The Orbital Structure of Triaxial Galaxies with Figure Rotation

7 MB·English

by  Alex T. Deibel

DRAFTVERSIONJANUARY6,2011
PreprinttypesetusingLATEXstyleemulateapjv.11/10/09
THEORBITALSTRUCTUREOFTRIAXIALGALAXIESWITHFIGUREROTATION
ALEXT.DEIBEL,1,2 MONICAVALLURI,1 ANDDAVIDMERRITT,3
DraftversionJanuary6,2011
ABSTRACT
Wesurveythepropertiesofallorbitfamiliesintherotatingframeofafamilyofrealistictriaxialpotentials
withcentralsupermassiveblackholes(SMBHs).Insuchgalaxies,mostregularboxorbits(vitalformaintaining
triaxiality)areassociatedwithresonanceswhichoccupytwo-dimensionalsurfacesinconfigurationspace. For
slow figure rotation all orbit families are largely stable. At intermediate pattern speeds a significant fraction
1 of the resonant box orbits as well as inner long-axis tubes are destabilized by the “envelope doubling” that
1 arises from the Coriolis forces and are driven into the destabilizing center. Thus, for pattern rotation periods
0 2×108yr(cid:46)T (cid:46)5×109yr,thetwoorbitfamiliesthataremostimportantformaintainingtriaxialityarehighly
p
2 chaotic. Aspatternspeedincreasesthereisalsoasharpdecreaseintheoverallfractionofprogradeshort-axis
n tubes and a corresponding increase in the retrograde variety. At the highest pattern speeds (close to that of
a triaxialbars),box-likeorbitsundergoasuddentransitiontoanewfamilyofstableretrogradeloop-likeorbits,
J whichresembleorbitsinthree-dimensionalbars,andcirculateabouttheshortaxis. Ouranalysisimpliesthat
4 triaxialsystems(withcentralcuspsandSMBHs)caneitherhavehighpatternspeedslikefastbarsorlowpatten
speedsliketriaxialellipticalgalaxiesordarkmatterhalosfoundinN-bodysimulations. Intermediatepattern

] speeds produce a high level of stochasticity in both the box and inner long-axis tube orbit families implying
A thatstabletriaxialsystemsareunlikelytohavesuchpatternspeeds.
G
h. 1. INTRODUCTION Bryan & Cress 2007) with slow pattern speeds following a
p Itiswidelyacceptedthatsincebothellipticalgalaxiesand lognormaldistributioncenteredonΩp=0.148hkms−1 kpc−1.
- darkmatterhalosformviahierarchicalmergers, theyshould Bailin & Steinmetz (2004) found that the pattern speed of
o
be triaxial. If a significant amount of gas is present in the the figure rotation is correlated with the cosmological halo
r
t progenitorgalaxies,itsdissipativecondensationgenerallyre- spin parameter λ (Peebles 1969), but is independent of halo
as sults in systems that are more oblate than those produced mass. Figure rotation of triaxial dark matter halos has been
[ in purely collisionless collapse (Dubinski 1994; Kazantzidis suggestedasamechanismfordrivingspiralstructure, warps
et al. 2004; Debattista et al. 2008). However, recent stud- andbarsinspiralandellipticalgalaxieswithlowmassdisks
2 ies have shown that two orbit families that characterize tri- (Bureauetal.1999;Dubinski&Chakrabarty2009).Theonly
v axial systems (boxes and long-axis tubes) persist in signifi- family of triaxial stellar systems with strong figure rotation
3 cantnumbersevenwhenatriaxialsystemlooksalmostoblate is the bars of spiral and lenticular galaxies. Rapidly rotating
5 (vandenBosch&deZeeuw2010;Vallurietal.2010). Since (triaxial)barscanbealmostcompletelyregular(Pfenniger&
7 merging systems generally have angular momentum (either Friedli 1991) and have a complex orbital structure (Skokos
2 internal angular momentum of the individual progenitors or etal.2002a,b;Patsisetal.2002;Harsoula&Kalapotharakos
.
8 theangularmomentumoftheirrelativeorbitorboth),merger 2009).
0 remnantsgenerallyhaveangularmomentum. Iftheremnants Emsellem et al. (2007) introduced a luminosity weighted
0 aretriaxial,theycanexhibitfigurerotation: apropertythatis measureofthespecificline-of-sightangularmomentum,λR,
1 independentofthestreamingmotionsofindividualparticles. to quantify the angular momentum content of E/S0 galaxies
v: Althoughitislogicaltoassumethattriaxialellipticalgalax- observed in the SAURON sample. They found that this pa-
i iescanhavefigurerotation,itiscurrentlydifficult(ifnotim- rameter allows E/S0 galaxies to be classified into two sub-
X possible) to observationally distinguish between figure rota- groups−the“fastrotators”withλR>0.1and“slowrotators”
r tion (tumbling) and orbital streaming. Although figure rota- with λR <0.1. The slow rotators are likely to be triaxial el-
a tion was first proposed to explain “anomalous dust lanes” in lipticalssincetheyfrequentlydisplayisophotalandkinematic
triaxialellipticalgalaxies(vanAlbadaetal.1982),therehave twists,includingkinematicallydecoupledcores. Theytendto
beenonlyafewobservationalattemptstomeasurethepattern bemoreluminousandhaveshallowcentralcusps(Emsellem
speeds of either elliptical galaxies or dark matter halos (Bu- etal.2007;Cappellarietal.2007). Incontrast,“fastrotators”
reau et al. 1999; Jeong et al. 2007). Cosmological N-body aremorenumerousandclosetoaxisymmetric,butmayretain
simulations(withoutdissipation)predictthatdarkmatterha- asignificantpopulationoftriaxialorbitfamiliessuchasbox
losaresignificantlytriaxialandthatamajorityofdarkmatter andlong-axistubeorbits(vandenBosch&deZeeuw2010).
halos(∼90%)havefigurerotation(Bailin&Steinmetz2004; Theseauthorsfindthatevenasmallfractionofbox-likeorbits
can significantly alter the dynamical estimates of supermas-
siveblackhole(SMBH)masses. RecentN-bodysimulations
1DepartmentofAstronomy, UniversityofMichigan, AnnArbor, MI of mergers of gas rich disk galaxies that include star forma-
48109,[email protected]
tionanddissipationshowthatbothfastandslowrotatorscan
2Department of Physics and Astronomy, Michigan State University,
EastLansingMI,48824,USA beformedinmergersbetweendiskgalaxies. Thelatterclass
3DepartmentofPhysicsandCenterforComputationalRelativityand primarily form in major mergers with smaller gas fractions
Gravitation, RochesterInstituteofTechnology, 84LombMemorialDr., and are more triaxial (Jesseit et al. 2009). (Recently, Bois
Rochester,NY14623,USA
2 Deibel,Valluri,&Merritt
etal.(2010)haveshownthatduetonumericalresolutionef- showed that self-consistent triaxial models with figure rota-
fectsinN-bodysimulations,thefractionofmergerremnants tioncouldbeconstructedsuchthattheyhadeithernostream-
thataretriaxialslowrotatorsmayhavebeenunderestimated). ingmotionsormaximalstreaming.
