# A 16 Hour Transit of Kepler-167 e Observed by the Ground-based Unistellar
Telescope Network

Amaury Perrocheau SETI Institute, Carl Sagan Center, 339 Bernardo Ave, Suite
200, Mountain View, CA 94043, USA  Unistellar, 5 allée Marcel Leclerc,
bâtiment B, Marseille, 13008, France  Thomas M. Esposito SETI Institute, Carl
Sagan Center, 339 Bernardo Ave, Suite 200, Mountain View, CA 94043, USA
Unistellar, 5 allée Marcel Leclerc, bâtiment B, Marseille, 13008, France
Astronomy Department, University of California, Berkeley, CA 94720, USA  Paul
A. Dalba Heising-Simons 51 Pegasi b Postdoctoral Fellow  SETI Institute, Carl
Sagan Center, 339 Bernardo Ave, Suite 200, Mountain View, CA 94043, USA
Department of Astronomy and Astrophysics, University of California, Santa
Cruz, CA 95064, USA  Franck Marchis SETI Institute, Carl Sagan Center, 339
Bernardo Ave, Suite 200, Mountain View, CA 94043, USA  Unistellar, 5 allée
Marcel Leclerc, bâtiment B, Marseille, 13008, France  Arin M. Avsar Lunar and
Planetary Laboratory, University of Arizona, Tucson, AZ 85721, USA  Ero
Carrera  Unistellar Citizen Scientist  Michel Douezy  Unistellar Citizen
Scientist  Keiichi Fukui  Unistellar Citizen Scientist  Ryan Gamurot
Unistellar Citizen Scientist  Tateki Goto  Unistellar Citizen Scientist  Bruno
Guillet Unistellar Citizen Scientist  Petri Kuossari  Unistellar Citizen
Scientist  Jean-Marie Laugier  Unistellar Citizen Scientist  Pablo Lewin The
Maury Lewin Astronomical Observatory, Glendora, CA, USA  Exoplanet Watch
Citizen Scientist  Margaret A. Loose  Unistellar Citizen Scientist  Laurent
Manganese  Unistellar Citizen Scientist  Benjamin Mirwald  Unistellar Citizen
Scientist  Hubert Mountz  Unistellar Citizen Scientist  Marti Mountz
Unistellar Citizen Scientist  Cory Ostrem  Unistellar Citizen Scientist  Bruce
Parker  Unistellar Citizen Scientist  Patrick Picard  Unistellar Citizen
Scientist  Michael Primm  Unistellar Citizen Scientist  Justus Randolph
Unistellar Citizen Scientist  Jay Runge  Unistellar Citizen Scientist  Robert
Savonnet  Unistellar Citizen Scientist  Chelsea E. Sharon Unistellar Citizen
Scientist  Yale-NUS College, 16 College Ave West 01-220, 138527, Singapore
Jenny Shih  Unistellar Citizen Scientist  Masao Shimizu  Unistellar Citizen
Scientist  George Silvis  Exoplanet Watch Citizen Scientist  Georges Simard
Unistellar Citizen Scientist  Alan Simpson  Unistellar Citizen Scientist
Thusheeta Sivayogan  Unistellar Citizen Scientist  Meyer Stein  Unistellar
Citizen Scientist  Denis Trudel  Unistellar Citizen Scientist  Hiroaki
Tsuchiyama  Unistellar Citizen Scientist  Kevin Wagner NASA Hubble Fellowship
Program – Sagan Fellow  Unistellar Citizen Scientist  Department of Astronomy
and Steward Observatory, University of Arizona, 933 N Cherry Ave, Tucson, AZ
85721, USA  Stefan Will  Unistellar Citizen Scientist

(Accepted in ApJL on November 1, 2022)

###### Abstract

More than 5,000 exoplanets have been confirmed and among them almost 4,000
were discovered by the transit method. However, few transiting exoplanets have
an orbital period greater than 100 days. Here we report a transit detection of
Kepler-167 e, a “Jupiter analog” exoplanet orbiting a K4 star with a period of
1,071 days, using the Unistellar ground-based telescope network. From 2021
November 18 to 20, citizen astronomers located in nine different countries
gathered 43 observations, covering the 16 hour long transit. Using a nested
sampling approach to combine and fit the observations, we detected the mid-
transit time to be UTC 2021 November 19 17:20:51 with a 1$\sigma$ uncertainty
of 9.8 minutes, making it the longest-period planet to ever have its transit
detected from the ground. This is the fourth transit detection of Kepler-167
e, but the first made from the ground. This timing measurement refines the
orbit and keeps the ephemeris up to date without requiring space telescopes.
Observations like this demonstrate the capabilities of coordinated networks of
small telescopes to identify and characterize planets with long orbital
periods.

