$({\bm{\mathcal{W}}}_{\rho}(t))_{t\in[0,1-2\varepsilon]}$. In particular, for
every $0<\varepsilon<1/2$ and for every integrable function
$h:\mathcal{C}([0,1-2\varepsilon],\mathbb{R}^{2})\to\mathbb{R}$,

$\mathbb{E}\left[h\left(({\bm{\mathcal{E}}}_{\rho}(\varepsilon+t)-{\bm{\mathcal{E}}}_{\rho}(\varepsilon))_{t\in[0,1-2\varepsilon]}\right)\right]=\mathbb{E}\left[h\left(({\bm{\mathcal{W}}}_{\rho}(t))_{t\in[0,1-2\varepsilon]}\right)\alpha_{\varepsilon}\left(-\inf_{[0,1-2\varepsilon]}{\bm{\mathcal{W}}}_{\rho}\;,\;{\bm{\mathcal{W}}}_{\rho}(1-2\varepsilon)\right)\right],$
(103)

where $\alpha_{\varepsilon}$ is a bounded positive continuous function on
$(\mathbb{R}_{+})^{2}\times\mathbb{R}^{2}$.

In addition, since $\alpha_{\varepsilon}>0$, we have that the two measures are
equivalent.

The theorem above was proved in [BM22, Proposition A.1] in the specific case
when $\rho=-1/2$ building on some specific results on a family of discrete
two-dimensional walks called _tandem walk_ [BMFR20]. In what follows we prove
the general case $\rho\in(-1,1)$ building on the more general results of
[DW15, DW20].

###### Proof.

We prove the proposition by considering a two-dimensional random walk
$(\bm{W}_{n})_{n\in\mathbb{Z}_{\geq 0}}$. Setting
$(\bm{X},\bm{Y})=\bm{W}_{1}-\bm{W}_{0}$, we assume that $\bm{X}$ and $\bm{Y}$
have finite moments, $\mathbb{E}[\bm{X}]=\mathbb{E}[\bm{Y}]=0$,
$\mathbb{E}[\bm{X}^{2}]=\mathbb{E}[\bm{Y}^{2}]=1$ and
$\operatorname{Cov}(\bm{X},\bm{Y})=\rho$. Under these assumptions, [DW20,
Theorem 4] guarantees222222The results in [DW20] are stated under the
assumption that $\operatorname{Cov}(\bm{X},\bm{Y})=0$. This does not restrict
the generality of the results and they remain valid (with the obvious
adaptations) when $\operatorname{Cov}(\bm{X},\bm{Y})=\rho$. See the bottom
part of page 996 in [DW15] for more explanations on this fact. that setting
$Q\coloneqq\mathbb{Z}_{\geq 0}^{2}$,

$\mathbb{P}_{x}\left(\left(\tfrac{1}{\sqrt{2n}}\bm{W}_{\lfloor
nt\rfloor}\right)_{t\in[0,1]}\in\cdot\;\middle|\;\bm{W}_{[0,n]}\subset
Q,\bm{W}_{n}=y\right)\xrightarrow[n\to\infty]{}\mathbb{P}(({\bm{\mathcal{E}}}_{\rho}(t))_{t\in[0,1]}\in\cdot),$
(104)

for all $x,y\in Q$. In [BM22, Lemma A.2] it was proved232323Note that the
proof of [BM22, Lemma A.2] does not use any specific property of the tandem
walk $(\bm{W}_{n})_{n\in\mathbb{Z}_{\geq 0}}$. that if
$h:(\mathbb{Z}^{2})^{n-2m+1}\to\mathbb{R}$ is a bounded measurable function,
$x,y\in Q$ and $1\leq m<n/2$, then

$\mathbb{E}_{x}\left[h((\bm{W}_{i+m}-\bm{W}_{m})_{0\leq i\leq
n-2m})\mid\bm{W}_{[0,n]}\subset Q,\bm{W}_{n}=y\right]\\\
=\mathbb{E}_{0}\left[h(\bm{W}_{i})_{0\leq i\leq
n-2m}\cdot\alpha_{n,m}^{x,y}\left(-\inf_{0\leq i\leq
n-2m}\bm{W}_{i}\;,\;\bm{W}_{n-2m}\right)\right],$ (105)

where

$\alpha_{n,m}^{x,y}(a,b)\coloneqq\sum_{z\in Q\colon z-a\in
Q}\frac{\mathbb{P}_{x}\left(\bm{W}_{m}=z,\bm{W}_{[0,m]}\subset
Q\right)\mathbb{P}_{\widehat{y}}\left(\bm{W}_{m}=\widehat{z}+\widehat{b},\bm{W}_{[0,m]}\subset
Q\right)}{\mathbb{P}_{x}\left(\bm{W}_{n}=y,\bm{W}_{[0,n]}\subset Q\right)}$
(106)

and $\widehat{(i,j)}=(j,i)$. Note that if we show that for fixed $x,y\in Q$
and for all $\varepsilon\in(0,1/2)$, there exists a bounded positive
continuous function $\alpha_{\varepsilon}$ on
$(\mathbb{R}_{+})^{2}\times\mathbb{R}^{2}$ such that

$\lim_{n\to\infty}\sup_{a\in\mathbb{Z}^{2}_{\geq
0},b\in\mathbb{Z}^{2}}\left\lvert\alpha^{x,y}_{n,\lfloor
n\varepsilon\rfloor}(a,b)-\alpha_{\varepsilon}\left(\tfrac{a}{\sqrt{n}},\tfrac{b}{\sqrt{n}}\right)\right\rvert=0,$
(107)

then Eq. 103 follows from Eqs. 104, 105 and 107. Therefore it just remains to
prove Eq. 107.

Fix $x\in Q$. From [DW15], there exists a positive function $V$ on $Q$ such
that as $n\to\infty$ the following asymptotics hold

$\displaystyle\mathbb{P}_{x}\left(\bm{W}_{[0,n]}\subset Q\right)\sim
c_{1}V(x)n^{-p/2}\text{ as }n\rightarrow\infty,$ (108)
$\displaystyle\delta_{1}(x,n)\coloneqq\sup_{y\in
Q}\left|n^{p/2+1}\cdot\mathbb{P}_{x}\left(\bm{W}_{n}=y,\bm{W}_{[0,n]}\subset
Q\right)-c_{2}V(x)g\left(\frac{y}{\sqrt{2n}}\right)\right|\rightarrow 0,$
(109) $\displaystyle\mathbb{P}_{x}\left(\bm{W}_{n}=y,\bm{W}_{[0,n]}\subset
Q\right)\sim c_{3}\cdot\frac{V(x)V(\widehat{y})}{n^{p+1}},$ (110)

where $c_{1},c_{2},c_{3}$ are three constants, $g$ is a positive bounded
integrable function on $\mathbb{R}^{2}_{\geq 0}$ and $p$ is a parameter
depending only on $\rho$ and $Q$. (We highlight that all these constants and
the function $g$ can be explicitly computed in specific cases, see for
instance [BM22, Lemma A.5].) More precisely, Eq. 108 is [DW15, Theorem 1], Eq.
109 is [DW15, Theorem 5], and Eq. 110 is [DW15, Theorem 6].

