The K2-138 System: A Near-resonant Chain of Five Sub-Neptune Planets Discovered by Citizen Scientists

The K2-138 System: A Near-resonant Chain of Five Sub-Neptune Planets

Discovered by Citizen Scientists

Jessie L. Christiansen

, Ian J. M. Cross

fi

eld

, Geert Barentsen

, Chris J. Lintott

, Thomas Barclay

Brooke . D. Simmons

, Erik Petigura

, Joshua E. Schlieder

, Courtney D. Dressing

, Andrew Vanderburg

Campbell Allen

, Adam McMaster

, Grant Miller

, Martin Veldthuis

, Sarah Allen

, Zach Wolfenbarger

, Brian Cox

Julia Zemiro

, Andrew W. Howard

, John Livingston

, Evan Sinukoff

, Timothy Catron

, Andrew Grey

Joshua J. E. Kusch

, Ivan Terentev

, Martin Vales

, and Martti H. Kristiansen

NASA Exoplanet Science Institute, California Institute of Technology, M

S 100-22,

770 S. Wilson Avenue, Pasadena, CA 91106 USA;

jessie.christiansen@caltech.edu

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

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

Bay Area Environmental Research Inst., 625 2nd Street Suite 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 Road, Greenbelt, MD 20771, USA

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

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 Road, Greenbelt, MD 20771, USA

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

The Adler Planetarium, 1300 S Lake Shore Drive, Chicago, IL 60605, USA

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

o Australian Broadcasting Corporation, Australia

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

‘

iatM

ā

noa, 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, DK-2800 Kgs. Lyngby, Denmark

Received 2017 May 11; revised 2017 November 14; accepted 2017 November 15; published 2018 January 11

Abstract

K2-138 is a moderately bright

(



=



12.2,



=



10.3

)

main-sequence K star observed in Campaign 12 of the

NASA

mission. It hosts

fi

ve small

(

1.6

–

3.3

)

transiting planets in a compact architecture. The periods of

the

fi

ve planets are 2.35, 3.56, 5.40, 8.26, and 12.76 days, forming an unbroken chain of near 3:2 resonances.

Although we do not detect the predicted 2

–

5 minute transit timing variations

(

TTVs

)

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 system for comparing radial velocity and TTV masses. K2-138 is the

fi

rst

exoplanet discovery by citizen scien

tists participating in the

Exoplanet Explorers project on the Zooniverse

platform.

Key words:

eclipses

–

stars: individual

(

K2-138

)

–

techniques: photometric

–

techniques: spectroscopic

1. Introduction

The NASA

mission

(

Howell et al.

2014

)

is in its third

year of surveying the ecliptic plane. The mission 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 calibrated pixel

fi

les and time-

series photometry on anywhere from 13000 to 28000 targets,

which are released to the public within three months of the end

of the observing campaign. This deluge of data is immediately

inspected by professional exoplanet hunters, producing rapid

announcements of interesting new planetary systems, e.g., the

recent examples of HD



106315

(

Cross

fi

eld et al.

2017

;

Rodriguez et al.

2017

)

and HD



3167

(

Vanderburg et al.

2016

)

There are many features in time-series data that 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,

fl

aring stars,

cosmic-ray pixel strikes

)

, or instrumental

(

e.g., apparent

variations in the brightness in a photometric aperture caused

by spacecraft pointing drift, or by focus drifts in response to the

changing thermal environment

)

. While these artifacts can

confuse automated procedures, the human brain is optimized

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

recognize the differences in the underlying phenomena causing

the putative signals and identify the transit signals.

This ability, along with the strong interest held by the public

in being involved in the process of scienti

fi

c discovery, has led

The Astronomical Journal,

155:57

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doi.org

10.3847

1538-3881

aa9be0

Sagan Fellow.

Einstein Fellow.

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

identify transit-like dips. Inspired by their success, we started

the Exoplanet Explorers

project in 2017 April. 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 resulting candidates to identify those most

closely resembling planetary transits. Here, we present K2-138,

the

fi

rst

planetary system discovered by the Exoplanet

Explorers project. In Section

, we describe the

data set. We

describe the Exoplanet Explorers project and the identi

fi

cation

of K2-138 in Section

, and the derivation of the stellar

parameters in Section

. In Section

, we describe the analysis

of the planet parameters. Finally, in Section

we place the

K2-138 system in context of other high-multiplicity systems

and discuss prospects for future characterization.

