curve leave their imprints in the mean flux of a slice. Furthermore, the
individual _slices_ have a different degree of coverage in $x^{\prime}$. This
complicates a polynomial fit to the data. We need an additional technical step
to proceed.

Figure 26:  Position dependence of the recorded raw flux
$f_{\text{raw}}(x^{\prime})$ and the correction thereof. Panel (a) shows the
raw flux from _Gaia DR3 604895948360165888_ as a function of the detector
coordinate and clearly shows the linear dependence. Flux values are plotted
for the same _slices_ as in Fig. 25. Each line represents one _slice_. Panel
(b) shows the same _slices_ but with the flux after the correction is applied.
Panels (c) + (d) are similar to (a) + (b), only for the full light curve of
_Gaia DR3 604895948360165888_ and the color coding representing the time as in
Fig. 20. Panels (e) + (f) are the same as (c) + (d) but for _Gaia DR3
604917629355039360_ for which the position dependency becomes nonlinear.

We designate the one _slice_ which has the most extensive coverage in
$x^{\prime}$ in each segment as the _prime slice_. Each _slice_ is then
rescaled to this prime slice. We proceed to fit a 5th-order polynomial
$p_{5}(x^{\prime})$ to the resulting distribution of fluxes, and apply it as a
correction to the individual _slice_ (their unscaled fluxes!) as

$f_{\text{cor1,slice}}(x^{\prime})=f_{\text{raw,slice}}(x^{\prime})/p_{5}(x^{\prime})\cdot\overline{f_{\text{raw,seg}}(x^{\prime})}$

to create the first step in the correction process $f_{\text{cor1}}$. We
multiply with the average flux of segment
$\overline{f_{\text{raw,seg}}(x^{\prime})}$ because the polynomial normalizes
the fluxes to the _segment_ mean. The result can be seen for a few selected
slices in Fig. 26. This processes is carried out for all _slices_ in a
_segment_ and for all _segments_. We note that this process conserves long
term flux changes and differences between individual _slices_. The resulting
light curve for our example star _Gaia DR3 604895948360165888_ is shown in
Fig. 27.

Figure 27:  Light curve of _Gaia DR3 604895948360165888_ after the application
of the first correction for the instrumental systematics. Both panels show the
same light curve. The color coding in the upper panel indicates the block as
in Fig. 23. In the lower panel, the colors indicate the segments as in Fig 22.

This process reveals that two regions of the light curve cannot be processed
in this way. The first $\approx 3\,$d and a central part ($t\approx 172$ – 177
d) correspond to exceedingly long _slices_ that exhibit behaviors that cannot
be found anywhere else and therefore those regions cannot be corrected this
way (cf. the same regions in Fig. 22). We mask those regions and cut them from
the final light curves. Furthermore, there are individual cadences in the
light curve from times of fast telescope motion (thruster firing). Those
cannot be processed as well. We mask them in the correction step and later,
when all other _segments_ are processed, recreate those points from a linear
interpolation in the corrected light curve. Those regions are entirely part of
the _blocks_ D and E (cf. the comment column Tab. 5).

Thanks to the processing described above, the relative fluxes between the
individual slices is conserved and the (fragmented) light curve of a _segment_
may act as a valid light curve all by itself. However, this is not true when
we compare the fluxes of the individual segments. They are not aligned. This
is, however, not a result of our processing but a consequence of the
instrumental systematics and different pixel masks that gives each segment a
slightly different recorded average flux (cf. Fig. 24). To align those fluxes,
we merge the segments successively with a scaling to the chronologically first
sector. The scaling factor is determined by minimizing the average standard
deviation in the merged light curve for a series of windows 15 cadences wide
along the light curve. This rescaling provides us with the next level of
correction $f_{\text{cor2}}$ which is shown for example star in Fig. 28. For
the chronologically first segment we set $f_{\text{cor2}}=f_{\text{cor1}}$.

Figure 28:  Light curve of _Gaia DR3 604895948360165888_ after the application
of the second correction, the realignment of the segments. Both panels show
the same light curve. The color coding in the upper panel indicates the
_block_. In the lower panel, the colors indicate the segments as in Fig 22. We
emphasize that the flux axis is identical between the Figs. 24, 27, and this
one.

