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

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 $37^{\prime} \times 28^{\prime}$ with a pixel scale of 172 per pixel and the IMX347 in the eVscope 2 has an FOV of $45^{\prime} \times 34^{\prime}$ with a pixel scale of 133 per pixel. Each eVscope is identified by a six-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.

Dalba & Tamburo (2019) predicted T₀ of this 2021 transit to be 2459538.2189 BJD_TDB ± 0.0021 (a 3.0 minutes uncertainty). They also ruled out TTVs greater than 34 minutes to 3σ significance. Using this information, the SETI Institute, the 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 hr 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° 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.

The observations were made from various sites spread around Earth, leading to variable data quality. At that time, the Moon was ∼ 100% illuminated and located ∼ 103° from the target. Some data were not usable because of poor weather, poor atmospheric seeing, or suboptimal 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 nonlinear 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.

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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 (S/N) 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. 2022). 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 $3^{\prime}$ 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^h30^m4031 +$38^\circ 19^{\prime} 34\buildrel{\prime\prime}\over{.} 14$), slightly fainter than Kepler-167 and located $1\buildrel{\,\prime}\over{.} 25$ away. Being similar in magnitude to the target star meant this comparison star was also the highest S/N option that was unsaturated. We then optimized the sizes of the circular flux apertures and annular background apertures for both the target and comparison star independently by minimizing the scatter in the individual star fluxes within each data set.

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 ${({G}_{\mathrm{BP}}-{G}_{\mathrm{RP}})}_{\mathrm{targ}}=1.227932$ and ${({G}_{\mathrm{BP}}-{G}_{\mathrm{RP}})}_{\mathrm{comp}}\,=1.190314$). 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.

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 & Gill 2019).

The parameters used to compute the transit model are given in Table 1. 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 × 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 h₁ and h₂. As a result, only the midtransit time T₀ 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 h₁ and h₂ are transformed parameters from the power-2 law described in Maxted et al. (2022).

Table 1. Transit Parameters Values

Parameter	Expected Value	Distribution Priors	Best Fit
T0 (BJDTDB)	2459538.2189 ± 0.0021	${ \mathcal U }[2459538.1793,2459538.2585]$	2459538.2225 ± 0.0068
D (%)	${1.540}_{\,-0.032}^{+0.027}$	${ \mathcal N }(1.540,4)$	1.77 ± 0.42
W × P (minutes)	975 ± 4
P (days)	${1071.23205}_{+0.00059}^{\,-0.00058}$	Fixed
b	${0.271}_{\,-0.073}^{+0.051}$
e	0
h1	0.684 ± 0.011
h2	0.434 ± 0.050
c1 (257)	1		0.997 ± 0.007
c2 (rcn)	1		1.001 ± 0.003
c3 (un8)	1		0.998 ± 0.006
c4 (24s)	1		0.990 ± 0.006
c5 (sdp)	1		0.992 ± 0.015
c6 (3pw)	1		1.001 ± 0.007
c7 (8cm)	0.993		0.992 ± 0.004
c8 (jpg)	0.978		0.982 ± 0.009
c9 (tuf)	0.978		0.978 ± 0.006
c10 (7bt)	0.979		0.982 ± 0.009
c11 (5m4)	0.979		0.987 ± 0.008
c12 (8zz)	0.979	${ \mathcal N }(1,0.0004)$	0.987 ± 0.007
c13 (etx)	0.979		0.980 ± 0.015
c14 (ujx)	0.988		0.991 ± 0.005
c15 (2mv)	0.993		0.993 ± 0.004
c16 (rp9)	0.993		0.989 ± 0.005
c17 (rcn-B)	0.996		0.994 ± 0.004
c18 (un8-B)	0.996		0.998 ± 0.004
c19 (rev)	0.999		0.993 ± 0.008
c20 (sdp-B)	1		0.998 ± 0.008
c21 (gau-B)	1		0.998 ± 0.005
c22 (p6r-B)	1		0.998 ± 0.003
c23 (2n9)	1		0.987 ± 0.011
c24 (fqe)	1		0.997 ± 0.009
c25 (33r)	1		1.003 ± 0.006
c26 (3pw-B)	1		0.987 ± 0.008
c27 (8cm-B)	1		0.998 ± 0.004

Note. The transit duration is W × P. Parameters h₁ and h₂ 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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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 midtransit time, we used a uniform prior for T₀. The prior bounds are chosen to be the expected T₀ ± the upper limit of the smallest TTV magnitude ruled out at 5σ significance by Dalba & Tamburo (2019), i.e., ±57 minutes. 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 1. 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 rms 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

$\begin{eqnarray}&&\mathrm{log}{ \mathcal L }=-\displaystyle \frac{1}{2}\sum _{i}\sum _{j}\displaystyle \frac{{\left({c}_{i}{y}_{i,j}-{f}_{j}\right)}^{2}}{{\sigma }_{i,j}^{2}}\end{eqnarray} \tag{ 1 }$

where σ_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 $\mathrm{log}{ \mathcal Z }\lt 0.1$, where ${ \mathcal Z }$ is the Bayesian model evidence.

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 Observers. ¹⁴ 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 1. The systematic "wiggle" in the light curve at X ≈ 388–407 corresponds to a temporary increase in image background level from environmental light pollution (a house light being turned on). Following this, increasing air mass (from 1.4 to 1.6) and a broadening PSF may have introduced additional systematic noise via imperfect background subtraction. The last 10 flux points were strongly affected by the local horizon vignetting the entrance pupil, a systematic error not encompassed by their error bars.

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Without conducting any additional fitting, the χ² and the reduced ${\chi }_{\nu }^{2}$ values between our transit model and the EW fluxes are χ² = 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 ${\chi }^{2}(\mathrm{flat})=146.6\ \mathrm{and}\ {\chi }_{\nu }^{2}(\mathrm{flat})=1.81$. We consider this an independent confirmation that the Unistellar Network data successfully detected the egress of the transit.

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, 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 1σ uncertainty in the midtransit time to less than half the transit duration) is crucial; for instance, most of the ephemerides coming from TESS will be outdated only 1 yr 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 midtransit time will produce an uncertainty of ∼15 hr in that planet's midtransit time 10 yr later. Thus, regular redetections with small ground-based telescopes are important to paving the way for detailed spectroscopic characterization by JWST and subsequent generations of exoplanet characterizing facilities.

Our timing measurement supports the previous finding by Dalba & Tamburo (2019) that Kepler-167 e's ephemeris does not contain large TTVs. The difference between our measured midtransit epoch and the initial transit epoch recomputed by Chachan et al. (2022) is 4284.9349 ± 0.0068 days, which when divided by four equates to an orbital period of 1071.2337 ± 0.0017 days. This is in line with the Kipping et al. (2016) value of P = 1071.23228 ±0.00056 days and $P={1071.23205}_{\,-0.00058}^{+0.00059}$ from Chachan et al. (2022), though about 2 minutes longer than both. Using our best-fit midtransit time as a reference epoch to compute future midtransit times, we obtain values that fall within the Dalba & Tamburo (2019) 1σ 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 TTVs over the planet's last orbit, but we can rule out 24 and 73 minutes TTVs to 1σ and 3σ significance, respectively. Our own measurement uncertainty will be the dominant factor in ruling out even smaller TTVs 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 TTVs 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.