Astronomy & Astrophysics manuscript no. K2201505350 c©ESO 2021November 4, 2021

One of the closest planet pairs to the 3:2 Mean Motion Resonance,confirmed with K2 observations and Transit Timing Variations:

EPIC201505350?

David J. Armstrong1, Dimitri Veras1, Susana C. C. Barros2, Olivier Demangeon2, James McCormac1, Hugh P.Osborn1, Jorge Lillo-Box3, Alexandre Santerne2, 4, Maria Tsantaki4, José-Manuel Almenara5, 6, David Barrado3,

Isabelle Boisse2, Aldo S. Bonomo7, François Bouchy2, 8, David J. A. Brown1, Giovanni Bruno2, Javiera Rey Cerda8,Bastien Courcol2, Magali Deleuil2, Rodrigo F. Díaz8, Amanda P. Doyle1, Guillaume Hébrard9, 10, James Kirk1, Kristine

W. F. Lam1, Don L. Pollacco1, Arvind Rajpurohit2, Jessica Spake1, and Simon R. Walker1

1 Department of Physics, University of Warwick, Gibbet Hill Road, Coventry, CV4 7AL, UKe-mail: d.j.armstrong@warwick.ac.uk

2 Aix Marseille Université, CNRS, LAM (Laboratoire d’Astrophysique de Marseille) UMR 7326, 13388, Marseille, France3 Departamento de Astrofísica, Centro de Astrobiología (CSIC-INTA), ESAC campus 28691 Villanueva de la Cañada (Madrid),

Spain4 Instituto de Astrofísica e Ciências do Espaço, Universidade do Porto, CAUP, Rua das Estrelas, PT4150-762 Porto, Portugal5 Univ. Grenoble Alpes, IPAG, F-38000 Grenoble, France6 CNRS, IPAG, F-38000 Grenoble, France7 INAF – Osservatorio Astrofisico di Torino, via Osservatorio 20, 10025 Pino Torinese, Italy8 Observatoire Astronomique de l’Université de Genève, 51 chemin des Maillettes, 1290 Versoix, Switzerland9 Institut d’Astrophysique de Paris, UMR7095 CNRS, Université Pierre & Marie Curie, 98bis boulevard Arago, 75014 Paris, France

10 Observatoire de Haute-Provence, Université d’Aix-Marseille & CNRS, 04870 Saint Michel l’Observatoire, France11 CNRS, Canada-France-Hawaii Telescope Corporation, 65-1238 Mamalahoa Hwy., Kamuela, HI 96743, USA

Received ; accepted

ABSTRACT

Aims. The K2 mission has recently begun to discover new and diverse planetary systems. In December 2014 Campaign 1 data fromthe mission was released, providing high-precision photometry for ∼22000 objects over an 80 day timespan. We searched these datawith the aim of detecting further important new objects.Methods. Our search through two separate pipelines led to the independent discovery of EPIC201505350, a two-planet system ofNeptune sized objects (4.2 and 7.2 R⊕), orbiting a K dwarf extremely close to the 3:2 mean motion resonance. The two planets eachshow transits, sometimes simultaneously due to their proximity to resonance and alignment of conjunctions.Results. We obtain further ground based photometry of the larger planet with the NITES telescope, demonstrating the presence oflarge transit timing variations (TTVs) of over an hour. These TTVs allows us to confirm the planetary nature of the system, and placea limit on the mass of the outer planet of 386M⊕.

Key words. planets and satellites: detection, dynamical evolution and stability, individual: EPIC201505350b, individual:EPIC201505350c, general

1. Introduction

With the steady release of data from the K2 satellite, severalprojects have begun to search for previously undiscovered plan-etary systems. A number of interesting systems have alreadycome to light (Crossfield et al. 2015; Vanderburg et al. 2014;Foreman-Mackey et al. 2015). For these systems we now havephotometry approaching the precision of the Kepler prime mis-sion, and crucially of host stars much brighter than the typicalKepler case. This promises the use of radial velocity and othertechniques to add to our knowledge of these already interestingobjects. In this work we present a two-planet system observedin K2 field 1. This system, EPIC201505350 (RA: 11:39:50.476,

? Using observations made with SOPHIE on the 1.93-m telescope atObservatoire de Haute-Provence (CNRS), France.

DEC: +00:36:12.87, Kepmag 12.8), lies exceptionally close tothe 3:2 mean motion resonance (MMR), and so has the poten-tial to show particularly large TTVs (a concept first suggestedby Agol et al. (2005); Holman (2005) for the general case). Interms of period ratio, only one object is yet known closer to thisresonance (and does not show TTVs, due to a large libration pe-riod). EPIC201505350 was originally presented as a candidateplanetary system in Foreman-Mackey et al. (2015), and is con-firmed here using further observations.

The 3:2 MMR holds a special significance in both Solar sys-tem and extrasolar planetary systems. For decades, Pluto andNeptune were classified as the only Solar system resonant planetpair, and their orbits evolve inside of a 3:2 MMR. The GrandTack model, a scenario proposed to explain the current archi-tecture of the inner Solar system, asserts that Jupiter and Sat-

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urn were once captured into a 3:2 MMR while embedded in thenascent Solar nebula (Walsh et al. 2011; Pierens et al. 2014). Fur-ther, the first extrasolar planetary system ever confirmed, aroundPSR 1257+12 (Wolszczan & Frail 1992; Wolszczan 1994), in-cludes two planets whose orbits are tightly coupled and are veryclose to residing within the 3:2 MMR (Malhotra et al. 1992;Gozdziewski et al. 2005; Callegari et al. 2006).

Transiting exoplanets orbiting main sequence stars representthe majority of known planets, but usually lack the necessaryconstraints to allow one to definitively assign membership to anindividual MMR1. A MMR between two planets occurs whena particular linear combination of mean longitudes, longitudesof pericentre, and sometimes longitudes of ascending node li-brate (or oscillate) about 0◦ or 180◦ over a given time interval(e.g. Murray & Dermott 1999). For EPIC201505350 and the 3:2MMR, this linear combination is represented by either the angleθ1 or θ2, where

θ1 = 3λc − 2λb −$b (1)θ2 = 3λc − 2λb −$c. (2)

with λ the mean longitude and $ the longitude of the ascendingnode. Here, and throughout the paper, we label the inner planetas b and the outer as c.

