# Radius Inflation at Low Rossby Number in the Hyades Cluster

Karl Jaehnig11affiliation: Department of Physics, Fisk University, Nashville,
TN, 37208, USA 22affiliation: Department of Physics and Astronomy, Vanderbilt
University, Nashville, TN 37235, USA , Garrett Somers22affiliationmark:
33affiliation: Vanderbilt Initiative in Data-intensive Astrophysics Fellow ,
and Keivan G. Stassun22affiliationmark:  11affiliationmark:
<EMAIL_ADDRESS>
###### Abstract

Radius inflation continues to be explored as a peculiar occurrence among
magnetically active, low-mass stars. Recently Somers & Stassun (2017) showed
that radius inflation among low-mass stars in the young open cluster M45
(Pleiades Cluster) is correlated to the rotation rate: faster rotators are
more inflated. Here we extend that work to a sample of 68 stars of the older
open Hyades Cluster. We derive the stars’ spectral energy distributions to
measure their bolometric fluxes. With spectroscopically defined
$T_{\mathrm{eff}}$ and Gaia distances we calculate stellar radii using the
Stefan-Boltzmann relation. We find numerous stars that exhibit significant
(3–4$\sigma$) radius inflation relative to a nominal cluster isochrone. We
compare these results to that of the younger Pleiades and consider radius
inflation as a function of open cluster evolution. We find that unlike the
Pleiades, there is not a statistically significant correlation between radius
inflation and stellar rotation period. However, we do find that most inflated
stars have (rapid) rotational Rossby numbers of 0.1–0.2, such that the
correlation of radius inflation with Rossby number is statistically
significant at 99.98% confidence. Because the canonical rotation-activity
relation of low-mass stars is understood to result from the connection between
magnetic activity and surface convection, our results imply that magnetic
activity within the convective layers of low-mass stars is what preferentially
drives radius inflation.

Astrophysics - Solar and Stellar Astrophysics

##  1 Introduction

A consensus is emerging that some low-mass stars ($M\lesssim 1M_{\odot}$) have
larger radii than expected from standard stellar theory. This so-called
“radius inflation” effect is typically of order 10-15% and is correlated with
a roughly 5–10% lower effective temperature than predicted. Radius inflation
has been observed in numerous studies using a variety of observational
techniques including eclipsing binary analysis (e.g Popper, 1997; Torres &
Ribas, 2002; López-Morales & Ribas, 2005), statistical projected radii (e.g.
Jackson et al., 2016, 2018), and spectral energy distributions (SEDs) (Somers
& Stassun, 2017, S2017 hereafter). Though the precise mechanism is still
debated, strong magnetic activity seems to play a role either though the
direct inhibition of convective energy transport, the influence of large
starspots on the photospheric pressure and temperature, or a combination of
both effects (e.g. Mullan & MacDonald, 2001; Chabrier, Gallardo & Baraffe,
2007; Macdonald & Mullan, 2010; Feiden & Chaboyer, 2013, 2014; Jackson &
Jeffries, 2014a, b; Somers & Pinsonneault, 2014, 2015b, 2015a). This
conclusion is based on observed correlations between radius inflation and
proxies of magnetic activity such as H$\alpha$ emission, X-rays, and rotation
rate  (e.g. López-Morales & Ribas, 2005; Stassun et al., 2012; Somers &
Stassun, 2017).

S2017 investigated whether any of the single K-dwarfs in the young Pleiades
open cluster showed evidence of radius inflation. By measuring the SEDs of the
stars to determine their luminosities, measuring their effective temperature
using color proxies, and solving the Stefan-Boltzmann equation, S2017
determined the radii of 80+ Pleiads and compared them to stellar evolution
models. They found that for stars rotating with a period slower than 2 d,
corresponding to a Rossby number111The Rossby number is defined as the ratio
of the rotation period to the convective overturn timescale (e.g. Noyes et
al., 1984). (RN) greater than $\sim 0.1$ in the mass range they studied ($\sim
0.7-0.9$$M_{\odot}$), the models predicted the $T_{\mathrm{eff}}$-radius
relation extremely well. However, stars rotating faster than 2 d
(R${}_{N}\lesssim 0.1$) were on average $\sim 12$% larger than predicted. This
is an interesting value of RN, as numerous other studies have found that the
correlation between rotation rate and magnetic proxies saturates at
approximately this value (e.g. Wright et al., 2011). This suggests that the
radius inflation mechanism may be connected to the as-yet unclear physics of
magnetic saturation. Moreover, these inflated Pleaids tended to show higher
lithium abundances as expected from stars experiencing radius inflation during
their early lives (e.g. Somers & Pinsonneault, 2014, 2015b, 2015a).

Following the results of S2017, we now wish to explore the evolution of radius
inflation with stellar age. If radius inflation is a consequence of magnetic
activity and rapid rotation, it stands to reason that the degree of radius
inflation at fixed spectral type should decline with increasing age as stars
both spin down and become less magnetically active (e.g. Skumanich, 1972). If
in fact radius inflation exists for stars with Rossby numbers below $\sim$0.1
as suggested by the results of S2017, then the masses of inflated stars should
be lower on average in older clusters, owing to the slower spin down rate of
less massive stars. As a complement to the Pleiades, this paper will focus on
the Hyades.

The structure of the paper is as follows: In section 2 we discuss the data we
collected to form a complete sample of the Hyades pre-main sequence stars as
well as any relevant data quality measures that were employed. In section 3 we
detail how we calculated the radius inflation for each star in the Hyades. In
section 4 we present the results we found between the derived radii inflation
within the Hyades and stellar rotation of the stars. In section 5 we discuss
the implications of our results with respect to stellar evolution and with
regard to previous work in radii inflation. We present a summary of our
findings in section 6

##  2 Data

###  2.1 The Hyades sample

In order to properly derive effective temperatures for the stars within the
Hyades, it is necessary to build a sample of known members and their magnitude
measurements in different bands. We start with the Hyades membership list from
the Goldman et al. (2013, G2013 hereafter) catalog. G2013 made use of Pan-
STARRS (Chambers et al., 2016) and PPMXL (Roeser et al., 2010) photometry,
complimented by photometry from 2MASS (Skrutskie et al., 2006), SDSS-III DR8
(Aihara et al., 2011), and UKIDSS DR8 (Lawrence et al., 2007), in order to
construct a complete stellar mass function. The G2013 final sample size of
Hyades member stars was 774. The G2013 catalog did not have rotational periods
for its final sample of Hyades stars and so it is necessary to complement the
G2013 photometry with other catalogues which included rotation measurements.

To this end we employ two catalogues which have Hyades member stars with
measured rotational periods. Douglas et al. (2016, D2016 hereafter) measured
rotation periods for 65 Hyades stars using the Kepler spacecraft during its K2
phase in order to study low-mass star gyrochronology. We cross-matched the
sample compiled by D2016 against the 2MASS catalog in order to find
complimentary stars within the membership catalog from G2013. We find that all
65 of the D2016 Hyades stars have cross-matches within the G2013 catalog.

Delorme et al. (2011, D2011 hereafter) also measured rotation periods for 63
Hyades stars in order to also study gyrochronology within the Hyades. We cross
matched the catalog from D2011 with the catalog from G2013 and find 59 Hyades
stars with period-of-rotation measurements. Within this set of 122 stars we
checked for stars with double observations, finding 13 stars that have
observations in both D2011 and D2016, thus reducing our initial sample to 109
stars with individual period-of-rotation measurements. For Hyades stars with
double observations, we employed the period-of-rotation measurements from the
more recent D2016 catalog. We plot the color-magnitude diagram of this sample
in Figure 1.

Having compiled a set of Hyades members with measured periods, we moved to
collect photometry in order to calculate the $T_{\mathrm{eff}}$. We queried
the SIMBAD database (Wenger et al., 2000) in order to gather photometry in the
Johnson pass-bands ($UBVR_{J}I_{J}JHK$).  We found that there is sufficient
$V$ and $K_{S}$ 2MASS photometry to calculate $T_{\mathrm{eff}}$ using the
$V-K_{S}$ color. We cross-matched 2MASS IDs with CDS to download the $K_{S}$
magnitudes and use the $V$ magnitudes from Johnson & Knuckles (1955) in order
to assemble our sample of colors for the Hyades stars.

With these data we assembled our initial sample of 109 Hyades stars with
$V-K_{S}$ photometry, and measured rotation periods. In section 3.2 we go into
detail on deriving $T_{\mathrm{eff}}$ values for our initial sample which
pruned our initial sample size of 109 stars to 68 due to constraints on
applicable color range for the derived color-$T_{\mathrm{eff}}$ polynomial
fits. In section 3.3 we detail our search of the previous literature for
confirmed spectroscopic/visual binaries within our Hyades sample, finding 23
binaries within our sample of 68 Hyades stars. Our final sample of Hyades
stars consists of this group of 45 confirmed single stars and 23 identified
binaries. We provide the 2MASS IDs, positions, photometric measurements, and
measured rotation period for these 68 Hyades stars in Table 1.

Figure 1: Color-magnitude diagram of the Hyades sample used in this work.
Triangles are sources taken from Delorme et al. (2011) and squares are sources
taken from Douglas et al. (2016). The points are all colored by their period
in $\log_{10}$ days.

##  3 Methods

###  3.1 Overview

The constraints applied to the data that we have just described largely follow
the same conventions employed by S2017 in their analysis of the Pleiades
cluster. In the following sections we outline our methods in analyzing our
sample of Hyads. We also introduce a novel method for calculating stellar
radius inflation that takes into account the relationship between
$T_{\mathrm{eff}}$ and radius (see section 3.6).

