Application of the Titius-Bode Rule to the 55 Cancri System: Tentative Prediction of a Possibly Habitable Planet

0.13 MB·

by  M. Cuntz

PASJ:Publ.Astron.Soc.Japan,1–??,
(cid:13)c 2012.AstronomicalSocietyofJapan.
Application of the Titius–Bode Rule to the 55 Cancri System:
Tentative Prediction of a Possibly Habitable Planet
ManfredCuntz
DepartmentofPhysics,UniversityofTexasatArlington,Box19059,
Arlington,TX76019,USA
[email protected]
(Received;accepted)
2
1 Abstract
0
2 Following the notion that the Titius–Bode rule (TBR) may also be applicable to some extra-solar planetary
systems,althoughthisnumbercouldberelativelysmall,itisappliedto55Cancri,whichisaG-typemain-sequence
n
star currentlyknownto hostfive planets. Followinga concise computationalprocess, we tentativelyidentifyfour
a
J new hypothetical planetary positions given as 0.081, 0.41, 1.51 and 2.95 AU from the star. The likelihood that
6 thesepositionsareoccupiedbyrealexistingplanetsissignificantlyenhancedforthepositionsof1.51and2.95AU
in the view of previoussimulationson planetformationand planetaryorbitalstability. For example, Raymondet

] al. (2008)arguedthat additionalplanetswouldbe possible between55Cnc f and55 Cnc d, whichwould include
P
planetssituatedat1.51and2.95AU.Iftwoadditionalplanetsareassumedtoexistbetween55Cncfand55Cncd,
E
thededuceddomainsofstabilitywouldbegivenas1.3–1.6and2.2–3.3AU.Thepossibleplanetnear1.5AUappears
.
h tobelocatedattheoutskirtsofthestellarhabitablezone,whichishowevernotablyaffectedbythestellarparameters
p aswellastheadoptedmodelofcircumstellarhabitability. Wealsocomputethedistanceofthenextpossibleouter
-
planet in the 55 Cnc system, which if existing is predicted to be located between 10.9 and 12.2 AU, which is
o
r consistentwithorbitalstabilityconstraints. Theinherentstatistical significanceoftheTBRisevaluatedfollowing
st themethodbyLynch(2003). Yetitisuptofutureplanetarysearchmissionstoverifyorfalsifytheapplicabilityof
a theTBRtothe55Cncsystem,andtoattaininformationonadditionalplanets,ifexisting.
[
Keywords:astrobiology—methods:statistical—stars: individual(55Cnc)—stars: planetarysystems

4
v
1. Introduction tistical analyses. For example, Lynch (2003) argued that it
8
3 is not possible to conclude unequivocallythat laws (or rules)
0 One of the more controversial aspects in the study of the of Titius–Bode type are, or are not, significant, a conclusion
5 Solar System as well as studies of selected extrasolar plane- that also appears to be consistent with the work by Hayes &
7. tarysystemsistheapplicationoftheTitius–Boderule(TBR). Tremaine (1998). Moreover, the main conclusion of Lynch
0 Historically,theTBRwasfirstderivedandappliedtotheSolar (2003)isalsosharedbyNeslusˇan(2004).Neslusˇan(2004)also
1 Systemwhereitplayedasignificantroleinthesearchfornew pointedoutthecuriositythatiftheEarth’sdistanceisexcluded
1 planetaryobjects(e.g.,Nieto1972). ThediscoveryofUranus from the sequence, then the probability of the occurrence of
:
v byHerschelin1781andthelargestobjectoftheMars–Jupiter thatsequencebychanceisradicallyreducedfrom95to100%
i asteroidbelt,Ceres,byPiazziin1801appearedtoconfirmthe downto3%.
X
applicabilityandsignificanceoftheTBR.TheTBRhistorically IfnoappropriatephysicalexplanationfortheTBRisidenti-
r givenbytheformula fied,itistypicallyarguedthattheTBRmaybemerelyaruleof
a
chance(“numerology”),fuelledbythepsychologicaltendency
r =0.4+0.3×2n, n=−∞,0,1,2,3,... (1)
n toidentifypatternswherenoneexist(Newmanetal.1994).On
(inAU)is,however,entirelyinapplicabletoNeptuneandtothe theotherhand,therearepreviousdetailedastrophysicalandas-
formerSolarSystemplanetPluto1,whichisstilloftenconsid- trodynamicalstudiesthatpotentiallypointtoaphysicalback-
eredtobeanappropriaterepresentativeofthesteadilyincreas- groundoftheTBR,atleastfortheSolarSystemand(byimpli-
ingnumberofKuiperBeltObjects(e.g.,Jewitt&Luu2001). cation)foraselectednumberofextrasolarplanetarysystemsas
A considerable amount of previous discussion revolves well. Forexample,White (1972)arguedthatjet streamsmay
around the interpretation of the TBR for the Solar System, developinarotatinggaseousdiskatdiscreteorbitaldistances
including its well-known insufficiency, see, e.g., Graner & given by a geometric progression. This result appears to be
Dubrulle (1994), Hayes & Tremaine (1998), Lynch (2003), broadlyconsistentwiththeworkbySchmitz(1984),whopur-
andNeslusˇan(2004)forpreviousstudieslargelybasedonsta- suedananalyticanalysisofthestructure,stability,andformof
marginally stable axisymmetric perturbationsof idealized ro-
1 Plutoismentionedmerelyforhistoricalreasons;itisoftenconsideredto
tatinggasclouds. Hisstudyshowedthattheradialpartsofthe
highlightthelimitationsoftheTBRfortheSolarSystem. Plutoisinthe
expansion solutions obtained from the governing differential
3:2mean-motionresonancewithNeptuneandisthusnotadynamically
independentobject.Moreover,long-termintegrationshaveuncoveredthat yieldsimpleoscillatingfunctionswithTBR-typefeatures.
itsorbitischaoticyetstableoverbillion-yeartimescales(Malhotra1993). A similar approach was undertaken by Graner & Dubrulle
2 AstronomicalSocietyofJapan [Vol.,
Table1.StellarParameters
Quantity Value Reference
SpectralType G8V Gonzalez(1998)
Distancea 12.5±0.13pc ESA(1997)
ApparentMagnitudeV 5.96 SIMBADwebsiteb
RACoordinate 08h52m35.811s SIMBADwebsiteb
DECCoordinate +28◦19′50.95′′ SIMBADwebsiteb
EffectiveTemperature 5280K seetext
Radiusc 0.925±0.023R Ribasetal.(2003)
⊙
Mass 0.92±0.05M Valenti&Fischer(2005)
⊙
Aged 5±3Gyr Fischeretal.(2008)
Metallicity[Fe/H] +0.31±0.04 Valenti&Fischer(2005)
aDatafromtheHipparcosCatalogue.
bSeehttp://simbad.u-strasbg.fr
cAlternativevalue:1.15±0.035R (Bainesetal.2008).