Because time series of planetary mean longitudes and longi-tudes of pericentre are typically not available in extrasolar sys-tems, a common practice is to use orbital period ratios by them-selves as a proxy for resonance. High frequencies of systemswith ratios of 1.5 and 2.0 (Lissauer et al. 2011) are suggestivethat the 3:2 and 2:1 commensurabilities represent a significanttracer of formation, regardless of whether those planetary can-didates are actually locked inside of a MMR. Recently, the fre-quency of systems just outside of the 3:2 period commensura-bility has exhibited a distinct asymmetry (e.g. Fabrycky et al.2014), which has led to substantial theoretical scrutiny (Batygin& Morbidelli 2013; Lee et al. 2013; Petrovich et al. 2013; Chat-terjee & Ford 2014; Delisle et al. 2014; Delisle & Laskar 2014).The period ratio of EPIC201505350 b and c as displayed in theK2 data is 1.503514+0.000052

−0.000057, among the closest systems to a 3:2commensurability so far detected.

We searched the Exoplanet Orbit Database (Han et al. 2014)for other systems close to this commensurability. The only sys-tem we could find with a closer normalised distance to reso-nance (defined in Lithwick et al. 2012), 4, was the Kepler-372cdpair (Rowe et al. 2014), where 4 is ∼0.0003, as compared toEPIC201505350 with ∼0.0023. However, neither Kepler-372cnor d exhibit TTVs during the Kepler observations due to a par-ticularly long predicted TTV libration period, ∼70 years. Alsoworth noting is the Kepler-342cd pair (Rowe et al. 2014), witha 4 of ∼0.0027, which also does not show TTVs due to a longerlibration period.

Interest in these special period ratios is motivated by boththe possibility of making deductions about a system’s formationchannel and its long-term future stability. Convergent migrationin protoplanetary discs is an effective and popular MMR forma-tion mechanism (Snellgrove et al. 2001), and can also achievethree-body resonances (Peale & Lee 2002; Libert & Tsiganis2011), although some MMRs are harder to lock into than others(Rein et al. 2012; Tadeu dos Santos et al. 2015). Forming the 3:2MMR in particular through energy dissipation has been widely

1 With limited constraints, one can more easily exclude systems fromexisting within MMRs (Veras & Ford 2012).

investigated (Papaloizou & Szuszkiewicz 2005; Hadjidemetriou& Voyatzis 2010; Emel’yanenko 2012; Ogihara & Kobayashi2013; Wang & Ji 2014; Zhang et al. 2014). Capture into MMRsthrough gravitational scattering alone – after the dissipation ofthe protoplanetary disc – occurs relatively rarely (Raymond etal. 2008).

In this work we characterise this multi-planet system, con-firm its planetary nature using observed TTVs, and discuss theimplications these have for future observations.

2. Observations

2.1. K2

Observations were made with the Kepler satellite as part of theK2 mission between BJD 2456811.57 and 2456890.33, span-ning ∼80 days. The K2 mission (Howell et al. 2014) is the sur-vey now being conducted with the repurposed Kepler space tele-scope, and became fully operational in June 2014. It is surveyinga series of fields near the ecliptic, returning continuous high-precision data over an 80 day period for each field. Despite thereaction wheel losses that ended the Kepler prime mission, K2has been estimated to be capable of 80ppm precision for V=12stars, close to the sensitivity of the primary mission. All datawill be public, although at the time of writing only campaigns0 and 1 have been released. Targets are provided by the EclipticPlane Input Catalogue (EPIC) which is hosted at the MikulskiArchive for Space Telescopes (MAST)2 along with the availabledata products.

Targets in K2 often display significant pointing drift over theK2 observations, typically on a timescale of 6 hours on whichthe spacecraft thrusters are fired. This leads to a major source ofsystematic noise in the lightcurve (Vanderburg & Johnson 2014),the removal of which is the key part of our detrending method.The full method is explained in Armstrong et al. (2015), butis summarised here for clarity. Initially a fixed aperture, of ra-dius 4 pixels in this case and shape as described in Armstronget al. (2015), was centred on the brightest target pixel. A rawlight curve is extracted directly from this aperture, with back-ground subtracted using the median out-of-aperture pixels. Rowand Column centroid variations are found for the time series. Atthis point points associated with spacecraft thruster firings areremoved, and the remaining points decorrelated from the cen-troid variations. We decorrelate from both row and column cen-troids simultaneously, as they are not statistically independent.This process can leave systematic or instrumental noise in place(see Foreman-Mackey et al. 2015). In this case this noise seemsto be weak compared to the intrinsic stellar variability, whichoccurs with a magnitude of ∼1%.

This stellar variability is removed, along with any longer pe-riod systematics, through the application of an iteratively fittedpolynomial. A 3D polynomial is fit to successive 2 day wideregions, with the fit repeated for 20 iterations clipping pointsgreater than 3σ from the best fit line at each iteration. This fit isthen used to detrend a 5 hour region at its centre, and the processrepeated for each 5 hour region. As the principal components ofthe instrumental noise show variations on order 10+ days in cam-paign 1 (Foreman-Mackey et al. 2015) they should be removedthrough this process, and crucially will not affect the transits. Wefound that for this target the best results were obtained by per-forming this polynomial flattening immediately after extractingthe lightcurve, before decorrelating the flux from the centroid

2 https://archive.stsci.edu/k2/

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Fig. 1. Data taken from the NITES telescope. The best fit transit derivedfrom K2 observations is also plotted, along with the binned light curve,and fits the shape seems with NITES well.

motion. The resulting light curve is shown in Figure 2. Note thatsome transits of each planet occur simultaneously with the otherthrough the dataset, although the final such simultaneous transitbegins to show signs of both planets individually.

2.2. NITES

Due to its proximity to MMR, EPIC201505350 had the potentialto show significant TTVs (see Section 5). As such we scheduledit for further ground based transit observations. The Near Infra-red Transiting ExoplanetS (NITES) Telescope is a semi-robotic0.4-m (f/10) Meade LX200GPS Schmidt-Cassegrain telescopeinstalled at the ORM, La Palma. The telescope is mounted witha Finger Lakes Instrumentation Proline 4710 camera, contain-ing a 1024 × 1024 pixels deep-depleted CCD made by e2v. Thetelescope has a FOV and pixel scale of 11 × 11 arcmin squaredand 0.66′′ pixel−1, respectively and a peak QE> 90% at 800 nm.For more details on the NITES Telescope we refer the reader toMcCormac et al. (2014).