The corrected and realigned flux $f_{\text{cor2}}$ now represents the light
curve in which we have removed all instrumental systematics. However, this is
not fully true for all light curves. A polynomial fit can only do so much and
an insufficient pixelmask may introduce effects that cannot be corrected. As
such, the process fails for stars in very crowded regions and on the edges of
the superstamp. The slices generally cover a few hours. Intrinsic variations
that occur on the same timescale may be misidentified by the polynomial.
However, such rapidly varying targets are not of interest to us here, allowing
us to ignore this problem. Despite this, we note that rapid signals are
generally still visible reasonably well in the final light curves, thanks to
their large amplitudes.

###  B.6 Cleaning the data

From this point onward, there is no further separation into blocks, segments,
or slices, and the light curves are always treated as a whole. In the next
step we apply $\sigma$-clipping to the data to clean it of outliers. This is
done only at this stage, because the instrumental systematics may create or
hide actual flux outliers and a $\sigma$ estimate is dominated by it. For each
point along the light curve we calculate the mean and standard deviation in a
1 d window around the point and replace it with a linear interpolation between
its neighbors if it exceeds a $3\sigma$ deviation from the mean. We note that
this may create artifacts for fast transients like transits or flares.
However, since we are not interested in such phenomena, we elect to ignore
those issues.

At this point we have constructed light curves for target stars that are
corrected for instrumental systematics and cleaned of outliers. The light
curves exhibit a gap of a few days around the middle and still include long
term trending effects. The removal of the latter will be performed using
Principal Component Analysis (PCA).

###  B.7 Principal Component Analysis

PCA is performed in a very similar manner to the light curve processing for
Ruprecht 147 (from _K2_ C07, see Gruner & Barnes 2020, Appendix B). However,
we modify the process slightly and describe the deviations below.

Because we do not have a large number of ($\sim 10$k) light curves from the
entire campaign for this dataset, we select stars for the PCA basis from our
processed sample. We identify 769 stars for the said basis, covering a large
range of brightnesses, colors, and locations on the detector (cf. Fig. 29).
From this basis, it becomes obvious that the trending signal is not universal
for all stars. It shows a strong brightness dependence and a weak location
dependence. We therefore limit the PCA basis applied for each target to stars
of similar brightness and location. This means we take only the 125 nearest
stars on the FFI within $\Delta G\pm 1$ mag. Both of these choices are the
result of testing different parameter combinations; these particular ones
provided the best results throughout the sample. We note that, depending on
the brightness, there could be fewer than 125 stars overall in the brightness
range and the PCA basis for such stars is correspondingly smaller.

Figure 29:  Star selection for the Principal Component Analysis, indicating
those used and those not used.

We smooth the light curves before entering them into the PCA to remove the
impact of the noise from the analysis. This smoothing is done by replacing
each flux value by the mean of a 1 d wide window around this point. We note
that this smoothing is only in place to create the PCA correction and is not
used beyond that. Furthermore, we normalize all light curves to their
respective means before entering them into the PCA. This smoothing removes
rapid-trending signals from the PCA input. This means that such signals are
not accounted for in the PCA correction and thus remain in the final light
curve. However, their impact is small with respect to our purposes and as such
we do not address them any further. Traces of this can be seen in the light
curve plots in Fig. 39 and are generally of two different types:

  1. 1.

A very rapid fluctuation, as seen for instance, between $t=150$ d and $t=160$
d for _Gaia DR3 604914880575997056_ and

  2. 2.

A distinct short term signal, as seen for instance, between $t=150$ d and
$t=170$ d for _Gaia DR3 604915773929008384_.

The (long-term) trending signals are dominant and the PCA can identify their
different shapes relatively easily. We use only two components for the
reconstruction; this is enough to remove the trending signals adequately. This
is a consequence of the limited PCA basis (brightness and position), which
preremoves very different trending signals from the basis. Figure 30 shows the
application of the PCA correction on the example star and on an essentially
constant star.

Figure 30:  Application of the PCA correction to the example star _Gaia DR3
604895948360165888_ (panels a and b), and the fainter and essentially constant
star _Gaia DR3 604909310002693632_ (panels c and d). Panels (a) and (c) each
show the result of the instrumental systematics correction (blue) and the PCA
correction (red) whereas panels (b) and (d) show the final light curve after
the PCA correction was applied.

