Escape from the vicinity of fractal basin boundaries of a star cluster

Mon.Not.R.Astron.Soc.000,1–11(2002)Printed 4February 2008

(MN L A T E X style ?le v2.2)

Escape from the vicinity of fractal basin boundaries of a

star cluster

A.Ernst 1,2 ,A.Just 1?,R.Spurzem 1?and O.Porth 1§

1Astronomisches

Rechen-Institut/Zentrum f¨u r Astronomie der Universit¨a t Heidelberg,M¨o nchhofstrasse 12-14,69120Heidelberg,Germany

2Max-Planck-Institut

f¨u r Astronomie,K¨o nigstuhl 17,69117Heidelberg,Germany

Accepted ...Received ...

ABSTRACT

The dissolution process of star clusters is rather intricate for theory.We investigate it

in the context of chaotic dynamics.We use the simple Plummer model for the gravi-tational ?eld of a star cluster and treat the tidal ?eld of the Galaxy within the tidal approximation.That is,a linear approximation of tidal forces from the Galaxy based on epicyclic theory in a rotating reference frame.The Poincar′e surfaces of section reveal the e?ect of a Coriolis asymmetry.The system is non-hyperbolic which has important consequences for the dynamics.We calculated the basins of escape with respect to the Lagrangian points L 1and L 2.The longest escape times have been mea-sured for initial conditions in the vicinity of the fractal basin boundaries.Furthermore,we computed the chaotic saddle for the system and its stable and unstable manifolds.The chaotic saddle is a fractal structure in phase space which has the form of a Cantor set and introduces chaos into the system.Key words:Star clusters –Stellar dynamics

1INTRODUCTION

The dissolution process of star clusters is an old problem in stellar dynamics.Once a star cluster has formed some-where in a galaxy,it tends to lose mass due to dynamical interactions until it has completely dissolved.It turned out that the physics behind the dissolution of star clusters is intricate and fascinating for theory.If a star cluster of ?-nite mass were isolated,in virial equilibrium (i.e. v 2

e =

12σ2

1D )and the velocity distribution given by a Maxwellian

f M (X )=(4/√π)X 2exp(?X 2

),where X =v/(√2σ1D )and v,v e and σ1D are the velocity,the escape velocity and the velocity dispersion,respectively,the fraction of stars which are faster than the rms escape speed were given by χe =R ∞

√6

f M (X )dX =2p 6/πexp(?6)+erfc √6 0.00738316.This simple analytical result was published by Ambartsum-ian (1938)and two years later,independantly by Spitzer (1940)who named this e?ect “evaporation”.The relevant process which brings stars above the escape speed and lets them evaporate,is two-body relaxation.The time scale of relaxation,which determines the rate of dynamical evolu-tion of a star cluster,yields thus an upper limit to the lifetime of any star cluster.However,since χe is so small

email:aernst@ari.uni-heidelberg.de ?email:just@ari.uni-heidelberg.de

?email:spurzem@ari.uni-heidelberg.de §email:oporth@ari.uni-heidelberg.de

(and relaxation time relatively long),the evaporation time is much longer than a Hubble time for typical globular clus-ters.Following a suggestion of Chandrasekhar (1942),King (1959)studied the e?ect of “potential escapers”.These are stars which have been scattered above the escape energy but which have not yet left the cluster.These may be scat-tered back to negative energies within a crossing time and remain bound.H′e non (1960,1969)stressed the importance of few close encounters between stars for the rate of mass loss of star clusters.The Fokker-Planck approximation,which is widely used to study the dynamical evolution of star clusters,neglects strong encounters by construction.Nev-ertheless,close encounters could still be interpreted statis-tically as a certain discontinuous Markov process (Tschar-nuter 1971).However,direct N -body models seem to be ide-ally suited to study this phenomenon in more detail.Spitzer &Shapiro (1972)estimate that “close encounters may pro-duce e?ects perhaps as great as 10percent of the “dominant”distant encounters”and ignore them.

Nature provides an environment for star clusters in which the escape rate is typically strongly enhanced as com-pared with the slow evaporation rate of isolated star clusters:The tidal ?eld of a galaxy induces saddle-like troughs in the walls of the potential well of a star cluster (cf.Figure 1).It therefore lowers the energy threshold in star clusters above which stars can escape from zero to a negative value (Wielen 1972,1974).Moreover,if we consider a star cluster in the

a r X i v :0710.4485v 1 [a s t r o -p h ] 24 O c t 2007

2 A.Ernst,A.Just,R.Spurzem&O.Porth tidal?eld of a galaxy as a dynamical system,the tidal?eld can change the system’s dynamics in a dramatical way as compared with an isolated system.

