The California Planet Survey III. A Possible 2:1 Resonance in the Exoplanetary Triple System HD 37124

2.2 MB·English

by  J. T. Wright

Draftversion January24,2011
PreprinttypesetusingLATEXstyleemulateapjv.11/10/09
THE CALIFORNIA PLANET SURVEY III. A POSSIBLE 2:1 RESONANCE IN THE EXOPLANETARY TRIPLE
SYSTEM HD 371241
J. T. Wright2,3, Dimitri Veras4, Eric B. Ford4, John Asher Johnson5, G. W. Marcy6, A. W. Howard6,7, H.
Isaacson6, D. A. Fischer, J. Spronck8, and J. Anderson9, J. Valenti9
Draft version January 24, 2011
ABSTRACT
1 We present new radial velocities from Keck Observatory and both Newtonian and Keplerian solu-
1 tionsforthetriple-planetsystemorbitingHD37124. Theorbitalsolutionforthesystemhasimproved
0 dramatically since the third planet was first reported in Vogt et al. (2005) with an ambiguous orbital
2 period. Wehaveresolvedthisambiguity,andshowthattheoutertwoplanetshaveanapparentperiod
commensurability of 2:1. A dynamical analysis finds both resonant and non-resonant configurations
n
consistent with the radial velocity data, and constrains the mutual inclinations of the planets to be
a
◦
J <∼30 .
WediscussHD37124inthecontextoftheother19exoplanetarysystemswithapparentperiodcom-
1
menserabilities,whichwesummarizeinatable. We showthatroughlyoneinthreewell-characterized
2
multiplanet systems has a apparent low-order period commensuribility, which is more than would

