# Statistics of BY Draconis Chromospheric Variable Stars

Deepak Chahal1,2, Richard de Grijs1,2, Devika Kamath1,2 and Xiaodian Chen3
1School of Mathematical and Physical Sciences, Macquarie University, Balaclava
Road, Sydney, NSW 2109, Australia
2Research Centre for Astronomy, Astrophysics and Astrophotonics, Macquarie
University, Balaclava Road, Sydney, NSW 2109, Australia
3CAS Key Laboratory of Optical Astronomy, National Astronomical Observatories,
Chinese Academy of Sciences, Beijing 100101, China E-mail:
<EMAIL_ADDRESS>
(Accepted 2022 June 9. Received 2022 June 2; in original form 2022 April 7)

###### Abstract

We present an extensive catalogue of BY Draconis (BY Dra)-type variables and
their stellar parameters. BY Dra are main-sequence FGKM-type stars. They
exhibit inhomogeneous starspots and bright faculae in their photospheres.
These features are caused by stellar magnetic fields, which are carried along
with the stellar disc through rotation and which produce gradual modulations
in their light curves (LCs). Our main objective is to characterise the
properties of BY Dra variables over a wide range of stellar masses,
temperatures and rotation periods. A recent study categorised 84,697 BY Dra
variables from Data Release 2 of the Zwicky Transient Facility based on their
LCs. We have collected additional photometric data from multiple surveys and
performed broad-band spectral energy distribution fits to estimate stellar
parameters. We found that more than half of our sample objects are of K
spectral type, covering an extensive range of stellar parameters in the low-
mass regime (0.1–1.3 M⊙). Compared with previous studies, most of the sources
in our catalogue are rapid rotators, and so most of them must be young stars
for which a spin-down has not yet occurred. We subdivided our catalogue based
on convection zone depth and found that the photospheric activity index,
$S_{\rm ph}$, is lower for higher effective temperatures, i.e., for thinner
convective envelopes. We observe a broad range of photospheric magnetic
activity for different spectral classes owing to the presence of stellar
populations of different ages. We found a higher magnetically active fraction
for K- than M-type stars.

###### keywords:

stars: main sequence — stars: rotational variables: chromospheric variables:
BY Dra — stars: magnetic field: stellar activity: starspots — stars: stellar
rotation — catalogues — surveys

††pubyear: 2022††pagerange: Statistics of BY Draconis Chromospheric Variable
Stars–Statistics of BY Draconis Chromospheric Variable Stars

##  1 Introduction

In the solar atmosphere, the layers of plasma located above the stellar
photosphere exhibit temperature inversion. The temperature increases to $\sim
10^{4}$ K in the chromosphere, and it peaks at $\sim 10^{6}$ K in the corona.
This heating of the solar atmosphere to such high temperatures is owing to the
dissipation of energy carried by magneto-hydrodynamic waves and/or from
reconnection of stressed magnetic field lines. The exact physics governing
this situation is not yet fully understood. Magnetic fields generated through
the solar dynamo process control the structure of the Sun’s atmosphere. This
leads to phenomena such as large and rapidly evolving starspots (cooler
regions where convection is suppressed by magnetic field lines) in the
photosphere, strong emission lines from the chromosphere, prominent flares or
superflares (sudden releases of magnetic energy) from the corona, etc. (de
Grijs & Kamath, 2021). Hence, both photospheric magnetic activity (originating
in the presence of starspots and bright faculae) and chromospheric magnetic
activity (originating in chromospheric emission lines) are indicators of
stellar magnetic activity in general.

Stellar magnetic activity is likely triggered by the interplay of rotation and
turbulent convection at the stellar surface. Stellar rotation plays an
important role in the dynamo efficiency and affects a star’s magnetic
activity. With increasing age, the loss of angular momentum induced by
magnetised winds and structural variations leads to a rotational spin-down
and, hence, it reduces the dynamo efficiency, in turn leading to reduced
magnetic activity (Parker, 1955; Skumanich, 1972). Therefore, we expect young,
rapidly rotating late-type stars to exhibit significant magnetic activity.
There exist strong correlations among age, stellar activity and rotation
period (Hall, 2008; de Grijs & Kamath, 2021). Magnetic activity is closely
linked to variations in the stellar magnetic field, and it is therefore
connected to the structure of the stellar subsurface zone, stellar rotation
and the regeneration of the magnetic field through self-sustaining dynamo
activity.

These regenerated magnetic field lines produce inhomogeneously distributed
dark spots and bright faculae in the stellar photosphere. These features move
across the stellar disc through rotation, giving rise to a gradual modulation
of the star’s luminosity. Such stars are generally referred to as rotational
modulators or rotational variables. Fitting a rotational variable’s light
curves (LCs) with a multiple-order Fourier function allows us to infer its
rotation period, which is the key parameter required to explain the stellar
activity. The rotation periods of thousands of main-sequence rotational
variables have been obtained from Kepler observations (Reinhold et al., 2013;
McQuillan et al., 2014; Reinhold & Hekker, 2020). The amplitude of the LC
fluctuations can be used as a proxy of their photospheric magnetic activity
(Mathur et al., 2014; Salabert et al., 2016). Larger LC amplitudes imply more
significant magnetic activity, i.e., larger fractions or sizes of starpots on
a star’s surface.

A special class of rotational variables that show strong emission lines in
their chromospheres are BY Draconis (BY Dra) variables. BY Dra stars are low-
mass main-sequence stars which exhibit low variability amplitudes with periods
of a few days; Ca ii H and K (and often hydrogen) emission lines are usually
present in their spectra (Bopp & Fekel, 1977). The variability amplitude of BY
Dra, caused by spots and faculae, can reach on the order of several tenths of
a magnitude for the most active objects, but it may only be on the order of
milli-magnitudes for solar-type stars. Recent observations obtained by the
Kepler Space Telescope (Reinhold et al., 2013; McQuillan et al., 2014;
Reinhold & Hekker, 2020), the Zwicky Transient Facility (ZTF; Chen et al.,
2020) and the Large Sky Area Multi-Object Fibre Spectroscopic Telescope
(LAMOST) Medium-Resolution Survey (Zhang et al., 2019) have provided large
numbers of high-quality photometric (with precisions as good as 0.01 mag) and
spectroscopic time-series observations of rotational modulators. Lanzafame et
al. (2018) presented a catalogue of 147,535 BY Dra candidates, including their
rotation periods and modulation amplitudes.

The exact mechanism which drives the stellar dynamo process—and, hence,
regeneration of the magnetic field in rotational variables—is not yet properly
understood. Therefore, understanding the relationship between stellar rotation
and tracers of magnetic activity is important to understand the physics of the
stellar dynamo. In addition, it is not yet clear either how the magnetic
activity varies across a range of stellar rotation rates, masses and ages.
Moreover, it is still uncertain how different the stellar dynamo process is
from the solar dynamo’s. Therefore, a comparison study with Sun-like active
stars offers significant promise to constrain the stellar dynamo process.
Statistical studies based on large data sets are still too few to be useful to
gain a full understanding of the rotation–activity–age and
activity–brightness–age relationships. More precise observations and estimates
are required to construct a ’chromospheric Hertzsprung–Russell (HR) diagram’.
The latter can be used to show where magnetically active stars are present and
how activity varies across the HR diagram.

Using the large ZTF database of BY Dra candidates, we aim to address and
quantify stellar magnetic activity across a wide range of stellar masses and
rotation periods. Our catalogue (Chen et al., 2020) contains stars of multiple
spectral types, ranging from F- to M-type. We have investigated and
characterised the physical properties of the BY Dra variables using a
combination of LCs and photometric data from multiple surveys. Here we present
an extensive catalogue of 78,954 BY Dra candidates with their stellar
parameters (effective temperature, radius, luminosity, mass), rotation period
and photospheric magnetic index in both the $g$ and $r$ bands. In Section 2,
we focus on the catalogue of ZTF BY Dra candidates. That section contains
information about the photometric data we adopted from multiple surveys and
our further analysis. In Section 3, we discuss the broad-band spectral energy
distribution (SED) fitting process we applied and the derived stellar
parameters. All photometric data and the derived stellar parameters are
included in Tables 1 and 2, respectively. In Section 4, we perform a
statistical analysis of the rotation periods and the photospheric magnetic
index, $S_{\rm ph}$. We discuss the relationships between multiple stellar
parameters derived from our SED fitting with the $S_{\rm ph}$ index and
rotation period for different spectral types. Finally, Section 5 summarises
our results.

##  2 Observations and Data Analysis

In this section, we present the target sample used in this study.

###  2.1 Target Sample

Our initial target sample is drawn from the ZTF data. The ZTF is an optical
time-domain survey undertaken with the 7-inch Oschin–Schmidt telescope at
Palomar Observatory (Masci et al., 2019). It is equipped with a 47 deg2 field
of view, which can scan the entire visible northern sky in one night to median
depths of $\emph{g}\sim 20.8$ mag and $\emph{r}\sim 20.6$ mag.

