# The Factory and the Beehive. V. Chromospheric and Coronal Activity and Its
Dependence on Rotation in Praesepe and the Hyades

Alejandro Núñez NSF MPS-Ascend Postdoctoral Research Fellow  Department of
Astronomy, Columbia University, 550 West 120th Street, New York, NY 10027, USA
Marcel A. Agüeros Department of Astronomy, Columbia University, 550 West 120th
Street, New York, NY 10027, USA  Laboratoire d’astrophysique de Bordeaux,
Univ. Bordeaux, CNRS, B18N, Allée Geoffroy Saint-Hilaire, 33615 Pessac, France
Jason L. Curtis Department of Astronomy, Columbia University, 550 West 120th
Street, New York, NY 10027, USA  Kevin R. Covey Department of Physics &
Astronomy, Western Washington University, Bellingham, WA 98225, USA  Stephanie
T. Douglas Department of Physics, Lafayette College, 730 High St, Easton, PA
18042, USA  Sabine R. Chu Department of Astronomy, Columbia University, 550
West 120th Street, New York, NY 10027, USA  Stanislav DeLaurentiis Department
of Astronomy, Columbia University, 550 West 120th Street, New York, NY 10027,
USA  Minzhi (Luna) Wang  Department of Astronomy, Columbia University, 550
West 120th Street, New York, NY 10027, USA  Jeremy J. Drake Center for
Astrophysics $|$ Harvard $\&$ Smithsonian, 60 Garden St, Cambridge, MA 02138,
USA

###### Abstract

Low-mass ($\lesssim$1.2 ${M_{\odot}}$) main-sequence stars lose angular
momentum over time, leading to a decrease in their magnetic activity. The
details of this rotation–activity relation remain poorly understood, however.
Using observations of members of the $\approx$700 Myr-old Praesepe and Hyades
open clusters, we aim to characterize the rotation–activity relation for
different tracers of activity at this age. To complement published data, we
obtained new optical spectra for 250 Praesepe stars, new X-ray detections for
ten, and new rotation periods for 28. These numbers for Hyads are 131, 22, and
137, respectively. The latter increases the number of Hyads with periods by
50%. We used these data to measure the fractional $\mathrm{H}\alpha$ and X-ray
luminosities, $L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$ and
$L_{\mathrm{X}}$/$L_{\mathrm{bol}}$, and to calculate Rossby numbers
$R_{\mathrm{o}}$. We found that at $\approx$700 Myr almost all M dwarfs
exhibit $\mathrm{H}\alpha$ emission, with binaries having the same overall
color–$\mathrm{H}\alpha$ equivalent width distribution as single stars. In the
$R_{\mathrm{o}}$–$L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$ plane, unsaturated
single stars follow a power-law with index $\beta=-5.9$$\pm$$0.8$ for
$R_{\mathrm{o}}$ $>0.3$. In the
$R_{\mathrm{o}}$–$L_{\mathrm{X}}$/$L_{\mathrm{bol}}$ plane, we see evidence
for supersaturation for single stars with $R_{\mathrm{o}}$ $\lesssim$ 0.01,
following a power-law with index $\beta_{\mathrm{sup}}=0.5^{+0.2}_{-0.1}$,
supporting the hypothesis that the coronae of these stars are being
centrifugally stripped. We found that the critical $R_{\mathrm{o}}$ value at
which activity saturates is smaller for $L_{\mathrm{X}}$/$L_{\mathrm{bol}}$
than for $L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$. Finally, we observed an
almost 1:1 relation between $L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$ and
$L_{\mathrm{X}}$/$L_{\mathrm{bol}}$, suggesting that both the corona and the
chromosphere experience similar magnetic heating.

Galaxy: open clusters and associations: individual (Praesepe) – Galaxy: open
clusters and associations: individual (Hyades) – stars: activity – stars:
chromospheres – stars: coronae – stars: rotation – stars: evolution – stars:
late-type

††facilities: CDS, CXO, Gaia, Hiltner (OSMOS, Modspec), LAMOST, MMT
(Hectospec), PO:1.2m (ZTF), ROSAT, Sloan, Swift, TESS, XMM††software: astropy
(Astropy Collaboration et al., 2022)

,

##  1 Introduction

The magnetic field of a low-mass, main-sequence star ($\lesssim$1.2
${M_{\odot}}$) is generated by a complex dynamo, which arises from
differential rotation and radial convective motions in the outer convective
envelope (cf. review in Fan, 2021). The magnetic field injects energy into the
stellar atmosphere (e.g., Vernazza et al., 1981; Nelson et al., 2013), and
produces magnetized winds. Through these winds, the star loses angular
momentum, and this weakens the magnetic dynamo that generates the magnetic
field (Parker, 1993).

While this picture is widely accepted, many unknowns remain about the nature
of stellar magnetic activity and its connection to rotation. For instance, the
intensity of different activity indicators is commensurate to the amount of
heat generated by the magnetic activity (Schrijver et al., 1989; Pevtsov et
al., 2003; Güdel, 2004; Reiners & Basri, 2007; Reiners & Mohanty, 2012), yet
how much energy is injected at different atmospheric heights is not fully
understood (e.g., Stelzer et al., 2013, 2016; Richey-Yowell et al., 2019). In
addition, the mechanism generating magnetic fields in solar-type stars has
long been thought to rely on the existence of the tachocline, the shear layer
between the radiative interior and the outer convective region (Ossendrijver,
2003; Miesch, 2005), but fully convective stars, which lack a tachocline,
nonetheless show strong magnetic activity (e.g., Reiners & Basri, 2007; Wright
et al., 2018).

The typical approach to quantifying the relationship between activity and
rotation is to examine the behavior of the fractional luminosity of an
activity indicator, i.e., the luminosity of the activity indicator divided by
the bolometric luminosity $L_{\mathrm{bol}}$ of the star, as a function of the
Rossby number $R_{\mathrm{o}}$, defined as the rotation period
$P_{\mbox{\scriptsize rot}}$ divided by the convective turnover time $\tau$
(e.g., Noyes et al., 1984; Randich, 1998; Cook et al., 2014).

X-rays, which originate in the coronae of low-mass stars (Vaiana et al.,
1981), and $\mathrm{H}\alpha$ emission, which originates in their
chromospheres (Campbell et al., 1983), are well-known activity indicators. In
the $R_{\mathrm{o}}$–$L_{\mathrm{X}}$/$L_{\mathrm{bol}}$ and
$R_{\mathrm{o}}$–$L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$ planes, low-mass
stars are generally in one of two regimes. For large $R_{\mathrm{o}}$
($\gtrsim$0.1), activity decreases as $R_{\mathrm{o}}$ increases (i.e., as
$P_{\mbox{\scriptsize rot}}$ increases). By contrast, for small
$R_{\mathrm{o}}$, activity is independent of $R_{\mathrm{o}}$ and appears to
saturate at a given level, which differs for each activity indicator (e.g.,
Stauffer et al., 1994; Randich, 2000; Pizzolato et al., 2003; Wright et al.,
2011; Núñez et al., 2015). Still, many details remain unexplained. For
example, it is unclear which magnetic properties define the power-law relation
in the unsaturated regime, or what sets the $R_{\mathrm{o}}$ value separating
unsaturated from saturated stars.

Some studies have also found that at very small $R_{\mathrm{o}}$
($\lesssim$0.01), X-ray activity levels decrease once again. This is the so-
called supersaturated regime (Randich et al., 1996; Stauffer et al., 1997a;
Jeffries et al., 2011; Argiroffi et al., 2016; Thiemann et al., 2020;
Alexander & Preibisch, 2012; Cook et al., 2014). Several hypotheses exist to
explain the transition between the saturated and supersaturated regimes (cf.
discussion in Wright et al., 2011). For example, Vilhu (1984) suggested that
the fraction of the stellar surface covered by star spots reaches a maximum at
some (high) magnetic field strength, thus effectively capping the amount of
activity. Solanki et al. (1997) proposed instead that the magnetic field in
ultrafast-rotating stars gets concentrated near the poles, thus decreasing the
amount of activity elsewhere (see also Stȩpień et al., 2001). In these
scenarios, supersaturation would affect all of the stellar atmosphere, and
therefore be observed in any tracer of magnetic activity.

Alternatively, Jardine & Unruh (1999) developed the idea of coronal stripping,
in which the outermost layers of the corona are centrifugally lost in
ultrafast rotators (see also Jardine, 2004). In this scenario, only the
corona, itself the outermost atmospheric layer, would display supersaturation.
Observationally, Marsden et al. (2009) found tentative evidence of coronal
supersaturation in a sample of $\approx$30-Myr-old solar-type stars that
showed no evidence of chromospheric supersaturation.

The samples used in the studies described above are usually heterogeneous,
including both young stars from open clusters and field-age stars (e.g.,
Jeffries et al., 2011; Wright et al., 2011; Stelzer et al., 2016). They may
also include binaries, which may have very different evolutionary histories
from their single counterparts, thanks to possible magnetic interactions with
their close stellar—or sub-stellar—companions (e.g., Stelzer & Neuhäuser,
2001; Wright et al., 2011). Or they are restricted to solar-type stars,
leaving gaps in our understanding of the magnetic behavior of their lower
mass, fully convective cousins. Indeed, persuasive evidence for coronal
supersaturation has only been found in G and K dwarfs (e.g., Prosser et al.,
1996; Stauffer et al., 1997b; Pizzolato et al., 2003; Jackson & Jeffries,
2010), and only tentative evidence for M dwarfs (e.g. James et al., 2000;
Reiners & Basri, 2010; Núñez et al., 2022b).

Open clusters are ideal laboratories for placing observational constraints on
the rotation–activity relation. Stars from the same open cluster have both the
same age and metallicity, allowing for a more robust characterization of the
$R_{\mathrm{o}}$–$L_{\mathrm{X}}$/$L_{\mathrm{bol}}$ or
$R_{\mathrm{o}}$–$L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$ planes. This paper
is the fifth in our study of rotation and activity in the $\approx$700-Myr-old
Praesepe and Hyades open clusters, which form a crucial bridge between the
studies of very young groups of stars and those with field ages.

Two of our previous papers are especially relevant to this one. In Douglas et
al. (2014, hereafter Paper II), we combined new and archival optical spectra
with literature $P_{\mbox{\scriptsize rot}}$ and X-ray data to show that
chromospheric and coronal activity depend differently on $P_{\mbox{\scriptsize
rot}}$. In Núñez et al. (2022b, Paper IV), we presented an in-depth analysis
of X-ray activity and rotation in both clusters, benefiting from the large
number of $P_{\mbox{\scriptsize rot}}$ measurements published by Douglas et
al. (2016, 2017, 2019) and Rampalli et al. (2021).

In this paper, we present our analysis of hundreds of low-mass members of the
two clusters with $L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$,
$L_{\mathrm{X}}$/$L_{\mathrm{bol}}$, and $R_{\mathrm{o}}$ measurements. We
begin by describing updates to our membership catalogs in Section 2, our
optical spectroscopic data in Section 3, our X-ray data in Section 4, and our
photometric light-curves and $P_{\mbox{\scriptsize rot}}$ measurements in
Section 5. We derive several parameters for the cluster stars in Section 6. We
present our results in Section 7 and conclude in Section 8.

##  2 Membership Updates

We adopted the original cluster membership catalog presented in Table 2 of
Paper IV for Praesepe and Hyades stars, updating the Gaia data to the values
published in the Data Release 3 (DR3; Gaia Collaboration et al., 2022). For
Praesepe, the catalog has 1739 members, 539 of which are candidate or
confirmed binaries. For Hyades, the numbers are 1315 and 298, respectively.

We updated the catalog entry for the Hyad 2MASS J05301288+2038486 (catalog )
to reflect the fact that there are two Gaia DR3 sources associated with it,
one of which was not included in the Gaia Data Release 2 (DR2; Gaia
Collaboration et al., 2018a).111The two objects are 3402090466142958464
(catalog Gaia DR3 3402090466142958464) ($G$ = 11.38 mag, $\varpi$ =
13.30$\pm$0.02 mas, $\mu_{\alpha}\cos\delta$ = 26.00$\pm$0.03 mas yr-1,
$\mu_{\delta}$ = $-$21.02$\pm$0.02 mas yr-1, RV = 0.88$\pm$0.28 km s-1) and
3402090466140560128 (catalog Gaia DR3 3402090466140560128) ($G$ = 14.84 mag,
$\varpi$ = 13.28$\pm$0.21 mas, $\mu_{\alpha}\cos\delta$ = 27.72$\pm$0.27 mas
yr-1, $\mu_{\delta}$ = $-$19.31$\pm$0.17 mas yr-1, RV = $-$0.70$\pm$6.32 km
s-1); the latter one is not in DR2. Whether these two DR3 sources are
gravitationally bound or unassociated remains to be confirmed, but for the
purpose of this study, we categorize the star as a binary and change its
binary flag from 0 (not binary) to 1 (candidate binary).