At this point we are in possession of the final light curves $f_{\text{fin}}$
which are corrected for instrumental systematics, cleaned for outliers, and
are free of trending. Figure 4 summarizes the individual steps for the example
star. This process is by no means perfect, and does not work for all targets.
We only use a carefully selected subsample of all potential targets, and
remove all light curves where residuals of the systematics or trending cause
uncertainties. Most removed light curves suffer from an insufficient
pixelmasking (mostly due to being in a crowded area in the superstamp FFI) or
because the corresponding star coincides with a region of bad pixels. Both
problems make corrections nearly impossible. Only for a few stars does the
correction itself directly fail. For those, the flux does not behave well
enough to be adequately modeled with a polynomial. However, their numbers are
small enough not to merit the effort required for an adjusted correction
procedure.

The above-described process is specifically designed for the _K2_ C05
superstamp around M67. However, it should be possible to adapt it for other
parts of the _K2_ survey as well. The principal idea should hold and, assuming
that the detector variability stays similar in its magnitude, provide good
results without modifying the core components. The parts that would need to be
modified are those that are fine-tuned to the superstamp. The pixelmasks for
the individual targets are the first among these. Secondly, the masking
(block, segment, slice) would also need to be adjusted to a different
campaign. However, we can reasonably assume that our masks work for other
regions of C05 without modification. Caveats such as the difficulties with
rapidly varying stars and crowded regions do, of course, remain. A
precondition for such work is the existence of a reference for the telescope
motion, and with that of motion of the stars on the detector. The _K2_ TPFs do
not provide this, but some of the other superstamps created by Cody et al.
(2018) do. We note that the SPICE kernels for the _K2_ mission only include
position and velocity of the telescope but not its orientation, a necessity to
be able to leverage those products.

###  B.8 Performance of the reduction

The correction process described above works well for stars that are isolated
and whose PSF does not exceed the FFI range, that is, stars for which we can
define a pixelmask that includes all of the stellar flux and none from
surrounding sources. This condition is not fulfilled on the edges of the
superstamp, and especially in the central region of the cluster. Figure 31
highlights some details that show the performance and limitations of our
corrections.

Figure 31:  Example light curves illustrating the capabilities and limitations
of our correction process. Panel (a) shows the light curve of an eclipsing
binary with a significant primary eclipse and a secondary eclipse that only
becomes really apparent after the correction was carried out. Panel (b) shows
a star where the instrumental systematics completely obscure the intrinsic
signal despite their being of comparable amplitude. Panel (c) shows the light
curve of a star located at the edge of the superstamp and which suffers from
artifacts created by the correction process as a consequence. Each panel
includes highlighted regions to show the details of the relevant effects.

Whenever the pixelmask cannot, for whatever reason, be created in a way that
includes all the stellar flux the recorded raw flux $f_{\text{raw}}$ includes
a second systematic effect. This effect invalidates the original assumption we
had to make, that the raw flux only changes because of variations in detector
sensitivity, and is otherwise somewhat constant on short time scales. With
that, it very much depends on the individual star’s location on the detector
whether our correction still works or whether we under- or over-correct the
apparent systematics. The star we used as an example for the correction
process is located on the edge of the superstamp (c.f. Fig. 23) but is well-
behaved enough to allow a good correction process. Meanwhile, _Gaia DR3
604896429396491136_ , whose light curve is depicted in panel (c) of Fig. 31,
shows remnants of an over-correction in the highlighted area due to proximity
to the superstamp edges and subsequent loss of flux.

###  B.9 Comparing our results with K2SC, SAP, EVEREST

We validate our light curves by comparing them with those produced as part of
the original survey and from two commonly used works that implemented a
systemtatics correction, k2sc, and everest. This comparison again highlights
one important fact that supports our approach with creating our own light
curves – not many light curves are available for targets within the
superstamp. Figure 32 shows the distribution of stars with available light
curves.

Figure 32:  Spatial distribution of stars with available light curves in the
archives for different data products. Panel (a) shows a map centered on M67
with the extent of the superstamp indicated (black box). Targets with archived
light curves within (red and blue) and outside (black) of superstamp are
overplotted. Panel (b) shows the same stars in a CMD.

Only 96 are within the field of view of the superstamp, as compared with GDR3
listing $\sim$2000 sources. This most likely originates from the differing
nature of the superstamp as compared with the normal TPFs. The latter are
typically associated with a proposed star and therefore are automatically
processed. This is not true for the superstamp. As far as we are aware, all
other projects that dealt with the _K2_ systematics only operate on the same
sample. Therefore, other extant work does not venture outside this 96-star
sample.