One transit of EPIC201505350 was observed on 2015 Feb28. The telescope was defocused slightly to 3.3′′ FWHM and814 images of 20 s exposure time were obtained with 5 s deadtime between each. Observations were obtained without a fil-ter. The data were bias subtracted and flat field corrected usingPyRAF3 and the standard routines in IRAF,4 and aperture pho-tometry was performed using DAOPHOT (Stetson 1987). Tennearby comparison stars were used and an aperture radius of6.6′′ was chosen as it returned the minimum RMS scatter in theout of transit data. Initial photometric error estimates were calcu-lated using the electron noise from the target and the sky and theread noise within the aperture. The data were normalised with afirst order polynomial fitted to the out of transit data. The result-ing lightcurve is shown in Figure 1.

3 PyRAF is a product of the Space Telescope Science Institute, whichis operated by AURA for NASA.4 IRAF is distributed by the National Optical Astronomy Observato-ries, which are operated by the Association of Universities for Researchin Astronomy, Inc., under cooperative agreement with the National Sci-ence Foundation.

2.3. SOPHIE

We observed the star EPIC201505350 with the SOPHIE spec-trograph mounted on the 1.93 m telescope at the Haute-ProvenceObservatory (France). We used the high-efficiency mode whichhas a spectral resolution of about 39 000 at 550nm. This modeis preferred for the observations of relatively faint stars (e.g.Santerne et al. 2014). For more information about the SO-PHIE spectrograph, we refer the interested readers to Perruchotet al. (2008) and Bouchy et al. (2009). We secured five epochsbetween 2015-01-23 and 2015-02-02 as part of our on-goingTRANSIT consortium5. The exposure time ranges between 400sand 2700s, which lead to signal-to-noise ratio per pixel in thecontinuum at 550nm ranging between 13.5 and 22.1.

We cross correlated the spectra with a numerical mask cor-responding of a G2 dwarf (Baranne et al. 1996; Pepe et al. 2002)and find a unique line profile with a width compatible with therotational period found in K2 photometry (see next section). Thederived radial does not exhibit variation at the level of 17m.s−1,which is compatible to our median uncertainty of 16m.s−1.

3. Stellar parameters

We obtained the parameters of the host star from the spectralanalysis of five co-added SOPHIE spectra. First, we subtractedfrom the spectra pointing to the source (in fiber A), any sky con-tamination using the spectra of fiber B, after correcting for therelative efficiency of the two fibers. The final spectrum has a S/Nof the order of 25 around 6070Å.

To derive the atmospheric parameters, namely the effectivetemperature (Teff), surface gravity (log g), metallicity ([Fe/H]),and microturbulence (vmic), we followed the methodology de-scribed in Tsantaki et al. (2013). This method relies on the mea-surement of the equivalent widths (EWs) of Fe i and Fe ii linesand by imposing excitation and ionization equilibrium. The anal-ysis was performed in local thermodynamic equilibrium (LTE)using a grid of (Kurucz 1993) model atmospheres and the radia-tive transfer code MOOG (Sneden 1973). Due to the low S/N ofour spectrum, the EWs were derived manually using the IRAFsplot task.

From the above analysis, we conclude that the host star is aslightly metal-rich K dwarf. The derived parameters are shownin Table 1. The stellar radius and mass were derived from the cal-ibration of Torres et al. (2010a), updated with the version fromSantos et al. (2013) and the atmospheric parameters describedabove. We also included in Table 1 the determination of surfacegravity from the transit fit parameters (see Section 4) and therespective results of stellar mass and radius. In this case, we pro-ceed using the transit derived parameters, as they are much moreaccurate.

We also study the stellar variability inherent in the lightcurveof EPIC201505350. This lightcurve may be contaminated byremnant instrumental noise, but we find that repeating patternsapparent across the entirety of the K2 observations do not gen-erally match the principal noise components seen (see Foreman-Mackey et al. 2015, for these components). A weighted, floatingmean Lomb-Scargle periodogram (Lomb 1976; Scargle 1982),following the method of Press & Rybicki (1989), gives a prin-cipal period of 20.3 days. If this peak is due to stellar rotationthen it represents the rotation period of the star. Errors on Protare derived from the FWHM of the periodogram peak.

5 OHP program: 14B.PNP.HEBR

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Table 1. Stellar Parameters for EPIC201505350

Parameter Value UnitsTeff 5230 ± 417 Klog g∗ 4.39 ± 0.79 dexvmic 0.92 ± 0.5 kms−1

[Fe/H] 0.38 ± 0.23 dexProt 20.3+3.7

−2.3 days

Derived Parameters:R∗ (spectroscopic) 1.03 ± 0.2 R�M∗ (spectroscopic) 0.92 ± 0.14 M�log g∗ (transit) 4.52 ± 0.22 dexR∗ (transit) 0.88 ± 0.06 R�M∗ (transit) 0.89 ± 0.06 M�

4. Light Curve fitting

To obtain the transit shape and parameters we limit ourselvesto the K2 data, as it is of significantly higher precision and theNITES data do not show the full transit. The data were detrendedas described in Section 2.1, then cut so that only data within a 7transit width region centred on each transit were used. We alsoremoved all simultaneous transits, along with two specific pointsin separate transits which showed clear evidence for being withina spot crossing (significant brightenings within transit relative totheir local transit shape). These points are highlighted in Figure2. We note that there are other apparently bad points which werenot removed - the decision to remove a point was based entirelyon its local transit, to avoid excessive bias, and so points whichonly appear bad when shown phase folded and against the fit willremain. The data were then fit using the JKTEBOP code (e.g.Southworth 2013; Popper & Etzel 1981), with numerical inte-gration used to account for the long cadence of K2 observations(splitting each point into 60 integrated sub-points).

We initialised the fits with a linear limb darkening coefficientof 0.56, suitable for a K dwarf, which was then allowed to vary.We then tested for eccentricity, but found no constraint for eitherobject. As such for the remaining tests the eccentricity of bothplanets was set to zero. To derive robust errors we used a MonteCarlo process whereby Gaussian observational errors are addedto each data point and the fit repeated 1000 times, producinga distribution of best fits. The medians and 68.27% confidencelimits are then taken to produce values and errors.