Table 1: Overview of Columns in the Praesepe and Hyades Membership Catalog

Column | DescriptionaaThis Table includes columns from Table 2 of Núñez et al. (2022b) that have been updated and new columns from this study.
---|---
1 | Name
2 | 2MASS designation
3 | Gaia DR3 designation
4 | Cluster to which the star belongs
5 | Binary flag: (0) no evidence for binarity; (1) candidate
| binary; (2) confirmed binary
6, 7 | X-ray energy flux $f_{\mathrm{X}}$ (0.1–2.4 keV) and 1$\sigma$ uncertainty
8 | Rotation period $P_{\mbox{\scriptsize rot}}$
9 | Source of $P_{\mbox{\scriptsize rot}}$bbPossible values: “TESS” (period measurement from TESS data); “ZTF;” “T&Z” (from both TESS and ZTF); “Legacy” (from Douglas et al. 2019 or Rampalli et al. 2021)
10, 11 | Measured $\mathrm{H}\alpha$ equivalent width EW and 1$\sigma$ uncertainty
12 | Number of spectra measured to obtain $\mathrm{H}\alpha$ EW
13 | Relative $\mathrm{H}\alpha$ EW
14, 15 | Effective temperature $T_{\mathrm{eff}}$ and 1$\sigma$ uncertainty
16, 17 |  $\chi$ and 1$\sigma$ uncertainty
18, 19 | Stellar radius $R_{\mathrm{\star}}$ and 1$\sigma$ uncertainty

Note. — (This table is available in its entirety in machine-readable form.)

We also updated the binary flag for Hyads 2MASS J02594633+3855363 (catalog )
and J04461522+1846294 (catalog 2MASS J04461522+1846294) from 0 to 1. Our TESS
light curve analysis (see Section 5) revealed multiple periodicity in these
two stars. As such, we considered them candidate binaries for this study (see
Section 5 for more details). The updated catalog for Hyades therefore now has
1312 single members and 301 candidate or confirmed binaries.

We present our updated catalog in Table 1, with our adopted name for each star
in Column 1. Columns 2 and 3 include 2MASS (Skrutskie et al., 2006) and Gaia
DR3 designations. Column 4 identifies the cluster to which the stars belongs.
Our updated binary flags are given in Column 5. In the following Sections we
describe the rest of the columns in the table.

Table 2: Spectra of Praesepe and Hyades Stars Obtained Since Douglas et al.
(2014)

|  | # of Spectra
---|---|---
Dates | Instrument | Hyades | Praesepe
2014 Nov 10–Nov 16 | ModSpec | 6 |  $\cdots$
2015 Feb 20–Feb 24 | ModSpec |  $\cdots$ | 20
2015 Nov 21–Nov 22 | Hectospec |  $\cdots$ | 164
2015 Dec 14–Dec 21 | ModSpec | 17 | 44
2016 Jan 29–Feb 03 | ModSpec | 26 | 40
2016 Nov 30–Dec 09 | ModSpec | 39 | 26
2017 Feb 15 | ModSpec |  $\cdots$ | 8
2017 Dec 15 | OSMOS |  $\cdots$ | 17
2018 Jan 11–Jan 28 | OSMOS | 5 | 276
2018 Feb 04–Feb 09 | OSMOS |  $\cdots$ | 71
2019 Jan 07–Jan 11 | OSMOS | 12 |  $\cdots$
2019 Feb 27–Mar 02 | OSMOS | 14 |  $\cdots$
2019 Nov 23–Nov 26 | OSMOS | 25 |  $\cdots$
2021 Sep 05–Sep 29 | OSMOS | 14 |  $\cdots$
2021 Oct 28 | OSMOS | 7 |  $\cdots$
2022 Sep 01–Sep 02 | OSMOS | 10 |  $\cdots$
2022 Sep 28–Oct 05 | OSMOS | 11 |  $\cdots$
2022 Nov 10–Nov 14 | OSMOS | 61 |  $\cdots$
2023 Mar 28–Mar 31 | OSMOS | 3 |  $\cdots$
| Total | 250 | 666

Note. — All dates are UT.

##  3 Optical Spectroscopy

Figure 1: Five representative Praesepe and Hyades spectra obtained with the
2.4-m Hiltner telescope at MDM (Modspec and OSMOS spectrographs) and the MMT
Observatory (Hectospec). Each spectrum is labeled with the instrument used,
2MASS designation of the target, and stellar mass $m$, and is normalized to
the flux at 6555 Å. The right panel shows a close-up of the $\mathrm{H}\alpha$
line. The vertical dotted line indicates the center of the $\mathrm{H}\alpha$
line.

In Paper II, we presented new spectra for 130 Hyads and 390 Praesepe members
obtained with the MDM Observatory 2.4-m Hiltner telescope and the WIYN 3.5-m
telescope, both on Kitt Peak, AZ, and with the Magellan 6.5-m Clay telescope,
Las Campanas Observatory, Chile. To these, we added archival spectra from
Allen & Strom (1995), Stauffer et al. (1997a), and Kafka & Honeycutt (2004,
2006), and from the Sloan Digital Sky Survey (SDSS; York et al., 2000)
archive. The resulting spectroscopic sample contained 720 spectra for 516
Praesepe members and 139 spectra for 130 Hyads.

###  3.1 New MDM Observations

We obtained additional spectra of Praesepe and Hyades stars over the course of
19 observing runs between 2014 Nov and 2023 Mar with the Modular Spectrograph
(ModSpec) and the Ohio State Multi-Object Spectrograph (OSMOS) on the MDM
Observatory 2.4-m Hiltner telescope (see Table 2). We configured ModSpec to
have a wavelength coverage of 4500–7500 Å with $\approx$1.8 Å sampling and R
$\approx$ 3600, which is the same configuration we used in Paper II.

With OSMOS, we used the blue 4K detector (OSMOS 4K) with a 1.2′′ inner slit,
for an approximate wavelength coverage of 4000–6800 Å, $\approx$0.7 Å
sampling, R $\approx$ 9300, and peak efficiency at 6400 Å. We also used the
red 4K detector (OSMOS R4K) with an OG-530 longpass filter and 1.2′′ center
slit, for an approximate wavelength coverage of 5500–10000 Å, $\approx$1.3 Å
sampling, R $\approx$ 5000, and peak efficiency near 9000 Å.

MDM spectra obtained before 2021 were reduced with a script written in
PyRAF,222https://pypi.org/project/pyraf/ the Python-based command language for
the Image Reduction and Analysis Facility (IRAF; Tody, 1986). Spectra obtained
in 2021 and later were processed with the Python package PypeIt (Version
1.10.1.dev3+g52d10edd; Prochaska et al., 2020, 2020). We tested the agreement
between the PyRAF and PypeIt pipelines by reducing a small sample of raw OSMOS
images with both pipelines; most of these data were for stars with
$\mathrm{H}\alpha$ absorption. We then measured the $\mathrm{H}\alpha$
equivalent width (see Section 6.1) in all spectra. The difference in
measurements between the pipelines were $<$10%.

All the spectra were trimmed, overscan- and bias-corrected, cleaned of cosmic
rays, flat-fielded, extracted, dispersion-corrected, and flux-calibrated.
Excluding poor quality spectra (e.g., $S$/$N$ $\lesssim$ 5 or acquisition
imperfections; 12% of Praesepe spectra and 8% of Hyades spectra), we collected
454 new spectra for 153 Praesepe members and 231 for 209 Hyads. The median
$S$/$N$ for these spectra is 76 at $\mathrm{H}\alpha$.

Spectra for four stars observed at MDM are shown in Figure 1 for illustrative
purposes. The MDM (and WIYN) spectra from Paper II and this work are available
online.333Available at https://doi.org/10.7916/8ag4-4c53.

###  3.2 New MMT Observations

We obtained additional spectra of Praesepe stars with the multiobject
spectrograph Hectospec (Fabricant et al., 2005) on the MMT 6.5-m telescope,
Mt. Hopkins, AZ. We used two fiber configurations over the course of two
consecutive nights (2015 Nov 21-22). The first configuration was centered near
$\alpha=08^{\mathrm{h}}41^{\mathrm{m}}10^{\mathrm{s}}$,
$\delta=+20^{\circ}09^{\prime}59\farcs 1$ (J2000) and targeted 57 Praesepe
stars. The second configuration was centered near
$\alpha=08^{\mathrm{h}}41^{\mathrm{m}}36^{\mathrm{s}}$,
$\delta=+19^{\circ}03^{\prime}50\farcs 5$ (J2000) and targeted 50 additional
Praesepe stars.

We used the 600 line grating centered at 6300 Å, which results in an
approximate wavelength coverage of 5030-7540 Å, and give $R\approx 11,000$ at
$\mathrm{H}\alpha$. Our targets had 14.9 $<G<$ 19.7 mag, and our integration
times were 3600 s (first night) and 5400 s (second night) with the first
configuration, and 4500 s (second night) with the second configuration. After
excluding spectra with $S$/$N$ $\lesssim$ 5, we have 126 MMT spectra for 47
Praesepe stars. The median $S$/$N$ at $\mathrm{H}\alpha$ is 48. All our MMT
spectra are also available online.

The data were reduced automatically by the Smithsonian Astrophysical
Observatory Telescope Data Center using the HSRED v2.0 pipeline. HSRED
performs the basic reduction tasks: bias subtraction, flat-fielding, arc
calibration, and sky subtraction.444See
http://mmto.org/~rcool/hsred/index.html for a description of HSRED. A
Hectospec spectrum is included in Figure 1 for illustrative purposes. The MMT
spectra are available online.3

Table 3: New Chandra Observations of Hyades Stars  Obs. ID | Nominal Aimpoint | Roll | Target(s) | Start | DurationaaExposure time before any filtering is applied.
---|---|---|---|---|---
| $\alpha_{\rm J2000}$ | $\delta_{\rm J2000}$ | (°) | (Gaia DR3 Desig.) | Date | (s)
27553 | 03:18:15.13 | +09:14.38.0 | 343.1 |  14143675198789504 (catalog Gaia DR3 14143675198789504)bbUndetected in observation. | 2022-11 | 10085
27566 | 03:13:03.29 | +32:53:55.2 | 210.1 |  125343573948444800 (catalog Gaia DR3 125343573948444800), | 2022-11 | 10083
|  |  |  | 125343608307015296 (catalog Gaia DR3 125343608307015296) |  |
27572 | 04:38:56.73 | +14:06:11.4 | 14.9 | 3309170875916905856 (catalog Gaia DR3 3309170875916905856) | 2022-12 | 9902
27573 | 04:33:41.90 | +19:00:38.0 | 28.5 | 3410453489022728576 (catalog Gaia DR3 3410453489022728576) | 2022-12 | 9903
27574 | 04:32:40.40 | +19:06:39.5 | 18.9 |  3410640887035452928 (catalog Gaia DR3 3410640887035452928), | 2022-12 | 9903
|  |  |  | 3410639993682264960 (catalog Gaia DR3 3410639993682264960) |  |
27612 | 04:48:50.99 | +15:56:57.9 | 299.8 | 3405127244241184256 (catalog Gaia DR3 3405127244241184256) | 2022-12 | 10080
27622 | 05:30:14.19 | +20:38:20.7 | 282.1 | 3402090466142958464 (catalog Gaia DR3 3402090466142958464) | 2022-12 | 10083
27623 | 06:03:26.87 | +24:02:26.8 | 265.5 | 3426209215771371648 (catalog Gaia DR3 3426209215771371648) | 2022-12 | 20085
28490 | 06:50:34.35 |  $-$17:11:50.5 | 119.4 | 2946050323261707648 (catalog Gaia DR3 2946050323261707648) | 2023-08 | 11082
29061 | 04:16:13.11 | +18:53:04.2 | 91.2 | 47804394753757056, (catalog Gaia DR3 47804394753757056) | 2023-11 | 9945
|  |  |  | 47803952373768960 (catalog Gaia DR3 47803952373768960) |  |

###  3.3 New Archival Spectroscopy

In Paper II, we found SDSS spectra for 66 Praesepe stars (as of 2013 Feb 14).
We repeated this search using the SDSS Science Archive
Server.555https://dr16.sdss.org/home SDSS spectra are sky-subtracted,
corrected for telluric absorption, spectrophotometrically calibrated, and
calibrated to heliocentric vacuum wavelengths. The wavelength coverage is
3800-9200 Å with R $\approx$ 4300 at $\mathrm{H}\alpha$. After excluding those
with $S$/$N$ $\lesssim$ 5, we found SDSS spectra for 102 Praesepe stars and 16
Hyads (as of 2023 May 21).

We also searched for spectra in the the Large Sky Area Multi-Object Fiber
Spectroscopic Telescope (LAMOST) Data Release 8
catalog.666http://dr8.lamost.org/v2/ These spectra are flux- and wavelength-
calibrated and sky-subtracted, and cover 3690-9100 Å with R $=1800$ at 5500 Å.
After excluding those with $S$/$N$ $\lesssim$ 5, we found 873 LAMOST spectra
for 324 Praesepe members and 535 for 252 Hyads. This includes 108 stars in
Praesepe and 146 in the Hyades that did not have any previously available
spectra.

Finally, J. Stauffer (2014, priv. comm.) shared 12 spectra of Hyads obtained
as part of the Stauffer et al. (1997a) survey of the cluster. In Paper II we
used 10 of these spectra to compare our equivalent width measurements to those
of Stauffer et al. (1997a); here we include all 12 in our spectroscopic
sample.