We extracted the 96 light curves in the FOV of the superstamp from the
_Kepler_ archive181818archive.stsci.edu/k2/data_products.html. These include
the light curves based on simple aperture photometry (SAP, $f_{\text{SAP}}$)
on the TPFs and corrected light curves that resulted from the Presearch Data
Conditioning (PDC, $f_{\text{PDCSAP}}$) module of the _Kepler_ pipeline (Smith
et al. 2020). The PDC is employed to remove data systematics and trending
while retaining astrophysical signals. This generally works reasonably well
but leaves traces of both artifacts. In some cases the trend correction
removes astrophysical signal. For our comparison we employ the SAP flux as
well as the PDCSAP with the aim of comparing the most similar products from
each. For the former we compare with our raw extracted flux. However, we
generally do not expect a large agreement in the shape of the raw light curves
on short term scales due to the differently shaped aperture mask used. Long
term effects are expected to be the same. The PDCSAP flux $f_{\text{PDCSAP}}$
is compared with our final light curve $f_{\text{fin}}$ (lower panel of Fig.
33) as both have the same level of processing (systematics and trend removal),
whereas the SAP flux $f_{\text{SAP}}$ is equivalent to our raw flux
$f_{\text{raw}}$ (upper panel of Fig. 33).

Figure 33:  Comparison between the _Kepler_ PDCSAP light curves and the ones
created by us. In both panels, our light curves are shown in blue and the
archival light curves are shown in red. Panel (a) shows the uncorrected
aperture photometry fluxes (flux_raw from ours and SAP_FLUX for theirs). Panel
(b) shows the final light curve product – flux_fin from ours and PDCSAP_FLUX
for theirs.

We can see that, as expected, the raw light curves generally do not agree with
respect to the short-term systematics while having the same long-term
behavior. However, after the correction is applied, we find very good
agreement between the _Kepler_ and our pipeline results. Despite our
relatively simple approach for an empirical correction, we not only do match
the quality of the official product but partially exceed it. Generally, our
approach is slightly better in the removal of the _instrumental systematics_.

The _K2 Systematics Correction_ (k2sc) pipeline implements Gaussian processes
to correct for the telescope jitter (Aigrain et al. 2015, 2016). It is geared
toward exoplanet detection; as such the pipeline is set up in a way that also
removes the astrophysical signal from the final product $f_{\text{k2sc}}$.
However, they provide the removed systematics as part of their data product,
differentiated in a position-dependent $f_{\text{posi}}$ and a time-dependent
part $f_{\text{time}}$. Long term variability is retained in the time-
dependent part, together with trending. Recombining their final light curve
with both those parts yields back the PDCSAP light curve
($f_{\text{PDCSAP}}=f_{\text{k2sc}}\cdot f_{\text{time}}\cdot
f_{\text{posi}}$) which was the starting point for the k2sc pipeline. Thus,
the k2sc light curves benefit from the long-term trend correction in the
_Kepler_ PDC pipeline. Technically, with their procedure they also implement a
correction for instrumental systematics twice. To obtain a light curve that is
corrected to a similar degree as our final product, we recombine the final
flux and the time-dependent trending part. Figure 34 shows the comparison. The
additional systematics removal pays off, as the light curves are improved as
compared with PDCSAP, and reach a fidelity similar to, or perhaps are even
superior to ours. All 96 PDCSAP light curves are processed with k2sc. The
final k2sc light curve ($f_{\text{k2sc}}$) is not comparable to any other, as
it blatantly removes any long term signal, including physical ones (cf. upper
panel of Fig. 34). A comparison makes sense between our light curves before
the PCA ($f_{\text{cor2}}$) and the k2sc one with the time-dependent signal
included ($f_{\text{k2sc}}\cdot f_{\text{time}}$, cf. lower panel of Fig. 34).

Figure 34:  Same as Fig. 33 but for the comparison between the k2sc light
curve of _Gaia DR3 604917491916095872_ and the ones created by us.

