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THE K2-138 SYSTEM: A NEAR-RESONANT CHAIN OF FIVE SUB-NEPTUNE PLANETS DISCOVERED BY

CITIZEN SCIENTISTS

Jessie L. Christiansen

1,2

, Ian J. M. Crossfield

3,4

, Geert Barentsen

5,6

, Chris J. Lintott

, Thomas Barclay

8,9

Brooke .D. Simmons

10,11

, Erik Petigura

, Joshua E. Schlieder

, Courtney D. Dressing

4,10

, Andrew

Vanderburg

, David R. Ciardi

, Campbell Allen

, Adam McMaster

, Grant Miller

, Martin Veldthuis

Sarah Allen

, Zach Wolfenbarger

, Brian Cox

, Julia Zemiro

, Andrew W. Howard

, John Livingston

Evan Sinukoff

17,20

, Timothy Catron

, Andrew Grey

, Joshua J. E. Kusch

, Ivan Terentev

, Martin Vales

and Martti H. Kristiansen

Caltech/IPAC-NASA Exoplanet Science Institute, M/S 100-22, 770 S. Wilson Ave, Pasadena, CA 91106 USA

jessie.christiansen@caltech.edu

Department of Astronomy & Astrophysics, University of California, Santa Cruz, CA, USA

Sagan Fellow

NASA Ames Research Center, Moffett Field, CA 94035, USA

Bay Area Environmental Research Inst., 625 2nd St. Ste 209 Petaluma, CA 94952, USA

Department of Physics, University of Oxford, Denys Wilkinson Building, Keble Road, Oxford, OX1 3RH, UK

NASA Goddard Space Flight Center, 8800 Greenbelt Rd, Greenbelt, MD 20771, USA

University of Maryland, Baltimore County, 1000 Hilltop Cir, Baltimore, MD 21250, USA

Einstein Fellow

Center for Astrophysics and Space Sciences (CASS), Department of Physics, University of California, San Diego, CA 92093, USA

Geological and Planetary Sciences, California Institute of Technology, Pasadena, CA, USA

NASA Goddard Space Flight Center, 8800 Greenbelt Rd, Greenbelt, MD 20771, USA

Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA

The Adler Planetarium, Chicago, IL 60605, USA

School of Physics & Astronomy, University of Manchester, Oxford Road, Manchester, M13 9PL, UK

c/o Australian Broadcasting Corporation

Department of Astrophysics, California Institute of Technology, Pasadena, CA 91125, USA

The University of Tokyo, 7-3-1 Bunkyo-ku, Tokyo 113-0033, Japan

Institute for Astronomy, University of Hawai‘i at M ̄anoa, Honolulu, HI 96822, USA

Arizona State University, Tempe, AZ 85281, USA

Citizen Scientists, c/o Zooniverse, Department of Physics, University of Oxford, Denys Wilkinson Building, Keble Road, Oxford, OX1

3RH, UK

Danmarks Tekniske Universitet, Anker Engelundsvej 1, Building 101A, 2800 Kgs. Lyngby, Denmark

ABSTRACT

K2-138 is a moderately bright (

= 12

= 10

3) main sequence K-star observed in Campaign 12

of the NASA

mission. It hosts five small (1.6–3.3R

⊕

) transiting planets in a compact architecture.

The periods of the five planets are 2.35 d, 3.56 d, 5.40 d, 8.26 d, and 12.76 d, forming an unbroken

chain of near 3:2 resonances. Although we do not detect the predicted 2–5 minute transit timing

variations with the

timing precision, they may be observable by higher cadence observations

with, for example,

Spitzer

or CHEOPS. The planets are amenable to mass measurement by precision

radial velocity measurements, and therefore K2-138 could represent a new benchmark systems for

comparing radial velocity and TTV masses. K2-138 is the first exoplanet discovery by citizen scientists

participating in the Exoplanet Explorers project on the Zooniverse platform.