††software: Astrometry.net (Lang et al., 2010), Astropy (Astropy Collaboration
et al., 2013, 2018), dynesty (Speagle, 2020), Photutils (Bradley et al.,
2020), PYCHEOPS (Maxted et al., 2022)

##  1 Introduction

Around 800 exoplanets are known to have an orbital period of more than 100
days, based on data extracted from the NASA Exoplanet Archive111As of 2022
September 2. (NEA; Akeson et al. 2013; NASA Exoplanet Archive 2022). Out of
these 800, only ${\sim}170$ have been detected using the transit method. This
is primarily because the probability of observing an exoplanet transit
decreases as the planet’s orbital semi-major axis increases, but also because
the relatively small fraction of its orbit that a planet spends in transit
requires almost continuous observation for detection (Beatty & Gaudi, 2008).
While the probability of detection does increase with orbital eccentricity,
with periastron becoming smaller, the time between transits remains long
(e.g., Kane, 2007; Dalba et al., 2021). Freed from the constraints of ground-
based observations, the Kepler (Borucki et al., 2010) and Transiting Exoplanet
Survey Satellite (TESS; Ricker et al., 2014) missions enabled great progress
in the search for long period exoplanets by providing long continuous baseline
transit surveys (Wang et al. 2015, Uehara et al. 2016, Herman et al. 2019),
although for TESS this is true only near the ecliptic poles.

Studying these long period planets is crucial to understanding our own solar
system. Jupiter has an orbital period of 12 years but dominates the planets’
gravitational effects and is partly responsible for the current architecture
of the solar system due to its migrations (e.g., Tsiganis et al. 2005, Walsh
et al. 2011). It is possible to observe some of these long period exoplanets
transiting only once every few years and it requires substantial observation
time on space telescopes to detect those with periods that are unknown or
poorly constrained (Foreman-Mackey et al., 2016). Observing them from the
ground is even harder because of the Sun’s daily interference and the long
durations of these transits; of the confirmed planets with period $>100$ days
and a measured duration, all but one have a duration $>3.5$ hours (NEA),
meaning at least a 7 hour observation is required to gather equal amounts of
in-transit and out-of-transit data. From the ground, therefore, only a network
of telescopes spanning the globe is capable of capturing enough of the transit
to lead to a detection (Von Braun et al., 2009). The longest-period transiting
planet previously detected was HIP 41378 f (542 days) and it required four
professional telescopes in Chile, Spain and Italy, only two of which actually
captured a combined ${\sim}3$ hours of in-transit data out of the 19 hour
event (Bryant et al., 2021). Similarly, two observing sites in Spain (Garcia-
Melendo & McCullough, 2009) and the United Kingdom (Fossey et al., 2009)
detected only egress of the first observed 12-hour-long transit by HD 80606 b
(111 day period); admirable considering the mid-transit date was only
predicted within a ${\sim}1$ day window (Laughlin et al., 2009). Re-detection
of a nearly complete transit by HD 80606 b by Pearson et al. (2022) via global
coordination of ground-based telescopes, including Unistellar Network
eVscopes, illustrates the value of geographically distributed observations.

Here, we present the first ground-based detection of the transiting “Jupiter
analog” Kepler-167 e with observations by the Unistellar Network (Marchis et
al., 2019). Kepler-167 e has a period of
$1071.23205\begin{subarray}{c}+0.00059\\\ -0.00058\end{subarray}$ days
(Chachan et al., 2022) and transit duration of $16.25\pm 0.07$ hours (taken as
the mean duration of 16,000 sample models drawn from the Chachan et al. 2022
posterior distribution). With a radius between that of Jupiter and Saturn (0.9
$R_{J}$), it has about the same mass as Jupiter (1.01 $M_{J}$) and a low
orbital eccentricity (0.06; Kipping et al. 2016). It orbits a K4V star with
$V$=14.284, along with three inner super-Earth planets that also transit
(Kipping et al., 2016). The Kepler mission enabled the first two observations,
in 2010 and in 2013 (Kipping et al., 2016), and half a transit was observed in
2018 with the Spitzer Space Telescope (Werner et al., 2004) to rule out
transit timing variations (TTV) of at least 34 minutes to 3$\sigma$ confidence
(Dalba & Tamburo, 2019). Thanks to these works, the uncertainty on the mid-
transit time ($T_{0}$) of the 2021 transit prediction was just
$\sigma_{T_{0}}=3.0$ min.

The Unistellar Network is composed of ${\sim}10,000$ digital telescopes called
Enhanced Vision Telescopes, or “eVscopes” (Marchis et al., 2020). Citizen
astronomers in possession of an eVscope can be found all over the world. These
telescopes have already been used to observe exoplanet transits (e.g., HD
80606 b by Pearson et al. 2022, WASP-148 b by Wang et al. 2022, TOI-2180 b by
Dalba et al. 2022), time asteroid occultations (Cazeneuve et al., 2022), and
monitor the James Webb Space Telescope (JWST) (Lambert et al., 2022).