We aim to compare our results to the Pleiades results from S2017, but their
method for determining $\Delta R$ did not account for this relationship.
Instead, they simply used the isochronal radius at the measured
$T_{\mathrm{eff}}$ of the star as a comparison point. A direct comparison with
their values would therefore be biased. Instead, we recompute radius inflation
for the Pleiades using the methods we have developed in section 3.6.

Using the well known Stefan-Boltzmann law

$L=4\pi R^{2}_{*}\sigma_{\rm SB}T_{\rm eff}^{4},$ (1)

the radius of a star can be derived if its luminosity and $T_{\mathrm{eff}}$
are known with acceptable precision. The luminosity can be determined using
the equation

$L=4\pi d^{2}f_{\rm bol},$ (2)

where $f_{\rm bol}$ is the bolometric flux and $d$ is the distance to the
star. The $T_{\mathrm{eff}}$ calculation remains the most crucial task in
studying radius inflation, as it is the biggest contributor to the final
uncertainty in the calculated radius ($R\propto\ L^{1/2}T^{-2}$). In the
subsections that follow, we briefly go over the derivation of
$T_{\mathrm{eff}}$ , the bolometric flux $f_{\rm bol}$, and the stellar radii
for our sample of Hyads. The methods used in the derivations of
$T_{\mathrm{eff}}$, and $f_{\rm bol}$ are the same used in S2017.

###  3.2  Effective Temperature

In order to calculate $T_{\mathrm{eff}}$ using the $V-K_{S}$ color we need to
account for any possible extinction present in the Hyades. Since the Hyades is
both within the local bubble and an older open cluster (650$\pm$70 Myr, from
Martín et al., 2018), there is very little extinction arising within the
cluster or in the intervening ISM. We adopt $E(B-V)=0.01$ (Gunn & Stryker,
1983), a typical value quoted for the Hyades. To convert this value to other
colors, we employ the standard reddening law from Cardelli et al. (1989) with
a selective reddening of RV = 3.1. This gives us an $E(V-K_{S})$ value of
0.027$\pm$0.01. With this we then proceed to use the empirical metallicity
dependent calibrations from Huang et al. (2015) to calculate
$T_{\mathrm{eff}}$ using $V-K$, adopting [Fe/H] = 0.13$\pm$0.01 (Paulson et
al., 2003). These color-$T_{\mathrm{eff}}$ calibrations are valid over a
$V-K_{S}$ color range of [0.85, 5.05].

We propogated the uncertainties on our $V-K_{S}$ color, the extinction
$E(V-K_{S})$ quoted for the Hyades, and the Hyades metallicity [Fe/H] to get
our overall uncertainties on $T_{\mathrm{eff}}$. These individual
uncertainties on $T_{\mathrm{eff}}$ are in the range of 5–34 K. We also had a
systematic spectroscopic uncertainty of 60 K calculated from the polynomial
fits of Huang et al. (2015). Our overall $T_{\mathrm{eff}}$ uncertainty ranges
over 65–94 K.

###  3.3 Accounting for Binaries within our Hyades Sample

When it comes to the study of radius inflation in single stars, binaries pose
a problem, as they can produce false indications of radius inflation
photometrically. This problem arises from the additional flux from an
unresolved binary system, causing observations to result in a higher measured
magnitude. Typically, photometric binaries are identified because they form
their own main sequence above the single star main sequence in the color-
magnitude diagram. This effect has already been explored by S2017 who searched
for and excluded binaries within their Pleiades sample. We summarize their
conclusions of the effects of binaries on radii inflation below:

  * •

Equal mass binaries will increase bolometric flux without significantly
affecting the measured $T_{\mathrm{eff}}$, thus leading to a significant
signal of radius inflation

  * •

For low-mass-ratio binaries, the total bolometric flux is barely affected but
the near-IR emission is significantly boosted, leading to a lower inferred
$T_{\mathrm{eff}}$ calculation. This can result in a false signal of radius
inflation because the lower inferred temperatures means the star will be
compared to the radius predicted for a lower-mass star.

We proceeded to look through our sample of Hyades stars for binary systems
that have been confirmed with previous surveys. We focused primarily on
finding confirmation either through spectroscopic surveys, which involve
measuring the radial velocity and possible calculation of the orbital
elements, or optical surveys, where the binary is resolved and its motions can
be calculated.

As the Hyades is among the most observed open clusters, it was not difficult
to find several surveys that considered both cluster membership and binarity.
For our sample of Hyades stars we consulted Kopytova et al. (2016) for an
initial list of their identified single stars, as well as the works they used
to exclude stars that were in binary systems. We constructed a list of
confirmed multiple systems using the catalogs of Mermilliod, Mayor & Udry
(2009), Patience et al. (1998), and Duchêne et al. (2013). With this compiled
list of catalogs  we found that 45 stars within our sample of 68 Hyades stars
had been previously identified as single stars.

For the other 23 stars within our sample we queried Vizier for any catalogues
in which the binary status had been confirmed visually or spectroscopically.
We found that 20 of our 23 Hyades stars are confirmed binaries, with
observations coming from the Tycho Double Star Catalog(Fabricius, Høg,
Makarov, Mason, Wycoff & Urban, 2002), the Washington Double Star
Catalog(Mason, Wycoff, Hartkopf, Douglass & Worley, 2001; Douglas, et al.,
2014). We could not find confirmation of binarity, or single star status for
three stars and decided to exclude them from our analysis. The final binary
status for our final sample can be found in the last column of Table 1, where

  * •

No = Confirmed Single Star

  * •

Yes = Confirmed Binary System

  * •

Yes? = Unconfirmed Binary/Single Star.

###  3.4  Bolometric Flux

In order to calculate the total bolometric flux of a star, $f_{\rm bol}$, we
interpolated the observed spectral energy distribution (SED) of each Hyades
star with a standard stellar atmosphere model from the model grid of Kurucz
(2013). This is the same SED interpolation procedure employed in Stassun &
Torres (2016) and was used in S2017.  Note that, as described by Stassun &
Torres (2016), the $f_{\rm bol}$ determined via this interpolation procedure
is virtually independent of $T_{\mathrm{eff}}$.

We adopted the Hyades metallicity and the $T_{\mathrm{eff}}$ derived in
Section 3.2. The multiple photometric observations were collected by querying
Vizier and cover a wide wavelength spectrum from the far-ultraviolet at $\sim
0.15\mu m$ to the far-infrared at $\sim 22\mu m$. The complete list of surveys
queried can be found in Section 2.3.1 of S2017. With these collected
photometric observations spanning a wide wavelength spectrum it was then
possible to construct an SED to calculate the total $f_{\rm bol}$. The
resulting SEDs have a goodness-of-fit, $\chi_{\nu}^{2}$, determined by
comparing the observed fluxes to the passband-integrated model fluxes, and
where the only free parameters in the fit are the reddening (limited to the
allowable range determined above) and the overall flux normalization. These
values are listed in Table 2, along with the calculated bolometric flux.

###  3.5  Stellar Radius

The angular radius can be derived using a rearranged version of the Stefan-
Boltzmann law, accomplished by replacing the luminosity in equation 1 with the
luminosity-flux relation given in equation 2. The resulting equation can be
rearranged such that we get

$\Theta=f_{\rm bol}^{0.5}\ \sigma_{\rm SB}^{-0.5}\ T_{\mathrm{eff}}^{-2}$ (3)

where $\Theta$ is the angular radius, given by $R_{*}/d$. As we have already
calculated the total bolometric flux, as well as the $T_{\mathrm{eff}}$, we
can simply calculate the angular radius for our Hyades Sample by inserting
these values into equation 3. We list the resulting angular radii in units of
milli-arcseconds in Table 2.

Our calculated angular radii for our identified single stars have a range of
[35-100] $\mu as$ with an average error on the calculated angular radius of
$\sim$2.5$\ \mu as$. Overall, the small magnitude of the angular radius error,
along with the relatively small values of $\chi^{2}_{\nu}$ indicate that the
SED fits were accurate.

To get the stellar radius from the angular radius, we multiplied the angular
radius $\Theta$ of a star by its distance from us. The distance to the Hyades
has been found to be about 46.75$\pm\ $0.46 pc, using TGAS data from Gaia Data
Release 1 (Gaia Collaboration, et al., 2017). S2017 employed a single star
cluster distance measurement for the Pleiades to convert their angular radii
derivations to stellar radii, with an uncertainty added to the distance to
reflect the depth of the Pleiades cluster.

With the recent release of Gaia Data Release 2 (Gaia Collaboration et al.,
2018), we now have individual parallax measurements with very high precision
($\sigma_{\pi}\sim 10^{-4}$ arcsec) for our entire Hyades sample. Instead of
simply inverting the Gaia DR2 parallaxes to derive distances, we employed the
distances calculated by Bailer-Jones et al. (2018) using Bayesian Inference
for our individual Hyades stars. We then derived the observed stellar radii by
multiplying our individual angular radii values by the associated individual
distances we have for the Hyades stars. We list these stellar radii values in
units of $R_{\odot}$ in Table 2.

The stellar radii are calculated using three other parmeters, each with their
associated uncertainties. These parameters are the derived $T_{\mathrm{eff}}$,
the total bolometric flux, and the distances for each star. We fitted a
gaussian kernel density estimator to the distributions of fractional errors
for the three aforementioned parameters and plot the resulting probability
density functions in Figure 3. From Figure 3 it is apparent that the error in
$\Theta$ and by extension, $\Delta R_{*}$, is dominated by the error on
$T_{\mathrm{eff}}$, though can be dominated by the error on $f_{\rm bol}$ in
some cases; the error contribution from the Gaia parallax is negligible in all
cases.