] na¨ıvely be expected if the periods of exoplanets in known multiplanet systems were drawn randomly
P from the observed distribution of planetary orbital periods.
E Subject headings: planetary systems — stars: individual (HD 37124)
.
h
p
1. INTRODUCTION ets with well-determined orbital paramaters: υ And
-
o Todate, over50exoplanetarysystems withmorethan (Butler et al. 1999), HIP 14810 (Wright et al. 2009a), µ
r one planet have been discovered,including: the extraor- Ara(Pepe et al.2007) andHD 37124(Vogt et al.2005).
st dinarydetectionsofthefirstexoplanetsorbitingthepul- HD 37124 (HIP 26381) is a 0.85 M⊙ metal-
a sar PSR B1257+12(Wolszczan & Frail 1992; Wolszczan poor ([Fe/H]=-0.44, Valenti & Fischer 2005) G4 dwarf
[ 1994); the imaged system orbiting HR 8799; those dis- (V=7.7). Vogt et al. (2000) announced the a Jovian,
3 coveredduringthemicrolensingeventOGLE-2006-BLG- P ∼ 150 d planet orbiting HD 37124 from HIRES
v 109L (Gaudi et al. 2008); several systems discovered by data taken at Keck Observatory as part of the Cali-
7 transit, including four or five10 multiply transiting sys- fornia and Carnegie Planet Search. Further monitor-
9 temsfromtheKeplermission(Steffen2010); and43sys- ing of the star revealed substantial long-term residuals.
0 temsdiscoveredbyradialvelocity(RV)searches(Wright Butler et al. (2003) fit these residuals with an eccentric,
1 1940dplanet,butnotedthatthesolutionwasnotunique
2009). The RV systems include the four-planet sys-
. (andGo´zdziewski(2003)showedthatthisfitwas,infact,
1 tems µ Ara (Santos et al. 2004; Pepe et al. 2007), GJ
unstable.)
0 581 (Mayor et al. 2009) and GJ 876 (Rivera et al. 2005,
After collecting two more years of data, Vogt et al.
1 2010) and the five-planet system orbiting 55 Cancri
(2005) was able to report the detection of a third planet
1 (Fischer et al. 2008). Of all these multplanet systems,
in the system, though with an ambiguity: while the b
: only four are known to host three or more giant11 plan-
v and c components had clearly defined periods, the d
i component could be fit nearly equally well with peri-
X
1Based on observations obtained at the W. M. Keck Obser- ods of either 2300 d or 29.32 d, the latter likely being
r vatory, whichisoperated jointlybytheUniversityofCalifornia an alias due to the lunar cycle.12 Wright (2009) re-
a and the CaliforniaInstitute of Technology. The Keck Observa-
ported that recent Keck velocities had resolved the am-
torywasmadepossiblebythegenerousfinancialsupportofthe
W.M.KeckFoundation. biguityqualitativelyinfavorofthelongerorbitalperiod.
2Department of Astronomy, 525 Davey Lab, ThePennsylva- Go´zdziewski,Konacki, & Maciejewski (2006) explored
niaStateUniversity,UniversityPark,PA16802 the many possible dynamical configurations consistent
3Center for Exoplanets and Habitable Worlds, The Pennsyl-
withtheVogt et al.(2005)velocities,includingmanyres-
vaniaStateUniversity,UniversityPark,PA16802
4DepartmentofAstronomy,UniversityofFlorida,211Bryant onantsolutions. Go´zdziewski,Breiter, & Borczyk(2008)
Space Science Center, P.O. Box 112055, Gainesville, FL 32611- used the system to demonstrate a fast MENGO algo-
2055
5Department of Astronomy, California Institute of Technol- rithm,buttheydidnotexplorethe 2:1resonance,asthe
data did not seem to favor it at the time.
ogy,MC249-17,Pasadena,CA
6DepartmentofAstronomy,601CampbellHall,Universityof WepresentnewKeckobservations,andthesedatapro-
California,Berkeley,CA94720-3411 vide a unique orbital solution for the outer planet. The
7Townes Postdoctoral Fellow, Space Sciences Laboratory,
UniversityofCalifornia,Berkeley
89ASpsatrcoenToemleyscDopepeaSrctmienencet,InYsatlietuUten,iv3e7r0s0ityS,aNneMwaHrtainveDnr,.C,BTal- 12 Time on the Keck telescopes dedicated to observing bright,
planetsearchtargetswithHIRESisusuallyassignedduringbright
timore,MD21218
orgraytime;theresultingscarcityofdatapointsduringnewmoon
10KOI877maybeablendoftwo,separatelytransitingsystems. can interact with planetary signals to create spurious, aliased so-
11 Msini>0.2MJup lutions.
2 Wright et al.
outer planet period we find is more consistent with the dom) errors, with no “jitter” included (Wright 2005).
originalperiod reportedby Butler et al. (2003) than the Thesevelocitiessupersedeourpreviouslypublishedve-
refined orbit of Vogt et al. (2005)13 (though we find a locities for this star, as we continue to refine our data
much lower eccentricity). Herein, we present the en- reduction pipeline. Our ever-evolving radial velocity
tire history of Keck velocities obtained for this star, and pipeline is descended in spirit and form from that de-
presentself-consistentorbitalsolutions showing that the scribed in Butler et al. (1996), but includes many small
outer two planets are in or very near a 2:1 mean-motion andlargetechnicalimprovements,athoroughdiscussion
resonance(MMR). This is the 20thexoplanetarysystem of which is beyond the scope of this manuscript. Some
to be found near an MMR, and only the tenth system details canbe found in § 4.1of Howard et al. (2010), § 3
with an apparent 2:1 commensurability. ofHoward et al.(2009),andinBatalhaetal.2011(ApJ,
Period commensurabilities (PCs) represent impor- accepted).
tant dynamical indicators in the Solar System and OnissueofinstantrelevanceisthatinAugust2004the
