The Stability of the Suggested Planet in the nu Octantis System: A Numerical and Statistical Study

1.8 MB·English

by  Billy Quarles

Mon.Not.R.Astron.Soc.000,1–7(2011) Printed12January2012 (MNLATEXstylefilev2.2)
ν
The stability of the suggested planet in the Octantis
system: a numerical and statistical study
2 B. Quarles,1⋆ M. Cuntz,1 and Z. E. Musielak1
1
0 1Universityof Texasat Arlington, Department of Physics, 502 YatesStreet, Arlington, TX 76019, USA
2

n
a
J Accepted ...Received...;inoriginalform...

1
1
ABSTRACT

We provide a detailed theoretical study aimed at the observational finding about the
]
P ν OctantisbinarysystemthatindicatesthepossibleexistenceofaJupiter-typeplanet
E in this system. If a prograde planetary orbit is assumed, it has earlier been argued
. that the planet, if existing, should be located outside the zone of orbital stability.
h
However,a previous study by Eberle & Cuntz (2010) [ApJ 721, L168] concludes that
p
the planet is most likely stable if assumed to be in a retrograde orbit with respect to
-
o the secondary system component. In the present work, we significantly augment this
r study by taking into account the observationally deduced uncertainty ranges of the
t
s orbitalparametersfor the stellar components and the suggestedplanet. Furthermore,
a
our study employs additional mathematical methods, which include monitoring the
[
Jacobiconstant,thezerovelocityfunction,andthemaximumLyapunovexponent.We

1 againfindthatthesuggestedplanetisindeedpossibleifassumedtobeinaretrograde
v orbit,butitisvirtuallyimpossibleifassumedinaprogradeorbit.Itsexistenceisfound
3 tobeconsistentwiththededucedsystemparametersofthebinarycomponentsandof
1 the suggestedplanet,including the associateduncertaintybarsgivenby observations.
3
2 Key words: methods: numerical — methods: 3-body simulations — techniques:
. Maximum Lyapunov Exponent — stars: individual (ν Octantis)
1
0
2
1
:
v 1 INTRODUCTION given as 0.2358±0.0003 (Ramm et al. 2009). Furthermore,
i the semimajor axis of the planet, with an estimated mini-
X The binary system ν Octantis, consisting of a K1 III star
mummass Mpsiniofapproximately2.5MJ,wasderivedas
r (Houk & Cowley 1975) and a faint secondary component, 1.2±0.1 AU (see Table 1). The distance parameters indi-
a identified as spectral type K7–M1 V (Ramm et al. 2009) is
cate that the proposed planet is expected be located about
located in the southernmost portion of the southern hemi-
halfway between the two stellar components. This is a set-
sphere, i.e., in close proximity to the Southern Celestial
upin stark contrast to otherobserved binary systemshost-
Pole.In2009,Ramm et al.foundnewdetailedastrometric–
ingoneormoreplanets(Eggenberger, Udry& Mayor2004;
spectroscopic orbital solutions, implying the possible exis-
Raghavan et al. 2006) and, moreover, difficult to reconcile
tenceofaplanet1aroundthemainstellarcomponent.Thus,
with standard orbital stability scenarios.
ν Octantis A is one of the very few cases where a planet is
A possible solution was proposed by Eberle & Cuntz
foundtoorbitagiantratherthanalate-typemain-sequence
(2010b). They argued that the planet might most likely be
star; the first case of this kind is the planetary-mass com-
stable if assumed to be in a retrograde orbit relative to the
panion to theG9 III star HD 104985 (Sato et al. 2003).
motion of the secondary stellar component. Previous cases
ThecompositionoftheνOctantissystemleadstomany
of planetary retrograde motion have been discovered, but
openquestionsparticularlyabouttheorbitalstabilityofthe
with respect to the spin direction of the host stars. Exam-
proposed planet. The semimajor axis of the binary system
plesinclude(mostlikely)HAT-P-7b(orKepler-2b),WASP-
has been deduced as 2.55±0.13 AU with an eccentricity
8b and WASP-17b that were identified by Winn et al.
(2009), Queloz et al. (2010), and Anderson et al. (2010),
respectively. These discoveries provide significant chal-
⋆ E-mail: [email protected] (BQ); [email protected] (MC);
lenges to the standard paradigm of planet formation
[email protected](ZEM)
1 Analternativeexplanationinvolvesahierarchicaltriplesystem, in protoplanetary disks as discussed by Lissauer (1993),
Lin, Bodenheimer & Richardson (1996) and others.