We adopted the data set from Chen et al. (2020), who used ZTF Data Release 2
(DR2), to search for and classify different types of variable stars. ZTF DR2
contains data acquired between 2018 March and 2019 June, a timespan of
$\sim$470 days. Photometric data are available in the g and r bands, obtained
with uniform exposure times of 30 s per snapshot observation. The search for
periodic variables was done using Lomb–Scargle periodograms (Lomb, 1976;
Scargle, 1982), while the classification was carried out using the DBSCAN
method (see Chen et al., 2020, and references therein) based on the
distribution of periods, LC parameters and luminosities. Chen et al. (2020)
classified 781,602 periodic variables into ${\delta}$ Scuti, EW- and EA-type
eclipsing binaries, fundamental-mode (RRab) and first-overtone (RRc) RR Lyrae,
classical and Type II Cepheids, semi-regular (SR) variables, Miras and
rotational variables, including BY Draconis (BY Dra) and RS Canum Venaticorum
(RS CVn) objects.

####  2.1.1 Classification of Rotational Variables

Figure 1: Example LCs of BY Dra variables. The blue and red points represent
LCs in the $g$ and $r$ bands, respectively. From top to bottom, the
variability amplitude increases; rotation periods are labelled for the
respective LCs.

Figure 2: Distribution (left: equatorial coordinates; right: Galactic
coordinates) of our sample of 78,954 ZTF BY Dra variables. The map on the
right is colour-coded based on variability amplitude.

To estimate the physical parameters associated with the LCs such as the
rotation periods and variability amplitudes, Chen et al. (2020) fitted the LCs
with a fourth-order Fourier function. Sources with poor LC fits were excluded
by optimisation of the $R^{2}$ goodness-of-fit parameter, i.e., they removed
sources with $R^{2}_{g}<0.4$ and $R^{2}_{r}<0.4$. Rotational variables such as
RS CVn and BY Dra exhibit periodic but non-characteristic LCs (see Figure 1).
BY Dra variables exhibit out-of-eclipse LC distortions arising from the
rotation of spotted regions, which leads to slow changes in their mean
brightness. We have found mean brightness variations in our data on the order
of 0.12 mag and 0.10 mag in the _g_ and _r_ bands, respectively. A few BY Dra
example LCs in the _g_ and _r_ bands are shown in Figure 1. The LCs in both
bands look similar. However, in most cases their overall shapes are not
symmetric about either phase $\phi=0.5$ or $\phi=1.5$. These variables have
small variability amplitudes compared with RR Lyrae, Cepheids or eclipsing
binary systems; their rotation periods range from a fraction of a day to well
over 100 days. BY Dra variables are generally late-type, main-sequence dwarf
stars exhibiting enhanced magnetic activity. Chen et al. (2020) observed
flaring features in several BY Dra LCs, but those data points were $\sigma$
clipped prior to LC fitting.

The classification by Chen et al. (2020) based on ZTF DR2 resulted in 84,697
and 81,293 BY Dra and RS CVn stars with purities of 94.6 per cent and 87.2 per
cent, respectively. RS CVn are more evolved active stars. RS CVn and BY Dra
objects exhibit similar rotation periods, although their luminosities differ
significantly, with BY Dra being fainter. Our knowledge of stellar
chromospheric activity will benefit greatly from a careful statistical
analysis of these rotational variables. In this paper, we focus on the sample
of 84,697 BY Dra variables. The full categorised ZTF catalogue is available
online.111http://variables.cn:88/ztf/ Based on a careful examination of the
number density distribution of periodic variable candidates, Chen et al.
(2020) imposed a cut-off at a period of 20 days, which seems a natural cut-off
for BY Dra variables. This cut-off also removed distant semi-regular variables
which may have been misclassified as BY Dra variables, since it is difficult
to distinguish between the two types of variables if Gaia parallaxes are
insufficiently accurate.

We have further refined our target sample by cross-matching our ZTF sample
with the Gaia early DR3 (EDR3; Gaia Collaboration et al., 2021) catalogue.
Stassun & Torres (2021) recently reported that a renormalised unit weight
error (RUWE) goodness-of-fit statistic $\gtrsim$ 1.4 indicates likely
unreliable astrometry. Objects with high RUWE values could be unresolved
binary stars or tight astrometric binaries. Therefore, we have ensured that
all stars in our catalogue have RUWE $<1.4$ and are thus characterised by
reliable Gaia parallaxes (as well as the corresponding Bailer-Jones geometric
distances, see below; Bailer-Jones et al., 2021)). This left a target sample
of 78,954 sources.

The distribution of the 78,954 BY Dra variables in the northern hemisphere is
shown in Figure 2. This distribution shows that the majority of our stars are
located in the Galactic disc, since ZTF has a higher cadence in the Galactic
plane ($\pm$ 7$\degree$). This is different from the Kepler sample, and so our
results may indeed differ compared with work resulting from Kepler data. Our
sample also consists of stars covering a much wider age range. Moreover, the
sample’s distribution outside the Galactic disc is confined to the solar
neighbourhood. The map in Figure 2 (right) shows that the majority of stars in
the solar neighbourhood have lower variability amplitudes (shown with red
colours in Figure 2), possibly because of our ability to detect fainter
magnitude variations at close proximity compared with more distant regimes.

###  2.2 Photometric Data

To estimate the stellar properties of our target sources we used photometric
data covering the optical, near- and mid-infrared wavelength ranges. We found
good cross-matches with Gaia EDR3 (Gaia Collaboration et al., 2021) for 78,954
sources within a tolerance radius of 1 arcsec. Cross-matching was done using
the CDS (Centre de données astronomiques)
X-match222http://cdsxmatch.u-strasbg.fr/ tool. Gaia EDR3 contains updated
astrometry and broad-band photometry in the $G$, $G_{\rm BP}$ and $G_{\rm RP}$
bands. We have used Bailer-Jones et al. (2021) distances—which were estimated
using a probabilistic approach—for our follow-up analysis. We had already
filtered out unreliable distances by removing sources with RUWE $\geq 1.4$.

To include photometric data from other surveys, we also cross-matched our
sample with the infrared AllWISE catalogue (Cutri et al., 2021) and the Two
Micron All-Sky Survey (2MASS) All-Sky Catalog of Point Sources (Cutri et al.,
2003). The 2MASS survey scanned the entire sky uniformly in three near-
infrared bands ($J,H,K$). AllWISE combines data from the Wide-field Infrared
Survey Explorer (WISE; Wright et al., 2010) and Near-Earth Objects WISE
(NEOWISE; Mainzer et al., 2011) surveys to provide the currently most
comprehensive view of the mid-infrared sky in four passbands ($W1,W2,W3,W4$).
Our cross-matching with the ZTF BY Dra sample resulted in 81,479 and 64,835
sources from the 2MASS and AllWISE surveys, respectively.

The photometric uncertainties in a few of our passbands are rather large
compared with the mean amplitude of variability, i.e., they exceed the mean
variability amplitude of 0.1 mag in the $r$ band. We applied 2$\sigma$
clipping to remove all photometric data with uncertainties greater than 0.1
mag ($\sim$2–3 per cent of our objects). Similarly, for the parallaxes and
Gaia distances, all sources with fractional uncertainties greater than
2$\sigma$ were removed. We corrected for extinction in multiple bands using
the 3D map of dust reddening of Green et al. (2019), combined with the
Galactic extinction law (Wang & Chen, 2019). The 2$\sigma$-clipped and
extinction-corrected data are included in Table 1, which contains source co-
ordinates, Gaia astrometry, multipassband photometry, rotation periods and
variability amplitudes. The entire table is available electronically in the
online journal. This photometry is used in our SED fitting, which is discussed
in Section 3.

Table 1: LC data from Chen et al. (2020) combined with photometric data of BY
Dra variable candidates from multiple surveys.