We describe here our observations and analyses that led to the detection of
the Kepler-167 e transit. In Section 2, we present the photometric data
gathered from the Unistellar Network. Section 3 describes the methods for
combining the observations and modeling the transit. Section 4 presents the
results and finalized light curve. Finally, Section 5 discusses the validity
of our results and relevance of a ground-based telescope network for observing
long duration exoplanet transits.

##  2 Observations

###  2.1 Unistellar eVscopes

The eVscopes are portable digital consumer telescopes manufactured and
distributed by Unistellar. Two hardware models are currently in use: the
eVscope 1 (same as the eQuinox model) and the eVscope 2. They both have an
aperture of 114.3 mm and a focal length of 450 mm. Both also use off-the-shelf
Sony CMOS sensors located at the prime focus with low read noise and dark
current, with a Bayer filter. The IMX224 sensor in the eVscope 1 has a field
of view (FOV) of $37\arcmin\times 28\arcmin$ with a pixel scale of 1$\farcs$72
per pixel and the IMX347 in the eVscope 2 has an FOV of $45\arcmin\times
34\arcmin$ with a pixel scale of 1$\farcs$33 per pixel. Each eVscope is
identified by a 6 digit alphanumerical string that is abbreviated in this
paper to the first three characters. A suffix “-B” is added when the citizen
scientist observed for a second time during the campaign with the same
telescope.

###  2.2 Observation Campaign

Dalba & Tamburo (2019) predicted $T_{0}$ of this 2021 transit to be
2459538.2189 $\text{BJD}_{\text{TDB}}\,\pm 0.0021$ (a 3.0 min uncertainty).
They also ruled out TTV’s greater than 34 min to 3$\sigma$ significance. Using
this information, the SETI Institute, scientific partner of Unistellar, made a
call to the Unistellar Network on 2021 November 4 to observe the upcoming
transit. In response, 31 citizen astronomers and two professional astronomers
from our team observed the target star between 2021 November 18 and 20 using
Unistellar eVscopes, producing 43 individual data sets. The exposure time was
set at 3.970 s and the gain at 0.02245 e- ADU-1 (40 dB). Detailed information
about the observations is given in Table 2. To have enough continuous dark
time to record the $>$16 hour transit duration (temporarily ignoring the time
also required to record important out-of-transit baseline data), a single
observing site would need an extreme latitude of ${>}76\arcdeg$ N; thus the
necessity to combine observations from multiple longitudes around the world.
Two citizen astronomers not affiliated with the Unistellar Network also
observed Kepler-167 during this campaign, discussed in Section 5.1.

Figure 1: Time frames for eVscope observations of Kepler-167 e. 27 out of the
43 observations (green) were used to construct the final light curve. Vertical
lines mark the predicted transit start, middle, and end times.

###  2.3 Observation Selection

The observations were made from various sites spread around Earth, leading to
variable data quality. At that time, the Moon was ${\sim}$100% illuminated and
located ${\sim}$103∘ from the target. Some data were not usable because of
poor weather, poor atmospheric seeing, or sub-optimal calibration of the
telescope focus or collimation. Moreover, the relatively high sensor gain used
to observe this $V$=14.3 target caused an unexpectedly large non-linear
increase in the background sky brightness that saturated some observations
made from light-polluted areas. To avoid these data, we excluded observations
where at least one pixel in the target star’s point-spread function was
saturated or the scatter in the relative flux residuals after model
subtraction was $>10$%. As a result, only 27 out of the 43 data sets were used
in the analysis. The time distribution of observations can be found in Figure
1 in relation to the predicted transit times.

##  3 Methods

###  3.1 Photometry

The raw images received from the eVscopes in FITS format are first dark
subtracted, plate solved using Astrometry.net (Lang et al., 2010), aligned,
and then stacked in groups of 30 representing 119.1 s of integration time.
This image stacking increases star signal-to-noise ratio (SNR) by reducing
random noise and further mitigates photometric noise by averaging over the
differing spectral responses of individual pixels caused by the Bayer filter
(e.g., Lambert et al., 2022).

Next we performed differential aperture photometry on the stacked images using
the Photutils Python package (Bradley et al., 2020). This allowed us to locate
every star in an image and measure its position, flux, and uncertainty on that
flux. For each data set, we identified several potential comparison stars that
satisfied our criteria of being located within $3\arcmin$ of the target star,
unsaturated in the individual images, and not blended with other sources. For
consistency between data sets, we chose to use the same single comparison star
to compute all of our relative fluxes: Gaia DR3 2051932862036924416
($19^{\text{h}}30^{\text{m}}40\fs 31$ +$38\text{\textdegree}19\arcmin 34\farcs
14$), slightly fainter than Kepler-167 and located $1\farcm 25$ away. Being
similar in magnitude to the target star meant this comparison star was also
the highest SNR option that was unsaturated. We then optimized the sizes of
the circular flux apertures and annular background apertures for both target
and comparison star independently by minimizing the scatter in the individual
star fluxes within each data set.