invokingaprecessionalmotionofthemainstellarcomponent,as
proposedbyMorais&Correia(2012). The previous theoretical work by Eberle & Cuntz
2 B. Quarles, M. Cuntz, and Z. E. Musielak
(2010b)exploredtwocasesoforbitalstabilitybehaviourfor the planet from its host star, the more massive of the two
ν Octantis based on a fixed mass ratio for the stellar com- stars with mass m1, and the initial distance between the
ponents and a starting distance ratio of ρ0 = 0.379 for the twostars,respectively.Inaddition,werepresentthesystem
putative planet (see definition below). They found that if inarotatingreferenceframe, whichintroducesCoriolis and
the planet is placed in a prograde orbit about the stellar centrifugal forces. Thefollowing equations of motion utilize
primary, it almost immediately faced orbital instability, as theseparameters (Szebehely 1967):
expected. However, if placed in a retrograde orbit, stability
isencounteredforatleast 1×107 yror3.4×106 binaryor-
x˙ = u (1)
bits, i.e., the total time of simulation. This outcome is also
consistent with theresultsfrom anearlier studybyJefferys y˙ = v (2)
(1974) based on a Henon stability analysis, which indicates z˙ = w (3)
significantly enlarged regions of stability for planets in ret- x−µ x+α
u˙ = 2v+Ω(x−α −µ ) (4)
rograde orbits. r3 r3
1 2
Nonetheless, there is a significant need for augment- y y
v˙ = −2u+Ω(y−α −µ ) (5)
ing the earlier study by Eberle & Cuntz (2010b). Notable r3 r3
1 2
reasons include: first, Eberle & Cuntz only considered one z z
w˙ = −z+Ω(z−α −µ ) (6)
possible mass ratio and one possible set of values for the r3 r3
1 2
semimajor axis and eccentricity for the stellar components,
where
thus disregarding their inherent observational uncertain-
ties. Hence, it is unclear if or how the prograde and ret- µ = m2 (7)
rograde stability regions for the planetary motions will be m1+m2
affected if other acceptable choices of system parameters α = 1−µ (8)
are made. Second, Eberle & Cuntz chose the 9 o’clock po- Ω = (1+ecosf)−1 (9)
sition as thestarting position for theplanetary component.
Thus, it is unclear if or how the timescale of orbital sta- r1 = (x−µ)2+y2+z2 (10)
bility will change due to other positional choices in the q
view of previous studies, which have demonstrated a sig- r2 = (x+α)2+y2+z2 (11)
nificant sensitivity in the outcome of stability simulations qR0
ontheadoptedplanetarystartingangle(e.g.,Fatuzzo et al. ρ0 = (12)
D
2006; Yeager, Eberle & Cuntz 2011). Third, and foremost,
The variables in the above equations describe how the
Eberle & Cuntz did not utilize detailed stability criteria
stateofaplanet,commensuratetoatestobject,evolvesdue
for the identification of the onset of orbital instability.
to the forces present. The state is represented in a Carte-
Possible mathematical methods for treating the latter in-
sian coordinate system {x, y, z, u, v, w}. We use dot no-
clude monitoring the Jacobi constant and the zero velocity
tation to represent the time derivatives of the coordinates
function (e.g., Szebehely 1967; Eberle, Cuntz& Musielak
x˙ = dx . Since the variables {u, v, w} represent the ve-
2008) as well as using the maximum Lyapunov ex- dt
locities,thetimederivativesofthesevariablesrepresentthe
ponent (e.g., Hilborn 1994; Smith & Szebehely 1993;
(cid:8) (cid:9)
Lissauer 1999; Murray & Holman 2001). In our previ- accelerations.Thevariables{r1, r2}denotethedistancesof
the planet from each star in the rotating reference frame.
ous work reported by Cuntz, Eberle & Musielak (2007),
Eberle, Cuntz& Musielak(2008)andQuarles et al.(2011), The value of f describes the true anomaly of the mass m1
and e refers to theeccentricity of the stellar orbits.
we extensively used these methods for investigating the or-
bitalstabilityinthecircularrestricted3-bodyproblem.The
same methods will be adopted in thepresent study. 2.2 Initial Conditions
Ourpaperisstructuredasfollows.InSect.2,wediscuss
our theoretical methods, including the initial conditions of ThecoplanarRTPindicatesthattheputativeplanetmoves
thestar-planetsystemandthestabilitycriteria.Theresults in the same orbital plane as the two stellar components;
and discussion, including detailed case studies, are given in additionally, its mass is considered negligible2. The initial
Sect. 3. Finally, Sect. 4 conveysour conclusions. orbital velocity of the planet is calculated assuming a Ke-
plerian circular orbit about its host star; see Fig. 1 for a
depiction of the initial system configuration. For simplicity
2 THEORETICAL APPROACH weassume the(u,w) components areequalto zero and the
magnitudeofthevelocitytobeinthedirectionofthevcom-
2.1 Basic Equations ponent.Themeanvalueoftheeccentricityofthebinary,e ,
b
isused duetoaccuracy of thevaluegiven. A Taylorexpan-
Using the orbital parameters attained by Ramm et al.
sion in initial orbital velocity of the binary shows that the
(2009), see Table 1, we consider the coplanar restricted 3-
firstordercorrectionwouldonlychangetheinitialorbitalve-
body problem (RTP) applied to the system of ν Octantis.
locityoftherespectivestarsbyafactorof10−8.Theinitial
Followingthestandardconventionspertainingtothecopla-
nar RTP, we write the equations of motion in terms of the eccentricityoftheplanet,ep,hasbeenchosentobeequalto
parametersµandρ0.Theparameterµandthecomplemen-
tary parameter α are defined by the two stellar masses m1 2 Negligiblemass means that although the body’s motion is in-
and m2 (see below). The planetary distance ratio ρ0 de- fluenced bythe gravity ofthe two massiveprimaries,its mass is
pends on R0 and D, which denote the initial distance of toolowtoaffectthemotionsoftheprimaries.