ID |  | RA (J2000)
---
(deg)
| Dec (J2000)
---
(deg)
| Parallax
---
(mas)
| Distance
---
(pc)
| $\langle g\rangle$
---
(mag)
| $\langle r\rangle$
---
(mag)
| $\langle G\rangle$
---
(mag)
| $\langle BP\rangle$
---
(mag)
| $\langle RP\rangle$
---
(mag)
| $\langle J\rangle$
---
(mag)
| $\langle H\rangle$
---
(mag)
| $\langle K\rangle$
---
(mag)
| $\langle W1\rangle$
---
(mag)
| $\langle W2\rangle$
---
(mag)
… |  | Period
---
(days)
| Ampg
---
(mag)
| Ampr
---
(mag)
…
ZTFJ000000.13+620605.8 | 0.00056 | 62.10163 | 1.58 | 636.77 | 16.79 | 15.73 | 15.77 | 16.56 | 14.95 | 13.83 | 13.24 | 13.06 | — | — | … | 1.945 | 0.113 | 0.078 | …
ZTFJ000002.20+480720.8 | 0.00918 | 48.12246 | 1.96 | 510.07 | 16.87 | 15.81 | 15.88 | 16.60 | 15.06 | 14.04 | 13.41 | 13.25 | 13.18 | 13.19 | … | 0.381 | 0.107 | 0.093 | …
ZTFJ000003.23+543605.4 | 0.01347 | 54.60151 | 1.15 | 846.92 | 17.20 | 16.22 | 16.33 | 17.03 | 15.52 | 14.45 | 13.73 | 13.73 | 13.56 | 13.63 | … | 0.786 | 0.166 | 0.112 | …
ZTFJ000003.76+532917.1 | 0.01568 | 53.48811 | 0.89 | 1116.84 | 15.84 | 15.32 | 15.35 | 15.73 | 14.79 | 14.20 | 13.74 | 13.59 | 13.66 | 13.66 | … | 0.592 | 0.164 | 0.159 | …
ZTFJ000004.34+522031.5 | 0.01809 | 52.34209 | 0.92 | 1078.8 | 17.04 | 16.28 | 16.41 | 16.94 | 15.73 | 14.98 | 14.32 | 14.26 | 14.16 | 14.22 | … | 4.083 | 0.087 | 0.095 | …
… | … | … |  |  | … | … | … | … | … | … | … | … | … | … | … | … | … | … | …
… | … | … |  |  | … | … | … | … | … | … | … | … | … | … | … | … | … | … | …

###  2.3 Colour–Magnitude Relations

Figure 3: Gaia absolute $G$-band magnitude versus ($G_{\rm BP}-G_{\rm RP}$)
diagram. Grey regions delineate the F, G, K and M spectral types (Pecaut &
Mamajek, 2013). The blue dashed lines represents the means in our colour bins;
equal-mass (equal-luminosity) binary systems are located 0.753 mag above the
mean single-star isochrone.

To be able to understand the evolutionary sequence of BY Dra variables, we
show the relevant Gaia colour–absolute magnitude diagram in Figure 3. We
compared our colours with the empirically calibrated main-sequence tables of
Pecaut & Mamajek (2013). We found that all of our ZTF sources fall within the
F-to-M spectral-type range, as shown in Figure 3. That figure shows that most
of our sources overlap with the main-sequence regime (shown as grey boxes),
except for the bifurcated sources at fainter magnitudes.

To understand this bifurcation, we compared our ZTF sample with the Kepler
catalogue of rotational modulators (McQuillan et al., 2014; Reinhold & Hekker,
2020). Very few Kepler sources lie along the bifurcation observed for our
sample. Since previous studies have shown that most BY Dra sources also have a
binary companion (for a recent review, see de Grijs & Kamath, 2021), we
conclude that this bifurcation is most likely caused by the presence of binary
systems in our sample, especially K- and M-type binaries. Since we do not have
radial-velocity or spectroscopic data, it is beyond the scope of this paper to
study the physical properties of these binaries. To classify the bifurcated
sources we eliminated any nearly equal-mass binaries which are typically
located 0.753 mag above the single-star main sequence (as shown by the blue
dashed lines in Figure 3).

##  3 Stellar Parameter Estimates

The colour–magnitude diagram offers a good handle on the stellar parameters.
However, better estimates of the stellar effective temperatures can be
obtained through SED fitting. In this section, we present the stellar
parameters derived from our photometry (see Section 2). We constructed the
stellar SEDs and performed broad-band SED fitting. The final stellar
parameters result from the specific SED model which best fits the
observations, based on $\chi^{2}$ minimisation.

###  3.1 Estimating Temperature, Luminosity and Radius based on SED Fitting

Figure 4: Example SEDs of BY Dra candidates. The black filled circles with
error bars are the observed photometric data points, whereas the blue line
represents the best-fitting BT-Settl (CIFIST) model (see the text).

The SEDs were constructed using the available photometric data from the ZTF,
Gaia EDR3, 2MASS and AllWISE surveys. We used the Virtual Observatory SED
Analyzer333http://svo2.cab.inta-csic.es/theory/vosa/ (VOSA; Bayo et al., 2008)
for our broad-band SED fitting. VOSA generates synthetic photometry from
theoretical spectra, which are then compared with the observed data and a
statistical (minimum reduced-$\chi^{2}$) test is performed to determine which
model best reproduces the observed data. We used the BT-Settl (CIFIST) stellar
atmosphere model (Allard et al., 2011). This model contains a grid of
theoretical spectra, which is valid across the entire parameter range of
interest here and uses the Caffau et al. (2011) solar abundances.

The results of the broad-band SED fits are included in Table 2. Table 2 can be
accessed electronically in its entirety in the online journal. The Table 2
contains the source coordinates and stellar parameters such as $T_{\rm eff}$,
$\log g$, luminosity, radius, mass, age and the photospheric magnetic index
($S_{\rm ph}$). We present a few example SED fits in Figure 4. That figure
shows a number of best-fitting SEDs characterised by different
reduced-$\chi^{2}$ values. It is evident that the sample SEDs with large
reduced-$\chi^{2}$ values nevertheless fit most of the data points well.
Therefore, we only removed sources for which we have fewer than seven
photometric data points since such sparse wavelength coverage results in
unacceptably poor SED fits. We note that many SED fits show a slight dip in
the ZTF $r$ band (central wavelength $\lambda=6250$ Å), as can be seen in
Figure 4. This dip might be related to the presence of an H$\alpha$ feature.

Table 2: Stellar Parameters of our sample BY Dra variable candidates

$\rm{ID}$ |  | RA (J2000)
---
(deg)
| Dec (J2000)
---
(deg)
… |  | $T_{\rm eff}$
---
(K)
| $\log g$
---
(cm s-2)
$\chi_{\rm red}^{2}$ |  | Luminosity
---
(L⊙)
| $\pm$
---
(L⊙)
| Radius
---
(R⊙)
| $\pm$
---
(R⊙)
| Mass
---
(M${}_{\odot})$
| …
---
| Age
---
(Gyr)
| …
---
| $S_{{\rm ph},g}$
---
($\rm ppm$)
| $S_{{\rm ph},r}$
---
($\rm ppm$)
ZTFJ000000.13+620605.8 | 0.00056 | 62.10163 | … | 4200 | 4.5 | 9.82 | 0.215 | 0.014 | 0.87 | 0.025 | 0.8 | … | 0.03 | … | 2121 | 1406
ZTFJ000002.20+480720.8 | 0.00918 | 48.12246 | … | 4300 | 5 | 8.65 | 0.117 | 0.005 | 0.61 | 0.01 | — | … | — | … | — | —
ZTFJ000003.23+543605.4 | 0.01347 | 54.60151 | … | 4200 | 2.5 | 5.15 | 0.224 | 0.018 | 0.90 | 0.03 | 0.81 | … | 0.028 | … | 3104 | 2324
ZTFJ000003.76+532917.1 | 0.01568 | 53.48811 | … | 5400 | 4.5 | 6.01 | 0.694 | 0.04 | 0.95 | 0.02 | 0.91 | … | 0.037 | … | 3056 | 2847
ZTFJ000003.34+522031.5 | 0.01809 | 52.34209 | … | 4700 | 3.5 | 2.95 | 0.278 | 0.032 | 0.795 | 0.043 | 0.78 | … | 0.05 | … | 2037 | 1438
… | … | … | … | … | … | … | … | … | … | … | … | … | … | … | … | …
… | … | … | … | … | … | … | … | … | … | … | … | … | … | … | … | …

We probed $T_{\rm eff}\in[2800-7000]$ K in steps of 100 K and allowed $\log g$
to vary between 3.5 and 5.0, i.e., covering the typical range of cool and low-
mass stars (including FGKM spectral types; see the colour–magnitude diagram in
Section 2) (Kubiak et al., 2021). However, note that the SED fitting procedure
is mostly insensitive to $\log g$. Therefore, the main parameter constrained
by our SED fitting is $T_{\rm eff}$. The luminsoities were derived from the
best-fitting observed flux and the radius were estimated from the
Stefan–Boltzmann equation (for more details, see Bayo et al., 2008). Since the
SED analysis is not sensitive to $\log g$, the resulting masses obtained from
using $\log g$ are not well-determined. The $T_{\rm eff}$ values imply that
more than half of our stars are of K spectral type, with the respective
fractions progressively decreasing for G, M and F types (see Table 3).

Table 3: Classification of BY Dra candidates based on their convection zone
depth.

Spectral Type | No. of Sources | Fraction
---|---|---
Class I ($\leqslant$ 3900 K) | 10,336 | 12 per cent
Class II (3900–5400 K) | 47,119 | 56 per cent
Class III ($\geqslant$ 5400 K) | 6879 | 8 per cent

Figure 5: Histogram of stellar parameters derived from the broad-band SED
fits. (right) Luminosities (in units of L⊙); (middle) radii (in units of R⊙);
(left) masses (in units of M⊙).