###  3.2 Detrending

With the photometry done, we began assembling the light curve of the complete
event from the individual light curves of each data set. By taking the
relative flux ratio between the target and the comparison star, long term
trends in the fluxes and image-to-image photometric variations were largely
removed. Trends due to differential atmospheric extinction (i.e., airmass), in
particular, were greatly mitigated by the nearly identical $G_{BP}-G_{RP}$
colors of the target and comparison stars, given by the Gaia DR3 catalog (Gaia
Collaboration et al., 2016, 2022) as $(G_{BP}-G_{RP})_{targ}=1.227932$ and
$(G_{BP}-G_{RP})_{comp}=1.190314$). No other detrending was applied. Finally,
the relative fluxes of each data set are divided by their medians to normalize
them to approximately 1.0 in preparation for further normalization during
modeling.

We found most data sets’ relative fluxes to be approximately flat over time
but some did contain significant slopes. The timing of the most steeply sloped
fluxes coincided with the transit’s expected ingress and egress times,
suggesting they were real physical features; however, we did not assume this
to be the case during our analysis.

Table 1: Transit Parameter Values

Parameter | Expected Value | Distribution Priors | Best Fit
---|---|---|---
$T_{0}$ ($\text{BJD}_{\text{TDB}}$) | $2459538.2189\pm 0.0021$ | $\mathcal{U}[2459538.1793,2459538.2585]$ | $2459538.2225\pm 0.0068$
$D$ (%) | $1.540_{\,-0.032}^{+0.027}$ | $\mathcal{N}(1.540,4)$ | $1.77\pm 0.42$
$W\times P$ (minutes) | $975\pm 4$ |  |
$P$ (days) | $1071.23205_{\,-0.00058}^{+0.00059}$ | Fixed
$b$ | $0.271_{\,-0.073}^{+0.051}$
$e$ | 0
$h_{1}$ | $0.684\pm 0.011$
$h_{2}$ | $0.434\pm 0.050$ |  |
$c_{1}$ (257) | 1 |  | $0.997\pm 0.007$
$c_{2}$ (rcn) | 1 |  | $1.001\pm 0.003$
$c_{3}$ (un8) | 1 |  | $0.998\pm 0.006$
$c_{4}$ (24s) | 1 |  | $0.990\pm 0.006$
$c_{5}$ (sdp) | 1 |  | $0.992\pm 0.015$
$c_{6}$ (3pw) | 1 |  | $1.001\pm 0.007$
$c_{7}$ (8cm) | 0.993 |  | $0.992\pm 0.004$
$c_{8}$ (jpg) | 0.978 |  | $0.982\pm 0.009$
$c_{9}$ (tuf) | 0.978 |  | $0.978\pm 0.006$
$c_{10}$ (7bt) | 0.979 |  | $0.982\pm 0.009$
$c_{11}$ (5m4) | 0.979 |  | $0.987\pm 0.008$
$c_{12}$ (8zz) | 0.979 | $\mathcal{N}(1,0.0004)$ | $0.987\pm 0.007$
$c_{13}$ (etx) | 0.979 |  | $0.980\pm 0.015$
$c_{14}$ (ujx) | 0.988 |  | $0.991\pm 0.005$
$c_{15}$ (2mv) | 0.993 |  | $0.993\pm 0.004$
$c_{16}$ (rp9) | 0.993 |  | $0.989\pm 0.005$
$c_{17}$ (rcn-B) | 0.996 |  | $0.994\pm 0.004$
$c_{18}$ (un8-B) | 0.996 |  | $0.998\pm 0.004$
$c_{19}$ (rev) | 0.999 |  | $0.993\pm 0.008$
$c_{20}$ (sdp-B) | 1 |  | $0.998\pm 0.008$
$c_{21}$ (gau-B) | 1 |  | $0.998\pm 0.005$
$c_{22}$ (p6r-B) | 1 |  | $0.998\pm 0.003$
$c_{23}$ (2n9) | 1 |  | $0.987\pm 0.011$
$c_{24}$ (fqe) | 1 |  | $0.997\pm 0.009$
$c_{25}$ (33r) | 1 |  | $1.003\pm 0.006$
$c_{26}$ (3pw-B) | 1 |  | $0.987\pm 0.008$
$c_{27}$ (8cm-B) | 1 |  | $0.998\pm 0.004$

###  3.3 Light Curve Normalization and Modeling

Extracting the transit’s signal and timing from these multiple data sets
requires combining them into one light curve and comparing that with a model.
Differential photometry does not establish the absolute brightness of the
target star, thus in order to properly construct a combined light curve we
need to normalize the individual light curves relative to some reference
baseline. We start with the assumption that relative fluxes measured outside
of the transit will have a mean of 1.0, which becomes our reference baseline.

