Reanalysis of radial velocity data from the resonant planetary system HD128311

0.77 MB·English

by  Hanno Rein

Mon.Not.R.Astron.Soc.000,000–000(0000) Printed7January2015 (MNLATEXstylefilev2.2)
Reanalysis of radial velocity data from the resonant planetary
system HD128311
Hanno Rein1
1UniversityofToronto,DepartmentofEnvironmentalandPhysicalSciences,1265MilitaryTrail,TorontoOntarioM1C1A4
e-mail:[email protected]
5
Submitted:25November2014—Revised:13December2014–Accepted:22December2014
1
0 ABSTRACT
2 The multi-planetary system HD128311 hosts at least two planets. Its dynamical formation

n history has been studied extensively in the literature. We reanalyse the latest radial velocity
a dataforthissystemwiththeaffine-invariantMarkovchainMonteCarlosamplerEMCEE.Us-
J ingthehighorderintegratorIAS15,weperformafullydynamicalfit,allowingtheplanetsto

5 interactduringthesamplingprocess.AstabilityanalysisusingtheMEGNOindicatorreveals
thatthesystemislocatedinastableislandoftheparameterspace.Incontrasttoaprevious

] study,wefindthatthesystemislockedina2:1meanmotionresonance.Theresonantangle
P
ϕ is librating with a libration amplitude of approximately 37◦. The existence of mean mo-
E 1
tionresonanceshasimportantimplicationforplanetformationtheories.Ourresultsconfirm
.
h predictionsofmodelsinvolvingplanetmigrationandstochasticforces.
p
- Keywords: methods:numerical—gravitation—planetsandsatellites:dynamicalevolution
o andstability—stars:individual(HD128311)
r
t
s
a
[