In what follows, $m=\lfloor n\varepsilon\rfloor$ for some $\varepsilon>0$. Let
us consider $\alpha_{n,m}^{x,y}(a,b)$ defined in Eq. 106. By Eq. 110, the
denominator (which is independent of $a,b$) is of order $n^{-p-1}$.

We now look at the numerator of $\alpha_{n,m}^{x,y}(a,b)$. We first cut the
sum at $t\sqrt{n}$ for some $t>0$ and bound the rest of the sum. Using Eq. 109
for the first factor (recall that $g$ is bounded) and Eq. 108 for the second
one, we can guarantee that there exists a constant $C>0$ depending only on
$x,y$ such that

$\displaystyle R_{n}\coloneqq$
$\displaystyle\sum_{|z|>t\sqrt{n}}\mathbb{P}_{x}(\bm{W}_{m}=z,\bm{W}_{[0,m]}\subset
Q)\mathbb{P}_{\widehat{y}}(\bm{W}_{m}=\widehat{z}+\widehat{b},\bm{W}_{[0,m]}\subset
Q)$ (111) $\displaystyle\leq
Cn^{-p/2}n^{-p/2-1}\sum_{|z|>t\sqrt{n}}\mathbb{P}_{x}(\bm{W}_{m}=z\mid\bm{W}_{[0,m]}\subset
Q)$ (112) $\displaystyle=Cn^{-p-1}\mathbb{P}_{x}(|\bm{W}_{\lfloor
n\varepsilon\rfloor}|>t\sqrt{n}\mid\bm{W}_{[0,\lfloor
n\varepsilon\rfloor]}\subset Q).$ (113)

Using [DW15, Theorem 3], it is possible to find a function
$\delta_{2}(x,y,\varepsilon,n,t)$ independent of $a,b$ such that

$n^{p+1}R_{n}\leq\delta_{2}(x,y,\varepsilon,n,t)\qquad\text{and}\qquad\lim_{t\to\infty}\limsup_{n\to\infty}\delta_{2}(x,y,\varepsilon,n,t)=0.$
(114)

Now set $S_{n,m}\coloneqq\sum_{z:z-a\in Q,|z|\leq
t\sqrt{n}}\mathbb{P}_{x}(\bm{W}_{m}=z,\bm{W}_{[0,m]}\subset
Q)\cdot\mathbb{P}_{\widehat{y}}(\bm{W}_{m}=\widehat{z}+\widehat{b},\bm{W}_{[0,m]}\subset
Q).$ Using Eq. 109, we have for fixed $x$ and $y$ that

$S_{n,m}=m^{-p-2}\cdot c_{2}^{2}\cdot V(x)V(\widehat{y})\sum_{z:z-a\in
Q,|z|\leq t\sqrt{n}}g\left(\frac{z}{\sqrt{2\varepsilon
n}}\right)g\left(\frac{\widehat{z}+\widehat{b}}{\sqrt{2\varepsilon
n}}\right)\\\
+O(1)\left(t\sqrt{n}\right)^{2}m^{-p-2}(\delta_{1}(x,m)+\delta_{1}(\widehat{y},m)).$
(115)

Putting together the latter estimate for the numerator of Eq. 106 with the
estimate in Eq. 110 for the denominator, both uniform in $(a,b)$, we have

$\displaystyle\alpha^{x,y}_{n,\lfloor\varepsilon n\rfloor}(a,b)$ (116)
$\displaystyle=O(1)n^{p+1}R_{n}+o(1)+\frac{1}{\varepsilon^{p+1}}\left(\frac{c_{2}^{2}}{c_{3}}+o(1)\right)\frac{1}{(\sqrt{n})^{2}}\sum_{z\geq
a,|z|\leq t\sqrt{n}}g\left(\frac{z}{\sqrt{2\varepsilon
n}}\right)g\left(\frac{\widehat{z}+\widehat{b}}{\sqrt{2\varepsilon n}}\right)$
(117)
$\displaystyle=O(1)n^{p+1}R_{n}+o(1)+\frac{1}{\varepsilon^{p+1}}\left(\frac{c_{2}^{2}}{c_{3}}+o(1)\right)\int_{w\geq\frac{a}{\sqrt{n}},|w|\leq
t}g\left(\frac{w}{\sqrt{2\varepsilon}}\right)g\left(\frac{\widehat{w}+\widehat{b}/\sqrt{n}}{\sqrt{2\varepsilon}}\right)dw+o(1),$
(118)

where the final $o(1)$ error term comes from the summation approximation. We
highlight that all the error terms are uniform in $a$ and $b$. Finally,
setting

$\alpha_{\varepsilon}\left(a,b\right)\coloneqq\frac{1}{\varepsilon^{p+1}}\cdot\frac{c_{2}^{2}}{c_{3}}\cdot\int_{w:w-a\in\mathbb{R}^{2}_{\geq
0}}g\left(\frac{w}{\sqrt{2\varepsilon}}\right)g\left(\frac{\widehat{w}+\widehat{b}}{\sqrt{2\varepsilon}}\right)dw,$
(119)

we obtain that

$\displaystyle\left|\alpha_{n,\lfloor
n\varepsilon\rfloor}(a,b)-\alpha_{\varepsilon}\left(\tfrac{a}{\sqrt{n}},\tfrac{b}{\sqrt{n}}\right)\right|$
$\displaystyle=O(1)n^{p+1}R_{n}+o(1)+O(1)\int_{|w|>t}g(w/\sqrt{2\varepsilon}).$
(120)

Since $g$ is integrable and we have the bounds in Eq. 114, we can conclude
that the latter term tends to zero first taking $n\to\infty$ and then
$t\to\infty$. ∎

##  Appendix B Simulations of the skew Brownian permuton

We briefly explain how we obtained the simulations of the skew Brownian
permuton $\bm{\mu}_{\rho,q}$ given in Fig. 3, p. 3.

  * •

The first step is to sample an approximation of a two-dimensional Brownian
excursion ${\bm{\mathcal{E}}}_{\rho}$ with correlations $\rho$. To do that,

    * –

we start with (any) two-dimensional walk in the non-negative quadrant, started
and ended at $(0,0)$, with (say) 10 points and linear interpolating among
these points. Then we run the following Glauber dynamics: we resample each
point of the walk in such a way that the conditional law of each point given
the neighboring points is that of the Gaussian distribution with correct mean
and variance to correspond to the desired Brownian excursion, then conditioned
to be in the non-negative quadrant.

    * –

To get a new walk with double points, we consider the walk obtained in the
previous step and then we add a new point in the middle point of each linear
segment. Then we run again the same Glauber dynamics used in the previous
step.

    * –

Iterating the previous step, we always get a new walk with double points,
which is a better approximation of two-dimensional Brownian excursion
${\bm{\mathcal{E}}}_{\rho}$.

  * •

The second step is to construct approximations of the solutions of the SDEs in
Eq. 8, p. 8, driven by the approximation of ${\bm{\mathcal{E}}}_{\rho}$
obtained in the previous step. This can be done by using some well-chosen
_discrete coalescent-walk processes_ (see [Bor22, Section 2.2]) whose walks
converge to the solutions of the SDEs in Eq. 8 (in the same spirit of [Bor22,
Proposition 4.8]).

  * •

The final step is the simplest one: once we have the approximated solutions of
the SDEs in Eq. 8, it is simple to construct the process
$\varphi_{{\bm{\mathcal{Z}}}_{\rho,q}}$ defined in Eq. 9. Finally, the skew
Brownian permuton $\bm{\mu}_{\rho,q}$ is well approximated by the graph of the
function of the process $\varphi_{{\bm{\mathcal{Z}}}_{\rho,q}}$.