Thusdespitetheabsenceofconcreteobservationalmeasure- Themainconclusionfromtheearlyworkonfigurerotation
ments, some fraction of triaxial elliptical galaxies and dark in triaxial galaxies with cores was that except for the Bin-
matterhaloscouldhavefigurerotation. ney instability associated with retrograde orbits confined to
There has been little work on the effects of figure rotation the equatorial plane, several stable sequences of orbits that
on triaxial galaxies in the last two decades − none on realis- parentedthemajororbitfamiliesexistedoverarangeofpat-
tic density profiles with central cusps and SMBHs. Binney ternspeedsandenergies,implyingthattriaxialgalaxieswith
(1981)studiedtheeffectoffigurerotationonthebehaviorof bothfast(rotationperiodT ∼106 yr)andmoderaterotation
p
theclosedperiodicorbitsintheequatorialplaneperpendicular speeds(T ∼109yr)couldexist. Ourfindingsherewillshow
p
totheaxisofrotation(z)offlattenedgalaxymodelsincluding that the presence of a central cusp or SMBH alters that con-
those with weak bars. He showed that closed retrograde or- clusion.
bitsinthex-yplanelyinginanannularregion(nowcalledthe Gerhard & Binney (1985) first proposed that the box or-
“Binney instability strip”) become unstable to perturbations bits which form the back bone of triaxial elliptical galaxies
perpendiculartotheplane. Healsoshowedthatthisinstabil- wouldbecomechaoticduetoscatteringbythedivergentcen-
ity strip moved inward as the speed of rotation of the figure tralforcearisingfromacentralblackhole. Thepresenceofa
increased. significantfractionofchaoticorbitsresultsinchaoticmixing
Heisleretal.(1982)studiedthestabilityofclosedorbitsin which can cause secular relaxation of orbits in phase space
atriaxialmodelwithacentraldensitycoresubjectedtoboth (Kandrup&Mahon1994)resultinginachangeintheshape
slow (pattern rotation period Tp ∼109 yr) and fast rotations of the galaxy from triaxial to axisymmetric on timescales of
(T ∼108 yr). They showed that the 1:1 periodic orbits that order∼30-50dynamicaltimes(Merritt&Valluri1996;Mer-
p
circulate about the long (x) axis of the model are stable to ritt&Quinlan1998). Gerhard&Binney(1985)hadalsoar-
figure rotation and are tipped about the y-axis by the Cori- gued, however, that if the triaxial potential had figure rota-
olis forces in a direction that depends on their sense of mo- tion,thenboxorbits(whicharecrucialtomaintainingtriaxi-
tion. Two such orbits exist: one rotates clockwise about the ality)wouldbelessaffectedbythecentralforceintherotating
x-axisandtheothercirculatesanti-clockwise. However,both framedueto“envelopedoubling”(deZeeuw&Merritt1983).
“tipped”orbitscirculatedabouttheshortaxisinthesamedi- Several early studies indicated that triaxial galaxies with
rection. (These orbits were termed “anomalous” by van Al- figurerotationcouldoftenhavesignificantfractionsofchaotic
bada et al. (1982).) The long-axis tube family that is “par- orbits. The studies by (Udry & Pfenniger 1988; Martinet &
ented” by the anomalous orbits was therefore also expected Udry 1990; Udry 1991; Tsuchiya et al. 1993) were however
to be stable and also “tipped” at an angle. They noted that restricted to limited numbers of orbits in models with cores
for orbits with very large energies (i.e., extending to large anddidnotpursuetheprimarycauseofthischaos.
radii) such orbits could be tipped by ∼90◦ into the equato- Itisnowbelievedthatallellipticalgalaxieshaveeithershal-
rialplane. Ourmoregeneralanalysisinmodelswithrealistic loworsteepcentraldensitycusps(Gebhardtetal.1996;Lauer
densityprofilesshowsthatwhileouterlong-axistubesindeed et al. 2007) and central SMBHs (Ferrarese & Merritt 2000;
behaveaspredicted, theinnerlong-axistubeswiththesmall Gebhardt et al. 2000). In realistic elliptical galaxy models
pericentricradiiareeasilydestabilizedbyfigurerotation(Sec- with cusps and black holes, a large fraction of phase space
tion3.2). is occupied by resonant and chaotic orbits (Miralda-Escude
Heisleretal.(1982)alsoshowedthatthenormalretrograde &Schwarzschild1989;Schwarzschild1993;Valluri&Mer-
orbits that lie in the equatorial plane were stable (except in ritt1998;Merritt&Valluri1999;Poon&Merritt2004). Al-
theBinneyinstabilitystrip,wheretheywereunstabletover- thoughthefractionofchaoticorbitsincreaseswithanincrease
ticalperturbations). Thesequenceofclosed,stableperiodor- inthestrengthofthedensitycusp(Schwarzschild1993;Mer-
bits identified by them were found to exist at both slow and ritt&Fridman1996;Merritt1997)orthemassofthecentral
fast pattern speeds and constituted one composite sequence blackhole(Merritt&Valluri1996),itisstillpossibletocon-
which was stable over the entire energy range. They specu- structtriaxialmodelsthatdonotevolverapidlyduetochaotic
lated that this implied that triaxial galaxies could have both mixingsolongasasignificantfractionoforbits(∼50%)are
fastandslowpatternspeeds. regular(Poon&Merritt2002,2004).
deZeeuw&Merritt(1983)complementedthiswork,with In a precursor to the present paper, Valluri (1999) first
a study of orbits in the principal planes of a rotating triax- showed that box orbits in triaxial galaxies with cusps and
ial galaxy with a central core. They found three prograde black holes are destabilized by moderate amounts of figure
sequences of stable orbits in addition to the retrograde se- rotation because the envelope doubling acts to further desta-
quences. Inside the core of the galaxy, they found that the bilizetheresonantboxorbits,ratherthanstabilizechaoticor-
x-axial orbit was stable and generated a family of box orbits bits as predicted by Gerhard & Binney (1985). This paper
thatunderwent“envelopedoubling”astheyloopedaroundthe further explores the cause of the destabilization of box-like
centerduetotheCoriolisforces. orbitsandinvestigatestheeffectsoffigurerotationonallma-
The first attempt to construct a self-consistent triaxial jororbitfamilies.
galaxywithslowfigurerotationwaspresentedinthepioneer- Our objective in this paper is to study the behavior of all
ingworkofSchwarzschild(1982). Basedonpreviousstudies themajorandminororbitfamilies,especiallythosethatform
ofperiodicorbits,Schwarzschildrestrictedhisrotatingtriax- thebackboneofrealisticrotatingtriaxialgalaxymodels. The
ialmodelstoamoderatepatternspeed(T ∼1.2×109yr)for goalistoidentifytherangeofrotationspeedsforwhichsuch
p
whichthemainresonances(corotation,outerinnerLindblad, orbitswillremainstable.
and the Binney instability strip) lay outside the model. He Thepaperisorganizedasfollows.InSection2,wedescribe
TriaxialGalaxieswithFigureRotation 3
thenumericalmodel,theselectionoforbitalinitialconditions, where−2Ω yand2Ω xareCoriolisforcetermsandΩ2xand
p p p
andtheLaskarmappingmethod(Laskar1990). InSection3, Ω2yarecentrifugalforceterms.4
p
wedescribetheresultsofouranalysisoftheeffectsoffigure
Most of the models used in this study are close to max-
rotationonmajororbitfamilies,asafunctionofpatternspeed,
imally triaxial with triaxiality parameter T =(a2−b2)/(a2−
orbitalenergy,andtheshapeofthetriaxialmodel,inmodels
c2)=0.58andminortomajoraxisratioc/a=0.5. Foralim-
withandwithoutcentralSMBHs. Wesummarizeourresults
itednumberofmodels,wealsoexploredtheeffectofchang-
anddiscusstheimplicationsofourfindingsinSection4.
ingtheshapeofthetriaxialfigurewithc/a=0.5,0.8,and0.7
withtriaxialityparametersT =0.1,0.9,and0.3,respectively.