In general,the escape process from star clusters in a tidal?eld proceeds in two stages:(1)Scattering of stars into the“escaping phase space”by two-body encounters and(2)leakage through openings in the equipotential su-faces around saddle points of the potential.The“escaping phase space”is de?ned as the subset of phase space,from which escape is possible.It is well understood that the time scale for a star to complete stage(1)scales with relaxation time.On the other hand,the time scale for a star to com-plete stage(2)depends mainly on its energy(but also on its location in phase space as we will see).When we neglect the e?ect of two-body relaxation for the consideration of stage (2),the motion of a single star in the star cluster is deter-mined between times t1and t2only by the smooth grav-itational potential in which the star moves.The potential itself is generated by the other stars in the star cluster dis-regarding their“grainyness”and by the superposed galactic gravitational?eld,which is due to the matter distribution of the galaxy.Within this framework,we will study stage (2)of the escape process in this paper.In this connection, the work of Fukushige&Heggie(2000)is of major interest. Their main result is an expression for the time scale of es-cape for a star in stage(2)which has just completed stage (1).The dependance of the escape process on two time scales which scale di?erently with the particle number N imposes a severe scaling problem for N-body simulations.The scaling problem is of relevance since the it is on today’s general-purpose hardware architectures not yet simply feasible to simulate the evolution of globular clusters with realistic par-ticle numbers of a few hundred thousands or even millions of stars by means of direct N-body simulations.The result of Fukushige&Heggie has been applied in Baumgardt(2001) to solve the important scaling problem for the dissolution time of star clusters in the special case of circular cluster

orbits.The obtained scaling law t dis∝t3/4

rh ,where t dis and

t rh are the dissolution and half-mass relaxation times,re-spectively has been veri?ed,e.g.in Spurzem et al.(2005).

The problem of escape has also a long history in the context of the theories of dynamical systems and chaos. It is well-known for a long time,that certain Hamiltonian systems allow for escape of particles towards in?nity.Such “open”Hamiltonian systems have been studied by Rod (1973),Churchill et al.(1975),Contopoulos(1990),Con-topoulos&Kaufmann(1992),Siopis et al.(1997),Navarro &Henrard(2001)and Schneider,T′e l&Neufeld(2002).The related chaotic scattering process,in which a particle ap-proaches a dynamical system from in?nity,interacts with the system and leaves it,escaping to in?nity,was investi-gated by many authors,as Eckhardt&Jung(1986),Jung (1987),Jung&Scholz(1987),Eckhardt(1987),Jung&Pott (1989),Bleher,Ott&Grebogi(1989)and Jung&Ziemniak (1992).Chaotic scattering in the restricted three-body prob-lem has been studied by Benet et al.(1997,1999).Typically, the in?nity acts as an attractor for an escaping particle, which may escape through di?erent exits in the equipoten-tial surfaces.Thus it is possible to obtain basins of escape(or “exit”basins),similar to basins of attraction in dissipative systems or the well-known Newton-Raphson fractals.Special types of basins of attraction(i.e.“riddled”or“intermingled”basins)have been explored by Ott et al.(1993)and Som-merer&Ott(1993,1996).Basins of escape have been stud-ied by Bleher et al.(1988),and they are discussed in Con-topoulos(2002).Reasearch on escape from the paradigmatic H′e non-Heiles system has been done by de Moura&Lete-lier(1999),Aguirre,Vallejo&Sanju′a n(2001),Aguirre& Sanju′a n(2003),Aguirre,Vallejo&Sanju′a n(2003),Aguirre (2004)and Seoane Seoane Sep′u lveda(2007).These papers served as the basis of this work.Relatively early,it was rec-ognized,that the key to the understanding of the the chaotic scattering process is a fractal structure in phase space which has the form of a Cantor set(Cantor1884)and is called the chaotic saddle.Its skeleton consists of unstable periodic or-bits(of any period)which are dense on the chaotic saddle (https://www.sodocs.net/doc/014054619.html,i1997)and introduce chaos into the system(e.g. Contopoulos2002).The properties of chaotic saddles have been investigated by di?erent authors,as Hunt(1996),Lai et al.(1993),Lai(1997)or Motter&Lai(2001).Both hyper-bolic and non-hyperbolic chaotic saddles occur in dynamical systems.In the?rst case,there are no Kolmogorov-Arnold-Moser(KAM)tori,which means that all periodic orbits are unstable.In the second case,there are both KAM tori and chaotic sets in the phase space(J.C.Vallejo,https://www.sodocs.net/doc/014054619.html,m. and https://www.sodocs.net/doc/014054619.html,i et al.1993).We note that all of the above ref-erences on the chaotic dynamics are exemplary rather than exhaustive since there exists a vast amount of literature on these topics.

The aim of this paper is to allude to the importance of this last-mentioned branch of research for the?eld of stellar dynamics of star clusters.We will study the escape process from star clusters within the framework of chaotic dynam-ics.In section2,we introduce the tidal approximation,i.e. approximate equations of motion for stellar orbits in a star cluster which is embedded in the tidal?eld of a galaxy.In section3,we describe our(very simple)model of the gravi-tational potential.In section4,we discuss Poincar′e surfaces of section,which show the e?ect of a Coriolis asymmetry. Furthermore,we discuss the basins of escape in Section5 and the chaotic saddle and its stable and unstable invariant manifolds in Section6.Section7contains the discussion and conclusions.