⊙
dAmorestringentdetermination,whichis4.5±1Gyr,hasbeengivenbyDonahue(1998)andHenryetal.(2000).
Table2.PlanetarySystema
Planet Distance(AU) Period(d) M sini(MJ) Discovery
55Cnce 0.01583 ±0.00020 0.7365400 ±0.0000030 0.027 ±0.0011 2004/2011
55Cncb 0.115 ±0.0000011 14.65162 ±0.0007 0.824 ±0.007 1996
55Cncc 0.240 ±0.000045 44.3446 ±0.007 0.169 ±0.008 2002
55Cncf 0.781 ±0.007 260.00 ±1.1 0.144 ±0.04 2007
55Cncd 5.77 ±0.11 5218 ±230 3.835 ±0.08 2002
aData given by Fischer et al. (2008) and references therein, except for 55 Cnc e, for which the data from Dawson &
Fabrycky(2010)andWinnetal.(2011)wereused.
(1994), who argued that a Titius–Bode type law emerges au- ened significance of the TBR, except that its meaning is that
tomaticallyasaconsequenceofthescaleinvarianceandrota- stable planetary systems tend to be spaced in a regular man-
tional symmetry of the protoplanetarydisc, and that such ge- ner. However, they indicate that their method could be used
ometrical relationshipsare a generic characteristic of a broad toidentifypotentiallyunstableplanetarysystems,especiallyif
range of physical systems. Li et al. (1995) investigated the appliedinconjunctionwithlong-termorbitintegrations.Afur-
nonlinear development and evolution of density disturbances thercontributiontotheinterpretationoftheTBRwasmadeby
in nebular disks on the basis of one of their previousstudies. Nottale(1996)andNottaleetal.(1997)basedonanapproach
Their analysis showed that the perturbed density is unstable ofscale relativityandquantizationofthe SolarSystem. Itat-
with respect to self-modulation, leading to the formation of temptstodescribetheSolarSystem intermsoffractaltrajec-
density field localization and collapse. In the case of a self- tories governedby a Schro¨dinger-likeequation. The physical
similar collapse, the perturbed density was found to increase backgroundandjustificationofthisapproach,however,remain
withtimeandtoformsteadyringswithverylargebutlimited highlyuncertain.Acontroversialbuttestableaspectofthisthe-
amplitudes.Thedistancesoftheseringswerefoundtofollowa oryisthatitproposestheexistenceofoneortwosmallplanets
geometricprogressionakintotheTBR. Also,inhissummary betweenthe Sun andMercury,whichto date haveno support
about the final stages of planetesimal accumulation, Lissauer throughobservations.Ontheotherhand,thesestudiespointto
(1993)pointedoutthat “[t]he self-limiting natureof runaway thegeneralpossibilityof“empty”orbits,whichmayalsobeof
growthstronglyimpliesthatprotoplanetsformatregularinter- interesttogeneralizedapplicationsoftheTBRaswell.
valsinsemimajoraxis.”Thisresultappearstoentailgeometri- Inthefollowing,wewillapplyageneralizedversionofthe
cal spacing of planetsin planetarysystems, potentiallymodi- TBR,withoutsubscribingtoaspecificphysicalinterpretation,
fiedbytheirlong-termorbitaldynamicalevolution,inpossible totheplanetarysystemof55Cancri(=ρ1 Cnc=HD75732=
agreementwith(amodifiedversionof)theTBR. HIP 43587 = HR 3522; G8 V) (see Table 1), which was his-
Another approach invoking statistical numerical experi- toricallythefourthstarotherthantheSunafter51Peg,70Vir,
mentswasadoptedbyHayes&Tremaine(1998). Theychose and47UMa(notcountingthepulsarPSR1257+12aswellas
to fit randomlyselected artificial planetarysystems to Titius– othercontroversialcases)thatwasidentifiedhostinga planet.
BodetypelawsconsideringadistanceruleinspiredbytheHill A previous effort to apply the TBR to 55 Cnc has been pur-
stability of adjacent planets. They did not identify a height- suedbyPoveda&Lara(2008),whichwillalsobediscussedin
No.] Usageofpasj00.cls 3
the following. Additionally, we will compareour results to a Table3.StellarEffectiveTemperature
modifiedversionoftheTBRgivenbyLynch(2003)previously
appliedtotheSolarSystem. Teff (K) Reference
55 Cnc is known to host five planets, named 55 Cnc b
5150±75 Gonzalez(1998)
to 55 Cnc f, which were discovered between 1996 and 2007
5336±90 Fuhrmannetal.(1998)
by Butler et al. (1997), Marcy et al. (2002), McArthur et al.
5250±70 Gonzalez&Vanture(1998)
(2004),andFischeretal.(2008);notethatallplanetshavebeen
5243±93 deStrobeletal.(2001)
detectedusingtheradialvelocitymethod.Thepositionsofthe
5338±53 Ribasetal.(2003)
planets,accordingtotheiroriginaldeterminations,aregivenas
5279±62 Santosetal.(2004)
0.038,0.115,0.240,0.781,and5.77AU,respectively,andtheir
5234±44 Fischer&Valenti(2005)
masses(Mpsini)rangefrom0.034to3.835MJ (seeTable2;
italsogivesinformationontherespectiveuncertainties).
However, a notable controversy emerged about the inner-
WealsocommentonapreviousapplicationoftheTBRtothe
most planet 55 Cnc e. Dawson & Fabrycky (2010) argued
55CncsystemgivenbyPoveda&Lara(2008). Moreover,we
that the position of 55 Cnc e is given as 0.016 rather than
providean analysis of the statistical significance for the TBR
0.038 AU based on an improved method that allowed to dis-
forwarded by this work, as well as of the previous TBR ver-
tinguish an alias from the true planetary frequencyin a more
sionbyPoveda&Lara(2008),followingthemethodofLynch
reliable manner. This reviseddistance was subsequentlycon-
(2003).Finally,wepresentoursummaryandconclusions.
firmedbyWinnetal.(2011)basedoncontinuousphotometric
monitoringwiththeMOST(Microvariability&Oscillationsof
2. MethodsandResults
STars)spacetelescope. Moreover,thereisanongoingdiscus-
sionabouttheprincipalpossibilityofadditionalplanetsinthe 2.1. ComputationoftheTBRParameters
55Cncsystem,particularlyinthegapbetween0.8and5AU.