The _EPIC Variability Extraction and Removal for Exoplanet Science Targets_
(everest) light curves (Luger et al. 2016, 2018) use a pixel-level
decorrelation (PLD) method that operates on pixel level light curves and,
since its update to v2.0, includes the telescope motion for the correction.
They are, as far as we know, the best available light curves based on _K2_
data and the study of variability longer than a few days. Only long-term
trending is a remaining issue; such trends are generally more pronounced than
in other light curves. However, with those retained, there is also no problem
with accidentally removed astrophysical signal. We have worked successfully
with them in the past (Gruner & Barnes 2020). However, they are even more
limited in their availability regarding the superstamp region. Only three
stars in the FOV of the superstamp have an everest light curve. And all three
of those stars are very faint. Two are white dwarfs, while the third is a very
faint red dwarf at the brightness limit of _Gaia_ (cf. panel (d) of Fig. 32).
All three stars are nearly constant in their light curves, and dominated by
noise and some weak long-term trending. We use the red dwarf for the
comparison in Fig. 35. The extent to which we can correct for instrumental
systematics is similar in both products, however, everest reaches a slightly
higher photometric precision. PDCSAP and k2sc fail to fully remove either
trends or systematics, and their photometric precision is lower compared with
ours or everest for the faint stars.

Figure 35:  Same as Fig. 33 but for the comparison between the everest,
PDCSAP, and k2sc light curves of _Gaia DR3 604898490980773888_ (panels a, b,
and c, respectively) and the ones created by us.

##  Appendix C Supplements

###  C.1 Supplementary tables

Table 6: Datapoints for the empirical cluster sequences.

$G-G_{\mathrm{RP}}$ | $G_{\mathrm{BP}}-G_{\mathrm{RP}}$ | $B-V$ | $V-K$ | $P_{\text{1}}$ | $P_{\text{2.5}}$ | $P_{\text{4}}$
---|---|---|---|---|---|---
[mag] | [mag] | [d] | [d] | [d] | [d] | [d]
0.32 | 0.50 | 0.38 | 0.99 | 1.5 | nan | nan
0.38 | 0.61 | 0.46 | 1.20 | 2.0 | 5.0 | nan
0.40 | 0.65 | 0.49 | 1.28 | 3.0 | 8.0 | 14.0
0.45 | 0.74 | 0.58 | 1.44 | 8.5 | 15.0 | 22.0
0.50 | 0.83 | 0.67 | 1.64 | 11.0 | 19.0 | 26.0
0.55 | 0.93 | 0.76 | 1.84 | 11.0 | 22.0 | 29.0
0.60 | 1.03 | 0.85 | 2.09 | 10.5 | 23.0 | 31.0
0.65 | 1.14 | 0.93 | 2.32 | 11.0 | nan | 31.0
0.70 | 1.25 | 1.02 | 2.57 | 12.0 | 23.0 | nan
0.75 | 1.37 | 1.11 | 2.83 | 12.5 | nan | 29.5
0.80 | 1.48 | 1.19 | 3.08 | 13.0 | 20.2 | 26.0
0.85 | 1.61 | 1.27 | 3.32 | 13.8 | nan | nan
0.90 | 1.74 | 1.33 | 3.55 | 14.3 | 18.0 | 25.5
0.95 | 1.87 | 1.37 | 3.79 | 15.0 | nan | nan
1.00 | 2.02 | 1.39 | 4.03 | nan | 22.0 | 32.0
1.10 | 2.33 | 1.44 | 4.55 | nan | 30.0 | 41.0
191919The manually drawn sequences used in the CPDs throughout this work are
plotted from cubic interpolation between the listed colors and periods. The
subscript to the period label indicates the corresponding age in Gyr.

Table 7: Results of the period analysis.