2THE TIDAL APPROXIMATION

The“tidal approximation”which is widely used in stellar and galactic dynamics for studies of stellar systems in a tidal ?eld is nothing else than a simple approximation which,his-torically,has been applied already in the19th century in “Hill’s problem”(e.g.Stump?1965,Szebehely1967,Siegel &Moser1971)in the context of the(rather intricate)lu-nar theory.The di?erence to the tidal approximation lies merely in the form of the gravitational potentials which are used.The assumption that a star cluster moves around the Galactic centre on a circular orbit allows to use the epicyclic approximation to calculate steady linear tidal forces acting on the stars in the star cluster.As in the circular restricted three-body problem the appropriate coordinate system is a rotating reference frame in which both the star cluster cen-tre and the Galactic centre(i.e.,the primaries)are at rest. Its origin is the star cluster centre,sitting in the minimum of the e?ective Galactic potential.The x-axis points away from

Escape from the vicinity of fractal basin boundaries of a star cluster3 the Galactic centre;the y-axis points in the direction of the

rotation of the star cluster around the Galactic centre;the

z-axis lies perpendicular to the orbital plane and points to-

wards the Galactic North pole.We de?ne scaled corotating

coordinates(x,y,z)as

x=(R?R g)/r t,y R g(φ?ωt)/r t,z=z /r t(1)

where(R,φ,z )are galactocentric cylindrical coordinates,

R g andωare the radius and the frequency of the circular

orbit,respectively,r t is a length scale(i.e.the tidal radius

de?ned in Equation(11)below)and t is time(in the context

of“Hill’s problem”cf.Glaschke2006).Since the coordinate

system is rotating,centrifugal and Coriolis forces appear ac-

cording to classical mechanics.In addition,tidal forces enter

the equations of motion for stellar orbits near the origin of

coordinates.To?rst order,we have in the rotating frame

¨x=??Φcl

?x

?

…

?2Φg

?R2

?

(R g,0)

x+ω2x+2ω˙y(2)

¨y=??Φcl

?y

?2ω˙x(3)

¨z=??Φcl

?z

?

…

?2Φg

?z2

?

(R g,0)

z(4)

whereΦcl(x,y,z)andΦg(R,z )are the star cluster potential and the axisymmetric galactic potential,respectively.The second-last term on the right hand side in(2)is the cen-trifugal force and the last terms in(2)and(3)are Coriolis forces.According to Binney&Tremaine(1987),the epicyclic frequencyκand the vertical frequencyνare given by

κ2=…

?2Φg

?R2

?

(R g,0)

+3ω2,ν2=

…

?2Φg

?z2

?

(R g,0)

(5)

Thus the equations of motion can be written as

¨x=f x?(κ2?4ω2)x+2ω˙y(6)¨y=f y?2ω˙x(7)¨z=f z?ν2z,(8) where(f x,f y,f z)=??Φcl is the(speci?c)force vector from the other cluster member stars which typically depends non-linearly on the coordinates.It is of interest for the following discussion that the equations of motion(6)-(8)are invariant under time reversal.1Note that under a time reversal the frequencies also change their sign.Also,the equations of motion(6)-(8)admit an isolating integral of motion,the Jacobian

C=1

2

`

˙x2+˙y2+˙z2

′

+Φe?(x,y,z),(9)

1The invariance under time reversal is related to the existence of a discrete group with only two elements,which acts on the space of solutions of the equations of motion(6)-(8).In our case,the e?ective potential(10)and the Coriolis forces are time-symmetric,which implies the same symmetry of the equations of motion.where

Φe?(x,y,z)=Φcl(x,y,z)+

1

2

(κ2?4ω2)x2+

1

2

ν2z2(10) is the e?ective potential,which is plotted in Figure1for the 2D case.Other isolating integrals are not given in the form of a simple analytical expression.However,some solutions of(6)-(8)are subject to a third integral(H′e non&Heiles 1964),as has been demonstrated numerically by Fukushige &Heggie(2000),who calculated a Poincar′e surface of sec-tion.In principle,one may obtain power series expansions of such third integrals,see,e.g.the original works by Gus-tavson(1966)and Finkler,Jones&Sowell(1990)and the review in Moser(1968).Also,third integrals can be related to the existence of Killing tensor?elds which are well-known in General Relativity(Clementi&Pettini2002).At last,the tidal radius(King1962)

r t=

…

GM cl

4ω2?κ2

?1/3

(11) where M cl is the star cluster mass,provides a fundamental length scale of the problem.It is the distance from the origin of coordinates to the Lagrangian points L1and L2(which lie on the x axis,see Figure1).