Stability in the ν Octantis system 3
Table 1.StellarandPlanetaryParametersofν Octantis
Parameter Value Reference
Spectraltype(1) K1III Houk&Cowley(1975)
Spectraltype(2) K7−M1V Rammetal.(2009)
R.A. 21h 41m 28.6463s ESA(1997)a,b
Decl. −77◦ 23′ 24.167′′ ESA(1997)a,b
Distance(pc) 21.20±0.87 vanLeeuwen(2007)
Mv (1) 2.10±0.13 Rammetal.(2009)c
Mv (2) ∼9.9 Drilling&Landolt(2000)
M1 (M⊙) 1.4±0.3 Rammetal.(2009)
M2 (M⊙) 0.5±0.1 Rammetal.(2009)
Teff,1 (K) 4790±105 AllendePrieto&Lambert(1999)
R1 (R⊙) 5.9±0.4 AllendePrieto&Lambert(1999)
Pb (d) 1050.11±0.13 Rammetal.(2009)
ab (AU) 2.55±0.13 Rammetal.(2009)
eb 0.2358±0.0003 Rammetal.(2009)
Mpsini(MJ) 2.5±... Rammetal.(2009)d
ap (AU) 1.2±0.1 Rammetal.(2009)d
ep 0.123±0.037 Rammetal.(2009)d
a DatafromSIMBAD,seehttp://simabd.u-strasbg.fr.
b AdoptedfromtheHipparcos catalog.
c Derivedfromthestellarparallax,seeRammetal.(2009).
d Controversial.
zeroduetotheredundancyitprovidesintheselectionofthe 2.3 Statistical Parameter Space
initial planetary position. Noting that the inclination angle
of the planet, ip, has not yet been determined, it is taken Inthepresentworkwewillexploretherole(s)ofsmallvaria-
to be 90◦ consistent with the requirement of the coplanar tions in the system parameters. Notably we will investigate
RTP. how the stellar mass ratio as well as the semimajor axis
and inherently the eccentricity of the stellar components
Tointegratetheequationsofmotion,asixthordersym-
canchangetheoutcomeofourorbitalstabilitysimulations.
plectic integration method is used (Yoshida 1990). In the
computationofFigs.2and3afixedtimestepof10−3 isas- Particularly when we consider the statistical probability of
prograde orbits for the suggested planet. These variations
sumed.Thisprovedtobeappropriatesincetheaveragerela-
are due to the observational uncertainties of the respective
tiveerrorintheJacobiconstantisfoundtobeontheorder
of 10−8. To increase the precision for the individual cases parameter; see Ramm et al. (2009) and Table 1 for details.
depicted in Figs. 5 to 9, a fixed time step of 10−4 is used, Equation (7) gives the definition of the mass ratio µ
which decreased the average relative error to the order of withm1=1.4±0.3M⊙ andm2 =0.5±0.1M⊙.Takingm1
10−10.However,thischoiceentailed aconsiderable increase and m2 as statistically independent from one another, the
of the computation time. Therefore, the models pertaining boundingvaluesforµare[0.220,0.306],i.e.,0.263±0.043.In
to Fig. 2 were integrated for 10,000 binary orbits (approx- thiscase, weassumed thereported uncertainty barsfor the
imately 29,000 years) and the models pertaining to Fig. 3 stellar masses to be 1.25 σ, noting that through this choice
wereintegratedfor1,000binaryorbits(approximately2,900 µisconsistentwiththemassratioq=0.38±0.03derivedby
years), noting that this figure is considerably less than uti- Ramm et al. (2009) that was used for computing m2 based
lizedinthemainframeofourstudyforthedeterminationof onthesumofthetwomasses.Formulasandmethodsforthe
the statistical probability of a prograde planetary configu- propagation-of-error, which in essence employ a first-order
ration.Moreover,themodelsofFigs.5to9wereintegrated Taylorexpansion,havebeengivenby,e.g.,Meyer(1975);see
for 80,000 binary orbits (approximately 232,000 years). alsoPress et al.(1986)fornumericalmethodsiftheerroris
Inthetreatmentofthisproblem,fourseparatetypesof calculated via a Monte-Carlo simulation.
initialconditionsofthesystemareconsidered.Theyencom- Equation (12) gives the definition for the relative ini-
pass the two different starting positions of the planet with tialdistance oftheplanet,expressed in termsofR0 andD,
respecttothemassivehoststar,whichare:the3o’clockand i.e., theinitial distance of the planet from its host star, the
9 o’clock positions (see Fig. 1). With each of these starting more massive of thetwo stars with mass m1 and theinitial
positions, weconsidered thepossibility of planetary motion distancebetweenthetwostars, respectively.Thisdefinition
in counter-clockwise and clockwise direction relative to the also allows to compute the bounding values for ρ0. Follow-
direction of motion of thesecondary stellar component; the ing the assumption made by Eberle & Cuntz (2010b), the
directions are referred to as prograde and retrograde, re- set up of the binary system will be initialized by assum-
spectively. The primary masses are considered to move in ing the secondary stellar component to be located in the
ellipses about thecentre of mass of thesystem. apastron starting position, entailing that D is modified as
4 B. Quarles, M. Cuntz, and Z. E. Musielak
D = ab(1+eb). Thus, the observation uncertainties of ap,
faobr, ρa0nd(seebewTialbllbee1u).tiAlizserdestuoltcowmepoubtteaitnhe[0b.3o4u4n,d0i.n4g18v],aliu.ee.s, x′1 = kxx11k (13)
0.381±0.037. These values will be considered in the sub- ′ ′
sequent set of detailed orbital stability studies by making ′ x2− x2, x1 x1
appropriate choices for µ and ρ0 when setting the initial x2 = x2−Dx2, x′1Ex′1 (14)
conditions. The outcome of the various simulations will, in
.