While producing good parameter and error estimates for theobservational errors given to it, this process does not properly ac-count for systematic errors. These are of particular concern forEPIC201505350, as as has been noted there is evidence withinsome transits for spot crossings. In the past such crossings haveproven useful in modelling starspots (e.g. Barros et al. 2013;Beky et al. 2014) but here they form a source of contamina-tion to our fits. We test for the effect of these spots by adoptinga prayer bead style residual permutation test. In this process, abest fit is acquired, and then the residuals of the data to this fit are‘rolled’ through the dataset, and a further best fit acquired eachtime. Due to the low cadence of K2 observations, there are notenough points near transit to get a distribution of parameter val-ues through this method (270 and 183 tests respectively for plan-ets b and c). However, the prayer-bead generated distribution atleast allows us to obtain an estimate of the systematic effect onour transit parameters. In all cases these systematic errors werecomparable to or smaller than the Monte Carlo generated errors.As such we present final values and errors from the Monte Carlotests. While we acknowledge that this method of testing for sys-

tematic errors is merely an estimate (especially as the full effectof spots only appears in transit), we note that as the errors gener-ated by the prayer-bead process are not significantly larger thatthose from the Monte Carlo tests, the effect of systematics on thetransit parameters is not particularly strong.

The resulting best fits are shown for each planet in Figure2. Note that the derived ephemeris are taken from only a smallpart of the TTV phase curve and so will require correction; seeSection 6 for detail. In particular, there are significantly largererrors on the period when TTVs are taken into account - finalvalues are found in Section 6.2.

5. Transit Timing Variations

Given the periods found in Section 4, it is immediately apparentthat the two planets in the EPIC201505350 system lie close tothe 3:2 MMR. It is common for systems close to MMR to showparticularly large TTVs (Lithwick et al. 2012; Xie 2014) and sowe searched the data in the hope of seeing variations. This searchwas carried out using the transit shape defined by our best fit pa-rameters. As before, simultaneous transits were ignored. We cutthe data to a region within 2 full transit widths of the approxi-mate transit centres, then passed the model transit over this re-gion with a resolution of 0.00015 days. The minimum χ2 of thistest series was recorded, at which point each datapoint was per-turbed by a random Gaussian with standard deviation equal tothe point error. The fit was then repeated, and this process un-dergone for 1000 iterations, to get a distribution of transit times.The mean of the distribution is then taken as the transit time. Aswhen fitting the transit shape, this process does not account forsystematic errors. This is particularly concerning for measuringtransit times, because due to the low cadence of K2 observationsonly a few points are seen within each transit. If one of thesepoints is significantly perturbed by a spot crossing (which oc-curs visibly for some transits) then the measured time would bestrongly affected. To estimate the effect of these systematics, werepeat the prayer-bead residual analysis of Section 4. In this casethough, as we are considering each transit independently, thereare even fewer data points near transit (typically ∼30). Also ofconcern is that the full effect of spots can generally only be seenwhen they are occulted in transit, where there are even fewerpoints. As such we perform this analysis and estimate the sys-tematic contribution to our error budget by taking the maximumand minimum parameter values which arise from the prayer beadtest, over the ∼30 iterations. We adopt these most pessimistic val-ues as our 1σ errors, to ensure that we do not underestimate theerrors on our transit times. The adopted values (from the meanof the monte carlo distribution) and errors (from the maximumand minimum of the prayer-bead residual test) are given in Table3.

We were fortunate enough to obtain an additional transit ofplanet b with the NITES telescope (Section 2.2). The time ofthis transit was obtained using the transit shape derived from theK2 observations, meaning the only fit parameter was the timeof transit centre. The same Monte Carlo test was performed, butnow adopting the standard deviation of the resulting distributionas the error. We did not repeat the prayer-bead analysis in thiscase, as the transit did not show evidence for systematics.

The observed-calculated times found from our TTV analysisare shown in Figures 3 and 4, using the K2-derived ephemeris.Planet b, in particular the NITES observation, show large TTVsof over an hour from the expected time (∼a quarter of the transitduration). Within the K2 data itself we do not find the TTVs tobe significant, beyond one transit for planet b. An initial anal-

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Fig. 2. Top: Extracted light curve for EPIC201505350, showing significant stellar variability. Middle: Flattened and detrended lightcurve, showingtransits of the inner planet (b, red), outer planet (c, green) and simultaneous transits of both planets (magenta). Some outlier points are not shownfor clarity. Bottom Left: The phase folded transits of planet b, excluding simultaneous transits and showing the best fit model. Some points (shownlighter than the others) displayed clear evidence of spot crossings by the planet and were excluded from the fit. Bottom Right: Same for planet c.Note the change in y-axis scale.

Table 2. System Parameters

Parameter Units b cModel Parameters:P days 7.919454+0.000081

−0.000078 11.90701+0.00039−0.00044

T0 BJDT DB − 2456000 813.38345+0.00036−0.00039 817.2759 ± 0.0012

Rp/R∗ 0.0753+0.0028−0.0015 0.0439+0.0011

−0.0012

(Rp + R∗)/a 0.0572+0.0084−0.0042 0.0414+0.0015

−0.0009

i deg 88.83+1.08−0.89 89.91+0.05

−0.32

e 0 (adopted) 0 (adopted)

Limb-Darkening 0.552 ± 0.041 0.57+0.14−0.13

Derived Parameters:Rp R⊕ 7.23+0.56

−0.51 4.21 ± 0.31

a AU 0.077+0.008−0.013 0.1032+0.0074

−0.0080

S inc S ⊕ 87.7+9.3−12.9 48.8+6.4

−6.2

Pc/Pb 1.503514+0.000052−0.000057

4 0.00234 ± 0.00002

Notes. 4 is defined in Section 6, and represents the normalised distance to resonance. Note that Pb,Pc, and parameters derived from them are onlyinstantaneous measurements, and will change over the course of the TTV phase curve (see Section 6). Transit based stellar parameters are usedfor derived quantities.

ysis of these TTVs is performed in Section 6.2. We note thatthis detection of TTVs implies that the ephemeris in Table 2 arelikely not the ‘true’ ephemeris, in the sense of the mean transitinterval over long timeframes. Readers should thus be careful inpredicting transit times. This is discussed further in Section 6.

6. TTV based Planet Confirmation and Stability

6.1. Transit Timing Variations - Overview

We leave a detailed study of the TTVs to future analysis (Barroset al in prep). It is however possible to place a number of con-straints on the system even with the limited coverage of the TTVphase curve which we obtain here. For this initial analysis, we

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Table 3. Detected Transit Times

Planet Time (BJDTDB-2456000) Error Sourceb 813.3841 0.0016 K2b 821.3039 0.0107 K2b 837.1382 0.0014 K2b 845.06176 0.00098 K2b 860.9000 0.0012 K2b 868.8196 0.0016 K2b 884.6597 0.0017 K2b 1082.6895 0.0022 NITESc 817.2741 0.0032 K2c 841.0942 0.0068 K2c 864.9105 0.0069 K2c 888.7136 0.0038 K2

Notes. Simultaneous transits are not shown here.