2. Observations and Photometric Reduction

data are downlinked from the spacecraft, processed into

calibrated pixel

fi

les and photometric time series, and released

to the public via the Mikulski Archive for Space Telescopes

(

MAST

)

Unlike the original

Kepler

mission, the target list is

entirely guest observer 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

fi

les.

We downloaded the raw pixel

fi

les from MAST for the

stellar targets that were proposed by K2

ʼ

s large exoplanet

search programs and calibrated these data using the

kadenza

software

(

Barentsen

2017

)

, generating calibrated pixel

fi

les.

We then used the publicly available

k2phot

photometry

code,

which generates aperture photometry and performs

corrections for the spacecraft pointing jitter using Gaussian

Processes

(

Rasmussen & Williams

2005

)

, to generate light

curves suitable for searching for periodic transit signals. Using

the publicly available TERRA algorithm

(

Petigura et al.

2013a

2013b

)

, we generated both a list of potential transiting

planet signals from the detrended light curves and a set of

accompanying diagnostic plots. TERRA identi

fi

ed a total of

4900 candidate transit signals in the C12 stellar data.

3. Transit Identi

fi

cation

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 examined these plots and

selected whether the putative signal looked like a true transiting

planet candidate. Figure

shows an example set of diagnostic

plots.

On 2017 April 4, the Exoplanet Explorers project was

featured on the Stargazing Live ABC broadcast in Australia.

Between 2017 April 4 01:00 UTC and 2017 April 6 19:48

UTC, the live project received 2,100,643 classi

fi

cations from

7270 registered Zooniverse classi

fi

ers and 7677 not-logged-in

IP addresses. Of these, 130,365 classi

fi

cations from 4325

Figure 1.

The set of diagnostic plots presented on the Exoplanet 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 alternating 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

fi

tted

planet model is overlaid in blue.

www.planethunters.org

www.exoplanetexplorers.org

https:

archive.stsci.edu

kepler

https:

github.com

petigura

k2phot

https:

github.com

petigura

terra

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Christiansen et al.

registered classi

fi

ers and 2012 not-logged-in IP addresses were

for candidate signals in the C12 data. C12 candidates received

a median of 26 classi

fi

cations each; the C12 candidate with

the lowest classi

fi

cation count received 14 classi

fi

cations, and the

most-classi

fi

ed C12 candidate received 43 classi

fi

cations. The

classi

fi

cations were aggregated for each candidate to provide

the fraction of classi

fi

ers who indicated they saw a transiting

planet signal.

Of the 4900 potential transiting signals identi

fi

ed in the C12

data, 72 were voted by more than 60% of users as looking like

transiting planet candidates. The dispositions of the full set of

C12 signals are shown in Figure

; the signals with the highest

con

fi

dence are unsurprisingly 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 identi

fi

ed 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 participants; the four

diagnostic plots that were voted on are shown in Figure

EPIC



245950175 was proposed for observations by four teams,

in Guest Observing Programs 12049, 12071, 12083, and 12122

(

PIs Quintana, Charbonneau, Jensen, and Howard

)

. The full,

unfolded light curve of K2-138 is shown in Figure

. After the

system was

fl

agged by the citizen scientists, additional

examination of the light curve revealed the signature of a

fi

fth

transiting signal, interior to the four signals identi

fi

ed by the

TERRA algorithm and on the same 3:2 resonant chain.

Characterization of the

fi

ve detected planet signals is detailed in

Section

In addition, two individual transit events were identi

fi

using LcTools

(

Kipping et al.

2015

)

, shown in Figure

separated by 41.97 days. The transits have consistent depths

and durations, and if con

fi

rmed, would correspond to an

additional

∼

2.8

sub-Neptune planet in the K2-138 system,

bringing the total to six planets. Additional observations are

required to secure a third epoch and con

fi

rm that the two

transits seen in the

data arise from a single planet and are

not individual transits of two similarly sized, longer-period

planets.

4. Stellar Characterization

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

. Following the procedure in Cross

fi

eld 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

)

fi

nds mass and

radius uncertainty

fl

oors for similar spectral types of 6% and

9% respectively, motivated by comparisons between stellar

radii derived using

isochrones

and spectroscopic para-

meters

(

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 & Mamjek

(

2013

)

. We adopt a

spectral type of K1V



±



1. We measure a log



value of

−

4.63, indicating a modestly magnetically active star, which

may present a challenge for precision radial velocity measure-

ments of the system.