ThecenterpieceofourstudyistheapplicationoftheTBRto
Thisspeculationisfuelledbydetailedorbitalstabilitystudies,
theplanetarysystemof55Cnc. Akintotheapplicationofthe
including studies involving putative Earth-mass planets (von
TBRtotheSolarSystem,weoriginallyselectedasapproach
Blohetal.2003;Rivera&Haghighipour2007;Raymondetal.
2008;Smith&Lissauer2009;Jietal.2009). Ifsuchaplanet TB A if i=1
a = (2)
exists, it would also possibly imply the presence of a poten- i A+B·Zi−2 if i>1
tially habitable planet if appropriate conditions are met (e.g., guidedby thne originalplanetarydata of Fischer et al. (2008).
Lammeretal.2009). HereA,B,andZ constitutefreeparameters,whichneedtobe
Thestar55Cncisamiddle-agedmain-sequencestarwitha determinedbasedontheknownpositionsofthepreviouslydis-
massofapproximately0.95M⊙(Valenti&Fischer2005),with coveredfiveplanets.Basedonthefiveknownplanets55Cnce,
a stellar effective temperature of about5250 K (see Table 3). 55Cncb, 55Cncc, 55Cnc f, and55Cncd, areidentifiedas
FromtheHipparcosparallaxof79.8±0.84mas(ESA1997), i=1,3,4,6,and9,respectively.
aluminosityof0.61±0.04L⊙ (seeTable1)canbededuced, This version of the TBR is consistentwith its originalver-
which is consistent with its spectral type G8 V (Gonzalez sioncustomarilyappliedtotheSolarSystem. NotethatB in-
1998); see also discussion by Marcy et al. (2002). The age dicates the compactnessof the system (i.e., the higher B, the
of the star can be obtained from the strength of the chromo- less densethe overallplanetarydistribution),whereasZ indi-
sphericCaIIH+Kemission,indicatinganageof4.5±1Gyr catestheprogressionoftherelativespacingoftheplanets. For
(Donahue 1998; Henry et al. 2000); see also Baliunas et al. theSolarSystem,thefreeparametersoftheabovegivenequa-
(1997)forpreviouswork. Anotherelementofourworkisthe tiontakethefollowingvalues:A=0.4,B=0.3,andZ=2.0.
assessment of 55 Cnc’s circumstellar habitability, which will However, in the meantime, a controversy emerged regarding
be pursued following the approach by Kasting et al. (1993) thelocationoftheinnermostplanet55Cnce,whichbasedon
andUnderwoodetal.(2003);additionally,itwillconsiderthe a re-analysis of existing data was identified to be positioned
possibleextensionoftheouterboundaryofhabitabilityasdis- at 0.016 AU rather than 0.038 AU. We therefore revised the
cussedby,e.g.,Forget&Pierrehumbert(1997),Mischnaetal. proposedversionof the TBR such thatthe position of the in-
(2000),andHalevyetal.(2009). nermost planet is changed to aTB = A/Z. Fortunately, test
1
Ourpaperisstructuredasfollows. Aftertheintroductionof simulationshaveshown thatthe key predictionsofthis work,
historicalaspectsoftheTBRand55Cncitself,asdone,wede- i.e.,thelocationofthehypotheticalplanetsat0.081,0.41,1.51
scribeourmethodsandresults. Emphasiswillbeplacedonthe and2.95AUaswellasthepossibleexteriorplanet,expectedto
calculationalprocessconcerningtheTBRbasedonfourdiffer- belocatedbetween10.9and12.2AU(ifexisting),areentirely
entmeasuresofdifference;theyareappliedtoassessthedevia- unchangedby this alteration. Moreover,the innermostplanet
tionsbetweenthepositionsofthefiveknownplanetsof55Cnc isalsonotpartofthepolynomialfitdeterminedbyZ,justlike
(i.e.,55Cncbto55Cncf)andthepredictionsoftheTBR,thus incaseoftheSolarSystem.
allowingto constrainthe unknownTBR parameters. In addi- Thepivotalaspectofourstudyistodetermineandoptimize
tion, we comment on the possibility of a habitable planet in theparametersA, B, andZ basedonthepositionsofthefive
the55CncsystemasimpliedbytheTBR.Thereafter,wecon- known planets of the 55 Cnc system, which are identified as
siderotherrelevantstudies,particularlystudiesregardingcon- i= 1, 3, 4, 6, and 9. The positions of these five planets are
straintsfromorbitalstabilitysimulationsandplanetformation. calculatedusingadequatetrialvaluesforA, B,andZ. These
4 AstronomicalSocietyofJapan [Vol.,
Table4.Titius–BodeParameters
MDIF A B Z
MDIF1 0.032±0.003 0.049±0.001 1.973±0.006
MDIF2 0.031±0.001 0.050±0.001 1.971±0.006
MDIF3 0.031±0.002 0.050±0.001 1.969±0.006
MDIF4 0.031±0.001 0.047±0.001 1.991±0.004
Table5.DistancesofRealandHypotheticalPlanetsa
Planet Observed Titius–BodeRule
MDIF1 MDIF2 MDIF3 MDIF4
55Cnce 0.016 0.016 0.016 0.016 0.015
... (HPL1) ... 0.082 0.081 0.081 0.078
55Cncb 0.115 0.130 0.129 0.130 0.125
55Cncc 0.240 0.224 0.224 0.225 0.218
... (HPL2) ... 0.411 0.411 0.413 0.403
55Cncf 0.781 0.779 0.781 0.783 0.771
... (HPL3) ... 1.506 1.509 1.511 1.504
... (HPL4) ... 2.940 2.945 2.945 2.965
55Cncd 5.77 5.770 5.774 5.768 5.872
aDistancesinunitsofAU.
parametersaresubsequentlyoptimizedthroughminimizingthe
numerical differences between the observed and “predicted”
6 distances (i.e., semimajor axes) of the observed five planets.
Thelateralsorequiresameasureofdifference. Inthefollow-
ing four different measures of difference, henceforth referred
5
toasMDIF,havebeenadopted,whichare: absolutelinearde-
viation(MDIF1),relativelineardeviation(MDIF2),absolute
U) 4 quadraticdeviation(MDIF3),andrelativequadraticdeviation
A (MDIF4). Theequationsforthedistancedifferenceparameter
(
ce 3 δri arethusgivenas
n
sta δri = rpl,i−rpTlB,i (3)
Di 2
(cid:12) (cid:12)
(cid:12)r −rTB(cid:12)
δr = (cid:12) pl,i pl,i(cid:12) (4)
1 i (cid:12) rpl,i (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)2
0 δri = (cid:12)(cid:12)rpl,i−rpTlB,i(cid:12)(cid:12) (5)
1 2 3 4 5 6 7 8 9
Number of Planet (cid:16) r −rTB(cid:17) 2
δr = pl,i pl,i , (6)
i
r
pl,i !