3THE MODEL

The characteristic frequenciesω,κandνarise as proper-ties of the galactic gravitational potential.Throughout the paper,we use the values of the characteristic frequencies in the solar neighborhood.All of them can be expressed in terms of Oort’s constants A and B(see,e.g.,Binney &Tremaine1987):ω2=(A?B)2,κ2=?4B(A?B),κ2?4ω2=?4A(A?B),ν2=4πGρg+2(A2?B2).The vertical frequencyνcan be derived from the Poisson equa-tion for an axisymmetric system(see Oort1965)andρg is the local Galactic density,which contributes to the dom-inant?rst term.We obtain both dimensionless parameters κ2/ω2 1.8andν2/ω2 7.6using the values of Oort’s con-stants given in Feast&Whitelock(1997)and the value for local Galactic density given in Holmberg&Flynn(2000).It is then convenient to choose the following system of units:

G=1,ω=1,M cl=2.2(12) The resulting length unit is the tidal radius r t and the for-mulation of the dynamical problem with its equations of motion is completely dimensionless.For the star cluster, we use a Plummer model,the most simple analytic model for a star cluster(see appendix A).The density pro?les of King models which have a cuto?radius,where the density drops to zero,?t the measured density pro?les of globu-lar clusters better than Plummer models(King1966).Since they are tidally limited by construction,they are at?rst glance ideally suited for our purpose.However,the grav-itational force?eld can only be tabulated from a numer-ical integration of a non-linear di?erential equation.We have made a compromise which is not relevant for the in-teresting physics:We choose the Plummer radius in such a way,that the Plummer model(see Appendix A)is the

4 A.Ernst,A.Just,R.Spurzem&O.

Porth

Figure1.E?ective potential in the tidal approximation(z=0plane).The Lagrangian points at L1=(?1,0)and L2=(1,0)can be seen.The escapers pass these saddle points while they leak out.The equipotential lines connecting them mark the tidal boundary of the star cluster.The details of the model are given in Section3.

best?t to a King model with W0=4(i.e.with concen-

tration c=log

10(r t/r K) 0.840),which completely?lls

the Roche lobe in the tidal?eld,i.e.the

density of the King

model approaches zero at the tidal radius(11).The?t of the density pro?les is quite acceptable for density contrasts of

log

10(ρc/ρ(r)) 3,whereρc andρ(r)are the central density

and the density as a function of radius,respectively.For the deviation between the King density pro?le and the Plummer ?t we obtain(ρP l(r)?ρK(r))/ρc<1.2%.The ratio of the Plummer radius to the King radius and the“concentration”of the Plummer model(which can only be de?ned because of the existence of a tidal radius)are then r P l/r K 1.257

and c P l=log

10(r t/r P l) 0.741,respectively.In our units,

the Plummer radius is therefore r P l 0.182.

As a more technical remark,we note that we used an 8th-order Runge-Kutta scheme for the orbit integrations. The relative error in the Jacobian C was always limited to ?C/C<10?12for all orbit integrations.

4POINCAR′E SURF ACES OF SECTION AND THE CORIOLIS ASYMMETRY

A critical Jacobi constant

C L=Φe?(r t,0,0)=Φe?(?r t,0,0)(13)Figure3.The two main types of orbits at b C=0.Left:Regular retrograde orbit,Right:Chaotic prograde orbit.The variable b C is de?ned in Equation(14).

is given by the value of the e?ective potential(10)at the Lagrangian points L1and L2.For our model we have C L=?3.264444506.For a Jacobian C>C L the equipotential surfaces are open and particles can escape.Furthermore,we de?ne the dimensionless deviation from C L by

b C=(C L?C)/C L,(14) where C is some other value of the Jacobian.The dimen-sionless deviation b C is positive for C>C L if C and C L are both negative,which is always the case in this paper.

Escape from the vicinity of fractal basin boundaries of a star cluster

5

Figure2.Poincar′e surfaces of section.Top left:At b C=0for orbits crossing y=0with˙y>0,Top right:Same as top left,but at b C=0.1,Bottom left:At b C=0for orbits crossing˙x=0with˙y>0,Bottom right:At b C=0for orbits crossing˙y=0with˙x>0.The variable b C is de?ned in Equation(14).

A?rst insight can be gained by calculating Poincar′e sur-faces of section which are shown in Figure2for two dif-ferent Jacobi constants:The upper left surface of section is at the critical Jacobian C L at which all orbits still re-main within a bounded area in phase space and we have no escapers.We can see that this is a system with divided phase space,i.e.we have both chaotic and regular orbits.It is striking that the left half of the surface of section is al-most completely occupied by regular,quasiperiodic orbits. These orbits are retrograde with respect to the orbit of the star cluster around the galactic centre(Fukushige&Heggie 2000),as can be seen by looking at the sketch in Figure4, and they are subject to a third integral of motion.On the other hand,most of the prograde orbits are chaotic apart from a few smaller regular islands.Since the e?ective poten-tial is mirror-symmetrical with respect to both the x-and y-axes(see Figure1),the asymmetry seen in the surfaces of section must be due to the Coriolis forces.Thus such a behaviour might be termed a“Coriolis asymmetry”(cf.In-nanen1980).The Coriolis forces are special in the sense that their direction is not perpendicular to the tangent plane to the equipotential surfaces but to the velocity of a particle. The upper right surface of section is at a higher Jacobi con-stant.The particles can leak out through the openings in the equipotential surfaces and escape towards in?nity(positive x-direction)or the galactic centre(negative x-direction).It is remarkable,that only the chaotic orbits escape,while the regular,quasiperiodic orbits remain within the tidal bound-ary of the star cluster,since the third integral restricts their accessible phase space and hinders their escape.In star clus-