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# Eclipsing Binaries Found by the EREBOS Project:
Gaia DR2 6097540197980557440 – A Deeply Eclipsing sdB+dM System

Kyle A. Corcoran  Brad N. Barlow  Veronika Schaffenroth  Uli Heber  Stephen
Walser  Andreas Irgang

###### Abstract

We present time-series spectroscopy and photometry of Gaia DR2
6097540197980557440, a new deeply–eclipsing hot subdwarf B (sdB) + M dwarf
(dM) binary. We discovered this object during the course of the Eclipsing
Reflection Effect Binaries from Optical Surveys (EREBOS) project, which aims
to find new eclipsing sdB+dM binaries (HW Vir systems) and increase the small
sample of studied systems. In addition to the primary eclipse, which is in
excess of $\sim$5 magnitudes in the optical, the light curve also shows
features typical for other HW Vir binaries such as a secondary eclipse and
strong reflection effect from the irradiated, cool companion. The orbital
period is 0.127037 d ($\sim$3 hr), falling right at the peak of the orbital
period distribution of known HW Vir systems. Analysis of our time–series
spectroscopy yields a radial velocity semi-amplitude of $K_{\rm sdB}=100.0\pm
2.0\,{\rm km\,s}^{-1}$, which is amongst the fastest line–of–sight velocities
found to date for an HW Vir binary. State-of-the-art atmospheric models that
account for deviations from local thermodynamic equilibrium are used to
determine the atmospheric parameters of the sdB. Although we cannot claim a
unique light curve modeling solution, the best–fitting model has an sdB mass
of $M_{\rm sdB}=0.47\pm 0.03\,M_{\odot}$ and a companion mass of $M_{\rm
dM}=0.18\pm 0.01\,M_{\odot}$. The radius of the companion appears to be
inflated relative to theoretical mass–radius relationships, consistent with
other known HW Vir binaries. Additionally, the M dwarf is one of the most
massive found to date amongst this type of binary.

###### Key Words.:

binaries: eclipsing – stars: fundamental parameters, subdwarfs

Kyle A. Corcoran

University of Virginia, Department of Astronomy, 530 McCormick Rd.,
Charlottesville, VA 22904, USA High Point University, Department of Physics,
One University Parkway, High Point, NC 27268, USA Universität Potsdam,
Institut für Physik und Astronomie, Haus 28, Karl-Liebknecht-Str. 24/25,
14476, Potsdam-Golm, Germany Dr. Karl Remeis-Observatory & ECAP, Astronomical
Institute, Friedrich-Alexander University Erlangen-Nuremberg (FAU),
Sternwartstr. 7, 96049 Bamberg, Germany Virginia Tech, Department of Physics,
850 West Campus Drive, Blacksburg, VA 24061, USA Dr. Karl Remeis-Observatory &
ECAP, Astronomical Institute, Friedrich-Alexander University Erlangen-
Nuremberg (FAU), Sternwartstr. 7, 96049 Bamberg, Germany

##  1 Introduction

Most hot subdwarfs are core He–burning extreme horizontal branch (EHB) stars
that formed from red giant progenitors that experienced mass loss near the tip
of the giant branch, due to binary interactions (see Heber, 2016, for a
detailed review). With temperatures from $20,000-45,000\rm\,K$ and spectra
dominated by broad H Balmer lines, they are classified either as sdB stars or
sdOB stars if they display the He ii 4686 Å line. They show a tight mass
distribution peaking near 0.47$M_{\odot}$ (the ‘canonical’ mass) and have
radii around 0.2$R_{\odot}$. Theoretical models such as those in Han et al.
(2002, 2003) describe sdB formation scenarios that account for the mass loss
in these systems, with three possible formation channels depending on the
initial configuration and mass ratio of the binary. One formation channel
produces an sdB via Roche lobe overflow (RLOF) to a MS companion of K type and
earlier. The binaries that form in this way are typically wide binaries
($P=10-1500\rm\,d$). These systems are often called “composite” binaries, as
both stars are seen in the spectrum and account for 30 – 40% of all sdBs (for
an overview see, Vos et al., 2019). The rest of the sdBs do not show any signs
of a companion in their spectra. Maxted et al. (2001) showed that a high
fraction of those sdBs do exist in short-period binaries leading to radial
velocity variations. Those can be formed by common envelope (CE) evolution,
which produces close binaries with periods as short as $\sim$1.5 hr with a hot
subdwarf primary and a cool, low-mass companion. In this scenario, an evolving
red giant and a companion object enter a CE, and the angular momentum resonant
in the orbit of the binary is deposited into the envelope, ejecting it from
the system. Typically, this companion is stellar in nature; however, Soker
(1998) proposed that sub-stellar and even planetary mass companions could be
sufficient to provide the orbital angular momentum necessary to eject the
envelope (e.g., Schaffenroth et al., 2015). The remaining sdBs do not show any
radial velocity variations and appear hence single. Such single sdBs could be
formed by the merger of two He WDs. Another possibility is that a substellar
companion was responsible for the mass loss, which was destroyed during the
common envelope phase.

The main challenge in studying close sdB binaries and their properties comes
from the single-lined nature of these systems, allowing only for mass limits
inferred based on the proposed inclination; however, some systems benefit from
the presence of an eclipse, which helps to constrain the inclination and
allows for more precise mass measurements. These so called HW Vir systems also
show photometric variation due to the reflection effect and have orbital
periods of $P<1\>\rm d$, making them vital tools for sdB studies due to the
relative ease in identifying them. The prototypical HW Vir is an sdB and M
dwarf (dM) binary. A few systems containing a brown dwarf (BD) have also been
discovered (e.g., Schaffenroth et al., 2014a).

The Eclipsing Reflection Effect Binaries from Optical Surveys (EREBOS) project
(Schaffenroth et al., 2019) is an effort to increase the sample of known HW
Vir systems and to measure orbital, atmospheric, and fundamental parameters of
those binaries. It is especially interested in finding the lower-mass limit of
an object able to remove the envelope in a CE phase and survive this phase in
order to investigate the effect that sub-stellar companions have on the late
stages of stellar evolution. Moreover, this project aims at studying post-CE
systems spanning the entire range of periods and companion masses for these
systems. For a better understanding of the poorly understood common envelope
phase, see Ivanova et al. (2013). Until recently, the number of HW Vir
binaries with known fundamental parameters was relatively small at 18 total
systems. The EREBOS project dramatically increased this number by inspecting
light curves from the Optical Gravitational Lensing Experiment (OGLE;
Pietrukowicz et al., 2013; Soszyński et al., 2015) and Asteroid Terrestrial-
impact Last Alert System (ATLAS; Tonry et al., 2018) surveys, finding over 150
new HW Vir candidates (Schaffenroth et al., 2019). With an extensive
spectroscopic and photometric follow up campaign we will dramatically increase
the number of systems with robust solutions.