With our newly obtained spectra and newly found archival spectra, we now have
a total of 2216 good quality (signal-to-noise ratio $S$/$N$ $\gtrsim$ 5)
spectra for 879 Praesepe members; for the Hyades, the numbers are 943 and 565,
respectively. Ninety-four of the Praesepe stars and 12 Hyads have five or more
spectra.

##  4 X-ray data

Paper IV explains in detail the origin of the X-ray data for our Praesepe and
Hyades stars. Briefly: we consolidated X-ray detections from the Röntgen
Satellite (ROSAT), the Chandra X-ray Observatory (Chandra), the Neil Gehrels
Swift Observatory (Swift), and the X-ray Multi-Mirror Mission Newton (XMM).
For faint X-ray sources, we converted instrumental counts to unabsorbed energy
fluxes $f_{\mathrm{X}}$ using WebPIMMS777https://heasarc.gsfc.nasa.gov/cgi-
bin/Tools/w3pimms/w3pimms.pl, and for bright X-ray sources, we performed
spectral analyses to extract unabsorbed $f_{\mathrm{X}}$. For sources with
flares in their X-ray light curves, we removed counts from the flare events
before calculating $f_{\mathrm{X}}$ to obtain a more representative
measurement of the quiescent X-ray activity level. We also homogenized all the
fluxes to the 0.1–2.4 keV energy band.

Finally, for stars with more than one X-ray detection, we calculated the
error-weighted average of the $f_{\mathrm{X}}$ values and adopted it as the
bona fide unabsorbed $f_{\mathrm{X}}$ for that star. We include unabsorbed
$f_{\mathrm{X}}$ values and their standard deviations (1$\sigma$
uncertainties) for Praesepe and Hyades stars in Columns 6 and 7 of Table 1.

Table 4: Overview of Columns in the Addendum to the Praesepe and Hyades X-ray
Source Catalog

Column | DescriptionaaThis Table has the same columns and formats as those in Table 3 of Núñez et al. (2022b).
---|---
1 | External catalog source ID
2 | Provenance of X-ray informationbb4XMM; CSC; CIAO: Reduction of Chandra observation with CIAO.
3 | IAU Name
4 | Observation ID
5 | Instrument
6, 7 | R.A., Decl. for epoch J2000
8 | X-ray positional uncertainty
9 | Off-axis angle $\theta$
10 | Detection likelihood $L$ccFor CIAO sources, it is the source significance; for all others, it is the maximum likelihood.
11 | Net counts in the broad band
12, 13 | Net count rate and 1$\sigma$ uncertainty in broad band
14, 15 | Net count rate and 1$\sigma$ uncertainty in soft band
16, 17 | Net count rate and 1$\sigma$ uncertainty in hard band
18$-$20 | Definition of broad, soft, and hard bands
21 | Hardness ratio: (hard band $-$ soft band) /
| (hard band + soft band)
22 | Exposure time
23 | Variability flag: (0) no evidence for variability;
| (1) possibly variable; (2) definitely variable
24, 25 | Unabsorbed energy flux and 1$\sigma$ uncertainty in the
| 0.1–2.4 keV band
26 | Source of energy flux: (ECF) from applying ECF;
| (SpecFit) from spectral fitting
27 | X-ray flare removed?
28 | Quality Flagddm: likely mismatch to optical counterpart; x: likely extended source.
29 | Name of the optical counterpart
30 | Separation between X-ray source and optical counterpart

Note. — (This table is available in its entirety in machine-readable form.)

For this work, we updated our X-ray data to reflect developments since the
publication of Paper IV, namely new Chandra observations, additions to the
Chandra Source Catalog (CSC), and the thirteenth data release of the XMM EPIC
Serendipitous Source Catalogue (4XMM-DR13; Webb et al., 2020).

As part of the Chandra Cool Targets
program888https://cxc.harvard.edu/proposer/CCTs.html (Proposal 20201075, PI:
Agüeros), we observed 13 Hyads with ten pointings. The details of these
observations are given in Table 3. For each observation, we used the ACIS-S3
chip in Very Faint telemetry mode. We processed the raw observations with the
Chandra Interactive Analysis of Observations (CIAO; Fruscione et al., 2006)
tools.999We used CIAO v.4.14 and CALDB v.4.10.2; see section 3.2.2 in Paper IV
for a full description of the data reduction. We include in Table 4 the X-ray
data for 12 of the targeted stars; one was undetected. We note that Table 4 in
this work is an addendum to Table 3 in Paper IV.

In addition, two Hyads were added to the CSC following the release of version
2.1 starting in late 2022. Data for these two new X-ray detections are also
included in Table 4.

Lastly, we found nine Praesepe low-mass members in 4XMM-DR13 that had no
previous X-ray detections and one more that had ROSAT and Swift X-ray
detections. We also found four Hyads with no previous X-ray detections, and
four Hyads that only had a ROSAT X-ray detection. Data for these 18 4XMM-DR13
detections are included in Table 4.

##  5 Rotation Period Measurements

Figure 2: $P_{\mbox{\scriptsize rot}}$ vs. ($G-K$) for Praesepe (left panel)
and Hyades (right panel) stars. Gray circles indicate single stars with
existing K2 $P_{\mbox{\scriptsize rot}}$ and other archival data collected in
Rampalli et al. (2021) and Douglas et al. (2019) for Praesepe and the Hyades,
respectively. Blue up triangles, red down triangles, and purple diamonds
indicate stars with new $P_{\mbox{\scriptsize rot}}$ values from TESS, ZTF, or
both, respectively. Black circles highlight new $P_{\mbox{\scriptsize rot}}$
values for known and candidate binaries.

The bulk of our rotational data for Praesepe and the Hyades came from the
catalogs published in Rampalli et al. (2021) and Douglas et al. (2019),
respectively. These catalogs consolidated $P_{\mbox{\scriptsize rot}}$
measurements made from light curves obtained by ground-based photometric
surveys and by K2 (Howell et al., 2014). For Praesepe, we have 1052 members
with these legacy $P_{\mbox{\scriptsize rot}}$ measurements; for the Hyades,
the number is 233.

More recent observations of the two clusters with the Transiting Exoplanet
Survey Satellite (TESS; Ricker et al., 2015), and continuing observations of
the clusters with the Zwicky Transient Facility (ZTF; Masci et al., 2018),
provided an opportunity to add to these totals. We showcase in Appendix A.1
the differing qualities of TESS and ZTF data and the benefit of using them
together to extract more reliable rotation periods. Accordingly, we searched
for light curves for stars in both clusters for which we have an optical
spectrum and/or an X-ray detection.101010Completing the $P_{\mbox{\scriptsize
rot}}$ census for all of the stars in either cluster, i.e., to obtain new
$P_{\mbox{\scriptsize rot}}$ values for stars without magnetic activity
measurements, is beyond the scope of this paper.

Our procedure for TESS followed the strategy employed in a number of recent
studies (e.g., McDivitt et al., 2022). We downloaded 40$\times$40 pixel
cutouts from the available full frame images using TESScut (Brasseur et al.,
2019), hosted by MAST.111111All the TESS data used in this paper can be found
in MAST: http://dx.doi.org/10.17909/0cp4-2j79 (catalog 10.17909/0cp4-2j79). We
extracted light curves for the pixel closest to each target using Casual Pixel
Modeling (Wang et al., 2016) as implemented in the package unpopular (Hattori
et al., 2022).

Using the interactive program tesscheck,121212https://github.com/SPOT-
FFI/tess_check we then inspected the TESS light curves for each star, selected
the optimal subset of sectors to search for a rotational signature, and
measured the period using Lomb–Scargle periodograms (Press & Rybicki, 1989).
The visual inspection allowed us to catch uncorrected systematics that can
introduce spurious signals into the periodogram and to flag and correct cases
where the periodogram favors the half-period harmonic. We measured periods for
19 Praesepe stars and 125 Hyads using TESS data.

Our ZTF procedure used the approach developed by Curtis et al. (2020) to
analyze Palomar Transient Factory (Law et al., 2010) data for the 2.7-Gyr-old
cluster Ruprecht 147. We downloaded 8′$\times$8′ cutout images from
IPAC131313All the ZTF data used in this paper can be found in IPAC:
http://dx.doi.org/10.26131/IRSA539 (catalog 10.26131/IRSA539). (IRSA, 2022)
and used simple aperture photometry to extract light curves for our targets
and neighboring reference stars identified with Gaia. We corrected the light
curves for systematics using the median-combined normalized light curves for
reference stars. We inspected the resulting light curves, isolated the segment
showing the cleanest periodic variability, and measured the period using
Lomb–Scargle periodograms. We measured periods for 18 Praesepe stars and 59
Hyads using ZTF data.

In total, we have new $P_{\mbox{\scriptsize rot}}$ measurements for 28
Praesepe stars and 137 Hyads; 56 of these 165 stars have periods determined
from both TESS and ZTF. For the Hyades, we have increased the sample of
cluster members with known $P_{\mbox{\scriptsize rot}}$ by $\approx$50%,
bringing the total to 370 stars. Figure 2 highlights our new
$P_{\mbox{\scriptsize rot}}$ measurements against the background of legacy
$P_{\mbox{\scriptsize rot}}$ for both clusters. In Table 1, we include the
$P_{\mbox{\scriptsize rot}}$ data for Praesepe and Hyades members in Column 8,
and we identify the source of the $P_{\mbox{\scriptsize rot}}$ measurement in
Column 9.

We measured $P_{\mbox{\scriptsize rot}}$ = 0.39 d from the TESS data for 2MASS
J05301288+2038486 (catalog ). This is an unusually short period for a star of
its color; the other single stars with ($G-K$) $\approx$ 2.5 mag in Figure 2
have $P_{\mbox{\scriptsize rot}}$ between 10 and 20 d. As mentioned in Section
2, we found two Gaia DR3 sources within $<$2′′ of each other and associated
with this 2MASS source. Given this, we consider 2MASS J05301288+2038486
(catalog ) a candidate binary and flag it as such in Figure 2.

Figure 3 compares our TESS- and ZTF-derived $P_{\mbox{\scriptsize rot}}$ for
the 56 cluster stars for which we have both. For 53 of the 56, the two
$P_{\mbox{\scriptsize rot}}$ disagree by $<$4%. Of the three stars for which
the disagreement is larger, two have TESS light curves that suggest multiple
periods (see Appendix A.2). Indeed, for these two stars, 2MASS
J02594633+3855363 (catalog ) and J04461522+1846294 (catalog 2MASS
J04461522+1846294), the Gaia re-normalized unit weight error (RUWE) values are
5.0 and 3.4, respectively. As discussed in Paper IV, stars with RUWE $>$ 1.4
have a high probability of being unresolved binaries (e.g., Deacon & Kraus,
2020; Ziegler et al., 2020; Kervella et al., 2022). For these two stars, we
adopted the TESS $P_{\mbox{\scriptsize rot}}$, assigned a binary flag = 1
(indicating they are candidate binaries), and flagged them as such in Figure
2.

For the third star, 2MASS J08412772+2103409 (catalog ), the TESS
$P_{\mbox{\scriptsize rot}}$ (4.1 d) is the half harmonic of the ZTF
$P_{\mbox{\scriptsize rot}}$ (8.2 d). We adopted the ZTF $P_{\mbox{\scriptsize
rot}}$ for this star.

Figure 3: $P_{\mbox{\scriptsize rot}}$ from ZTF vs. $P_{\mbox{\scriptsize
rot}}$ from TESS for the 56 Praesepe and the Hyades stars for which both
surveys yielded periods. The gray line is the 1:1 relation, and the gray area
the $\pm$10% difference range. The orange lines indicate the 1:2 and 2:1
harmonic lines. The two stars outside the 10% difference range have
indications of multiple periodicity (see Appendix A); we adopted the TESS
$P_{\mbox{\scriptsize rot}}$ and flag them as candidate binaries. For the star
falling on the 2:1 harmonic line, we adopted the ZTF $P_{\mbox{\scriptsize
rot}}$.

##  6 Other measurements and Derived Quantities

###  6.1 H$\alpha$ Equivalent Width Measurements

We measured the equivalent width (EW) of the $\mathrm{H}\alpha$ Balmer line in
all our optical spectra, both newly acquired and archival (see Section 3). For
this purpose, we used the tool PHEW (Núñez et al., 2022a), which automates the
EW measurement using PySpecKit (Ginsburg & Mirocha, 2011) to fit a Voigt
profile to the $\mathrm{H}\alpha$ line. We interactively defined continuum
regions to either side of the $\mathrm{H}\alpha$ line in each spectrum, each
between 5 and 35 Å in length.

PHEW performs 1000 Monte Carlo iterations by re-sampling the flux measurements
within the flux uncertainties, or, if flux uncertainties are unavailable, by
adding Gaussian noise to the flux spectrum. It then calculates the standard
deviation of the 1000 EWs, which we adopted as the 1$\sigma$ EW uncertainty.
We extracted the noise for each point from a Gaussian with width equal to the
associated uncertainty at that point. If a flux spectrum lacked an associated
uncertainty spectrum, we instead extracted the noise from a Gaussian with
width equal to the standard deviation of the flux in the continuum regions
defined for that spectrum.