2. MODELSANDNUMERICALMETHODS
Theshapesthereforerangefromnearlyoblate,throughoblate
2.1. Densitymodel triaxialtonearlyprolate.
Westudiedtriaxialgeneralizationsofsphericalmodelsfirst Following the standard practice, we adopt a set of units
presentedbyDehnen(1993)andTremaineetal.(1994). The where the total galactic mass M, the semimajor axis scale
model, which we will henceforth refer to as the “Dehnen- length a, and the gravitational constant G are set to unity.
model”, has a density law that is a good fit to the observed Whenacentralblackholeisaddedtothemodel,itsmassMbh
luminosityprofilesofellipticalsandthebulgesofspirals,and isexpressedasafractionofthetotalgalaxymassM. Inthis
isgivenby paperwerestrictourselvestostudyingmodelswitheitherno
centralpointmass(M =0)ormodelswithM =0.001(i.e.,
bh bh
ρ(m)= (3−γ)Mm−γ(1+m)−(4−γ), 0≤γ<3, (1) 0.1%ofthemassofthegalaxy). Thelattervalueisconsistent
4πabc with0.14%±0.04%themassfractioninacentralSMBHthat
isexpectedtobefoundinmostellipticalgalaxies(Häring&
where
Rix 2004). The potential and forces due to the central black
x2 y2 z2
m2= + + , a≥b≥c≥0, (2) holearethoseofasoftenedpointmasswithsofteninglength
a2 b2 c2 (cid:15)=10−5a.
The orbital structure of the stationary versions of triaxial
and M is the total mass of the model. The mass distribu-
Dehnenmodelswithvariouscuspslopesandarangeofblack
tion is stratified on concentric ellipsoids where a,b, and c
hole mass fractions have been previously studied (Merritt &
arethescalelengthsofthesemimajor,semi-intermediate,and
Fridman 1996; Valluri & Merritt 1998; Siopis & Kandrup
semiminor axes of the model, which are aligned with Carte-
2000). Inthispaperwerestrictourselvestopresentingmod-
sian coordinates x,y, and z, respectively. The parameter γ
elswithγ=1forthefollowingreasons. First,themostlikely
whichdeterminesthelogarithmicslopeofthecentraldensity
candidatesfortriaxialellipticalgalaxieswithslowfigurerota-
cusprangesobservationallyfromγ=0.5-1inluminousgalax-
tionarethemoreluminous“slowrotators”withshallowcen-
ieswith“shallowcusps”(sometimescalleda“core”)toγ=2
tral cusps (Emsellem et al. 2007). Second, a Dehnen model
inthelowerluminositygalaxieswith“steepcusps”(Gebhardt
with γ =1 is quite similar (at least in the inner regions) to
etal.1996;Laueretal.2007). Atlargeradiithedensitypro-
the density profiles of cosmological dark matter halos with
file of the model always falls as m−4. The model has a fi-
the main difference being that cosmological density profiles
nite density core for γ =0 and an infinite central density for
falloffmoreslowlyatlargeradii(r−3comparedtor−4forthe
γ >0. The potential (Φ(x)) and forces in the stationary tri-
Dehnenmodel). Modelswithγ=2werestudied(butarenot
axialDehnenmodelinellipsoidalcoordinatesaretakenfrom
presented)sincetheirdependenceonpatternspeedisqualita-
Merritt&Fridman(1996).
tivelysimilartothatoftheγ=1modelswithanSMBH.(We
Intherotatingframe,theenergyofanorbitisnotanintegral
notethatwhileourchoiceoftriaxialpotentialisrepresentative
ofmotionbuttheJacobiintegral(E )isaconservedquantity:
J ofatriaxialellipticalgalaxy,itisanincompleterepresentation
1 1 ofadarkmatterhalowhichcouldhaveasignificantpotential
EJ = |x˙|2+Φ− |Ωp×x|2, (3) contributionfromastellardisk.)
2 2
At the present time only a few measurements of the pat-
wherexandx˙ arethree-dimensionalspatialandvelocityvec- ternspeedsoffigurerotationintriaxialdarkmatterhalosand
tors,respectively. early-typegalaxiesexist. Thepatternspeedsoffastbarsmea-
In our models, figure rotation is about the short (z) axis, suredbyapplyingtheTremaine−Weinbergmethod(Tremaine
henceΩ(cid:126) =Ω eˆ andequationsofmotionintherotatingframe & Weinberg 1984; Meidt et al. 2008) typically indicate that
p p z
(BT08,Section3.3.2)become the ratio of the corotation radius to the length of the bar
R /a=[1,1.4] (for a bar of semi-major axis length a) (e.g.,
Ω
x¨ =−∇Φ−2(Ω(cid:126) ×x˙)−Ω(cid:126) ×(Ω(cid:126) ×x) (4) Debattistaetal.2002;Aguerrietal.2003;Corsini2010;Bin-
p p p
ney & Tremaine 2008). Hereafter we shall use the quantity
=−∇Φ−2(Ω(cid:126) ×x˙)+|Ω |2x. (5)
p p R /a tocomparethepatternspeedsoforbitslaunchedfrom
Ω i
different radial shells (of semimajor axis length a) to each
In Cartesian coordinates the equations of motion are then i
otherandtothepatternspeedsofbars.
givenby
Thepatternspeedofearly-typegalaxyNGC2974hasbeen
∂Φ measuredbyfittingthepropertiesofthreeringsofrecentstar
¨x=− −2Ω y+Ω2x, (6)
∂x p p formation (Jeong et al. 2007). This galaxy is normally clas-
∂Φ
¨y=− +2Ω x+Ω2y, (7) 4Followingcurrentconvention(e.g.,BT08)weusetheright-handedscrew
∂y p p rule (with positive angular momentum vector pointed up) for figure rota-
∂Φ tionandaright-handedcoordinatesystem. Notethatsomepreviousauthors
¨z=− , (8) (Schwarzschild1982)usedaleft-handedscrewruleforfigurerotationanda
∂z right-handedcoordinatesystem.
4 Deibel,Valluri,&Merritt
sified as E4, but the authors argue that all three rings can
TABLE1
be accounted for as occurring at resonances if this galaxy is
PATTERNFREQUENCIESOFFIGUREROTATIONOFTRIAXIALDEHNEN
a S0 galaxy with an extended stellar bar and pattern speed, MODELSWITHγ=1,Mbh=0
Ω ∼78±6kms−1 kpc−1. Thispatternspeedisonlyslightly
p
slowerthanthatoffastbars. Figurerotationofatriaxialdark RΩ Ωp Ω∗p Tp∗
matterhalowasproposedasthecauseoftheextendedspiral (a) (programunits) (kms−1kpc−1) (yrs)
arms in the blue compact dwarf galaxy NGC 2915 (Bureau
etal.1999;Bekki&Freeman2002)whosemodelingrequired 2 2.57×10−1 33.8 1.82×108
ahalopatternspeedofΩ =7±1kms−1 kpc−1. Thispattern 5 7.45×10−2 9.80 6.28×108
p 10 2.87×10−2 3.78 1.63×109
speedisanorderofmagnitudelargerthanthemaximumpat-
20 1.09×10−2 1.44 4.29×109
tern speed measured for dark matter halos produced in cos-
40 3.85×10−3 0.51 1.21×1010
mologicalN-bodysimulations(Ωp=1.01hkms−1kpc−1)and 60 2.12×10−3 0.28 2.21×1010
nearlytwoordersofmagnitudelargerthanthemedianpattern ∞ 0.0 0.0 ∞
speed(Ω =0.148hkms−1kpc−1)(Bailin&Steinmetz2004).