Keywords:

eclipses, planetary systems: individual(K2-138), techniques: photometric, techniques:

spectroscopic

INTRODUCTION

The NASA

mission (Howell et al. 2014) is in its

third year of surveying the ecliptic plane. The mission

arXiv:1801.03874v2 [astro-ph.EP] 20 Jan 2018

Christiansen et al.

uses the repurposed

Kepler Space Telescope

to tile the

ecliptic, and consists of successive

∼

80-day observations

of 12

12 degree regions of sky known as campaigns.

Each campaign yields high-precision, high-cadence cal-

ibrated pixel files and time-series photometry on any-

where from 13,000 to 28,000 targets, which are released

to the public within three months of the end of the ob-

serving campaign. This deluge of data is immediately

inspected by professional exoplanet hunters, producing

rapid announcements of interesting new planetary sys-

tems, e.g. the recent examples of HD 106315 (Crossfield

et al. 2017; Rodriguez et al. 2017) and HD 3167 (Van-

derburg et al. 2016).

There are many features in time-series data which

can be matched to potential transit signals by signal-

processing algorithms. These features can be either

astrophysical in origin (e.g. pulsating variable stars,

eclipsing binaries, flaring stars, cosmic-ray pixel strikes),

or instrumental (e.g. apparent variations in the bright-

ness in a photometric aperture caused by spacecraft

pointing drift, or by focus drifts in response to the chang-

ing thermal environment). While these artifacts can

confuse automated procedures, the human brain is op-

timised for pattern matching, and is remarkably good

at discriminating these artifacts from a train of planet

transits. This ability is exploited in information security

technology for instance, with the CAPTCHA algorithm

(von Ahn et al. 2003) being a well-known example. In

light curve analysis, humans can readily recognise the

differences in the underlying phenomena causing the pu-

tative signals and identify the transit signals.

This ability, along with the strong interest held by

the public in being involved in the process of scientific

discovery, has led to the ongoing success of the Planet

Hunters

project (e.g. Fischer et al. 2012).

Hosted

by the Zooniverse platform (Lintott et al. 2008), the

project displays to users the publicly available

Kepler

and

time series photometry, and asks them to iden-

tify transit-like dips.

Inspired by their success, we

started the Exoplanet Explorers

project in April 2017.

In this project we run a signal detection algorithm to

identify potential transit signals in the

time series

photometry, and ask the users to sift through the result-

ing candidates to identify those most closely resembling

planetary transits. Here we present K2-138, the first

planetary system discovered by citizen scientists. In

Section 2, we describe the

data set. We describe

the Exoplanet Explorers project and the identification

of K2-138 in Section 3, and the derivation of the stel-

www.planethunters.org

www.exoplanetexplorers.org

Figure 1

. The set of diagnostic plots presented on the Exo-

planet Explorers project, for the target K2-138. From top to

bottom the four plots are for the putative signals for K2-138

c, d, e, and f. In each case, the left panel shows the individual

transit events, with an arbitrary vertical offset and alternat-

ing color for visual clarity. The top right panel shows the

entire light curve folded at the period of the putative transit

signal. The black points are the original

data, and the

yellow circles are binned data. The bottom right panel shows

the same phase-folded light curve, zoomed in on the transit

event itself. An initial fitted planet model is overlaid in blue.

lar parameters in Section 4. In Section 5 we describe

the analysis of the planet parameters. Finally, in Sec-

tion 6 we place the K2-138 system in context of other

high-multiplicity systems and discuss prospects for fu-

ture characterisation.

OBSERVATIONS AND PHOTOMETRIC

REDUCTION

data are downlinked from the spacecraft, pro-

cessed into calibrated pixel files and photometric time

The K2-138 System

series, and released to the public via the Mikulski

Archive for Space Telescopes (MAST)

. Unlike the orig-

inal

Kepler

mission, the target list is entirely guest ob-

server driven: all observed targets are proposed to the

project by the community. Campaign 12 (C12), which

was observed for 79 days from 2016 December 15 to 2017

March 4, contained the target star TRAPPIST-1 (Gillon

et al. 2016, 2017; Luger et al. 2017; Wang et al. 2017). In

order to facilitate rapid analysis of the

observations

of TRAPPIST-1, the data were released to the public

immediately after downlink from the spacecraft on 2017

March 9 as raw pixel files.