Figure 2: Calculated stellar radii for the Hyades stars against their
calculated $T_{\mathrm{eff}}$ values. The diagonal error bars are the
correlated errors between the stellar radius and the $T_{\mathrm{eff}}$. The
red squares are stars with measured rotation periods less than 10 day. The
purple triangles are stars with measured rotation periods greater than 10
days. The empty gray circles are the identified binaries within the sample.
The solid blue line is the isochrone with a cluster age of 625 Myr. Figure 3:
Normalized probability distribution functions of the fractional uncertainties
that $f_{\rm bol}$, $T_{\mathrm{eff}}$, and Distances from Gaia DR2 that
contribute to the error on the calculated stellar radius. These PDFs have been
generated from Kernel Density estimators that were fitted to the distributions
for the final sample of single stars. Here we see that the typical
contributions to the uncertainty on $R_{*}$ are $\lesssim 1\%$, $\lesssim
4\%$, and $\lesssim 3\%$, for distance, $T_{\mathrm{eff}}$, and $f_{\rm bol}$,
respectively.

###  3.6 Calculating Radius Inflation

In order to determine if a star has a larger radius than expected, we compared
our derived values to stellar isochrones from the literature. Isochrones are
frequently used to predict theoretical properties of stellar populations under
the assumption that they are all the same age and composition. In modeling the
Hyades, we decided to use isochrones generated by Somers & Pinsonneault (2014)
with a cluster age of 625 Myr and a metallicity of [Fe/H] = 0.13. We plot this
isochrone alongside our Hyades sample in Figure 2.

The method employed in calculating $R_{iso}$ for the Pleiades in S2017 (see
their section 3.3) is summarized here. Calculating the amount of radius
inflation occurring with a particular star requires the measured stellar
radius and the expected stellar radius based on $T_{\mathrm{eff}}$ from a
representative isochrone. To calculate the isochronal stellar radius,
$R_{iso}$, a spline was fitted to the isochrone relating radius and
$T_{\mathrm{eff}}$. Then $R_{iso}$ is calculated for a particular star by
using the fitted spline to calculate the radius at its calculated
$T_{\mathrm{eff}}$.

We introduced a new method for calculating the expected stellar radius
$R_{iso}$ for a star at a measured $T_{\mathrm{eff}}$.  As we assume that the
luminosity will remain the same whether or not a star is undergoing radius
inflation, we took advantage of this by calculating the luminosity, as given
by equation 1 of our Hyades stars using $R_{*}$ and $T_{\mathrm{eff}}$.

We fit a univariate spline to the isochrone radius and luminosity from the 625
Myr isochrone. With this spline, we can then calculate $R_{iso}$ for each of
our Hyades stars. We can now proceed in calculating the radius inflation that
might be taking place within the Hyades. We employ the same equation from
S2017, which is written as:

$\Delta R_{*}=\frac{R_{*}-R_{iso}}{R_{iso}}$ (4)

Where $R_{*}$ is the measured stellar radius we derived in section 3.5 and
$R_{iso}$ is the radius we calculated using isochrones and the calibrated
$T_{\mathrm{eff}}$. Thus the $\Delta R_{*}$ value can be more simply stated as
the fractional height above the fitted isochrone.

##  4 Results

We list the calculated overall radius inflation, $\Delta R$, for each star
within our final sample in table 2. We also re-plot the color-magnitude
diagram of the Hyades within Figure 4 to include the radius inflation
information. confirmed single stars within our sample are colored by the
$\chi^{2}_{\nu}$ values of the fitted SED and sized according to the
calculated radius of inflation, using equation 4. Confirmed binary stars
within the Hyades are plotted as red x’s and not sized according to their
degree of radii inflation. S2017 showed how their properties affect the
calculation of radius inflation, causing false signals of radius inflation.
Within Figure 4 we see the presence of the binary main-sequence, as well as
the location of two inflated single stars within this binary main-sequence.
Single stars that have undergone radius inflation are thought to exist within
what is considered the binary main-sequence and are incorrectly classified as
photometric binaries. We will comment on this further with respect to our
results in section 5.

###  4.1 Relationship to Rotation Period

As mentioned in $\S$1, radius inflation may be connected to rapid rotation. We
therefore considered the connection between radius inflation and stellar
rotation, as was considered within the younger Pleiades open cluster in S2017,
by looking at the observed rotation period of our Hyades sample and the radius
inflation. In Figure 5 (left panel), we plot the observed rotation periods of
our single stars against their calculated percentage of radii inflation.
Overall we found that our Hyades sample periods are clustered around 10–15
days.

We binned the Hyades stars into two bin: the slow rotating stars (period $>$
10 days) and the fast rotating stars (period $<$ 10 days). Within these two
bins we found the average percentage of radius inflation and plot the two
average values as the orange squares within Figure 5 (left panel).

We explored the degree of monotonicity between radius inflation and rotation
period by calculating the Spearman’s Rank correlation coefficient, denoted by
$\rho_{spearman}$ and the Kendall Tau correlation coefficient, denoted by
$\tau_{kendall}$. These coefficients describe the degree of correlation
between two variables, where a value of 0 is no correlation, values $>$0 are
positive correlations, and values $<0$ are negative correlations. In addition
to the actual correlation coefficient, we calculated their p-values, to test
against the null-hypothesis that these correlations could be due to random
chance.

For period and $\Delta R$, we calculate a value of $\rho_{spearman}=-0.032$
and $\tau_{kendall}=-0.022$, with p-values of 0.832 and 0.829, respectively.
This suggests that there is little to no correlation between period and radius
inflation within this sample.

This result suggests that the relation between rotation and radius inflation
found by S2017 becomes undetectable via observed rotation periods or
disappears as the cluster ages and the star’s magnetic fields become less
active due to age related spin-down. To explore the likelihood of the second
possibility, we considered also the Rossby number (Noyes et al., 1984) of the
stars within the Hyades sample.

###  4.2 Relationship to Rossby Nubmer

The Rossby number ($R_{\rm N})$ is the ratio of the rotation period of a star
over the convective zone overturn time, denoted by $\tau_{cz}$, which is a
timescale to describe how long it takes convective motions to traverse the
convective envelope. Noyes et al. (1984) argued that the Rossby number is the
preferential metric to study magnetic activity because it appears to correlate
with magnetic proxies such as Ca II emission more strongly that just rotation
itself.

In order to calculate the Rossby number for our Hyades sample, we employed the
empirically calculated relationship between $T_{\mathrm{eff}}$ and the
convective overturn time, $\tau_{cz}$ from Wright et al. (2011). We fitted a
cubic spline to this empirical relationship and calculated the expected
convective overturn rate for our Hyades sample, using their individual
$T_{\mathrm{eff}}$ from section 3.2. We calculated the Rossby number for our
Hyades sample by dividing the rotation period of our stars by their calculated
$\tau_{cz}$ value.

We plot the Hyades Rossby numbers against the calculated percentage of radius
inflation in Figure 5 (right panel). We bin this sample by either being slowly
convective (R0 $>$ .25) or rapidly convective (R0 $<$ .25) We see there there
is a greater spread in the distribution of Rossby numbers within the Hyades
sample than with period, indicative of the fact that most of the rapidly
spinning stars are lower mass. We calculated the $\rho_{spearman}$ and
$\tau_{kendall}$ coefficients and attain values of $-0.549$ and $-0.404$,
respectively. The associated p-value are both $\sim$0.0001. This indicates
that there is significantly non-random negative trend between the Rossby
number and the percentage of radius inflation.

Figure 4: Temperature-magnitude diagram of the Hyades sample. The points are
sized proportionally to their calculated $\Delta R_{\star}/R_{\star}$ and
colored according to the $\chi^{2}$ values for their SED fits. The gray dashed
line is a 625Myr isochrone from an atmospheric model from Somers et al. 2019
(in preparation).  Figure 5: Left Panel: Measured rotation period of the
single stars in the Hyades final sample against the calculated percentage of
radius inflation from Section 3.6. The purple points are the confirmed single
stars. The orange squares are the average of the sample, binned into slow
rotators ($P_{\rm rot}$ $>$ 10 d) and fast rotators ($P_{\rm rot}$ $<$ 10 d).
Right Panel: Same as left panel but with the calculated Rossby number ($R_{\rm
N}$) on the x-axis. The orange squares are the average of the sample, binned
into slowly convective ($R_{\rm N}>$ 0.25) and rapidly convective ($R_{\rm
N}<$ 0.25). Figure 6: Top Row: $\Delta R$ versus rotation period for slowly
rotating single Hyades stars (left panel) and the confirmed binary stars
(right panel). Bottom Row: $\Delta R$ versus Rossby number ($R_{\rm N}$) for
“slowly convective” ($R_{\rm N}$ $>$ 0.25) single stars (left panel) and the
confirmed binary stars (right panel). The orange square in each panel is the
average of the plotted sample.  Figure 7: Inferred distances from Gaia DR2
parallaxes (see Bailer-Jones et al., 2018) versus $T_{\mathrm{eff}}$. Error
bars are included for both the distances and $T_{\mathrm{eff}}$. The gray
solid line at 46.75 pc is the typically quoted distance to the center of the
Hyades cluster. The two black horizontal dashed lines delineate the tidal
radius of 10 pc for the Hyades; this region is shaded.
Figure 8: Left plot: Plot of the rotation period and $T_{\mathrm{eff}}$ for
the Hyades single stars (filled-triangles) and the Pleiades single stars
(filled-circles). Right plot: Same as the left plot but now plotting Rossby
number($R_{\rm N}$) and $T_{\mathrm{eff}}$. All points within the two panels
are sized proportionally to their calculated value of radius inflation
($\Delta R$). The three red crosses in the right panel serve as reference
baselines. All points in the two panels are colored according to the sigma
significance of the $\Delta R$.