The effective temperature, $T_{\rm eff}$, controls the width of the convective
envelope in a star. As $T_{\rm eff}$ increases, the convective envelope
becomes thinner. The depth of the convective envelope increases towards lower
effective temperatures and masses; below $\sim 0.3$ M⊙, stars are fully
convective. The presence of a convective envelope plays a significant role in
the regeneration of magnetic fields through dynamo physics (de Grijs & Kamath,
2021, and references therein). Hence, stars with larger convective envelopes
will be more magnetically active. Our sample, which contains predominantly
K-type stars, is dominated by convection. The fraction of fainter, M-type
stars is smaller. The latter are fully convective. The lower fraction of
M-type stars is owing to the fact that our observations are limited by
instrument capabilities. The fraction of F-type stars is just 2 per cent. This
is not unexpected, since these stars have very thin outer convection zones.

To properly understand our full data set of 78,954 objects, we classified all
sources into three classes based on the depths of their convective zones (see
van Saders & Pinsonneault, 2012). Figure 1 of van Saders & Pinsonneault (2012)
shows the variation of the convective zone depth with $T_{\rm eff}$. It shows
steep slopes for lower and higher $T_{\rm eff}$ values and a roughly constant
plateau in between. The statistics of our classification are summarised in
Table 3. Class I objects have $T_{\rm eff}\leqslant 3900$ K. Here, the depth
of the convective zone increases exponentially for progressively lower
effective temperatures. Class II objects have $T_{\rm eff}$ between 3900 and
5400 K, a temperature regime where the convective zone depth is approximately
constant (within 5–10 per cent). Finally, Class III sources have $T_{\rm
eff}\geqslant 5400$ K, where the depth of the convective zone decreases
exponentially with increasing effective temperature.

The ranges of luminosities and radii for different classes of objects are
shown in Figure 5. This figure shows that there is a significant overlap in
radii among the different classes, whereas the luminosities are mostly
distinct. Our derived radius and luminosity estimates suggest that most of our
sample stars are of F to M spectral types.

###  3.2 Estimating Masses and Ages from Isochrone Fitting

With increasing age, a star spins down and, hence, its stellar activity
decreases (Skumanich, 1972). Therefore, information related to masses and ages
is important to understand the stellar activity. We estimated our stellar
masses and ages by fitting a ‘Stellar Parameters Of Tracks with Starspots’
(SPOTS) model (Somers et al., 2020). This model contains a grid of solar-
metallicity stellar evolutionary tracks and isochrones which include treatment
of the structural effects of starspots. The model covers the stellar mass
range of 0.1–1.3 M⊙ and a surface covering fraction from 17 to 85 per cent. In
this model, stars are evolved from the pre-main-sequence to the red-giant-
branch phase.

The mean of the variability amplitude of the ZTF data is approximately 0.14
mag in the $g$ band, which results in a spot fraction of 12 per cent (assuming
that the LC variability is entirely caused by starspots). Therefore, for our
isochrone fitting we have adopted the model with a spot fraction of 17 per
cent, i.e., the SPOT-f017 model. The estimated mass values do not have large
errors, but they show clustering owing to the discrete values of $T_{\rm eff}$
contained in the model. Meanwhile, the age estimates from the isochrone fits
are subject to large errors, and hence they are unreliable.

Figure 5 shows a histogram of the stellar masses for the different classes of
objects, ranging from 0.1 to 1.3 M⊙. The mass distributions are different for
the different classes of objects, with only small overlaps. Figure 5 shows
that our ZTF catalogue contains sources spanning a wide variety of stellar
parameters in the low-mass regime, especially for Class II ($T_{\rm
eff}\in[3900-5400$] K) objects.

###  3.3 Rotation Periods

The rotation periods of our sample objects were estimated by Chen et al.
(2020), based on fits to their LCs, as discussed in Section 2. We have adopted
the latter authors’ periods. There is a possibility that some of the rotation
periods in our catalogue, particularly those with values near 1 day, might be
aliased periods owing to the Earth’s rotation. However, Chen et al. (2020)
were clearly aware of this potential bias. They attempted to avoid it from
occurring as evidenced by, e.g., the distinct dearth of periods near 1 and 0.5
days evident in Figure 6 (and also in other figures in this paper). The
rotation periods of ZTF BY Dra objects range from 0.15 days to 61 days. A
histogram of the rotation periods for 65,735 BY Dra candidates is shown in
Figure 6; the mean of the distribution is 1.86 days. We found that 32 per
cent, 62 per cent and 5 per cent of sources have periods of less than one day,
between 1 and 10 days, and longer than 10 days, respectively. Hence, a large
fraction of our sample objects rotate rapidly compared with the objects in the
Kepler catalogue of rotational modulators (McQuillan et al., 2014; Reinhold &
Hekker, 2020). This means that a large fraction of our sample stars are young
(probably in the early main-sequence phase) and, hence, the spin-down effect
has not yet occurred (Skumanich, 1972). The statistics of the rotation periods
of our ZTF BY Dra sample agree well with those pertaining to the Gaia
catalogue of rotational variables, covering late-type dwarfs (Lanzafame et
al., 2018).

Figure 6: Histogram of the rotation periods of our 65,735 BY Dra candidates,
where different colours correspond to different classes of objects.

In Figure 6, we see a clear bimodality at a rotation period of $\sim$2 days,
for different classes. This bimodal period distribution at $\sim$2 days has
not been observed in previous studies based on Kepler data (McQuillan et al.,
2014; Reinhold & Hekker, 2020; Gordon et al., 2021). The rotation
distributions of the different classes show a lack of intermediate-period
rotators. A similar conclusion was reached by Reinhold & Hekker (2020).
However, we did not find any significant decrease in the variability amplitude
at $\sim$2 days. Therefore, its less likely that the bimodality arises from
the non-detection of rotation periods because of the temporal increase in
bright faculae, which dilutes the starspot effect (Reinhold & Hekker, 2020).

###  3.4 The Photospheric Magnetic Index

LC variability is associated with the rotation of starspots, and thus it can
be used as a proxy of stellar activity. Surface magnetic activity arises owing
to the regeneration of magnetic fields through the dynamo process in the
stellar interior, which is linked to the stellar rotational period. By taking
into account a star’s rotation period, the derived activity index, $S_{\rm
ph}$, can be adopted as a proxy for its magnetic activity (Mathur et al.,
2014). The $S_{\rm ph}$ index is defined as the mean value of the standard
deviation of the LC estimated over $5P_{\rm rot}$. The measured photospheric
magnetic index is only related to magnetism (i.e., starspots) and not to other
sources of variability such as stellar oscillations, convective motions or
instrumental issues (Mathur et al., 2014). Since this index depends on the
angle of inclination of the rotation axis relative to the line of sight, it
must be considered a lower limit to the photospheric activity. Salabert et al.
(2016) compared the $\langle S_{\rm ph}\rangle$ index with the spectroscopic
$S$ index (the usual chromospheric activity index, derived by measuring
chromospheric Ca ii H and K line fluxes) (Wilson, 1968) and found a good
correlation.

Figure 7: Histograms of the photospheric magnetic index ($S_{\rm ph}$) in
(top) the $g$ and (bottom) $r$ bands for different classes of objects.

Histograms of the $S_{\rm ph}$ index in both the $g$ and $r$ bands are shown
in Figure 7, where different colours correspond to different classes, as
specified in Section 3. We see a clear distinction in the distributions of the
different classes, although the distributions are very wide, i.e., spread over
a broad range of the activity index (see Figure 7). The means of the $S_{\rm
ph}$ histograms in the $g$ and $r$ bands are 2486 and 1750 ppm (parts per
million), respectively. Since the ZTF $r$ band is highly sensitive to
chromospheric emission from active regions, the amplitude arising from the
rotation of starspots is diluted by chromospheric emission in the $r$ band.

##  4 Discussion

BY Dra represent a population of magnetically active K–M-type main-sequence
stars which show low amplitude variability and strong chromospheric emission.
Because of their low-amplitude variability, BY Dra variables have not
previously been studied statistically, until the recent study by Lanzafame et
al. (2018). We are interested in understanding the stellar properties and
evolutionary nature of this class of stars. These low-mass main-sequence stars
exhibit low to high magnetic activity in the form of starspots, chromospheric
activity, flares, etc. The observed photospheric and chromospheric activity
indicators have a similar origin. However, the exact origin of these magnetic
events is not yet properly understood.

It has been argued that convection and differential rotation may play a
significant role in the generation of stellar magnetic activity. Turbulent
convection and differential rotation induce dynamical instabilities and,
hence, play a key role in the regeneration of magnetic fields through dynamo
activity. Therefore the presence of a convective envelope (which is related to
stellar mass), as well as differential rotation, are the key ingredients that
induce stellar magnetic activity through the dynamo process. Hence, in this
section we explore the mutual dependencies of different stellar parameters
such as rotation period, effective temperature, mass, etc. on stellar
activity. By virtue of our large catalogue, we have managed to statistically
characterise the relationships of these stellar parameters with the
photospheric magnetic activity index.