6 A.Ernst,A.Just,R.Spurzem&O.Porth

Figure4.Sketch of the coordinate system.The escapers leak out through the openings in the equipotential surfaces passing either L1or L2.Only schematically,two orbits are shown which cross the x axis with˙y>0.

ters,two-body relaxation may scatter stellar orbits beyond the critical Jacobi constant.However,if the orbits are retro-grade,the stars will remain bound to the star cluster with high probability until two-body relaxation further scatters them into the escaping phase space.The two lower surfaces of section in Figure2indicate the orbital structure in po-sition space at C=C L,where all orbits are restricted to the region within the almond-shaped tidal boundary of the star cluster(cf.Figure1).One notes that certain parts of the chaotic regions of the lower surfaces of section are less densely?lled by stellar orbits which is a common property of dynamical systems.

Figure3shows typical examples of the two main types of orbits.The regular orbit resembles a rosette orbit in an axisymmetric potential or a loop orbit which suggests that the third integral is some sort of generalization of angular momentum(Binney&Tremaine1987).We found indeed that the angular momentum is approximately conserved for the left orbit of Figure3,while it is not at all conserved for the right orbit.Other types of orbits can be found which are associated with the smaller regular islands in the sur-faces of section and which may look rather interesting.At last,we remark that the di?erence between retrograde and prograde orbits appears prominently in N-body models of dissolving rotating star clusters within the framework of the tidal approximation:Where either the regular or the chaotic domains in phase space are more strongly occupied by stel-lar orbits due to the existence of a net angular momentum of the cluster in one direction(see Ernst et al.2007).

5THE BASINS OF ESCAPE

Figure5shows the basins of escape for a tidally limited star cluster within the framework of the tidal approximation. For the plots on the left-hand side,～3×105orbits have been integrated;for the plots on the right-hand side,it were ～6?8×105orbits,depending on the area of the surface of section containing initial conditions.The phase space is di-vided into the escaping phase space(red and yellow regions) and the non-escaping phase space(black regions).The red regions denote initial conditions,where the escaping stars pass L1while the yellow regions represent initial conditions,

Plot(Figure5)Black Red Yellow

Top left25.436.638.0

Top right12.519.368.2

Middle left36.931.531.5

Middle right21.436.841.8

Bottom left40.929.529.6

Bottom right27.236.236.6

Table1.Fraction of orbits(in percent)belonging to the inter-

section of the basins of escape with Poincar′e surfaces of section

which are shown in Figure5.

where the escaping stars pass L2.The black regions show

those initial conditions,where stars do not escape.These

are?rst and foremost the regular regions where a third inte-

gral is present.Note that the stable manifold of the chaotic

saddle is not marked in black,since it is of Lebesgue measure

zero(cf.Section6).Since there exist regular regions(with

KAM tori),the system is non-hyperbolic,i.e.there exist sta-

ble periodic orbits with corresponding elliptic points in the

surfaces of section(cf.Section4and Figure2).We can see

that in the escaping phase space there exist regions,where

we have a very sensitive dependance of the escape process

on the initial conditions,i.e.a slight change of the initial

conditions makes the star escape through the opposite La-

grangian point.This is the classical indication of chaos.It

is interesting to note that these regions arise from imme-

diate vicinity of the black regions where orbits are regular.

In these domains of phase space the red and yellow regions

are completely intertwined with respect to each other:The

boundary between these regions is fractal.The volume in

phase space occupied by these regions(with sensitive depen-

dance on the initial conditions)increases as the Jacobian C

approaches the critical Jacobi constant C L(i.e.in the limit b C→0)and the exits become smaller.At b C→0there is a maximal“fractalization”of phase space;when the sys-

tem approaches the limit of small exits the basins become

uncertain(Aguirre&Sanju′a n2003).The term“uncertain”

means that we become unable to follow the real trajectory

of a particle by means of numerical integration.2Moreover,

the following theorem can be formulated:“For all points P

in the escaping phase space of an open Hamiltonian system,

and for allδ>0(precision of the experiment),there exists

a critical size of the exits w c>0such that for all w w c

we can?nd a point P in a ball centreed in P and radiusδ

that belongs to a di?erent basin than P”(Aguirre&Sanju′a n

2003).

Table1shows the fraction of orbits(in percent)belong-

ing to the intersection of the basins of escape with Poincar′e

surfaces of section which are shown in Figure5.It can be

seen that in the limit b C→0the fractions of particles pass-ing L1and L2tend to be equal while this must not be the

case if there are large areas without sensitive dependance of

the escape process on the initial conditions.