Despite this unprecedented increase, HW Virs still represent only a small
fraction of the sdB population. Given the typical radii of both components,
these systems have to be relatively edge–on to show any eclipse. For example,
the smallest, grazing eclipses occur in systems such ASAS 102322$-$3737
(Schaffenroth et al., 2013), a sdB+dM, at $i=65.9^{\circ}$; however,
inclinations do range up to perfectly edge-on systems such as AA Dor (Kilkenny
et al., 1978), a sdOB+dM/BD. One HW Vir system, Konkoly J064029.1+385652.2
(Derekas et al., 2015), is an sdO+dM binary that even shows a total eclipse
due to a relatively small ($R=0.096\,R_{\odot}$) sdO being in a nearly edge on
($i=87.11$) orbit with an inflated dM. Total eclipses are sometimes seen in
WD+dM binaries such as NN Ser (Parsons et al., 2010), where a high inclination
angle allows the dM to completely block the smaller WD along our line of
sight. Due to the similarity in size between typical sdBs and dMs, even edge
on systems struggle to achieve geometries sufficient to produce a total
eclipse.

Here we present system parameters for the first deeply eclipsing sdB+dM system
Gaia DR2 6097540197980557440, which exhibits an eclipse in excess of $\sim$5
magnitudes in the optical. We discovered Gaia DR2 6097540197980557440 in the
course of the EREBOS project while obtaining follow–up observations of known
HW Virs using the Goodman spectrograph (Clemens et al., 2004) on the 4.1–m
Southern Astrophysical Research (SOAR). In §2 we describe the initial
observations leading to its discovery. In §3 we present time–series
spectroscopic observations as well as the radial velocities and atmospheric
parameters derived from them. In §4 we present multi–color, time–series
photometric observations and the details of our light curve modeling solution.
§5 presents system parameters derived from the best–fitting light curve
modeling solution. In §6, we discuss how the system compares to the EREBOS
sample as well as potential follow-up studies. Finally, we summarize our work
in §7.

##  2 Discovery Run

(a)  (b)  (c)  (d)

(e)

(f)

Figure 1: Discovery data for Gaia DR2 6097540197980557440 from SOAR/Goodman.
Top: Raw Johnson $V$ filter frames from the discovery data set obtained on
2019 June 9. We highlight five select frames corresponding to the marked
locations on the light curve in the bottom panel. These frames represent
phases (a) just prior to ingress, (b) shortly before the systems drops below
detection limits, (c) during primary eclipse totality, (d) shortly after the
system returns above detection limits, and (e) just after egress. Bottom: The
corresponding light curve of Gaia DR2 6097540197980557440 in the Johnson $V$
filter.

During a small amount of down time between SOAR/Goodman observations of EREBOS
targets on 2019 June 9, we discovered Gaia DR2 6097540197980557440 with
approximately 45 minutes of time-series photometry using a Johnson $V$ filter.
We had previously identified Gaia DR2 6097540197980557440 as a strong
candidate variable hot subdwarf from its anomalously high Gaia DR2 photometric
uncertainty, and its inclusion in the Geier et al. (2019) catalog of candidate
hot subdwarf stars (see Barlow et al. in prep, Guidry et al. (2020) for
details). We unwittingly began observing just before primary eclipse and, upon
noticing the star disappear from the raw image frames111At this moment,
Stephen Walser, who was monitoring the frames as they came in, apologetically
informed us he had ‘lost our star.’ (shown in Figure 1), continued observing
long enough to safely capture egress. Using the processes described in §4.1,
we constructed a light curve and determined that the primary eclipse lasted
approximately 25 minutes. The shape of the eclipse stuck out to us immediately
as being different than in other HW Vir binaries. Whereas the ingress and
egress segments of most primary eclipses have positive second time derivatives
(i.e. concave up), Gaia DR2 6097540197980557440’s second derivatives are
negative during ingress and egress (i.e. concave down). This can only be
explained by the geometry of a nearly perfectly edge–on eclipse, so we were
eager to obtain photometry and spectroscopy over the full orbit to solve for
all system parameters.

Unfortunately, we were unable to determine a precise orbital period for the
system using our exploratory time–series photometry. However, Gaia DR2
6097540197980557440 was also observed by TESS in Sector 11 through full-frame
image (FFI), 30–min cadence observations. The data were downloaded from the
Mikulski Archive for Space Telescopes (MAST) web portal, and the lightkurve
(Lightkurve Collaboration et al., 2018) Python package was used to extract
time-series photometry from the FFIs. A Lomb-Scargle periodogram (Scargle,
1982) was computed and yielded an initial estimate of the system’s orbital
period of $P=3.0614\>\rm hr$. This estimate helped guide subsequent
observations.

##  3 Time–Series Spectroscopy

###  3.1 Observations & Reductions

We obtained 53 spectra using SOAR/Goodman on 2019 July 25 and 46 spectra on
2020 February 17, both in an uninterrupted series of back–to–back exposures.
Each of these data sets covered roughly 75% of the $\sim$3–hr orbital period.
We used the $0.8^{\prime\prime}$ long slit, 2$\times$2 binning, and the 930
$\rm mm^{-1}$ VPH grating (0.84 Å per binned pixel dispersion), giving us
average spectral resolutions of 2.38 Å and 2.04 Å over the wavelength range
3600-5300 Å for the 2019 and 2020 data, respectively. We note that the
spectral resolutions are different despite using the same instrumental
configuration due to the camera–collimator focus values not being set to their
optimal values during the 2019 observations. On both observing nights, we
aligned the slit axis to a position angle of $278.3^{\circ}$ E of N in order
to place a bright star222Gaia DR2 6097528446950034944 $23.5^{\prime\prime}$
away on the slit and monitor any drifts in the wavelength solution due to
instrumental flexure. Individual spectra in each series were integrated for
120–s, yielding an average S/N of $\sim$30 per resolution element. We also
obtained spectra of FeAr lamps immediately following each series for
wavelength calibration purposes.

Reduction of the frames was carried out using the ccdproc task in IRAF (Tody,
1986, 1993). After bias-subtracting and flat-fielding all spectral images, we
ran the apall task to extract a one-dimensional spectrum for each frame and
remove a fit to the sky background. For the 2020 data, a wavelength solution
was generated from the FeAr lamp spectra and applied to all individual
spectra. We note that slow drifts in the wavelength solution over the course
of the series are expected due to instrumental flexure, and thus the FeAr
wavelength solution does not provide an accurate zero–point — only an accurate
dispersion solution. For the 2019 data, an intermittent issue with the FeAr
lamp prevented us from obtaining an accurate dispersion solution with it.
Instead, we created a self template from the combined 2019 spectra and use the
Balmer and He I lines to determine the wavelength solution. Once again, this
only provides a dispersion solution and not an absolute RV zero point.
Consequently, we are unable to report on the binary’s systemic velocity. The
spectrum of Gaia DR2 6097540197980557440, shown in Figure 3, is dominated by
strong H Balmer absorption features and weaker He I lines (4026, 4471, 4921,
5015 Å). The absence of the He II 4686 Å line rules out an sdOB
classification.

(a)

(b)

Figure 2: Radial velocity curves constructed from spectra obtained with
SOAR/Goodman, plotted twice for better visualization. Left: Best fitting model
for the data taken in 2019. Right: Best fitting model for the data taken in
2020. Both of these solutions agree within the error bars with the weighted
average $K_{\rm sdB}=100.0\pm 2.0\rm\>km\,s^{-1}$.