1 THEPLANETARYSYSTEMHD128311 a system in a MMR favours the idea that giant planets migrated
1
when they were still embedded in a protoplanetary disc (Lee &
v
ThefirstplanetaroundHD128311wasdiscoveredbyButleretal. Peale2002;Rein&Papaloizou2009).
0
8 (2003). A second planet was found two years later (Vogt et al. Inthisletterwereanalysethecombinedradialvelocity(RV)
9 2005). Both planets are most likely gas giants with minimum datasets including the recalibrated data from the Hobby-Eberly
0 masses of 1.8 Mjup and 3.2 Mjup. A third signal has been discov- Telescope(HET)andtheLickObservatory.OurRVdataisthereby
0 eredbyMcArthuretal.(2014)butitsplanetarynaturehasyettobe equivalent to that of McArthur et al. (2014) but we do not take
. confirmed. Soon after its discovery HD128311 began to emerge intoaccounttheastrometricobservationsinourmodelwhichonly
1
0 as a test bed of planet formation scenarios within the scientific constrain the system’s inclination. In contrast to previous stud-
5 community. A large number of groups studied the formation and ies,weuseafullydynamicalmodel,allowingplanetstointeract,
1 evolutionofthissystemwithparticularfocusonthesystem’sdy- ratherthanbeingonfixedKeplerianorbits.Weuseauseamod-
: namicalhistory(Beauge´etal.2006;Quillen2006;Sa´ndor&Kley ernMarkovchainMonteCarlosamplertoexplorethehighdimen-
v
2006;Sa´ndoretal.2007;Michtchenkoetal.2008;Meschiarietal. sional parameters space and converge on a new set of orbital pa-
i
X 2009; Crida et al. 2008; Lee & Thommes 2009; Voyatzis et al. rameters.ThedetailsaredescribedinSection2.Wethenperform
r 2009; Barnes & Greenberg 2006; Raymond et al. 2008; Rein & longtermorbitintegrationstostudythestabilityinSection3us-
a Papaloizou2009;Lecoanetetal.2009;Gayon-Markt&Bois2009; ing the fast chaos indictor MEGNO (Cincotta & Simo´ 2000). In
Goz´dziewski & Konacki 2006; E´rdi et al. 2007; Giuppone et al. Section4weconcludebydiscussingtheresonantnatureofthesys-
2010). temandtheimplicationsforplanetformationscenariosinvolving
The most important dynamical feature of HD128311 is its stochasticmigration.
proximitytoa2:1meanmotionresonance(MMR).Earlyobserva-
tionssupportedtheideathatHD128311isinresonance,although
some of the orbital solutions were dynamically unstable on short
2 MARKOVCHAINMONTECARLO
timescales(Vogtetal.2005).ThemostrecentworkonHD128311
byMcArthuretal.(2014)includesnewradialvelocitydata,are- We use a Markov chain Monte Carlo method to fit the observed
calibrationofolderdatasetsandastrometricandphotometriccon- radial velocity datasets. In comparison to the fit performed by
straints.Theauthorsfoundthattheplanetsintheirbestfitmodel McArthuretal.(2014)weuseafullydynamicaltwoplanetmodel.
arenotinaMMR.WhetherthissystemisinaMMRornotisan Forthispurpose,wecouplethehighorderIAS15(Rein&Spiegel
important constraint for planet formation scenarios. For example 2015) integrator which is available within the REBOUND package
(cid:13)c 0000RAS
2 HannoRein
150
m/s] 100
city [ 50
o
vel 0
al
adi 50
r
100
s]
m/ 20
al [ 0
u
sid 20
e
r
0 1000 2000 3000 4000
time [days]
Figure1. Radialvelocitydataandmodel.ThetimeisrelativetoJD2450983.83.Toppanel:RVdatapoints,MCMCmean(red),MCMCsamples(gray).
Bottompanel:residualerrorsnotincludingjitters.
(Rein&Liu2012)totheEMCEEpackage.EMCEEisanopen-source, linescorrespondto50randomlydrawnsamplesfromtheMCMC
parallelized affine-invariant Marcov chain Monte Carlo sampler posteriordistribution.
writteninpython(Foreman-Mackeyetal.2013).
We assume co-planarity of the planets but allow the orbital
planetobeinclinedfromthelineofsightbyananglei.Thecen- 3 LONGTERMSTABILITY
tral object has a fixed mass of M = 0.828M . We have four or-
∗ (cid:12)
bital elements and one mass parameter, msin(i), for each planet. We focus on parameters close to those of the converged MCMC
TheorbitalparametersaretheperiodP,theeccentricitye,thear- samples and run a total of 57600 realisations of HD128311 for
gumentofperiapsisωandthemeananomalyM.Forthesampling tmax = 105 years to map out the structure of the phase space. A
process, we perform a coordinate transformation to h = esin(ω) smaller subset was integrated for tmax = 106 years, but no qual-
andk = ecos(ω)variables.Thisallowsustoavoidthecoordinate itative differences could be observed. As in the previous section
singularityate=0andspeedupconvergence(foradiscussionsee weuseREBOUNDandthehighorderIAS15integratorfortheselong
Houetal.2012).WeuseaJacobicoordinates,i.e.theouterplanet’s term integrations. We declarea system unstable if at least one of
orbitalparametersaregivenwithrespecttothecentreofmassof the planets gets ejected, if the planets have a close encounter, or
thestarandtheinnerplanet.BuildingupontheworkofMcArthur ifthesemi-majoraxisofatleastoneplanetchangesbymorethan
etal.(2014),wefurtherincludeanindividualoffsetγandjitterpa- 50%.WerefertothisdefinitionasLagrangestabilityandcallthe
rameter sforeachRVdataset(fourforLick,oneforHET).This timeuntilinstabilityLagrangetimescale.
allowsthemodeltoaccountforvariationsintheoffsetsfordiffer- For systems that are Lagrange stable, we measure the Mean