###  4.1 Period–Luminosity Relation

To investigate the relationships among our photometric data, our objects’
rotation periods and their variability amplitudes, we plotted the rotation
periods and the ZTF absolute Wesenheit magnitudes (which are a proxy of
extinction-free luminosity) in Figure 8. The Wesenheit magnitude was obtained
through $m_{W_{gr}}=3.712\langle r\rangle-2.712\langle g\rangle$ (Chen et al.,
2020). It is evident that sources corresponding to long rotation periods (slow
rotators) and high luminosities (absolute magnitudes) have lower variability
amplitudes (shown in red in Figure 8) and, hence, relatively lower fractions
of starspots. Meanwhile, more rapidly rotating sources exhibit lower
luminosities and, hence, higher variability amplitudes (shown in blue in
Figure 8), i.e., they have relatively larger fractions of starspots.
Therefore, rapidly rotating lower-luminosity stars of late spectral types,
i.e., possessing thicker convective envelopes, exhibit higher magnetic
activity and, hence, larger fractions of starspots.

Figure 8: Rotation period versus absolute Wesenheit magnitude ($M_{W_{gr}}$)
diagram, colour-coded with the variability amplitude in the g band. Figure 9:
Rotation period versus Gaia ($G_{\rm BP}-G_{\rm RP}$) colour diagram, colour-
coded with the photospheric magnetic index ($S_{\rm ph}$) in the g band.

Figure 10: Spearman correlation coefficient ($\rho$) matrix of stellar
parameters (rotation period, $T_{\rm eff}$, radius, luminosity and mass) and
stellar activity (variability amplitude and photospheric magnetic index).

Figure 11: Rossby Number versus $S_{\rm ph}$ index for different classes of
objects, with the temperature distribution colour-coded. (Top left) Class I;
(top right) Class II; (bottom) Class III. The black point at the bottom is the
position of the Sun.

###  4.2 Period Gap

As highlighted in Section 3, we observe a bimodal distribution at a rotation
period of $\sim$2 days. We also observe a bimodal distribution at the period
of $\sim$10 days (see Figure 6). The latter bimodal distribution has been
found previously in Kepler and K2 data (McQuillan et al., 2014; Reinhold &
Hekker, 2020; Gordon et al., 2021). In order to understand this specific
bimodality in detail, in Figure 9 we have plotted the Gaia colour–rotation
period distribution. In this figure, we observe a small clump, and a gap below
it, at a period of $\sim$10 days (see Figure 9, top left). The clump and gap
are also visible for a constant Rossby number ($\sim$0.5–0.7; see below) in
Class II objects: see Figure 11. The gap is prominent for G–K-type stars. It
appears to be a continuation of the gap previously observed in Kepler and K2
data (McQuillan et al., 2014; Reinhold & Hekker, 2020; Gordon et al., 2021).
Gordon et al. (2021) found that the gap extends from 15–25 days, whereas the
gap in our data occurs at shorter periods and redder colours, as if it were an
extension of their gap. Several explanations of the observed bimodality and
associated gap have been proposed. McQuillan et al. (2014); Davenport & Covey
(2018) suggested that the bimodality may be owing to a bimodal star-formation
history, with an older population of stars forming a slowly rotating branch
and a more recent burst of star formation corresponding to the rapidly
rotating branch. Independently, (Reinhold & Hekker, 2020) suggested that these
features may be owing to the non-detection of a range in rotation periods
because of a temporary increase in bright faculae and, hence, dilution of the
starspot effect.

Gordon et al. (2021) suggested that since the stellar core and envelope are
initially not coupled, the envelope slows down because of magnetic braking,
while the core may continue to spin rapidly. Magnetic braking results in a
spin-down, as the star loses angular momentum through magnetic activity. After
a period of time, the core and envelope start to exchange angular momentum. In
turn, this counteracts the spin-down, thus resulting in an overdensity of
periods below the gap. If this is indeed the case, we should have observed
only a low level of activity at the bottom of the period gap, given that the
spin-down would have been halted. Although we do see lower activity for
F–G-type stars, we also observe a higher level of activity for K–M-type stars
at the bottom of the gap (see Figure 9).

Gordon et al. (2021) also suggested that the underdensity within the gap might
have been caused by an additional stage of accelerated spin-down immediately
following the stalled spin-down phase. This accelerated spin-down could be
owing to a temporary increase in stellar magnetic activity. If such an
accelerated spin-down could have caused the observed underdensity, then the
activity above the period gap should be higher relative to that in the
overdense region at the bottom of the gap. However, we do not see enhanced
activity. Rather, in Figure 9 we observe stars with a relatively low level of
activity above the period gap. Since the photospheric magnetic index is prone
to dilution owing to the presence of bright faculae, it tracks the minimum
stellar activity. A more detailed analysis using the chromospheric activity
index of the period–colour gap will offer greater constraints on the gap’s
origin.

###  4.3 Correlation Study

The mutual dependencies among rotation period, stellar activity and stellar
parameters are complex and, hence, not yet fully understood. Since we have a
large data set of magnetic variables, we have explored correlations among
multiple parameters: see Figure 10. We derived the Spearman correlation
coefficients ($\rho$) for different stellar parameters and provide a
correlation matrix in Figure 10. However, we obtained low correlation
coefficients for most parameter combinations because of the complex
interdependencies. Still, negative correlations are discernible between
certain stellar parameters ($T_{\rm eff}$, luminosity and mass) on the one
hand and, on the other, the amplitude of variability and photospheric magnetic
index (shown in red in Figure 10). This is expected, since with the increase
in stellar parameters (i.e., an increase in $T_{\rm eff}$ or mass), the width
of the convective envelope decreases and, hence, stellar activity decreases.
The stellar radii show a modest anti-correlation with stellar activity.

We expect the stellar activity to increase for rapidly rotating stars. In
Figure 10, we see a modest negative correlation of rotation period with
variability amplitude, as expected. However, we see a slight positive
correlation of rotation period with $S_{\rm ph}$ index, which might be related
to how the $S_{\rm ph}$ index is estimated (i.e., from the mean values of the
standard deviation over $5P_{\rm rot}$). The poor correlation of rotation
period with stellar activity might be because of the relatively narrow range
in rotation period covered by our sample, given that 87 per cent of our
sources have rotation periods between 0.4 and 9 days. In Figure 10 we also
notice the lack of clear correlations between rotation period and certain
stellar parameters ($T_{\rm eff}$, luminosity, radius and mass).

###  4.4 Rossby number–Activity–Temperature

Stellar chromospheric activity correlates very well with the so-called Rossby
number (Noyes et al., 1984), $Ro\equiv P_{\rm rot}/\tau_{\rm c}$, where
$\tau_{\rm c}$ is the convective turnover time. The latter is estimated from
the analytical equation derived from the parametrised fit as a function of
effective temperature (for more details, see Cranmer & Saar, 2011). The Rossby
number probes the interplay between rotation and convection which is
responsible for driving dynamo action. Figure 11 shows the relationships among
the Rossby number, $S_{\rm ph}$ index and $T_{\rm eff}$ for the different
classes of objects.

Among Class III objects, the width of the convective envelope becomes
increasingly thinner, and therefore most of these objects are less active and,
hence, have lower $S_{\rm ph}$ values. Few F-type stars (especially, $T_{\rm
eff}>$ 6500 K) have large Rossby numbers because of very small values of the
convective turnover time, derived from the analytical equation by Cranmer &
Saar (2011) (can be seen as black data-points in Figure 11). Class II objects
show a clear variation with $T_{\rm eff}$; as $T_{\rm eff}$ increases, i.e.,
as the convective envelope becomes thinner, we see a clear decrease in the
$S_{\rm ph}$ index. However, for Class I objects, which are M-type stars, a
similar trend with $T_{\rm eff}$ is not so clear, possibly owing to the
increasingly thicker envelopes and eventually the absence of a tachocline,
resulting in a turbulent dynamo (Strassmeier, 2009). The convective envelope
in this class of stars is so deep that the resulting dynamo detaches from the
surface or loses its connection (as for tachocline stars; Strassmeier, 2009).

We see a broad range in stellar activity among Classes I and II. This might be
caused by the presence of a wide range of ages in the different classes. As
for age, the rotation period of a star increases, i.e., the star rotates
increasingly slowly owing to spin-down. Hence, stellar activity decreases with
increasing age. The broad ranges in the photospheric stellar activity also
arise because of the more even distribution of starspots across the stellar
disc or the presence of starspots away from the line of sight, or both. Hence,
we see a decrease in the variability amplitude in the LCs. Therefore, the
photospheric magnetic index $S_{\rm ph}$ underestimates the magnetic activity
of a star. The Sun’s position is shown as the black filled circle in Figure
11. The $S_{\rm ph}$ index of the Sun was taken from Salabert et al. (2016).
The Sun belongs to a category of less active stars; all of our sources have
activity levels in excess of that of the Sun.