##  5 discussion

We have employed several measures to ensure that our data are free of spurious
values and that the relationship we have found between radius inflation and
rotational period is not random. In the following section we discuss possible
sources of error and the impact of these sources of error on our findings
within the Hyades cluster. The rest of the section will be dedicated to a
discussion of our findings in the Hyades with respect to the findings of S2017
in the Pleiades. We will also discuss radius inflation as a function of age
using these two clusters.

###  5.1 Potential Sources of Errors

####  5.1.1 Binaries

As discussed above, binaries are a potential source of false-positive signals
of radius inflation since the bolometric flux enhancement from the presence of
a secondary would appear as higher luminosity – thus larger radius at fixed
temperature – in our analysis. While it is nearly impossible to prove that a
star does not have a binary companion, we can assess how efficiently we have
excluded binaries by comparing the derived radii of presumed-single and
confirmed-binary stars rotating slower than $R_{N}\sim 0.1$. As these stars
are not expected to be inflated by magnetic activity, stars with anomalously
large derived radii can be confidently marked as binaries. S2017 did precisely
this, finding significantly disparate distributions between the two samples.
This provided confidence that most, if not all, binaries had been excluded
from the full sample.

We repeat here this same exercise. In the top half of Fig.6, we plot the
periods of single stars with a rotation period greater than 10 days and
binaries against their measured radius inflation. In addition to period, we
plot the Rossby number ($R_{\rm N}$) of the same two sub-samples of stars
against their measured radius inflation in the bottom panel of Figure 6. Like
S2017, we found that there is a significant contribution to calculated radius
inflation due to binarity. We also found that in considering both rotation
period and Rossby number, that the distribution of slowly rotating single
stars and binaries are different enough to claim here that we have effectively
filtered out binaries from our Hyades sample.

####  5.1.2 Distance Errors

The calculation of the stellar radius, which is important as the baseline by
which a star is judged to be radially inflated or not is vulnerable to either
spurious distance measurements or high errors on the distance itself. High
error arises from the typical employment of a cluster distance $\pm$ cluster
depth, where the cluster depth is then taken as the error on the distance of
all the stars. For the Pleiades, this was taken to be 134$\pm$3 pc (from
Soderblom et al., 2005).

We are able to consider the individual distances to all of our Hyades stars
with the Gaia DR2 catalog. The Gaia DR2 catalog measured proper-motions and
parallaxes for over $10^{9}$ stars down to a magnitude limit of $M_{G}\sim
20$, with parallax errors approaching 40 $\mu as$. For our final sample of
Hyades single stars, we found that the Gaia DR2 parallax errors span a range
of [37.8, 511.9] $\mu as$ with an average of 86 $\mu as$. Using the distance
derivations in parsecs from Bailer-Jones et al. (2018) we have an error on the
individual distances of our single stars spanning the range [0.064, 2.435] pc,
with an average distance error of 0.30 pc. We plot the distances and
$T_{\mathrm{eff}}$ for the single stars in our Hyades sample, along with the
relevant errors in Figure 7.

The plotted single stars are all grouped around the typically quoted cluster
distance to the Hyades of 46.75 pc (see Reino et al., 2018). The scatter
within this group exhibits the cluster depth along our line of sight and is
well within the typically cited tidal radius of the Hyades of $\pm$10 pc
(Reino et al., 2018). From the Figure 7, as well as the relatively small
contribution of distance errors shown in Figure 3, it is apparent that the
distance measurements from Gaia DR2 for the Hyades are precise enough to
employ for our conversion from angular radius to stellar radius.

Figure 9: Right Panel: $\Delta R$ versus rotation period for both the Pleiades
(circles) and Hyades (triangles) samples. The blue and red squares correspond
to means of the two bins within the Pleiades and Hyades samples respectively.
The Hyades is binned into slow rotators ($P>10d$) and fast rotators ($P<10$
d). The Pleiades is binned into slow rotators ($P>2$ d) and fast rotators
($P<2$ d). Left panel: Plot of Rossby number ($R_{\rm N}$) against $\Delta R$
for both the Hyades (triangles) and Pleiades (circles) single star samples.
The two red squares are the mean values of $\Delta R$ for the rapidly
convective ($R_{\rm N}<0.2$) bin and the slowly convective ($R_{\rm N}>0.2$)
bin of Hyades stars. The blue squares are means for the Pleiades stars, binned
the same way.

###  5.2 Radius Inflation and Cluster Evolution: From the Pleiades to the
Hyades

We now have two sets of measurements of radius inflation occurring within open
clusters of different ages: The Pleiades cluster which has an estimated age of
about 125 Myr and the Hyades cluster which has an estimated age of 625 Myr.
With this set of data covering a range of about 500 Myr we now have the
opportunity to consider radius inflation in the broader sense of cluster
evolution.

One of the aspects that need to be addressed in comparing the Hyades and
Pleiades is the inherent mass difference we find between the samples. The
Hyades cluster, being about 625 Myr old has spun down considerably more than
the Pleiades. We find the effect of this spin down appear as a lower
$T_{\mathrm{eff}}$ distribution when compared to the younger Pleiades (125
Myr). Our Hyades sample has an average $T_{\mathrm{eff}}$ of 4300 K while the
Pleiades sample has an average $T_{\mathrm{eff}}$ of 5300 K. We test the
similarity of these two distributions using the two sample KS test, finding a
non-random difference between the two samples. We argue that this does not
affect our results as presented, as we are comparing the overall presence of
radius inflation as a function of cluster evolution here and not the relative
degree of radius inflation.

In the right panel of Fig. 8, we plot the rotation rate of single stars in
both the Pleiades (circles) and Hyades (triangles) against their
$T_{\mathrm{eff}}$. Each datum is colored according to its inferred radius
inflation. As shown before, the stars with convincing evidence of inflation
(darkest points) are rotating faster than average for their mass range.
However, the inflated Hyades stars rotate slower than the inflated Pleiades
stars. In the left panel, we show the same plot with the Rossby number
substituted for rotation rate. As low-mass stars have a longer convective
overturn timescale, the Rossby numbers of the inflated Hyades are similar to
those of the faster-rotating Pleiads. In this panel, the inflated members of
both cluster now occupy a similar range, namely $\lesssim 0.2$. That the onset
of inflation seems more closely tied to Rossby number than raw rotation rate
strongly suggests a link with magnetic properties.

An alternative way to view these data is shown in Figure 9. Here we plot the
rotation rate and Rossby number of each star directly against the inferred
radius inflation. While there seems to exist a connection between rotation and
inflation in both clusters, in the sense that below a threshold rotation rate
stars can show significant inflation, the transition rotation rate does not
align for the two clusters. However, in Rossby space we see again an apparent
transition around R${}_{N}\sim 0.2$, above which stars cluster around a low
$\Delta R$ value and above which stars may show large radius inflation values.
The most likely explanation for this alignment is that the same magnetic
mechanism is operating in both clusters, but sets in at different mass
thresholds given the progressive spindown of stars with age.

Many previous studies have considered correlations between Rossby number and
the magnetic properties of stars (Noyes et al., 1984; Pizzolato et al., 2003;
Wright et al., 2011; Douglas, et al., 2014). A notable result is that around a
Rossby number of 0.1–0.2, there is a break in correlation between rotation
rate and magnetic proxies (e.g., H$\alpha$, X-rays, Ca II H&K). Towards larger
Rossby numbers, the magnetic proxies decrease in strength with decreasing
rotation rate, but at lower Rossby numbers the correlation ceases and the
strength of magnetic proxies are approximately constant regardless of the
rotation rate. A similar connection between Rossby number and starspot
properties has also been noted (O’dell et al., 1995) SS17 noted that radius
inflation set in around this Rossby value in the Pleiades, suggesting a
connection. Figure 9 shows that this Rossby threshold appears to hold in the
much older Hyades as well, even though the low-Rossby stars are in a different
mass range. This strongly suggests that the still-unclear physics of magnetic
saturation are related to the onset of radius inflation at high rotation rate.

##  6 conclusions

The causes of radius inflation—the tendency of some low mass stars to be
physically larger by $\sim$10% than both theoretical predictions and other
stars of equal mass, age, and composition—remains an active area of research.
In a recent paper, S2017 searched for the presence of radius inflation among
the low mass members of the Pleiades, finding a significant correlation
between rapid rotation and radius inflation. This strongly implicated dynamo-
generated magnetic fields, which are strong in rapidly-rotating young stars,
as the driver of inflation. In this work, we have expanded the reach of this
investigation to the older Hyades cluster in order to test how the inflation
pattern within clusters evolves as a function of age.

We used archival photometry, distances, and rotation rates for a sample of 68
single-star Hyads between $\sim 0.5-1$ $M_{\odot}$, and empirically determined
their $T_{\mathrm{eff}}$s and radii. By assuming a correlation between
inflated radius and reduced surface temperature (see $\S$3.6) we calculated
the fractional deviation of each star’s radius from theoretical expectations.
From this exercise, we found several Hyades members exhibiting radius
inflation.

These values reveal a strong correlation between the degree of radius
inflation and Rossby number (RN), a robust indicator of the strength of
dynamo-generated magnetic fields. Thus we concluded that Hyades members with
RN below 0.1 (implying very rapid rotation) tend to be physically larger than
their siblings with higher RNs (slower rotation).

Finally, we compared these Hyades results to the findings of S2017 for the
Pleiades. Those authors only reported a correlation with rotation rate, so we
re-analyzed their results with the methods laid out in this paper. We find
that the S2017 Pleiades sample also shows a strong correlation between radius
inflation and RN. Moreover, and remarkably, the Pleiades exhibits the same
transition point at R${}_{\rm N}\sim 0.1$ as the Hyades, with inflated members
below this value and non-inflated members above. This strongly suggests that
this RN threshold for radius inflation is universal, at least over the range
of cluster ages spanned by the Pleiades ($\sim$100 Myr) to the Hyades
($\sim$700 Myr). Future work should focus on characterizing inflation over a
larger range of open cluster ages, in order to assess the full age domain in
which this Rossby-inflation correlation persists.