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 Cross

fi

eld

)

. We observed at

fi

dither positions and used the dithered images to remove sky

background and dark current; we then aligned,

fl

at-

fi

elded and

stacked the individual images. The inset in Figure

shows the

fi

nal stacked image, and the plot shows the detection limits of

the

fi

nal image. The limits were determined by injecting

simulated sources into the

fi

nal 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

fl

ux in the AO image within



, 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

Cross

fi

eld et al.

(

2016

)

. This method is sensitive to secondary

stars that lie within



of the primary star

(

one half of the slit

width

)

and that are up to 5 mag 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

3400 6100 K

eff

–

and

Δ

(

)



>



10 kms

−

We further discuss the possibility of the observed periodic signals

originating from a faint star

∼



away in Section

5.1

,butforthe

following analysis, we assume the putative planet signals arise

from K2-138.

Figure 2.

The distribution of user votes on the C12 candidate 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.

”

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5. Planet Parameters

We analyzed the

fi

ve transit signals independently in the

K2-138 light curve, using the same modeling,

fi

tting, and

MCMC procedures as described in Cross

fi

eld et al.

(

2016

)

.In

summary, we

fi

t the following model parameters: mid-transit

time

(

)

; the candidates orbital period and inclination

(

and

)

; the scaled semimajor axis

(

)

; the fractional candidate

size

(

*

)

; the orbital eccentricity and longitude of periastron

(

and

)

, the fractional level of dilution

(

)

from any other

sources in the aperture; a single multiplicative offset for the

absolute

fl

ux level; and quadratic limb-darkening coef

fi

cients

(

u1 and u2

)

. We explore the posterior distribution using the

emcee

software

(

Foreman-Mackey et al.

2013

)

.We

fi

nd that

the signals correspond to

fi

ve sub-Neptune-sized planets

ranging from 1.6 to 3.3

;

the best-

fi

tting transit models are

shown in Figure

. As a self-consistency check, we note that

the stellar density values derived from the independent transit

fi

ts are consistent across all

fi

ve planets and also consistent with

the direct calculation from the stellar mass and radius.

All

fi

ve 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

, is that each successive pair of

planets is just outside the

fi

rst-order 3:2 resonance.

5.1. Validation

Lissauer et al.

(

2012

)

analyzed the distribution of

Kepler

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

“

multi-

plicity boost

”

to the statistical validation of candidates in multi-

planet systems. Here, we validated each candidate individually

using the publicly available

vespa

code,

which computes the

Table 1

K2-138 Stellar Parameters

EPIC ID

245950175

2MASS ID

J23154776-1050590

R.A.

(

J2000.0

)

23:15:47.77

Decl.

(

J2000.0

)

−

10:50:58.91

(

mag

)

12.21

(

mag

)

10.305

Spectral type

K1V



±



eff

(

)

5378



±



log

(

cgs

)

4.59



±



0.07

[

]

0.16



±



0.04



(



)

0.86



±



0.08



(



)

0.93



±



0.06

Distance

(

)

183



±



sin

(

km s

−

)

2.7



±



1.5

Note.

EPIC classi

fi

cation, see Huber et al.

(

2016

)

and

https:

github.com

danxhuber

galclassify

Figure 5.

Inset: the Gemini

NIRI AO image of K2-138. We detect no additional

sources of

fl

ux. Plot: the 5

contrast limits for additional companions, in

Δ

magnitude, are plotted against angular separation in arcseconds; the black

points represent one step in the FWHM resolution of the images.

Figure 3.

The time series of the K2 data, with the

fi

ve-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

are shown as

“

”

The 5-day gap two-thirds of the way through the campaign was

the result of a spacecraft safe-mode event.

Figure 4.

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 demonstrate the consistency in depth and duration of the two events,

which correspond to a

∼

2.8

planet.

https:

github.com

timothydmorton

VESPA

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likelihood of various astrophysical false-positive scenarios. We

use as input a photometric exclusion radius of



,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 planets b, c, d, e, and f, respectively. As

we have multiple candidates orbiting a single star, the

“

multi-

plicity boost

”

(

and an additional

“

near-resonance boost

”

)

further

suppresses these FPPs

(

Lissauer et al.

2012

; Sinukoff et al.

2016

)

. Applying the

multiplicity boost derived by Sinukoff

et al.

(

2016

)

,we

fi

fi

nal FPPs of

8.3 10

.6 10

.17 10

1.1 10

,and

1.0 10

Recently, Cabrera et al.