Thus,attphepresenttime,observationalconstraintsonthepat- ∗Fora=5kpc,M=5×1011M(cid:12)
tern speeds of triaxial halos and elliptical galaxies are quite FollowingMerritt&Fridman(1996),themodel’smassdis-
uncertain. OurfastestpatternspeedΩ =34kms−1kpc−1(for tribution is stratified into 20 ellipsoidal shells dividing the
p
R /a=2)isslightlyslowerthanthespeedsobservedinfast model into 21 sections of equal mass. In this paper, we fo-
Ω
bars, our slowest Ω =0.58 km s−1 kpc−1(for R /a=40) is cus mostly on orbits with energy equal to the potential en-
p Ω
comparable to fastest measured in simulations of dark mat- ergy at the point the 8th shell intersects the major axis of
terhalos. WeexplorearangeofvaluesforΩ betweenthese the model (x = 1.6a). (Note that shell 10 corresponds to
p
limits. the half-mass radius of the model). In the rotating poten-
The triaxial Dehnen models studied here rotate about the tial, orbits were launched from the equi-effective-potential
short axis (z). Simulations of collisionless dark matter halos surface which is analogous to launching all orbits in a non-
showthattheangularmomentumaxeshaveameanmisalign- rotating model from an equipotential surface. Thus all or-
mentof∼25◦withtheminoraxis(Bailin&Steinmetz2005). bitslaunchedfromagivenshellhavethesameJacobiintegral
However, the dissipative collapse accompanying galaxy for- (EJ =E−12|Ωp×r|2),whereE isthetotalenergyofanorbit.
mation is likely to induce angular momentum transport and Westudyorbitson6energyshellsforonepatternfrequency
a higher degree of alignment between the spin axis and the butdeferafullinvestigationoforbitsinself-consistentpoten-
shortaxisofthegalaxy. Thepatternspeedofthetriaxialfig- tialstoafuturepaper(Valluri2011).
ureisgivenintermsofthe“corotationradius”,hereafterR . Initial conditions for the orbits were selected in two dif-
Ω
Inanearlyaxisymmetricpotential,thecorotationradiusisthe ferentwayssuchthatorbitsfromallfourmajorfamilies(the
radiusatwhichthefrequencyΩ ofaclosed(almostcircular) boxorbitsandthreefamiliesoftubeorbits)wererepresented
c
orbitintheequatorial(x-y)planeofthepotentialisthesame (de Zeeuw 1985). Although models are rotating, the orbits
asthepatternfrequency(generallycalled“patternspeed”)Ω : were launched, integrated, examined, and classified only in
p
the frame that is corotating with the figure, allowing for a
(cid:114)
(cid:104) 1 (cid:105) moredirectcomparisonwiththestationarymodelsthathave
Ω =Ω = ∇Φ . (9)
p c R R=RΩ,z=0 beIennastustdaiteiodnianrythpeoptaesntt.ial, box orbits are characterized by a
Working in the equatorial plane of our triaxial model we stationarypointontheequipotentialsurface. Inarotatingpo-
set R2 =(x/a)2+(y/b)2. In our models RΩ is given in units tentialtherearenotrueboxorbits(i.e.,orbitswithstationary
ofa,thescalelengthofthesemimajoraxis,andrangesfrom pointsinaninertialframe). However, orbitsthatarecharac-
RΩ=60(slowlyrotating)toRΩ=2(veryrapidlyrotating)(see terizedbyastationarypointontheeffectivepotentialsurface
Table 1). In this nomenclature the stationary (non-rotating) (surface of constant Jacobi integral) look and behave much
modelhasitscorotationradiusatinfinityandislabeledwith likeboxorbitsintheframethatiscorotatingwiththefigure.
RΩ=∞. Wedonotdiscusstheorbitalstructureofmorepro- Accordingly, we launch orbits at zero velocity on a regular
latetriaxialstructuressimilartofastbarswithhigherpattern gridononeoctantoftheeffectivepotentialsurfacetoobtain
frequencies(RΩ≤2)sincetheyhavebeenpreviouslystudied box-likeorbits. Inanon-rotatingtriaxialmodel, atubeorbit
(Skokosetal.2002a,b;Harsoula&Kalapotharakos2009).To is characterized by a finite angular momentum (about either
give the reader a better physical appreciation of the pattern thelongorshortaxis)whichoscillatesbetweentwovaluesof
frequencies implied by the co-rotation radii, we convert our the same sign. Thus the magnitude of the angular momen-
model units to physical units in an elliptical galaxy. For a tum is not a conserved quantity, but the sign of the angular
semi-major axis scale radius a=5 kpc and a galaxy mass of momentumofatubeorbitremainsconstant5. Consequently,
M=5×1011M ,theunitoftimeforthemodelisgivenby tubeorbitsavoidthecenter.
(cid:12)
For exploring the phase space structure of a stationary tri-
(cid:114) a3 (cid:20) a (cid:21)3/2(cid:20) M (cid:21)−1/2 axial model it is customary (Schwarzschild 1993; Merritt &
T = =1.49×106 yr. (10)
GM kpc 1011M Fridman 1996; van den Bosch et al. 2008) to launch orbits
(cid:12)
uniformly from the “x-z start space” i.e., from one quadrant
FortheseparametersTable1givesthepatternfrequency(Ωp) ofthex-zplanewithvx=vz=0withvy>0determinedbythe
and rotation period T =2π/Ω (in years) for each value of
p p
corotationradiusRΩthatwasstudied. 5Werefertotubeorbitsas“anti-clockwise”or“clockwise”dependingon
whethertheirtimeaverageangularmomentumvector(Jxforx-tubesandJz
forz-tubes)ispositiveornegative,respectively(inaright-handedcoordinate
2.2. Orbitstartspaces
system)
TriaxialGalaxieswithFigureRotation 5
energy of the equipotential surface, this is adequate for sta-
tionary models since the properties of orbits with v <0 are
y
simply obtained by relying on the symmetries of the model.
However, in a triaxial model with figure rotation about the
shortaxis,orbitsthatarelaunchedwithv >0inthex-zstart
y
spacearedifferentfromthosewithv <0. Sinceorbitsofall
y
majorfamiliesintersecttheintermediatey-axisweuseinstead
the “Y-α start space” (Schwarzschild 1982) where all four
families of orbits are launched from the y-axis of the model
with v =0 and at an angle α between the starting velocity
y
vector (perpendicular to the y-axis) and the x-y plane. The
initialconditionsforanorbitaregivenby
x=0 y=Y z=0
v =Vcosα v =0 v =Vsinα,
x y z
where V is the magnitude of the total velocity of the orbit
atthatpositiondeterminedfrompotentialenergyatthestart-
ingpointandtheJacobiintegralofallorbitsonthatsurface.
