π Earth: a 3.14-day Earth-sized Planet from K2's Kitchen Served Warm: Analysis and Validation

Table of Links

Abstract & Introduction

Observation

Analysis and Validation

Future Prospects & References

3. ANALYSIS AND VALIDATION

3.1. Stellar Host Characterization

3.1.1. Semi-empirical Stellar Parameters

We constructed the spectral energy distribution (SED) of EPIC 249631677 using photometric magnitudes from Gaia (GBP and GRP ; Gaia Collaboration et al. 2018) and the AllWISE source catalog (J, H, Ks, W1, W2, and W3; Cutri et al. 2013). The corresponding fluxes for these magnitudes are tabulated on VizieR (Ochsenbein et al. 2000) and shown in Figure 5 and Table 1. The parallax of EPIC 249631677 is π = 17.61 ± 0.09 mas, which yields a distance of 56.8 ± 0.3 pc (Gaia Collaboration et al. 2018; Stassun & Torres 2018). We then derived the stellar luminosity L∗ by integrating over the SED, which yielded L∗ = 0.0041 ± 0.0001L .

Two independent methods were applied to obtain stellar mass. First, we used the empirical M∗−MKs relation (applying the metallicity obtained in Section 3.1.2) from Mann et al. (2019) to obtain M∗ = 0.1721 ± 0.0044M . We also applied stellar evolution modeling, using the models presented in Fernandes et al. (2019), using as a constraint the luminosity inferred above and the metallicity derived in Sect. 3.1.2). We considered the stellar age to be >1 Gyr in the absence of signs of youth, such as presence of prominent flares (Ilin et al. 2019, see Section 3.1.3). We obtained with this method M∗ = 0.176 ± 0.004 M . This uncertainty reflects the error propagation on the stellar luminosity and metallicity, but also the uncertainty associated with the input physics of the stellar models. We combined these two mass estimates as in Van Grootel et al. (2018) to obtain M∗ = 0.174±0.004M as our best estimate for the stellar mass of EPIC 249631677. Given its proximity, we expect minimal extinction (Av) for the target; the SED fitting analysis described in Section 3.2.1 similarly constrains it to be 0.02 at 3σ confidence and we adopt that upper limit here. Finally, we note that given its luminosity, mass, and Gaia colors this star is likely to be fully convective (Jao et al. 2018; Rabus et al. 2019).

Due to the absence of a strong constraint on the stellar density from the transits, we further obtained stellar radius, surface gravity, effective temperature and density from our evolutionary models. Table 1 summarizes the results from this analysis, along with other properties of the star. Our values are consistent with those listed in the TESS Input Catalog (Stassun et al. 2019), and we adopt them for the remainder of this analysis.

3.1.2. Reconaissance Spectroscopy

To confirm EPIC 249631677’s stellar properties and better characterize the system, we acquired an optical spectrum using Keck/HIRES (Vogt et al. 1994) on UT 30 May 2020. The observation took place in 0.6 00 effective seeing and using the C2 decker without the HIRES iodine gas cell, giving an effective resolution of λ/∆λ ≈ 55, 000 from 3600 ˚A to 7990 ˚A. We exposed for 1800 s and obtained SNR of roughly 23 per pixel. Data reduction followed the standard approach of the California Planet Search consortium (Howard et al. 2010).

We used our Keck/HIRES radial velocity and Gaia DR2 data to estimate the 3D galactic (UV W) space velocity using the online kinematics calculator[2] of Rodriguez (2016). Following Chubak et al. (2012), our Keck/HIRES spectrum gives a barycentric radial velocity of 6.25±0.17 km s−1 . With the Gaia-derived coordinates, proper motion, and distance listed in Table 1, we find (U, V, W) values of (−17.02, −9.06, +33.66) km s−1 , indicating a likely membership in the Milky Way’s thin disk (Bensby et al. 2014).