Notably, the mass range of stars with low RN is lower in the Hyades than in
the Pleiades due to the progressive spin-down of stars below $\sim
1.3$$M_{\odot}$ as they age. This implicates rotation, and not the mass range,
as the primary corollary of radius inflation. More fundamentally, because the
canonical rotation-activity relation of low-mass stars is understood to result
from the connection between magnetic activity and surface convection, our
results imply that magnetic activity within the convective layers of low-mass
stars is what preferentially drives radius inflation.

We thank the referee for their comments and suggestions which greatly improved
the quality and scope of this paper. KOJ and GS acknowledge the support of the
Vanderbilt Office of the Provost through the Vanderbilt Initiative in Data-
intensive Astrophysics (VIDA). KOJ thanks the LSSTC Data Science Fellowship
Program, which is funded by LSSTC, NSF Cybertraining Grant #1829740, the
Brinson Foundation, and the Moore Foundation; their participation in the
program has benefited this work. KGS acknowledges NASA grant 17-XRP17 2-0024
for partial support. This research made use of the cross-match service
provided by CDS, Strasbourg.

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Table 1: Observed Stellar Properties  2MASS | R.A. | Decl. | Distance | Vmag | $B-V$ | $V-K_{S}$ | $P_{\rm rot}$ | Binary?
---|---|---|---|---|---|---|---|---
|  |  | (PC) |  |  |  | (Days) |
03373495+2120355 | 54.3956 | 21.3432 | 45.81${}^{+0.13}_{-0.13}$ | 9.362$\pm$0.008 | 0.92$\pm$0.004 | 2.240$\pm$0.021 | 10.57 | No
03434706+2051363 | 55.946 | 20.86 | 45.21${}^{+0.13}_{-0.13}$ | 14.54$\pm$0.008 | 0.9$\pm$0.004 | 4.927$\pm$0.023 | 12.3 | No
03510309+2354134 | 57.7628 | 23.9037 | 40.83${}^{+0.08}_{-0.08}$ | 10.118$\pm$0.008 | 1.2819$\pm$0.004 | 2.723$\pm$0.021 | 12.57 | Yes
03524101+2548159 | 58.1708 | 25.8044 | 44.41${}^{+0.12}_{-0.12}$ | 11.21$\pm$0.008 |  | 3.317$\pm$0.027 | 14.66 | No
03550142+1229081 | 58.7559 | 12.4855 | 45.98${}^{+0.09}_{-0.09}$ | 10.094$\pm$0.008 | 1.07$\pm$0.004 | 2.519$\pm$0.022 | 11.66 | No
03583581+1306182 | 59.6492 | 13.105 | 61.24${}^{+0.19}_{-0.19}$ | 8.951$\pm$0.008 |  | 1.576$\pm$0.022 | 22.26 | Yes?
03590972+2628340 | 59.7905 | 26.4761 | 35.82${}^{+0.09}_{-0.09}$ | 11.47$\pm$0.008 |  | 3.585$\pm$0.021 | 15.25 | No
03591417+2202380 | 59.809 | 22.0439 | 40.55${}^{+0.08}_{-0.08}$ | 13.066$\pm$0.008 |  | 4.362$\pm$0.018 | 20.73 | No
04033902+1927180 | 60.9125 | 19.455 | 47.97${}^{+0.11}_{-0.11}$ | 10.095$\pm$0.008 | 1.08$\pm$0.004 | 2.495$\pm$0.021 | 11.45 | No
04052565+1926316 | 61.3568 | 19.4421 | 47.23${}^{+0.12}_{-0.12}$ | 11.4$\pm$0.008 | 1.35$\pm$0.004 | 3.292$\pm$0.027 | 13.51 | No
04063463+1332566 | 61.6443 | 13.549 | 49.84${}^{+0.38}_{-0.38}$ | 13.409$\pm$0.008 | 1.47$\pm$0.004 | 4.318$\pm$0.019 | 16.68 | No
04070122+1520062 | 61.7551 | 15.335 | 45.03${}^{+0.24}_{-0.24}$ | 10.473$\pm$0.008 | 1.18$\pm$0.004 | 2.808$\pm$0.027 | 11.98 | No
04074319+1631076 | 61.9299 | 16.5187 | 46.5${}^{+0.11}_{-0.11}$ | 9.924$\pm$0.008 | 1.02$\pm$0.004 | 2.412$\pm$0.027 | 12.3 | No
04081110+1652229 | 62.0462 | 16.873 | 40.09${}^{+0.07}_{-0.07}$ | 11.51$\pm$0.008 | 1.44$\pm$0.004 | 3.577$\pm$0.025 | 13.63 | No
04082667+1211304 | 62.1111 | 12.1918 | 46.38${}^{+0.15}_{-0.15}$ | 11.269$\pm$0.008 | 1.33$\pm$0.004 | 3.223$\pm$0.028 | 12.96 | No
04083620+2346071 | 62.1508 | 23.7686 | 47.45${}^{+0.13}_{-0.12}$ | 9.41$\pm$0.008 | 0.9$\pm$0.004 | 2.087$\pm$0.016 | 9.35 | No
04084015+2333257 | 62.1673 | 23.5571 | 46.02${}^{+0.17}_{-0.17}$ | 12.861$\pm$0.008 |  | 4.332$\pm$0.025 | 21.02 | Yes
04115620+2338108 | 62.9841 | 23.6363 | 40.3${}^{+1.19}_{-1.13}$ | 9.392$\pm$0.008 |  | 2.982$\pm$0.025 | 2.309 | Yes
04151038+1423544 | 63.7932 | 14.3984 | 47.95${}^{+0.09}_{-0.09}$ | 11.585$\pm$0.008 | 1.38$\pm$0.004 | 3.457$\pm$0.018 | 13.91 | No
04153367+1542226 | 63.8902 | 15.7062 | 45.52${}^{+0.14}_{-0.14}$ | 10.931$\pm$0.008 |  | 3.077$\pm$0.024 | 47 | Yes
04163346+2154269 | 64.1394 | 21.9074 | 51.02${}^{+0.11}_{-0.11}$ | 9.125$\pm$0.008 | 0.81$\pm$0.004 | 1.849$\pm$0.024 | 10.26 | No
04172512+1901478 | 64.3547 | 19.0299 | 47.79${}^{+0.18}_{-0.18}$ | 10.8$\pm$0.008 | 1.22$\pm$0.004 | 2.891$\pm$0.024 | 12.95 | No
04172811+1454038 | 64.3671 | 14.901 | 49.45${}^{+0.13}_{-0.13}$ | 14.47$\pm$0.008 | 1.55$\pm$0.004 | 4.849$\pm$0.020 | 2.35 | No
04174767+1339422 | 64.4486 | 13.6617 | 47.39${}^{+0.12}_{-0.12}$ | 12.54$\pm$0.008 |  | 4.012$\pm$0.020 | 8.685 | Yes
04175061+1828307 | 64.4609 | 18.4751 | 46.65${}^{+0.31}_{-0.31}$ | 13.954$\pm$0.008 |  | 4.962$\pm$0.019 | 22.94 | Yes
04175555+1432464 | 64.4814 | 14.5462 | 52.38${}^{+0.2}_{-0.2}$ | 14.89$\pm$0.008 |  | 4.873$\pm$0.018 | 10.11 | No
04181077+2317048 | 64.5449 | 23.2846 | 53.92${}^{+0.13}_{-0.13}$ | 9.471$\pm$0.008 |  | 2.534$\pm$0.017 | 1.862 | Yes
04223953+1816097 | 65.6647 | 18.2693 | 41.66${}^{+0.09}_{-0.09}$ | 13.04$\pm$0.008 |  | 4.313$\pm$0.019 | 23.12 | No
04232283+1939312 | 65.8451 | 19.6586 | 45.71${}^{+0.34}_{-0.34}$ | 9.381$\pm$0.008 | 0.91$\pm$0.004 | 2.230$\pm$0.021 | 9.9 | No
04232526+1545474 | 65.8552 | 15.7631 | 41.47${}^{+0.15}_{-0.15}$ | 10.472$\pm$0.008 | 1.24$\pm$0.004 | 2.977$\pm$0.019 | 12.38 | No
04235070+0912193 | 65.9612 | 9.20538 | 44.83${}^{+0.11}_{-0.11}$ | 12.896$\pm$0.008 | 1.51$\pm$0.004 | 4.67$\pm$0.019 | 5.33 | No
