20
0.05%	66	50	36	16	16	13	6

Note. Total number of injections per period–amplitude bin: 70; σ = ±12.0%.

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Table 3.  Oxford SAP Completeness (%)

${a}_{\mathrm{inj}}\setminus {P}_{\mathrm{inj}}$	5 days	10 days	15 days	20 days	25 days	30 days	35 days
3.00%	99	97	92	86	85	79	65
1.00%	93	81	75	65	65	56	28
0.50%	82	71	58	43	43	30	8
0.30%	73	59	40	24	21	12	3
0.10%	44	21	5	3	2	1	1
0.05%	21	4	1	1	0	0	0

Note. Total number of injections per period–amplitude bin: 1639; σ = ±2.5%.

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Table 4.  CfA SAP Completeness (%)

${a}_{\mathrm{inj}}\setminus {P}_{\mathrm{inj}}$	5 days	10 days	15 days	20 days	25 days	30 days	35 days
3.00%	97	92	89	82	74	72	68
1.00%	91	81	73	66	61	53	39
0.50%	83	69	63	53	47	33	23
0.30%	74	58	52	41	37	30	18
0.10%	54	33	23	18	14	11	7
0.05%	35	19	9	5	3	2	2

Note. Total number of injections per period–amplitude bin: 1587; σ = ±2.5%.

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Table 5.  Oxford Superstamp Completeness (%)

${a}_{\mathrm{inj}}\setminus {P}_{\mathrm{inj}}$	5 days	10 days	15 days	20 days	25 days	30 days	35 days
3.00%	100	98	93	88	89	77	74
1.00%	91	81	74	65	66	50	46
0.50%	78	66	54	46	46	37	34
0.30%	68	50	41	38	37	27	24
0.10%	39	30	23	15	13	2	2
0.05%	28	14	8	3	2	0	0

Note. Total number of injections per period–amplitude bin: 843; σ = ±3.4%.

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Table 6.  CfA Superstamp Completeness (%)

${a}_{\mathrm{inj}}\setminus {P}_{\mathrm{inj}}$	5 days	10 days	15 days	20 days	25 days	30 days	35 days
3.00%	93	86	83	74	63	56	51
1.00%	82	70	61	52	45	36	27
0.50%	69	57	50	40	34	27	23
0.30%	58	47	41	34	31	26	16
0.10%	42	31	21	15	15	11	7
0.05%	29	15	10	5	4	4	2

Note. Total number of injections per period–amplitude bin: 844; σ = ±3.4%.

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The reliability results for the Oxford and CfA pipelines are shown in the top and bottom rows of Figure 17, respectively. The bins with red dots are those with relatively few detections. We consider bins to be insignificant if they have 100 detections or less in the SAP and SS samples and 30 or less in the hot star sample. All three samples, from both pipelines, have very few detections in the longest-period, lowest-amplitude bins. The corresponding (generally very high) reliability values therefore have large uncertainties and should not be treated as very meaningful. The actual values with uncertainties are presented in Tables 7 through 12 in Appendix A.

Finally, Figure 18 shows the injected versus detected amplitudes for valid detections from the Oxford (top) and CfA (bottom) pipelines. The detected amplitudes can differ significantly from the injected ones for several reasons. First, the light curves into which the signals were injected may already contain some power at the corresponding period. Second, the intrinsic noise levels of the light curve will affect the amplitude of the flux as a whole. Lastly, the systematics correction steps can alter the injected signal, in some cases leading to measured amplitudes that are smaller than the injected ones (e.g., the Oxford SS).

The median periodograms shown in Figure 6 of Section 3.3.1 tell us a lot about the residual trends present in the light curves after full processing from both the Oxford and CfA pipelines. For the SAP and SS samples, the median periodogram rises steeply for periods above 20 days, peaking around ∼50–55 days and ∼40–45 days, respectively, before decreasing again and flattening off for periods >80 days (to which we expect little or no sensitivity in K2 light curves anyhow). While evident in both pipelines, this feature is much more dramatic in the Oxford pipeline. Importantly, the power present at periods relevant to a study of M67 in both hot star samples shows that this is largely non-astrophysical, despite our steps for systematic correction. Furthermore, the differences between the SAP and SS samples are significant: our a priori expectation was that the SS sample might be more problematic than the SAP one, due to increased crowding in the densest parts of M67. However, the median periodograms tell a different story, with considerably more residual power in the 30–70 day period range in the SAP than the SS light curves for both pipelines.