particular,allowadetailedstatisticalanalysisofthemathe- . (cid:13) (cid:10) (cid:11) (cid:13)
. (cid:13) (cid:13)
maticalprobabilityfortheexistenceofthesuggestedplanet
′ ′ ′ ′
in a prograde orbit; see Sect. 3.3. ′ x6− x6, x5 x5−...− x6, x1 x1
x6 = x6−Dx6, x′5Ex′5−...−Dx6, x′1Ex′1 (15)
2.4 Stability Criteria
With(cid:13)this n(cid:10)ew set (cid:11)of tangent v(cid:10)ectors, (cid:11)the (cid:13)spectrum of
(cid:13) (cid:13)
Lyapunovexponentsis computed by using
In our approach we employ two independent criteria. One
method deals with identifying events that can cause insta- i
1 ′ ′
bility,whereastheothermethodallowsustodeterminesta- λi = τ λi−1+log xi− xi, xj−1 xj−1 (16)
bility.First,wemonitortherelativeerrorintheJacobicon- " (cid:13)(cid:13) Xj=1D E (cid:13)(cid:13)#
stant to detect instabilities. By monitoring this constant (cid:13) (cid:13)
we can determine when the integrator loses its accuracy where λo =x′o =0. (cid:13)(cid:13) (cid:13)(cid:13)
or fails completely. It has been shown that this method is The Lyapunov exponents have extensively been used
particularly sensitive toclose approach eventsof theplanet instudiesof orbital stability in differentsettings pertaining
with any of the stellar components. However, it has previ- to the Solar System (e.g., Lecar, Franklin & Murison 1992;
ously been shown by Eberle, Cuntz& Musielak (2008) and Milani & Nobili 1992; Lissauer 1999; Murray & Holman
Eberle & Cuntz(2010a)thatthesecloseapproacheventsare 2001). Typically, themaximum Lyapunovexponent (MLE)
preceded by an encounter with the so-called “zero velocity isadoptedasameasureoftheratewithwhichnearbytrajec-
contour” (ZVC) for thecoplanar circular RTP. toriesdiverge.ThereasonisthattheMLEofferstheunique
Generalizing this criteria to the elliptic RTP, an anal- measure of the largest divergence. We previously used the
ogous “zero velocity function” (ZVF) can be found that is, MLEcriterion toprovideacutoffforwhichstability canbe
however,dependentontheinitialdistancebetweenthestel- assured given the behaviour of the MLE in time. This cut-
lar components (Szebehely 1967). TheZVF can also be de- off value has previously been determined by Quarles et al.
scribed by a pulsating ZVC; see Szenkovits& Mak´o (2008) (2011);itisgivenaslogλmax =−0.82onalogarithmicscale
forupdatedresults.NotethattheZVCwillbelargestwhen of base 10.
the binary components are at apastron and smallest when Recalling that the cutoff in MLE is inversely related
theyareatperiastron.Notingthatwechoosethestarstobe to the Lyapunov time tL, the latter provides an estimate
at the apastron starting position, the largest ZVC is inher- of the predictive period for which stability is guaranteed.
ently defined as well. When the planetary body encounters The lower (i.e., more negative) the logarithmic value of the
the ZVC, the orbital velocity decreases dramatically, which MLE, thelonger thetime for which orbital stability will be
reducestheu˙ andv˙-relatedforces.Asaresult,theplanetary ensured.Dueto thefinitetime of integration, thisestimate
body follows a trajectory that results in a close approach. is only valid for the integration timescale considered. But
This behaviour provides an indication that the system will thenatureofhowtheMLEchangesintimewillprovidethe
eventually become unstable, resulting in an ejection of the evidence to whether the MLE is asymptotically decreasing
planetary body from the system (most likely outcome) or andtherebyallowingtoidentifyorbitalstability.Hencethis
its collision with one of the stellar components (less likely cutoff will be used for establishing orbital stability of the
outcome). suggested planet in the ν Octantis system.
The second criterion utilizes the method of Lya-
punov exponents (Lyapunov 1907), which is commonly
used in nonlinear dynamics for determining the onset of 3 RESULTS AND DISCUSSION
chaos in different dynamical systems (e.g., Hilborn 1994;
3.1 Model Simulations
Musielak & Musielak 2009). A numerical algorithm that is
typically adoptedfor calculating thespectrumof Lyapunov In the following we report the general behaviour of sta-
exponentswasoriginallydevelopedbyWolf et al.(1985).In bility for the set of orbital stability simulations pursued
this approach, a set of tangent vectors xi and their deriva- in our study. Our initial focus is on the MLE. In Figs. 2
tives x˙i with i = 1,...,6 are introduced and initialized. In and3,thevariousvaluesoftheMLEasafunctionofµand
our approach, we choose all tangent vectors to be unit vec- ρ0 are denoted using a colour code. The range of colours
torsforsimplicity,anduseastandardintegratorakintothe shows that dark red represents a relatively high (less nega-
Runge-Kuttaschemesforcomputingchangesinthetangent tive) logarithmic MLE value, whereas dark blue represents
vectors within each timestep. After the tangent vectors be- a relatively low (more negative) logarithmic MLE value.
come oriented along the flow, it is necessary to perform a The white background represents conditions, which failed
Gram-SchmidtRenormalization(GSR)toorthogonalizethe to complete the respective allotted run time; see Sect. 2.4
tangentspace.Inthisspecificprocedure,wecalculateanew foradiscussionofpossibleconditionsforterminatingasim-
orthonormal set of tangent vectors here denoted by primes ulation.
(′),and obtain Figure 2 shows the result of our simulations for 10,000
Stability in the ν Octantis system 5
Table 2.SurvivalCountsoftheSuggested Planetinν Octantis for1,000binaryorbits.