0 5 10 15 20 25 30 35Orbit

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Fig. 3. Observed-Calculated transit times for planet b. Calculated timesare taken from the ephemeris of Table 2, and show large variations froma constant period.

use the analytical representation of the TTVs derived by Lith-wick et al. (2012), hereafter L12, which has been shown to bevalid for systems near MMR (Deck & Agol 2014), a conditionstrongly met in this case. This allows us to obtain a more intu-itive description of the parameter space than is generally possibleusing N-body simulations. Given the potential for spots or othersystematic errors to affect the K2 transit times, and the other-wise limited coverage of the TTV phase curve, we defer such ananalysis to future work.

The TTV phase curve described by L12 is a sinusoid withtwo key parameters: (1) an amplitude |V | given as a function ofplanetary mass, stellar mass, 4 (the normalised distance to reso-nance), and the free eccentricity Z f ree (a complex number), and(2) a period given by

Psuper =Pouter

j|4|(3)

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4 =Pouter

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For the 3:2 MMR j = 3. This leads to a phase curve of theform

TTV = |V | sin(t − t0Psuper

+ φ) (5)

where φ is the phase of the curve and changes over the seculartimescale (hence is constant for our purposes). In our case, wecan set t0 to be the time of first transit to acceptable accuracy,due to the alignment of planetary conjunctions demonstrated bythe simultaneous transits observed. Both the amplitude and pe-riod depend strongly on 4. The closer a system is to resonance,the larger the amplitude becomes, but the longer the period. Fora system as close to resonance as EPIC201505350, the periodis particularly long, of the order several years. This means thatwithin the 80 days of K2 observations, we would not expect tosee large variation. With the later NITES transit however, we arestarting to see the high amplitude TTV curve that these planetsexhibit.

The period Psuper depends only on 4, and the period of theouter body. The TTV amplitude however generally shows a de-generacy between the free eccentricity Z f ree and planetary mass(L12). Hadden & Lithwick (2014) break this degeneracy statis-tically, but for individual objects it can be difficult to circum-vent (although so-called synodic chopping signals can help, seeNesvorný & Vokrouhlický (2014)). Furthermore with our limitedobservations the current phase of the TTV curve φ is unclear. Inthe low free eccentricity case (|Z f ree| << |4|), φ must be zero(L12), however there is no guarantee that this is the case here.

6.2. Transit Timing Variations - Analysis

We here explore the parameter space allowed by this analyti-cal representation, given the transit times we have observed. Theanalysis is most strongly constrained by the NITES transit, asthis transit allows much more coverage of Psuper. The process iscomplicated by a correction term which must be added to our de-rived periods. As they are derived from a limited part of the TTVphase curve, they do not represent the overall ‘true’ period, inthe sense of the mean transit interval over long timescales. Thismakes determination of Psuper non-trivial. Small corrections tothe periods can change 4 significantly, which has a strong effecton the TTV period and amplitude. We circumvent this problem

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Armstrong et al: EPIC201505350

0 100 200 300 400 500 600Mc (M⊕)

0

100

200

300

400

500

600

Mb (

M⊕)

1.2

1.8

2.4

3.0

3.6

4.2

4.8

5.4

Log χ

square

Fig. 5. The χ2 surface given by the assumption that |Z f ree| << |4|.Agood fit is obtained for limited combinations of the two masses.

by utilising the fact that our derived periods are in fact measure-ments of the gradient of the TTV curve at the time of the K2observations. The sensitivity to 4 exhibited by the TTV periodand amplitude cancels out when calculating the gradient, allow-ing the correction to the periods to be made using only the initialperiods. Using this, we can fit our transit times using the follow-ing process:

1. Take values for the input parameters, Mb, Mc, Re(Zfree),Im(Zfree), φ

2. Calculate the correction to make to the derived periods3. Use the corrected periods to find the true 4 and Psuper

4. Use the input parameters and the newly corrected values tocalculate the TTV amplitude

5. Compare the now fully defined TTV curve to the observa-tions

We begin with the case where |Z f ree| << |4|, and the freeeccentricity can be ignored. This reduces the input parameters tosolely the two planetary masses, as φ must also be zero in thiscase (L12). We set t0 to be zero at the time of first transit - this isaccurate to within ∼20 days, and given the long Psuper of severalyears, this does not need to be more accurate for this analysis.

The parameter space of the low eccentricity case is bestshown in Figure 5, which shows the log χ2 surface seen. Thekey points are a maximum mass for the outer planet c, of 350M⊕,found when the mass of the inner planet b goes to zero. The massof planet b is less constrained (to constrain it properly using thisanalysis would require observations of planet c’s TTV curve),but can only take high values in the event that planet c’s massdrops much lower. Similarly Psuper, excepting very high masses(over ∼600M⊕) for planet b, is constrained to be greater than∼480 days, and less than ∼3050 days. The accompanying surfacefor the ‘true’ 4 is shown in Figure 6, and makes clear that allzero eccentricity best fitting planetary mass combinations implya ‘true’ 4 that is in fact slightly below the 3:2 resonance. As suchwhile we cannot confirm this is the case from these observationsalone, it is possible that the the apparent planetary periods oscil-late around the resonance over the course of the TTV curve.

Extending our analysis to the case where there is significantfree eccentricity, we can immediately constrain φ. Because theNITES transit arrived late rather than early, φ must be in the

0 100 200 300 400 500 600Mc (M⊕)

0

100

200

300

400

500

600

Mb (

M⊕)

0.012

0.010

0.008

0.006

0.004

0.002

0.000

0.002

Norm

alis

ed d

ista

nce

to r

eso

nance

Fig. 6. As Figure 5, but showing the normalised distance to resonance4 over the same mass ranges. In this case it can be seen that 4 becomesnegative for all good fits, implying that the planets are in fact below the3:2 resonance if this assumption held.