If the raw periodograms were used for period detection, these residual trends would lead to numerous false detections at mid-to-long periods (see Appendix B.1). Our normalization procedure designed to avoid this seems successful, as we record consistently high reliability wherever we have a significant number of detections (see Figure 17 and Tables 7–12 in Appendix A). However, normalization also suppresses the detection of real signals at longer periods, as is apparent in Figure 16. While we may avoid the trap of lingering systematics by normalizing the periodograms, this comes at the potential cost of real astrophysical signals.

The set of 89 hot stars is an ideal test case, as they should not show significant rotational modulation beyond about five days. Therefore, we can be confident that, if we see a long period where we did not inject one, it is most likely the result of lingering systematics (as we alluded previously). For the Oxford pipeline, we were left with 66 stars after removing stars found to have intrinsic variability (i.e., those likely to have an astrophysical signal that could interfere with the injection test results) using our detection threshold on the non-injected versions of the light curves. Of the 23 stars marked as variable, 19 had detected periods less than 10 days and the rest had periods less than ∼16 days, apart from one (see Figure 1). In the CfA pipeline, we used 70 stars in the analysis. All 19 variables had periods under 12 days. These numbers further enforce the fact that our normalization works reasonably well for this particular data set in both pipelines, though the presence of flagged variables with periods longer than five days shows that the set of hot stars may also be contaminated by background stars, or that there are a number of cooler F stars in the sample, but it is good enough for our purposes.

As shown in Figures 10 and 11, we do a good job at recovering the injected periods from both pipelines in the hot star sample. Some of the invalid detections likely come from variables—possibly pulsators—not removed from the sample but with low amplitudes. Where these start to become detections at the lowest injected amplitudes, the injected signal may be "boosting" (i.e., increasing the amplitude of the intrinsic signal) them just enough to be detected. Even for very regularly sampled data like K2, injecting a signal at a given period affects the periodogram at other frequencies in a complex way. However, a number of the invalid detections could also be the result of the normalization, which promotes shorter periods at the expense of longer ones: when dividing out the median periodograms, the much lower power seen at short periods increases short-period significance relative to that of longer periods.

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The hot star completeness values for both pipelines can be seen in the leftmost panels of Figure 16 and in Tables 1 and 2 in Appendix A. As expected from simple signal-to-noise arguments, we see completeness decrease with increasing injected period and decreasing injected amplitude for both pipelines. We point out that the best-case scenario for recovering a solar-like signal of 25 days and 0.1% amplitude is ∼45% from the hot star data set. However, the limited size of the hot star sample means that the individual completeness values are somewhat uncertain (±∼12%) for both pipelines.

The reliability in the hot star sample from either pipeline is consistently high (>90%). High reliability means that we can typically trust a detection made within a measured amplitude and period range. We can see from Figure 17 that, where we have a significant number of detections (more than 30 for the hot star samples), we can be fairly confident in our results, though again, the uncertainties are relatively large for the hot star samples. The high reliability also indicates that normalization did not introduce low-frequency signals in light curves that did not have a strong intrinsic signal in the first place, again reinforcing the validity of this step.

We have shown that the hot star sample is important for validating our injection test procedures and highlighting the presence of non-astrophysical power around 25 days and longer in the K2 Campaign 5 light curves. We now discuss the results from the SAP and SS samples, which specifically illustrate the complexity of a period search in M67. Figures 12–15, which show the detected versus injected periods for the SAP and SS data sets from both pipelines, are a lot messier than their hot star counterparts, most likely due to the larger sample sizes and more diverse stellar populations within. However, like the hot stars, detections become more difficult and less reliable as the injected period increases and the injected amplitude decreases. In addition, lingering power around 10–20 days seems to exist in the SAP and SS light curves. This power was either not originally strong enough to mark the light curves as variables until boosted by a lower-amplitude, injected signal, or it could be the result of half-period harmonic measurements from the Lomb–Scargle periodogram.

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The completeness results for both the SAP and SS samples are in the middle and far right panels, respectively, in Figure 16. As with the hot stars, we see the same general trend of diminishing completeness with increasing injected period and decreasing amplitude, but it has been exacerbated. For both samples from either pipeline, collapsing the figures onto either axis, the completeness falls around or below 50% for amplitudes ≤0.50% and periods ≥20 days, showing just how difficult it is to detect long-period, low-amplitude signals in either case. Of particular importance is the solar case, highlighted in blue in Figure 16, where the injected period is 25 days and the amplitude is 0.10%. Critically, the completeness here is ∼15% or lower for both the SAP and SS samples in either pipeline. Even with the best-case scenario of perfect sinusoidal signals, solar variability, which we expect in M67, is clearly difficult to find with our detection criteria.