have been linked with observables and formation HIRES CCD detector was upgraded to a CCD mosaic.
mechanisms (Goldreich 1965). The near-5:2 PC TheoldTektronix2048EB2engineering-gradeCCDdis-
of Jupiter and Saturn, also known as “The Great playeda variable instrumental profile asymmetry due to
Inequality”, might be the remnant of a divergent a charge transfer inefficiency which manifested itself as
resonant crossing that produced the current archi- small changes in a star’s measured radial velocity as a
tecture of the outer Solar System, the Late Heavy function of exposure time (i.e. raw counts on the chip.)
Bombardment, and the Trojan Asteroids (Gomes et al. We apply an emperical, spectral-type dependent model
2005; Morbidelli et al. 2005; Tsiganis et al. 2005; to correct this effect for velocities measured prior to the
Tsiganis, Varvoglis, & Dvorak 2005). The populations detector change. The new CCD mosaic shows no evi-
of the asteroid belt and the Kuiper Belt, exemplified dence of this effect, but as a consequence of the switch
by the PC and near-PC-populated Kirkwood Gaps thereisasmallvelocityoffsetbetweendatasetsthatspan
(e.g. Tsiganis, Varvoglis,& Hadjidemetriou 2002) the the two detector sets similar to the detector-to-detector
Plutinos (3:2 PCs with Pluto and Neptune) and offsetsdiscussedinGregory & Fischer(2010). Theseoff-
the twotinos (2:1 PCs with Pluto and Neptune; e.g. sets could, in principle, be different for every target.
Murray-Clay & Chiang 2005; Chiang & Jordan 2002), AnalysisofRVstandardsandknownplanetarysystems
have implications for the migratory history of Jupiter show that such an offset is usually small – of order 5
and Neptune and the prospect of, e.g. secular reso- m/s – and very often consistent with zero. As a result,
nant sweeping (e.g., Nagasawa,Ida, & Bessho 2008). we report two independent data sets for this system in
Near-PCs found in satellite and ring systems have Table1,onefromeachofthetwodetectors. Wesolvefor
had direct observational consequences; the Saturnian the detector offset as an unconstrained free parameter.
satellite Pandora was ∼19◦ behind its predicted orbital The times of observation are given in JD-2440000.
longitude in a 1995 ring plane crossing (French et al. We fitted the data using the publicly available multi-
2003) due to its 121:118 PC with neighboring satellite planet RV-fitting IDL package RV FIT MP, described in
Prometheus. Wright & Howard (2009). In Table 2 we present our 3-
Byextension,wemayanticipatesimilarimportancein planet Keplerian (kinematic) fit,14 which yields r.m.s.
the growing number of exoplanetary systems exhibiting residuals of 4.4 ms−1, and we plot the fit and veloci-
PCs. In extrasolar systems, Mean Motion Resonances ties in Figure 1. We find a best-fit offset between CCDs
(MMRs) have been interpreted as the indication of of 4.8 ms−1. The orbital parameters and their uncer-
convergent migration in multi-planet systems, (e.g. tainties were determined from 100 bootstrapped trials
Thommes & Lissauer2003;Kley, Peitz, & Bryden2004; (as described in Marcy et al. 2005; Butler et al. 2006;
Papaloizou& Szuszkiewicz 2005). Several subsequent Wright et al.2007). Theorbitalfitsanddynamicalanal-
studies (Beaug´e, Michtchenko, & Ferraz-Mello 2006; ysis herein are put forth under the assumption that the
Terquem & Papaloizou 2007; Pierens & Nelson 2008; velocities are not detectably influenced by additional,
Podlewska & Szuszkiewicz2008,2009;Libert & Tsiganis unmodeled planets in the system. We have integrated
2009; Rein & Papaloizou 2009; Papaloizou& Terquem these orbital parameters for 10 Myr using the methods
2010; Rein, Papaloizou, & Kley 2010; Zhang & Zhou describedin§3assumingcoplanarity,andfoundthemto
2010a,b) exploring convergent migration for a variety of yield a stable configuration.
masses,separationsanddiskpropertieshavefoundmany The residuals to this fit have an RMS 4.03 m/s and
regionsofmassandorbitalelementphasespaceinwhich show with no significant periodogram peak at any pe-
planets are easily captured through this mechanism. riod. The tallest peak is at 3.81 days. We have run a
Monte Carlo FAP analysis on these residualsof our best
2. VELOCITIES AND ORBITAL SOLUTION fit for this tallest peak, and find similarly good fits in
50% of velocity-scrambled trials, consistent with noise.
Table 1 contains radial velocity measurements for HD
We thus conclude that our model is sufficient to explain
37124fromtheHIRESspectrograph(Vogt et al.1994)at
KeckObservatoryobtainedbytheCaliforniaPlanetSur- the data and that there are no other statistically signifi-
cant planetary signals detected.
vey consortium using the iodine technique (Butler et al.
1996). Notethatthe quotederrorsareourinternal(ran-
3. NEWTONIAN FITS AND STABILITY ANALYSIS
13 Vogtetal. (2005) opted to refer to the new, 840 d signal as
theccomponent,despitetheprior1940dfitofButleretal.(2003), 14 This solution is of similar quality to the best fit Newtonian
becausethatpriorfitwassospeculative,andbecausetheirnewfit solution, and is dynamicallystable. We consider itrepresentative
puttheveryexistenceofa1940dperiodicityinsomedoubt. oftheensembleofgoodNewtoniansolutions.
Possible 2:1 Resonance for HD 37124 3
Fig.1.—RadialvelocitycurvesfortheHD37124triplesystem.
60
)
s 40
/
m
(