Configuration UnstableCount StableCount StablePercentage
Prograde 3o’clock 17544 1776 9.2%
Prograde 9o’clock 19200 120 0.6%
Retrograde 3o’clock 1040 19184 94.9%
Retrograde 9o’clock 0 20224 100%
binary orbits, i.e., about 29,000 years, in the prograde con- starting position. The natureof this region can be deduced
figuration for starting conditions given by the 3 o’clock or from the parameters given by Ramm et al. (2009); see Ta-
9 o’clock planetary position. The results were obtained us- ble 1. There exists a 5:2 resonance of the planet with the
ing values of µ between 0.220 and 0.306 in increments of binarythatplacesthesuggestedplanetmovingtowardsthe
0.001. Concerning the initial conditions in ρ0, denoting the barycentreattheperiastronpointofthebinaryinducingan
relative starting position of the planet, we use values be- instability. For the 9 o’clock position the suggested planet
tween 0.272 and 0.502 in increments of 0.001. We show the would be movingaway which would providealarge pertur-
1σ regioninρ0 between0.344and0.418whereν Octispro- bation butinsufficient for producing an instability.
posedtoexistbyagreyshadedregion;seeSect.2.3.Between Figure 4 provides the histograms of simulations per-
the two bounding values regarding ρ0, there exists a region taining to both starting positions for the prograde configu-
of stability as well as a marginal stability limit. The limit ration. These histograms indicate the number of surviving
of marginal stability is consistent with previous findingsby cases related to the array of initial conditions for the pa-
Holman & Wiegert (1999) for the case with the planetary rameter space detailed in Fig. 2. The bin width is 0.01 in
eccentricitytakenintoaccount.Thisregionofmarginalsta- thelogarithmic MLEscale, correspondingto1.47×10−4 in
bility is more pronounced in the 3 o’clock starting configu- thelinearMLEscale.Althoughthereisnotacasethatsur-
ration;itbeginsatρ0≈0.315.Themarginalstabilityregion vives10,000binaryorbitsinthe1σregion,weprovidethese
decreasesasµincreases,whichisexpectedbecausetheper- histograms to demonstrate what different initial conditions
turbingmassisbecomingmoresignificantasthemassratio entailnearthelowerandupperuncertaintylimits.Figure2
µ increases. There is a noticeable favoritism towards the 3 shows that all thesurvivingcases cluster towards thelower
o’clock startingposition asindicatedbythesurvivalcounts bound in ρ0. Furthermore, it is obtained that the 9 o’clock
giveninTable2.Thisoutcomeisexpectedastheperturbing starting position has far fewer surviving initial conditions
massexertsalesser force ontheplanetif initially placed at than the 3 o’clock starting position. This is also reflected
the3 o’clock position. in the histograms. The histogram of the 9 o’clock position
Figure 3 shows the result for the retrograde configura- is sparse; without the aid of the 3 o’clock histogram there
tion for 1,000 binary orbits considering both the 3 o’clock wouldbetoofewcountstoprovidesubstantialinformation.
and9o’clockplanetarystartingpositions.Duetothelarger But inspecting the 3 o’clock histogram reveals that there
numberofconditionsthatsurvivedduringthistimeinterval is a Gaussian-type distribution about the logarithmic MLE
we have reduced the bounds of initial conditions following valueof−2.2.Thisindicatesthatforthelimitedsimulation
the bounding values of Eberle & Cuntz (2010b). Here the timeadoptedforthispartofourstudy,thereisaverysmall
bounding values for µ are given as 0.2593 and 0.2908 with number of cases indicating stability for the prograde con-
increments of 0.0001. The increment in µ was reduced to figuration. However,these cases are nearthe2σ or3σ limit
allow better insight into the role of chaos in this type of concerningthepermittedvaluesoftheplanetarypositionρ0
configuration. The initial conditions of ρ0 begin with 0.353 as discussed in Sect.3.3.
and end with 0.416 in increments of 0.001, which allows to
cover the most pronounced features of the retrograde con-
3.2 Case Studies
figuration.
Figure3demonstratesabroadregionofstabilitywithin Inthefollowing weprobesomeorbitsofretrogradeconfigu-
the selected parameter space. The 9 o’clock starting posi- rationsinmoredetail;seeFigs.5to9forinformation.These
tion has many regions of bounded chaos. These regions of figuresshowfromlefttorighttherespectiveorbitaldiagram
boundedchaosexhibitlocally lessnegativevaluesforMLE, inarotatingreferenceframe,theassociatedLyapunovspec-
whichhoweverdonotexceedthepreviouslydeterminedcut- trum,andtheFourierperiodogram.Throughthesediagrams
off value of log λmax = −0.82; see Sect. 2.4. The nature of wecanassess thenatureoftheseorbitstodeterminestabil-
these orbits and the connection to the MLE will be dis- ity and measure thechaos within thesystem.
cussed in Sect. 3.2. The greatest region of bounded chaos Figure 5 with (µ,ρ0) = (0.2825,0.400) is provided to
is obtained for ρ0 ≈ 0.41. This region decreases in ρ0 as µ showacaseofinstability.Inthiscasewefindthattheorbit
increases for thesamereason as thechangein themarginal presents itself as a wide annulusand already at first glance
stabilitylimitinFig.2.The3o’clockstartingpositionshows appears chaotic. The Lyapunov spectrum reveals that the
similar features but there exists a region of instability with MLE levels off at log λ = −1 fairly quickly, which is near
this starting position at ρ0 ≈ 0.41. This region coincides theproposedcutoffvalueforunstablechaoticbehaviour.For
with the extended region of bounded chaos in the 9 o’clock the timespan between 300 and 800 binary orbits, the MLE
6 B. Quarles, M. Cuntz, and Z. E. Musielak
Table 3.Statistical ProbabilityofProgradePlanetaryOrbitsa. massratioµandtheinitialplanetarydistanceparameterρ0.
The bounding values for µ were given as [0.220,0.306], i.e.,
0.263±0.043, andforρ0 as[0.344,0.418],i.e.,0.381±0.037.