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The middle and far right panels of Figure 17 present the reliability for the SAP and SS samples. As with the hot stars, where there is a significant number of detections (>100 for the larger data sets), the reliability for the SAP and SS samples rarely drops below 90% for both pipelines. This is encouraging, as it shows that the procedure we have developed should lead to relatively few false alarms, i.e., where we do get a detection, we can generally trust the period measurement.

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There are a few striking features in Figure 17, however. Detecting long periods (particularly at ∼35 days) in the Oxford SAP sample appears to be not only more difficult than the CfA pipeline, but also less reliable. The low reliability in this period range could either be a result of periodogram normalization or a failure to correct the residual long-term trends in the data, which the CfA pipeline does better. The Oxford SAP sample also has lower reliability in the shortest-period, lowest-amplitude bin. This could be an effect of promoting short-period signals when dividing out the median periodogram. We experimented with adding a "floor" to the Oxford median periodograms, via which we set any value below a given threshold to that floor in an attempt to reduce this effect. We tested floor values of 0.005 and 0.01. While these did improve reliability in the short-period, low-amplitude bin, the floors tended to reduce reliability overall in all three Oxford samples, particularly the hot star and SS sets, and especially around mid-range periods. Thus, we decided to exclude a floor from the study to avoid further artificial effects.

One surprising feature in the SS samples is the lack of detections across all periods at amplitudes of 2.0% and greater for the Oxford pipeline, even though many 3.0% signals were injected. This can be seen in both Figures 17 and 18. If we then compare the number of SS detections from one pipeline to another in the amplitude ranges 0.75–2.00% and 0.075–0.25% in Tables 11 and 12 of Appendix A, there are systematically more detections in the Oxford pipeline for these ranges than in the CfA pipeline. This indicates that the amplitudes of the Oxford SS sample are suppressed, primarily after the PCA. Though the PCA may also suppress amplitudes in the Oxford SAP sample, it is more evident in the SS sample due to the high noise amplitude of the SS PCs compared to the SAP PCs. The amplitude suppression could also indicate that the Oxford pipeline removes some intrinsic variability in addition to the systematics. However, it seems that, despite the suppression, the signals remain intact enough to be detected at a similar rate, just at lower amplitudes.

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In both pipelines, the hot star sample has the highest overall completeness and reliability. This is unsurprising, given its smaller sample size and lack of diversity in terms of variability. Due to the general absence of rotational modulation and other competing signals in the original light curves, the hot stars should also better preserve the injected sinusoids. Finally, the average brightness of the hot star data set is K_p = 11.0, while the combined average of the SAP and SS data sets is K_p = 15.3, and the number of valid detections generally decreases with increasing magnitude.

The Oxford SS completeness is slightly lower than the Oxford SAP at short periods and large amplitudes, but it is better for periods ≥15 days and amplitudes below about 0.50%. In addition, the SS sample is more reliable for both the longest and shortest detected periods. These differences are likely due to the separate light curve extraction methods in the Oxford pipeline. The SAP sample is extracted with pixelated apertures, which may not be optimal for a crowded field, even in the outer regions of the cluster. The SS sample, however, uses deblended, circular apertures, making it better able to deal with crowding.

On the other hand, the CfA SAP completeness and reliability are quite comparable (or slightly higher) across most bins when compared against the CfA SS sample. The slight advantage to the SAP sample is probably due to less crowding, as the two data sets are extracted and processed the same way in the CfA pipeline. In addition, the larger size of the SAP sample means that there is a larger matrix from which to characterize and correct the common-mode trends via PDC–MAP (or PCA, in the case of the Oxford pipeline). This could help to partially explain the generally higher completeness values from the SAP sample in either pipeline, as well as the higher reliability in the case of the CfA pipeline.