Gaia DR3 | EPIC | $P$ | $P_{\text{err}}$ | $G$ | $(G-G_{\mathrm{RP}})_{0}$ | $(G_{\mathrm{BP}}-G_{\mathrm{RP}})_{0}$ | $(B-V)_{0}$ | $(V-K)_{0}$ | HPS
---|---|---|---|---|---|---|---|---|---
|  | [d] | [d] | [mag] | [mag] | [mag] | [mag] | [mag] |
598903678707639296 | 211398541 | $32.4$ | 1.3 | 15.93 | 0.65 | 1.16 | 0.94 | 2.27 |
604896498115959296 |  | $23.5$ | 1.5 | 17.50 | 0.88 | 1.73 | 1.43 | 3.27 |
604896635554924672 | 211397512 | $32.5$ | 1.4 | 15.87 | 0.65 | 1.15 | 0.96 | 2.26 |
604897558972113024 | 211400106 | $29.1$ | 2.6 | 14.75 | 0.55 | 0.93 | 0.69 | 1.86 |
604897833850019328 | 211400500 | $24.3$ | 1.6 | 14.37 | 0.48 | 0.79 | 0.58 | 1.50 |
604898490980772352 |  | $27.4$ | 2.1 | 16.61 | 0.79 | 1.48 | 1.20 | 2.93 |
604899831010539904 |  | $18.1$ | 0.4 | 13.29 | 0.42 | 0.69 | 0.42 | 1.42 | x
604900071528704128 |  | $34.7$ | 2.5 | 16.18 | 0.73 | 1.30 | 1.04 | 2.61 |
604900651348634240 |  | $23.1$ | 2.1 | 17.60 | 0.88 | 1.73 | 1.38 | 3.56 |
604903331408222208 |  | $29.5$ | 1.6 | 18.46 | 1.02 | 2.12 | 1.50 | 3.84 |
604903438783070208 |  | $37.0$ | 3.0 | 18.64 | 1.02 | 2.19 | 1.61 | 4.17 |
604906840397139584 |  | $29.1$ | 1.1 | 17.76 | 0.95 | 1.92 | 1.47 | 3.80 | x
604907046555568896 |  | $27.2$ | 1.4 | 16.84 | 0.79 | 1.50 | 1.27 | 2.93 |
604908004332577152 | 211400002 | $22.9$ | 2.2 | 14.31 | 0.47 | 0.77 | 0.55 | 1.52 |
604909206923484160 |  | $27.6$ | 1.1 | 17.23 | 0.83 | 1.62 | 1.30 | 3.31 | x
604909756679296640 |  | $30.2$ | 2.2 | 15.65 | 0.61 | 1.06 | 0.90 | 2.17 |
604911204083987584 |  | $7.37$ | 0.14 | 15.08 | 0.55 | 0.92 | 0.78 | 1.89 |
604913540546033024 |  | $31.1$ | 1.5 | 16.31 | 0.70 | 1.30 | 1.05 | 2.45 |
604913952863073920 |  | $26.5$ | 2.2 | 16.82 | 0.80 | 1.52 | 1.24 | 3.05 |
604914880575997056 |  | $29.1$ | 1.8 | 16.37 | 0.72 | 1.32 | 1.08 | 2.60 |
604915773929008384 |  | $29.0$ | 0.5 | 15.16 | 0.55 | 0.94 | 0.77 | 1.84 |
604916117526551680 | 211405832 | $23.4$ | 1.5 | 14.42 | 0.48 | 0.79 | 0.55 | 1.62 |
604917148318674816 |  | $30.8$ | 2.7 | 15.49 | 0.60 | 1.04 | 0.95 | 1.98 |
604919587860083328 |  | $29.5$ | 2.6 | 17.42 | 0.88 | 1.74 | 1.39 | 3.34 |
604920549932807296 |  | $24.6$ | 1.4 | 17.33 | 0.87 | 1.70 | nan | nan |
604920549932809344 |  | $26.1$ | 1.6 | 15.16 | 0.56 | 0.95 | 0.81 | 1.83 |
604922130480588544 |  | $27.2$ | 2.0 | 16.73 | 0.77 | 1.44 | 1.21 | 2.89 |
604922229264424448 |  | $9.72$ | 0.25 | 17.07 | 0.84 | 1.61 | 1.27 | 3.19 |
604922817675316096 |  | $27.6$ | 3.2 | 14.97 | 0.54 | 0.91 | 0.77 | 1.71 |
604923333071376512 |  | $32.1$ | 2.5 | 15.66 | 0.62 | 1.09 | 0.94 | 2.22 |
604923848467470976 |  | $26.2$ | 2.3 | 17.37 | 0.87 | 1.69 | 1.40 | 3.44 |
604930681760054656 |  | $27.6$ | 2.3 | 17.73 | 0.92 | 1.85 | nan | nan |
604943674036665472 | 211418998 | $26.8$ | 2.6 | 14.60 | 0.51 | 0.85 | 0.49 | 1.82 |
604944120713062784 |  | $31.7$ | 2.7 | 15.47 | 0.60 | 1.04 | 0.83 | 2.15 |
604963362166649856 |  | $22.8$ | 1.4 | 16.78 | 0.80 | 1.52 | 1.14 | 3.07 |
604969061592133376 |  | $1.84$ | 0.01 | 19.57 | 1.11 | 2.52 | nan | nan |
604969237681908480 |  | $26.3$ | 2.0 | 16.78 | 0.78 | 1.46 | 1.13 | 2.96 |
604969267746267520 |  | $26.5$ | 1.2 | 17.24 | 0.85 | 1.65 | 1.22 | 3.34 |
604969267746269696 |  | $30.7$ | 2.2 | 15.56 | 0.61 | 1.07 | 0.89 | 2.13 |
604969306401373824 |  | $30.2$ | 1.1 | 15.87 | 0.65 | 1.15 | 0.96 | 2.33 |
604969306401373952 |  | $24.6$ | 1.7 | 16.93 | 0.81 | 1.54 | 1.26 | 3.08 |
604969336465748352 |  | $24.9$ | 1.7 | 17.28 | 0.87 | 1.69 | 1.43 | 3.36 |
604970131035099008 |  | $24.1$ | 2.3 | 16.98 | 0.81 | 1.57 | 1.18 | 3.12 |
604971466769552128 |  | $27.9$ | 2.0 | 16.51 | 0.75 | 1.40 | 1.06 | 2.94 |
604972467498966400 |  | $32.3$ | 1.5 | 15.53 | 0.60 | 1.04 | 0.97 | 1.92 |
604973979325779328 |  | $28.7$ | 1.1 | 14.88 | 0.53 | 0.90 | 0.71 | 1.76 | x
604974151124482944 |  | $31.6$ | 2.9 | 15.81 | 0.64 | 1.13 | 0.88 | 2.23 |