y
20
t
i
c
o
l
e
v 0

l
a
i
d
a −20
R
−40
1996 1998 2000 2002 2004 2006 2008 2010
Date
s)
m/ 40 HD 37124 b, other components removed
P = 154.4 d
y ( 20
ocit 0
el
V −20
dial −40
a
R
−0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2
Phase
s)
m/ 40 HD 37124 c, other components removed
y ( 20
cit
o
el 0
V
al −20
di
a
R
1996 1998 2000 2002 2004 2006 2008 2010
Year
m/s) 30 HD 37124 d, other components removed
y ( 1200
cit
o 0
el
V −10
al −20
adi −30
R
1996 1998 2000 2002 2004 2006 2008 2010
Year
4 Wright et al.
TABLE 1 TABLE 2
Radial Velocities for HD37124 Best-Fit KinematicOrbitalElements forExoplanets inthe HD
37124System
Time Velocity Uncertainty CCD
JD-2440000 ms−1 ms−1 Parameter b c d
10420.04655 54.81 1.5 1 P (d) 154.378±0.089 885.5±5.1 1862±38
10546.73646 28.74 1.2 1 Tp (JD-2440000) 10305±11 9534±11 8558±11
10837.76625 6.32 1.6 1 e 0.054±0.028 0.125±0.055 0.16±0.14
10838.94868 6.38 1.7 1 ω (degrees) 130a 53±17 0a
10861.80464 17.89 1.4 1 K (ms−1) 28.50±0.78 15.4±1.2 12.8±1.3
11069.03617 -3.99 1.4 1 Msini(MJup) 0.675±0.017 0.652±0.052 0.696±0.059
11070.13190 -1.66 1.3 1 a(AU) 0.53364±0.00020 1.7100±0.0065 2.807±0.038
11071.11494 1.28 1.6 1 R.M.S. 4.03
11072.12947 -11.58 1.5 1 jitter 4m/s
11073.02962 -8.72 1.3 1 χ2 0.8
11172.89571 39.40 1.6 1 ν
11226.78065 -0.48 1.4 1 Nobs 66
11227.78167 -2.05 1.4 1
11228.74293 -11.01 1.3 1 a Orbit is consistent with circular, so errors in ω are large; see (see
11412.14161 -33.76 1.6 1 Butleretal.2006, forafullerexplanation.)
11543.98278 -31.36 1.4 1
11550.94262 -44.57 1.4 1
11551.94008 -46.20 1.5 1
3.1. MCMC analyses
11552.89162 -47.97 1.5 1
11580.76121 -36.28 1.8 1
We studied the dynamical stability of HD 37124 by
11581.83559 -36.17 1.4 1
11582.78849 -37.36 1.4 1 combining the radial velocity data with Markov Chain
11583.72387 -35.69 1.5 1 Monte Carlo (MCMC) analyses to obtain ensembles of
11884.04436 -14.73 1.6 1 masses, semimajor axes, eccentricities, and orbital an-
11900.03518 3.01 1.4 1
gles consistent with the RV data. These ensembles were
11974.80019 39.56 1.4 1
12007.74522 4.48 1.5 1 generated without regard to dynamical stability consid-
12242.99064 48.20 1.5 1 erations. We then imposed line-of-sight and relative in-
12333.94545 -19.13 1.7 1 clination distributions on these sets of parameters. By
12334.78556 -12.82 1.7 1
incorporating the unknown inclination parameters with
12536.12848 10.64 1.8 1
12537.08597 10.28 1.7 1 observation-derived parameters, we sampled the entire
12573.03767 27.38 1.6 1 phase space of orbital parameters. We subsequently ran
12574.99934 27.48 1.7 1 N-body simulations on each element in these ensembles
12576.02212 23.58 1.6 1
in order to assess each system’s stability and resonant
12600.99996 8.06 1.7 1
12602.03213 7.02 1.7 1 evolution. Our treatment follows that of Ford (2005,
12925.01639 13.03 1.7 1 2006); Veras & Ford (2009, 2010).
13044.77359 36.56 1.6 1 Inparticular,wecalculated5Markovchains,eachcon-
13045.74638 32.78 1.5 1
taining over 106 states. Each state includes the orbital
13072.85948 -2.85 1.7 1
13240.13983 -34.52 1.5 2 period (P), velocity amplitude (K), eccentricity (e), ar-
13302.13193 -4.28 1.6 2 gumentofpericentermeasuredfromtheplaneofthesky
13302.97959 -5.69 1.4 2
(ω),andmeananomalyatagivenepoch(u)forplanetsb,
13338.96446 15.77 1.2 2
13340.09520 16.45 1.5 2 c and d. The MCMC uses a standard Gaussian random
13368.88930 -6.51 1.0 2 walk proposal distribution and the Metropolis-Hastings
13369.78156 -6.60 1.0 2 algorithmfor accepting or rejecting eachproposalfor all
13425.87197 -39.98 1.4 2
modelparametersexceptcos(i ) andΩ. Since the ra-
13426.82980 -39.12 1.3 2 LOS
13428.77030 -36.85 1.3 2 dialvelocitysignatureisonlyweaklydependantonthese
13483.72749 19.49 1.0 2 values, cos(iLOS) and Ω were drawn randomly for each
13723.90630 -7.28 1.6 2 state. ThiscanstillbeconsideredaMarkovchain,asthe
13841.76427 34.09 1.4 2
proceduresatisfiestheMarkovcondition,i.e. thatatrial
14544.83023 25.40 1.6 2
14545.78169 26.99 1.4 2 state not depend on states other than the current state,
14546.78977 24.28 1.3 2 as well as the other conditions (time-homogeneous, irre-
14718.08322 43.60 1.6 2 ducible, aperiodic) to prove that the Markov chain will
14806.91704 -7.74 1.5 2
(eventually) converge to the posterior distribution.
14810.89031 -9.50 1.7 2
14838.94681 0.39 1.8 2 We imposed an isotropic distribution of line-of-sight
14864.95362 29.45 1.8 2 inclinations(i ) anda uniformsample oflongitude of
LOS
14929.76349 -5.49 1.7 2 ascending nodes (Ω) on our MCMC-derived initial con-
15135.00085 -21.29 1.6 2
ditions. The planet masses, m, and semimajor axes,
15172.99171 12.60 1.6 2
15229.78574 -12.32 1.6 2 a, were obtained from each set of (P,K,e,ω,i,Ω,u)
values from relations derived with a Jacobi coordinate
system (Lee & Peale 2003). The approximate range
of minimum masses obtained, in Jupiter masses, were:
0.60 . m sini < 0.72, 0.40 . m sini < 0.75, and
b b c c
0.55.m sini <0.90.
d d
We treated the both the offset between the chips and
Possible 2:1 Resonance for HD 37124 5
the jitter as free parameters.15 The 5th percentile, me- tire duration of our simulations. Below, we refer to this
dian,and95th percentileoffsetsbetweenthechipsinour value as a “libration RMS”.
ensemble were 3.16, 3.78, 4.62, and the median jitter we HD 37124 presents a clear initial choice of angles to
find to be 4 m/s. test for libration. As indicated by Fig. 2, the semimajor
axis ratioof planets c and d is suggestive ofa 2:1MMR.
Therefore,we sampled the following angles for libration:
3.2. Coplanar, Prograde Systems
Weintegrated850setsofinitialconditionsinthecopla-
φ ≡2λ −λ −̟ (1)
nar case with all three planets in prograde orbits by us- 1 d c c