To normalize fluxes within our model, we multiply a scalar coefficient $c_{i}$
with the relative fluxes $y_{i,j}$ of every individual light curve $i$, where
$j$ denotes the time of each measurement. Thus, every individual light curve
is free to move up and down relative to the reference baseline. We then
construct a transit light curve model using the PYCHEOPS package (Maxted et
al., 2022) and fit it to the combination of all normalized fluxes. Stellar
limb darkening effects are taken into account using the power-2 law as
implemented in the qpower2 algorithm (Maxted, P. F. L. & Gill, S., 2019).

The parameters used to compute the transit model are given in Table 3.2. In
our approach, which aims to retrieve the transit times and depth assuming the
other transit properties are well known, we fix the following model parameters
to the latest values published by Chachan et al. (2022): the planet’s orbital
period $P$, transit duration (parameterized as $P\times W$ where $W$ is the
length of the transit in orbital phase units), impact parameter $b$,
eccentricity $e$ (assumed to be 0), and stellar limb darkening coefficients
$h_{1}$ and $h_{2}$. As a result, only the mid-transit time $T_{0}$ and the
transit depth $D$ can vary. The eccentricity measurements from Kipping et al.
(2016) and Chachan et al. (2022) are both consistent with zero, so we assume
the simplest case of a circular orbit. The limb darkening coefficients $h_{1}$
and $h_{2}$ are transformed parameters from the power-2 law described in
Maxted et al. (2022).

###  3.4 Dynamic Nested Sampling

To fit our model to the data we used Bayesian statistics via a dynamic nested
sampling approach (Higson et al., 2018). This method lets us estimate
parameters from a highly multidimensional model. The sampling algorithm was
implemented with the dynesty Python package (Speagle, 2020), which required
two functions: the prior transformation and the likelihood function.

The prior transformation converts a hypercube of values between 0 and 1 to
physical scales. To avoid biasing the result toward a particular mid-transit
time, we used a uniform prior for $T_{0}$. The prior bounds are chosen to be
the expected $T_{0}$ $\pm$ the upper limit of the smallest TTV magnitude ruled
out at 5$\sigma$ significance by Dalba & Tamburo (2019), i.e., $\pm 57$ min.
For $D$ and normalization coefficients $c_{i}$, we chose normal priors with
the expected value as the mean and a 2% standard deviation. The values chosen
are listed in Table 3.2. These priors were specifically chosen to be loose so
the intrinsic likelihood would primarily drive the fit.

We assume that the flux measurements are independent and therefore drawn from
a normal distribution. We consider this assumption to be justified because we
confirmed that the time-averaged root mean squared of the full light curve’s
binned flux residuals (using the best-fit model) decreased with bin size at
the rate that theory predicts for residuals dominated by uncorrelated white
noise (Cubillos et al., 2017). The overall likelihood is then obtained by
multiplying together the individual measurement likelihoods. Finally, because
it is easier in practice to maximize the logarithm of the likelihood, we are
left with an effective likelihood function of:

$\log\mathcal{L}=-\frac{1}{2}\sum_{i}\sum_{j}\frac{\big{(}c_{i}y_{i,j}-f_{j}\big{)}^{2}}{\sigma_{i,j}^{2}}$
(1)

where $\sigma_{i,j}$ are the uncertainties associated with each relative flux
measurement $y_{i,j}$, and $f_{j}$ are the modeled light curve fluxes at
measurement time $j$.

We consider that the model fitting procedure has converged when
$\log\mathcal{Z}<0.1$, where $\mathcal{Z}$ is the Bayesian model evidence.

Figure 2: Transit light curve of Kepler-167 e (top) and residuals (middle) as
observed by the Unistellar Network. Relative fluxes from individual data sets
are plotted with different colors and symbols. The maximum likelihood model
corresponding to the values in Table 3.2 is the red line and the time axis is
defined as time since that model’s mid-transit time. The blue squares are
weighted average fluxes of the combined data. The expected transit model and
its transit start, middle, and end times are plotted as black broken lines.
The bottom panel is the same as the top panel but zooms in on the times the
transit occurred and only shows the weighted average fluxes.

##  4 Results

Using the nested sampling approach described in Section 3, we obtained the
maximum likelihood (“best-fit”) light curve at the top of Figure 2, along with
an estimation of the transit parameters. We measure $T_{0}$ to be UTC 2021
November 19 17:20:51 $\pm$ 9.8 min ($2459538.2225\pm 0.0068$
$\text{BJD}_{\text{TDB}}$) within the 68% confidence interval. This is
statistically consistent with the predicted mid-transit time of UTC 2021
November 19 17:15:35 $\pm$ 3.0 min ($2459538.2189\pm 0.0021$
$\text{BJD}_{\text{TDB}}$). The depth of the transit is measured to be
$D=1.77\pm 0.42$% compared to the predicted $1.540^{+0.027}_{\,-0.032}$%. The
computed values for the normalization coefficients $c_{i}$ are listed by data
set in Table 3.2.