202020For convenience, we have limited the sample shown here to data used to
create Fig. 9. _HPS_ indicates whether are star is found on the half-period
sequence. A star indicated as such is listed here with its period doubled,
that being assumed to be the actual period. EPIC ids in italics denote stars
for which a PDCSAP light curve is available.

Table 8: Selected stars from the Gonzalez (2016b, a) sample.

Gaia DR3 | $P$ | $P_{\text{err}}$ | $(G-G_{\mathrm{RP}})_{0}$ | binary
---|---|---|---|---
| [d] | [d] | [mag] |
604907454576711040 | $27.2$ | 0.5 | 0.42 |
598689411379091328 | $24.9$ | 0.5 | 0.42 |
605000024007420416 | $12.1$ | 0.5 | 0.43 | PB
598692675554477056 | $22.6$ | 0.5 | 0.48 |
604960235430488960 | $26.8$ | 0.5 | 0.53 |
598901788922041728 | $32.4$ | 0.9 | 0.58 |
604949961868553856 | $32.6$ | 0.8 | 0.58 |
598689926775182208 | $29.4$ | 0.5 | 0.59 |
604895325589137920 | $31.5$ | 0.5 | 0.61 |
598902716634970240 | $30.4$ | 0.5 | 0.66 |
604987139105593344 | $28.9$ | 1.0 | 0.68 |
604898731498904704 | $29.0$ | 0.5 | 0.71 |
604964010706281856 | $32.4$ | 0.6 | 0.74 |
604901239759778176 | $26.3$ | 0.5 | 0.76 |
604965178937376512 | $29.0$ | 0.5 | 0.76 |
604704633336193152 | $30.8$ | 0.5 | 0.77 |
598955729416267264 | $27.3$ | 0.7 | 0.79 |
604895948360165888 | $12.4$ | 0.5 | 0.81 | PB
604946693397490816 | $28.8$ | 0.5 | 0.83 |
598889213257785984 | $29.5$ | 0.8 | 0.83 |
604899448757770624 | $13.5$ | 0.5 | 0.83 |
604894535315152512 | $28.6$ | 0.5 | 0.84 |
605000058367154176 | $13.7$ | 0.5 | 0.85 |
604901686436367232 | $25.6$ | 0.5 | 0.85 |
604980713834545664 | $15.6$ | 0.5 | 0.85 | PB
598903128951834496 | $29.2$ | 0.5 | 0.88 | PB
604930509961364352 | $26.8$ | 0.5 | 0.88 |
604930445537379712 | $23.3$ | 0.5 | 0.90 |
598956485330513536 | $24.7$ | 0.5 | 0.91 |
604966518966784256 | $21.3$ | 0.7 | 0.92 |
604713669947377024 | $22.0$ | 0.5 | 0.93 |
598901995079871744 | $24.0$ | 0.5 | 0.98 | PB
604926043198690176 | $23.9$ | 0.5 | 1.04 | PB
212121Only those stars from the Gonzalez (2016b, a) sample that have been
classified by us as having reliable periods and subsequently adopted in this
study.

###  C.2 The data provided

Table 7 gives an overview of the most relevant sample of the stars for which
we found periodic signals. As a supplement to this work we provide the extend
version containing all 136 stars with additional columns in machine-readable
form. The columns included are described in Table 9. Additionally, we provide
the light curves, including, raw fluxes, the intermediate steps, the PCA, and
all masking.