ing the Burlish-Stoer integrator of Mercury (Chambers
1999) for 107 yr with an output interval of 104 yr. φ ≡2λ −λ −̟ (2)
2 d c d
We also incorporated the effects of general relativity in
the code, which could have profound consequences for and found that φ1 librates in 28/850 = 3.3% of cases,
multi-planet system stability (Veras & Ford 2010), al- while φ2 librates in 9/850=1.1% of cases. Further, the
thoughthe effectis likelyto be negligible in this system. systemsforwhichφ2 isresonantareasubsetofthosefor
We classified systems as “unstable” if, for any planet, which φ1 is resonant.
|a −a |/a > τ, where a , a and a repre- If we tighten our definition of resonance to include
max min 0 max min 0
sent the maximum, minimum and initial values of the only those systems with RMS resonant angles under
◦
semimajor axis, and τ =0.9. 70 , then no φ2 arguments are resonant. Under this
One may visualize a representativearchitectureof HD stricter definition, the φ1 arguments are only resonant
37124by comparing the semimajor axis and eccentricity in 14/850=1.6% of cases, and if we further tighten the
◦
ranges of all three planets. Figure 2 plots the observed libration criterion to an RMS of or 50 , then this num-
eccentricity vs. derived semimajor axis for all planets in ber decreases to 4/850 = 0.5%. The lowest libration
◦ ◦
theprogradecoplanarstate. Blackdotsindicateunstable RMS detected is 23.0 . All RMS’s under 75 were for a
◦
systemswhilegreensquaresandredcrossesindicatesta- librationcenterof0 . Figure3illustratesthreeexamples
ble systems, and red crosses indicate systems which are of “resonant”systems from this, eachwith a different li-
ina2:1meanmotionresonance(MMR)betweenplanets bration RMS.
c and d, according to our definition below. The figure We additionally sampled all 3 pairs of apsidal angles
indicates i) a closely packed system, with the inner and (the difference between two longitudes of pericenter) in
outer planets separated by no more than six times the the coplanar prograde state, and found only two in-
◦
innermost planet’s semimajor axis. ii) a relatively circu- stancesoflibration,bothathigh(>70 )librationRMS’s
◦ ◦
lar innermost planet (with e . 0.1 in most cases) that and around the “asymmetric” centers 90 and 270 for
b
is likely too far from the parent star to be classified as the inner and outer planet apsidal angle. Inspection re-
a “Hot Jupiter”, iii) the greater the number of orbital veals,however,thattheseinstancesoflibrationaremore
periods sampled by RV, the greater the constraint on indicative of long period (>10 Myr) circulation.
the planet’s likely semimajor axes and eccentricities, iv) Additionally,thesemimajoraxisratioofplanetsband
most(664/850=78%)currentorbitalfitspredictanun- c could indicate the presence of a 6:1 MMR. Therefore,
stablesystem,v)themajorityofinitialconditionswhich we sampled all angles of the form6λd−λc−t̟c−s̟d,
produce stable orbits containan outer planet with a low where t + s = 5. None of the coplanar prograde sys-
(< 0.2) eccentricity, and a middle planet with a semi- temsexhibited librationofanyofthe 6:1anglesbetween
major axis > 1.695 AU and eccentricity less than about planets “b” and “c” over 10 Myr. However, preliminary
0.2., vi) systems containing a 2:1 resonance occur only samplingof these angles overintervals of2 Myr does oc-
◦
when 2.7 AU . a . 2.8 AU. We emphasize that this casionally exhibit libration RMS’s close to 90 . Because
3
approximate semimajor axis range appears to be neces- the period ratios between planets c and d may skirt the
sarybutnotsufficientforresonanceto occur. Thefigure 7:3PC,we alsotestedthe 7λd−3λc−t̟c−s̟d angles,
demonstrates that other MCMC fits with outer planet where t+s=4, but found no instances of libration.
periods inthe resonantrangeareeither unstable, orsta-
ble but non-resonant. The architecture of these systems 3.3. Mutually Inclined Systems
(as defined by, e.g., the mean longitude and longitude Having analyzed the coplanar prograde bin, we can
of pericenter) do not allow them to settle into resonance now consider the case where the planets have nonzero
even though the outer planet period might favor reso- mutual inclinations. We used rejection sampling to ob-
nance. taintripletsofi valuessuchthatthesystemisplaced
LOS
Because of finite sampling, our definition of “reso- into one of 144 bins according to the relative inclination
nance” in this analysis comes from consideration of the between planets b and c (i ) and planets c and d
rel,b,c
RMS deviation of each resonant angle about each of (i ). In no two bins were the same ensemble of ini-
◦ ◦ ◦ ◦ rel,c,d
(0 ,90 ,180 ,270 ), which includes common libration tial conditions used. We binned relative inclinations in
centers. We flag systems as “resonant” if at least one intervals of 15◦, and used stratified sampling in order to
◦
of these angles has RMS under 90 for 10 Myr, the en- obtainauniformnumberofsamplesperbin. Weinitially
sampled100initialsystemstatesperbin. Forthosebins
15Weadoptedasinglevalueofjitterforallobservations;inprin- where we found more than one system to be stable, we
ciplethe twoCCDsmaydisplaydifferingamounts of“instrumen- added 200 additional ensembles of initial conditions
tal jitter”, such as that due to insufficient modeling of the charge
By considering the fraction of stable systems in the
transfer inefficiencies. The RMS residuals to our fit for the two
detectors were3.67and4.11m/s,suggestingthatourassumption non-coplanar cases, we can obtain a broader dynamical
ofasinglejittervalueisvalid. portrait of this system. Fig. 4 illustrates the fraction of
6 Wright et al.