To test that our normalization of the individual light curves is not biasing
our timing measurements, we also fit a transit model to a light curve that
excluded the fully in-transit data, which involved removing fluxes from the
following data sets: jpg, tuf, 7bt, 5m4, 8zz and etx. This left only data sets
containing at least some out-of-transit data (based on our maximum likelihood
model). The resulting $T_{0}$ computed without in-transit data is UTC 2021
November 19 17:07:45 $\pm$ 11 min ($2459538.2134\pm 0.0073$
$\text{BJD}_{\text{TDB}}$) and the depth is $1.20\pm 0.48$%. This transit
depth is still consistent with the prediction and the two configurations —
with and without the in-transit data — are consistent with each other at the
$1\sigma$ level.

##  5 Discussion

###  5.1 Exoplanet Watch Observation

Two citizen astronomers (Lewin and Silvis) participating in NASA’s Exoplanet
Watch program (EW; Zellem et al. 2020) observed Kepler-167 during the same
campaign in November 2021. We acquired their data from the public database
hosted by the American Association of Variable Star
Observers222https://www.aavso.org/. Lewin observed a flat light curve in data
taken well before the expected transit ingress, but Silvis observed during the
expected egress with a 12 inch (30 cm) telescope (Table 2). Those relative
fluxes, after being detrended with a standard airmass correction function, are
plotted in Figure 3 and overlaid with the maximum likelihood transit model
computed from our “best-fit” parameters of Table 3.2. The systematic “wiggle”
in the light curve at $X\approx 388$–407 corresponds to a temporary increase
in image background level from environmental light pollution (a house light
being turned on). Following this, increasing airmass (from 1.4 to 1.6) and a
broadening PSF may have introduced additional systematic noise via imperfect
background subtraction. The last ten flux points were strongly affected by the
local horizon vignetting the entrance pupil; a systematic error not
encompassed by their error bars.

Without conducting any additional fitting, the $\chi^{2}$ and the reduced
$\chi^{2}_{\nu}$ values between our transit model and the EW fluxes are
$\chi^{2}=65.9$ and $\chi_{\nu}^{2}=0.82$, respectively. In contrast, the
corresponding values when comparing the EW fluxes with a flat model normalized
to the median of the detrended data are $\chi^{2}\text{(flat)}=146.6\text{ and
}\chi_{\nu}^{2}\text{(flat)}=1.81$. We consider this an independent
confirmation that the Unistellar Network data successfully detected the egress
of the transit.

Figure 3: Exoplanet Watch observation of the Kepler-167 e egress. Relative
fluxes are the blue dots, with corresponding $1\sigma$ error bars. The best-
fit transit model from fitting only Unistellar Network data is overlaid in
red. This model was not fit to the Exoplanet Watch observations.

###  5.2 Observing Exoplanet Transits with a Network of Small Ground-Based
Telescopes

This successful measurement of Kepler-167 e proves that it is possible to
detect a long period, long duration exoplanet transit with a network of small
ground-based telescopes. Even though we knew the exoplanet parameters
precisely before the transit occurred in this case, we can reproduce this
result for planet candidates that have their properties less constrained.
Using small telescopes such as the eVscope therefore permits the detection and
confirmation of planet candidates with imprecisely known orbital periods
without requiring observation time on in-demand space telescopes such as JWST
or the Hubble Space Telescope (HST). A mission like ESA’s current
CHaracterising ExOPlanet Satellite (CHEOPS), which can do long duration
follow-up, could also benefit from a small-telescope network because it has
its own observability limitations owing to its orbit and Earth occultations
(Benz et al., 2020).

Keeping transiting planet ephemerides “fresh” (i.e the 1$\sigma$ uncertainty
in the mid-transit time to less than half the transit duration) is crucial;
for instance, most of the ephemerides coming from TESS will be outdated only
one year after the last observation (Dragomir et al., 2020). This concern is
not limited to transiting planets, as radial velocity-detected planets also
need to be monitored (Kane et al., 2009). Zellem et al. (2020) proved that
uncertainties of just 1 minute in a planet’s orbital period and mid-transit
time will produce an uncertainty of ${\sim}15$ hours in that planet’s mid-
transit time ten years later. Thus, regular re-detections with small ground-
based telescopes are important to paving the way for detailed spectroscopic
characterization by JWST and subsequent generations of exoplanet
characterizing facilities.