If a star had multiple spectra available, we adopted the error-weighted mean
EW (and the weighted mean standard error) as the representative EW value (and
1$\sigma$ uncertainty) for that star. Columns 10 and 11 in Table 1 include our
measured EW and its 1$\sigma$ uncertainty, respectively. Negative EWs indicate
emission, and an EW value of zero indicates that the spectrum for the star
does not display a measurable $\mathrm{H}\alpha$ feature at that spectral
resolution. Column 12 indicates the number of spectra we used to calculate
each star’s EW value. The output figures from PHEW showing our EW measurements
are available online.3

Figure 4: Measured $\mathrm{H}\alpha$ EW vs. ($G-K$) for Praesepe (left panel)
and Hyades (right panel) members. Single stars are indicated with gray circles
and binaries with orange triangles. Most of the EW error bars are smaller than
the symbols. For clarity, we excluded from the right panel three outlier
stars, one with ($G-K$) $>$ 5.5 mag and two with $\mathrm{H}\alpha$ EW $<-18$
Å. We also excluded stars with ($G-K$) $<$ 1.0 (spectral types earlier than
$\approx$F5), as they are not relevant to our analysis. Black symbols indicate
stars for which $\mathrm{H}\alpha$ is immeasurable in our spectra, and for
which we set EW = 0. We consider one of these stars, annotated with its 2MASS
designation, to be a potential non-member based on its unusual inactivity (see
Section 6.1). The EWs shown here were not corrected for the quiescent
$\mathrm{H}\alpha$ absorption present in these stars.

Figure 4 shows our EW measurements as a function of ($G-K$) for single and
binary members (gray circles and orange triangles, respectively) of Praesepe
(left panel) and of the Hyades (right panel). To better visualize the overall
pattern, we omitted from the figure three Hyades outliers, one with ($G-K$)
$>$ 5.5 mag and two with EW $<-18$ Å. We also excluded stars with ($G-K$) $<$
1 (spectral type earlier than $\approx$F5) from the figure, as their lack of
significant convective envelopes implies a different rotational evolution than
the solar-like stars we focus on. However, we include in Table 1 values for
all stars with at least one spectrum, regardless of spectral type.

In Figure 4, we also highlight with black symbols stars for which we find no
measurable $\mathrm{H}\alpha$ (EW = 0 Å). Among these stars is 2MASS
J08391960+2017306 (catalog ), a Praesepe star with ($G-K$) = 4.4 mag. All of
its M5-M6 cousins exhibit some level of $\mathrm{H}\alpha$ emission, which
makes its $\mathrm{H}\alpha$ inactivity unusual. In Paper IV, we considered
this star a plausible Praesepe member based on its Kraus & Hillenbrand (2007)
membership probability of $\approx$70%. However, none of the Gaia-based
membership studies included in Paper IV considered it a cluster member, as its
Gaia data do not include parallax and proper motion information (as of DR3).
We therefore believe this star to be a likely contaminant in our membership
catalog for Praesepe.141414We have no X-ray detection or period for this star,
so it does not appear elsewhere in our analysis.

###  6.2 H$\alpha$ Relative to Quiescence

We corrected our measured EW values for each star to account for the quiescent
photospheric $\mathrm{H}\alpha$ absorption naturally present in low-mass stars
(cf. discussion for M dwarfs in Stauffer & Hartmann, 1986). As stars become
more magnetically active, the line fills in and eventually transitions to
emission. To report more accurately the level of chromospheric activity, we
therefore need to consider this quiescent absorption level, which is a
function of stellar mass $m$.

We used the empirical model of Newton et al. (2017), valid for stars with $m$
< 0.8 ${M_{\odot}}$, to calculate the quiescent photospheric absorption EW for
cluster stars in that $m$ range.151515In principle, stars with $m$ ¿ 0.8
${M_{\odot}}$ also exhibit quiescent photospheric $\mathrm{H}\alpha$
absorption. However, the main focus of our study is on stars with
$\mathrm{H}\alpha$ in emission, and none of our stars with spectra and $m$ ¿
0.8 ${M_{\odot}}$ fall in that category. We then determined the _relative_ EW
by subtracting the quiescent EW from our measured EW. Column 13 in Table 1
indicates the relative EW value for each star with a measured EW and within
the $m$ range of the empirical model.

###  6.3 The $\chi$ Factor and $L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$

To obtain $L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$ for stars with
$\mathrm{H}\alpha$ in emission, we used the relation

$\frac{L_{\mathrm{H}\alpha}}{L_{\mathrm{bol}}}=-\mathrm{EW}_{\mathrm{H}\alpha}\
\chi,$ (1)

where EW${}_{\mathrm{H}_{\alpha}}$ is the relative $\mathrm{H}\alpha$ EW
calculated in Section 6.2, and $\chi$ is the ratio of the continuum flux near
the $\mathrm{H}\alpha$ line and of the apparent bolometric flux.

In Paper II, we presented several empirical $\chi$–photometric color relations
based on PHOENIX ACES model spectra (Husser et al., 2013). We measured $\chi$
in the model spectra with surface gravity log($g$) = 5.0, solar metallicity,
and in the effective temperature ($T_{\mathrm{eff}}$) range 2500–5200 K. For
this work, we extended this calculation of $\chi$ to include the
$T_{\mathrm{eff}}$ range 2300–6500 K by following the methodology described in
Paper II (see Table 5).

Figure 5: $\chi$ values for single Praesepe and Hyades stars calculated in
Paper II vs. our calculations in this work, with their respective 1$\sigma$
errors. The gray dashed line is the 1:1 relation. Stars are color-coded
according to their $T_{\mathrm{eff}}$ (as derived in this work; see Section
6.3), following the colorbar at the top left. The two stars represented with
star symbols are objects for which we identified erroneous or unreliable
$r^{\prime}$ photometry, which was used in Paper II to estimate their $\chi$.

To calculate $\chi$ for our cluster stars, we first derived their
$T_{\mathrm{eff}}$ using the empirical $T_{\mathrm{eff}}$–$M_{G}$ relation of
E. Mamajek.161616Version 2022.04.16. Available at
http://www.pas.rochester.edu/~emamajek/EEM_dwarf_UBVIJHK_colors_Teff.txt. Much
of this table comes from Pecaut & Mamajek (2013). We linearly interpolated
between the $M_{G}$ values in the empirical relation to obtain
$T_{\mathrm{eff}}$. Columns 14 and 15 in Table 1 include our derived
$T_{\mathrm{eff}}$ values and 1$\sigma$ uncertainties, respectively, for each
main sequence cluster star.

Next, we calculated $\chi$ using $T_{\mathrm{eff}}$ by linearly interpolating
between the $T_{\mathrm{eff}}$ values in Table 5. We assumed an intrinsic 10%
error in our $T_{\mathrm{eff}}$–$\chi$ relation (identified in Paper II) and
we added this error in quadrature to produce the 1$\sigma$ of the $\chi$
values we calculated for our cluster stars. Columns 16 and 17 include our
$\chi$ values and 1$\sigma$ uncertainties for each cluster star. Lastly, we
applied Equation 1 for all stars with relative $\mathrm{H}\alpha$ EW and
$\chi$ values to obtain $L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$.

In Figure 5, we compare our new $\chi$ values to those in Paper II for the
sample of single Praesepe and Hyades stars in both studies. The $\chi$ values
from the earlier work were derived using the log($\chi$)–($r^{\prime}-K$)
relation. We identified two stars for which an erroneous or unreliable
$r^{\prime}$ photometry was assigned in Paper II (highlighted in Figure 5 with
stars symbols), which explains their significant deviation from the general
trend.

Our new $\chi$ values are systematically larger than those in Paper II by a
factor of $\approx$1.3. This discrepancy is mostly driven by the
$T_{\mathrm{eff}}$–($r^{\prime}-K$) relation in Paper II, which produces
cooler $T_{\mathrm{eff}}$ values than those derived from the E. Mamajek table.

Table 5: $T_{\mathrm{eff}}$ and $\chi$ Values from PHOENIX Model Spectra

$T_{\mathrm{eff}}$ | $\chi$ | $T_{\mathrm{eff}}$ | $\chi$ | $T_{\mathrm{eff}}$ | $\chi$
---|---|---|---|---|---
(K) | ($\times 10^{-5}$) | (K) | ($\times 10^{-5}$) | (K) | ($\times 10^{-5}$)
6500 | 8.695 | 5000 | 9.581 | 3500 | 5.019
6400 | 8.797 | 4900 | 9.409 | 3400 | 4.477
6300 | 8.907 | 4800 | 9.279 | 3300 | 3.913
6200 | 9.038 | 4700 | 9.195 | 3200 | 3.328
6100 | 9.188 | 4600 | 9.127 | 3100 | 2.714
6000 | 9.299 | 4500 | 9.071 | 3000 | 2.181
5900 | 9.426 | 4400 | 8.658 | 2900 | 1.702
5800 | 9.510 | 4300 | 8.197 | 2800 | 1.252
5700 | 9.667 | 4200 | 7.632 | 2700 | 0.886
5600 | 9.777 | 4100 | 7.144 | 2600 | 0.618
5500 | 9.351 | 4000 | 6.825 | 2500 | 0.473
5400 | 8.961 | 3900 | 6.523 | 2400 | 0.603
5300 | 8.597 | 3800 | 6.201 | 2300 | 0.564
5200 | 9.160 | 3700 | 5.858 |  |
5100 | 9.622 | 3600 | 5.494 |  |

Note. — The methodology used to calculate $\chi$ is described in the appendix
of Paper II.

###  6.4 Bolometric Luminosities and Rossby Numbers

We used the bolometric luminosities and $R_{\mathrm{o}}$ derived in Paper IV
for our cluster stars. Briefly: to obtain $L_{\mathrm{bol}}$, we used the
empirical log($L_{\mathrm{bol}}$)–$M_{G}$ relation of E. Mamajek.

To calculate $R_{\mathrm{o}}$, we first calculated $m$ using the empirical
$m$–$M_{G}$ relation of E. Mamajek. Next, we found the convective turnover
time $\tau$ using the empirical $m$–log($\tau$) relation of Wright et al.
(2018). Finally, we computed $R_{\mathrm{o}}$ = $P_{\mbox{\scriptsize
rot}}$/$\tau$.

In Paper II, we used the $m$–$M_{K}$ relation of Kraus & Hillenbrand (2007) to
calculate $m$ and the $m$–log($\tau$) relation of Wright et al. (2011) to
calculate $\tau$. Compared to the $R_{\mathrm{o}}$ values in Paper II, our new
$R_{\mathrm{o}}$ values for the same stars are between 45% smaller and 20%
larger, the median being 14% smaller. The largest discrepancies are mostly due
to differences in the two $m$ calculation methods and to the distances used to
calculate absolute magnitudes, as Paper II relied mostly on individual
Hipparcos parallaxes or Hipparcos-derived cluster distances (van Leeuwen,
2009).

##  7 Results and Discussion

###  7.1 Chromospheric Activity

Figure 4 shows that in both clusters, all the late F, G, and K dwarfs have
converged onto a tight sequence of $\mathrm{H}\alpha$ absorption (EW $>$ 0 Å),
which is independent of magnetic activity. On the other hand, most M dwarfs
exhibit some level of $\mathrm{H}\alpha$ emission. The transition between
$\mathrm{H}\alpha$ absorption and emission in the two clusters occurs
essentially at the same color (corresponding to a spectral type M0-M1),
suggesting that both clusters are indeed of very similar ages.

Figure 6: Color-binned median measured $\mathrm{H}\alpha$ EWs for single stars
(gray circles) and binaries (orange triangles) in Praesepe and Hyades, with
the 16th and 84th percentiles represented by whiskers

. The binary sample includes all stars with RUWE > 1.4. Numbers next to
symbols indicate the number of stars in each bin.

Save for a handful of late K Hyads, the binaries in Figure 4 appear to follow
the same distribution as their single-star counterparts in both clusters. To
compare the two distributions more carefully, we binned our EWs by ($G-K$).
Figure 6 shows the median EW values for single stars (gray circles) and
binaries (orange triangles) in both Praesepe and the Hyades for 0.3 or 0.5 mag
color bins, with their 16th and 84th percentiles represented by whiskers. The
median EW values of single and binary stars are almost identical in most color
bins, and the difference in median EW between single stars and binaries is
$\buildrel<\over{\sim}$1$\sigma$ in all color bins.

We do note a slightly higher median $\mathrm{H}\alpha$ EW for binaries
compared to single stars in the ($G-K$) = 3.0–3.3 mag bin (spectral types
$\approx$M0–M2). However, these differences are not statistically significant,
and we do not consider them to be evidence of enhanced chromospheric activity
in binary systems in our sample. Lastly, the reddest color bin shows a
$\approx$38% difference in median between single and binary stars. We
attribute this discrepancy to the small sample size in those bins.