By allowing 0◦ ≤α≤360◦ we obtain tube orbits with both
clockwise and anti-clockwise motions. Figure 1 shows the
Y-α start space with 225 different initial conditions marked FIG.1.—LocationofdifferentorbitfamiliesontheY-αstartspaceina
by their orbital types. (We plot −30<α<330 to illustrate sTthateiolinnaersymtraiarkxiaaplpDreohxnimenatmeobdoeulnwdaitrhiecs/bae=tw0e.e5n,Tm=aj0o.r5f8a,mγil=ie0s,.aSnqduMarbehs=((cid:50)0).
that the retrograde z-tubes wrap around the top and bottom denoteboxorbits,asterisks(∗)denotechaoticorbits,diamonds((cid:5))indicate
boundaries.) The maximum value ofY is determined by the innerlong-axis(x)tubes,triangles((cid:52))denoteouterlong-axistubes,andplus
signs(+)indicatez-tubes.
intersectionoftheequipotentialsurfacewiththey-axis.
Theorbitaltypesweredeterminedvisuallybyplottingthree
different projections in coordinate space (x−y, y−z, and
2.3. NumericalAnalysisofOrbitalFrequencies
x−z) and by examining their time-averaged normalized an-
gular momenta about the three principal axes (i.e., (cid:104)J /|J |(cid:105), InfullyintegrablepotentialsliketheStäckelpotentials,all
x x
(cid:104)J /|J |(cid:105), and (cid:104)J/|J|(cid:105)). For a box orbit (or strongly chaotic orbits are confined to N-dimensional tori, with N the num-
y y z z
orbit), all three components of normalized angular momen- ber of degrees of freedom (dof), i.e., the number of spatial
tum are approximately zero. For x-axis tubes and z-axis dimensions. Motion on the torus is defined in terms of the
tubes,theabsolutevaluesofthexandzcomponentsareequal fundamental frequencies ωi, i=1,...,N, the rates of change
to unity (within 0.005%). Note that a weakly chaotic or- of the corresponding angle variables. Expressed in terms of
bit associated with a tube family will have (cid:104)J /|J |(cid:105)∼1 or Cartesiancoordinates,themotionisquasi-periodic,e.g.,
x x
(cid:104)J/|J|(cid:105)∼1overafinitenumberoforbitalperiods. Inathree
z z ∞
dimensionalpotential,Arnolddiffusionwouldcausesuchan x(t)=(cid:88)Akeiνkt, (11)
orbit to eventually ergodically fill the entire energy surface
availabletoitandwouldthereforenotbeidentifiableasatube k=1
orbit(Lichtenberg&Lieberman1992). Hereafter,wereferto where the ν ’s are linear combinations, with integer coeffi-
k
such orbits as “tube-like”. Note that this criterion for classi- cients,ofthethreefundamentalfrequencies:
fyingorbitsintothreemajorfamiliesreliespurelyontherela-
ν =n ω +n ω +n ω (12)
tivemagnitudesofthethreecomponentsofthetime-averaged k 1,k 1 2,k 2 3,k 3
angular momenta but does not indicate that these orbits are andtheA arethecorrespondingamplitudes.Thesameistrue
k
regular. for regular orbits in arbitrary potentials, i.e., for orbits that
Figure1showshoworbitsofdifferenttypesaredistributed respect at least three isolating integrals of the motion (e.g.,
ontheY-αstartspace. Squaresindicateboxorbits,diamonds Lichtenberg&Lieberman1992).
indicate inner long-axis tubes, triangles indicate outer long- Whenaregularorbitisfollowedformany(∼100)dynam-
axistubes,plussignsindicateshort-axistubeswhileanaster- icaltimes,aFouriertransformofthetrajectoryyieldsaspec-
isk indicates an orbit identified as stochastic by its high dif- trum with discrete peaks. The locations of the peaks in the
fusionrate(seeSection2.3). Notethatsincethetriaxialfig- spectrum correspond to the frequencies ν in Equation (11)
k
urerotatesintheanti-clockwisedirectionaboutthez-axis,we andcanbeusedtocomputethethreefundamentalfrequencies
refertoanti-clockwisez-tubesas“prograde”andtheirclock- andtheintegercoefficients(n ,n ,n )thatcorrespondto
1,k 2,k 3,k
wisecounterpartsas“retrograde”(evenwhenaspecificmodel eachpeak(e.g.,Binney&Spergel1982).
is stationary). We will see in Figure 3 that stochastic orbits Laskar (1990) developed a fast and accurate numerical
arefrequentlyfoundalongtheboundaries(orseparatrixlay- technique (“Numerical Analysis of Fundamental Frequen-
ers)betweenregionsoccupiedbydifferentorbitfamilies.The cies,” hereafter NAFF) to decompose a complex time se-
general location of the different regular orbit families in this ries representation of the phase space trajectory of the form
start space will remain roughly the same as the potential is x(t)+iv (t). Our own implementation of Laskar’s algorithm
x
perturbed with non-zero cusp slope γ, the addition of a cen- usesintegerprogrammingtoobtainthefundamentalfrequen-
tralpointmass,orasthepatternfrequency,orbitalenergy,or cies from the spectrum (Valluri & Merritt 1998, hereafter
theshapeofthefigurearealtered. VM98). We integrated orbits using an explicit Runge Kutta
6 Deibel,Valluri,&Merritt
integrator (DOP853) of order eight (5,3) due to Dormand & componentisgivenby
Prince with stepsize control and dense output by Harier and
(cid:12) (cid:12)
Wsifiacnanteiorn(,Haanriderfreetqaule.n1c9y93a)n.aTlyhseisowrbeirteinatlelgcraartiroiend,oorubtitincltahse- log(∆fx)=log(cid:12)(cid:12)(cid:12)ωx(tω1)−(tω)x(t2)(cid:12)(cid:12)(cid:12), (14a)
x 1
framethatiscorotatingwiththepatternfrequencyofthefig- (cid:12) (cid:12)
ure. log(∆fy)=log(cid:12)(cid:12)(cid:12)ωy(tω1)−(tω)y(t2)(cid:12)(cid:12)(cid:12), (14b)
Evenforregularorbits,thecharacterofthemotiondepends y 1
(cid:12) (cid:12)
cinridteicpaelnlydeonnt owrhwethheetrhtehreththeyreesaftuisnfdyaomneentoarlmfreoqreuennocnietrsivairael log(∆fz)=log(cid:12)(cid:12)(cid:12)ωz(t1ω)−(tω)z(t2)(cid:12)(cid:12)(cid:12). (14c)
z 1
linearrelationsoftheform
Wedefinethediffusionparameterlog(∆f)tobethevalueas-
(cid:96)ω +mω +nω =0 (13)
1 2 3 sociated with the largest of the three amplitudes A ,A , and
x y
where((cid:96),m,n)areintegers,notallofwhicharezero. Gener- A . The larger the value of the diffusion parameter the more
z
allythereexistsnorelationlikeEquation(13);thefrequencies chaotic the orbit. Our second representation of phase space,
are incommensurable, and the trajectory fills its torus uni- adiffusionmap, isobtainedbyplottingtheinitialconditions
formly and densely in a time-averaged sense. When one or ofmanythousandsoforbitsandaddingagrayscale(orcolor
moreresonancerelationsaresatisfied,however,thetrajectory intensity scale) corresponding to the diffusion parameter. In
is restricted to a phase space region of lower dimensionality a diffusion map (e.g., Figure 2(b) and Figure 3(b)) the gray
thanN. scale(orintensityofthecolor)correspondstolog(∆f)such
In 2 dof systems, a resonance implies a closed, or peri- thatregionsofinitialconditionspaceoccupiedbyregularor-
odic,orbit,e.g.,a“boxlet”(Miralda-Escude&Schwarzschild bitsarewhiteandthoseoccupiedbychaoticorbitsaredark.