Figure6shows how the escape times are distributed

on surfaces of section.The longest escape times correspond

to initial conditions near the boundaries between the basins

2In other words,the computer fails here to be a Laplacian demon (Laplace1814).

Escape from the vicinity of fractal basin boundaries of a star cluster7

Figure5.The basins of escape.Top left:At b C=0.1for orbits crossing y=0with˙y>0,Top right:At b C=0.1for orbits crossing ˙y=0with˙x>0,Middle row:As the upper row,but at b C=0.01,Botton row:As the middle row,but at b C=0.001.The red regions denote initial conditions,where the escaping stars pass L1while the yellow regions represent initial conditions,where the escaping stars pass L2.The black regions show those initial conditions,where stars do not escape.The variable b C is de?ned in Equation(14).

8 A.Ernst,A.Just,R.Spurzem&O.

Porth Figure6.Distribution of escape times t e on surfaces of

section for b C=0.01.Top:For the x?v x surface of section of Figure5, Bottom:For the x?y surface of section of Figure5.The darker the color,the longer the escape time.

of escape of Figure5.The shortest escape times have been measured for the ordered regions without sensitive depen-dance on the initial conditions,i.e.those far away from the fractal basin boundaries.

Figure7shows the fraction of remaining(non-escaped) orbits N e(t e>t)/N e,0after time t corresponding to the basins of escape shown in Figure5.Only the escaping orbits have been used for the statistics.The orbits corresponding to the regions without sensitive dependance on the initial con-ditions shown in Figures5have short escape times as can be seen in Figure6.For these orbits,the decay law is a power law as shown in the inlays for the solid curve(b C=0.1).On the other hand,the decay law is exponential for the chaotic orbits near the fractal basin boundaries(i.e.the orbits with Figure7.Histogram of the fraction of remaining(non-escaped)

orbits N e(t e>t)/N e,0after time t.Top:For the x?v x surfaces

of section of Figure6,Bottom:For the x?y surfaces of section of

Figure6,Solid:b C=0.1,Dashed:b C=0.01,Dotted:b C=0.001. The inlays with two logarithmic axes show the early phase for the

solid line(i.e.for b C=0.1).The variable b C is de?ned in Equation (14).

long escape times which correspond to the regions with sen-sitive dependance on the initial conditions).The slopes of the exponentials(i.e.the decay constants)depend on the value of b C but are identical for both surfaces of section.The exponential decay law indicates that the underlying process is of a statistical nature similar to the radioactive decay of unstable nuclides or that of bubbles in beer foam.

6THE CHAOTIC SADDLE AND ITS

INV ARIANT MANIFOLDS

The stable manifold of the chaotic saddle is shown in the top row of Figure8for two Poincar′e surfaces of section. The stable manifold coincides with the fractal basin bound-aries of Figure5and therefore acts as a separatrix between the exit basins..With data points of?nite size,the top row shows orbits which do not escape for time t→∞,although, strictly speaking,their Lebesgue measure is zero.The un-stable manifold of the chaotic saddle is shown in the middle row of Figure8.These are orbits which do not escape for time t→?∞.Note that the stable and unstable manifolds are symmetric with respect to each other,since the equa-

Escape from the vicinity of fractal basin boundaries of a star cluster9

Figure8.The non-hyperbolic invariant set and its stable and unstable manifolds at b C=0.01.Top row:Stable manifold,Middle row: Unstable manifold,Bottom row:Invariant set.The variable b C is de?ned in Equation(14).

10 A.Ernst,A.Just,R.Spurzem&O.Porth

tions of motion(6)-(8)are time-symmetric.For the plots

in the middle row of Figure8,the sign of the time step in

the Runge-Kutta integrator has been reversed,The bottom

row of Figure8shows the intersection of the chaotic sad-

dle(i.e.a non-hyperbolic chaotic invariant set)with the two

Poincar′e surfaces of section.It is the invariant set of non-

escaping orbits for time t→∞and t→?∞.The chaotic

saddle has the form of a Cantor set(Cantor1884)which is

formed by the intersection of its stable and unstable man-

ifolds.The fact that the system is non-hyperbolic implies

that there are tangencies between the stable and unstable

manifolds,i.e.that their angle is not always bounded away

from zero(Lai et al.1993).The unstable(hyperbolic)points

in the intersection of the Poincar′e surface of section with the

chaotic saddle correspond to unstable periodic orbits.As is

well-known(e.g.Contopoulos2002),these introduce chaos

into the system since they repel the orbits in their neigh-

bourhood in the direction of their unstable eigenvectors.