Fig.1. Distancesofobserved(fullcircles)andpredicted(opencircles)
planets (orplanetary positions)inthe55Cncsystem. Thepredicted wherer denotestheobserveddistance(semi-majoraxis)of
pl,i
planetshavebeencomputedfollowingtheTBRbasedonMDIF1(see planeti and rTB denotesthe distance valueobtainedthrough
Table2). pl,i
theapplicationoftheTBR.
Table4givesdetailedinformationontheTBRparametersA,
B,andZ,includingtherespectiveuncertaintybars,whichare
duetothedistanceuncertaintiesof55Cncbtof(seeTable1)
givenbyobservations. ItisfoundthatA≃0.031,B≃0.049,
and Z ≃ 1.98. There is very little impact by the particular
choiceofmeasureofdifference. Forexample,thespacingpa-
rameter Z is found to vary between 1.969±0.006 (MDIF 3)
No.] Usageofpasj00.cls 5
A, B, and Z, the smaller the resulting residual. Second, the
residual cannot approach zero, even if an increasingly higher
0.03 0.2 precisionforA, B, andZ ispermitted, owingtothe factthat
55 Cnc f
HPL 3 theTBR(ifapplicable)shouldbeviewedasa“rule”ratherthan
55 Cnc d a(physical)“law”. ThereisnowaythattheTBRasproposed
0.015 0.1 (oranymodificationthereof)willbeabletopreciselyrepresent
PL 3] d] thepositionsofthecurrentlyknownfiveplanetsinthe55Cnc
nc f, H 0 0 5 Cnc sSyosltaermSyassteamls.o encounteredin applications of the TBR to the
C 5
55 δr [ NextwefocusonthecaseofthehypotheticalplanetHPL3
δr [ (see Table 5) positioned near 1.51 AU; see Table 7 for de-
−0.015 −0.1
tails. Concerningthe four measuresof difference, MDIF, the
position of HPL 3 is found to range between 1.504±0.025
(MDIF 4) and 1.511±0.028 (MDIF 3). Note again that the
−0.03 −0.2
11 22 33 44 55 66 77 88 991100111111221133114411551166117711881199220022112222223322442255 uncertaintyinthepositionofHPL3duetothechoiceofMDIF
Serial Number of Simulation isinsignificant. However,thishypotheticalplanetisofpoten-
tiallyhighrelevancebecauseitmaybelocatedin55Cnc’shab-
Fig.2. Distancevariationsδrfor55Cncf,55Cncd,andthehypothet-
itablezone(HZ).Therefore,itwillbediscussedinmoredetail
icalplanet3foraseriesof25statisticalsimulations. For55Cncfand
below(seeSect.2.3).
55Cncd, thesevariations areduetotheobservational uncertainties,
whichare0.007and0.11AU,respectively. ForHPL3,thedistance ConsideringthepotentialimportanceofHPL 3, ifexisting,
variations werecomputedfollowing theTBRbasedonMDIF1(see we also investigatedthe distancevariationδr duetothe mea-
Table2). surementuncertaintiesfor55Cncfand55Cncd(seeTable1)
aspublished;theuncertaintymeasurementismostsubstantial
and1.991±0.004(MDIF4).Furthermore,thedistanceuncer- for55Cncd(bothinabsoluteandrelativeunits). Theresults
taintiesfor55Cncbtof,givenbytheobservations,havealso are givenin Fig.2; here theimplementedversionof theTBR
arelativelysmalleffectonthevaluesofA,B,andZ;notethat was based on MDIF 1. Various positions for 55 Cnc f and
theassociateduncertaintyis0.3%orlessincaseofZ. 55CncdwereassumedusingaMonte–Carloapproach(Press
etal.1989)consideringtheirobservationallydeduceddistance
2.2. ApplicationsandTests
measurements. We pursued a total of 25 simulations. In our
The derivationof the TBR parametersA, B, andZ can be seriesofsimulations,theabsolutevariationδr forHPL3was
used to identify possible additional planetary positions in the lessthan0.03AU.
55 Cnc system (see Table 5), potentially occupied by plan- Another application of the TBR is the computation of the
ets. Specifically,fourplanetarypositionsarefound,whichare: distanceofthenextpossibleouterplanetofthe55Cncsystem,
0.081, 0.41, 1.51 and 2.95 AU. As expected from our previ- i.e.,locatedbeyond55Cncd,whichisatadistanceofapprox-
ousdiscussion,theinfluenceofthedifferentchoicesofMDIF imately5.77AU.Thishypotheticalplanetispredictedtobelo-
is only of very minor importance. Figure 1 offers a detailed catedbetween10.9and12.2AU(seeTable8).Moreprecisely,
depiction of the observed and predicted hypothetical planets. its position is predicted in the range between 11.33±0.43
Notetheveryhighqualityofthefitfori>1. (MDIF3)and11.66±0.39(MDIF4).
Prior to discussing further results of our study, we want to
2.3. TheHabitableZoneof55Cnc
presentthefindingsfromdetailedtestsabouttheconvergence
oftheTBRparameters,notablyZ. Again,theTBRparameters Inthefollowingwediscussthepossiblelocationofanewly
A, B, and Z are obtained by trying to fit the positionsof the predictedplanetintheHZ of55Cnc. Theextentof55Cnc’s
known planets 55 Cnc b to 55 Cnc f by using the TBR. This HZcanbecalculatedfollowingtheformalismbyUnderwood
requires the choice of a measure of difference, MDIF; addi- etal.(2003)basedonpreviousworkbyKastingetal.(1993).
tionally,italsorequiresthechoiceofaprecisionparameterfor Underwoodet al. (2003)supplieda polynomialfit depending
A,B,andZ thatwaschosenas10−k withk=1,2,3,and4. on the stellar luminosity and the stellar effective temperature
Thecontrolparameterofconvergencewasthe summedresid- that allows to calculate the extent of the conservativeand the
ualofthedifferenceinthepositionsbetweentheobservedand generalizedHZ. Noting that55Cnc is less luminousthanthe
predicted (fitted) distances of the five known 55 Cnc system Sun, it is expected that its HZ is less extended than the solar
planets. HZ,forwhichthelimitsofthegeneralizedHZhavebeengiven
Our results are given in Table 6. As expected, it is found as0.84and1.67AU,respectively(Kastingetal.1993).