0 100 200 300 400 500 600Mc (M⊕)

0

100

200

300

400

500

600

Mb (

M⊕)

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

Log χ

square

Fig. 7. As Figure 5, but showing the χ square surface for Re(Zfree)=0.2,and φ = π/4 over the same mass range. A degeneracy can be seenbetween positive 4 (top left) and negative 4 (down and towards theright).

range 0 ≤ φ ≤ π. Within this range however a number of differ-ent effects can occur. We test these cases by repeating the analy-sis with the real component of Zfree set to be 0.2, at various val-ues of φ. We hold the imaginary component at zero. Althoughthe imaginary component can affect the amplitude and phase ofthe TTV curve, as we are trialling different values of φ it is prin-cipally the amplitude of Zfree which is important. The results ofthese tests can be summed up simply: while Mb remains poorlyconstrained, Mc can only rise above its zero eccentricity value invary rare cases, and then not by much. The worst case we foundwas for φ = π/4, where for Mb = 0 the maximum mass for Mcwas 386M⊕. The particular χ2 square surface varies for differentinput φ values. One example surface is shown in Figure 7, whichalso demonstrates an interesting degeneracy that arises betweenpositive and negative 4 in that case.

Given the poorly mapped TTV curve, the sensitivity of ouranalysis to small corrections to the planetary periods, and thefree eccentricity degeneracy it would be premature to make de-

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terminations of the planetary masses at this point. We can how-ever limit both them and the corrections to the periods whichwould have to be made. As has been stated, Mc < 350M⊕ in thezero free eccentricity case. At this point (Mb = 0, Mc = 350M⊕)we also obtain the maximum period correction to make to planetb. This is +0.029 days to Pb in the zero free eccentricity case.As planet b’s mass is less constrained placing a limit on the pe-riod correction for planet c is harder, but limiting Mb to 600M⊕gives a maximum amplitude for the period correction to Pc of-0.12 days. At the zero eccentricity case (φ = 0) we are alreadyat the steepest gradient found on the TTV phase curve - as thissets the period correction, these amplitudes cannot go higher.They can however change sign (at φ = π for example), and sowe constrain |Correction(Pb)| < 0.029d and |Correction(Pc)| <0.12d immediately. The correction to Pb can be further limitedby noting that not all of the masses which provide a fit at φ = 0do so at other values for φ. In particular, when π/2 < φ < π (thecase for a negative correction to Pb), the allowed range for Mc ismuch smaller (Mc . 30M⊕), which corresponds to a maximumnegative correction of Correction(Pb)>-0.002. As such, the finallimits for Pb are −0.002 < Correction(Pb) < 0.029, where Cor-rection(P) is to be made to the periods found in Table 2. Thislimits −0.011 < 4 < 0.013 in the extreme case, confirming thatthe system remains very close to resonance.

At this stage it is worth stating the now better understoodperiods of these two objects. Using the period implied by ourlatest NITES transit measurement and the T0 from K2, we obtainPb = 7.921+0.028

−0.003 days , and Pc = 11.91 ± 0.12 days, where theerrors are ranges rather than 1σ errors, and account for TTVrelated period corrections. When predicting transit times, theseperiods and the T0 values of Table 2 should be used. Note thatthere will also be possible TTVs of magnitude up to at least anhour.

6.3. Hill stability

Stable main sequence evolution of a two-planet system may beguaranteed by residing in a 3:2 MMR, although the Kirkwoodgaps demonstrate that this MMR can instead harbour unstableorbits. Also, as-yet-undetected planets perturbing the 3:2 MMRcan cause complex dynamical structures (Fuse 2002) and po-tentially instability. The potential protection afforded by the 3:2MMR becomes more important when the two planets are Hillunstable. Two planets are Hill stable if their orbits are guaran-teed to never cross; hence Neptune and Pluto are Hill unstable,but protected from each other by the 3:2 MMR. Hill stability isa function of masses, semimajor axes, eccentricities and incli-nations. Veras et al. (2013) outline an algorithm for computingthe Hill stability limit; no explicit formula exists for arbitraryeccentricities and inclinations.

In order for EPIC201505350b and c to be Hill unstable, theirmasses and/or eccentricities must be sufficiently large. The mu-tual inclination between the planets of just about a degree neg-ligibly affects the Hill stability limit (Veras & Armitage 2004)6.Constraining the eccentricities and masses based on orbital peri-ods alone with Hill stability is a useful exercise but requires as-sumptions. A commonly-made assumption for transiting planetsis that those planets are on circular orbits; the closer the planet isto the star, the better that assumption, based on tidal circulariza-tion arguments. We need not make such assumptions here.

6 Just the existence of a nonzero mutual inclination between the planetsindicates that the planets might instead reside in an inclination-based6:4 MMR (see equation 1 of Milani et al. 1989).

10-6 10-5 10-4 10-3 10-2

Mc/M¯

10-6

10-5

10-4

10-3

10-2

Mb/M

¯

eb=0, ec=0eb=0, ec=0.1eb=0.1, ec=0eb=0.1, ec=0.1eb=0.2, ec=0eb=0, ec=0.2

Fig. 8. Hill stability limits for the EPIC201505350 system. We assumeM? = 0.9M�, the mutual inclination between the planets is 1 degree,and ab/ac is within 0.1 per cent of (3/2)2/3 based on the planetary or-bital period ratios. Eccentricites are measured in Jacobi coordinates. IfEPIC201505350 is Hill unstable, then the 3:2 MMR might act as a cru-cial protection mechanism to ensure the system’s long-term stability onthe main sequence.

We can use the green curves in Fig. 1 of Veras & Ford (2012)to roughly estimate Hill stability limits for EPIC201505350.Broadly, the plot shows that the system will be Hill stable if both(i) the sum of the eccentricities of both bodies (measured in Ja-cobi coordinates) does not exceed about 0.2, and (ii) that eachplanet is less massive than Jupiter. These relations help motivatethe setup for Figure 8. The figure plots the pairs of planet massesfor different eccentricities (measured in Jacobi coordinates) thatwould place the system on the edge of Hill stability, assuming amutual inclination of 1 degree and a semimajor axis ratio whichis just 0.1 per cent within (3/2)2/3. We generated each set ofcoloured points by sampling 350 different values of each of Mband Mc uniformly in log space between 10−6M� and 10−2M�, arange which covers both Earth masses and Jupiter masses. Theplot axes span the entire range of masses that we sampled.

Even if EPIC201505350 is Hill stable, then planet c mighteventually escape the system or planet b might crash into thestar through Lagrange instability (Barnes & Greenberg 2006,2007; Raymond et al. 2009; Kopparapu & Barnes 2010; Decket al. 2013; Veras & Mustill 2013). No analytical Lagrange un-stable boundary is known to exist. Regardless, the 3:2 MMR maythen provide a protection mechanism not only for Hill unstablesystems, but also for Lagrange unstable systems. The upcomingspace mission PLATO (Rauer et al. 2014) will provide accurateenough stellar age constraints to potentially detect a decreasingtrend in stable multi-planet systems with time due to Lagrangeinstability (Veras et al. 2015).