WefoundthatbasedontheMLEaswellasanumbercount
MassRatio 9o’clockPosition 3o’clockPosition of surviving cases (see Sect. 3.1) that the existence of the
suggested planet in a retrograde orbit is almost certainly
µ σ Probability(%) σ Probability(%) possible, whereasinaprogradeorbit itisvirtuallyimpossi-
ble.
0.306 ... ... 2.30 1.1 Forprogradeorbitsbasedonthe“landscapeofsurvival”
0.263 ... ... 2.11 1.7 (see Fig. 2), another type of analysis can be given, see Ta-
0.220 2.95 0.16 1.89 2.9
ble3.Weconclude that for the9 o’clock planetary starting
position and for a relatively high stellar mass ratios, i.e.,
a Basedon10,000binaryorbits;seeFig.2 µ ∼> 0.25, the planet must start very close to the star, i.e.,
thattheassociatedprobabilityforρ0 iswayoutsidethesta-
tistical3σlimit.Onlyforrelativelysmallstellarmassratios,
fluctuatesuntilitfinallypushesupwardtoλmax.TheFourier i.e., µ ≃ 0.22, the planetary starting distance ρ0 would be
periodogramshowsthatthiscaseistrulychaoticduetothe barelyconsistentwiththe3σ limit,amountingtoastatisti-
amount of noise presented by the periodogram. There are cal probability of 0.16%. For a 3 o’clock planetary starting
manyperiodswhichmaypromotetheoutcomeofinstability position,theseprobabilitiespertainingtothepermittedval-
via resonance overlap (e.g., Mudryk & Wu 2006; Mardling ues of ρ0 are moderately increased. It is found that the as-
2007). The proposed planet is approaching the barycentre sociated probabilities for µ given as 0.306, 0.263, and 0.220
when the binary is at its periastron position, which should are 1.1%, 1.7%, and 2.9%, respectively.
beconsidered themain cause of thisinstability. Moreover, it should be noted that this statistical anal-
Figures 6 and 7 with (µ,ρ0) = (0.2805,0.358) and ysis refers to a “landscape of survival” that has been ob-
(0.2630,0.374),respectively,exhibitsimilarfeaturesbetween tainedforatimespanof10,000binaryorbits,corresponding
eachotherintheorbit diagram andperiodogram. However, to 29,000 years. An analysis of the evolution of that land-
thereisadifferenceintherespectiveLyapunovspectra.Both scapeshowsthatittendstodissipateiflargertimespansare
casesshowtrendsofstability,althoughFig.6demonstrates adopted. Even for timespans of 10,000 binary orbits, there
bounded chaos. It presents a different nature than the pre- aremanyparametercombinations(µ,ρ0)where,concerning
viouscaseofinstability(seeFig.5)astheMLEsettlesnear progradeorbits,noorbitalstabilityisfoundevenifρvalues
log λ=−1.8 and does not fluctuate enough to reach λmax. are chosen at or beyondthe statistical 3σ level.
BothcasesgivenasFigs.6and7representorbitsthatform Another perspective is offered through Fig. 10, which
annuli; however, the annulus widths are less than that of providesadditional insight intothestatistical natureof our
the unstable case depicted in Fig. 5. They are thus classi- simulations.Itaddressestheviewoflongtermstability.Fig-
fied as stable with Fig. 6 being chaotic and Fig. 7 being ure 10 depicts two curves representing the number of sur-
non-chaotic. vivingconfigurations as afunction of timefor each starting
Figures 8 and 9 with (µ,ρ0) = (0.2696,0.401) and position in the prograde configuration. The features of this
(0.2780,0.388), respectively, reveal similar features com- logarithmic plot show us that for the first 10 binary orbits
pared to the previous set of figures regarding all three di- thereis not a preference towards stability of either starting
agrams. These cases are provided to demonstrate the simi- position.After10binaryorbitsthe9o’clockpositionshows
larities between the 3 o’clock starting positions depicted in a significantly steeper trend of instability as indicated by
Figs. 6 and 7 and the 9 o’clock starting positions depicted theslopeoftherespectivecurve.The3o’clockpositionalso
in Figs. 8 and 9. The comparison between Figs. 6 and 8 in- showsatrendtowardinstabilitybutitisasymptoticallyap-
dicates a similar orbit diagram and periodogram, but the proaching a limit of 103 much more smoothly. With these
limitingvaluesoftheMLEsettleataslightlymorenegative limits, we can estimate the likelihood for a prograde plane-
value in Fig. 6. Figures 7 and 9 supply similar orbital dia- tary configuration to be extremely small in the framework
grams but there are minor differences in the periodograms oflong-term simulations. However,foraverylargerangeof
and Lypaunov spectra. These figures display the same be- (µ,ρ0) combinations, the stability of retrograde planetary
haviour in MLE but there is a difference in the periodic configuration appearsalmost certainly guaranteed,evenfor
nature of the decrease in MLE, which is also revealed by timescales beyond 105 binary orbits.
the difference in the number of peaks in the corresponding
periodograms.