that the higher the selected precision, the more accurate are Theluminosityof55Cnchasbeendeducedas0.61±0.04
the obtainedvaluesfor A, B (notshown), and Z deducedby L basedonthestellardistanceandapparentmagnitude;note
⊙
fittingthepositionsofthefiveobservedplanets. However,the that this value is fully consistent with the standard values for
summed residuals are found to never approach exactly zero, the stellar effectivetemperatureand luminosity (see Table 1).
regardlessofMDIFandtheprecisionparameter,althoughthe However, if its stellar radius is 1.15 ± 0.035 R , obtained
⊙
residualwillconvergetowardaverysmalllimit(i.e.,between byBainesetal.(2008)basedonnewinterferometricmeasure-
10−2 and 5×10−4), especially if MDIF 3 is selected. This ments,insteadofR=0.925R (Ribasetal.2003),therevised
⊙
resultis expected. First, the higherthe intendedprecisionfor stellarluminosityisthusgivenas0.90L . Thisincreasedlu-
⊙
6 AstronomicalSocietyofJapan [Vol.,
Table7.DistanceoftheHypotheticalPlanet3
MDIF Distance(AU)
s
et MDIF1 1.506±0.027
n MDIF2 1.509±0.029
a
Pl MDIF3 1.511±0.028
MDIF4 1.504±0.025
0.5 1 1.5 2 2.5
Distance (AU)
minosity is, however,difficult, thoughnot impossible, to rec-
oncilewiththeotherparametersof55Cnc,suchasitsapparent
s
et magnitude,spectraltypeanddistance.
n
a Based on the standard values for the stellar parameters of
Pl 55 Cnc, the limits of the conservative HZ are given as 0.75
and 1.08 AU, whereas the limits of generalizedHZ are given
0.5 1 1.5 2 2.5
as 0.68 and 1.34 AU (see Fig. 3 and Table 9). Surely, larger
Distance (AU)
limitsareattainedifthehighervalueforthestellarluminosity
is adopted. The underlyingdefinition of habitability is based
ontheassumptionthatliquidsurfacewaterisaprerequisitefor
Fig.3. Distancesof55Cncf(fullcircle)andthehypotheticalplanet
3(opencircles)inrelationshiptotheHZof55Cnc,wheredarkgray, life, akeyconceptthatisalsothebasisofongoingandfuture
mediumgrayandlightgraydenotetheconservative, general,andex- searchesforextrasolarhabitableplanets(e.g.,Catanzariteetal.
tremeHZ,respectively(seeTable9).ThepositionofHPL3wascom- 2006;Cockelletal.2009). Thenumericalevaluationofthese
putedfollowingtheTBRbasedonMDIF1,2,3,and4(fromtopto
bottom). Thedepicteduncertaintybarsare3σ,andaremostlydueto limits is based on an Earth-type planet with a CO2/H2O/N2
atmosphere.
thesmallobservationaluncertaintiesof55Cncfand55Cncd.Thetop
figureshowstheHZof55CncbasedonR=0.925R⊙(Ribasetal. We pointoutthatconcerningthe outeredgeofhabitability,
2003),whereasthebottomfigureassumesR=1.15R⊙(Bainesetal. evenlessconservativelimitshavebeenproposedinthemean-
2008),implyingarelativelyhighstellarluminosityof0.90L⊙. time(e.g.,Forget&Pierrehumbert1997;Mischnaetal.2000).
Theyarebasedontheassumptionofrelativelythickplanetary
CO2 atmospheresinvokingstrongbackwarmingthatisfurther
enhancedby the presence of CO2 crystals and clouds. These
types of limits can be as large as 2.4 AU in case of the Sun;
however, they depend on distinct properties of the planetary
atmosphere, and are thus subject to ongoingstudies and con-
Table6.ConvergenceofTitius–BodeParameters troversies (e.g., Halevy et al. 2009). Concerning55 Cnc, the
revisedouterlimitcanbeaslargeas1.85AUforthestandard
MDIF Precision Za Residual stellar parameters and up to 2.31 AU for the increased value
MDIF1 10−1 2.1 4.41×10−1 of the stellar luminosity. These limits can be considered in
10−2 1.98 3.90×10−2 the assessment of the hypothetical planet near 1.51 AU (see
10−3 1.973 3.31×10−2 Table 7). Note that the position of this hypotheticalplanet is
10−4 1.9704 3.03×10−2 notsignificantlyaffectedbythechoiceofMDIF.Basedonthe
MDIF2 10−1 2.0 3.93×10−1 standardparametersfor55Cnc,thisplanetwouldbelocatedin
10−2 1.98 2.04×10−1 theextremelyextendedHZ.Iftheincreasedvalueofthestellar
10−3 1.971 1.91×10−1 luminosity of 0.90 L⊙ is assumed, it would be located in the
10−4 1.9704 1.90×10−1 generalizedHZ.Adepictionofthedifferentscenariosisgiven
MDIF3 10−1 2.1 6.16×10−2 inFig.3.
10−2 1.98 5.94×10−4 Another important aspect regarding the circumstellar hab-
10−3 1.969 4.56×10−4 itability of 55 Cnc concerns the planet 55 Cnc f. It is a hot
10−4 1.9691 4.52×10−4 Neptunewithamassofabout15%ofJupiter’smass;therefore,
MDIF4 10−1 2.0 3.80×10−2 itisalmostcertainlyunfittohostlife. 55Cncfislocatedator
10−2 1.99 1.68×10−2 neartheinneredgeofthe55Cnc’sHZ,especiallyifthestan-
10−3 1.991 1.68×10−2 dard value for the 55 Cnc’s luminosity is adopted. However,
10−4 1.9908 1.68×10−2 55Cncfmaybeanappropriateobjectforhostingoneormore
habitablemoons.Previousstudiesonhabitablemoonsofextra-
aAnequivalentincreaseinprecisionisalso attainedfor solar planets have been given by, e.g., Williams et al. (1997)
theparametersAandB. andDonnison(2010).