We downloaded the raw pixel files from MAST for the

stellar targets that were proposed by K2’s large exo-

planet search programs, and calibrated these data using

the

kadenza

software (Barentsen 2017), generating cal-

ibrated pixel files.

We then used the publicly available

k2phot

pho-

tometry code

, which generates aperture photometry

and performs corrections for the spacecraft pointing jit-

ter using Gaussian Processes (Rasmussen and Williams

2005), to generate light curves suitable for searching for

periodic transit signals. Using the publicly available

TERRA algorithm

(Petigura et al. 2013a,b), we gen-

erated both a list of potential transiting planet signals

from the detrended light curves, and a set of accom-

panying diagnostic plots. TERRA identified a total of

4,900 candidate transit signals in the C12 stellar data.

TRANSIT IDENTIFICATION

For each signal from C12 and the earlier campaigns

already processed, we uploaded a subset of the standard

TERRA diagnostic plots to the Exoplanet Explorers

project. The plots included a phase-folded light curve

and a stack of the individual transit events; users exam-

ined these plots and selected whether the putative signal

looked like a true transiting planet candidate. Figure 1

shows an example set of diagnostic plots.

On 2017 April 4, the Exoplanet Explorers project was

featured on the Stargazing Live ABC broadcast in Aus-

tralia. Between April 4, 2017 01:00 UTC and April 6,

2017 19:48 UTC, the live project received 2,100,643 clas-

sifications from 7,270 registered Zooniverse classifiers

and 7,677 not-logged-in IP addresses. Of these, 130,365

classifications from 4,325 registered classifiers and 2,012

not-logged-in IP addresses were for candidate signals in

the C12 data. C12 candidates received a median of 26

classifications each; the C12 candidate with the lowest

https://archive.stsci.edu/kepler/

https://github.com/petigura/k2phot

https://github.com/petigura/terra

0        5        10       15       20       25       30       35       40

Candidate orbital period (days)

10

1

10

2

Candidate SNR (

)

All candidates

>60% votes

>75% votes

>90% votes

Figure 2

. The distribution of user votes on the C12 can-

didate transiting planet signals. All signals received 14 or

more votes, and the percentage of ‘yes’ votes is shown: the

small blue dots represent signals for which fewer than 60%

of users voted ‘yes’. The green stars, yellow + symbols and

red circles show signals for which greater than 60%, 75% and

90% of users voted ‘yes’.

classification count received 14 classifications, and the

most-classified C12 candidate received 43 classifications.

The classifications were aggregated for each candidate to

provide the fraction of classifiers who indicated they saw

a transiting planet signal.

Of the 4,900 potential transiting signals identified in

the C12 data, 72 were voted by more than 60% of users

as looking like transiting planet candidates. The dispo-

sitions of the full set of C12 signals are shown in Fig-

ure 2; the signals with the highest confidence are un-

surprisingly at shorter periods (with a higher number

of individual events contributing to the signal for the

users to assess) and higher signal-to-noise values. The

highly voted signals were inspected visually and K2-138

(EPIC 245950175) was rapidly identified as a promising

multi-planet system. The initial automated search of the

light curve produced four distinct transiting signatures,

each with a high (

90%) fraction of votes from the par-

ticipants; the four diagnostic plots that were voted on

are shown in Figure 1. EPIC 245950175 was proposed

for observations by four teams, in Guest Observing Pro-

grams 12049, 12071, 12083, and 12122 (PIs Quintana,

Charbonneau, Jensen, and Howard). The full, unfolded

light curve of K2-138 is shown in Figure 4. After the

system was flagged by the citizen scientists, additional

examination of the light curve revealed the signature of

a fifth transiting signal, interior to the four signals iden-

tified by the TERRA algorithm and on the same 3:2

resonant chain. Characterisation of the five detected

planet signals is detailed in Section 5.