###  5.3 Transit Timing Consistency and Future Studies

Our timing measurement supports the previous finding by Dalba & Tamburo (2019)
that Kepler-167 e’s ephemeris does not contain large TTV’s. The difference
between our measured mid-transit epoch and the initial transit epoch
recomputed by Chachan et al. (2022) is $4284.9349\pm 0.0068$ days, which when
divided by four equates to an orbital period of $1071.2337\pm 0.0017$ days.
This is in line with the Kipping et al. (2016) value of $P=1071.23228\pm
0.00056$ days and $P=1071.23205_{\,-0.00058}^{+0.00059}$ from Chachan et al.
(2022), though about 2 min longer than both. Using our best-fit mid-transit
time as a reference epoch to compute future mid-transit times, we obtain
values that fall within the Dalba & Tamburo (2019) 1$\sigma$ predictions, but
we do not improve upon their precision so we do not report new values here.
The relatively large uncertainty on the previous epoch’s measurement from
Spitzer data also prevents us from tightly constraining TTV’s over the
planet’s last orbit, but we can rule out 24 and 73 min TTV’s to $1\sigma$ and
$3\sigma$ significance, respectively. Our own measurement uncertainty will be
the dominant factor in ruling out even smaller TTV’s over the next orbit, thus
improved timing precision for the next transit (e.g., with greater telescope
coverage from our network) will be a future objective.

The lack of large TTV’s means that future observations by telescopes like HST
or JWST will not run the risk of missing substantial portions of valuable in-
transit time for Kepler-167 e. It is rare for such a long period (and low
temperature) planet to be so well characterized with an unambiguous ephemeris
and measured mass and radius. Therefore, we encourage additional work to
explore this planet’s suitability for expanded characterization of its
atmosphere and potential exomoons.

##  6 Acknowledgements

The authors thank all the citizen astronomers that participated in this
observation campaign and contributed their precious data. We also thank Rob
Zellem for his contributions to this research effort and the anonymous
reviewer for their thoughtful feedback.

This research has made use of the NASA Exoplanet Archive, which is operated by
the California Institute of Technology, under contract with the National
Aeronautics and Space Administration under the Exoplanet Exploration Program.
This work has also made use of data from the European Space Agency (ESA)
mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data
Processing and Analysis Consortium
(DPAC,https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the
DPAC has been provided by national institutions, in particular the
institutions participating in the Gaia Multilateral Agreement.

This publication makes use of data products from Exoplanet Watch, a citizen
science project managed by NASA’s Jet Propulsion Laboratory on behalf of
NASA’s Universe of Learning. This work is supported by NASA under award number
NNX16AC65A to the Space Telescope Science Institute, in partnership with
Caltech/IPAC, Center for Astrophysics—Harvard & Smithsonian, and NASA Jet
Propulsion Laboratory. We acknowledge with thanks the variable star
observations from the AAVSO International Database contributed by observers
worldwide and used in this research.

T.M.E., F.M., and the Unistellar Network’s science campaigns are supported in
part by a grant from the Gordon and Betty Moore Foundation. P.D. acknowledges
support from a National Science Foundation (NSF) Astronomy and Astrophysics
Postdoctoral Fellowship under award AST-1903811 and a 51 Pegasi b Postdoctoral
Fellowship from the Heising-Simons Foundation.

##  Appendix A Summary of Observations

The times, locations, telescopes, and observer names for the Kepler-167 e
observations analyzed in this work are given in Table 2. The time given is the
length of the full observation but useful data may not have been acquired
during that entire period.