Table 9: Sample table overview.

Column | Unit | Description
---|---|---
`Star` |  | Running catalog number
`Sample` |  | =`Y` if in M 67 single MS star sample
`Gaia DR3` |  | Gaia DR3 identifier
`EPIC` |  | EPIC identifier
`2MASS` |  | 2MASS identifier
`period` | [d] | derived period
`Xmag` | [mag] |  $X$-band magnitude for
|  | $X\in\left\\{B,V,R,I,J,H,K,G,G_{\mathrm{BP}},G_{\mathrm{RP}}\right\\}$
`M67` |  | member `M`, candidate `C`, field star `F`
`M67_pmu` |  | proper motions cluster member
`M67_plx` |  | parallax cluster member
`M67_vrad` |  | radial velocity cluster member
`M67_phot` |  | photometric cluster member
`G_RP` | [mag] |  $G-G_{\mathrm{RP}}$ color
`G_RP_0` | [mag] |  $(G-G_{\mathrm{RP}})_{0}$ color
`BP_RP` | [mag] |  $G_{\mathrm{BP}}-G_{\mathrm{RP}}$ color
`BP_RP_0` | [mag] |  $(G_{\mathrm{BP}}-G_{\mathrm{RP}})_{0}$ color
`B_V` | [mag] |  $B-V$ color
`B_V_0` | [mag] |  $(B-V)_{0}$ color
`V_K` | [mag] |  $V-K$ color
`V_K_0` | [mag] |  $(V-K)_{0}$ color
`ra_icrs` | [∘] | GDR3 Right Ascension $\alpha$
`de_icrs` | [∘] | GDR3 Declination $\delta$
`plx` | [mas] | GDR3 parallax $\varpi$
`pmra` | [mas/yr] | GDR3 proper motion $\mu_{\text{RA}}$
`pmde` | [mas/yr] | GDR3 proper motion $\mu_{\text{Dec}}$
`vrad` | [km/s] | radial velocity $v_{\text{rad}}$
`otype` |  | Simbad object type
`sptype` |  | Simbad spectral type
`binary` |  | =`Y` if binary/multiple
`phot_bin` |  | =`Y` if photometric binary
`MS` |  | =`Y` if star is main sequence
`HPS` |  | =`Y` if star is half-period sequence
`DD` |  | =`Y` if star shows ¿1 spot
`x_stamp` | [pxl] | mean $x$ coordinate on the superstamp
`y_stamp` | [pxl] | mean $y$ coordinate on the superstamp
`kepler` |  | =`Y` if star has Kepler light curve
`k2sc` |  | =`Y` if star has k2sc light curve
`everest` |  | =`Y` if star has everest light curve
222222Measured and derived values such as photometry, astrometry, and the
rotation period have an additional column with a e_ prefix which denote the
associated errors. Those are not listed here separately.

###  C.3 Supplementary figures

Figure 36:  Same as Fig. 3 but for $B-V$ and $V-K$ colors.  Figure 37:  Color-
period diagram (CPD, panel a) and color-magnitude diagram (CMD, panel b) of
the complete superstamp sample. Blue symbols indicate the M 67 members as
discussed above. Overplotted in red are the field stars for which we found a
periodic signal. Note that the colors here are not reddening corrected. Open
symbols denote binaries.  Figure 38:  Same as Fig. 15 but for
$(G_{\mathrm{BP}}-G_{\mathrm{RP}})_{0}$, $(B-V)_{0}$, and $(V-K)_{0}$ colors.

###  C.4 Sample light curves

Figure 39:  Light curves for our sample stars. Each row in the figure is a
star, with the name indicated. The left panel shows the light curve, the right
panel their phase folded counterparts. The period used for the phase folding
is shown in the upper right. The light curve panel displays the _Gaia_ DR3 id
in the lower left corner. Additional labels indicate whether a star’s light
curve exhibits multiple spot features, i.e., is _double dipping_ (DD, upper
right corner in the left panel) or is a member of the _half-period sequence_
(HPS, lower left corner of the right panel). The orange line indicates a 0.01
binning. Note that double dipping refers to all stars whose light curves
exhibit signs of more than one spot. Stars are sorted as in Table 7, by their
_Gaia_ Id.  Figure 40: continued. Figure 41: continued. Figure 42: continued.
Figure 43: continued.