6.4. Planet Confirmation

Candidates from Kepler or K2 can be confirmed in a numberof ways. These include radial velocity observations, BLENDERand PASTIS analyses analysing the probability of false posi-tive scenarios (Torres et al. 2010b; Díaz et al. 2014), high res-olution photometry to search for close companions (e.g. Ev-erett et al. 2015; Lillo-Box et al. 2012, 2014; Law et al. 2014)or as is relevant in this case, TTVs (e.g. Steffen et al. 2012;

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Armstrong et al: EPIC201505350

Nesvorný et al. 2014). The presence of significant TTVs, sug-gested by the K2 observations and confirmed by the NITES tran-sit, demonstrate that both objects in this system orbit the samehost star. Although we do not have high resolution images ofEPIC201505350, SDSS images do not show evidence for moder-ately close companions. Furthermore, it has already been shownthat objects in multiple planetary systems are extremely unlikelyto be false positives (Lissauer et al. 2012), and the probability oftwo chance aligned binaries falling into MMR is even lower.

Beyond these points however, we have a mass constraint onthe outer planet c, of Mc < 386M⊕ (strictly this limit assumes|Zfree| < 0.2. This confirms the planetary nature of planet c. If c isa planet, and the objects orbit the same host star as the presenceof TTVs shows, it is extremely unlikely that b could exhibit thetransits it does without also being planetary in nature. As suchwe are able to confirm EPIC201505350 as a multi-planet system.

7. Conclusion

We have presented and confirmed EPIC201505350, a system oftwo Neptune sized planets orbiting close to the 3:2 MMR, viatransit observations with K2 and the NITES telescope. The inner,larger planet b shows high amplitude TTVs of ∼an hour, and willlikely show larger amplitudes when further transits are observed.The outer, smaller planet c can be expected to show even greaterTTVs (scaled up by the mass ratio of the two planets), althoughthese have not yet been observed. The precise ephemeris of theseplanets is still in doubt, see Sections 4 and 6 for the fit values andlimits to the possible corrections to them.

Future observations of EPIC201505350 have the potential tolead to interesting discoveries. The system is bright enough toobserve from the ground, leading to great potential for futurework. The observation of more transits of either planet will leadto fully characterised TTV phase curves, as well as possibly be-ing able to fully solve for the planetary masses via full dynamicalanalysis. Radial velocity observations have the potential to inde-pendently characterise the planetary masses, providing a windowon the discrepancies that sometimes exist between radial veloc-ity and TTV derived masses. We expect that this system is oneof many more which will arise from the K2 mission.

Acknowledgements. The data presented in this paper were obtained from theMikulski Archive for Space Telescopes (MAST). STScI is operated by the As-sociation of Universities for Research in Astronomy, Inc., under NASA contractNAS5-26555. Support for MAST for non-HST data is provided by the NASAOffice of Space Science via grant NNX13AC07G and by other grants and con-tracts. A.S. is supported by the European Union under a Marie Curie Intra-European Fellowship for Career Development with reference FP7-PEOPLE-2013-IEF, number 627202. The authors would like to thank Thomas Marshfor use of his periodogram generating Python code. We acknowledge all theobservers who attempt to detect the transits: Luc Arnold, Maurice Audejean,Mathieu Bachschmidt, Rahoul Behrend, Laurent Bernasconi, the C2PU team,Jean-Christophe Dalouzy, André Debackere, Serge Golovanow, João Gregorio,Patrick Martinez, Federico Manzini, Thierry Midavaine, Jacques Michelet, Ro-main Montaigut, Gérald Rousseau, René Roy, Dominique Toublanc, MichaelVanhuysse, and Daniel Verilhac. SCCB thanks CNES for the grant 98761.

ReferencesAgol, E., Steffen, J., Sari, R., & Clarkson, W. 2005, Monthly Notices of the Royal

Astronomical Society, 359, 567Armstrong, D. J., Kirk, J., Lam, K. W. F., et al. 2015, eprint arXiv:1502.04004Barnes, R., & Greenberg, R. 2006, ApJ, 647, L163Barnes, R., & Greenberg, R. 2007, ApJL, 665, L67Baranne, A., Queloz, D., Mayor, M., et al. 1996, A&AS, 119, 373Barros, S. C. C., Boué, G., Gibson, N. P., et al. 2013, MNRAS, 430, 3032Batygin, K., & Morbidelli, A. 2013, AJ, 145, 1

Beky, B., Kipping, D. M., & Holman, M. J. 2014, Monthly Notices of the RoyalAstronomical Society, 442, 3686

Bouchy, F., Hébrard, G., Udry, S., et al. 2009, A&A, 505, 853Callegari, N., Ferraz-Mello, S., & Michtchenko, T. A. 2006, Celestial Mechanics

and Dynamical Astronomy, 94, 381Chatterjee, S., & Ford, E. B. 2014, In Press, ApJ, arXiv:1406.0521Crossfield, I. J. M., Petigura, E., Schlieder, J., et al. 2015, eprint

arXiv:1501.03798Deck, K. M. & Agol, E. 2014, eprint arXiv:1411.0004Deck, K. M., Payne, M., & Holman, M. J. 2013, ApJ, 774, 129Delisle, J.-B., Laskar, J., & Correia, A. C. M. 2014, A&A, 566, AA137Delisle, J.-B., & Laskar, J. 2014, A&A, 570, LL7Díaz, R. F., Almenara, J. M., Santerne, A., et al. 2014, MNRAS, 441, 983Emel’yanenko, V. V. 2012, Solar System Research, 46, 321Everett, M. E., Barclay, T., Ciardi, D. R., et al. 2015, The Astronomical Journal,

149, 55Fabrycky, D. C., Lissauer, J. J., Ragozzine, D., et al. 2014, ApJ, 790, 146Foreman-Mackey, D., Montet, B. T., Hogg, D. W., et al. 2015, eprint

arXiv:1502.04715Fuse, T. 2002, PASJ, 54, 493Gozdziewski, K., Konacki, M., & Wolszczan, A. 2005, ApJ, 619, 1084Hadden, S. & Lithwick, Y. 2014, The Astrophysical Journal, 787, 80Hadjidemetriou, J. D., & Voyatzis, G. 2010, Celestial Mechanics and Dynamical