04235911+1643178 | 65.9963 | 16.7216 | 46.0${}^{+0.12}_{-0.12}$ | 12.514$\pm$0.008 | 1.49$\pm$0.004 | 3.952$\pm$0.019 | 17.14 | No
04241691+1800107 | 66.0704 | 18.0029 | 46.66${}^{+0.09}_{-0.09}$ | 9.966$\pm$0.008 | 1.06$\pm$0.004 | 2.441$\pm$0.019 | 11.6 | No
04250024+1659057 | 66.251 | 16.9849 | 54.61${}^{+0.19}_{-0.19}$ | 10.248$\pm$0.008 | 1.03$\pm$0.004 | 2.417$\pm$0.017 | 11.77 | No
04251456+1858250 | 66.3106 | 18.9736 | 52.87${}^{+0.2}_{-0.2}$ | 12.728$\pm$0.008 | 1.48$\pm$0.004 | 4.022$\pm$0.019 | 10.84 | Yes
04252501+1754552 | 66.3542 | 17.9153 | 46.85${}^{+0.09}_{-0.09}$ | 11.128$\pm$0.008 |  | 3.151$\pm$0.019 | 70 | Yes
04254754+1801022 | 66.448 | 18.0172 | 42.11${}^{+1.05}_{-1.0}$ | 8.9889$\pm$0.008 | 0.94$\pm$0.004 | 2.221$\pm$0.022 | 8.46 | Yes
04264825+1052160 | 66.701 | 10.8711 | 46.97${}^{+1.01}_{-0.97}$ | 9.432$\pm$0.008 | 1.04$\pm$0.004 | 2.549$\pm$0.021 | 10.4 | Yes
04272532+1415384 | 66.8555 | 14.2606 | 52.59${}^{+0.51}_{-0.5}$ | 10.313$\pm$0.008 | 1.08$\pm$0.004 | 2.600$\pm$0.024 | 12.77 | No
04274701+1425041 | 66.9459 | 14.4178 | 49.32${}^{+0.1}_{-0.1}$ | 9.475$\pm$0.008 | 0.92$\pm$0.004 | 2.127$\pm$0.021 | 9.7 | No
04275895+1830009 | 66.9956 | 18.5002 | 51.06${}^{+0.27}_{-0.27}$ | 10.129$\pm$0.008 |  | 2.596$\pm$0.019 | 11.13 | Yes
04282878+1741453 | 67.1199 | 17.6959 | 46.83${}^{+0.58}_{-0.57}$ | 12.122$\pm$0.008 | 1.49$\pm$0.004 | 4.411$\pm$0.027 | 2.42 | Yes
04303385+1444532 | 67.641 | 14.7481 | 49.98${}^{+1.17}_{-1.12}$ | 14.72$\pm$0.008 | 1.56$\pm$0.004 | 5.008$\pm$0.019 | 18.41 | Yes
04303486+1544023 | 67.6452 | 15.7339 | 57.07${}^{+0.9}_{-0.87}$ | 8.84$\pm$0.008 | 0.84$\pm$0.004 | 2.022$\pm$0.019 | 8.73 | Yes
04315244+1529585 | 67.9685 | 15.4995 | 45.73${}^{+0.08}_{-0.08}$ | 11.0$\pm$0.008 | 1.31$\pm$0.004 | 3.105$\pm$0.025 | 13.13 | No
04322565+1306476 | 68.1068 | 13.1132 | 46.72${}^{+0.1}_{-0.1}$ | 11.0$\pm$0.008 | 1.19$\pm$0.004 | 3.346$\pm$0.021 | 1.48 | Yes
04332699+1302438 | 68.3624 | 13.0455 | 43.09${}^{+0.1}_{-0.1}$ | 13.328$\pm$0.008 | 1.57$\pm$0.004 | 4.505$\pm$0.020 | 16.29 | No
04333716+2109030 | 68.4048 | 21.1508 | 43.74${}^{+0.09}_{-0.09}$ | 10.726$\pm$0.008 | 1.23$\pm$0.004 | 3.040$\pm$0.027 | 12.69 | No
04341113+1133285 | 68.5464 | 11.5579 | 47.78${}^{+0.5}_{-0.49}$ | 11.25$\pm$0.008 | 1.39$\pm$0.004 | 3.191$\pm$0.030 | 11.03 | Yes
04354850+1317169 | 68.9521 | 13.288 | 49.17${}^{+0.14}_{-0.14}$ | 14.91$\pm$0.008 | 1.63$\pm$0.004 | 5.042$\pm$0.019 | 13.36 | No
04360525+1541026 | 69.0218 | 15.684 | 50.15${}^{+0.12}_{-0.12}$ | 9.345$\pm$0.008 | 0.87$\pm$0.004 | 1.972$\pm$0.016 | 9.47 | No
04395095+1243426 | 69.9623 | 12.7285 | 43.63${}^{+0.09}_{-0.09}$ | 9.992$\pm$0.008 | 1.07$\pm$0.004 | 2.512$\pm$0.024 | 10.85 | No
04412780+1404340 | 70.3658 | 14.0761 | 49.2${}^{+1.28}_{-1.22}$ | 13.395$\pm$0.008 |  | 4.683$\pm$0.022 | 1.28 | No
04412876+1200337 | 70.3698 | 12.0093 | 47.16${}^{+0.1}_{-0.09}$ | 12.898$\pm$0.008 | 1.5$\pm$0.004 | 4.144$\pm$0.021 | 18 | No
04431568+1704088 | 70.8153 | 17.0691 | 45.88${}^{+0.41}_{-0.4}$ | 9.92$\pm$0.008 | 1.0$\pm$0.004 | 2.524$\pm$0.017 | 10.31 | No
04461879+0338108 | 71.5783 | 3.63636 | 45.1${}^{+0.08}_{-0.08}$ | 10.922$\pm$0.008 | 1.27$\pm$0.004 | 3.096$\pm$0.021 | 13.25 | No
04463036+1528194 | 71.6265 | 15.472 | 49.05${}^{+0.21}_{-0.2}$ | 8.28$\pm$0.008 | 0.66$\pm$0.004 | 1.541$\pm$0.033 | 7.95 | No
04471851+0627113 | 71.8271 | 6.45315 | 40.54${}^{+0.07}_{-0.06}$ | 11.348$\pm$0.008 | 1.42$\pm$0.004 | 3.473$\pm$0.027 | 14.44 | No
04480086+1703216 | 72.0036 | 17.056 | 44.13${}^{+0.9}_{-0.87}$ | 11.085$\pm$0.008 | 1.41$\pm$0.004 | 3.711$\pm$0.024 | 10.77 | Yes
04483062+1623187 | 72.1276 | 16.3885 | 48.12${}^{+0.13}_{-0.13}$ | 12.427$\pm$0.008 | 1.47$\pm$0.004 | 3.884$\pm$0.018 | 15.69 | No
04484211+2106035 | 72.1754 | 21.1009 | 45.37${}^{+0.23}_{-0.23}$ | 9.057$\pm$0.008 | 0.85$\pm$0.004 | 1.985$\pm$0.019 | 9.69 | No
04491296+2448103 | 72.304 | 24.8028 | 49.59${}^{+0.14}_{-0.14}$ | 9.492$\pm$0.008 | 1.04$\pm$0.004 | 2.536$\pm$0.038 | 6.9 | Yes
04500069+1624436 | 72.5028 | 16.4121 | 49.34${}^{+0.13}_{-0.13}$ | 10.69$\pm$0.008 | 1.16$\pm$0.004 | 2.851$\pm$0.017 | 11.98 | No
04510241+1458167 | 72.76 | 14.9713 | 40.77${}^{+0.75}_{-0.72}$ | 11.675$\pm$0.008 |  | 3.708$\pm$0.021 | 13.14 | No
04522352+1859489 | 73.098 | 18.9969 | 50.75${}^{+0.12}_{-0.12}$ | 10.28$\pm$0.008 | 1.07$\pm$0.004 | 2.537$\pm$0.021 | 11.34 | No
04522385+1043099 | 73.0994 | 10.7194 | 52.04${}^{+0.14}_{-0.14}$ | 12.816$\pm$0.008 |  | 4.044$\pm$0.023 | 9.88 | Yes?
05054038+0627545 | 76.4182 | 6.46515 | 65.44${}^{+1.43}_{-1.37}$ | 9.88$\pm$0.008 | 0.95$\pm$0.004 | 2.362$\pm$0.022 | 10.41 | Yes
05110971+1548574 | 77.7904 | 15.8159 | 57.09${}^{+2.44}_{-2.25}$ | 12.08$\pm$0.008 |  | 3.977$\pm$0.018 | 14.94 | Yes?
Table 2: Derived Stellar Properties ($V-K_{s}$)  2MASS | $T_{\mathrm{eff}}$ | $\sigma_{teff}$ | $\mathcal{F}_{bol}$ | $\chi_{\nu}^{2}$ | Angular Diameter | Radius |  $\Delta$Radius
---|---|---|---|---|---|---|---
|  | (Systematic) | $\times 10^{-10}$ |  | $\times 10^{-2}$ |  |
| (K) | (K) | (erg cm-2 s-1) |  | (mas) | ($R_{\odot})$ | (%)
3373495+2120355 | 4967$\pm$14.29 | 60 | 62.8${}^{+2.0}_{-2.52}$ | 1.22 | 8.8${}^{+0.15}_{-0.18}$ | 0.87${}^{+0.01}_{-0.02}$ | 7.7108${}^{+3.663}_{-3.663}$