The remarkable similarity of our system with the H′e non-

Heiles system(see Aguirre,Vallejo&Sanju′a n2001)is that

the fractal dimension of the chaotic saddle tends to three

(i.e.the black areas in Figure8grow until we have a maxi-

mal fractalization of the phase space,cf.Aguirre&Sanjuan

2003),which is the dimension of the hypersurface of phase space with constant Jacobian.At b C=0there is a sudden transition where the non-hyperbolic invariant set abruptly

?lls the whole non-regular subset of phase space within the

last closed equipotential surface and no escape is possible

any more.This situation can be seen in the Poincar′e sur-

faces of section in Figure2in Section4which shows the

chaotic domains of phase space as a dotted area.

7DISCUSSION AND CONCLUSIONS

We studied the chaotic dynamics within a star cluster which

is embedded in the tidal?eld of a galaxy.We calculated

within the framework of the tidal approximation Poincar′e

surfaces of section,the basins of escape,the chaotic saddle

and its stable and unstable manifolds as well.The system

is non-hyperbolic which has important consequences for the

dynamics,i.e.there are orbits which do not escape if relax-

ation is neglected.These are mainly the retrograde orbits

as has been shown earlier by Fukushige&Heggie(2000)

and,more recently,in the N-body study by Ernst et al.

(2007).The escape times are longest for initial conditions

near the fractal basin boundaries.The decay law is a power

law for those stars which escape from the regions without

sensitive dependance on the initial conditions in Figure5

(i.e.with short escape times,as can be seen in Figure6).

On the other hand,the decay law is exponential for orbits

which escape from the regions with sensitive dependance on

the initial conditions(i.e.with long escape times).The ef-

fect of relaxation(i.e.a di?usion in the Jacobian and the

third integral among di?erent stellar orbits)on the chaotic

dynamics which we investigated in this work may be a very

interesting topic for future research.8ACKNOWLEDGEMENTS

We thank Dr.Juan Carlos Vallejo and Prof.Burkhard Fuchs for pointing out some references to us,which were necessary for this work.Also,we thank Prof.Rudolf Dvorak for his comments on the manuscript.AE would like to thank his father for a discussion and Dr.Patrick Glaschke for many inspiring discussions.Furthermore,he gratefully acknowl-edges support by the International Max Planck Research School(IMPRS)for Astronomy and Cosmic Physics at the University of Heidelberg.

REFERENCES

Ambartsumian V.A.,1938,Ann.Leningrad State Univ., 22,19

Aguirre J.,Vallejo J.C.,Sanju′a n M.A.F.,2001,Phys. Rev.E,64,066208

Aguirre J.,Sanju′a n M. A. F.,2003,Phys.Rev.E,67, 056201

Aguirre J.,Vallejo J.C.,Sanju′a n M.A.F.,2003,Int.J. Mod.Phys.B,17,4171

Aguirre J.,2004,PhD thesis,Universidad Rey Juan Carlos, Spain

Baumgardt H.,2001,MNRAS,325,1323

Benet L.,Trautmann D.,Seligman T.H.,1997,Cel.Mech. Dyn.Astron.,66,203

Benet L.,Seligman T.H.,Trautmann D.,1999,Cel.Mech. Dyn.Astron.,71,167

Binney J.,Tremaine S.,1987,Galactic Dynamics,Prince-ton Univ.Press

Bleher S.,Grebogi C.,Ott,E.,Brown R.,1988,Phys.Rev. A,38,930

Bleher S.,Ott E.,Grebogi C.,1989,Phys.Rev.Lett.,63, 919

Cantor G.,1884,Acta Mathematica,4,381.English transl. in ed.Edgar G.A.,1993,Classics on Fractals,Addison-Wesley

Chandrasekhar S.,1942,Principles of Stellar Dynamics, Univ.Chicago Press,Chicago

Churchill R.C.,Pecelli G.,Rod D.L.,1975,J.Di?.Eq., 17,329

Clementi C.,Pettini M.,2002,Cel.Mech.Dyn.Astron., 84,263

Contopoulos G.,1990,Astron.Astrophys.,231,41 Contopoulos G.,Kaufmann D.,1992,Astron.Astrophys., 253,379

Contopoulos G.,2002,Order and Chaos in Dynamical As-tronomy,Berlin(Springer)

De Moura A.P.S.,Letelier P.S.,1999,Phys.Lett.A,256, 362

Eckhardt B.,Jung C.,1986,J.Phys.A:Math.Gen.,19, L829

Eckhardt B.,1987,J.Phys.A:Math.Gen.,20,5971 Ernst A.,Glaschke P.,Fiestas J.,Just A.,Spurzem R., 2007,MNRAS,377,465

Feast M.,Whitelock P.,1997,MNRAS,291,683

Finkler P.,Jones C.E.,Sowell G.A.,1990,Phys.Rev.A, 42,1931

Fukushige T.,Heggie D.C.,2000,MNRAS,318,753 Glaschke P.,2006,PhD thesis,University of Heidelberg, http://www.ub.uni-heidelberg.de/archiv/6553