1989). In three dimensions, a single resonance relation like Severalpropertiesofthephasespacecanbeinferredfrom
Equation (13) does not imply that an orbit will be closed; frequencyanddiffusionmaps. Webeginwithadiscussionof
rather,itrestrictstheorbittoaspaceofdimensiontwo(Mer- phase space maps for the stationary (here after "box orbit")
ritt & Valluri 1999). An orbit satisfying one such relation start space for a baseline model, namely a triaxial Dehnen
is“thin,”confinedforalltimetoa(possiblyself-intersecting) modelwithc/a=0.5,T =0.58,γ =0,andM =0. Thefre-
bh
membrane.Inorderforanorbitina3dofsystemtobeclosed, quency map in Figure 2(a) shows a plot of the ratios of fun-
itmustsatisfytwosuchindependentrelations;suchorbitsare damental frequencies ω /ω versus ω /ω for each of 9408
x z y z
likelytobemuchrarerthanthinorbits. Inwhatfollows, we boxorbitsdroppedwithzerovelocityfromtheequipotential
willusetheterm“resonant”toreferbothtothinorbits(satis- surfacecorrespondingtothe8thshell.
fyingonerelationlikeEquation13)andtoclosedorbits(sat- Note that although we focus primarily on the behavior of
isfying two such relations), e.g., boxlets. Stable resonances orbitslaunchedfromshell8th,thebehaviorofanorbitinre-
of both sorts generate new families of regular orbits whose sponsetofigurerotationdependsbothonthe(radial)energy
shapemimicsthatoftheparentorbit. Unstableresonanttori shellfromwhichitislaunchedaswellastheco-rotationra-
aretypicallyassociatedwithabreakdownofintegrabilityand dius. However,sincethedensityprofilehasanapproximately
withchaos. power-law profile outside the inner cusp, the behavior of or-
Oncethefundamentalfrequenciesandtheiramplitudesare bitsdepends primarilyon R /a, where a isthe semi-major
Ω i i
obtained,itispossibletoobtaintwocomplementaryrepresen- axis of the ith shell. Consequently, the behavior of orbits
tationsofphasespaceatagivenenergy. Afrequencymapis launchedfromanoutershellatamoderatepatternspeedcan
obtainedbyplottingratiosofpairsoffrequencies(e.g.,ω /ω resemblethatoforbitslaunchedataninnerradiusandafaster
x z
versus ω /ω ) for many thousands of orbits in the potential. pattern speed. Consequently, we will frequently give values
y z
Such a representation of phase space (e.g., Figure 3(a) and ofR /a aswellasR .
Ω i Ω
Figure2(a))isparticularlyusefulforidentifyingthemostim- The frequency map shows that most points at the bottom
portantorbitalresonances. Stableresonancesappearasfilled left corner of the plot lie on a fairly regular grid which re-
lineswithanincreaseddensityoforbitsclusteredalongthem. flectstheregulardistributionofinitialconditions. Awayfrom
This is because these orbits have been trapped by the stable this region the regular, grid-like structure is disrupted by the
resonance. Incontrast,unstableresonancesappearas“blank” appearance of dark lines corresponding to stable resonances
ordepopulatedlinesandareassociatedwithstochasticorbits. andtheirassociatedregularorbits. Alltheorbitslyingalong
Laskar(1990)demonstratedthatsincemostorbitsinrealis- suchlinesobeyasingleresonantconditionlikethatofEqua-
ticgalacticpotentialsareonlyweaklychaotic,theyliecloseto tion(13). Intheplot,severalsuchresonancesarehighlighted
regularorbitsinphasespacemimickingtheirregularbehavior withdashedlinesandarelabeledwiththeirdefiningintegers
over finite time intervals. Consequently, frequency analysis ((cid:96),m,andn).
canbeusedtodistinguishbetweenregularandchaoticorbits. Figure3(a)showsafrequencymapfor10,000orbitsinitial-
Thefrequencydriftcangiveameasureofthedegreeofchaos izedontheY−αstartspacewiththeenergycorrespondingto
inanorbit. Thefrequencydriftofanorbitcanbedetermined the x-axial orbit on shell 8 in this model. In this frequency
fromthechangeinitsfundamentalfrequenciesmeasuredover map,allfourmajororbitfamiliesarerepresented: boxorbits
twoconsecutivetimeintervals. areblack, anti-clockwisex-tubesareblue, clockwisex-tubes
In applying Laskar’s formalism, we integrated each or- are cyan, clockwise z-tubes are ochre, and anti-clockwise z-
bit for 100T , where T is the period of the long-axis or- tubes are red. Note that a significant fraction of the z-tubes
D D
bit of the same energy in the stationary model. The time (redandochrepoints)andx-tubes(blueandcyanpoints)ap-
interval was divided into two equal segments labeled t and pear clustered along straight lines. These are orbits that lie
1
t andthreefundamentalfrequenciesω (t ),ω (t ),ω (t )and closetothethin-shelltubeorbitparentofthefamilyandap-
2 x 1 y 1 z 1
ω (t ),ω (t ),ω (t )werecomputedineachtimesegment.The pearas“resonancelines”inthefrequencymapalthoughthey
x 2 y 2 z 2
“frequencydrift”or“diffusionparameter”ineachfrequency arenottraditionallyviewedasresonances. Toaccuratelyrep-
TriaxialGalaxieswithFigureRotation 7
FIG.2.—Tworepresentationsofphasespacefor9408boxorbitslaunchedfromshell8inanon-rotatingtriaxialDehnenmodel(c/a=0.5,T =0.58,γ=0,
andMbh=0). (a)Afrequencymap:ωx/ωzversusωy/ωzforthefundamentalfrequenciesgivenbytheNAFFalgorithm. Dashedlinesmarkvariousimportant
resonanceslabeledbytheirintegers((cid:96),m,andn). (b)Diffusionmap: grayscalecorrespondstothediffusionparameter(log(∆f))oforbitsatvariousinitial
positionsononeoctantoftheequipotentialsurface(the“stationary(boxorbit)startspace”). Darkregionsonthemapcorrespondtochaoticorbitsandwhite
regionscorrespondtoregularorbits.Thelabels"X","Y",and"Z"marktheintersectionsoftheequipotentialsurfacewiththethreeprincipalaxes.
FIG.3.—(a)Frequencymapand(b)diffusionmapfor10,000orbitsinthe“Y-αstartspace”foramodelwithc/a=0.5,T=0.5,andγ=0.Orbitfamiliesare
colorcodedsothatboxorbitsareblack,anti-clockwisex-tubesareblue,clockwisex-tubesarecyan,clockwise(retrograde)z-tubesareochre,andanti-clockwise
(prograde)z-tubesarered.Inthediffusionmaponlythechaoticorbitsappearcoloredsincetheintensityofthecolordependsontheorbitaldiffusionparameter.