With respect to completeness, the CfA pipeline seems to perform slightly better than the Oxford pipeline, particularly in the hot star and SAP samples. Thus, Figure 19 provides the average completeness (and associated reliability in smaller text) from the CfA SAP and SS samples, as a summary of the best we can do with the K2 Campaign 5 M67 data, with the solar case highlighted in red. The differences between the Oxford and CfA pipelines are rooted in the approaches each takes to produce corrected light curves from raw K2 data. The moving apertures and deblending procedures during light curve preparation give the Oxford pipeline an edge in certain cases, particularly with the SS sample, but the larger, more optimized aperture selection from the CfA pipeline is superior elsewhere. While the Oxford pipeline typically better removes systematics from strongly variable stars, most M67 targets won't be sufficiently variable on short timescales, so the advantage is minimal. Finally, the more sophisticated PDC–MAP outperforms the relatively crude PCA of the Oxford pipeline in the common-mode systematic removal. While an ideal pipeline would combine the best elements of the two, the performance of the existing pipelines is comparable, and both are valid options regarding the analysis of K2 light curves. Most critically, however, is that for both pipelines, the completeness around the solar case in the SAP and SS samples is ∼15% at best, illustrating that it is very difficult to detect 0.1% amplitude, 25 days signals in K2 Campaign 5 M67 data.

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The injection test results indicate that, if the Sun is considered typical, measuring rotation periods for solar-like stars in M67 based on their K2 light curves is challenging. Specifically, for the best-case scenario of solar-like rotational variability—0.1% amplitude, 25 days non-evolving sinusoidal signal—our completeness maxes out at 14% and 15% for the SAP and SS samples, respectively. Considering that detecting non-sinusoidal, evolving spot-modulation will be more difficult, these results suggest that both current and future rotation period measurements in M67, based solely on the K2 Campaign 5 data, must be carefully examined before they are used to guide models of stellar angular momentum evolution and/or calibrate gyrochronology relations.

We consider the 20 stars with measured rotation periods from Barnes et al. (2016) and the stars from Gonzalez (2016a) where we had detections from either the CfA or Oxford pipeline. Recall that all of these light curves come from the SAP sample. Because we do not have amplitude measurements for the versions of the light curves used in either paper, we estimated them by binning the flux of each light curve based on the reported period and taking the median of the difference between the 95th and 5th percentiles of each bin. We did this for both the Oxford and CfA versions of each light curve and averaged the values. We then estimated the completeness and reliability based on the corresponding SAP results from this study. We averaged the Oxford and CfA completeness for the injected period and amplitude bins closest to the determined values. If the calculated amplitude was halfway between the injected amplitudes, we used the higher amplitude bin. We also used the average Oxford and CfA reliability from the appropriately ranged bins into which the reported periods and estimated amplitudes fell.

The results are shown in Tables 13 and 14. The completeness for the reported periods from Barnes et al. (2016) is low, averaging at about 27%, with only one period falling into a range above 50% (EPIC 211394185). The corresponding reliability is high, only dipping below 95% in one instance (EPIC 211397512). However, these values are a bit misleading, as they only really apply to those cases where we would actually get a detection according to our criteria, i.e., using a threshold factor of C = 4. If we have a detection in an area with low completeness, then the result is generally trustworthy. Therefore, in Table 13, we also show the period from the normalized Lomb–Scargle for the corresponding, non-injected light curves from the Oxford and CfA pipelines where we had a detection based on our threshold (i.e., where the star was classified as a "variable"). For this sample of stars, the Oxford pipeline only has three detections, while the CfA has four. Almost all of these detections appear to be rough harmonics of the published result from Barnes et al. (2016), while the CfA period for EPIC 211423010 is close to the Barnes value.

Both the Oxford and CfA pipelines have detections for EPICs 211394185 and 211411621 from the Barnes sample. Figures 20 and 21 show the Oxford and CfA versions of these two light curves, respectively, along with the original periodograms in the middle panels and the normalized versions in the bottom panels. While obviously variable, a clear period is difficult to find by eye in either case, though it appears that the normalization suppressed any power at a period that would be comparable to the Barnes result. The normalization step is important for avoiding false positives, however, as we have illustrated with the presence of long-term trends in the median periodogram of the hot star samples. We recognize that Barnes et al. (2016) used several different period detection methods other than the Lomb–Scargle periodogram. However, the concern here is obviously the underlying signal. The lack of detections from the Oxford and CfA pipelines, as well as the lack of agreement with the Barnes result where we do have one, show just how challenging it is to measure solar-like signals in the K2 Campaign 5 data.