0.70
0.56
0.42
y
cit
ri
nt
e
c
c
E

0.28
0.14
0.00
0.5328 0.5344 2.69 2.94

1.674 1.742
Semimajor Axes (AU)
Fig.2.— Representative eccentricities and semimajor axes of the planets in HD 37124. The three planets are partitioned by panels,
eachwithadifferenthorizontalscale. TheseensemblesofparametersarederivedfromRVobservations usingMarkovChainMonteCarlo
(MCMC)techniques,andrepresenttheinitialconditionsforasubsetofournumericalsimulations(here,thecoplanarprogradesimulations).
Note that the semimajor axis ratio of the middle and outer planets roughly correspond to a 2:1 period commensurability, and the inner
andmiddleplanetstoa6:1PC.Notethechangingscaleonthex-axesinthethreepanels: theinnermostplanetisverywellconstrainedin
a,theoutermostplanetmuchlessso. TheBlackdotsindicateunstablesystems,greensquaresrepresentnon-resonantstablesystems,and
redx’sarestableresonant systems. Note that system stabilityisstronglydependent onthe eccentricity of themiddleandouter planets,
andtheouterplanetofresonantsystemstendtoharborthesmallestinitialsemimajoraxesoftheensembleofouterplanetICs.
◦
stable systems in each bin overall (top panel) and with libration of φ and φ under 90 for 10 Myr is given by
1 2
respectto allsystemsfor whichthe initial e <0.2(bot- Fig. 5. 2:1 resonant systems occur, therefore, generally
d
tom panel). This cutoff was motivated by the rightmost atthefewpercentlevel,andmostlikelywhenallplanets
panel in Fig. 2 and could suggest a constraint on the are coplanar with prograde orbits.
orbital properties of the system in order to ensure that
it remains stable. Fig. 4 shows that the system must 4. DISCUSSION
be roughly coplanar, with relative inclinations less than As Rivera et al. (2010) showedfor the GJ 876 system,
◦ ◦
∼ 30 −45 , in order to be stable. This constraint al- eventhemostwell-establishedanddeepestmean-motion
lowsvariouspairsofplanets to harborretrogradeorbits. resonances can prove illusory if additional planets are
We also performed limited resonant testing for systems found in the system (although in that case it appears
inthesebins. Thefractionoftotalsystemswhichexhibit thatthe resonancestillpresent,albietconsiderablyshal-
Possible 2:1 Resonance for HD 37124 7

360
300
)
g
e
d 240
(
e
gl
n
A 180
nt
a
n
o 120
s
e
R
60
0
0.00 2.00 4.00 6.00 8.00 10.00
Time (Myr)

360
300
)
g
e
d 240
(
e
gl
n
A 180
nt
a
n
o 120
s
e
R
60
0
0.00 2.00 4.00 6.00 8.00 10.00
Time (Myr)

360
300
)
g
e
d 240
(
e
gl
n
A 180
nt
a
n
o 120
s
e
R
60
0
0.00 2.00 4.00 6.00 8.00 10.00
Time (Myr)
Fig.3.— Three examples of systems that we find to be resonant, according to our definition requiringan librationof under 90◦ for 10
Myr. Plottedisthetimeevolutionoftheresonantangle2λd−λc−̟c forasystemwithacomputedlibrationRMSof(upperpanel)23.0◦
about0◦,(middlepanel)70.2◦ about0◦,and(lowerpanel)86.1◦ about180◦.
8 Wright et al.

165-180
150-165

135-150
150-165
120-135
g)
e 9999999999900000000000%%%%%%%%%%% ----------- 111111111110000000000000000000000%%%%%%%%%%%
d] (d105-120 135-150
n [c, 90-105 120-135 8888888888800000000000%%%%%%%%%%% ----------- 9999999999900000000000%%%%%%%%%%%
o
ati
clin 75-90 g)105-120 7777777777700000000000%%%%%%%%%%% ----------- 8888888888800000000000%%%%%%%%%%%
Relative In 4650--6705 n [c,d] (de 90-105 6666666666600000000000%%%%%%%%%%% ----------- 7777777777700000000000%%%%%%%%%%%
o 5555555555500000000000%%%%%%%%%%% ----------- 6666666666600000000000%%%%%%%%%%%
ati 75-90
30-45 nclin 4444444444400000000000%%%%%%%%%%% ----------- 5555555555500000000000%%%%%%%%%%%
15-30 ve I 60-75
ati 3333333333300000000000%%%%%%%%%%% ----------- 4444444444400000000000%%%%%%%%%%%
0-15 Rel 45-60
2222222222200000000000%%%%%%%%%%% ----------- 3333333333300000000000%%%%%%%%%%%
30-45
0-15 30-45 60-75 90-105 120-135 150-165
Relative Inclination [b,c] (deg)
1111111111100000000000%%%%%%%%%%% ----------- 2222222222200000000000%%%%%%%%%%%
15-30