###  3.2 Radial Velocity Curve

Radial velocity (RV) shifts were determined from non–linear, least–squares
fits of Gaussian profiles to the sdB H Balmer lines, which was carried out
using the curve_fit function in the Python package scipy (Virtanen et al.,
2020). The He I profiles were too noisy in individual spectra for this
purpose. In order to correct for drifts in the wavelength solution (and thus
drifts in the RVs) due to instrumental flexure during the observations, we
also measured the relative velocity shifts of the absorption features of the
second star on the slit. This object displayed spectral features consistent
with a G/K–type star, and so we used the crosscorrRV function in the
PyAstronomy library (Czesla et al., 2019) to measure velocity shifts via cross
correlation. The second star’s RV curves revealed gradual, nearly–linear
shifts on the order of $\sim$75 km s-1 over $\sim$2 hours, in both the 2019
and 2020 data sets. The magnitude and pattern of these shifts — slightly
different on the two nights — were consistent with expectations given the
target’s RA, DEC, average hour angle during each run, and associated Nasmyth
cage rotations. We are confident they are due to instrumental flexure and not
intrinsic RV variations of the second star on the slit. To remove this flexure
drift from the target RV curves, we fitted low–order polynomials to the
comparison star’s RV curves and subtracted this fit from the raw target star
RV curves. The resulting RV curves are shown in Figure 2.

In order to determine the RV semi–amplitude of the sdB ($K_{\rm sdB}$), we
fitted sine waves to each of the data sets separately, with the orbital period
and phase fixed to the values described in §4.2. From the 2019 July 25 data,
we find $K_{\rm sdB}=97.9\pm 2.6\rm\>km\,s^{-1}$, and from the 2020 February
17 data, we derive $K_{\rm sdB}=102.1\pm 3.0\rm\>km\,s^{-1}$. These results
agree within their 1-$\sigma$ uncertainties, and we adopt as our final RV
semi-amplitude their weighted average: $K_{\rm sdB}=100.0\pm
2.0\rm\>km\,s^{-1}$. The residuals in the bottom panels of Figure 2 are
consistent with random noise and show the data are consistent with a circular
orbit, as expected for post-common-envelope HW Vir binaries.

###  3.3 Atmospheric Parameters

For use in the spectroscopic analysis, model spectra are computed following
the so-called hybrid approach (Przybilla et al., 2006b, a; Nieva & Przybilla,
2008). In this approach, deviations from local thermodynamic equilibrium (LTE)
can be treated very efficiently using a combination of updated versions of the
Atlas12 (Kurucz, 1996), Detail (Giddings, 1981; Butler & Giddings, 1985), and
Surface (Giddings, 1981; Butler & Giddings, 1985) codes. The Atlas12 code, for
which the mean metallicity for hot sdB according to Naslim et al. (2013) is
used here, is initially used to compute the temperature/density structure of a
line-blanketed, plane-parallel, and chemically homogeneous atmosphere in
hydrostatic and radiative equilibrium. This LTE atmosphere is then used as
input for the Detail code, which solves the coupled radiative transfer and
statistical equilibrium equations to obtain occupation numbers in NLTE for
hydrogen and helium. Finally, the Surface code is used to compute the final
synthetic spectrum using the atmosphere from Atlas12 and the occupation
numbers from Detail as well as more sophisticated line-broadening data. Also
taken into consideration are the recent improvements to all three codes
(Irrgang et al., 2018) concerning NLTE effects on the atmospheric structure,
the implementation of the occupation probability formalism (Hubeny et al.,
1994) for hydrogen and neutral helium, and new Stark broadening tables for
hydrogen (Tremblay & Bergeron, 2009) and neutral helium (Beauchamp et al.,
1997). The application of these models to sdBs is also shown in Schaffenroth
et al. (2020).

The observed spectra are matched to the model grid by $\chi^{2}$ minimization
as described by Saffer et al. (1994) as implemented by Napiwotzki et al.
(1999). We use six H Balmer lines and four He I lines. H $\epsilon$ is
excluded because of contamination by interstellar Ca II. Since the binary
orbit is so tight, tidal forces probably have spun up the sdB star, which
causes extra line broadening. However, the resolution of the spectra is
insufficient to measure the projected rotational velocity v $\sin\,i$. We
assume that the rotation of the sdB is tidally locked to the binary orbit and
convolve the model spectrum with a rotational broadening profile with a
corresponding v $\sin i$=87 km s-1 in the fitting procedure.

Previous studies have shown that some sdBs with reflection effects have
atmospheric parameters that can vary with phase when analyzing spectra of
sufficiently high S/N taken at different phases of the orbit (e.g.,
Schaffenroth et al., 2013, 2014b). These variations can be explained by the
companion’s phase-variable contributions to the spectrum from only the
reflection effect, causing apparent variations of order $1000-1500$ K and 0.1
dex in the sdB temperature and surface gravity, respectively.

To account for any of these variations, we derived the atmospheric parameters
from the single-radial velocity corrected spectra. Exemplary fits are shown
for individual spectra from the 2019 and 2020 observing runs for similar
orbital phases in Fig. 3. Results from both observing runs are consistent. The
variations of the atmospheric parameters, which are consistent with previous
determinations, can be seen in Fig. 4. The effective temperature appears to
increase slightly near the secondary eclipse. Any variations in the surface
gravity or helium abundance remain below detection limits. In order to
determine the atmospheric parameters of the sdB we averaged the parameters
near the primary eclipse, where only the dark side of the companion is
visible: $T_{\rm eff}=26100\pm 400$K, $\log\left(g\right)=5.50\pm 0.07$, and
$\log\left(y\right)=-2.32\pm 0.10$.

(a)  (b)

Figure 3: Line fits to the hydrogen Balmer and neutral helium lines in
individual SOAR/Goodman spectra from 2019 (left hand panel) and 2020 (right
hand panel). Listed in the upper right of each panel is the orbital phase and
the resulting set of best fitting atmospheric parameters. Figure 4: Apparent
$T_{\mathrm{eff}}$ (bottom), $\log g$ (middle), $\log y$ (top) variations with
1 $\sigma$ error bars as a function of the orbital phase. Results from spectra
taken in 2019 are shown in black while those from 2020 in blue.

##  4 Time–Series Photometry

###  4.1 Observations & Reductions

Follow–up time–series photometry was obtained on 2020 February 18 using
SOAR/Goodman in imaging mode. In an effort to obtain multi-color photometry
for more precise modeling, the filter wheel was manually switched between the
Johnson $B$ and $R$ filters every few minutes when not in primary or secondary
eclipse, and every 30 seconds during eclipses. The integration time was fixed
to 5 seconds for both filters in order to minimize dead time and errors
associated with changing this value back and forth every few minutes. We used
2 $\times$ 2 binning and read out only a 350 $\times$ 175 binned pixel subset
of the image to minimize the readout time between exposures. This relatively
small field still provided several nearby comparison stars through which to
track sky transparency variations. We achieved a duty cycle of roughly 54%
over the course of our observations, which covered a little more than one full
orbital period. A more efficient duty cycle would have required either
decreasing the subframe region further and sacrificing comparison stars, or
increasing the exposure time and risk saturating the target and comparison
stars.