.
resenttheresonancesinthetwotubeorbitfamiliesitisessen- ofthe diffusionplotconsistsof regularboxorbits thatorigi-
tialtoobtainthefundamentalfrequenciesincylindricalcoor- nateclosetothex-axis. Inaddition,severalwhitebandsmark
dinatesabouttheappropriatesymmetryaxis. VM98showed regular islands of resonant boxlet families. Each white band
that such a representation reveals several unstable tube orbit correspondstoadarkresonancelineinthefrequencymapin
resonances. ThesecanbeseenasdarkbandsinFigure3(b). Figure 2(a). Most white bands are flanked by narrow dark
Figure2(b)isadiffusionmapfor9408boxorbitslaunched regions (occupied by stochastic orbits) which lie along the
with zero initial velocity from the equipotential surface (at “separatrix”(transitionlayer)betweendifferentorbitfamilies.
shell8)ofastationarymaximallytriaxialDehnenmodelwith Stochastic orbits (dark regions) are also seen at the intersec-
γ =0. The large swath of regular orbits at the bottom left tionsofstableresonances(whitebands),whentheparentpe-
8 Deibel,Valluri,&Merritt
riodic orbit is unstable. The prominent dark band of chaotic tainsasomewhatlargerfractionofstochasticorbits. Almost
orbitsthatrunsalongtherightedgeofthediffusionmapcor- all the regular orbits (white regions on the map) are now as-
responds to orbits originating close to the y-z plane. Orbits sociatedwithresonantislandscontainingboxletfamilies. All
in this region are chaotic due to the well-known instabilities orbits that qualitatively resemble box orbits in cored poten-
ofthey-axialandz-axialorbits(Heiligman&Schwarzschild tialsaremildlytostronglychaotic.
1979;Heisleretal.1982;deZeeuw&Merritt1983;Adams AsmentionedinSection1,Gerhard&Binney(1985)first
etal.2007)aswellasduetotheinstabilityoforbitsconfined proposedthattheboxorbitsrenderedchaoticduetoscattering
in the y-z plane to perturbations perpendicular to this plane byamassivecentralpointmasscouldbestabilizedinarotat-
(Adamsetal.2007). Thisregionwillbeseentoexpandwith ingframeduetotheenvelopedoublingeffectoftheCoriolis
increasing figure rotation and is referred to hereafter as the force. Since a box orbit (in a stationary model) has no net
“y-zinstabilityband”. Mostofthesefeaturesweredescribed angularmomentumitoscillatesbetweenstationarypointsre-
in detail in previous papers (Valluri & Merritt 1998; Merritt versing its sense of progression around the center each time
&Valluri1999). itreachesaturningpoint. Inarotatingframethismeansthat
Figure 3(b) shows a diffusion map for 10,000 orbits thepathdescribedduringtheprogradesegmentoftheorbitis
launched from the Y-α start space. The orbits were classi- notretracedduringtheretrogradesegmentbecausetheCorio-
fied into three different orbit families based on their time- lisforceonthetwosegmentsdiffers(Schwarzschild1982;de
averaged normalized angular momentum values. The boxes Zeeuw&Merritt1983).
arecoloredblack,anti-clockwisex-tubesareblue,clockwise We see in Figure 4 that this prediction does not hold up
x-tubesarecyan,clockwise(retrograde)z-tubesareochre,and inrealistictriaxialgalaxymodels. Infact, asthepatternfre-
anti-clockwise (prograde) z-tubes are red. Note that regions quenciesoffigurerotationincrease(R decreases)theareaof
Ω
of more intense color (occupied by stochastic orbits) gener- thediffusionmapoccupiedbyregularorbits(white)decreases
allyappearattheseparatrixbetweenthemajororbitfamilies. and the resonant islands shrink until only a small fraction of
Withintheregionsoccupiedbyx-axistubes(blueandcyanre- orbits remain regular at R =5. When a central point mass
Ω
gions),thetransitionbetweentheinnerandouterx-axistubes isadded,theincreaseinthefractionofchaoticorbitsiseven
is also marked by a weakly stochastic layer. Thick stochas- more significant (Figure 5) and in addition the delineation
tic layers separate the z-tubes from the box orbits that lie at between the various resonant box orbit families is even less
valuesofY <0.5. Inaddition, stochasticorbitsarefoundat clear−pointingtoincreasedresonanceoverlap. Weseefrom
α∼90◦±30◦andα∼270◦±30◦mostlyatverysmallY val- acomparisonoftheFigures 4and5thathighpatternspeed
ues(theseorbitsarelaunchednearlyalongthezaxiswhichis increasesthefractionofchaoticorbitsmoresignificantlythan
knowntobeunstable). thepresenceofacentralblackhole.
Themainfeaturesidentifiedonthefrequencyanddiffusion Wenoteinpassingthatanewregularorbitfamilyappears
mapsabove,fortheγ=0andM =0case,aremeanttogive as a white region in the lower right hand corner of the dif-
bh
readers insight into how they should interpret the results for fusion maps in Figures 4 and 5 for R =10,5. In the sta-
Ω
modelswithfigurerotation. tionary model, box orbits originating here are highly unsta-
ble since they lie close to the unstable y-axis. Figure 11(top
3. RESULTS
row)showstwoprojectionsofonesuchorbitinamodelwith
We now discuss the effects of increasing the pattern fre- R =5. Althoughitislaunchedwithnonetangularmomen-
Ω
quencyoffigurerotationonbox-likeandtube-likeorbitsus- tumintherotatingframe,thisorbitisashort-axistube. This
ingthephase-spacemappingtechniquesdiscussedinthepre- tubeorbit(likeothersintriaxialpotentials)doesnotconserve
vioussection. angularmomentum,butitsvalueoscillatesbetweentwoval-
ues (in this case zero and a large negative value) indicating
3.1. Effectsofslow/Moderatepatternfrequencyonboxlike
thatitsmotionintherotatingframeisretrogradetothatofthe
orbits
figure. This new family of orbits lies close to the x-y plane.
Box orbits were integrated, from start spaces such as the This family appears to arise when the pattern frequency of
one in Figure 2(b), for various model parameters and in- thefigureishighenoughthatCoriolisandcentrifugalforces
creasing pattern speeds. We generally present results for causeittolooparoundthecenter.Orbitsinthisregionofstart
a default model which is close to maximally triaxial, with spacewerechaoticinthestationarymodelandareindeedsta-
c/a=0.5,T =0.58andcentraldensitycuspofγ=1. Wealso bilized by figure rotation as predicted by Gerhard & Binney
presentresultsformodelswithacentralpointmassrepresent- (1985). However, this stabilization results from a complete
ing a central SMBH with M =0.001 (0.1% of the galaxy transformationoftheorbitalcharacterfrombox-liketotube-
bh
mass). (Models with γ = 2 were studied but are not pre- like, rather than due to a small deflection of the chaotic box
sented since their dependence on R is qualitatively similar orbitaroundthedestabilizingcenter. Wewillseelater(Sec-
Ω
althoughtheyhaveahigherfractionofchaoticorbits,VM98. tion 3.3) that as the corotation radius moves inward, this re-
Higherblackholemassfractionswerepreviouslystudiedby gion of stable tube-like orbits grows steadily till most of the
VM98 and were found to induce more chaos.) In each fig- box start space is occupied by regular orbits, similar to this
ure, we show phase space maps for a non-rotating model one.