Escape from the vicinity of fractal basin boundaries of a star cluster11

Gustavson F.,1966,AJ,71,670

H′e non M.,1960,Ann.Astrophys.,23,668

H′e non M,Heiles C.,1964,AJ,69,73

H′e non M.,1969,A&A,2,151

Holmberg J.,Flynn C.,2000,MNRAS,313,209

Hunt B.R.,Ott E.,Yorke J.A.,1996,Phys.Rev.E,54, 4819

Innanen K.A.,1980,Ap.J.,85,81

Jung.C.,1987,J.Phys.A:Math.Gen.,20,1719

Jung C.,Scholz H.J.,1987,J.Phys.A:Math.Gen.,20, 3607

Jung C.,Pott S.,1989,J.Phys.A:Math.Gen.,22,2925 Jung C.,Ziemniak E.,1992,J.Phys.A:Math.Gen.,25, 3929

King I.R.,1959,AJ,64,351

King I.R.,1962,AJ,67,471

King I.R.,1966,AJ,71,64

Lai Y.-C.,Grebogi C.,Yorke J.A.,Kan I.,1993,Nonlin-earity,6,779

Lai Y.-C.,1997,Phys.Rev.E,56,6531

Laplace,1814,Th′e orie analytique des probabilit′e s.In: Oeuvres Compl′e tes de Laplace,Volume VII.,Paris,1820 (Gauthier-Villars)

Moser J.K.,1968,Lectures on Hamiltonian Systems,Mem. AMS,81,1.Reprinted in eds.MacKay R.S.and Meiss J.

D.,1987,Hamiltonian Dynamical Systems,Bristol(Adam Hilger)

Motter https://www.sodocs.net/doc/014054619.html,i Y.,2001,Phys.Rev.E,65,015205 Navarro J.F.,Henrard J.,2001,A&A,369,1112

Oort J.H.,1965,Stellar Dynamics,in:Galactic Structure, eds.A.Blaauw&M.Schmidt,Univ.Chicago Press,p.455 Ott E.,Sommerer J.C.,Alexander J.C.Kan I.,Yorke J.

A.,1993,Phys.Rev.Lett.,71,4134

Plummer H.C.,1911,MNRAS,71,460

Rod D.L.,1973,J.Di?.Eq.,14,129

Schneider J.,T′e l T.,Neufeld Z.,2002,Phys.Rev.E,66, 066218

Seoane Sep′u lveda J.M.,2007,PhD thesis,Universidad Rey Juan Carlos,Spain

Siegel C.L.,Moser J.K.,1971,Lectures on Celestial Me-chanics,Springer Edition,Berlin,Heidelberg,New York Siopis C.,Kandrup H.E.,Contopoulos G.,1997,Cel.Mech. Dyn.Astron.,65,57

Sommerer J.C.,Ott E.,1993,Nature,365,138 Sommerer J.C.,Ott E.,1996,Phys.Lett.A,214,243 Spitzer L.Jr.,Shapiro S.L.,1972,Ap.J.,173,529 Spurzem R.,Giersz M.,Takahashi K.,Ernst A.,2005,MN-RAS,364,948

Stump?K.,1965,Himmelsmechanik,Vol.2,VEB Deutscher Verlag der Wissenschaften,Berlin

Szebehely V.,1967,Theory of orbits.The restricted prob-lem of three bodies,New York(Academic Press) Tscharnuter W.M.,1971,Ap&SS,14,11

Wielen R.,1972,in Lecar M.,ed.,Gravitational N-body problem,Proceedings of IAU Colloquium10,Reidel Pub-lishing Company,Dordrecht

Wielen R.,1974,in Mavridis L.N.,ed.,Stars and the Milky Way System.Proceedings of the First European As-tronomocal Meeting,Athens,Sept.4-9,1972.,Vol.2,p. 326APPENDIX A:THE ISOLATED PLUMMER MODEL

The most well-known self-consistent model of a spherically symmetric star cluster is the Plummer model(Plummer 1911).It is given by

Φ(y)=?Φ0

1

p

1+y2

=?Φ0cosφ(A1) M(y)=M

y3

(1+y2)

=M sin3φ(A2)

ρ(y)=ρ0

1

(1+y2)5/2

=ρ0cos5φ(A3) with y=r/r Pl 0,φ=arctan(y)and the Plummer radius

r Pl=

GM

Φ0

=

…

3M

4πρ0

?1/3

(A4) where M is the total mass,?Φ0is the central potential and ρ0the central density,all of them being?nite.From(A2) we?nd the half mass radius

y h=tan

h

arcsin

“

2?1/3

”i

1.30476603(A5) The velocity dispersion can be obtained by integrating the Jeans equation of hydrostatic equilibrium.It is given by

σ2(y)=

Φ0

6

1

p

1+y2

=

Φ0

6

cosφ(A6) Therefore the relation

v2e(y)=12σ2(y),(A7) where v e=

p

2|Φ(y)|is the escape speed from an isolated Plummer model,strictly holds at any radius(cf.Section1 of this paper).One can show that the Plummer model is the only model which has this property.

This paper has been typeset from a T E X/L A T E X?le prepared by the author.