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Looking at Table 14, we have a list of 18 EPICs from either the "PDCSAP" or "K2SC" samples given in Gonzalez (2016a) where we had a detection from at least one of our pipelines. The average completeness is ∼60%, which is higher than the Barnes sample, and the reliability is ≥95% (except in one case, EPIC 211430648), but the stars listed here are only a small fraction of those published by Gonzalez (2016a). The PDCSAP sample in total has 98 reported periods; we detected 16 from this sample. The K2SC sample had 40 periods; we detected 9. In addition, notice that none of the stars in Table 13 show up in 14, meaning that neither the Oxford nor the CfA pipeline records a detection of the nine stars reported by Gonzalez (2016a) that overlap with Barnes et al. (2016). However, as with the Barnes detections, the CfA and Oxford results agree, and they are either in agreement with the Gonzalez value, or they are rough harmonics. The harmonics again highlight ambiguity in the light curves, while the general lack of detections from the Gonzalez sample—especially where reported from Barnes—reinforces the conclusions we have already made about finding solar signals in M67 with K2 data.

We can use the completeness and reliability statistics from this work to guide future re-evaluations of M67 periods from K2 data. Reliability helps establish amplitude and period thresholds for detecting true signals in M67. We can probably trust detections in a measured period and amplitude bin with reliability of 80% or higher, suggesting that one or fewer out of every five measurements is likely to be unreliable. However, we need to take completeness into account: we simply cannot say anything about the true number of stars with periods and amplitudes corresponding to a given bin without it, because completeness tells us what fraction of potential detections we miss. For example, if we have three detections in a bin where the completeness was 50%, we know that there were approximately three other stars in that range that we missed.

It is important to recall that these sinusoidal injection tests represent a "best case" scenario: real rotation signals are non-sinusoidal and evolve over time, and both of these factors are likely to affect their detectability in a negative way. Therefore, although it is of course possible that better light curve extraction, detrending, and period search methods might be devised than those we have used here, it seems rather unlikely that the detectability of true rotation signals will improve much beyond what we have shown. In short, the K2 Campaign 5 data may allow the measurement of some rotation periods for main sequence stars in M67, but only if they are relatively active and display amplitudes somewhat larger than the active Sun.

To see the effect of using the regular Lomb–Scargle periodogram (i.e., without normalization) on the completeness and reliability for the K2 Campaign 5 M67 light curves, we computed the periods for the Oxford hot star and SAP injected data sets without using the normalized power. For a detection threshold, we followed the example of Nielsen et al. (2013) and set the threshold at four times the root mean square (rms) of the zero-mean time series. We removed the same stars marked as variables as in the main set of injection tests. Figure 22 shows the completeness and reliability results for the solar amplitude (0.1% for completeness, 0.07–0.25% for reliability), along with the number of detections with measured amplitudes ranging from 0.07% to 0.25% (i.e., similar to the numbers in brackets in Tables 7 through 12). The blue lines show the results with normalization, while the red lines show the results without normalization. Due to the lower detection threshold criteria and the lingering presence of the long-term trends seen in the median periodograms, the number of detections for the un-normalized samples increases significantly, compared to the normalized hot star and SAP samples. This allows for an increase in completeness, including at the solar case, which improved from 17% with the normalized periodograms to 39% for the regular Lomb–Scargle with the hot stars, and from 2% to 17% for the SAP sample. In addition, completeness begins to increase with increasing period within an injected amplitude, starting around 25 days, countering the general trends of the normalized sample. However, as expected, the reliability at periods of 25 days and longer decreases drastically. There is also a disproportionate number of detections at these longer periods, showing just how much of an effect the lingering long-term systematic features have on periodic measurements.

Table 15 shows the minimum detection threshold factors for the periods found by the normalized Lomb–Scargle periodogram to count as detections using the light curves from the Oxford and CfA pipelines for the stars in Barnes et al. (2016). We present the EPICs from Barnes et al. (2016), the published periods, the periods computed from the normalized periodograms of the corresponding non-injected Oxford and CfA pipeline light curves, the threshold factor C required to count the Oxford and CfA periods as detections, and the associated reliabilities. In most cases, the value of C had to be reduced to one or two in order for the measured period from the Oxford and CfA pipelines to be counted as a detection, which means the reliability falls considerably; the average Oxford reliability is 65%, and the average CfA reliability is 61%. In addition, the Oxford and CfA periods rarely match the published values (though in some cases they are harmonics), and they do not always match each other. This further illustrates how difficult it is to find accurate rotation periods in the K2 Campaign 5 M67 light curves.