(

2017

)

showed that stars within the

Kepler

photometry aperture but outside the small area surveyed

by high-resolution imaging were responsible for several falsely

validated planets. Here, we examine the possibility that the

fi

periodic signals do not arise from the brightest star in the

aperture. Given that the signals form an unbroken chain of near

fi

rst-order resonances, we consider the possibility that some

number of the signals arise on one star and the remainder on a

star coincident with the line of sight to be unlikely, and

consider the

fi

ve signals as a related set. The brightest nearby

star is 2MASS



J23154868-1050583, which is

∼



from

EPIC



245950175, and 5.6 mag fainter in the

-band. This star is

shown to the west of EPIC



245950175 in Figure

. Following

from Equation

(

)

of Ciardi et al.

(

2015

)

,we

fi

nd that the putative

planets would be 13.2 times larger if they orbited the fainter target,

increasing to 1.9

–

4.0

. 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

(

Smalley

et al.

2012

)

. Therefore, we conclude that the

fi

ve putative planets

are extremely unlikely to orbit 2MASS



J23154868-1050583.

6. 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

; Howard et al.

2012

;

Fabrycky et al.

2014

; Winn & Fabrycky

2015

)

. This has

continued in the

mission, including the discoveries of the

K2-3

(

Cross

fi

eld et al.

2015

)

, K2-37

(

Sinukoff et al.

2016

)

, and

K2-72

(

Cross

fi

eld 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

Figure 6.

The folded transits of K2-138 b, c, d, e, and f overplotted with the

best-

fi

tting transit model in red. Binned data points are shown in blue. The

planets range in size from 1.57 to 3.29

Figure 7.

A60





60 arcsec image from the SDSS DR7

-band. The companion

to the west is

∼

14 arcsec away, and is 5.6 mag fainter than EPIC 245950175 in

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of resonances. The

fi

ve validated planets of K2-138 lie close to

fi

rst-order resonant chain. We

fi

nd 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 of multi-transiting

planet systems in the

Kepler

data, showed that pair-wise period

ratios pile-up just outside of the

fi

rst- 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

fi

rst-order resonance for a pair of planets due to

Type I migration. Their Figure

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

–

c and c

–

d pairs may be captured in the 3:2 resonance, but

unlikely that the d

–

eore

–

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

shows, for the con

fi

rmed

multi-planet systems, the distance of each period 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 con

fi

guration

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 contain four planets in

different con

fi

gurations of

fi

rst-order resonances, also high-

lighted in Figure

. 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 ratios of 1.3333, 1.5021, and 1.3338 for the b

–

c, c

–

d, and

–

e pairs respectively. Kepler-223 demonstrates signi

fi

cant

transit timing variations

(

TTVs

)

, 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 resonant 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

Figure 8.

The distribution of distances from 3:2 period ratios in con

fi

rmed

multi-planet systems that have three or more planets in a compact geometry

(

fi

ned as having three planets with Period

Shortest Period

<

)

. Planetary

systems with multiple near-3:2 resonances are highlighted in red. K2-138 is the

only system near an unbroken chain of four near-3:2 resonances. Kepler-79 and

Kepler-223

(

shown in blue

)

both have four planets in or near a chain of

resonances. The vertical lines indicate the positions of successive 3:2 period

ratios.

The Astronomical Journal,

155:57

(

9pp

)

, 2018 February

Christiansen et al.

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

fi

ve-planet system where the four outer planets

form a tightly linked pair of three-body resonances

(

Lissauer et al.

2011

; MacDonald et al.

2016

)

; Kepler-223, described above, also

contains a pair of three-body resonances. One other system,

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

resonance or a chain of two-planet mean motion resonances

(

ź

dziewski et al.

2016

)

. A three-body resonance satis

fi

es the

condition that

pP p q P qP

123

-+ + »

(

)[(

) ]( )

,where

and

are integers and

the period of the

th planet. For K2-138,

fi

nd that the three consecutive sets of three planets

(

bcd, cde,

and def

)

all satisfy this condition with

,2,3

(

)( )

, resulting in

values of 4.2



±



1.7





−

days

−

1.6

-

0.9





−

days

−

and

2.4

-

4.9





−

days

−

respectively, all close to zero. In

simulating the Kepler-80 system, MacDonald et al.

(

2016

)

fi

that their migration simulations can naturally describe the

fi

nal

system architecture, with dissipa

tive 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 relati

vely modest population of known

systems with four or more planets in or close to a resonant chain,

and a very small population of systems with interlocking chains of

three-body resonances, making it an ideal target to study for TTVs.