###  7.2 The Dependence of Chromospheric Activity on Rotation

To characterize the rotation–chromospheric activity relation, we followed
previous authors in parametrizing the relation, in this case
$R_{\mathrm{o}}$–$L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$, as a flat region
connected to a power-law. For stars with $R_{\mathrm{o}}$
$\leq{R_{\mathrm{o}}}_{\mathrm{,sat}}$, activity is saturated—i.e.,
constant—and equal to ($L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$)sat. Above
${R_{\mathrm{o}}}_{\mathrm{,sat}}$, activity declines as a power-law with
index $\beta$, and is, therefore, unsaturated. Functionally, this corresponds
to

$\frac{L_{\mathrm{H}\alpha}}{L_{\mathrm{bol}}}=\left\\{\begin{array}[]{l
l}\left(\frac{L_{\mathrm{H}\alpha}}{L_{\mathrm{bol}}}\right)_{\mathrm{sat}}&\quad\textrm{if
$R_{\mathrm{o}}\leq R_{\mathrm{o,sat}}$}\\\
CR_{\mathrm{o}}^{\beta}&\quad\textrm{if $R_{\mathrm{o}}$ >
$R_{\mathrm{o,sat}}$}\end{array}\right.$ (2)

where $C$ is a constant. This model has been widely used in the literature
(e.g., Randich 2000, Wright et al. 2011, Paper II, Núñez et al. 2015, Paper
IV).

Table 6: Rotation–Activity Relation Fitting Results |  $R_{\mathrm{o}}$–$L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$ |  $R_{\mathrm{o}}$–$L_{\mathrm{X}}$/$L_{\mathrm{bol}}$
---|---|---
Sample | $N_{\star}$ | ($L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$)sat | $R_{\mathrm{o,sat}}$ | $\beta$ | $N_{\star}$ | $\beta_{\mathrm{sup}}$ | $R_{\mathrm{o,sup}}$ | ($L_{\mathrm{X}}$/$L_{\mathrm{bol}}$)sat | $R_{\mathrm{o,sat}}$ | $\beta$
|  | ($10^{-4}$) |  |  |  |  |  | ($10^{-3}$) |  |
Single stars
Praesepe | 196 | 1.76$\pm$0.09 | 0.28${}^{+0.02}_{-0.03}$ | $-5.19^{+1.32}_{-0.94}$ | 124 | 0.70${}^{+0.60}_{-0.32}$ | 0.011$\pm$0.005 | 1.14$\pm$0.12 | 0.19$\pm$0.02 | $-3.48^{+0.34}_{-0.39}$
Hyades | 116 | 1.53$\pm$0.08 | 0.31${}^{+0.01}_{-0.02}$ | $-7.07^{+1.40}_{-1.57}$ | 162 | 0.54${}^{+0.19}_{-0.15}$ | 0.014${}^{+0.004}_{-0.005}$ | 1.15${}^{+0.13}_{-0.12}$ | 0.17$\pm$0.02 | $-3.04^{+0.27}_{-0.28}$
All | 312 | 1.65$\pm$0.06 | 0.29$\pm$0.01 | $-5.85^{+0.81}_{-0.80}$ | 286 | 0.53${}^{+0.16}_{-0.12}$ | 0.015${}^{+0.003}_{-0.005}$ | 1.17$\pm$0.09 | 0.17$\pm$0.01 | $-3.18^{+0.20}_{-0.21}$
Binaries & stars with RUWE $>$ 1.4
Praesepe | 134 | 1.84${}^{+0.11}_{-0.12}$ | 0.24$\pm$0.03 | $-2.77^{+0.51}_{-0.70}$ | 112 | 0.13${}^{+0.22}_{-0.11}$ | 0.009${}^{+0.007}_{-0.004}$ | 1.17$\pm$0.15 | 0.12${}^{+0.03}_{-0.02}$ | $-2.20^{+0.26}_{-0.38}$
Hyades | 92 | 1.63$\pm$0.14 | 0.16${}^{+0.05}_{-0.04}$ | $-1.51^{+0.39}_{-0.59}$ | 138 | 0.08${}^{+0.15}_{-0.07}$ | 0.009${}^{+0.007}_{-0.005}$ | 1.02${}^{+0.11}_{-0.10}$ | 0.14$\pm$0.02 | $-2.36^{+0.27}_{-0.31}$
All | 226 | 1.76$\pm$0.09 | 0.20${}^{+0.03}_{-0.04}$ |  $-2.05\pm$0.50 | 250 | 0.06${}^{+0.11}_{-0.05}$ | 0.009${}^{+0.007}_{-0.005}$ | 1.07$\pm$0.08 | 0.13${}^{+0.02}_{-0.01}$ | $-2.26^{+0.19}_{-0.24}$

We used the open-source Markov-chain Monte Carlo (MCMC) package emcee
(Foreman-Mackey et al., 2013) to fit this three-parameter model to our data.
Following the emcee implementation by Magaudda et al. (2020), we allowed for a
nuisance parameter $f$ to account for underestimated errors.171717See
https://emcee.readthedocs.io/en/develop/user/line/. We assumed flat priors
over each parameter and used 300 walkers, each taking 5000 steps in their MCMC
chain, to infer maximum likelihood parameters. Our results are presented in
Figure 7 for several subsamples. The posterior distributions for each
parameter and 2D correlations between pairs of parameters from each fit are
included in a figure set in Appendix B; 200 random samples from these
distributions are shown in Figure 7, along with the maximum a posteriori
model.

In Table 6, we present the ($L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$)sat,
$R_{\mathrm{o}}$,sat, and $\beta$ parameters corresponding to the maximum a
posteriori model for the six subsamples we show in Figure 7, and we also
annotate them in each panel in the Figure. For each parameter, we assumed the
50th percentile of the results to be the mean value, and the 16th and 84th
percentiles, their approximate 1$\sigma$ uncertainties. In all cases, the
nuisance parameter $f$ converged to $\approx$0.06, suggesting that our
$L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$ uncertainties are underestimated by
no more than $\approx$6%.

We applied the model in Equation 2 to single members separately from binary
members and members with RUWE $>$ 1.4. Without a more detailed study of the
characteristics of the known and candidate binaries, it is not possible to
determine whether gravitational and magnetic interactions may have altered
their spin-down evolution.181818Gaia cannot resolve separations
$\lesssim$0$\farcs$7 (Ziegler et al., 2018), which corresponds to a semimajor
axis $a\approx 130$ au for the average Praesepe star and $\approx$35 au for
the average Hyades star. Most of the candidate binaries in our sample with
high RUWEs are therefore likely intermediate binaries rather than tight,
tidally interacting binaries, for which $a\ \lesssim\ 0.1$ au. Still,
intermediate binaries can have small enough separations ($0.1\ \lesssim\ a\
\lesssim\ 80$ au) for the binary components to have affected each other’s
protoplanetary disks in the first 10 Myr (Rebull et al., 2006; Meibom et al.,
2007; Kraus et al., 2016; Messina et al., 2017), thereby impacting their
rotation–activity relation. Furthermore, for binaries, the $\chi$ and $\tau$
parameters—ultimately derived from $M_{G}$ and used to calculate
$L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$ and $R_{\mathrm{o}}$,
respectively—have dubious validity, as we expect them to be overestimated to
varying degrees for binaries, the effects of which are difficult to track in
our analysis. In our discussion below, we therefore distinguish between the
nominally single stars and those flagged as either known or candidate
binaries.

We also applied the model to members of each cluster separately and together.
Combining the stars from both clusters to create a larger sample is reasonable
given the very similar ages and metallicities of Praesepe and the Hyades
(e.g., Cummings et al., 2018; Gaia Collaboration et al., 2018b; Douglas et
al., 2019). We consider the results from the combined sample, which we
nicknamed HyPra, to be most statistically meaningful. In any case, the values
for the parameters obtained from applying the model to the clusters
individually almost always agree to within 1$\sigma$ (and always to within
2$\sigma$).

Figure 7: $L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$ vs. $R_{\mathrm{o}}$ for
Praesepe stars (left panels), for Hyads (middle), and for the two clusters
combined (right). The top panels show single stars, and the bottom panels show
confirmed and candidate binaries. This latter set includes nominally single
stars with RUWE $>$ 1.4 (indicated with solid black circles). Single stars are
color-coded by their $m$ according to the colorbar in the top left panel. The
solid black line in each panel is the maximum a posteriori fit from the MCMC
algorithm, and the gray lines are 200 random samples from the posterior
probability distributions. We assumed a flat saturated regime described by
($L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$)sat and $R_{\mathrm{o}}$,sat, and
an unsaturated regime described by a power-law of index $\beta$. The results
of the fit for these three parameters are given in each panel. We show in
Appendix B the marginalized posterior probability distributions from the MCMC
analysis for each fit.

#### The Saturated Regime

For single HyPra stars, ($L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$)sat =
(1.65$\pm$0.06)$\times$10-4, with only a handful of outliers in the Hyades
deviating from the narrow distribution around this
$L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$ level (see top panels, Figure 7).
This value of ($L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$)sat is consistent
with what Newton et al. (2017) found for their sample of saturated field M
dwarfs, for which $L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$ =
(1.49$\pm$0.08)$\times$10-4, within 2$\sigma$ of our result.

On the other hand, in Paper II, we found that, for single members in both
clusters, ($L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$)sat =
(1.26$\pm$0.04)$\times$10-4. Similarly, Núñez et al. (2017) found
($L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$)sat = (1.27$\pm$0.02)$\times$10-4
for single members of the $\approx$500 Myr-old cluster M37. Both of these
values are statistically discrepant with our new result at the
$\approx$4$\sigma$ level. However, in neither of these studies were the EW
measurements corrected to account for the quiescent photospheric
$\mathrm{H}\alpha$ absorption. In addition, our new $\chi$ values used to
calculate $L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$ are $\approx$1.3$\times$
larger than those used in the two studies (see Section 6.3).

Accounting for quiescent absorption and using updated larger $\chi$ values
results in slightly enhanced $L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$ values,
which explains our larger best-fit value for
($L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$)sat compared to Paper II and Núñez
et al. (2017). In Appendix C, we repeated our fitting to the
$R_{\mathrm{o}}$–$L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$ data when the EW
data are not corrected, which provides a clearer comparison to previous
studies that did not apply any correction to the EW values.

Binaries and stars with RUWE $>$ 1.4 (bottom panels, Figure 7) exhibit a
spread around the saturated level similar to that observed in single cluster
stars. Their ($L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$)sat value,
(1.76$\pm$0.09)$\times$10-4, is within 1$\sigma$ of that of their single
counterparts.

#### The Rossby Threshold Between Saturated and Unsaturated Regimes

For single HyPra stars, the transition between the saturated and unsaturated
regimes occurs at $R_{\mathrm{o}}$,sat = 0.29$\pm 0.01$. We note that the
quoted 1$\sigma$ uncertainties for $R_{\mathrm{o}}$,sat in all of the studies
under consideration, including this one, are unrealistically small, as
$R_{\mathrm{o}}$ uncertainties are difficult to estimate and therefore not
included when running the MCMC fit. As such, we do not expect our result to
statistically agree with results in similar studies. Indeed, Newton et al.
(2017) found $R_{\mathrm{o}}$,sat = 0.21$\pm 0.02$, Paper II,
0.11${}^{+0.02}_{-0.03}$, and Núñez et al. (2017), 0.03$\pm 0.01$. All of
these results are statistically discrepant by $\geq$3$\sigma$.

As we show in Appendix C, however, re-running our MCMC fit without applying
the quiescent absorption correction to our measured EW values results in
$R_{\mathrm{o}}$,sat values that do agree statistically with that from Paper
II, but are still statistically discrepant from that in Núñez et al. (2017).

Lastly, for known and candidate binaries,
$R_{\mathrm{o}}$${}_{\mathrm{,sat}}=0.20^{+0.03}_{-0.04}$, which is within
2$\sigma$ of the value of single members, notwithstanding the unaccounted for
uncertainties in $R_{\mathrm{o}}$ mentioned above.

Figure 8: Same as Figure 7, but for $L_{\mathrm{X}}$/$L_{\mathrm{bol}}$. In
addition to the saturated and unsaturated regimes in Figure 7, we also assumed
the existence of a supersaturated regime described by a power-law of index
$\beta_{\mathrm{sup}}$ and $R_{\mathrm{o}}$,sup. The result of the fit for the
five parameters $\beta_{\mathrm{sup}}$, $R_{\mathrm{o}}$,sup,
($L_{\mathrm{X}}$/$L_{\mathrm{bol}}$)sat, $R_{\mathrm{o}}$,sat, and $\beta$,
are given in each panel. We show in Appendix B the marginalized posterior
probability distributions from the MCMC analysis for each fit.

#### The Unsaturated Regime

For single HyPra stars, we found that $\beta=-5.85^{+0.81}_{-0.80}$. This
result is $>$4$\sigma$ away from that of Newton et al. (2017), who found
$\beta=-1.7$$\pm$$0.1$. Although our methods are similar to those used by
these authors, our unsaturated stars are significantly different from those in
Newton et al. (2017) in three ways.