In addition, two individual transit events were identi-

fied, shown in Figure 3, separated by 41.97 days. The

Christiansen et al.

Figure 3

. The two transits of the putative 42-day period

planet candidate that fall in the

C12 data. The time of

the second transit has been offset by 41.97 days to demon-

strate the consistency in depth and duration of the two

events, which correspond to a

∼

2.8R

⊕

planet.

transits have consistent depths and durations, and if

confirmed, would correspond to an additional

∼

2.8R

⊕

sub-Neptune planet in the K2-138 system, bringing the

total to six planets. Additional observations are required

to secure a third epoch and confirm that the two tran-

sits seen in the

data arise from a single planet, and

are not individual transits of two similarly-sized, longer-

period planets.

STELLAR CHARACTERISATION

On 2017 June 1 we obtained a spectrum of K2-138

using Keck/HIRES, without the iodine cell as is typical

of the precision radial velocity observations. We derive

stellar parameters using SpecMatch (Petigura et al.

2015), given in Table 1. Following the procedure in

Crossfield et al. (2016), we estimate the stellar radius

and mass using the publicly available

i

sochrones

Python package (Morton 2015) and the Dartmouth

stellar evolution models (Dotter et al. 2008).

The

California Kepler Survey (CKS; Petigura et al. 2017)

finds mass and radius uncertainty floors for similar

spectral types of 6% and 9% respectively, motivated

by comparisons between stellar radii derived using

isochrones

and spectroscopic parameters (as is done

here for K2-138) and asteroseismic radii (Johnson et

al. 2017). We therefore adopt these uncertainties on

the mass and radius of K2-138 to incorporate the

isochrone model uncertainties. Additional estimates of

the parameters of K2-138 are available in the Ecliptic

Plane Input Catalog (EPIC, Huber et al. 2016) on the

MAST and are consistent with those from the RAVE

spectrum and

isochrones

. The HIRES/

isochrones

parameters and the EPIC parameters are consistent

with a solar-metallicity, main-sequence, early-K type

star at a distance of

∼

180 pc when comparing to the

color-temperature relations of Pecaut & Mamajek

(2013). We adopt a spectral type of K1V

1. We mea-

sure a log

′

value of

−

63, indicating a modestly

magnetically active star, which may present a challenge

for precision radial velocity measurements of the system.

Table 1

. K2-138 stellar parameters

EPIC ID

245950175

2MASS ID

J23154776-1050590

RA (J2000.0)

23:15:47.77

Dec (J2000.0)

-10:50:58.91

(mag)

12.21

(mag)

10.305

Spectral type

K1V

eff

(K)

5378

log

(cgs)

4.59

0.07

[Fe/H]

0.16

0.04

)

0.86

0.08

)

0.93

0.06

Distance (pc)

183

sin

(km/s)

2.7

1.5

EPIC classification, see Huber et al.

(2016) and

https://github.com/danxhuber/galclassify

On 2017 May 31 we obtained a high-resolution image

of K2-138 in

-band using the Altair AO system on

the NIRI camera at Gemini Observatory (Hodapp et al.

2003) under program GN-2105B-LP-5 (PI Crossfield).

We observed at five dither positions, and used the

dithered images to remove sky background and dark

current; we then aligned, flat-fielded and stacked the

individual images. The inset in Figure 5 shows the

final stacked image, and the plot shows the detection

limits of the final image. The limits were determined

by injecting simulated sources into the final image,

with separation from K2-138 determined by integer

multiples of the FWHM, as in Furlan et al. (2017). We

see no other source of contaminating flux in the AO

image within 4

′′

, the size of one

pixel. In addition

to the AO data, we examine the HIRES spectrum

for evidence of additional stellar lines, following the

procedure of Crossfield et al. (2016). This method is

sensitive to secondary stars that lie within 0.4

′′

of the

primary star (one half of the slit width) and that are

up to 5 magnitudes fainter than the primary star in the

- and

-bands (Kolbl et al. 2015), complementing the

sensitivity limits of the NIRI observations. We are able

to rule out companions with T

eff

= 3400–6100 K and

∆(RV)

10 kms

−

. We further discuss the possibility

of the observed periodic signals originating from a faint

star

∼

′′

away in Section 5.1, but for the following

analysis we assume the putative planet signals arise

from K2-138.