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

shown in Figure

. For each transit, we

fi

x the model transit

parameters to the best-

fi

t values given in Table

, and allow

only the mid-transit time to vary. To calculate the uncertainties,

we compute the residuals from the best-

fi

t model and perform a

bootstrap analysis using the closest 100 timestamps, re-

fi

tting

the mid-transit time at each timestamp permutation

(

Wall

et al.

2003

)

. Examining the resulting transit times, we do not

fi

nd evidence of signi

fi

cant variations at the level of the average

–

10 minute timing precision from the

data. The individual

transits of K2-138



b have insuf

fi

cient signal-to-noise for robust

transit time calculation.

In order to estimate the amplitude of potential TTVs, we use

the mass

–

radius relation of Weiss & Marcy

(

2014

)

for planet

radii in the range 1.5

–

(

2.69

0.93

= ́

)

, predicting

that the

fi

ve planets have masses between 4 and 7

⊕

. Near

resonance, TTV amplitudes depend on planet masses, proxi-

mity to resonance, and orbital eccentricities. Using the

TTVFaster

code

(

Agol & Deck

2016

)

, we estimate

potential TTV amplitudes of 2.5, 5.1, 7.1, 6.9, and 4.8 minutes

for planets b, c, d, e, and f, respectively, assuming circular

orbits; for eccentric orbits, these amplitudes could be higher.

We can also estimate the

“

super-period

”

of the planets: when

two planets are close to resonance, their TTVs evolve on a

larger timescale referred to as the super-period. Using Equation

(

)

from

(

Lithwick et al.

2012

)

, we calculate super-periods of

139.4 day, 148.1 day, 144.7 day, and 144.2 day for the b

–

d, d

–

e, and e

–

f pairs, respectively. The

observations span

slightly more than half of this amount of time, but as shown,

the uncertainties on the measured transit times with the

processed photometry are large enough to swamp the

amplitude of the expected signal. However, with careful

sampling over a longer observing baseline and higher precision

photometry, the TTVs may be accessible. One possibility is the

NASA

Spitzer

telescope. For K2-18b, Benneke et al.

(

2017

)

measure a transit timing precision of

∼

0.9 minutes with

Spitzer

. Using the error approximation of Carter et al.

(

2008

)

and scaling for the properties of the K2-138 planets, we

estimate that

Spitzer

would achieve transit timing precision of

∼

2 minutes, which would be suf

fi

cient to measure the TTVs of

the outer planets. Another possibility for measuring TTVs is

the ESA CHEOPS mission

(

Broeg et al.

2013

)

, although

K2-138

(



=



12.2,



=



10.3

)

is at the faint end of their target

range.

The empirical relation of Weiss & Marcy

(

2014

)

disguises a

large scatter in the measured masses for planets ranging from

–

, spanning nearly an order of magnitude from roughly

–

⊕

(

see Figure 11 of Christiansen et al.

2017

)

. This

diversity is due to a wide, degenerate mix of rock, volatile, and

gas compositions that can comprise this size of planet.

Although the K2-138 planets do not demonstrate signi

fi

cant

TTVs in the

data, their masses may be accessible to radial

velocity observations. By comparing to the ensemble of mass

–

radius measurements to date, we estimate a minimum mass of

for the four outer planets, and therefore RV semi-

amplitudes of







−

. Achieving this precision on K2-138

may be a challenge, given the aforementioned stellar activity

level. If any of the planets are measured to be lower density,

and therefore likely volatile rich

(

such as the resonant planets in

Kepler-79

)

, they may be interesting yet challenging prospects

for atmosphere characterization, given the moderate brightness

of the host star.

7. Conclusions

We have presented K2-138, the

fi

rst discovery from the

citizen scientists participating in the Exoplanet Explorers

project. K2-138 is a compact system of

fi

ve sub-Neptune-

sized planets orbiting an early-K star in a chain of successive

near-

fi

rst-order resonances; in addition, the planets are locked

in a set of three-body Laplace resonances. The planets may be

accessible to mass measurement via dedicated radial velocity

monitoring, and possibly via TTVs with improved timing

Figure 9.

Examining the transit times of K2-138 c, d, e, and f. There are no

signi

fi

cant variations observed at the timing precision of the

30-minute

cadence observations.

https:

github.com

ericagol

TTVFaster

The Astronomical Journal,

155:57

(

9pp

)

, 2018 February

Christiansen et al.