3434706+2051363 | 3419$\pm$6.36 | 60 | 2.67${}^{+0.245}_{-0.215}$ | 7.21 | 3.83${}^{+0.18}_{-0.15}$ | 0.37${}^{+0.02}_{-0.02}$ | 5.5743${}^{+6.351}_{-6.351}$
3510309+2354134 | 4565$\pm$11.71 | 60 | 36.5${}^{+1.3599}_{-0.975}$ | 2.8 | 7.94${}^{+0.15}_{-0.11}$ | 0.7${}^{+0.01}_{-0.01}$ | -0.044${}^{+3.655}_{-3.655}$
3524101+2548159 | 4169$\pm$12.35 | 60 | 19.6${}^{+0.571}_{-0.547}$ | 0.95 | 6.98${}^{+0.11}_{-0.11}$ | 0.67${}^{+0.01}_{-0.01}$ | 4.1394${}^{+3.929}_{-3.929}$
3550142+1229081 | 4725$\pm$13.34 | 60 | 34.5${}^{+1.97}_{-1.820}$ | 5.98 | 7.2${}^{+0.21}_{-0.19}$ | 0.71${}^{+0.02}_{-0.02}$ | -1.172${}^{+4.171}_{-4.171}$
3583581+1306182 | 5686$\pm$20.52 | 60 | 71.5${}^{+3.6400}_{-3.39}$ | 6.11 | 7.16${}^{+0.19}_{-0.18}$ | 0.94${}^{+0.03}_{-0.02}$ | -0.048${}^{+3.818}_{-3.818}$
3590972+2628340 | 4012$\pm$8.59 | 60 | 18.2${}^{+0.364}_{-0.527}$ | 1.18 | 7.26${}^{+0.08}_{-0.11}$ | 0.56${}^{+0.01}_{-0.01}$ | -2.101${}^{+3.496}_{-3.496}$
3591417+2202380 | 3637$\pm$5.89 | 60 | 7.090${}^{+0.4479}_{-0.458}$ | 8.71 | 5.51${}^{+0.18}_{-0.18}$ | 0.48${}^{+0.02}_{-0.02}$ | 2.9676${}^{+4.954}_{-4.954}$
4033902+1927180 | 4745$\pm$12.84 | 60 | 34.9${}^{+1.66}_{-1.55}$ | 2.5 | 7.19${}^{+0.18}_{-0.16}$ | 0.74${}^{+0.02}_{-0.02}$ | 1.0510${}^{+3.930}_{-3.930}$
4052565+1926316 | 4183$\pm$12.45 | 60 | 15.5${}^{+0.552}_{-0.524}$ | 2.04 | 6.16${}^{+0.12}_{-0.11}$ | 0.63${}^{+0.01}_{-0.01}$ | 0.1523${}^{+3.908}_{-3.908}$
04063463+1332566 | 3656$\pm$6.25 | 60 | 5.020${}^{+0.29}_{-0.303}$ | 6.28 | 4.59${}^{+0.13}_{-0.14}$ | 0.49${}^{+0.01}_{-0.02}$ | 3.0635${}^{+4.840}_{-4.840}$
04070122+1520062 | 4506$\pm$14.91 | 60 | 28.80${}^{+0.8029}_{-0.771}$ | 0.49 | 7.24${}^{+0.11}_{-0.11}$ | 0.7${}^{+0.01}_{-0.01}$ | 1.2427${}^{+3.690}_{-3.690}$
04074319+1631076 | 4819$\pm$17.5 | 60 | 39.5${}^{+1.44}_{-1.69}$ | 5.32 | 7.41${}^{+0.15}_{-0.17}$ | 0.74${}^{+0.01}_{-0.02}$ | -0.111${}^{+3.700}_{-3.700}$
04081110+1652229 | 4018$\pm$10.36 | 60 | 17.1${}^{+0.8260}_{-0.917}$ | 5.33 | 7.01${}^{+0.17}_{-0.19}$ | 0.61${}^{+0.01}_{-0.02}$ | 1.8593${}^{+4.336}_{-4.336}$
04082667+1211304 | 4227$\pm$13.31 | 60 | 17.7${}^{+0.33}_{-0.162}$ | 0.29 | 6.45${}^{+0.07}_{-0.05}$ | 0.64${}^{+0.01}_{-0.01}$ | 0.8483${}^{+3.636}_{-3.636}$
04083620+2346071 | 5109$\pm$11.96 | 60 | 55.89${}^{+1.56}_{-1.49}$ | 1.55 | 7.84${}^{+0.12}_{-0.11}$ | 0.8${}^{+0.01}_{-0.01}$ | 0.3837${}^{+3.166}_{-3.166}$
04084015+2333257 | 3652$\pm$8.17 | 60 | 8.68${}^{+0.534}_{-0.489}$ | 4.8 | 6.05${}^{+0.19}_{-0.17}$ | 0.6${}^{+0.02}_{-0.02}$ | 11.581${}^{+5.412}_{-5.412}$
04115620+2338108 | 4381$\pm$12.79 | 60 | 88.1${}^{+2.73}_{-3.87}$ | 2.85 | 13.39${}^{+0.22}_{-0.3}$ | 1.16${}^{+0.04}_{-0.04}$ | 41.846${}^{+6.679}_{-6.679}$
04151038+1423544 | 4081$\pm$7.75 | 60 | 14.80${}^{+0.669}_{-0.381}$ | 2.83 | 6.32${}^{+0.14}_{-0.08}$ | 0.65${}^{+0.02}_{-0.01}$ | 4.7336${}^{+4.210}_{-4.210}$
04153367+1542226 | 4317$\pm$11.8 | 60 | 21.29${}^{+1.16}_{-1.08}$ | 7.22 | 6.78${}^{+0.19}_{-0.18}$ | 0.66${}^{+0.02}_{-0.02}$ | 1.0256${}^{+4.354}_{-4.354}$
04163346+2154269 | 5365$\pm$19.71 | 60 | 66.8${}^{+1.29}_{-2.46}$ | 4.05 | 7.78${}^{+0.09}_{-0.15}$ | 0.85${}^{+0.01}_{-0.02}$ | 0.0777${}^{+3.134}_{-3.134}$
04172512+1901478 | 4445$\pm$12.66 | 60 | 21.8${}^{+0.9770}_{-0.915}$ | 6.27 | 6.47${}^{+0.15}_{-0.14}$ | 0.67${}^{+0.02}_{-0.01}$ | -1.043${}^{+3.939}_{-3.939}$
04172811+1454038 | 3446$\pm$5.68 | 60 | 2.900${}^{+0.154}_{-0.123}$ | 4.14 | 3.93${}^{+0.11}_{-0.08}$ | 0.42${}^{+0.01}_{-0.01}$ | 6.8703${}^{+4.971}_{-4.971}$
04174767+1339422 | 3794$\pm$7.16 | 60 | 8.690${}^{+0.581}_{-0.528}$ | 8.58 | 5.61${}^{+0.19}_{-0.17}$ | 0.57${}^{+0.02}_{-0.02}$ | 4.8220${}^{+5.110}_{-5.110}$
04175061+1828307 | 3405$\pm$5.28 | 60 | 4.979${}^{+0.287}_{-0.264}$ | 5.82 | 5.27${}^{+0.15}_{-0.14}$ | 0.53${}^{+0.02}_{-0.01}$ | 16.098${}^{+5.621}_{-5.621}$
04175555+1432464 | 3437$\pm$5.16 | 60 | 1.870${}^{+0.1750}_{-0.037}$ | 7.03 | 3.17${}^{+0.15}_{-0.03}$ | 0.36${}^{+0.02}_{-0.0}$ | 3.9298${}^{+6.272}_{-6.272}$
04181077+2317048 | 4710$\pm$10.37 | 60 | 66.1${}^{+2.06}_{-1.97}$ | 0.81 | 10.04${}^{+0.16}_{-0.16}$ | 1.17${}^{+0.02}_{-0.02}$ | 33.526${}^{+4.510}_{-4.510}$
04223953+1816097 | 3658$\pm$6.26 | 60 | 7.02${}^{+0.3}_{-0.282}$ | 3.41 | 5.42${}^{+0.12}_{-0.11}$ | 0.49${}^{+0.01}_{-0.01}$ | 2.5476${}^{+4.319}_{-4.319}$
04232283+1939312 | 4977$\pm$14.36 | 60 | 59.0${}^{+2.0100}_{-3.13}$ | 8.35 | 8.49${}^{+0.15}_{-0.23}$ | 0.84${}^{+0.02}_{-0.02}$ | 5.1025${}^{+3.700}_{-3.700}$
04232526+1545474 | 4381$\pm$9.63 | 60 | 32.8${}^{+0.6310}_{-0.912}$ | 0.6 | 8.17${}^{+0.09}_{-0.12}$ | 0.73${}^{+0.01}_{-0.01}$ | 5.9528${}^{+3.538}_{-3.538}$
04235070+0912193 | 3513$\pm$5.68 | 60 | 9.959${}^{+0.7859}_{-0.591}$ | 8.15 | 7.0${}^{+0.28}_{-0.21}$ | 0.68${}^{+0.03}_{-0.02}$ | 22.958${}^{+6.690}_{-6.690}$
04235911+1643178 | 3822$\pm$6.95 | 60 | 8.42${}^{+0.66}_{-0.591}$ | 5.06 | 5.44${}^{+0.21}_{-0.19}$ | 0.54${}^{+0.02}_{-0.02}$ | 1.2178${}^{+5.327}_{-5.327}$
04241691+1800107 | 4788$\pm$11.88 | 60 | 38.8${}^{+0.6990}_{-0.679}$ | 0.65 | 7.44${}^{+0.08}_{-0.07}$ | 0.75${}^{+0.01}_{-0.01}$ | 0.8544${}^{+3.168}_{-3.168}$
04250024+1659057 | 4806$\pm$10.87 | 60 | 28.00${}^{+1.01}_{-0.954}$ | 1.91 | 6.27${}^{+0.12}_{-0.11}$ | 0.74${}^{+0.01}_{-0.01}$ | -0.287${}^{+3.463}_{-3.463}$
04251456+1858250 | 3789$\pm$6.81 | 60 | 7.25${}^{+0.54}_{-0.485}$ | 4.89 | 5.14${}^{+0.19}_{-0.17}$ | 0.58${}^{+0.02}_{-0.02}$ | 6.0036${}^{+5.451}_{-5.451}$