4 CONCLUSIONS
3.3 Statistical Analysis of Orbital Stability
This study offers detailed insights into the principal possi-
Asignificantcomponentofthepresentstudyistoprovidea bility of prograde and retrograde orbits for the suggested
statistical analysis regarding whether or not the suggested planetintheν Octantissystem,ifconfirmedthroughfuture
planet in the ν Octantis system is able to exist in a pro- observations. A previous study by Eberle & Cuntz (2010b)
grade or retrograde orbit. Previously, we reported on the concludedthattheplanetismostlikelystableifassumedto
general behaviour of orbital stability for the set of simula- be in a retrograde orbit with respect to the secondary sys-
tionswithfocusontheMLE(seeSect.2.4).InFigs.2and3, temcomponent.Inthepresentwork,wewereabletoconfirm
thevariousvaluesoftheMLEweregivenasfunction ofthe thistheoreticalfinding,whiletakingintoaccounttheobser-
Stability in the ν Octantis system 7
vationallydeduceduncertaintyrangesoftheorbitalparam- Jefferys W. H., 1974, AJ, 79, 710
etersforthestellarcomponentsandthesuggestedplanetas Lecar M., Franklin F., Murison M., 1992, AJ, 104, 1230
well as different mass ratios of thestellar components. Lin D. N. C., Bodenheimer P., Richardson D. C., 1996,
In addition, wealso employed additional mathematical Nature,380, 606
methods, particularly the behaviour of the maximum Lya- LissauerJ.J.,1993,Ann.Rev.Astron.Astrophys.,31,129
punov exponent (MLE). We found that virtually all cases Lissauer J. J., 1999, Rev.Mod. Phys., 71, 835
of prograde orbit became unstable over a short amount of LyapunovM.A.,1907,Ann.Fac.Sci.,UniversityToulouse,
time;thebestcasesforsurvival(althoughbarelysignificant 9, 203
in a statistical framework) were found if a relatively small Meyer S. L., 1975, Data Analysis for Scientists and Engi-
mass ratiofor thetwo stellar componentswas assumed and neers, Wiley, Hoboken,NJ
if theinitial configuration was set for theplanet to start at Mardling R. A., 2007, in Vesperini E., Giersz M., Sills A.,
the3o’clockposition.Thislaterpreferenceisalsoconsistent eds,DynamicalEvolutionofDenseStellarSystems,Proc.
withpreviousgeneralresultsbyHolman & Wiegert (1999) IAUSymp.246, p.199
and others. Milani A., Nobili A. M., 1992, Nature, 357, 569
Thus,ourresultsoftheretrogradesimulationsindicate MoraisM.H.M.,CorreiaA.C.M.,2012,MNRAS,inpress
a high probability for stability within this type of configu- (arXiv:1110.3176)
ration. Moreover, this configuration displays regions in the Mudryk L.R., Wu Y., 2006, ApJ, 639, 423
systemwhereboundedchaoscanexistduetotheresonances Murray N.,Holman M., 2001, Nature,410, 773
present. Concerning retrograde orbits, there is little prefer- Musielak Z. E., Musielak D. E., 2009, Int. J. Bifurcation
ence for the planetary starting position. In summary, we and Chaos, 19, 2823
concludethattheν Octantisstar–planet systemisaninter- Press W. H., Flannery B. P., Teukolsky S. A., Vetterling
esting case for future observational and theoretical studies, W. T., 1986, Numerical Recipes, Cambridge University
including additional long-term orbital stability analyses if Press, Cambridge, UK
thesuggestedplanetisconfirmedbyfollow-upobservations. Quarles B., Eberle J., Musielak Z. E., Cuntz M., 2011,
A&A,533, A2
Queloz D., et al., 2010, A&A,517, L1
Raghavan D., et al., 2006, ApJ, 646, 523
ACKNOWLEDGMENTS
Ramm D. J., Pourbaix D., Hearnshaw J. B., Komonjinda
This work has been supported by the U.S. Department of S.,2009, MNRAS,394, 1695
EducationunderGAANNGrantNo.P200A090284 (B.Q.), Sato B., et al., 2003, ApJ,597, L157
theSETIinstitute(M.C.)andtheAlexandervonHumboldt SmithR.H.,SzebehelyV.,1993,Cel.Mech.Dyn.Astron.,
Foundation (Z. E. M.). 56, 409
SzebehelyV.,1967,TheoryofOrbits.AcademicPress,New
Yorkand London
Szenkovits F., Mak´o Z., 2008, Cel. Mech. Dyn. Astron.,
REFERENCES
101, 273
Anderson D. R., et al., 2010, ApJ, 709, 159 van Leeuwen F., 2007, A&A,474, 653
Allende Prieto C., Lambert D.L., 1999, A&A,352, 555 Winn J. A., et al., 2009, ApJ,703, L99
Baker G. L., Gollub J. P., 1990, Chaotic Dynamics: An Wolf A., Swift J. B., Swinney H. L., Vastano J. A., 1985,
Introduction.Cambridge Univ.Press, Cambridge Physica D,16, 285
BenettinG.,GalganiL.,Giorgilli A.,StrelcynJ.-M.,1980, YeagerK.E.,EberleJ.,CuntzM.,2011,Int.J.Astrobiol.,
Meccanica, 15, 9 10, 1
CuntzM.,EberleJ.,Musielak,Z.E.,2007,ApJ,669,L105 Yoshida H.,1990 Phys.Lett. A, 150, 262
DrillingJ.S.,LandoltA.U.,2000,inA.N.Cox,ed.,Allen’s
Astrophysical Quantities, 4th edn. Springer-Verlag, New
York,p.381
Eberle J., Cuntz M., 2010a, A&A,514, A19
Eberle J., Cuntz M., 2010b, ApJ, 721, L168
EberleJ.,CuntzM.,MusielakZ.E.,2008,A&A,489,1329
Eggenberger A., Udry S.,Mayor M., 2004, A&A,417, 353
European Space Agency, 1997, SP-1200, The Hipparcos
and Tycho Catalogues. ESA,Noordwijk
Fatuzzo M., Adams F. C., Gauvin R., Proszkow E. M.,
2006, Publ. Astron. Soc. Pac., 118, 1510
Gonczi R., Froeschl´e C., 1981, Cel. Mech. Dyn. Astron.,
25, 271
Hilborn R. C., 1994, Chaos and Nonlinear Dynamics, Ox-
ford Univ.Press, Oxford
Holman M. J., Wiegert P. A.,1999, AJ, 117, 621
Houk N.,Cowley A.P.,1975, University of Michigan Cat-
alogue of Two-Dimensional Spectral Types for the HD
Stars, Vol. 1. Univ.of Michigan, AnnArbor
8 B. Quarles, M. Cuntz, and Z. E. Musielak
3
v
A
)
U
A 0
(

*
Y
v
B
−3
−3 0 3
*
X (AU)
Figure1.Exampleofthepossibleinitialstartingconfigurations.Thedirectionsoftheinitialvelocitiesofthestellarcomponents(νOctA
andν OctB)areshownwithblackcolouredarrowsandappropriatelabels.Theproposedplanetisrepresentedbytheredcoloureddots
inthe3o’clockposition(right)andthe9o’clockposition(left).Thepossibledirectionsoftheinitialvelocityfortheplanetarycomponent
areeitherprograde(purple)orretrograde(blue).Thearrowsindicatethedirectionsintheinitialconfigurations ofthesystem.