entinstrumentsaswellasvariationsintheLickdetectorsbetween ExponentialGrowthofNearbyOrbits,MEGNO(Cincotta&Simo´
observingruns.Thetotalnumberoffreeparametersisthus21. 2000). MEGNO is a fast chaos indicator similar to the tradi-
tionalLyapunovexponent,butgivesmoreusefulresultsonshorter
Atotalof500walkersareevolvedforseveralthousandgener-
timescales.WecomputetheMEGNOvalue(cid:104)Y(cid:105)byintegratingthe
ations.Bothanglesωand Maswellastheeccentricityeforeach
variationalequations(Mikkola&Innanen1999)usingIAS15.Fora
planethaveauniformpriors.WefollowHouetal.(2012)anduse
definitionof(cid:104)Y(cid:105)andadetaileddiscussionoftheMEGNOindicator
uninformativeJefferyspriorsfortheperiod,theplanetmassaswell
seeCincotta&Simo´(2000)andGoz´dziewskietal.(2001)1.
asthejitterparameters.Weinitializethewalkerswithmassandpe-
In Figure 2 we show a slice of the parameter space in the
riodparametersthatareroughlythoseofpreviousresultstospeed
planeoftheouterplanet’speriodandeccentricity, P ande .All
upconvergence.Ourexperimentsshowedthattheprecisedetailson 2 2
other orbital parameters are those listed in Table 1. All simula-
howthewalkersareinitializeddonotmatter.Finally,weperform
tions in red regions are Lagrange unstable. The shade of red in-
asimpledeclusteringalgorithmafter1000generationsbyremov-
dicatestheLagrangetimescale.Simulationsinblueregionsremain
ingthosesamplesfromtheensemblethathavealikelihoodsignifi-
stablefortheentireintegration.Theshadeofbluecorrespondsto
cantlysmallerthanthebestsamples(Houetal.2012).
theMEGNO(cid:104)Y(cid:105).Avalueof(cid:104)Y(cid:105)=2(darkblue)indicatesastable
AftertheMCMChasconvergedandhasbeendeclustered,we quasi-periodicorbit,whereasavaluelargerthan2indicateschaotic
sampleitover1250generationsandcalculatethemeanofallpa- behaviour.Onecanseeinthefigurethatseeminglystablesystems
rametersaswellasthetwosigmaconfidenceintervals.Theresults
are listed in Table 1. In Figure 1 we plot the observed radial ve-
locitydatapointswiththeoffsetadjustedaccordingtoourmodel. 1 Note that there is a typo in the denominator of δ˙ in Equation 5 of
TheredlinecorrespondstothemeansolutionofTable1.Thegray Goz´dziewskietal.(2001).
(cid:13)c 0000RAS,MNRAS000,000–000
ReanalysisofradialvelocitydatafromtheresonantplanetarysystemHD128311 3
0.6 105 4
0.5
104
rs] Y fi
y ›
0.4 e [ or
m at
103 y ti dic
e2 0.3 stabilit 3 bility in
n a
102 e i st
g O
0.2 ran GN
g E
a M
L
101
0.1
2
100
700 800 900 1000 1100 1200
P [days]
2
Figure2. LagrangestabilityplotinthevicinityoftheMCMCsolution.Theaxescorrespondtotheperiodandeccentricityoftheouterplanet.Redregionsare
Lagrangeunstable.Blueregionsarestableforthedurationoftheintegration.Darkblueregionsarestablequasi-periodicsolutionsaccordingtotheMEGNO
indicator.ThewhitecrossshowsthenominalmeanorbitalparameterslistedinTable1.TheblackdotsshowsamplesoftheMCMCposterior.
whichareclosetothestabilityboundaryareinfactchaoticandthus TheexistenceofaMMRisanimportantobservableinmany
mightbecomeunstableoverlongertimescales. planetformationscenarios.AsysteminaMMRsupportstheidea
ThewhitecrossinFigure2showsthemeanorbitalparameters that giant planets migrated into their current position while they
oftheMCMCsample.TheblackdotsshowtheindividualMCMC werestillembeddedinaprotoplanetarydisc(Lee&Peale2002).
samplesfromtheposteriordistribution.Boththemeansolutionand Furthermore,Rein&Papaloizou(2009)predictedthatifthesys-
allMCMsamplesarelocatedwellwithinastableisland.Notethat tem did undergo a phase of stochastic migration, then ϕ should
1
theeccentricitye isnotwellconstrained.However,theobserved librate at moderate amplitudes whereas ϕ should be close to the
2 2
RVdataisinconsistentwiththeassumptionofacircularorbit. separatrix of libration. Our results are in perfect agreement with
The observational data is currently not good enough to con- thisprediction,thussupportingtheideathatbothplanetmigration
strainthemutualinclinationofthetwoplanets.Totesttheeffects andstochasticforcesoccurredduringthesystem’sevolution.
ofmutuallyinclinedorbits,werananadditional14400simulations
withthesameinitialconditionsasbeforebutperturbedeachsystem
witharandommutualinclinationof∼2◦.Ourresultsindicatethat
thesystem’sstabilityisnotsignificantlyaffectedbyasmallamount ACKNOWLEDGMENTS
ofmutualinclinationoveramillionyeartimescale.
We thank Barbara McArthur for sharing the latest radial velocity
dataset.DanielTamayoandananonymousrefereeprovidedvalu-
able feedback. This research has been supported by the NSERC
4 MEANMOTIONRESONANCEANDCONCLUSIONS DiscoveryGrantRGPIN-2014-04553.
TheperiodratioinourMCMCsample,P /P ,iswithin2%ofa2:1
2 1
meanmotionresonance.Theperiodratioaloneisnotasufficient
criteria for a MMR. To test whether the system is in a MMR or
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jitterb s5 2.2+−12..90m/s
Notes:
aOffsetsarerelativetothosefoundbyMcArthuretal.(2014)andareall
consistentwithzero.
bJitterisaddedontopofthereportedRVerrorswhichdonotincludestellar
variabilityandadditionalinstrumentalnoise.
(cid:13)c 0000RAS,MNRAS000,000–000