PLANET PARAMETERS

The K2-138 System

Figure 4

. The time series of the K2 data, with the five-planet transit model shown in blue. The planets of the individual transits

are marked with the appropriate letter; the times of the two transits of the putative planet candidate discussed in Section 3 are

shown as ‘g’. The 5-day gap two-thirds of the way through the campaign was the result of a spacecraft safe-mode event.

Figure 5

Inset:

The Gemini/NIRI AO image of K2-138.

We detect no additional sources of flux.

Plot:

The 5-

con-

trast limits for additional companions, in ∆magnitude, are

plotted against angular separation in arcseconds; the black

points represent one step in the FWHM resolution of the

images.

We analyzed the five transit signals independently in

the K2-138 light curve, using the same modeling, fitting,

and MCMC procedures as described in Crossfield et

al. (2016). In summary, we fit the following model pa-

rameters: mid-transit time (T

); the candidates orbital

period and inclination (P and i); the scaled semimajor

axis (R

/a); the fractional candidate size (R

∗

/a); the

orbital eccentricity and longitude of periastron (e and

), the fractional level of dilution (

) from any other

sources in the aperture; a single multiplicative offset for

the absolute flux level; and quadratic limb-darkening

coefficients (u1 and u2).

We explore the posterior

distribution using the

emcee

software (Foreman-Mackey

et al. 2013). We find that the signals correspond to five

sub-Neptune-sized planets ranging from 1.6–3.3R

⊕

; the

best fitting transit models are shown in Figure 6. As a

self-consistency check, we note that the stellar density

values derived from the independent transit fits are

consistent across all five planets, and also consistent

with the direct calculation from the stellar mass and

radius.

All five planets have periods under 13 days, making

K2-138 an example of a tightly-packed system of small

planets. One particularly interesting aspect to the K2-

138 architecture, discussed further in Section 6, is that

each successive pair of planets is just outside the first-

order 3:2 resonance.

5.1.

Validation

Lissauer et al. (2012) analysed the distribution of

Ke-

pler

planet candidates and showed that systems with

multiple candidate signals were substantially more likely

to be true planetary systems than false positives. This

provides a ‘multiplicity boost’ to the statistical valida-

tion of candidates in multi-planet systems. Here, we

validated each candidate individually using the publicly

available

vespa

code

, which computes the likelihood of

various astrophysical false positive scenarios. We use

as input a photometric exclusion radius of 13

′′

, the

band and

Kepler

magnitudes, and the HIRES stellar

parameters. The results are false positive probabilities

of 0.20%, 0.11%, 0.76%, 0.027%, and 0.24% for plan-

ets b, c, d, e, and f respectively. Since we have multiple

candidates orbiting a single star, the ‘multiplicity boost’

(and an additional ‘near-resonance boost’) further sup-

presses these FPPs (Lissauer et al. 2012; Sinukoff et al.

2016). Applying the

multiplicity boost derived by

https://github.com/timothydmorton/VESPA

Christiansen et al.

Figure 6

. The folded transits of K2-138 b, c, d, e, and f

overplotted with the best fitting transit model in red. Binned

data points are shown in blue. The planets range in size from

1.57 to 3.29 R

⊕

Sinukoff et al. (2016), we find final FPPs of 8.3

−

4.6

−

, 3.17

−

, 1.1

−

, and 1.0

−

Recently, Cabrera et al. (2017) showed that stars

within the

Kepler

photometry aperture but outside the

small area surveyed by high-resolution imaging were re-

sponsible for several falsely validated planets. Here we

examine the possibility that the five periodic signals do

not arise from the brightest star in the

aperture.