First, the majority of stars with $R_{\mathrm{o}}$ $>$ $R_{\mathrm{o}}$,sat in
our sample have masses $\gtrsim$0.5 ${M_{\odot}}$ (see the colormap in Figure
7), whereas their sample does not have any stars with masses $\gtrsim$0.5
${M_{\odot}}$ (see their Figure 6). Second, our largest $R_{\mathrm{o}}$
values are $\approx$ 0.5, whereas most unsaturated stars in their sample have
$R_{\mathrm{o}}$ $>$ 0.5 and up to 2.0. And third, all of our stars are
$\approx$700 Myr old, whereas their sample mostly included field-age dwarfs,
which presumably have ages $>>$1 Gyr. The $\beta$ discrepancy between these
two samples may partly be evidence for a steeper decay in chromospheric
activity for the more massive, partly convective dwarfs versus for fully or
almost fully convective dwarfs. On the other hand, the $\beta$ discrepancy may
just reflect different dominant chromospheric radiative coolants for stars at
different $T_{\mathrm{eff}}$: in M dwarfs, Balmer lines emission dominates,
whereas in G and K dwarfs, Ca ii and Mg ii emission dominates (Linsky et al.,
1982; Reid & Hawley, 2005).

In Paper II, we found that $\beta=-0.73^{+0.16}_{-0.12}$, and in Núñez et al.
(2017), $\beta=-0.51$$\pm$$0.02$. Both of these results are also statistically
inconsistent with our new result. However, as noted earlier, these two studies
must be compared to our results when we do not apply the quiescent correction
to our EWs (see Appendix C).

For binaries and candidate binaries, we found that $\beta=-2.05$$\pm$$0.50$,
which is within 3$\sigma$ of our result for single stars. The shallower
$\beta$ for binaries partly reflects the slightly higher—although
statistically insignificant—$\mathrm{H}\alpha$ emission in binaries compared
to single stars in the ($G-K$) = 3.0–3.3 mag bin ($\approx$M0–M2 stars; see
Figure 6).

However, as we mentioned earlier, using $M_{G}$ to derive $T_{\mathrm{eff}}$
and $m$, from which we then calculated $\chi$ and $\tau$, leads to
overestimated $L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$ and $R_{\mathrm{o}}$
values to varying degrees for binaries. Therefore, we do not consider our
shallower $\beta$ result to be evidence for higher chromospheric activity in
unsaturated binaries compared to their single counterparts.

###  7.3 The Dependence of Coronal Activity on Rotation

In Paper IV, we presented a comprehensive study of
$L_{\mathrm{X}}$/$L_{\mathrm{bol}}$ as a coronal activity indicator and of its
dependence on $R_{\mathrm{o}}$ in Praesepe and the Hyades. We used a sample of
114 Praesepe and 63 Hyades single stars to characterize the saturated and
unsaturated regimes in the
$R_{\mathrm{o}}$–$L_{\mathrm{X}}$/$L_{\mathrm{bol}}$ plane, using the same
parametrization given in Equation 2 (we also had 107 Praesepe and 98 Hyades
binary stars or with RUWE $>$ 1.4).

In that study, we found weak evidence for supersaturation (see appendix B of
Paper IV), the $R_{\mathrm{o}}$ regime in which super-fast rotators
($R_{\mathrm{o}}$ $\lesssim$ 0.01) show a decrease in activity level relative
to their saturated cousins. To characterize this behavior, we modified the
$R_{\mathrm{o}}$–$L_{\mathrm{X}}$/$L_{\mathrm{bol}}$ relation parametrization
presented in Equation 2 by adding a secondary power-law at small
$R_{\mathrm{o}}$: below $R_{\mathrm{o,sup}}$, activity declines as a power-law
with index $\beta_{\mathrm{sup}}$.

Since that study, we have added 154 stars to our sample of cluster stars with
both $R_{\mathrm{o}}$ and $L_{\mathrm{X}}$/$L_{\mathrm{bol}}$ measurements.
These are primarily Hyads; we have an additional 99 single stars and 40 known
and candidate binaries in that cluster with these measurements (the numbers
for Praesepe are 10 and five, respectively). Figure 8 shows the updated
$R_{\mathrm{o}}$–$L_{\mathrm{X}}$/$L_{\mathrm{bol}}$ relation for single stars
(top panels) and binary stars (bottom panels) for Praesepe (left panels),
Hyades (middle panels), and both clusters combined (right panels).

With this update, we found more compelling evidence of supersaturation in
single stars in both clusters. In this regime, single stars in Praesepe follow
a power-law of slope $\beta_{\mathrm{sup}}=0.70^{+0.60}_{-0.32}$, 2$\sigma$
away from a flat relation, while in Hyades, where the number of supersaturated
stars is larger, $\beta_{\mathrm{sup}}=0.54^{+0.19}_{-0.15}$, 3$\sigma$ away
from being flat. Meanwhile, for the combined HyPra sample
$\beta_{\mathrm{sup}}=0.53^{+0.16}_{-0.12}$, which is at least 4$\sigma$ away
from a flat relation (top row, Figure 8).

On the other hand, for known and candidate binaries, the supersaturated regime
is almost indistinguishable from a flat relation ($\beta_{\mathrm{sup}}$ = 0)
and $R_{\mathrm{o}}$,sup remains poorly constrained (bottom panels, Figure 8).
The lack of supersaturation in binaries may be partly explained by magnetic
interactions between the binary components increasing their quiescent activity
levels and/or increasing the frequency of flaring activity. The updated
results for the other four parameters, namely, $R_{\mathrm{o}}$,sup,
($L_{\mathrm{X}}$/$L_{\mathrm{bol}}$)sat, $R_{\mathrm{o}}$,sat, and $\beta$,
only change marginally compared to our results in Paper IV, with $\beta$ being
now more constrained for both single and binary members. We include these
updated parameters in Table 6.

###  7.4 Chromospheric Versus Coronal Activity

Figure 9: $L_{\mathrm{X}}$/$L_{\mathrm{bol}}$ vs.
$L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$ for single stars in Praesepe and the
Hyades, color-coded by their spectral type. The dashed black line is the
power-law relation found with a least-squares bisector regression. The slope
and Pearson $r$ of the regression is noted at the top left. Green edges
highlight stars in the supersaturated regime in the
$R_{\mathrm{o}}$–$L_{\mathrm{X}}$/$L_{\mathrm{bol}}$ plane.

Several studies have shown differences in the dependence of $\mathrm{H}\alpha$
and X-ray emission on rotation (e.g., Hodgkin et al., 1995; Núñez et al.,
2017; Preibisch & Feigelson, 2005; Stelzer et al., 2013). These differences
could point to differences in magnetic heating mechanisms acting on different
layers of the stellar atmospheres. At the same time, some positive correlation
between $L_{\mathrm{X}}$/$L_{\mathrm{bol}}$ and
$L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$ is expected, partly because a
fraction of the coronal X-rays will inevitably heat the underlying
chromosphere (Mullan, 1976; Cram, 1982). We directly compare
$L_{\mathrm{X}}$/$L_{\mathrm{bol}}$ and
$L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$ for single stars in both clusters in
Figure 9 to characterize their relationship.

Using a least-squares bisector regression, we found a power-law relation such
that $L_{\mathrm{X}}$/$L_{\mathrm{bol}}$ $\propto$
($L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$)α, with $\alpha$ = 1.23$\pm 0.09$
(dashed line, Figure 9), with a correlation coefficient $r$ = 0.77, which
suggests a strong positive correlation between
$L_{\mathrm{X}}$/$L_{\mathrm{bol}}$ and
$L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$.

Most of the stars are concentrated near
$L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$ $\approx~{}10^{-4}$ and
$L_{\mathrm{X}}$/$L_{\mathrm{bol}}$ $\approx 10^{-3}$. These two values
correspond to the saturation levels in both activity indicators. The tail-like
structure that goes from this locus to smaller values in both
$L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$ and
$L_{\mathrm{X}}$/$L_{\mathrm{bol}}$ corresponds to stars in the unsaturated
regime of both indicators. Finally, we highlight in Figure 9 stars in the
supersaturated regime in the
$R_{\mathrm{o}}$-$L_{\mathrm{X}}$/$L_{\mathrm{bol}}$ plane, most of which lie
below the power-law relation.

In Núñez et al. (2017), we found for single members of M37 a weaker
correlation ($r$ = 0.63) and a slope closer to 1:1 ($\alpha$ = 1.05$\pm$0.01).
In that $\approx$500-Myr-old cluster, our sample included stars in the
spectral range K0–M1 that were almost all saturated in both
$L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$ and
$L_{\mathrm{X}}$/$L_{\mathrm{bol}}$. By contrast, our Praesepe and Hyades
sample includes K6–M6 stars with $L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$ and
$L_{\mathrm{X}}$/$L_{\mathrm{bol}}$ measurements (see the colorbar in Figure
9), and a significant number of these are unsaturated. In addition, the
$L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$ values in Núñez et al. (2017) did
not account for the quiescent correction described in Section 6.2, the effect
of which is difficult to quantify in this analysis.

By contrast, He et al. (2019) found $\alpha$ = 1.12$\pm$0.30 for a sample of
field-age K and M dwarfs. This result agrees with ours at the 1$\sigma$ level.
Also, for a sample of M dwarfs within 10 pc, Stelzer et al. (2013) found
$\alpha$ = 1.90$\pm$0.31, implying a steeper slope for the unsaturated
rotation–activity relation—but this value is in 2$\sigma$ agreement with our
value for $\alpha$. The former study accounted for quiescent
$\mathrm{H}\alpha$ absorption, while the latter did not.

It is more informative to directly compare relations for
$L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$ and
$L_{\mathrm{X}}$/$L_{\mathrm{bol}}$ as a function of $R_{\mathrm{o}}$. We re-
create the top right panel of Figures 7 and 8, i.e., the HyPra sample, as the
top and bottom panels (respectively) in Figure 10. We highlight the results
from the MCMC algorithm with solid lines and shaded regions, corresponding to
the maximum a posteriori and 1$\sigma$ MCMC results. We also highlight with
vertical dashed lines the threshold Rossby values, namely,
$R_{\mathrm{o}}$,sat for the
$R_{\mathrm{o}}$–$L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$ relation and
$R_{\mathrm{o}}$,sup and $R_{\mathrm{o}}$,sat for the
$R_{\mathrm{o}}$–$L_{\mathrm{X}}$/$L_{\mathrm{bol}}$ relation.

Figure 10: $L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$ vs. $R_{\mathrm{o}}$ (top
panel) and $L_{\mathrm{X}}$/$L_{\mathrm{bol}}$ vs. $R_{\mathrm{o}}$ (bottom
panel) for single members of Praesepe and the Hyades combined (gray circles).
The maximum a posteriori fits from the MCMC algorithm (see Sections 7.2 and
7.3) and their approximate 1$\sigma$ uncertainties are indicated with solid
lines and shaded regions, respectively. The Rossby threshold values
($R_{\mathrm{o}}$,sup and $R_{\mathrm{o}}$,sat) and their 1$\sigma$
uncertainties are indicated with vertical dashed lines and shaded regions,
respectively, and are annotated next to each line. We extend these vertical
dashed lines along both panels to more easily compare the different regimes
(supersaturated, saturated, and unsaturated) in both chromospheric
($L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$) and coronal
($L_{\mathrm{X}}$/$L_{\mathrm{bol}}$) activity indicators.

In the top panel of Figure 10 the lack of supersaturation in
$L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$ is evident. If the fastest spinners
($R_{\mathrm{o}}$ $\lesssim$ 0.01) appear supersaturated in X-rays but not in
$\mathrm{H}\alpha$, then whatever mechanism is curtailing the magnetically
driven X-ray emission is present in the coronae of these stars, but not in
their chromospheres.

Of the two most invoked mechanisms to explain supersaturation, centrifugal
stripping of the corona (Jardine & Unruh, 1999) and reduction of the filling
factor (Stȩpień et al., 2001), our evidence favors the former, echoing the
conclusions of, e.g., Marsden et al. (2009); Jackson & Jeffries (2010); Wright
et al. (2011). In the centrifugal stripping scenario, the chromospheric layers
would not be affected by the stars’ super rapid rotation, whereas in the
reduced filling factor scenario, all atmospheric layers would be impacted. The
centrifugal stripping scenario would not conflict with the expectation that
some of the chromospheric heating comes from X-rays emitted in the corona. It
is reasonable to expect that most of the X-rays heating the chromosphere
originate in the denser inner layers of the corona. Thus, it is possible for
$L_{\mathrm{X}}$/$L_{\mathrm{bol}}$ to decrease due to plasma loss at the
outermost layers of the corona, while maintaining
$L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$ mostly unaffected.

Figure 11: $L_{\mathrm{X}}$ (top panel) and
$L_{\mathrm{X}}$/$L_{\mathrm{bol}}$ (bottom panel) vs. centrifugal
acceleration for single members of Praesepe and the Hyades that are in the
saturated or supersaturated regimes in the top right panel of Figure 8. Stars
are color coded by their spectral type. Green edges highlight stars in the
supersaturated regime in the
$R_{\mathrm{o}}$–$L_{\mathrm{X}}$/$L_{\mathrm{bol}}$ plane. The x axis is
inverted (i.e., acceleration increases toward the left) for easier comparison
to Figure 8.