Using the SpecMatch-Empirical algorithm (Yee et al. 2017), we derive from our HIRES spectrum stellar parameters of Teff = 3195 ± 70 K, R∗ = 0.23 ± 0.10R , and [Fe/H]= −0.24 ± 0.09, consistent with the values tabulated in Table 1. The three best-matching stars in the SpecMatch-Empirical template library are GJ 15B, GJ 447, and GJ 725B, which have spectral types of M3.5V, M4V, and M3.5V, respectively. Given the close match between the spectra of these stars and our target (see Fig. 4), we therefore classify EPIC 249631677 as an M dwarf with subclass 3.5±0.5. We see no evidence of emission line cores at Hα, consistent with our determination that our target is not a young star. We see no evidence of spectral broadening compared to these three stars (which all have v sin i < 2.5 km s−1 ; Reiners et al. 2012), so we set an upper limit on EPIC 249631677’s projected rotational velocity of < 5 km s−1 , comparable to the spectral resolution of HIRES.

3.1.3. Stellar Variability

The long-term variations apparent in the everest light curve (Figure 1) are not evident in light curves from other reduction pipelines (e.g., K2SFF). These variations likely arise from systematics and are not reliable for estimating the stellar rotation period (Esselstein et al. 2018). Similarly, no flares are apparent either in the K2 or SPECULOOS data. Flare rates peak for ∼M3.5 stars in TESS data (G¨unther et al. 2020). However, given the long integration time of 29.4 minutes as well as a need for data processing which corrects for the saw-tooth pattern, flare signals, unless very prominent, are expected to be difficult to detect in K2 long cadence data.

3.2. Vetting

In order to produce a transit depth on the level of 0.2% in the light curve of the primary target, a background eclipsing binary producing eclipses with depths of 25% to 50% would have to be 5.25 to 6.0 mag fainter than the target, respectively. Qualitatively, the odds of EPIC 249631677 hosting a planet are higher than the odds of such magnitude contrast eclipsing binary being present within the SPECULOOS aperture, given occurrence rates of M-dwarf planets (Dressing & Charbonneau 2013; Mulders et al. 2015; Hardegree-Ullman et al. 2019). A stringent quantitative constraint can be placed using the ingress/egress duration (T12/T34) compared to the total transit duration (T14) (Seager & Mall´enOrnelas 2003, Equation 21). Such a test yields an upper limit on the relative radius of the transiting body. By assuming equal effective surface temperatures, the lower magnitude limit ∆m (corresponding to a flux difference

∆F) for a blended binary mimicking a signal of depth δ is given by:

Using the posterior for the transit fit (See Section 3.3), we find for EPIC 249631677 that such a background object can be fainter at most by 1.73 mag at the 3σ level. Fortunately, EPIC 249631677 has a significant proper motion, ∼140 mas yr−1 , which allows us to investigate the presence of background sources at its current sky position. We looked at archival imaging of EPIC 246331677 going back to 1953[3] . A POSS I plate from 1953 is the publicly available oldest image of EPIC 249631677, and it does not show any background source at the current position of the target as shown in Figure 6. The plate is sensitive to objects at least 3.5 magnitudes fainter than the target. Similarly, the Hubble Guide Star Catalogue (GSC), with a limiting magnitude of 20 (Lasker et al. 1990), does not show any background source. While POSS II would go the deepest in terms of limiting magnitude (20.8, Reid et al. 1991), the star has moved appreciably closer to its current location, precluding a definitive measurement from this image. Overall, using archival images we can rule out the possibility of the transit signal originating from background star at a high level of confidence.

3.2.1. Binarity of the Host Star

Despite the lack of background sources, the host star could produce a false-positive transit signal if it were a grazing eclipsing binary or a hierarchical eclipsing binary. We investigated the evidence for host star binarity using the isochrones software package (Morton 2015), which performs isochrone fitting in the context of the MESA (Paxton et al. 2011, 2013, 2015) Isochrones and Stellar Tracks database (Dotter 2016; Choi et al. 2016). Single-star and binary evolutionary models are available within isochrones, and the inference is performed via the nested sampling algorithm MULTINEST (Feroz et al. 2009) (as implemented in the PyMultiNest software package (Buchner et al. 2014)), which allows for direct comparisons of the Bayesian evidence ln Z.