11111111111%%%%%%%%%%% ----------- 1111111111100000000000%%%%%%%%%%%
0-15
165-180
0-0 00000000000%%%%%%%%%%%
45-60 75-90 105-120
150-165
Relative Inclination [b,c] (deg)
135-150
120-135
g)
e
d
d] (105-120
c,
n [ 90-105
o
ati
clin 75-90
n
ve I 60-75
ati
el
R 45-60
30-45
15-30
0-15
0-15 30-45 60-75 90-105 120-135 150-165
Relative Inclination [b,c] (deg)
Fig.4.—TwostabilityportraitsofHD37124. Eachbinindicatesthefractionofstablesystemsafter10Myrforallsystems(toppallete)
andforsystemswithaninitialed <0.2(bottom pallete). Notethatnearlyallnon-coplanar systemsareunstable (indicatedbythewhite
spaces),inbothvariousprogradeandretrogradecases. Theseportraitscanprovideausefulconstraintontheviablerelativeinclinationsin
HD37124. Note,however,thatthemutualinclinationofplanetsbanddarenotrepresentedonthisplotandareonlyweaklyconstrained
bytheothertwoinclinationpairs(e.g. iftwopairsaremutuallyinclinedby30degreeseach,thenthethirdpairmaybemutuallyinclined
anywherebetween0and60degrees).
Possible 2:1 Resonance for HD 37124 9

165-180
150-165

135-150
150-165
120-135
g)
e 33333333333...........3333333333333333333333%%%%%%%%%%%
d] (d105-120 135-150
n [c, 90-105 120-135 33333333333...........0000000000000000000000%%%%%%%%%%%
o
ati
clin 75-90 g)105-120 22222222222...........6666666666677777777777%%%%%%%%%%%
Relative In 4650--6705 n [c,d] (de 90-105 22222222222...........3333333333333333333333%%%%%%%%%%%
o 22222222222...........0000000000000000000000%%%%%%%%%%%
ati 75-90
30-45 nclin 11111111111...........6666666666677777777777%%%%%%%%%%%
15-30 ve I 60-75
ati 11111111111...........3333333333333333333333%%%%%%%%%%%
0-15 Rel 45-60
11111111111...........0000000000000000000000%%%%%%%%%%%
30-45
0-15 30-45 60-75 90-105 120-135 150-165
Relative Inclination [b,c] (deg)
00000000000...........6666666666677777777777%%%%%%%%%%%
15-30