No.] Usageofpasj00.cls 7
Table8.DistanceofanHypotheticalExteriorPlanet Table9.HabitableZoneof55Cancri
MDIF Distance(AU) Description Distance(AU)
MDIF1 11.35±0.45 ... Standard Alternative
MDIF2 11.35±0.44
HZ-i(general) 0.68 0.84
MDIF3 11.33±0.43
HZ-i(conservative) 0.75 0.93
MDIF4 11.66±0.39
HZ-o(conservative) 1.08 1.35
HZ-o(general) 1.34 1.66
HZ-o(extreme) 1.85 2.31
3. ConsiderationofOtherStudies
3.1. Constraints from Orbital Stability Simulations and
CommentsonPlanetaryFormation planetsis relativelyinsensitive tostellar mass. Thisworkhas
been expanded by Raymond et al. (2006) and applied to the
Anyplanetproposedtoexistinastar–planetsystembythe
special case of 55 Cnc. They found that the majority of the
meansoftheTBRorotherwiseneedstopassthetestoforbital
simulationsresultedintheformationofterrestrialplanetsposi-
stability.Previously,detailedsimulationsoforbitalstabilityfor
tionedin55Cnc’sHZ,includingplanetsnear1.5AUfromthe
thesystemof55Cnchavebeengivenby,e.g.,Raymondetal.
star.
(2008)takingintoaccountthefiveknownplanetsofthe55Cnc
The existence of additional planets, including habitable
system. Raymond et al. (2008) presented a highly detailed
planets,inthegapbetween55Cncfand55Cncd,i.e.,between
studywithparticularfocusontheeffectsofmeanmotionreso-
0.781and5.77AU,alsoappearstobeconsistentwiththeexis-
nancesduetotheplanets55Cncfand55Cncd. Theyshowed
tenceofaprotoplanetarydiskestimatedtohavearadialsurface
that additional planets could exist between these two known densityprofilesuchasσ(r)=r−3/2(Fischeretal.2008).They
planets, including hypothetical planets located near 1.51 and
alsodiscussedtheoverallsurfacedensityofthedisk,whichis
2.95AU.Specifically,iftwoadditionalplanetsareassumedto
estimated to be considerably higher than in the solar nebula.
exist between 55 Cnc f and 55 Cnc d, the domains of stabil-
Theformationofterrestrialplanets2 in thegapisfurthermore
ity aregivenas1.3–1.6and2.2–3.3AU. Thisis an important
indirectlyimpliedby therelativelyhighcontentofheavyele-
findingregardingthehypothetical,potentiallyhabitableplanet
mentsasindicatedbytheenhancedmetallicityof55Cncitself
HPL3,proposedat1.51AU(seeTable7).
givenas[Fe/H]=+0.31±0.04(Valenti&Fischer2005).
A subsequent study of orbital stability was given by Ji et
al. (2009). They also considered the impact of mean motion 3.2. PreviousApplicationoftheTBRto55Cnc
resonancesandidentifiedvariousunstablelocations.However,
A previous attempt to predict additional planets in the
they identified a wide region of orbital stability between 1.0
55CncsystembasedontheTBRhasbeengivenbyPoveda&
and2.3AU,apotentialhomesteadofhabitableterrestrialplan-
Lara(2008).Atthattimeallcurrentlyknownplanets55Cncb
ets. Thesepotentialplanetswerealsoidentifiedtohavearela-
to 55 Cnc f were already discovered. Poveda & Lara (2008)
tivelyloworbitaleccentricity.Furthermore,theorbitalstability
assumeda“law”oftheformof
simulationsby Raymondet al. (2008)and Ji et al. (2009)are
also lendingindirectsupportto the principalpossibility of an aTB = α·eλi (7)
i
exterior planet, which, if existing, should be located between
(inAU)withαandλasfreeparameters. Takingintoaccount
10.9and12.2AUfromthestar(seeTable8).
theplanets55Cnce,55Cncb,55Cncc,and55Cncf,located
Although orbital stability for the hypothetical planet near
between 0.038 (originally determined position for 55 Cnc e)
1.51AUappearstobewarranted,itisstillimportanttogaugeif
and0.781AU,andrepresentedbyi=1to4,αandλwerede-
suchaplanetcouldhaveformedinthefirstplacenotingthatthe
ducedas0.0148and0.9781,respectively. ThismodifiedTBR
existenceofclose-ingiantplanet(s)havebeenviewedtohave
is successful to also “predict” the outermost planet 55 Cnc d
an adverse effect (Armitage et al. 2003). Subsequent studies
at5.77AU,countedasi=6,whichhoweverwaswell-known
byRaymondetal. (2005,2006)havepointedoutthatthefor-
whenPoveda&Lara(2008)proposedthenewlaw.
mationandhabitabilityofterrestrialplanetsinthepresenceof
Making 55 Cnc d also part of the fit results in a small ad-
close-ingiantplanetsisindeedpossible.
justment of the parameters α and λ, which then acquire the
Raymond et al. (2006) gave detailed simulations also en-
valuesof0.0142and0.9975,respectively. Themainaspectof
compassing the system of 55 Cnc. Assuming that the giant
the work by Poveda& Lara (2008)is to predicta new planet
planetsformedandmigratedquickly,theyfoundthatterrestrial
fori=5andpossiblyalsoanexteriorplanetfori=7aswell.
planetsmaybeabletoformfromasecondgenerationofplan-
The correspondingdistancesof these hypotheticalplanetsare
etesimals. For55Cnc,objectswithmassesupto0.6M and,
⊕ given as 2.08 and 15.3 AU; the correspondingorbitalperiods
insomecases,substantialwatercontentareabletoform;these
oftheplanetsareapproximately3.1and62years,respectively.
objectsare foundin orbitin 55 Cnc’s HZ. This type of result
Noneoftheseputativeplanetsareexpectedtobelocatedinthe
isalsoconsistentwithfindingsfromthequantitativenumerical
programbyWetherill(1996),whodemonstratedthathabitable 2 FollowingthestudybySmith&Lissauer(2009),anyadditionalplanetsin
terrestrialplanetscanformforawiderangeofmain-sequence the55Cncsystemareexpectedtohavearelativelysmallmasstowarrant
stars,notingthatthenumberanddistancedistributionofthose thesystem’sorbitalstability.Thelowmassofanypossibleplanetcanalso
readilyexplainwhysofartheywereabletoescapedetection.
8 AstronomicalSocietyofJapan [Vol.,
55 Cnc’s HZ. Moreover, the work by Poveda & Lara (2008) oftheplanetarysequence. However,thenewlyproposedTBR
assumesthe planet55Cnc e to belocatedat0.038AU rather has the attractive feature that it is of virtually the same form
than0.01583AUasidentifiedbyWinnetal.(2011);notethat as that of the Solar System; see Eq. (1). Here the parameter
Poveda&Larapublishedtheirworkbeforetheupdatedresult B describes the compactness of the system3, which is much
became available. Thus, it is unclear if or how the “law” by higherthanforthe SolarSystem. Furthermore,theparameter
Poveda & Lara (2008) could be altered to accommodate this Z describes the relative geometricalspacing of the system; it
finding.Aless-than-idealapproachcouldbetoassigni=0.11 isvirtuallyidenticaltothatoftheSolarSystem,i.e.,Z=2.0.