The distribution of Rossby number versus $S_{\rm ph}$ index shows a kink in
both Classes I and II (see Figure 11). The kink corresponds to a change in the
slope of the distribution from a saturated to a linear dependence of the
photometric activity index on Rossby number. This kink corresponds to
$Ro\approx 0.2$–0.3 and $S_{\rm ph}\approx 10^{4}$ ppm, which is in agreement
with the kink’s locus found by Corsaro et al. (2021) and See et al. (2021). In
coronal X-ray emission, Pizzolato et al. (2003); Reiners et al. (2014) found a
similarly saturated regime at smaller Rossby numbers and subsequent linear
dependence. However, we do not observe a clearly saturated region for smaller
Rossby numbers as found by Corsaro et al. (2021). Instead, we see a slight
decline of the $S_{\rm ph}$ index for smaller Rossby numbers. Corsaro et al.
(2021) and See et al. (2021) explored the flat, saturated region to $Ro=0.1$,
whereas our data reach $Ro=0.01$. Therefore, we can say that there is slight
decline in the saturation limit for very small Rossby numbers. The slope of
the decline is steeper for Class I objects, which are highly active, and
decreases for the other classes. This decline might be caused by bright
faculae partially cancelling out dark spots, resulting in a reduction in the
overall variability amplitude. Alternatively, this trend might have originated
from the way the $S_{\rm ph}$ index is estimated (i.e., from the mean values
of the standard deviation over $5P_{\rm rot}$), because we did not find any
such decline in variability amplitude for lower Rossby numbers.

We also found that all categories of stars (i.e., slow, intermediate and rapid
rotators) exhibit a wide range of stellar activity. To fully understand the
context, age is an important parameter that must be taken into consideration.
With increasing age, stars lose angular momentum and spin down, which leads to
a decrease in stellar activity. Age estimates from isochrone fits have large
associated errors. We will be able to constrain rotation periods, stellar
parameters and magnetic activity more optimally once accurate age estimates
have been obtained. We will explore ages in a future paper.

###  4.5 Chromospheric Hertzsprung–Russell diagrams

We have some evidence that stellar magnetic activity is generated through a
process akin to that governing the solar dynamo. Yet, it is still questionable
whether or not the stellar dynamo which generates stellar magnetic activity is
similar to that operating in the Sun. Therefore, studying stellar magnetic
activity with reference to solar activity could help in understanding how
different the stellar dynamo process may be from the solar dynamo for
different classes of stars.

Figure 12: Gaia absolute colour–magnitude diagram (CMD), colour-coded with the
photospheric magnetic index ($S_{\rm ph}$) in the g band. The black point is
the position of the Sun in the Gaia CMD (Casagrande & VandenBerg, 2018). Cross
sign are the probable binary systems classified in Section 2

We show the Gaia colour–absolute magnitude diagram of BY Dra candidates,
including the density distribution of the $S_{\rm ph}$ index, in Figure 12. It
shows that as the temperature increases, the convective envelope becomes
thinner and, hence, the $S_{\rm ph}$ index decreases. Salabert et al. (2016)
estimated the Sun’s maximum $S_{\rm ph}$ index at 314.5 ppm, which is well
below the $S_{\rm ph}$ index measured for the majority of the stars in our
catalogue. As we have already noted, our catalogue contains very young stars
(since most are rapidly rotating), and therefore we observe higher activity
levels for our sample stars compared with the Sun. The Sun belongs to a class
of low-to-moderately magnetically active stars. This result is also in
agreement with Reinhold et al. (2020), who compared 369 solar-like stars with
the Sun.

Using our large catalogue of chromospherically active variables we constructed
a ‘chromospheric HR diagram’, which shows where Sun-like chromospheres are
expected to be found. Figure 12 shows that F- and late-G-type stars exhibit
very low levels of stellar activity and that the activity increases towards
K–M-type stars. By virtue of our large database, we conclude that most of the
rotationally and magnetically active main-sequence stars in our sample are of
later spectral type than F6V. We also found that late-K-type stars are most
magnetically active, even more so than M-type stars. The cross signs in Figure
12 show that the probable binary systems are highly active stars (visible in
blue above the main sequence region at the bottom of Figure 12), which is
expected since the presence of a companion increases the magnetic activity in
the primary star (de Grijs & Kamath, 2021).

###  4.6 The Putative Vaughan–Preston Gap

Figure 13: Gaia ($G_{\rm BP}-G_{\rm RP}$) colour versus photospheric magnetic
index diagram for F- and G-type stars, with the temperature distribution
colour-coded.

An apparent dearth of F- and G-type stars of intermediate activity was
discovered by Vaughan & Preston (1980) based on a sample of 486 stars. We also
investigated this putative ‘Vaughan–Preston gap’ for our sample of active F-
and G-type stars. We present the $S_{\rm ph}$ index versus Gaia colour diagram
in Figure 13 to investigate the dearth of intermediate-activity stars. Figure
13 shows clear signatures of the presence of stars of intermediate activity,
and we observe a good spread in the magnetic activity (500–5000 ppm) of F- and
G-type stars. We do not even observe a bimodal distribution in the
photospheric activity index, as was seen by Boro Saikia et al. (2018). By
virtue of our large data set, we conclude that the putative Vaughan–Preston
gap may have been caused by a lack of data from stars of intermediate activity
and, hence, this gap might not be a true manifestation of stellar dynamo
activity.

##  5 Conclusions

In this paper, we have compiled a catalogue of low-mass main-sequence stars
which exhibit varying levels of stellar magnetic activity. We present a large
catalogue of 78,954 BY Dra candidates, including their photometric
measurements, rotation periods, variability amplitudes, etc. This large
catalogue has aided us in understanding the evolutionary nature of BY Dra
variables. Through broad-band SED fitting, we estimated their stellar
parameters, including the effective temperatures, radii, luminosities, masses,
etc. We found that more than half of our sample objects are K-type stars, and
the respective fraction decreases for G (12 per cent), M (8 per cent) and F (2
per cent) types. The ZTF catalogue of BY Dra candidates contains an extensive
range of stellar parameters in the low-mass regime (0.1–1.3 M⊙).

Our ZTF catalogue contains rapidly rotating stars (94 per cent have rotation
periods $\leqslant$ 10 days) compared with the Kepler catalogue of rotational
modulators (see Reinhold & Hekker, 2020; McQuillan et al., 2014). This high
rotation rate might have resulted because most of our sample stars are young
(probably in the early phases of the main sequence) and, hence, the inevitable
spin-down might not yet have occurred (Skumanich, 1972). To investigate the
dependence of stellar magnetic activity, we have studied their correlation
with multiple stellar parameters. We observed a negative correlation between
the stellar activity (variability amplitude and $S_{\rm ph}$ index) and a
number of stellar parameters ($T_{\rm eff}$, luminosity and mass), i.e., the
activity decreases for hotter stars. This is expected, since with increasing
$T_{\rm eff}$ or mass, the convective envelope becomes thinner, hence stellar
activity decreases. We also noted a negative correlation between variability
amplitude and rotation period, which is expected as rotation induces
instabilities in the convection zone for the regeneration of magnetic fields
through the stellar dynamo.

To understand the different types of stars we classified our catalogue into
three classes based on the slope of the convection zone depth. Class III
objects ($\geqslant$5400 K) have very thin convective envelopes and, hence,
yield low values of the $S_{\rm ph}$ index. Class II (3900–5400 K) objects
show clear signatures of a decrease in the $S_{\rm ph}$ index as $T_{\rm eff}$
increases, i.e., the convective envelope becomes thinner. However, the trend
breaks down for Class I ($\leqslant$3900 K) objects, which show a broad range
of magnetic activity. This is possibly owing to the absence of a tachocline
and, hence, a turbulent dynamo. The extensive range of stellar activity among
all classes might be owing to the presence of a range of stellar ages for all
classes. We observed a declining saturated regime in both Class I and II
objects at lower Rossby numbers, and the kink is the inflexion point between
the saturated and unsaturated regimes. The kink locus we observed is in
agreement with that found by Corsaro et al. (2021); See et al. (2021). Instead
of a flat, saturated region for lower Rossby numbers (down to $Ro=0.01$), we
observed a minor decline of the saturation limit. This might be caused by
bright faculae partially cancelling out dark spots and, hence, reducing the
corresponding activity index.

We constructed a chromospheric HR diagram, which tells us that stellar
activity decreases for stars hotter than early F-type, owing to the lack of a
convective envelope. The activity increases towards K–M-type stars. We found
that K-type stars are more magnetically active even than M-type stars. The
probable binary systems in our catalogue are the most magnetically active
stars. The Sun belongs to a class of low-to-moderately magnetically active
stars, and the majority of our sample stars have an $S_{\rm ph}$ index greater
than the Sun’s. We have not found tightened relationships among stellar
magnetic activity, rotation period and/or a number of other stellar
parameters. Instead, we have found weak correlations among them, likely
because of complex interdependencies. To fully understand these
interdependencies, stellar age must be taken into consideration, since it is a
key parameter that shows strong connections with both activity and rotation
period.

