The Roles of Discs for Planetary Systems

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by  Li-Chin Yeh

Proceedingsof9thAsian-PacificRegionalIAUMeeting2005, 1–2(2005) Printed5February2008 (APRIMLATEXstylefilev1.0)
The Roles of Discs for Planetary Systems
Li-Chin Yeh1 and Ing-Guey Jiang2
1Department of Applied Mathematics, National Hsinchu Universityof Education, Hsin-Chu, Taiwan
2Institute of Astronomy, National Central University,Chung-Li, Taiwan
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ABSTRACT
n
a
J It is known that the discs are detected for some of the extra-solar planetary
systems. It is also likely that there was a disc mixing with planets and small bodies
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2 while our Solar System was forming. From our recent results, we conclude that the
discs play two roles: the gravity makes planetary systems more chaotic and the drag

1 makes planetary systems more resonant.
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5 Key words: celestial mechanics – planetary systems – stellar dynamics
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1 INTRODUCTION merical surveys to explore the possible chaotic and regu-
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0 lar orbits. They found that, in general, discs with different
Morethan160extra-solarplanetarysystemshavebeendis-
/ masses might change the sizes and locations of chaotic re-
h coveredsofarandthenumberofdetectedexoplanetsislikely
gion. Some sample orbits are further studied and the plots
p toincreaserapidly.Thisdevelopmenthasopenedanewwin-
ofPoincar´esurfaceofsectionareconsistentwiththeresults
- dow for astronomers to investigate the planetary formation
o ofLyapounovExponentIndicator.Ontheotherhand,they
and evolution in detail. There are many numerical studies
r pointedoutthattheinfluenceofthedisccanchangethelo-
t onthedynamicalevolutionofthesedetectedsystems(please
s cationsofequilibriumpointsandalsotheorbitalbehaviors.
seeLaughlin & Adams1999, Rivera& Lissauer 2000, Jiang
a Thispointisparticularly interesting.Withouttheinfluence
: & Ip 2001, Ji et al. 2002, Benest 2003).
v from thedisc, Wisdom (1983) showed that oneof theKirk-
Amongthesesystems,theexistenceofdiscsseemstobe
i woodgapsmightbeexplainedbythechaoticboundary.This
X ageneralphenomenon.Forinstance,thereareasteroid belt
is probably due to that the chaotic orbits are less likely to
and Kuiper belt for the Solar System, the discs of dust for
r become resonant orbits. Indeed, the chaotic orbits in the
a extra-solar planetary systems and also circumbinary rings
disc-star-planet system shall play essential roles during the
for binary systems. The disc or belt-like structure should
formation of Kirkwood gaps. For example, the disc might
influencethedynamicalevolutionofthesesystems.Asmen-
make some test particles to transfer from chaotic orbits to
tionedinJiang&Ip(2001),theoriginoforbitalelementsof
regularorbitsorfromregularorbitstochaoticorbitsduring
theplanetarysystemofupsilonAndromedaemightrelateto
theevolution.
the disc interaction. Moreover, Yeh & Jiang (2001) studied
theorbital migration of scattered planets. They completely Thediscsareoftenobservedwithvariousmassesinboth
classifytheparameterspaceandsolutionsandconcludethat theyoungstellarsystemsandextra-solarplanetarysystems.
the eccentricity always increases if the planet, which moves It is believed that the proto-stellar discs are formed during
on circular orbit initially, is scattered to migrate outward. theearlystageofstarformationandtheplanetsmightform
Thus,theorbitalcircularizationmustbeimportantforscat- on the discs through the core-accretion or disc instability
tered planets if they are now moving on nearly circular or- mechanism. The planetary formation timescale is not clear
bits. butthegaseousdiscwillbesomehowdepletedandthedust
To concentrate on the dynamical role of discs for the particleswill gradually form.Theabovepictureisplausible
planetarysystems,Jiang&Yeh(2003,2004a)didsomeanal- but unfortunately the timescales are not completely under-
ysis on the orbital evolution for systems with planet-disc stood.Forasystemwithastar,planets,discsandasteroids,
interaction. itisnotclearwhattheorderofdifferentcomponent’sforma-
tionisandwhatthedurationofeachcomponent’sformation
process would be.
2 THE CHAOTIC ORBITS Nevertheless,onethingwecanconfirmisthatthediscs
are there and can have different masses at different stages.
Jiang&Yeh(2004b)studiedthechaoticorbitsforthedisc- From our results, it is clear that the dynamical properties
star-planet systems through the calculations of Lyapounov of orbits, i.e. chaotic or regular, can be determined quickly
Exponent. Different initial conditions are used in the nu- for different disc masses or models. Because those particles
2 Yeh and Jiang
moving on chaotic orbits might have smaller probability to Nakai(2001), Jietal.(2002) andGozdziewski &Maciejew-
become the resonant objects in the system, the Kirkwood ski(2001).
gaps, which are associated with the location of mean mo-
tionresonance,intheSolarSystemmightbeformed bythe
dynamical effect we have discussed here. The detail pro- 4 CONCLUDING REMARKS
cesses would depend on the evolutionary timescale of the
star, planet and also the disc. To conclude, it is important The conventional models used in celestial mechanics con-
siderthedynamicalinteractions between particles only.We
thattheeffectofdiscsonthechaoticorbitsmightinfluence
take a first step to include the dynamical effect from the
theformationofthestructuresofplanetarysystemssuchas
disc in the conventional celestial mechanics. In general, the
theKirkwood gaps in the Solar System.
disc shall vary with time. Nevertheless, in these studies,
we only consider the orbital behaviors during a timescale
shorter than the lifetime of the disc, the models shall give
3 THE RESONANCE
goodimplications forthefirsttrendoforbitalevolutionun-
der the disc potential. We find that when we consider the
Themeanmotionresonantrelationsbetweendifferentceles-
gravity only, there are more chaotic orbits when the disc
tialbodiesareconfirmedtobecommonphenomenaforboth
is included. We also consider the effect of an artificial drag
Solar System and extra-solar planetary systems. For exam-
andfindthatthedragcouldbringparticlesintoresonances.
ple,thegapsinthedistributionofasteroidswereconnected
Our results’ general conclusion is consistent with the ones
tomeanmotion resonancesbyKirkwood(1867) andknown
in Armitage et al. (2002) and Trilling et al. (2002), though
as Kirkwood gaps. Greenberg & Scholl (1979) classified the
theyhavedifferentfocuses,baseondifferentapproaches,and
proposed explanations into four groups: the statistical hy-
mainly studythegas-dynamical effects.
pothesis,thecollisional hypothesis,thecosmogonichypoth-
esis, and the gravitational hypothesis. Dermott & Murray
(1983) showedthatthereisagood correspondencebetween
ACKNOWLEDGMENT
thewidth of theKirkwood gapsand thewidth of thelibra-
tion zone when thedistribution of asteroid orbits is plotted WearegratefultotheNationalCenterforHigh-performance
in the a−e plane. They concluded that the gravitational Computing for computer time and facilities.
hypothesis was likely to be correct. This is completely con-
sistentwiththeresultsonthechaoticorbitsaswementioned
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