Reduction of the SOAR frames was once again carried out using the ccdproc
procedure in IRAF. Each raw image frame was first bias-subtracted and flat-
fielded. We then extracted aperture photometry using a range of aperture sizes
with a custom code utilizing the photutils (Bradley et al., 2019) Python
package. Sky counts were removed using sky annuli drawn around the apertures.
Apertures were chosen to maximize the signal-to-noise ratio (S/N) in each
light curve. This process was repeated on multiple nearby, bright comparison
stars to remove sky transparency variations and flux-normalize the light
curves. Multiple cycles of observing are typically needed to remove
airmass–related changes in the flux, therefore, any of these slight flux
variations were not removed during the reduction process. The resulting
differential light curves are shown in Figure 5 and used for modeling the
binary.

###  4.2 Binary Light Curve Modeling

The Gaia DR2 6097540197980557440 light curve exhibits all the typical HW Vir
features. The amplitude of the reflection effect is noticeably stronger in the
$R$ filter ($\sim 30\%$) than in the $B$ filter ($\sim 20\%$), and it is quite
strong in general compared to other reflection effect systems. Initially, this
led us to believe that either the sdB was slightly hotter than in typical HW
Virs, or the companion was slightly larger than usual. The deep primary
eclipse, implying a nearly edge–on inclination, lent credence to the latter
explanation. The shape of the eclipse itself sticks out amongst other HW Vir
binaries. As mentioned in Section 2, the ingress and egress segments of the
primary eclipse have negative second derivatives (more V–shaped) instead of
the more frequently observed positive second derivatives (more U–shaped). This
implies the eclipse geometry is nearly perfectly edge–on and that the
companion might be slightly larger than the primary. Secondary eclipses are
also present in the light curve, during which the sdB is blocking irradiated
light from the cool companion. Notably, the flux at the center of the
secondary eclipse returns to its exact value immediately preceding and
following ingress and egress, respectively — further implying that the
inclination must be nearly edge-on.

To model the light curves, we use the code LCURVE (for details, see Appendix A
in Copperwheat et al., 2010). In addition to recreating deep eclipses, LCURVE
was designed for binaries with WDs and has been used to fit WD+dM systems
exhibiting the reflection effect (e.g., Parsons et al., 2010); therefore, HW
Vir binaries are naturally suited to be modeled in a similar fashion (see
Schaffenroth et al., 2020, for an example and further details). To form full
solutions for these systems, there are many parameters that are not all
independent, so we can greatly improve our ability to constrain each solution
by fixing as many parameters as possible. We fixed the sdB temperature to the
value determined in our spectroscopic fit (described in §3.3). We also fix the
gravitational limb darkening coefficients to values expected of a primary with
a radiative atmosphere (von Zeipel, 1924) and a companion with a convective
atmospheres (Lucy, 1967) by calculating the resulting intensities using a
blackbody approximation. Then we adopted a quadratic limb darkening law for
the primary using the values in the Claret & Bloemen (2011) closest to the
parameters derived in our spectroscopic fits.

It is important to note that there is a large degeneracy in the light curve
solutions of HW Vir binaries, even when fixing all of the above parameters.
The orbit is certainly almost circular, so each model is not sensitive to the
mass ratio ($q$) of the system. For this reason we calculated different
solutions over a range of various, fixed mass ratios. We then use a SIMPLEX
algorithm (Press et al., 1992) to vary parameters such as the inclination,
both radii, the companion’s temperature, albedo, and limb darkening, and even
the period and primary eclipse time to help localize the solutions.
Additionally, we allow for linear trends due to airmass–related changes in
flux over the course of the observations.

Figure 5: SOAR/Goodman light curves of Gaia DR2 6097540197980557440 in both
the Johnson $B$ (blue points) and $R$ (red points) filters, along with their
respective best-fitting models from §4.2. The $B$–filter light curve is offset
by 0.4 for better visualization. Residuals are shown in the bottom panel with
offsets of 0.05 and $-$0.05 for the $B$ and $R$ curves, respectively.

Next we tested the degeneracy of each light curve solution and determined the
parameter errors by performing Markov-Chain Monte-Carlo (MCMC) computations
using emcee (Foreman-Mackey et al., 2013). We used the best-fit solution from
our SIMPLEX algorithm for initial values, and then we again varied the
inclination angle, both radii, the limb darkening coefficient assuming a
linear limb darkening law for the companion, and the companion’s temperature
and albedo for the mass ratio of our most probable solution (see §5). In all
cases, the temperature of the companion is not well constrained as its
fractional luminosity contribution to the system — outside of the reflection
effect — is negligible. We therefore constrained the companion’s temperature
to the range $2500-3500\,$K (the expected range for the low-mass companion).
The results and errors from our emcee run (shown in Table 1) then form the
basis for our most probable solution.

Figure 6: Photometric surface gravities plotted against their respective sdB masses for different mass ratio solutions ranging from q = 0.3 – 0.45, in 0.01 increments. The solid horizontal line (black) and shaded area represent the spectroscopically–derived $\log(g)$ and associated 1-$\sigma$ error, respectively. The vertical dashed line (yellow) represents the canonical sdB mass of $0.47\,M_{\odot}$. The intersection of these two lines shows that our most probable solution is the one with a sdB mass just below the canonical mass. Table 1: Parameters used to model the light curve for both the SOAR/Goodman $B$ and $R$ data. Parameter |  SOAR/Goodman – $B$ |  SOAR/Goodman – $R$ | Unit | Description
---|---|---|---|---
Fixed Parameters
$q\;(M_{\rm dM}/M_{\rm sdB})$ | 0.375 | 0.375 |  | mass ratio
$P$ | $0.127037$ | $0.127037$ | d | orbital period
$T_{\rm sdB}$ | 26100 | 26100 | K | primary temperature from spectroscopy
$\textsl{g}_{1}$ | $0.25$ | $0.25$ |  | gravitational darkening exponent
$\textsl{g}_{2}$ | $0.08$ | $0.08$ |  | gravitational darkening exponent
$x_{1,a}$ | $0.097$ | $0.070$ |  | primary linear limb darkening coefficient
$x_{1,b}$ | $0.285$ | $0.222$ |  | primary quadratic limb darkening coefficient
Adjusted Parameters
$i$ | $90^{+0.0}_{-0.3}$ | $90^{+0.0}_{-0.4}$ | ∘ | inclination angle
$x_{2,a}$ | $0.2992$ | $0.2734$ |  | companion linear limb darkening coefficient
$R_{\rm sdB}/a$ | $0.2180\pm 0.0007$ | $0.2174\pm 0.0007$ |  | primary radius
$R_{\rm dM}/a$ | $0.2402\pm 0.0006$ | $0.2407\pm 0.0006$ |  | companion radius
$T_{\rm dM}$ | $2800\pm 500$ | $3100\pm 500$ | K | companion temperature
$A_{2}$ | $1.14\pm 0.01$ | $1.4\pm 0.01$ |  | companion albedo (absorb)
$m$ | $0.00026\pm 0.00001$ | $0.00026\pm 0.00001$ |  | slope

###  4.3 Orbital Ephemeris

To aid in future observations of Gaia DR2 6097540197980557440, we have listed
its orbital ephemeris ($T_{0}$, $P$) in Table 2. We adopt the orbital period
from our best fit emcee solution in §4.2. To construct an initial eclipse time
($T_{0}$), we fit inverted Gaussian profiles to both the $B$ and $R$ time-
series data using curve_fit. We then adopt the weighted average of the central
times from both filter series as our $T_{0}$ value.