We tested both single-star and binary models using the priors on photometric magnitudes and stellar distance described in Section 3.1.1. The inferred properties from the single-star model fit are consistent with those given in Table 1 at the 2σ level. The ln Z for the single-star model is −213.86±0.04, whereas the ln Z for the binary model is −229.6 ± 0.2. According to Kass & Raftery (1995), the corresponding Bayes factor of 16 indicates “decisive” evidence in favor of the single-star model.

We also examined our Keck/HIRES spectrum for secondary lines that would indicate the presence of another star following the approach of Kolbl et al. (2015). We found no evidence of additional lines down to the method’s standard sensitivity limit of ∆V = 5 mag for ∆v > 10 km s−1 , consistent with EPIC 249631677 being a single, isolated star. We therefore conclude that the available data strongly support EPIC 246331677 being a single star.

3.2.2. Photometric Tests

We performed a series of tests on the photometric data to rule out false-positive scenarios. First, we performed an even-odd test on the target using K2 photometry. The even and odd transits are consistent with one another in transit depth to within 1σ. We also looked for secondary eclipses in the phase-folded light curve and found none to be present. Note that since we observe consistent signals in both the K2 and SPECULOOS data sets, we can rule out the signal originating from systematics. The transit depth in SPECULOOS observations with I+z filter, which is redder than Kepler bandpass, are consistent to K2 transit depths within 1σ level, keeping up with the expectation of the achromatic nature of planetary transit. Furthermore, a massive companion, such as a faint white dwarf, can be ruled out using the ellipsoidal variation, which puts a 3σ upper limit on the mass of any companions at the given orbital period of the transit signal as ∼100 MJup (Morris 1985; Niraula et al. 2018). From the transit fit, we can rule out a grazing eclipse originating from a larger transiting object (i.e. ≥ 2R⊕) at >3σ confidence. Together, these tests rule out the object at 3.14 days being a massive companion.

3.3. Transit Fitting

We used the refined estimates of the host properties together with a joint analysis of the K2 and SPECULOOS light curves to derive the planetary properties. In order to calculate the transit model, we used batman (Kreidberg 2015). We simultaneously model both the K2 observation as well as the ground-based observations with 21 parameters in a Monte Carlo Markov Chain (MCMC) framework using the emcee package (ForemanMackey et al. 2013). We use a Gaussian prior on the scaled semi-major axis of the orbit a/R∗ of N (25.72, 0.27), derived using Eq. 30 from Winn (2010) along with the stellar density of 32.6 ± 1.0 g cm−3 (see Section 3.1.1) and orbital period of 3.1443 days from the TLS search. As for the limb darkening, we use the non-informative q1, q2 parameterization of the quadratic limb-darkening law as suggested by Kipping (2013). We fixed the eccentricity to 0, given that the expected time of circularization is 50 Myr (assuming a quality factor Qp ∼ 500, Goldreich & Soter 1966; Patra et al. 2017), which is at least an order of magnitude smaller than the estimated age of the system. For K2 data, we supersample the transits by a factor of 15 in batman to take into account the effect of non-negligible integration time. As for the ground-based data, we use second-order polynomials to detrend against the observables airmass and FWHM.

We ran the MCMC for 50,000 steps with 150 walkers performing a combined fit of K2 and SPECULOOS

References— (1) This work. (2) Huber et al. (2016). (3) Stassun et al. (2019). (4) Cutri et al. (2003). (5) Gaia Collaboration et al. (2018). (6) Cutri et al. (2013). (7) This work, evolutionary model analysis. (8) This work, Keck/HIRES analysis. (9) This work, SED analysis.

data, and use the last half of the run to build the parameter posteriors. We assessed the convergence of walkers using the suggested autocorrelation test for emcee. The resulting median values from the fit with 1σ deviation are reported in Table 2, while the best-fit transit models are shown in Figure 2 and Figure 3.

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[2] //kinematics.bdnyc.org/query

[3] //stdatu.stsci.edu/cgi-bin/dss form