## DATA AVAILABILITY

The ZTF data analysed in this paper are publicly available from the Chen et
al. (2020) and also from the ZTF Archive. The data underlying this article are
available in the article and in its online supplementary material.

## Acknowledgements

D.C. acknowledges funding support from the International Macquarie Research
Excellence Scheme (iMQRES). D.K. acknowledges support from the Australian
Research Council (ARC) through DECRA grant number DE190100813. This research
was also supported in part by the ARC Centre of Excellence for All Sky
Astrophysics in 3 Dimensions (ASTRO 3D), through project number CE170100013.
X.C. acknowledges funding support from the National Natural Science Foundation
of China (NSFC) through grants 12173047 and 11903045. This publication is
based on observations obtained with the Samuel Oschin 48-inch Telescope at
Palomar Observatory as part of the Zwicky Transient Facility project. ZTF is
supported by the U.S. National Science Foundation through grant AST-1440341
and a collaboration including Caltech, IPAC, the Weizmann Institute for
Science, the Oskar Klein Center at Stockholm University, the University of
Maryland, the University of Washington, Deutsches Elektronen-Synchrotron and
Humboldt University, Los Alamos National Laboratories, the TANGO Consortium of
Taiwan, the University of Wisconsin at Milwaukee and Lawrence Berkeley
National Laboratories. Operations are conducted by Caltech Optical
Observatories, IPAC and the University of Washington.

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# TartuNLP @ AXOLOTL-24: Leveraging Classifier Output for New Sense Detection
in Lexical Semantics

Aleksei Dorkin  Kairit Sirts
Institute of Computer Science
University of Tartu
{aleksei.dorkin<EMAIL_ADDRESS>

###### Abstract

We present our submission to the AXOLOTL-24 shared task. The shared task
comprises two subtasks: identifying new senses that words gain with time (when
comparing newer and older time periods) and producing the definitions for the
identified new senses. We implemented a conceptually simple and
computationally inexpensive solution to both subtasks. We trained adapter-
based binary classification models to match glosses with usage examples and
leveraged the probability output of the models to identify novel senses. The
same models were used to match examples of novel sense usages with Wiktionary
definitions. Our submission attained third place on the first subtask and the
first place on the second subtask.

TartuNLP @ AXOLOTL-24: Leveraging Classifier Output for New Sense Detection in
Lexical Semantics

Aleksei Dorkin and Kairit Sirts Institute of Computer Science University of
Tartu {aleksei.dorkin<EMAIL_ADDRESS>

##  1 Introduction

The subject of the AXOLOTL-24 shared task Fedorova et al. (2024) is diachronic
semantic change detection and explanation. Diachronic semantic change is
understood as the change in word meanings (i.e., words losing old senses and
obtaining new ones) over shorter or longer periods. Accordingly, given a
dataset containing usage examples from different periods (old and new), the
task is to identify and define the new senses that words gain in the new time
period compared to the old one.

The goal of the shared task is to implement a semantic change modeling system
for two tasks:

  1. 1)

Correctly assigning existing senses to target word usages and identifying
novel, previously unseen senses;

  2. 2)

Describing the identified novel senses.

The data in the shared task is provided in three languages: Finnish, Russian,
and German (the surprise language for which only the test split is available).
For each language, examples from old and new periods are given. Each data
point consists of a target word and its usage example, gloss (target word
definition), the period the example comes from, and usage and sense IDs. The
data includes glosses for both time periods in the training and validation
splits, while glosses for the new time period are not provided in the test
splits. The “old” and “new” periods differ for each language. For Finnish, old
texts are dated before 1700, and new ones are dated after 1700. For Russian,
the old target word usages are from the 19th century, and the new data
represents modern usages of words. For German, the old period is from 1800 to
1899, and the new period is from 1946 to 1990 Schlechtweg (2023).

Team | ARI | F1
---|---|---
deep-change | 0.413 | 0.750
Holotniekat | 0.312 | 0.641
TartuNLP (ours) | 0.310 | 0.590
IMS_Stuttgart | 0.287 | 0.487
ABDN-NLP | 0.221 | 0.431
WooperNLP | 0.187 | 0.316
Baseline | 0.041 | 0.207
Table 1: Overall results on the Subtask 1. Team | Overall | BLEU | BERTScore
---|---|---|---
TartuNLP (ours) | 0.467 | 0.208 | 0.726
WooperNLP | 0.340 | 0.020 | 0.660
ABDN-NLP | 0.253 | 0.045 | 0.461
baseline | 0.218 | 0.013 | 0.423
Table 2: Overall results on the Subtask 2.

Although we participated in both subtasks, we were primarily interested in the
second subtask of producing definitions for new senses. We implemented a
solution that matches identified novel sense usages with definitions from an
external resource (Wiktionary). Our approach is based on a binary
classification task to predict whether a proposed definition matches the sense
under consideration. We reused this binary classification model for the second
subtask of describing the identified novel senses.

Our system attained the first place on the second subtask (Table 2) and
obtained competitive results on the first subtask (Table 1).

##  2 Methodology

We propose a simple classification-based solution for both subtasks. We adopt
the GlossBERT approach Huang et al. (2019) that treats word sense
disambiguation as a sentence pair classification task, where each pair
comprises a usage example and a sense definition. In turn, we frame the
problem of new sense identification as the problem of matching between usage
examples and sense definitions. Accordingly, the matching problem can be
solved with a binary classification model that, given a usage example and a
sense definition, outputs the probability of the sense definition correctly
describing the usage example.

We adopt the cross-encoder model that simultaneously processes the usage
examples and the sense definitions with the same model. Given a usage example
and a sense inventory, we apply the classification model to predict binary
probabilities for each example/sense definition combination. If the highest
probability over all candidate pairs exceeds a predefined threshold, the
system assigns the highest probability sense to the usage example. Otherwise,
the sense used in the example is deemed to be new.

###  2.1 Subtask 1: Bridging Diachronic Word Uses and a Synchronic Dictionary

This subtask aims to assign a sense ID to every usage example from the new
period; the sense ID may come either from the senses in the old period or, if
the system identifies a novel sense, a unique new sense ID is created.

The data for the first subtask contains sense definitions and correct usages,
which can be used to construct positive examples for our task formulation.
However, having only positive examples for a classification model is generally
insufficient. To produce negative examples, we employ a simple algorithm. We
only consider words associated with at least two distinct sense IDs. For a
given sense ID and its associated gloss, we create all possible combinations
of the gloss with the usage examples associated with the other sense IDs of
the same word (consult Appendix A for additional details). We expect that the
negatives obtained with this algorithm are hard and, as such, are more useful
for training that could be obtained, for instance, via random sampling.

We transform every split of every language by extending it with negative
examples created with the procedure described above. For each language, we
train a separate classification model on the train split and evaluate on the
development split. When training and evaluating the classifier, we do not
consider the period (old or new) from which the examples come. The best
checkpoint for each model is selected based on the development F1 score.

Having trained the classification models, we perform inference on the test set
and transform the output into the expected format. During inference, the usage
examples from the old period are ignored and the classification is performed
only on pairs of usage examples from the new period and sense definitions from
the old period. If the highest predicted probability for a usage example is
above a threshold, we assign the sense ID of the most probable sense
definition to the usage example. Otherwise, a new sense ID is created. The
final result submitted for evaluation contains both the predicted senses for
the examples from the new period as well as the positive examples in the test
split from the old period.111Since the test examples for the old period were
already annotated, we simply copied their sense definitions to the submitted
result file.

For the surprise language—German—the training process is slightly different.
No training or validation data is provided, so we train and validate the
classification model on the positive and negative examples obtained from the
old period in the test data. Inference, however, is exactly the same.

###  2.2 Subtask 2: Definition Generation for Novel Word Senses

Subtask 2 aims to define each novel sense identified in the first subtask.
Despite the name of the subtask, our approach does not generate any new
definitions. We also do not train any additional models. As previously
mentioned, we consider this task a matching problem, except that the
definitions for the novel senses are not present in the data provided in the
shared task. To solve this problem, we scrape the definitions of the surface
forms, for which we identified at least one example as the usage of a novel
sense, from the Wiktionary. More specifically, we scrape the definitions from
the language-specific Wiktionary versions for each language (i.e.
Finnish,222https://fi.wiktionary.org/ Russian,333https://ru.wiktionary.org/
and German444https://de.wiktionary.org/ Wiktionaries).

Having scraped the necessary definitions, we head straight to inference on the
test set. We reuse the models and predictions from the first subtask. We
collect the examples identified as the usages of the novel senses from the
predictions and match them with the Wiktionary definitions using the
classifier models trained in the first subtask. After that, we add the matched
definitions to the predicted new senses.