04252501+1754552 | 4267$\pm$9.04 | 60 | 17.9${}^{+0.9440}_{-1.01}$ | 8.25 | 6.36${}^{+0.17}_{-0.18}$ | 0.64${}^{+0.02}_{-0.02}$ | -0.111${}^{+4.173}_{-4.173}$
04254754+1801022 | 4985$\pm$15.13 | 60 | 86.2${}^{+3.0900}_{-4.8}$ | 3.69 | 10.23${}^{+0.19}_{-0.29}$ | 0.93${}^{+0.03}_{-0.03}$ | 11.693${}^{+4.799}_{-4.799}$
04264825+1052160 | 4700$\pm$12.55 | 60 | 67.1${}^{+1.63}_{-3.82}$ | 6.11 | 10.16${}^{+0.13}_{-0.29}$ | 1.03${}^{+0.03}_{-0.04}$ | 24.501${}^{+4.923}_{-4.923}$
04272532+1415384 | 4662$\pm$14.19 | 60 | 29.8${}^{+0.532}_{-0.518}$ | 0.91 | 6.88${}^{+0.07}_{-0.07}$ | 0.78${}^{+0.01}_{-0.01}$ | 5.5282${}^{+3.632}_{-3.632}$
04274701+1425041 | 5075$\pm$15.03 | 60 | 51.4${}^{+0.9609}_{-1.39}$ | 2.81 | 7.62${}^{+0.08}_{-0.11}$ | 0.81${}^{+0.01}_{-0.01}$ | 1.5360${}^{+3.155}_{-3.155}$
04275895+1830009 | 4662$\pm$11.16 | 60 | 35.8${}^{+1.06}_{-1.02}$ | 2.62 | 7.54${}^{+0.12}_{-0.11}$ | 0.83${}^{+0.01}_{-0.01}$ | 9.7850${}^{+3.769}_{-3.769}$
04282878+1741453 | 3620$\pm$8.69 | 60 | 16.9${}^{+1.0}_{-0.920}$ | 5.68 | 8.59${}^{+0.26}_{-0.24}$ | 0.87${}^{+0.03}_{-0.03}$ | 36.433${}^{+6.779}_{-6.779}$
04303385+1444532 | 3389$\pm$5.22 | 60 | 2.59${}^{+0.134}_{-0.124}$ | 4.94 | 3.84${}^{+0.1}_{-0.09}$ | 0.41${}^{+0.01}_{-0.01}$ | 9.2007${}^{+5.673}_{-5.673}$
04303486+1544023 | 5176$\pm$14.26 | 60 | 87.4${}^{+3.15}_{-2.99}$ | 8.41 | 9.56${}^{+0.18}_{-0.17}$ | 1.17${}^{+0.03}_{-0.03}$ | 22.461${}^{+4.575}_{-4.575}$
04315244+1529585 | 4300$\pm$12.22 | 60 | 19.9${}^{+0.771}_{-0.904}$ | 8.11 | 6.61${}^{+0.13}_{-0.15}$ | 0.65${}^{+0.01}_{-0.02}$ | 0.0410${}^{+3.882}_{-3.882}$
04322565+1306476 | 4148$\pm$9.31 | 60 | 22.9${}^{+1.5}_{-1.150}$ | 5.87 | 7.62${}^{+0.25}_{-0.19}$ | 0.77${}^{+0.03}_{-0.02}$ | 13.762${}^{+5.328}_{-5.328}$
04332699+1302438 | 3578$\pm$6.21 | 60 | 6.11${}^{+0.3500}_{-0.322}$ | 5.88 | 5.29${}^{+0.15}_{-0.14}$ | 0.49${}^{+0.01}_{-0.01}$ | 5.9090${}^{+4.961}_{-4.961}$
04333716+2109030 | 4344$\pm$13.65 | 60 | 26.5${}^{+0.758}_{-0.962}$ | 2.3 | 7.47${}^{+0.12}_{-0.14}$ | 0.7${}^{+0.01}_{-0.01}$ | 4.1654${}^{+3.839}_{-3.839}$
04341113+1133285 | 4248$\pm$14.58 | 60 | 17.80${}^{+0.7159}_{-0.838}$ | 1.99 | 6.4${}^{+0.14}_{-0.16}$ | 0.66${}^{+0.02}_{-0.02}$ | 1.7683${}^{+4.250}_{-4.250}$
04354850+1317169 | 3377$\pm$nan | 60 | 2.199${}^{+0.147}_{-0.133}$ | 6.86 | 3.56${}^{+nan}_{-nan}$ | 0.38${}^{+nan}_{-nan}$ | 7.9629${}^{+nan}_{-nan}$
04360525+1541026 | 5225$\pm$12.6 | 60 | 55.79${}^{+1.05}_{-2.02}$ | 4.13 | 7.49${}^{+0.08}_{-0.14}$ | 0.81${}^{+0.01}_{-0.02}$ | -0.844${}^{+2.918}_{-2.918}$
04395095+1243426 | 4733$\pm$14.7 | 60 | 38.3${}^{+0.684}_{-0.993}$ | 3.26 | 7.56${}^{+0.08}_{-0.11}$ | 0.71${}^{+0.01}_{-0.01}$ | -1.518${}^{+3.237}_{-3.237}$
04412780+1404340 | 3509$\pm$6.49 | 60 | 6.42${}^{+0.4870}_{-0.437}$ | 4.78 | 5.63${}^{+0.21}_{-0.19}$ | 0.6${}^{+0.03}_{-0.03}$ | 16.672${}^{+6.953}_{-6.953}$
04412876+1200337 | 3733$\pm$7.21 | 60 | 6.88${}^{+0.468}_{-0.425}$ | 9.4 | 5.15${}^{+0.18}_{-0.16}$ | 0.52${}^{+0.02}_{-0.02}$ | 2.7593${}^{+5.093}_{-5.093}$
04431568+1704088 | 4718$\pm$10.41 | 60 | 41.7${}^{+2.59}_{-1.609}$ | 2.13 | 7.95${}^{+0.25}_{-0.16}$ | 0.78${}^{+0.03}_{-0.02}$ | 5.1217${}^{+4.621}_{-4.621}$
04461879+0338108 | 4303$\pm$10.17 | 60 | 22.5${}^{+0.634}_{-0.408}$ | 0.55 | 7.01${}^{+0.1}_{-0.07}$ | 0.68${}^{+0.01}_{-0.01}$ | 2.8151${}^{+3.657}_{-3.657}$
04463036+1528194 | 5745$\pm$34.28 | 60 | 127.0${}^{+5.5100}_{-1.32}$ | 1.61 | 9.35${}^{+0.23}_{-0.12}$ | 0.99${}^{+0.02}_{-0.01}$ | 1.1597${}^{+4.002}_{-4.002}$
04471851+0627113 | 4077$\pm$11.7 | 60 | 18.2${}^{+1.12}_{-1.02}$ | 5.98 | 7.03${}^{+0.22}_{-0.2}$ | 0.61${}^{+0.02}_{-0.02}$ | 1.2485${}^{+4.733}_{-4.733}$
04480086+1703216 | 3946$\pm$9.48 | 60 | 26.0${}^{+1.89}_{-1.71}$ | 6.7 | 8.97${}^{+0.33}_{-0.3}$ | 0.85${}^{+0.04}_{-0.03}$ | 26.207${}^{+6.890}_{-6.890}$
04483062+1623187 | 3855$\pm$6.77 | 60 | 8.709${}^{+0.7509}_{-0.727}$ | 5.15 | 5.44${}^{+0.24}_{-0.23}$ | 0.56${}^{+0.02}_{-0.02}$ | 2.2890${}^{+5.663}_{-5.663}$
04484211+2106035 | 5215$\pm$14.51 | 60 | 74.9${}^{+2.98}_{-4.82}$ | 5.97 | 8.71${}^{+0.18}_{-0.28}$ | 0.85${}^{+0.02}_{-0.03}$ | 2.3311${}^{+3.602}_{-3.602}$
04491296+2448103 | 4726$\pm$25.52 | 60 | 60.6${}^{+2.37}_{-2.96}$ | 2.95 | 9.55${}^{+0.21}_{-0.25}$ | 1.02${}^{+0.02}_{-0.03}$ | 23.383${}^{+5.086}_{-5.086}$
04500069+1624436 | 4468$\pm$9.16 | 60 | 24.5${}^{+1.38}_{-1.27}$ | 4.67 | 6.79${}^{+0.19}_{-0.18}$ | 0.72${}^{+0.02}_{-0.02}$ | 3.6727${}^{+4.347}_{-4.347}$
04510241+1458167 | 3945$\pm$8.25 | 60 | 16.3${}^{+0.655}_{-0.617}$ | 2.0 | 7.1${}^{+0.15}_{-0.14}$ | 0.62${}^{+0.02}_{-0.02}$ | 5.2541${}^{+4.629}_{-4.629}$
04522352+1859489 | 4710$\pm$12.61 | 60 | 30.4${}^{+1.2200}_{-1.150}$ | 2.89 | 6.81${}^{+0.14}_{-0.13}$ | 0.74${}^{+0.02}_{-0.01}$ | 1.7520${}^{+3.751}_{-3.751}$
04522385+1043099 | 3780$\pm$8.15 | 60 | 7.040${}^{+0.368}_{-0.406}$ | 2.92 | 5.08${}^{+0.13}_{-0.15}$ | 0.57${}^{+0.02}_{-0.02}$ | 5.0041${}^{+4.684}_{-4.684}$
05054038+0627545 | 4859$\pm$14.25 | 60 | 41.7${}^{+0.4100}_{-0.404}$ | 0.26 | 7.49${}^{+0.06}_{-0.06}$ | 1.06${}^{+0.02}_{-0.02}$ | 22.910${}^{+4.659}_{-4.659}$
05110971+1548574 | 3809$\pm$6.58 | 60 | 13.69${}^{+0.3629}_{-0.349}$ | 1.35 | 6.99${}^{+0.1}_{-0.09}$ | 0.86${}^{+0.04}_{-0.04}$ | 30.224${}^{+7.385}_{-7.385}$