As an additional test of whether we are seeing evidence of centrifugal
stripping, we compared X-ray activity to the stellar centrifugal acceleration,
defined as the square of the angular rotation frequency (i.e., the reciprocal
of $P_{\mbox{\scriptsize rot}}$) times the stellar radius:
$\omega^{2}R_{\mathrm{\star}}$. We derived $R_{\mathrm{\star}}$ and 1$\sigma$
uncertainties for main sequence cluster stars using the empirical
$R_{\mathrm{\star}}$–$M_{G}$ relation of E. Mamajek, and they are included in
Table 1. In Figure 11 we plot $L_{\mathrm{X}}$ and
$L_{\mathrm{X}}$/$L_{\mathrm{bol}}$ vs. centrifugal acceleration for Praesepe
and Hyades single members with $R_{\mathrm{o}}$ $<$ $R_{\mathrm{o}}$,sat, and
we highlight with green circles those stars with $R_{\mathrm{o}}$ $<$
$R_{\mathrm{o}}$,sup. We find all stars in the supersaturated regime to have
$\omega^{2}R_{\mathrm{\star}}$ $\buildrel>\over{\sim}$ 1 cm s-2. We see an
indication of supersaturation at $\omega^{2}R_{\mathrm{\star}}$
$\buildrel>\over{\sim}$ 3 cm s-2, although not as clear as in the
$L_{\mathrm{X}}$/$L_{\mathrm{bol}}$–$R_{\mathrm{o}}$ plane.

Also evident in Figure 10 is the smaller $R_{\mathrm{o}}$,sat for
$L_{\mathrm{X}}$/$L_{\mathrm{bol}}$ compared to
$L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$—6$\sigma$ away from each
other.191919As discussed in Section 7.2, our $R_{\mathrm{o}}$,sat
uncertainties are likely underestimated. Therefore, the difference between the
two $R_{\mathrm{o}}$,sat values may not be as pronounced. This difference
indicates that the transition from the saturated to unsaturated regimes does
not occur in tandem in these two layers of the stellar atmosphere. Our HyPra
sample suggests that as stars spin down (i.e., their $R_{\mathrm{o}}$
increases), saturation ends in the corona before it ends in the
chromosphere.202020 In Núñez et al. (2017), we also found a difference in the
two $R_{\mathrm{o}}$,sat values for our sample of M37 stars, but the result
was the opposite: $R_{\mathrm{o}}$,sat was smaller for
$L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$ than for
$L_{\mathrm{X}}$/$L_{\mathrm{bol}}$. However, as we describe in Appendix C,
the M37 sample was significantly smaller and our $\mathrm{H}\alpha$
measurements were contaminated by emission from a foreground nebula, both of
which undermined our analysis. This difference in timing could indicate a
difference in the sensitivity to field components of the magnetic field at
different atmospheric altitudes, which would not be surprising. For example,
See et al. (2019) found that $R_{\mathrm{o}}$,sat for an activity indicator
derived from Zeeman-Doppler imaging, which is particularly sensitive to large-
scale components of the magnetic field (e.g. Brown et al., 1991), is smaller
than that of other activity indicators. Based on our results, we infer that
$L_{\mathrm{X}}$/$L_{\mathrm{bol}}$ is more sensitive to large-scale
components than $L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$.

##  8 Conclusion

We have performed an analysis of chromospheric and coronal activity in low-
mass stars in the Praesepe and Hyades open clusters. These two coeval groups
of stars, with a crucial age between that of very young clusters and that of
field stars, are pivotal in our understanding of the dependence of stellar
magnetic activity on rotation and of the evolution of this dependence.

We used the Praesepe and Hyades membership catalogs of Paper IV, which include
several stellar parameters such as mass, distance, $L_{\mathrm{bol}}$,
$P_{\mbox{\scriptsize rot}}$, $\tau$, and binarity identification, as well as
Gaia and 2MASS photometry. We updated these quantities when appropriate (e.g.,
to Gaia DR3 values), and added to the catalogs the ratio of the continuum flux
near the $\mathrm{H}\alpha$ line to the apparent bolometric flux, $\chi$, and
$T_{\mathrm{eff}}$ for most stars.

We gathered several hundred new optical spectra using the MDM and MMT
Observatories to complement our sample of existing spectra, published nearly a
decade ago in Paper II. We complemented these new spectra with spectra from
the public SDSS and LAMOST catalogs. For a few hundred cluster stars we have
multiple high-quality spectra. We also obtained new X-ray detections and
$L_{\mathrm{X}}$ measurements for an additional ten Praesepe stars and 22
Hyads.

To complement the existing rotational data for Praesepe and Hyades stars, we
measured $P_{\mbox{\scriptsize rot}}$ values using TESS and ZTF light curves
for an additional 28 Praesepe stars and 137 Hyads.

From our optical spectra, we measured the $\mathrm{H}\alpha$ EW and then
estimated a relative EW value after accounting for the quiescent photospheric
$\mathrm{H}\alpha$ absorption present in low-mass stars. We then estimated
$L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$ by using our relative EW values and
an expanded version of our previously published $\chi$–$T_{\mathrm{eff}}$
relation based on PHOENIX model spectra. In the color–EW plane, we find that
at $\approx$700 Myr all late F, G, and K type dwarfs have converged onto a
tight sequence of $\mathrm{H}\alpha$ absorption, and that by contrast nearly
all M dwarfs exhibit some level of $\mathrm{H}\alpha$ emission. In both
clusters, the transition between $\mathrm{H}\alpha$ absorption and emission
occurs at the same spectral type, approximately M0–M1. We also find that
binaries follow the same EW distribution as their single counterparts,
suggesting negligible enhancement of chromospheric activity in binary systems
in the two clusters.

In the $R_{\mathrm{o}}$–$L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$ plane for
the combined sample of single stars from both clusters, we found a saturated
regime for stars with $R_{\mathrm{o}}$ $\lesssim$ 0.3, with a saturation level
($L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$)${}_{\mathrm{sat}}\approx 1.7\times
10^{-4}$. We found an unsaturated regime described by a power-law with slope
$\beta\approx-5.8$ for single members and $\approx$$-2.0$ for binaries; the
former is significantly steeper than the slopes found in similar studies in
the literature. This difference may partly be explained by the quiescent
photospheric correction we implemented and by the updated $\chi$ values we
used. Nonetheless, our unsaturated stars include many more massive stars
($\gtrsim$0.5 ${M_{\odot}}$) than samples in the literature, which may be
driving the steepness of the power-law fit. This steeper slope may be evidence
for a more rapid decay in chromospheric activity for partly convective stars
compared to their fully or almost fully convective counterparts.
Alternatively, the steeper slope may just reflect a shift in chromospheric
radiative cooling mechanism from Balmer lines in the cooler M dwarfs to Ca ii
and Mg ii lines in the hotter G and K dwarfs. Finally, we found no evidence
for supersaturation in $L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$.

We updated the $R_{\mathrm{o}}$–$L_{\mathrm{X}}$/$L_{\mathrm{bol}}$ analysis
in Paper IV by including our expanded sample of new stars with
$P_{\mbox{\scriptsize rot}}$ and $L_{\mathrm{X}}$ measurements. This resulted
in compelling evidence for supersaturation in
$L_{\mathrm{X}}$/$L_{\mathrm{bol}}$ in single stars. At $R_{\mathrm{o}}$
$\lesssim$ 0.01, $L_{\mathrm{X}}$/$L_{\mathrm{bol}}$ decreases following a
power-law with slope $\beta_{\mathrm{sup}}\approx 0.5$. For binaries, on the
other hand, we found no evidence for supersaturation.

A comparison of $L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$ and
$L_{\mathrm{X}}$/$L_{\mathrm{bol}}$ of Praesepe and Hyades single members
revealed a close to 1:1 relation. However, stars are less well-defined by this
1:1 relation at $L_{\mathrm{X}}$/$L_{\mathrm{bol}}$ $\approx 10^{-3}$ and
$L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$ $\approx 10^{-4}$, which correspond
to the activity levels of saturated stars in the two activity indicators.

As Praesepe and Hyades stars show supersaturation at $R_{\mathrm{o}}$
$\lesssim$ 0.01 in the coronal activity indicator
($L_{\mathrm{X}}$/$L_{\mathrm{bol}}$) and not in the chromospheric indicator
($L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$), our data favor centrifugal
stripping as the most likely explanation for this supersaturation. Estimating
the centrifugal acceleration in these stars also provides some evidence for
centrifugal stripping. Also, a smaller $R_{\mathrm{o}}$,sat for the coronal
activity indicator compared to the chromospheric indicator may be evidence for
a higher sensitivity of $L_{\mathrm{X}}$/$L_{\mathrm{bol}}$ to large-scale
magnetic field components.

## Acknowledgments

We thank the anonymous referee for their helpful comments. We thank Matthew
Browning for insightful discussions on stellar magnetic fields. We thank
Elliana S. Abrahams, Emily C. Bowsher, Theron Carmichael, Victoria DiTomasso,
Rose K. Gibson, David Jaimes, Andy Lawler, Mark Popinchalk, Justin Rupert, Ian
Weaver, and José M. Zorrilla for their assistance in obtaining spectra for
this work at the MDM Observatory.

A.N. acknowledges support provided by the NSF through grant 2138089. M.A.A.
acknowledges support from a Fulbright U.S. Scholar grant co-funded by the
Nouvelle-Aquitaine Regional Council and the Franco-American Fulbright
Commission and from a Chrétien International Research Grant from the American
Astronomical Society. M.A.A. also acknowledges support from NASA TESS grant
80NSSC19K0383 (program ID G011197) and NASA grant 80NSSC21K0989. J.L.C. is
supported by NSF grant AST-2009840 and NASA TESS grant 80NSSC22K0299 (program
ID G04217). K.R.C. acknowledges support provided by the NSF through grant
AST-1449476. J.J.D. was supported by NASA contract NAS8-03060 to the Chandra
X-ray Center.

This work is based on observations obtained at the MDM Observatory, operated
by Dartmouth College, Columbia University, Ohio State University, Ohio
University, and the University of Michigan. The authors are honored to be
permitted to conduct astronomical research on Iolkam Du’ag (Kitt Peak), a
mountain with particular significance to the Tohono O’odham.

This paper includes data collected by the TESS mission. Funding for the TESS
mission is provided by the NASA’s Science Mission Directorate.

This work is based on observations obtained with the Samuel Oschin Telescope
48-inch and the 60-inch Telescope at the Palomar Observatory as part of the
Zwicky Transient Facility project (Masci et al., 2019). ZTF is supported by
the National Science Foundation under Grants No. AST-1440341 and AST-2034437
and a collaboration including current partners Caltech, IPAC, the Weizmann
Institute for Science, the Oskar Klein Center at Stockholm University, the
University of Maryland, Deutsches Elektronen-Synchrotron and Humboldt
University, the TANGO Consortium of Taiwan, the University of Wisconsin at
Milwaukee, Trinity College Dublin, Lawrence Livermore National Laboratories,
IN2P3, University of Warwick, Ruhr University Bochum, Northwestern University
and former partners the University of Washington, Los Alamos National
Laboratories, and Lawrence Berkeley National Laboratories. Operations are
conducted by COO, IPAC, and UW.

Some of the data presented in this paper were obtained from the Mikulski
Archive for Space Telescopes (MAST). STScI is operated by the Association of
Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555.
Support for MAST for non-HST data is partly provided by the NASA Office of
Space Science via grant NNX09AF08G.

Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P.
Sloan Foundation, the U.S. Department of Energy Office of Science, and the
Participating Institutions. SDSS acknowledges support and resources from the
Center for High-Performance Computing at the University of Utah.

This work has benefited from the public data released from the Guoshoujing
Telescope (Large Sky Area Multi-Object Fiber Spectroscopic Telescope, or
LAMOST), a National Major Scientific Project built by the Chinese Academy of
Sciences. Funding for the project has been provided by the National
Development and Reform Commission. LAMOST is operated and managed by the
National Astronomical Observatories, Chinese Academy of Sciences.

This research has made use of data obtained from the 4XMM XMM-Newton
Serendipitous Source Catalog compiled by the 10 institutes of the XMM-Newton
Survey Science Centre selected by ESA, and of data obtained from the Chandra
Source Catalog, provided by the Chandra X-ray Center (CXC) as part of the
Chandra Data Archive.

This work has made use of data from the European Space Agency (ESA) mission
Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing
and Analysis Consortium (https://www.cosmos.esa.int/web/gaia/dpac/consortium),
and funded by national institutions. Gaia data was accessed via the VizieR
database of astronomical catalogs (Ochsenbein et al., 2000).

This research has made use of the NASA/IPAC Infrared Science Archive, which is
operated by the Jet Propulsion Laboratory, California Institute of Technology,
under contract with the National Aeronautics and Space Administration. The Two
Micron All Sky Survey was a joint project of the University of Massachusetts
and IPAC.

This research has made use of NASA’s Astrophysics Data System Bibliographic
Services and the SIMBAD database, operated at CDS, Strasbourg, France.

PyRAF is a product of the Space Telescope Science Institute, which is operated
by AURA for NASA. IRAF is distributed by the National Optical Astronomy
Observatories, which are operated by the Association of Universities for
Research in Astronomy, Inc., under cooperative agreement with the National
Science Foundation.