Astronomy, 107, 3Han, E., Wang, S. X., Wright, J. T., et al. 2014, Publications of the Astronomical

Society of the Pacific, 126, 827Holman, M. J. 2005, Science, 307, 1288Howell, S. B., Sobeck, C., Haas, M., et al. 2014, Publications of the Astronomical

Society of the Pacific, 126, 398Kopparapu, R. K., & Barnes, R. 2010, ApJ, 716, 1336Kurucz, R. 1993, ATLAS9 Stellar Atmosphere Programs and 2 km/s grid. Ku-

rucz CD-ROM No. 13. Cambridge, Mass.: Smithsonian Astrophysical Ob-servatory, 1993., 13

Law, N. M., Morton, T., Baranec, C., et al. 2014, ApJ, 791, 35Lee, M. H., Fabrycky, D., & Lin, D. N. C. 2013, ApJ, 774, 52Libert, A.-S., & Tsiganis, K. 2011, Celestial Mechanics and Dynamical Astron-

omy, 111, 201Lillo-Box, J., Barrado, D., & Bouy, H. 2012, A&A, 546, A10Lillo-Box, J., Barrado, D., & Bouy, H. 2014, A&A, 566, A103Lissauer, J. J., Ragozzine, D., Fabrycky, D. C., et al. 2011, ApJS, 197, 8Lissauer, J. J., Marcy, G. W., Rowe, J. F., et al. 2012, The Astrophysical Journal,

750, 112Lithwick, Y., Xie, J., & Wu, Y. 2012, The Astrophysical Journal, 761, 122Lomb, N. R. 1976, Astrophysics and Space Science, 39, 447Malhotra, R., Black, D., Eck, A., & Jackson, A. 1992, Nature, 356, 583McCormac J., Skillen I., Pollacco D., et al. 2014, MNRAS, 438, 3383Milani, A., Nobili, A. M., & Carpino, M. 1989, Icarus, 82, 200Murray, C. D., & Dermott, S. F. 1999, Solar system dynamics by Murray, C. D.,

1999,Nesvorný, D., Kipping, D., Terrell, D., & Feroz, F. 2014, The Astrophysical Jour-

nal, 790, 31Nesvorný, D. & Vokrouhlický, D. 2014, The Astrophysical Journal, 790, 58Ogihara, M., & Kobayashi, H. 2013, ApJ, 775, 34Papaloizou, J. C. B., & Szuszkiewicz, E. 2005, MNRAS, 363, 153Peale, S. J., & Lee, M. H. 2002, Science, 298, 593Pepe, F., Mayor, M., Galland, F., et al. 2002, A&A, 388, 632Perruchot, S., Kohler, D., Bouchy, F., et al. 2008, in Society of Photo-Optical

Instrumentation Engineers (SPIE) Conference Series, Vol. 7014, Society ofPhoto-Optical Instrumentation Engineers (SPIE) Conference Series, 0

Petrovich, C., Malhotra, R., & Tremaine, S. 2013, ApJ, 770, 24Pierens, A., Raymond, S. N., Nesvorny, D., & Morbidelli, A. 2014, ApJL, 795,

LL11Popper, D. M. & Etzel, P. B. 1981, The Astronomical Journal, 86, 102Press, W. H. & Rybicki, G. B. 1989, The Astrophysical Journal, 338, 277Rauer, H., Catala, C., Aerts, C., et al. 2014, Experimental Astronomy, 38, 249Raymond, S. N., Barnes, R., Armitage, P. J., & Gorelick, N. 2008, ApJL, 687,

L107Raymond, S. N., Barnes, R., Veras, D., et al. 2009, ApJL, 696, L98Rein, H., Payne, M. J., Veras, D., & Ford, E. B. 2012, MNRAS, 426, 187Rowe, J. F., Bryson, S. T., Marcy, G. W., et al. 2014, The Astrophysical Journal,

784, 45Santerne, A., Hébrard, G., Deleuil, M., et al. 2014, A&A, 571, A37Santos, N. C., Sousa, S. G., Mortier, A., et al. 2013, Astronomy and Astro-

physics, 556, A150Scargle, J. D. 1982, The Astrophysical Journal, 263, 835Sneden, C. A. 1973, PhD thesis, The University of Texas at Austin.Snellgrove, M. D., Papaloizou, J. C. B., & Nelson, R. P. 2001, A&A, 374, 1092Southworth, J. 2013, Astronomy and Astrophysics, 557, A119Steffen, J. H., Fabrycky, D. C., Ford, E. B., et al. 2012, Monthly Notices of the

Royal Astronomical Society, 421, 2342

Article number, page 9 of 10

A&A proofs: manuscript no. K2201505350

Stetson P. B., 1987, PASP, 99, 191Tadeu dos Santos, M., Correa-Otto, J. A., Michtchenko, T. A., & Ferraz-Mello,

S. 2015, A&A, 573, AA94Torres, G., Andersen, J., & Giménez, A. 2010a, The Astronomy and Astro-

physics Review, 18, 67Torres, G., Fressin, F., Batalha, N. M., et al. 2010b, The Astrophysical Journal,

727, 24Tsantaki, M., Sousa, S. G., Adibekyan, V. Z., et al. 2013, A&A, 555, A150Vanderburg, A. & Johnson, J. A. 2014, Publications of the Astronomical Society

of the Pacific, 126, 948Vanderburg, A., Montet, B. T., Johnson, J. A., et al. 2014, eprint arXiv:1412.5674Veras, D., & Armitage, P. J. 2004, Icarus, 172, 349Veras, D., & Ford, E. B. 2012, MNRAS, 420, L23Veras, D., Mustill, A. J., Bonsor, A., & Wyatt, M. C. 2013, MNRAS, 431, 1686Veras, D., & Mustill, A. J. 2013, MNRAS, 434, L11Veras, D., Brown, D. J. A., Mustill, A. J., Pollacco, D. 2015, Submitted to MN-

RASWalsh, K. J., Morbidelli, A., Raymond, S. N., O’Brien, D. P., & Mandell, A. M.

2011, Nature, 475, 206Wolszczan, A., & Frail, D. A. 1992, Nature, 355, 145Wolszczan, A. 1994, Science, 264, 538Wang, S., & Ji, J. 2014, ApJ, 795, 85Xie, J.-W. 2014, The Astrophysical Journal Supplement Series, 210, 25Zhang, X., Li, H., Li, S., & Lin, D. N. C. 2014, ApJL, 789, LL23

Article number, page 10 of 10