−1 0.3
0.3
−1
−1.5 0.28
0.28
−1.5
µ
µ 0.26
0.26 −2
−2
0.24
0.24
−2.5 −2.5

0.3 0.4 0.5 0.3 0.4 0.5
ρ ρ
0 0
Figure 2. Detailed results of simulations for the parameter space for the case of prograde planetary motion. The left and right panel
represent the simulations withthe planet initiallyplaced at the 9o’clock and 3o’clock position, respectively. The greyarea represents
thestatistical 1σ rangeinρ0,whichis0.381±0.037.
0.29 0.29 −1
−1.6
0.285 0.285
−1.8 −1.5
0.28 0.28
−2
µ 0.275 µ 0.275
−2.2 −2
0.27 0.27
−2.4
0.265 −2.6 0.265 −2.5
0.26 0.26

0.36 0.38 0.4 0.36 0.38 0.4
ρ ρ
0 0
Figure 3.SameasFig.2,butnowforretrogradeplanetarymotion.
Stability in the ν Octantis system 9
Full Space Histogram of Nu Oct at 9 o clock starting position
6
5
s
e
s4
a
C
of 3
r
e
b
m2
u
N
1
0
−2.4 −2.2 −2 −1.8 −1.6 −1.4 −1.2 −1 −0.8
Maximum Lyapunov Exponent (log λ)
Full Space Histogram of Nu Oct at 3 o clock starting postition
80
s60
e
s
a
C
of 40
r
e
b
m
u
N20
0
−2.5 −2 −1.5 −1 −0.5
Maximum Lyapunov Exponent (log λ)
Figure 4. Histograms of configurations that survived 1000 binary orbits in prograde motion. Note the significant difference in scale
regardingboththexandy-axis.
λ) 1 λ λ λ 0.1
3 og 0 1 2 3 e0.08
AU) 0 onent (l−1 mplitud0.06
*Y ( Exp−2 ve A0.04
ov −3 ati
−3 un−4 Rel0.02
p
a
y
L−5 0
−3 0 3 0 2 4 6 8 10 12
0 0.5 1 1.5
X* (AU) Time (binary orbits x 102) Time (binary orbits)
Figure 5.Casestudyofplanetarymotionwiththeplanetplacedinthe3o’clockpositionandinretrogrademotion.Thiscasedisplays
theconditions forµ=0.2825andρ0=0.400.
10 B. Quarles, M. Cuntz, and Z. E. Musielak
) 0
λ λ λ λ
3 g −1 1 2 3 0.4
o e
(l d
AU) 0 onent −−32 mplitu0.3
*Y ( Exp−4 e A0.2
ov −5 ativ
−3 un el0.1
p−6 R
a
y
L−7 0
−3 0 3 0 2 4 6 8
0 0.5 1 1.5 2
* 4
X (AU) Time (binary orbits x 10) Time (binary orbits)
Figure 6.Casestudyofplanetarymotionwiththeplanetplacedinthe3o’clockpositionandinretrogrademotion.Thiscasedisplays
theconditions forµ=0.2805andρ0=0.358.
λ) 0 λ λ λ
3 g −1 1 2 3 0.4
o e
AU) 0 onent (l−−32 mplitud0.3
*Y ( Exp−4 e A0.2
ov −5 ativ
−3 un el0.1
p−6 R
a
y
L−7 0
−3 0 3 0 2 4 6 8
0 0.5 1 1.5 2
X* (AU) Time (binary orbits x 104) Time (binary orbits)
Figure 7.Casestudyofplanetarymotionwiththeplanetplacedinthe3o’clockpositionandinretrogrademotion.Thiscasedisplays
theconditions forµ=0.2630andρ0=0.374.
) 0
λ λ λ λ
3 g −1 1 2 3
o e0.4
(l d
AU) 0 onent −−32 mplitu0.3
*Y ( Exp−4 e A0.2
ov −5 ativ
−3 pun−6 Rel0.1
a
y
L−7 0
−3 0 3 0 2 4 6 8
0.2 0.4 0.6 0.8 1 1.2 1.4
* 4
X (AU) Time (binary orbits x 10) Time (binary orbits)
Figure 8.Casestudyofplanetarymotionwiththeplanetplacedinthe9o’clockpositionandinretrogrademotion.Thiscasedisplays
theconditions forµ=0.2696andρ0=0.401.