Given that the signals form an unbroken chain of near

first-order resonances, we consider the possibility that

some number of the signals arise on one star and the

Figure 7

. A 60

60 arcsecond image from the SDSS DR7

-band. The companion to the west is

∼

14 arcseconds away,

and is 5.6 magnitudes fainter than EPIC 245950175 in

remainder on a star coincident with the line of sight to

be unlikely, and consider the five signals as a related

set. The brightest nearby star is 2MASS J23154868-

1050583, which is

∼

′′

from EPIC 245950175, and 5.6

magnitudes fainter in

-band. This star is shown to the

west of EPIC 245950175 in Figure 7. Following from

Eq. (5) of Ciardi et al. (2015), we find that the putative

planets would be 13.2 times larger if they orbited the

fainter target, increasing to 1.9–4.0 R

. These would be

as large or larger than the largest planet known to date

with a radius measured by the transit method, WASP-

79b with a radius of 2.09

0.14 R

(Smalley et al. 2012).

Therefore we conclude that the five putative planets are

extremely unlikely to orbit 2MASS J23154868-1050583.

DISCUSSION

One of the interesting discoveries from the NASA

Kepler

mission is the prevalence of compact, highly

co-planar, and often dynamically packed systems of

small (

⊕

) planets (Latham et al. 2011; Lissauer

et al. 2011; Fabrycky et al. 2014; Howard et al.

2012; Winn & Fabrycky 2015).

This has continued

in the

mission, including the discoveries of the

K2-3 (Crossfield et al. 2015), K2-37 (Sinukoff et al.

2016), and K2-72 (Crossfield et al. 2016) systems.

Multi-planet systems are crucial laboratories for testing

planetary formation, migration and evolution theories.

A further interesting subset of these systems are those

demonstrating resonances, or chains of resonances.

The five validated planets of K2-138 lie close to a

first-order resonant chain.

We find period ratios of

1.513, 1.518, 1.528 and 1.544 for the b-c, c-d, d-e, and

e-f pairs respectively, just outside the 3:2 resonance.

Fabrycky et al. (2014), examining the large population

The K2-138 System

of multi-transiting planet systems in the

Kepler

data,

showed that pair-wise period ratios pile-up just out-

side of the first- and second-order resonances. They

offer several possible explanations for this, including

gravitational scattering slightly out of resonance by the

additional bodies in the system, or tidal dissipation

preferentially acting to drag the inner planets inward

from the resonance. Lithwick & Wu (2012) and Batygin

& Morbidelli (2013) investigate the suggestion of tidal

dissipation as a mechanism for keeping individual pairs

of planets just outward of the resonance; they note

that in systems with more than two planets, where the

planets can inhabit multiple resonances, the planets

can remain close to resonance despite tidal dissipation.

Recently, Ramos et al. (2017) analytically derived the

expected offset from a first-order resonance for a pair

of planets due to Type I migration. Their Fig. 3 shows

that for periods shorter than

∼

10 days, the resonance

period ratio is 1.505–1.525, depending on the mass of

the inner planet and the mass ratio of the two planets,

with higher period ratios expected as the mass ratio

approaches unity.

Therefore it is possible that the

K2-138 b-c and c-d pairs may be captured in the 3:2

resonance, but unlikely that the d-e or e-f pairs are in

resonance.

It is illustrative to compare K2-138 to the other known

systems with multiple planets, and to examine whether

3:2 period ratios are common. Figure 8 shows, for the

confirmed multi-planet systems, the distance of each pe-

riod ratio in the system from a 3:2 period ratio. K2-

138 is the only system with an unbroken chain of four

period ratios near 3:2. There are seven systems with

planets in consecutive 3:2 pairs: Kepler-23 (Ford et al.

2012); Kepler-85 and Kepler-114 (Xie 2013; Rowe et

al. 2014); Kepler-217 (Rowe et al. 2014; Morton et al.