00000000000...........3333333333333333333333%%%%%%%%%%%
0-15
165-180
0-0 00000000000...........0000000000000000000000%%%%%%%%%%%
45-60 75-90 105-120
150-165
Relative Inclination [b,c] (deg)
135-150
120-135
g)
e
d
d] (105-120
c,
n [ 90-105
o
ati
clin 75-90
n
ve I 60-75
ati
el
R 45-60
30-45
15-30
0-15
0-15 30-45 60-75 90-105 120-135 150-165
Relative Inclination [b,c] (deg)
Fig. 5.— Two resonant portraits of HD 37124. The legend indicates the fraction of systems for which the angles φ1 ≡2λd−λc−̟c
(upper panel) andφ1≡2λd−λc−̟d (lowerpanel) areresonant. Wedefine “resonant” asthe situationwheretheRMSdeviation ofan
angle about a fixed value is less than 90◦ over 10 Myr. In most cases, this deviation is between 70◦ - 90◦, but goes as low as 23◦. Note
thatonlyforthenear-coplanarcasesareanysystemsresonant.
10 Wright et al.
lowerandmorecomplexthanpreviouslythought). Even (again subject to the constraint that no pair of planets
fortrulyresonantsystems,ademonstrationofresonance in the system have r < 1.3). We found 16% of these
can be difficult. For instance, triple-planet systems may artificial systems passed our apparent PC criterion, re-
feature two planets with a mostly-librating resonant ar- flecting the higher number of planet pairs available to
gument that occasionally circulates due to interactions test per star compared to our first test. Despite this in-
with the third planet. Near separatrix behavior (as in flation, the actual value of 33% among all multiplanet
thecaseofυAnd;Malhotra2002;Ford, Lystad, & Rasio systems is still significantly higher. 18 These results
2005) can also make libration and circulation essentially underscore that the orbital periods of the population of
indistinguishable. planets known to be in multiplanet systems is inconsis-
Wenotethatnear-resonantbehaviorcanitselfdynam- tent with the apparently-singletonsample (Wright et al.
ically interesting: the 5:2 near-resonance of Jupiter and 2009b).
Saturn (the Great Inequality) has major consequences This apparently high percentage of known multi-
for the dynamics of the Solar System. Given the above- planet systems with an apparent PC might favor par-
mentioned difficulties in proving that a resonant argu- ticular formation mechanisms. Planet-planet scatter-
ment for a given system of planets satisfies some pre- ing, planetesimal disk migration, and gas disk migra-
cisely specified definition of libration given the typical tion have all been shown to produce systems with at
uncertainties in radial velocity measurements, we sug- least one pair of planets that not only are commensu-
gest that studies of resonant interactions would benefit rateinperiod,but alsoresonant. Raymond et al.(2008)
fromidentifyingsystemsthatappeartobeinornearres- found that planet-planet scattering produced MMRs in
onance (apparent period commensuribilities). With that roughly 5% of the systems that they simulated, and
in mind, we note that in addition to HD 37124, there Raymond, Armitage, & Gorelick (2010) discovered that
are 21 other systems in the Exoplanet Orbit Database between 50% - 80% of systems undergoing planetesimal
(Wrightetal.2010,PASPsubmitted)ofpeer-reviewed16 disk migration yielded resonant capture. Convergent
literature with well-established apparent PCs, which we gas disk migration, the thrust of the numerous papers
present in Table 3. This list includes all pairs of plan- cited in §1 can occur with near 100% efficiency for cer-
ets for which the period ratio r is less than 6 and within tain initial planetary and disk parameters. As observed
0.05ofanintegerorhalfinteger(neglectinguncertainties by Thommes & Lissauer (2003) and Libert & Tsiganis
in periods), and other exoplanetary pairs whose likely (2009), the inclination may be excited as well as the ec-
MMRs are discussed in the literature. centricityinmanyresonantcases. Ifaresonanceexistsin
The fraction of known multiplanet systems exhibiting HD 37124, it could have been produced by any of these
at least one apparent PC is high. Of the 43 well de- methods. IftheRMSlibrationofsucharesonanceisrep-
termined multiplanet systems discovered by radial ve- resentativeofthe bottompanelofFig.3andhasavalue
locities around normal stars, 15 appear in Table 3, or that approaches 90 degrees, then planet-planet scatter-
35%, including 9 of the 30 apparent double-planet sys- ingisalikelyoriginofthisresonance. Alternatively,disk
tems 30%.17 To determine if this is more than would be or gas migration would likely produce a system that is
expectedsimplybychance,wehaveperformedtwotests. ”deeper” in resonance, with a smaller variation in reso-
In the first test, we randomly drew pairs of periods nant angle, similar to the top panel of Fig. 3. Resonant
fromthe340RV-discoveredplanetsintheExoplanetOr- librating angles need not involve the eccentricities and
bit Database (EOD) (Wright et al. 2010, PASP submit- pericenters, but instead the inclinations and longitudes
ted), and rejected those pairs with period ratios r <1.3 ofascending nodes, similar to the 4:2 Mimas-Tethys res-
(corresponding to the smallest r among real multiplanet onance in the Saturnian system (Champenois & Vienne
systems). We counted the fraction of remaining systems 1999b,a).
with r within 0.05 of an integer or half integer ≤5 (cor-
responding to the largest apparent PC in Table 3). We 5. CONCLUSIONS
found that only 4% of our random pairs satisfy our ap-
We have resolved the period ambiguity of HD 37124
parentPC criterion,far smaller than the 30%of double-
d from Vogt et al. (2005) and find that HD 37124 c and
planet systems actually found in apparent PCs.
d are in an apparent 2:1 period commensurability. Our
Inthesecondtest,weincludedtheeffectsoftripleand
numericalintegrationsshowthatbothresonantandnon-
higher-multiple systems by randomly assigning periods
resonantconfigurationsareconsistentwiththeradialve-
from the EOD to all planets in real multiple systems
locity data, and that stability requires a nearly circular
16 WehaveincludedtheKeplermultiplanetsystems,whichhad orbit (e <0.3) for the d component. Our stability anal-
ysisshowsthatthe systemmustbe nearlycoplanar,and
notyetbeenaccepted forpublicationatthetimeofthiswriting.
17 We have excluded in this statistic the Kepler systems, the that the three planets have identical minimum masses
pulsarsystem,themicrolensingsystem,planets fromdirectimag- within the errors (of 3–10%).
ing, and the Solar System. We acknowledge that a more rigor- We show that the roughly one in three well-
ous statistic would be valuable, but note that it would need to
characterizedmultiplanetsystemsshowsanapparentpe-
address some strong detectability and selection effects regarding
planets in multiple planet systems, and to assess these detection riod commensurability, which is more than a na¨ıve esti-
thresholds across multiple, heterogeneous surveys. To give just matebasedonrandomlydrawingperiodsfromtheknown
one example, we note that in addition to a radial velocity sur- exoplanetpopulationwouldsuggest. Thisoffersevidence
vey’s decreasing sensitivity to planets in longer periods, it can
be difficult to detect an interior planet in a 2:1 resonance due
to approximate degeneracy with eccentricity in a single planet 18Wesuspectthatthereasontheobservedvalueisnotsimilarly
model (e.g., Anglada-Escud´e, L´opez-Morales,&Chambers 2010; inflatedwithrespecttodoubleplanetsystemsisthatourrandom-
Moorhead&Ford 2010). Such an analysis is beyond the scope ization did not include the requirement of dynamical stability, as
ofthismanuscript. realsystemsimplicitlydo.