, emcee (Foreman-Mackey et al., 2013), Matplotlib (Hunter, 2007), NumPy
(Harris et al., 2020), PHEW (Núñez et al., 2022a), PypeIt (Prochaska et al.,
2020), PyRAF (Science Software Branch at STScI, 2012), PySpecKit (Ginsburg &
Mirocha, 2011), SciPy (Virtanen et al., 2020), tesscheck (Curtis et al.,
2021), tesscut (Brasseur et al., 2019), unpopular (Hattori et al., 2022)

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##  Appendix A Rotation Analysis with TESS and ZTF Light Curves

###  A.1 An Example Showcasing the Differing Qualities of TESS and ZTF Data

Figure 12 presents the TESS and ZTF light curve data for the Hyad 2MASS
J05512353+1533043 (catalog ) (Gaia DR3 3348035553945613952 (catalog )),
spectral type $\approx$M3.5. At the time our analysis was performed, TESS had
observed this target in Sectors 6, 33, 44, and 45 ($\Delta t\approx$ 1082
days, $N=$ 10,864 observations); ZTF had observed this star over four seasons
($\Delta t\approx$ 1469 days, $N=$ 708 observations). The second column of
Figure 12 zooms in on a representative $\approx$27-d segment (approximately
the length of one TESS sector); this highlights the vast difference in cadence
between ZTF (approximately nightly) and TESS (collected every 30 min during
Cycle 6, and every 10 min for the later sectors operating during the first
extended mission).

The third panel shows Lomb–Scargle periodograms for the individual
sectors/seasons (color-coded according to the light curves plotted in the
first column) and for each full data set (black). Due to the $\sim$nightly
cadence, periodograms for ZTF light curves often show high-frequency peaks
near the 1-day sampling alias. Fortunately in this case, the true period has a
higher power and is corroborated by the TESS periodogram, which is immune to
such aliasing. The extended baseline for each ZTF season, and the consistency
between seasons, ensures that the period recovered is likely the true period
and not a half-period harmonic. Together, ZTF and TESS provide a powerful
opportunity for deriving accurate and precise rotation periods than can be
derived from either survey alone. However, as ZTF saturates at $G\approx$13
mag, and measuring periods with TESS for stars fainter than $G\gtrsim$ 16 and
$P_{\rm rot}>$ 12 d becomes challenging, they also complement each other and
enable the derivation of a more complete rotational census than can be done
with either alone.

In this example, we measured $P_{\mbox{\scriptsize rot}}$ = 7.39$\pm$0.11 d
with TESS and $P_{\mbox{\scriptsize rot}}$ = 7.43$\pm$0.03 d with ZTF, where
the uncertainties are the standard deviations among the Sectors/seasons. In
other cases, we found evidence for longer periods (15–30 d) with TESS but
could not determine the period due to the Sector duration; with ZTF, however,
we were able to clearly determine the long period, while ruling out the
nightly alias periods thanks to TESS.

Figure 12: TESS and ZTF light curves and period analysis for the Hyad 2MASS
J05512353+1533043 (catalog ) (Gaia DR3 3348035553945613952). Top panels show
TESS data and bottom panels show ZTF data. The leftmost panels show all
available light curve data (at the time of our analysis), with each survey’s
time reference to their first epochs (i.e., both start at zero time). The
center-left panels show representative segments of length equal to a single
TESS sector ($\approx$27 days). The center-right panels show Lomb–Scargle
periodograms for each segment (color-coded for individual sectors for TESS;
different seasons for ZTF), and the full data set (black). The rightmost
panels present the phase-folded light curves. Despite differences in data
quality (total baseline, duration of each sector/season, cadence, precision,
angular resolution), the periods precisely agree.

###  A.2 Two Rapidly Rotating Hyads With Discrepancies Between TESS and ZTF

We present the TESS and ZTF light-curve analyses for two Hyads that have
discrepant TESS and ZTF $P_{\mbox{\scriptsize rot}}$ values discussed in
Section 5: 2MASS J02594633+3855363 (catalog ) (Gaia DR3 143558461530827264) in
Figure 13 and J04461522+1846294 (catalog 2MASS J04461522+1846294) (Gaia DR3
3409867964719693824) in Figure 14.

For the first star, the TESS periodogram shows multiple rapid peaks; the most
prominent has a period of 0.3 d. The ZTF periodogram shows some weak peaks in
the 0.1–1.0 d range—the highest peak corresponds to 0.2 d and it looks
convincingly periodic in the phase-folded light curve. This ZTF period appears
to be represented in the TESS periodogram by the second highest peak. Given
the high RUWE for this star (=5.0), we conclude it is likely a binary and TESS
is detecting periods from both binary components. Perhaps the ZTF periodogram
does not show the other significant periods because of the cadence. We adopt
the primary TESS period for this star.

For the second star, the TESS periodogram shows two peaks, which are not
harmonics. The primary TESS period is 0.24 days, whereas the primary ZTF
period is 0.38 d. Although the ZTF periodogram and phase-folded light curves
are not convincing on their own, the ZTF period is consistent with the
secondary peak in the TESS light curve. For that reason, we consider the two
periods detected by TESS to be hosted by the same target (and not caused by an
unrelated star blended in the large TESS pixel). As in the first case, this
target also boasts an elevated RUWE of 3.4, which indicates that the target is
likely a binary. We assign the period for the primary peak in the TESS
periodogram as the period for the primary star of the binary, although it is
also possible that we have attributed the period to the wrong binary
component. However, if that is the case, it will not impact the conclusions of
our work: first, the Rossby number for either period places this star in the
saturated regime; second, we flag all candidate and confirmed binaries and
analyze them separately from the single-star cohort, the latter being the
focus of our work.

Figure 13: TESS and ZTF light curve analysis for the Hyad 2MASS
J02594633+3855363 (catalog ). Top panel: TESS light curve calibrated with
Causal Pixel Modeling showing the characteristic gap midway through the 27-day
observations. Middle top left: a TESS cutout indicating the position of the
star. Middle top center: the Lomb-Scargle periodogram for the TESS light curve
shows at least four significant peaks; the one with the highest power is
indicated with a red triangle (0.30 d). Middle top right: the phase-folded
TESS light curve for the 0.30 d period. Middle bottom: ZTF $r$-band light
curve for three seasons. Bottom left: Lomb-Scargle periodogram for ZTF shows a
weak peak at 0.21 days. Bottom right: the phase-folded ZTF light curve. We
adopt the primary TESS period for this star.

Figure 14: Same as Figure 13 but for the Hyad 2MASS J04461522+1846294 (catalog
). The TESS periodogram shows two significant peaks, the one with the most
power being 0.24 d. The ZTF periodogram shows a peak at 0.38 days, but
presents an unconvincing phase-folded light curve, so we reject the ZTF period
and adopt the primary TESS period for this star.

##  Appendix B Marginalized Posterior Probability Distributions for the MCMC
Analysis of Our $R_{\mathrm{o}}$–$L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$ and
$R_{\mathrm{o}}$–$L_{\mathrm{X}}$/$L_{\mathrm{bol}}$ Models

We present the marginalized posterior probability distributions from the MCMC
analysis we performed on six different subsamples of Praesepe and Hyades
stars: single members of each cluster, binary members of each cluster, single
members of both clusters combined, and binary members of both clusters
combined (see Section 7.2 and Table 6). The binary samples include candidate
and confirmed binaries, which includes stars with RUWE $>$ 1.4. Figure 15
shows an example of the marginalized posterior probability distributions for
the combined sample of single members from both clusters for the
$R_{\mathrm{o}}$–$L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$ model.

Fig. Set15. Marginalized posterior probability distributions from the MCMC
analysis of Our $R_{\mathrm{o}}$–$L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$ and
$R_{\mathrm{o}}$–$L_{\mathrm{X}}$/$L_{\mathrm{bol}}$ Models

Figure 15: Marginalized posterior probability distributions from the MCMC
analysis of our $R_{\mathrm{o}}$–$L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$
model using emcee for single members in both Praesepe and the Hyades. The
parameter values of the a posteriori model are the peaks of the one-
dimensional distributions; the vertical dashed lines approximate the median
and 16th, 50th, and 84th percentiles. The two-dimensional distributions
illustrate covariances between parameters; the contour lines approximate the
1$\sigma$ and 2$\sigma$ levels of the distributions. The complete figure set,
which includes an image for each of the six subsamples in Table 6 and for both
$R_{\mathrm{o}}$–$L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$ and
$R_{\mathrm{o}}$–$L_{\mathrm{X}}$/$L_{\mathrm{bol}}$, is available in the
online journal.

##  Appendix C Re-Analysis of the
$R_{\mathrm{o}}$–$L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$ Relation Without
Applying the Quiescent $\mathrm{H}\alpha$ Absorption Correction

In this Appendix, we include the results we obtained after re-analyzing the
$R_{\mathrm{o}}$–$L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$ relation in Section
7.2 without correcting our $\mathrm{H}\alpha$ EW measurements for the
quiescent $\mathrm{H}\alpha$ absorption present in these stars (see Section
6.2). This allows us to compare our findings to those of previous studies that
did not include this correction, such as Paper II and Núñez et al. (2017).

In Paper II, we found $\beta=-0.73^{+0.16}_{-0.12}$ for the combined sample of
Praesepe and Hyades single members. Our new result in this Appendix,
$\beta=-1.27^{+0.15}_{-0.17}$, is steeper, but within 2$\sigma$. In that
previous study, we also noted a sharp decrease in
$L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$ over a small range in
$R_{\mathrm{o}}$ for stars with $R_{\mathrm{o}}$ $\gtrsim$ 0.45. Whereas this
claim was speculative given the small sample size in Paper II, our current
expanded sample allows us to more confidently confirm it.

Our new $R_{\mathrm{o}}$${}_{\mathrm{,sat}}=0.14$$\pm$$0.01$ agrees within
1$\sigma$ with that found in Paper II,
$R_{\mathrm{o}}$${}_{\mathrm{,sat}}=0.11^{+0.02}_{-0.03}$. Notably, however,
our new ($L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$)sat =
(1.73$\pm$0.06)$\times 10^{-4}$ disagrees with our previous result,
($L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$)sat = (1.26$\pm$0.04)$\times
10^{-4}$, at the 5$\sigma$ level. This discrepancy is caused by the updated
$\chi$ values that we used to calculate
$L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$ in this work. As shown in Figure 5,
our updated $\chi$ values are $\approx$1.3$\times$ larger than those used in
Paper II.

In Núñez et al. (2017), we found $\beta=-0.51$$\pm$$0.02$ for a sample of
single cluster members in the $\approx$500-Myr-old open cluster M37. Our
$\beta$ in this Appendix for the combined sample of Praesepe and Hyades
disagrees with the M37 result at the 5$\sigma$ level. We point out two issues
that may be driving this large difference. First, the sample of
$\mathrm{H}\alpha$ EW measurements in the M37 study was partly contaminated by
$\mathrm{H}\alpha$ emission from a foreground nebula. In that work, an attempt
was made to mitigate the impact of the $\mathrm{H}\alpha$ nebular emission by
excluding stars for which [N ii] emission $\leq-3$ Å. However, this still left
stars with mildly contaminated $\mathrm{H}\alpha$ EW values in the sample,
leading to artificially higher $L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$
values for those stars and potentially a shallower value for the best-fit
$\beta$.

Second, the largest $R_{\mathrm{o}}$ value for a M37 member in that study is
$\approx$0.4, while our study shows the sharpest
$L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$ decline at $R_{\mathrm{o}}$
$\gtrsim$ 0.4. In addition, most of the M37 stars with the lowest
$L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$ and largest $R_{\mathrm{o}}$ values
in the unsaturated regime have $L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$
uncertainties larger than those in our sample. The MCMC algorithm that
calculates $\beta$ incorporates the uncertainty associated with each
measurement by assigning weights to individual data points. Therefore, these
larger $L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$ uncertainties in the M37
sample probably resulted in a shallower $\beta$.

Our new $R_{\mathrm{o}}$,sat for the combined sample of Praesepe and Hyades
stars is larger than the M37 $R_{\mathrm{o}}$${}_{\mathrm{,sat}}=0.03\pm 0.01$
by a factor of almost 5. Also, our new
($L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$)sat disagrees with the M37
($L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$)sat = (1.27$\pm$0.01)$\times
10^{-4}$ at the $>$5$\sigma$ level. Although the latter discrepancy is mostly
explained by the aforementioned differences in $\chi$ values between the two
studies, we found no evident explanation for the discrepancy in
$R_{\mathrm{o}}$,sat. Outdated M37 stellar parameters, including cluster
membership, may be partly driving the large differences in the
characterization of the
$R_{\mathrm{o}}$–$L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$ relation between
our study and that of M37.

Figure 16: Same as Figure 7, but with
$L_{\mathrm{H}\alpha}$/$L_{\mathrm{bol}}$ calculated using our measured EW
values instead of relative EW values from Section 6.2. The latter account for
the $\mathrm{H}\alpha$ quiescent photospheric absorption present in these
stars.