##  5 System Parameters

In Figure 6, we plot the surface gravity and sdB mass for each of the
potential solutions, and we compare the photometric surface gravities to our
spectroscopically derived surface gravity.

Based on the spectroscopic surface gravity, we get a consistent solution for
an sdB mass of $0.3-0.64\,\rm M_{\odot}$. All possible solutions fit the light
curve data nearly equally well; thus, we cannot claim a unique solution
without additional data (e.g., velocity measurements from the dM). The most
probable solution is the one with an sdB mass consistent with the canonical
mass of $0.47\,M_{\odot}$. The adopted best–fitting light curve solution and
all relevant parameters are given in Table 1, and both of these best-fit
models are shown together with their respective observations and residuals in
Figure 5. All possible solutions are given in Table 3 of the Appendix.

We compute the binary mass function for Gaia DR2 6097540197980557440 using the
expression

$f=\frac{K_{\rm sdB}^{3}P}{2\pi G}=\frac{M_{\rm
sdB}q^{3}\sin^{3}i}{\left(1+q\right)^{2}},$ (1)

finding $f=0.0132\pm 0.0008M_{\odot}$ using the period and sdB velocity semi-
amplitude. Combining this with the adopted mass ratio derived before, we find
the sdB and dM masses to be $M_{\rm sdB}=0.47\pm 0.03\,M_{\odot}$ and $M_{\rm
dM}=0.18\pm 0.01\,M_{\odot}$, respectively. Using Kepler’s third law, we then
find the orbital separation to be $a=0.921\pm 0.018\,R_{\odot}$. We also find
$R_{\rm sdB}=0.199\pm 0.004\,R_{\odot}$ and $R_{\rm dM}=0.222\pm
0.004\,R_{\odot}$. Table 2 gives an overview of the adopted parameters for
Gaia DR2 6097540197980557440.

In Figure 7, we show each set of parameters for the companion and the
theoretical mass-radius relations for low-mass main sequence stars from
Baraffe et al. (2015) as an additional check. It is clear that each solution
yields a companion radius that is inflated relative to what is predicted by
theory, which is a trend commonly seen in close binaries with M dwarf
components (Parsons et al., 2018). For our most probable solution we get a
companion inflation of about $\sim$13%.

Figure 7: Mass–radius diagram for the dM companion illustrating the degeneracy in model solutions (orange squares). Theoretical mass-radius relations of low-mass stars (Baraffe et al., 2015) for a 1 Gyr (dashed blue line), 5 Gyr (dash-dotted green line), and 10 Gyr (dotted pink line) are also included. The vertical yellow line and shaded region represent the most probable dM mass and 1-$\sigma$ error, respectively, associated with the adopted $0.47\,M_{\odot}$ sdB solution. Table 2: Overview of derived parameters for Gaia DR2 6097540197980557440 that represent the most probable solution from the set of potential solutions. Parameter | Value | Unit
---|---|---
Basic Information
$\alpha^{a,b}$ | 213.577775581303 | deg
$\delta^{a,b}$ | $-43.552249057309$ | deg
$G^{a}$ | 16.358994 | mag
$G_{\rm bp}-G_{\rm rp}\\!^{a}$ | $-0.27529526$ | mag
System Properties
$T_{0}$ | $2458898.85724\pm 0.00003$ | BJD
$P$ | $0.127037\pm 0.000001$ | d
$i$ | $90^{+0.0}_{-0.3}$ | ∘
$q$ | $0.375\pm 0.003$ |
$a$ | $0.921\pm 0.018$ | $R_{\odot}$
sdB Properties
$M_{\rm sdB}$ | $0.47\pm 0.03$ | $M_{\odot}$
$R_{\rm sdB}$ | $0.199\pm 0.004$ | $R_{\odot}$
$T_{\mathrm{eff}}$ | $26100\pm 400$ | K
$\log\left(g\right)$ | $5.50\pm 0.07$ |
$\log\left(y\right)$ | $-2.32\pm 0.10$ |
$K_{\rm sdB}$ | $100.0\pm 2.0$ | $\mathrm{km\,s^{-1}}$
dM Properties
$M_{\rm dM}$ | $0.177\pm 0.010$ | $M_{\odot}$
$R_{\rm dM}$ | $0.222\pm 0.004$ | $R_{\odot}$
$T_{\mathrm{eff}}$ | $3000\pm 500$ | K
Notes
a From Gaia DR2 (Gaia Collaboration et al., 2018)
b Epoch J2015.5.

##  6 Discussion

Our analysis of Gaia DR2 6097540197980557440 represents the first EREBOS case
study following the Schaffenroth et al. (2019) report of newly discovered
sdB+dM systems. With each additional system that is solved, EREBOS comes one
step closer to achieving one of its goals to make statistical statements about
a homogeneously selected population of close sdB systems. While one new system
by itself might not push the boundaries of key parameters in these studies,
each system provides self-consistent feedback about the methodology used to
study the overall population. It is only through these self-consistent
measures that EREBOS can eventually make statements regarding the effects
stellar and sub-stellar companions have on the late stages of stellar
evolution.

Our light curve and atmospheric modeling solutions imply Gaia DR2
6097540197980557440 is a fairly typical sdB+dM system, aside from the chance
alignment of its orbital plane nearly perfectly along our line–of–sight. The
peak of the EREBOS orbital period distribution for both new and
previously–published systems from Schaffenroth et al. (2019) is at $P=0.1\,$d,
meaning that Gaia DR2 6097540197980557440 falls at the typical period for HW
Virs. The most probable solution is a sdB with a mass of the canonical mass
$M_{\rm sdB}=0.47\pm 0.03\,M_{\odot}$. Additionally, our derived
$\log\left(g\right)$ and $\log\left(y\right)$ values are also fairly typical
of sdBs in HW Virs, but it is worth noting that our $T_{\rm eff}$ value is
slightly lower than is typically found (for comparison, see Fig. 6 in
Schaffenroth et al., 2019).

There are also noteworthy aspects of the system that are somewhat atypical
among HW Virs, namely the derived companion mass and sdB velocity semi-
amplitude. The companion mass is tied for the most massive in an HW Vir
binary, along with that of Konkoly J064029.1+385652.2 – also a
deeply–eclipsing HW Vir–type (sdO+dM) binary. Gaia DR2 6097540197980557440 has
an orbital period that is $\sim$1.5 hr shorter than Konkoly J064029.1+385652.2
and will one day evolve into a more rapid analog of Konkoly J064029.1+385652.2
when the sdB evolves into an sdO after the He in the core is exhausted and
then, inevitably, into a WD. The sdB semi-amplitude we derive from the two
sets of RV data make Gaia DR2 6097540197980557440 the fastest line–of–sight
sdB velocity semi-amplitude reported to date for an HW Vir binary.

The most striking aspect of Gaia DR2 6097540197980557440 is the total eclipse
of the sdB by its companion. Due to this system being relatively bright
($G\sim 16.4$ mag), a large eclipse depth means future eclipse timing (O$-$C)
analyses to search for changes in the orbital period ($\dot{P}$) and even
Rømer delay studies should be possible using telescopes with a variety of
aperture sizes (e.g., Barlow et al., 2012). Additionally, Gaia DR2
6097540197980557440 will be observed at 2-min cadence in Sector 38 of TESS
Cycle 3 through the Guest Investigator program (proposal #G03221), providing
space-quality data spanning 27 d of observations. This is a unique opportunity
to explore a relatively novel parameter space with one of the most accurate
astrophysical clocks known (e.g., Kilkenny, 2014).