(orroundedtoi=0.1)totheinnermostplanet55Cnce. Perhapssurprisingly,theTBRappearstobeaboutasappli-
cable to the 55 Cnc system as to the Solar System gaugedby
4. EvaluationofStatisticalSignificance whether or not the degreeof similarity between the predicted
and observed distance sequences can occur by chance (see
Animportantaspectofthisstudyistoevaluatethestatistical Introduction for data on the Solar System). Thus, the appli-
significance of the TBR as deduced. In this case, we largely cationoftheTBRto55Cncappearstobejustified. However,
followthemethodpreviouslyappliedbyLynch(2003)toboth considerablecareisrequiredtodrawfinalconclusionsbecause
theSolarSystemandtheUraniansatellite system. Inthefirst thefittingoffiveknownplanetsassumingfourgapsinthese-
step, the mean square deviation is calculated. For the TBR quenceis mucheasierthanfitting the continuoussequenceof
proposedinthepresentwork,wefind ninedistances,i.e.,withthemeanasteroidaldistanceincluded
andPlutoomitted; see Neslusˇan(2004). Itwillbe thetask of
χ2fit = 1 log(ai−A)− logB+(i−2)·logZ 2(8) futureplanetarysearchmissionstoinquireiftheplanetarysys-
4
i temof55Cncfollows(1)theTBRofthiswork,(2)theTBR
X(cid:2) (cid:0) (cid:1)(cid:3)
wherea constitutetheobserveddistancesfortheplanetsi=3, byPoveda&Lara(2008),(3)anotherversionofTBR-typese-
i
4,6,and9. ForA,B, andZ,weadopttheaveragevaluesat- quencing,or(4)noTBR-typesequencingafterall,asfoundfor
tainedbythesetsofrunspertainingtothefourdifferentmea- thevastmajorityofthecurrentlyknownplanetarysystems.
suresofdifference,MDIF(seeTable4). Furthermore,wepur-
sued a sufficently large number N of statistical model runs, 5. SummaryandConclusions
whereforeachmodelj weobtain
The goal of our study is to provide an application of the
χ2 = 1 log(a −A)− logB+(i−2+ky)·logZ 2(9) TBR to the 55 Cancri star–planet system. It is clearly un-
j 4 i i derstoodthattheTBRdoesnothavethesamestandingasthe
i
X(cid:2) (cid:0) (cid:1)(cid:3) variouswell-establishedmethodsforthesearchandidentifica-
withy asa randomnumberintherange[−1/2,+1/2]andk
i tionofextrasolarplanets;see,e.g.,Jones(2008)foradetailed
takenask=2/3(Lynch2003). FortheTBRasproposed,we
figinvdenχafist1=.586.·4170·−110−w2it;hmNor=eo1v0e7r,.the average value of χj is oapvperevaireswto.Hboewsuevcceer,sassfuplo(iwntiethdionultiminitths)efIonrtraodlaurcgteiosne,gtmheeTntBoRf
theSolarSystem. Basedonprincipalconsiderations,itwould
Equivalent expressions hold if the method suggested by
therefore be highly extremely unlikely if there was no other
Lynch(2003)isappliedtotheversionofTBRpreviouslygiven
planetarysystem forwhicha TBR-typeplanetspacingwould
byPoveda&Lara(2008).Inthiscase,wefind
berealized,althoughthenumberofthosesystemsmayberel-
χ2 = 1 log a − logα+iλ 2 (10) ativelysmall.
fit 4 i Thus, the predictionsconveyedin thisstudy followthe hy-
i
X(cid:2) (cid:0) (cid:1)(cid:3) pothesis that the TBR is applicable to the 55 Cnc planetary
fori=2,3,4,and6,and
system. Thisapproachentailsthepredictionoffourplanetary
χ2 = 1 log a − logα+(i+ky )λ 2. (11) positionsinteriortotheoutermostplanet55Cncd, whichare
j 4 i i given as 0.081, 0.41, 1.51, and 2.95 AU. Whether these po-
Xi (cid:2) (cid:0) (cid:1)(cid:3) sitions are “empty”, filled with asteroid / comet belts (like in
For the TBR by Poveda & Lara (2008), we find χfit =9.67· one instant of the Solar System, where Ceres is customarily
10−2andtheaveragevalueofχj isgivenas2.15·10−1. viewed as substitute for a planet)or occupiedby a developed
Evidently,eachindividualvalueofχjcaneitherbeχj≤χfit planetisbeyondthescopeofthiswork;inthiscaseadditional
or χj > χfit. The amount of χj values with χj ≤χfit gives astrophysicalrequirementsneedtobemet.
theprobabilitythatthesequenceascalculatedbytheTBRand Pertaining to this latter point, the strongest case for addi-
theobservedsequencecanoccurbychange. Basedonthisap- tionalplanets around55 Cnc is given for the possible planets
proach,itisfoundthatthenewlyproposedTBRissignificantat near 1.51 and 2.95 AU, notingthat they would be most com-
alevelof91.8%,whereastheTBRproposedbyPoveda&Lara patiblewithdetailedstudiesoforbitalstability(Raymondetal.
(2008)issignificantatalevelof96.5%. Thus,theanalysisof 2008;Jietal.2009)aswellaspreviousstudiesabout55Cnc’s
statistical significance of these two differentversionsof TBR proposedprotoplanetarydisk(Fischeretal.2008). Forexam-
for55Cncshowsthatbothofthemareinherentlyhighlysignif- ple,Raymondetal.(2008)arguedthatadditionalplanetscould
icantalbeitthelevelofsignificancefortheversionpreviously
given by Poveda & Lara (2008)is slightly higher. Moreover, 3 Theincreasedcompactnessoftheplanetarysystemof55Cnccompared
totheSolarSystemisalreadyimpliedbytheknownfiveplanets,which
theTBRattainedbyPoveda&Lara(2008)isbasedontwofree
inspired coining the term “packed planetary systems” hypothesis (e.g.,
parametersinsteadofthreefreeparametersforthedescription
Raymondetal.2005).