###  2.3 Implementation Details

Different from GlossBERT Huang et al. (2019), which is based on the BERT
Devlin et al. (2019) model, we instead use XLM-RoBERTa Conneau et al. (2020)
as the base model for our classifiers. XLM-RoBERTa is a multilingual model
that includes Finnish, Russian, and German in its training data. We expect our
system to benefit from the multilinguality. Instead of full fine-tuning, we
opt for parameter-efficient fine-tuning. More specifically, we train
bottleneck adapter Houlsby et al. (2019) classifiers for each language. We
adopted this approach because it makes our solution computationally
lightweight and easily reproducible.

In GlossBERT, Huang et al. (2019) differentiate between training setups with
and without weak supervision, with the former including the defined word
itself in the gloss, as well as highlighting it in the usage example.
According to the experimental results reported by Huang et al. (2019), weak
supervision appears to bring minor improvements in sense prediction. However,
we do not use weak supervision in our submission. The reason is that Finnish
and Russian are substantially more morphologically rich languages than
English; thus, the target words rarely appear in their dictionary forms in the
usage examples. Moreover, in some cases, the orthography also differs between
old and new periods.

To delimit context and gloss, Huang et al. (2019) use the special [SEP] token
that is pre-trained into the BERT model via the next sentence prediction task.
However, RoBERTa Zhuang et al. (2021), and by extension XLM-RoBERTa, omitted
the next sentence prediction task in the pre-training. As a result of that,
the </s> token that is used by RoBERTa in place of [SEP] does not have the
same classification-oriented meaning. For this reason, we employed the
tabulation symbol as the delimiter instead.

For each language, we employed different variations of the base model and
varying training setups. For Finnish, we used the large version of XLM-RoBERTa
and trained for ten epochs in half-precision with a batch size of 128 and 3
steps of gradient accumulation. We observed that the training did not converge
with a smaller effective batch size. For Russian, we trained the classifier
adapter with the base version of XLM-RoBERTa for 50 epochs with a batch size
of 144. We also experimented with the large version of the model for the
Russian language; however, it showed no improvements compared to the base
version. For German, we did not train the classifier from scratch. Instead, we
continued training from the best checkpoint trained on the Finnish data. The
motivation is that there is considerably more data in Finnish than in Russian
in the shared task, so we assume the Finnish model to be stronger. We
continued training the Finnish classifier for 20 epochs in half-precision with
a batch size of 48 and 6 steps of gradient accumulation. All models were
trained with a 5e-4 learning rate.

The threshold value for the classifier’s probability to identify novel senses
was selected as the highest scoring option in the first subtask using the
evaluation script provided by the organizers. We tested a small number of
values in the range of of 0.2 to 0.5 on Russian and determined the best value
to be 0.35. The same value was used for all languages without additional
testing due to time limitations.

The models were trained on the University High-Performance Cluster University
of Tartu (2018). We used a single Tesla V100 GPU for Russian and German, while
for Finnish, we used a single A100 80GB GPU. The time elapsed on training is 9
hours for Finnish, 3 hours for Russian, and 9 minutes for German. We
implemented our solution using the
transformers555https://github.com/huggingface/transformers and the
adapters666https://github.com/adapter-hub/adapters libraries. The source code
and the data are available on
GitHub777https://github.com/slowwavesleep/ancient-lang-adapters/tree/axolotl
and HuggingFace Hub,888https://huggingface.co/datasets/adorkin/axolotl-
wiktionary-definitions respectively.

Team | Fi-BLEU | Ru-BLEU | De-BLEU | Fi-BERTScore | Ru-BERTScore | De-BERTScore
---|---|---|---|---|---|---
TartuNLP (ours) | 0.028 | 0.587 | 0.01 | 0.679 | 0.869 | 0.63
WooperNLP | 0.023 | 0.027 | 0.01 | 0.675 | 0.656 | 0.65
ABDN-NLP | 0.107 | 0.027 | 0.0 | 0.706 | 0.677 | 0.0
baseline | 0.033 | 0.005 | 0.0 | 0.403 | 0.377 | 0.49
Table 3: Language specific results for the Subtask 2.

##  3 Results

The overall results of both subtasks are presented in Tables 1 and 2. For
subtask 1, the metrics reported are the average macro-F1 score and the average
Adjusted Rand Index (ARI) Hubert and Arabie (1985) across target words per
language. The overall F1 and ARI scores are computed as the mean across all
languages. For subtask 2, the evaluation metrics are the BERTScore Zhang et
al. (2020) and BLEU Papineni et al. (2002) averaged across target words per
language. BLEU and BERTScore values for the entire subtask are the respective
averages across all languages. The overall score is the mean of BLEU and
BERTScore.

Our submission attained the third place out of eight participants in the first
subtask and the first place out of four participants in the second subtask
(Table 2). This aligns with our expectations since we focused on the second
subtask from the beginning and applied the system developed for the second
subtask to the first subtask. When looking at the language-specific measures
of subtask 2 (Table 3), one can see considerable differences between
languages. Our system works the best in Russian while also performing well in
German in terms of BERTScore (although the BLEU score is close to 0 for all
systems). In Finnish, our system is competitive in terms of BERTScore but
underperforms compared to the baseline in terms of BLEU.

##  4 Discussion

Our submission to the second subtask is well ahead of the other participants
in the overall leaderboard (Table 2) despite the simplicity of our approach.
However, the language-specific results show that it is not so clear-cut (Table
3). Some of the success can be attributed to accidentally matching the source
of definitions for the Russian language, which is the Russian Wiktionary. We
believe so because the value of the BLEU metric of our submission in the
Russian language is higher than that of the other teams and in the other
languages by an order of magnitude. However, we do not consider this a
critical issue because the BERTScore metric is reasonably high and well above
the baseline for all languages, suggesting that the matched definitions
capture the expected senses well. However, the corresponding low BLEU scores
highlight the inadequacy of the BLEU metric for this task.

Secondly, our approach to the second subtask has limitations. More
specifically, matching usage examples only against the definitions of the
target word, while efficient, considerably limits the system’s ability to
describe completely new senses. Intuitively, a definition associated with a
different word may be a more suitable description of a new sense. A more
robust solution would involve matching usage examples against all available
definitions. However, that would likely require using a bi-encoder
architecture (as proposed by Blevins and Zettlemoyer (2020), for instance)
instead of a cross-encoder due to the computational complexity of matching
every example with every definition.

Wiktionary language | Number of unique pages
---|---
Finnish | 586,439
German | 1,314,597
Russian | 2,877,010
Table 4: The number of unique Wiktionary pages per language.

Accessing the definitions for all the words in a given language-specific
Wiktionary is time-consuming because the layout, article structure, and
templates used are completely different for each Wiktionary version. While
there is a resource providing Wiktionary dumps in a much more convenient
format,999https://kaikki.org/ it is mostly limited in its support to the
English language, with the support for some languages, such as Russian and
German, being work in progress, and for others, such as Finnish, completely
missing at the time of writing. Moreover, the Finnish, German, and Russian
Wiktionaries differ in size and the fullness of their coverage. A rough
estimate can be made by accessing the Special:Statistics page of each
Wiktionary and examining the total number of unique pages (Table 4). We note
the correlation between the smaller sizes of the Finnish and German
Wiktionaries and the lower performance of our system on these languages.

Lastly, although we did not focus on the first subtask, we believe the results
of the sense prediction task obtained with our systems could also be improved.
For instance, the choice of the threshold value for determining a new sense
could be done in a more systematic manner or made learnable. Similarly,
adjusting the training data or the hyperparameters might bring further
improvements.

##  5 Conclusion

This paper described our solution to both subtasks of the AXOLOTL-24 shared
task based on leveraging classifier probabilities for usage example/sense
definition pairs. The developed system is conceptually simple, adopting a
binary classification approach to predict the probability of a sense
definition matching the usage example and employing the adapters framework to
reduce computation resource requirements. Our submission attained the third
place in the first subtask and the first place in the second subtask, showing
the feasibility of our approach.

## Acknowledgements

This research was supported by the Estonian Research Council Grant PSG721.

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##  Appendix A Training Examples

Table 5 presents two training examples for the Russian word “Перо” (Feather).
In the first row, we have a gloss and a matching usage example for the
figurative meaning of the word (a symbol of the writer’s art), which is
denoted by the label 1. Each usage example of the word in its other senses is
paired with this gloss and used as a negative example labeled 0. For instance,
in the second row, the same gloss is paired with a mismatching usage example
in the literal sense of the word. We omit the rest of the negative examples
and the other senses for brevity.

Gloss | Usage example | Label
---|---|---
“Символ искусства писателя, писательского труда, его ремесла.” | “У него бойкое, острое перо.” | 1
“Символ искусства писателя, писательского труда, его ремесла.” | “Перья зверя.” | 0
Table 5: A subset of training examples for a single word.