##  7 Summary

We have presented photometric and spectroscopic observations of Gaia DR2
6097540197980557440, the first deeply eclipsing sdB+dM binary. Other than the
remarkably striking nature of the eclipse, the system is a rather typical
sdB+dM system. We find an orbital period of $P=0.127037$ d and an sdB velocity
semi-amplitude of $K_{\rm sdB}=100.0\rm\>km\,s^{-1}$ which combined with the
most probable light curve solution yields masses of $M_{\rm
sdB}=0.47\,M_{\odot}$ and $M_{\rm dM}=0.18\,M_{\odot}$, respectively. This
gives a radius of $R_{\rm dM}=0.222\,R_{\odot}$ for the companion, which is
slightly inflated relative to theoretical mass–radius relationships of low-
mass main sequence stars. Gaia DR2 6097540197980557440 represents the first HW
Vir solved as part of the EREBOS project. Eventual solutions for the more than
100 new HW Vir binaries uncovered by EREBOS will help improve our
understanding of the common envelope channel leading to sdBs, and help
determine the effects nearby low-mass stellar and substellar objects can have
on stars climbing the giant branch.

## Acknowledgements

K.A.C., B.N.B., and S.W. acknowledge the support of the National Science
Foundation under grant AST-1812874. BB also acknowledges partial support of
this work by NASA through the TESS Guest Investigator program, under grant
80NSSC19K1720. V.S. is supported by the Deutsche Forschungsgemeinschaft (DFG)
through grant GE 2506/9-1. A.I. acknowledges funding by the DFG through grant
HE1356/70-1.

This research draws upon data provided by NOIRLab 2019A-0255 and 2020A-0268;
PI B. Barlow as distributed by the NOIRLab Astro Data Archive. NOIRLab is
managed by the Association of Universities for Research in Astronomy (AURA)
under a cooperative agreement with the National Science Foundation.

Based on observations obtained at the Southern Astrophysical Research (SOAR)
telescope, which is a joint project of the Ministério da Ciência, Tecnologia,
Inovações e Comunicações (MCTIC) do Brasil, the U.S. National Optical
Astronomy Observatory (NOAO), the University of North Carolina at Chapel Hill
(UNC), and Michigan State University (MSU).

This paper includes data collected with the TESS mission, obtained from the
MAST data archive at the Space Telescope Science Institute (STScI). Funding
for the TESS mission is provided by the NASA Explorer Program. STScI is
operated by the Association of Universities for Research in Astronomy, Inc.,
under NASA contract NAS 5–26555.

This research made use of Lightkurve, a Python package for Kepler and TESS
data analysis (Lightkurve Collaboration et al., 2018). This research made use
of Photutils, an Astropy package for detection and photometry of astronomical
sources (Bradley et al., 2019).

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Shown in Figures 8 and 9 are the corner plots for the SOAR/Goodman $B$ and $R$
light curve solutions, respectively, using the Python package corner (Foreman-
Mackey, 2016) for visualization. Also, we give the full set of possible
solutions from the light curve modeling in Table 3.

Figure 8: Corner plot of the most probable light curve solution for the SOAR/Goodman – $B$ data. Figure 9: Corner plot of the most probable light curve solution for the SOAR/Goodman – $R$ data. Table 3: All of the possible light curve solutions output by LCURVE. $q$ | $a$ | $M_{\rm sdB}$ | $M_{\rm dM}$ | $R_{\rm sdB}$ | $R_{\rm dM}$ | $\log(g)$
---|---|---|---|---|---|---
| [$R_{\odot}$] | [$M_{\odot}$] | [$M_{\odot}$] | [$R_{\odot}$] | [$R_{\odot}$] |
0.30 | $1.090\pm 0.022$ | $0.826\pm 0.050$ | $0.248\pm 0.015$ | $0.234\pm 0.005$ | $0.268\pm 0.005$ | $5.616\pm 0.009$
0.31 | $1.063\pm 0.021$ | $0.760\pm 0.046$ | $0.236\pm 0.014$ | $0.231\pm 0.005$ | $0.261\pm 0.005$ | $5.591\pm 0.009$
0.32 | $1.038\pm 0.021$ | $0.701\pm 0.042$ | $0.224\pm 0.013$ | $0.224\pm 0.004$ | $0.253\pm 0.005$ | $5.585\pm 0.009$
0.33 | $1.014\pm 0.020$ | $0.649\pm 0.039$ | $0.214\pm 0.013$ | $0.219\pm 0.004$ | $0.246\pm 0.005$ | $5.571\pm 0.009$
0.34 | $0.992\pm 0.020$ | $0.603\pm 0.036$ | $0.205\pm 0.012$ | $0.214\pm 0.004$ | $0.240\pm 0.005$ | $5.558\pm 0.009$
0.35 | $0.971\pm 0.019$ | $0.561\pm 0.034$ | $0.196\pm 0.012$ | $0.209\pm 0.004$ | $0.234\pm 0.005$ | $5.547\pm 0.009$
0.36 | $0.951\pm 0.019$ | $0.523\pm 0.031$ | $0.188\pm 0.011$ | $0.205\pm 0.004$ | $0.229\pm 0.005$ | $5.531\pm 0.009$
0.37 | $0.932\pm 0.019$ | $0.489\pm 0.029$ | $0.181\pm 0.011$ | $0.201\pm 0.004$ | $0.224\pm 0.004$ | $5.521\pm 0.009$
0.38† | $0.914\pm 0.018$ | $0.458\pm 0.027$ | $0.174\pm 0.010$ | $0.197\pm 0.004$ | $0.219\pm 0.004$ | $5.509\pm 0.009$
0.39 | $0.897\pm 0.018$ | $0.430\pm 0.026$ | $0.168\pm 0.010$ | $0.194\pm 0.004$ | $0.215\pm 0.004$ | $5.497\pm 0.009$
0.40 | $0.881\pm 0.018$ | $0.404\pm 0.024$ | $0.162\pm 0.010$ | $0.190\pm 0.004$ | $0.210\pm 0.004$ | $5.486\pm 0.009$
0.41 | $0.865\pm 0.017$ | $0.380\pm 0.023$ | $0.156\pm 0.009$ | $0.187\pm 0.004$ | $0.206\pm 0.004$ | $5.473\pm 0.009$
0.42 | $0.851\pm 0.017$ | $0.359\pm 0.022$ | $0.151\pm 0.009$ | $0.184\pm 0.004$ | $0.203\pm 0.004$ | $5.464\pm 0.009$
0.43 | $0.837\pm 0.017$ | $0.339\pm 0.020$ | $0.146\pm 0.009$ | $0.181\pm 0.004$ | $0.199\pm 0.004$ | $5.453\pm 0.009$
0.44 | $0.824\pm 0.016$ | $0.321\pm 0.019$ | $0.141\pm 0.008$ | $0.178\pm 0.004$ | $0.195\pm 0.004$ | $5.443\pm 0.009$
0.45 | $0.811\pm 0.016$ | $0.304\pm 0.018$ | $0.137\pm 0.008$ | $0.175\pm 0.004$ | $0.192\pm 0.004$ | $5.433\pm 0.009$
Notes
† Most probable solution as outlined in the text