Table 2: Observations Summary

eVscope ID | eVscope | UTC Start | Start Time | End Time a | Observer | Location
---|---|---|---|---|---|---
(abbreviated) | Model | Date | (hh:mm:ss) | (hh:mm:ss) |  |
257 | 1 | 2021 Nov 18 | 19:13:08 | 20:43:19 | Kuossari | Tervakoski, Finland
ydh | 1 | 2021 Nov 18 | 22:41:19 | 23:11:10 | Stein | Lakewood, NJ, USA
ka5 | 1 | 2021 Nov 18 | 23:12:51 | 23:16:53 | Mountz | Louisville, KY, USA
rcn | 2 | 2021 Nov 18 | 23:14:21 | 03:46:22 | Randolph | Athens, GA, USA
un8 | 1 | 2021 Nov 18 | 23:20:08 | 02:01:20 | Randolph | Athens, GA, USA
24s | 1 | 2021 Nov 19 | 00:23:36 | 04:10:57 | Ostrem | Ames, IA, USA
p6h | 1 | 2021 Nov 19 | 00:27:55 | 02:16:24 | Sivayogan | Raleigh, NC, USA
sdp | 1 | 2021 Nov 19 | 01:01:20 | 05:12:44 | Wagner | Tucson, AZ, USA
gau | 1 | 2021 Nov 19 | 01:35:23 | 04:47:26 | Loose | San Diego, CA, USA
p6r | 1 | 2021 Nov 19 | 04:58:17 | 05:23:01 | Dalba | Santa Cruz, CA, USA
3pw | 1 | 2021 Nov 19 | 06:02:37 | 07:20:14 | Gamurot | Kapolei, HI, USA
66d | 1 | 2021 Nov 19 | 08:27:40 | 12:03:20 | Tsuchiyama | Tokyo, Japan
8cm | 1 | 2021 Nov 19 | 08:44:17 | 11:57:34 | Shimizu | Sayo-cho, Hyogo, Japan
y6x | 1 | 2021 Nov 19 | 09:05:34 | 13:21:17 | Goto | Toyonaka, Osaka, Japan
26e | 1 | 2021 Nov 19 | 09:47:06 | 12:25:55 | Fukui | Tsukuba, Ibaraki, Japan
zqq | 1 | 2021 Nov 19 | 12:50:32 | 13:52:24 | Sharon | Singapore, Singapore
jpg | 1 | 2021 Nov 19 | 16:58:59 | 18:32:02 | Simpson | Thirsk, UK
tuf | 1 | 2021 Nov 19 | 17:10:31 | 21:52:12 | Laugier | Simiane Collongue, France
zh6 | 1 | 2021 Nov 19 | 17:22:46 | 18:55:27 | Mirwald | Munich, Germany
p4t | 1 | 2021 Nov 19 | 17:35:37 | 21:29:06 | Carrera | Barcelona, Spain
7bt | 1 | 2021 Nov 19 | 17:52:08 | 21:32:19 | Manganese | Cavaillon, France
nsp | 1 | 2021 Nov 19 | 17:59:18 | 18:29:08 | Douezy | Serignan, France
5m4 | 1 | 2021 Nov 19 | 18:05:36 | 21:57:31 | Picard | Courthezon, France
8zz | 1 | 2021 Nov 19 | 18:49:42 | 21:55:37 | Savonnet | Trevignin, France
etx | 1 | 2021 Nov 19 | 19:24:08 | 22:19:38 | Guillet | Caen, France
ujx | 1 | 2021 Nov 19 | 22:16:59 | 01:03:12 | Simard | Mirabel, Québec, Canada
2mv | 1 | 2021 Nov 19 | 22:45:55 | 02:52:39 | Parker | High Bridge, NJ, USA
vwh | 1 | 2021 Nov 19 | 22:52:53 | 23:22:44 | Trudel | Saint-Lazare, Québec, Canada
qer | 1 | 2021 Nov 19 | 23:06:02 | 03:12:28 | Will | Raleigh, NC, USA
rp9 | 1 | 2021 Nov 19 | 23:12:10 | 02:24:10 | Runge | Durham, NC, USA
rcn-B | 2 | 2021 Nov 19 | 23:13:07 | 03:38:29 | Randolph | Athens, GA, USA
un8-B | 1 | 2021 Nov 19 | 23:13:50 | 03:38:47 | Randolph | Athens, GA, USA
rev | 1 | 2021 Nov 20 | 00:34:44 | 03:10:17 | Primm | Austin, TX, USA
sdp-B | 1 | 2021 Nov 20 | 01:01:12 | 04:32:26 | Wagner | Tucson, AZ, USA
gau-B | 1 | 2021 Nov 19 | 01:25:18 | 05:03:18 | Loose | San Diego, CA, USA
p6r-B | 1 | 2021 Nov 19 | 01:41:35 | 05:28:08 | Dalba | Santa Cruz, CA, USA
2n9 | 1 | 2021 Nov 20 | 03:06:56 | 05:33:22 | Esposito | Telegraph City, CA, USA
fqe | 1 | 2021 Nov 20 | 04:01:11 | 05:41:29 | Esposito | Telegraph City, CA, USA
33r | 1 | 2021 Nov 20 | 04:44:09 | 07:34:08 | Shih | Kahului, HI, USA
3pw-B | 1 | 2021 Nov 20 | 04:44:38 | 07:09:16 | Gamuro | Kapolei, HI, USA
8cm-B | 1 | 2021 Nov 20 | 08:40:41 | 10:43:37 | Shimizu | Sayo-cho, Hyogo, Japan
y6x-B | 1 | 2021 Nov 20 | 08:41:44 | 10:44:37 | Goto | Toyonaka, Osaka, Japan
$\cdots$ |  $\cdots$ | 2021 Nov 19 | 01:27:41 | 05:41:08 | Lewinb | Glendora, CA, USA
$\cdots$ |  $\cdots$ | 2021 Nov 19 | 22:12:20 | 01:10:13 | Silvisb | Bourne, MA, USA

aafootnotetext: Some end times occur on the following day.

bbfootnotetext: Citizen astronomers affiliated with Exoplanet Watch, not using
eVscopes. See Section 5.1.

Note. — The transit duration is $W\times P$. Parameters $h_{1}$ and $h_{2}$
are stellar limb darkening coefficients within the PYCHEOPS model (Maxted et
al., 2022). The normal prior on the normalization coefficients $c_{i}$ has a
mean of 1 and a standard deviation of 2%. The expected value of each $c_{i}$
is the mean of the relative flux during the times of observation $i$ as
measured from a model built from the “expected” values for all parameters. The
distribution priors are written as $\mathcal{U}[a,b]$ and
$\mathcal{N}(\mu,\sigma^{2})$ for the uniform and normal distributions,
respectively.

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