(R =∞)andthreeadditionalvaluesofthecorotationradius Figures4(a)and5(a)showthatmost(butnotall)theregular
Ω
(R =40,10, and 5). In all cases orbits were launched with orbitsinthestationarymodel,whichappearaswhiteregions
Ω
energy E equal to that of the x-axial orbit started from the on the diffusion maps, are associated with resonant orbits.
8thshell. Therefore,foranorbitlaunchedfromthe8thshell, Therearetworeasonswhythemajorityofboxorbits(which
thesepatternspeedscorrespondtoR /a =24.8,6.2,and3.1. are associated with two-dimensional resonance surfaces) are
Ω i
Figure 4 shows phase space diffusion maps for box orbits notstabilizedbyslowtomoderatefigurerotationaspredicted
in our default model with no central point mass. The non- byGerhard&Binney(1985).
rotatingmodel(panel(a))issimilarto(Figure2(b))butcon- First, the increase in the fraction of chaotic orbits can be
TriaxialGalaxieswithFigureRotation 9
FIG.4.—Diffusionmapsoforbitslaunchedfromtheboxorbitstartspaceat8thenergyshellinmodelswithγ=1andMbh=0andvaryingpatternfrequencies
asindicatedbylabels.ThevalueofRΩ=∞referstoamodelwithnorotation,whileRΩ=40,10,and5refertothecorotationradiiofmodelswithincreasing
patternfrequencies.ThesepatternspeedscorrespondtoRΩ/ai=24.8,6.2,and3.1.
attributedtothedistinctconfigurationspacestructureofres- centerisaconsequenceofthedivergentcentraldensitycusp
onant orbits. As described by Merritt & Valluri (1999), the orcentralblackholepresentinallrealisticgalaxymodels.
parent of a resonant orbit family satisfies a condition like Intherotatingframeofaslowlyrotatingmodel, theCori-
lω +mω +nω =0. This additional resonant condition con- olis force can produce “envelope doubling” which broadens
x y z
finesaresonantorbittothesurfaceofatwo-dimensionaltorus a nearly resonant box orbit driving it into the divergent cen-
inthree-dimensionalphasespace. Ingeneral,suchorbitsare tral cusp (or central black hole), reducing the sizes of stable
not periodic. The more commonly known (but less numer- resonantislands, andeventuallydestroyingthem. Theeffect
ous) near-periodic versions of such resonances were identi- ofenvelopedoublingcanbeseeninFigure6,whichshowsa
fied by Miralda-Escude & Schwarzschild (1989) and given resonantboxletorbitassociatedwiththe(3,−1,−1)resonance.
names such as “bananas”, “pretzels”, “fish” etc. based on ThetoprowshowstwoCartesianprojectionsoftheorbit,x-y
their projected shapes. A non-periodic resonant orbit there- (left)andx-z(right). Inprojection(toprow)theorbitappears
fore, in comparison, appears similar to a three-dimensional similar to three-dimensional box orbits. However, the sec-
box orbit in projection but occupies only a two-dimensional ond row shows intersections of the orbit with the x-y plane
surface in configuration space. Consequently, all families of (left) and the x-z plane (right) shows its perfectly sheet-like
stableresonantorbitsarecentrophobicandavoidthecenterof structure with a clear “hole” in the center. The next three
thepotentialduetotheirthinsheet-likestructure. Theparent rows show cross sections of the same orbit in models with
oftheresonantfamilyissurroundedbyaregionoccupiedby R =40,20,and10. WhenR =20(4throw)thecrosssec-
Ω Ω
nearly resonant orbits which are characterized by two of the tion shows that the Coriolis force broadening has caused it
samefrequenciesastheresonantparent,butwithanincreas- to nearly fill the center in the x-z projection. When R =10
Ω
ing third frequency. As the third frequency increases from (5throw)theorbitisthickenoughtopassthroughthecenter
zero, thethicknessoftheorbitincreasesandthenearestdis- renderingitcompletelychaotic.
tanceofapproachtothecenterofthepotential(thepericenter While the majority of the regular orbits on the stationary
distance)decreases. Whentheorbitbecomesthickenoughto start space in Figure 4 and Figure 5 are associated with res-
passthroughthecenteritisdestabilized.Theinstabilityatthe onances, a few regular box orbits are non-resonant. These
10 Deibel,Valluri,&Merritt
FIG.5.—Diffusionmapsoforbitslaunchedfromtheboxorbitstartspaceonthe8thenergyshellofamodelwithc/a=0.5,T=0.58,γ=1,andMbh=0.001.
ThevalueofRΩ=∞referstoamodelwithnorotationwhileRΩ=40,10,and5refertothecorotationradiiofmodelswithincreasingpatternfrequencies.
tooappeartobedestabilizedbyfigurerotation. Additionally, model with large y values, the change in orbital frequencies
(Gerhard&Binney1985)predictedthatorbitsthatarechaotic islargeenoughtoconvertthemtoresonantz-tubes(e.g.,Fig-
inthestationaryframeshouldbecomeregularintherotating ure11(toprow)).
frame,butthiswasnotseen. Thefrequencymapsshowtheincreasedwidthofthestable
Thesecondreasonfortheincreaseinthefractionofchaotic resonance lines (which occur due to the envelope doubling
boxlike orbits can be seen by plotting frequency maps. Fre- discussedabove). Inadditionafewnewresonancesappearas
quencymapsforthedefaultmodel(c/a=0.5,T =0.58,γ=1, the pattern speed increases. The shrinking of the frequency
andM =0)showresonantorbitsclusteredalongresonance map, the broadening of resonances, and the appearance of
bh
lines, and non-resonant regular orbits and stochastic orbits newresonancestogethercontributetoasignificantincreasein
scattered over the rest of the map (Figure 7(a)). The points theoverlapofresonances.Resonanceoverlapisawellknown
scatteredalongalinewithaslopeofapproximatelyunityon causeofglobalchaosinHamiltoniansystems(Chirikov1979)
therighthandsideofthemapcorrespondtochaoticboxorbits andmaybethoughtofasoccurringwhenseveraldifferentres-
which lie at the separatrix boundary with a family of short- onancescompetetotrapthesameorbit(BT08). Thisimplies
axistubeswithω ∼ω . that even if figure rotation could allow chaotic box orbits to
x y
For the slowest rotation frequency (R =40), we observe avoid the center as suggested by Gerhard & Binney (1985),
Ω
thedisappearanceofsomeoftheweakestresonancesbutthe the compression of the range of frequencies resulting from
overallstructureofthefrequencymapremainsintact. Asthe themodulationoforbitfrequencieswiththepatternfrequency
corotationradiusofthemodeldecreases(higherrotationfre- results in such significant overlap in resonant orbits that the
quencies) the entire frequency map appears to shrink with effectsofglobalchaosdominatethebehaviorofboxorbits.
the boundaries moving toward the bottom right toward the
short-axis tube “resonance”. This is because as the pattern 3.2. Effectofmoderatefigurerotationontubelikeorbits
frequency of the model increases, each orbit experiences a
We now explore the effect of figure rotation on tube-like
centrifugal force that changes both ω and ω . For a small
x y
orbits in the default model. The diffusion maps in Figure 8
fraction of orbits that lie close to the equatorial plane of the
showorbitslaunchedfromtheY-αstartspace,withthegray