No.] Usageofpasj00.cls 9
existbetween55Cncfand55Cncd,includingplanetslocated Cockell,C.S.,etal.2009,Astrobiology,9,1
near1.51and2.95AUintheviewoflong-termorbitalstability deStrobel,G.C.,Soubiran,C.,&Ralite,N.2001,A&A,373,159
simulations.Specifically,iftwoadditionalplanetsareassumed Dawson,R.I.,&Fabrycky,D.C.2010,ApJ,722,937
toexistbetween55Cncfand55Cncd,thedomainsofstabil- Donahue, R. A. 1998, in ASP Conf. Ser., 154, Cool Stars, Stellar
Systems, and the Sun, Tenth Cambridge Workshop, ed. R. A.
itywerederivedas1.3–1.6and2.2–3.3AU.Thesetworanges
Donahue&J.A.Bookbinder(SanFrancisco:ASP),CD1235
arefullycompatiblewiththeresultsofthepresentstudy.
Donnison,J.R.2010,MNRAS,406,1918
Moreover, the possible planet near 1.51 AU is of particuar
EuropeanSpaceAgency1997,TheHipparcosandTychoCatalogues
interest as it appearsto be located at the outskirts of the stel-
(SP-1200)(Noordwijk:ESA)
larHZ.Theexistenceofthisplanetisconsistentwithbothor-
Fischer,D.A.,&Valenti,J.2005,ApJ,622,1102
bital stability simulations and studies of the formationof ter- Fischer,D.A.,etal.2008,ApJ,675,790
restrialplanetsfor55Cnc. Forexample,calculationsofterres- Forget,F.,&Pierrehumbert,R.T.1997,Science,278,1273
trialplanetformationbyRaymondetal.(2006)showedthatthe Fuhrmann,K.,Pfeiffer,M.J.,&Bernkopf,J.1998,A&A,336,942
majorityoftheirsimulationsleadtothebuild-upofterrestrial Gonzalez,G.1998,A&A,334,221
planetsin55Cnc’sHZ,includingplanetsnear1.5AUfromthe Gonzalez,G.,&Vanture,A.D.1998,A&A,339,L29
star. Nonetheless, the final verdictof potentialhabitabilityof Graner,F.,&Dubrulle,B.1994,A&A,282,262
Halevy, I., Pierrehumbert, R. T., &Schrag, D.P.2009, J. Geophys.
thispossibleplanet(ifexisting)dependsonasizeablenumber
Res.,114,D18112
of factors, including the definition of the zone of circumstel-
Hayes,W.,&Tremaine,S.1998,Icarus,135,549
larhabitability,consideringthattheouterboundaryofthestel-
Henry,G.W.,Baliunas,S.L.,Donahue,R.A.,Fekel,F.C.,&Soon,
lar HZ is notablyimpactedby a variety of astrobiologicalas-
W.2000,ApJ,531,415
pects(e.g.,Lammeretal.2009)includingthestructure,density Jewitt,D.C.,&Luu,J.X.2001,AJ,122,2099
andcompositionsof the planetaryatmosphere(e.g., Forget& Ji, J.-H., Kinoshita, H., Liu, L., & Li, G.-Y. 2009, Res. Astron.
Pierrehumbert1997;Mischnaetal.2000;Halevyetal.2009). Astrophys.,9,703
TheTBR asimplementedfollowscloselythe previousver- Jones,B.W.2008,Int.J.Astrobiol.,7,279
sion as appliedto the Solar System with A, B, and Z as free Kasting,J.F.,Whitmire,D.P.,&Reynolds,R.T.1993,Icarus,101,
parameters,whichwereobtainedthroughcarefulstatisticalfit- 108
Lammer,H.,etal.2009,A&ARev.,17,181
tingconsideringthefiveknownplanetsofthe55Cncsystem.
Li,X.Q.,Zhang,H.,&Li,Q.B.1995,A&A,304,617
Here we obtain A≃ 0.031, B ≃ 0.049, and Z ≃ 1.98. The
Lissauer,J.J.1993,ARA&A,31,129
parameter B indicates the compactness of the planetary sys-
Lynch,P.2003,MNRAS,341,1174
tem, which is much higher than for the Solar System as also
Malhotra,R.1993,Nature,365,819
indicated by the conventionalversion of Solar System’s TBR
Marcy,G.W.,etal.2002,ApJ,581,1375
(B =0.3). The parameter Z indicates the relative geometri- McArthur,B.E.,etal.2004,ApJL,614,L81
calspacing ofthe planetarysystem; itis virtuallyidenticalto Mischna, M. A., Kasting, J. F., Pavlov, A., & Freedman, R. 2000,
thatoftheSolarSystem(Z =2.0). Wealsocomputedtheor- Icarus,145,546
bitaldistanceofathenextpossibleouterplanetofthe55Cnc Neslusˇan,L.2004,MNRAS,351,133
system, i.e., exterior to 55 Cnc d. Our study predicts that if Newman,W.I.,Haynes,M.P.,&Terzian,Y.1994,ApJ,431,147
existing it would be located between 10.9 and 12.2 AU from Nieto,M.M.1972,TheTitius–BodeLawofPlanetaryDistances: Its
HistoryandTheory(Oxford:Pergamon)
thestar. Thisdistanceisconsistentwithpreviousorbitalstabil-
Nottale,L.1996,A&A,315,L9
ity analyses, which predict the position of a possible exterior
Nottale,L.,Schumacher,G.,&Gay,J.1997,A&A,322,1018
planetasbeyond10AU(Raymondetal.2008).
Poveda,A.,&Lara,P.2008,Rev.Mex.Astron.Astrof´ısica,44,243
In summary, it is obviously up to future observational ef-
Press, W. H., Flannery, B. P., Teukolsky, S.A., & Vetterling, W. T.
fortstoverifyorfalsifytheexistenceofanypossibleadditional 1989,NumericalRecipes.TheArtofScientificComputing(New
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in gauging the applicability of the TBR outside of the Solar Raymond,S.N.,Quinn,T.,&Lunine,J.I.2005,Icarus,177,256
System, and will help to discriminate between different ver- Raymond,S.N.,Barnes,R.,&Kaib,N.A.2006,ApJ,644,1223
sionsoftheTBRiffoundapplicabletothe55Cncstar–planet Raymond,S.N.,Barnes,R.,&Gorelick,N.2008,ApJ,689,478
system. Ribas, I., Solano, E.,Masana, E., &Gime´nez, A.2003, A&A,411,
L501
Theauthoracknowledgesvaluablecommentsbyananony- Rivera,E.,&Haghighipour,N.2007,MNRAS,374,599
Santos,N.C.,Israelian,G.,&Mayor,M.2004,A&A,415,1153
mousreferee,whichalsoresultedintheadditionofSection4.
Schmitz,F.1984,A&A,131,309
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