2016); Kepler-339 and Kepler-402 (Rowe et al. 2014);

and Kepler-350 (Rowe et al. 2014; Xie 2014). There are

an additional nine systems with a ‘broken’ chain of 3:2

pairs, i.e. a configuration like K2-138 but where one

planet is missing, or perhaps undiscovered: GJ 3293

(Astudillo-Defru et al. 2015, 2017); K2-32 (Dai et al.

2016); Kepler-192 and Kepler-304 (Rowe et al. 2014;

Morton et al. 2016); Kepler-215, Kepler-254, Kepler-

275 and Kepler-363 (Rowe et al. 2014); and Kepler-

276 (Rowe et al. 2014; Xie 2014). These systems are

highlighted in red. A small number of systems con-

tain four planets in different configurations of first or-

der resonances, also highlighted in Figure 8. Kepler-223

is comparable to K2-138 : a compact system of four

sub-Neptune-sized planets with periods shorter than 20

days, in a 3:4:6:8 resonant chain (Rowe et al. 2014; Mills

et al. 2016). In the case of Kepler-223, the period ratios

are much closer to resonance than for K2-138, with ra-

tios of 1.3333, 1.5021, and 1.3338 for the b-c, c-d, and

d-e pairs respectively. Kepler-223 demonstrates signifi-

cant transit timing variations, allowing for robust mass

constraints to be placed. Kepler-79 (Jontof-Hutter et

al. 2014) is a scaled-up version of K2-138 and Kepler-

223, with four sub-Saturn-sized planets in a 1:2:4:6 res-

onant chain with periods from 13–81 days. Finally, the

benchmark TRAPPIST-1 system hosts seven planets in

a resonant chain, with successive period ratios of 8:5,

5:3, 3:2, 3:2, 4:3, and 3:2 (Gillon et al. 2017; Luger et

al. 2017). Like TRAPPIST-1, K2-138 may represent a

pristine chain of resonances indicative of slow, inward

disk migration.

Another notable feature of the TRAPPIST-1 system

is that the seven known planets form a complex chain

of linked three-body Laplace resonances (Luger et al.

2017). Similarly, Kepler-80 (KOI-500) is a five-planet

system where the four outer planets form a tightly linked

pair of three-body resonances (Lissauer et al. 2011; Mac-

Donald et al. 2016); Kepler-223, described above, also

contains a pair of three-body resonances. One other sys-

tem, Kepler-60, appears to be in either a true three-body

Laplace resonance or a chain of two-planet mean motion

resonances (Go ́zdziewski et al. 2016). A three-body res-

onance satisfies the condition that (

p/P

)

−

[(

)

(

q/P

)

≈

0, where

and

are integers and

the period

of the

th planet. For K2-138 we find that the three con-

secutive sets of three planets (bcd, cde, and def) all sat-

isfy this condition with (

p,q

) = (2

3), resulting in values

of 4.2

1.7

−

days

−

0.9

−

days

−

, and

−

4.9

−

days

−

respectively, all close to zero.

In simulating the Kepler-80 system, MacDonald et al.

(2016) find that their migration simulations can natu-

rally describe the final system architecture, with dissi-

pative forces pushing the interlocked planets out of two-

body resonances and into three-body resonances; the

K2-138 system may have undergone something similar.

K2-138 joins a relatively modest population of known

systems with four or more planets in or close to a reso-

nant chain, and a very small population of systems with

interlocking chains of three-body resonances, making it

an ideal target to study for transit timing variations.

We calculated the transit times of K2-138 c, d, e, and

f shown in Figure 9. For each transit, we fix the model

transit parameters to the best-fit values given in Table

2, and allow only the mid-transit time to vary. To cal-

culate the uncertainties, we compute the residuals from

the best-fit model and perform a bootstrap analysis us-

ing the closest 100 timestamps, re-fitting the mid-transit

time at each timestamp permutation. (Wall et al. 2003).

Examining the resulting transit times, we do not find ev-

idence of significant variations at the level of the aver-

age 8–10 minute timing precision from the

data.The

individual transits of K2-138 b have insufficient signal-