Continuum Models of Globular Cluster Dynamics

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Continuum Models of Globular Cluster Dynamics

Rainer Spurzem, Astronomisches Rechen-Institut Heidelberg, Germany

[email protected]://www.ari.uni-heidelberg.de/mitarbeiter/spurzem/

CGWP WorkshopCGWP WorkshopOct. 2003Penn State

Continuum models....Continuum models....

IntroductionIntroduction

RotationRotation

BinariesBinaries

Black Holes, Grav. Rad.Black Holes, Grav. Rad.

Main Collaborators:M. Giersz (CAMK Warsaw, Poland)E. Kim, H.M. Lee (Seoul Nat. Univ., Korea)S. Aarseth (Inst. Of Astron. Cambridge UK)...

...and students...:P. Amaro-Seoane, S. Deiters, J. Fiestas, E. Khalisi...

HYDRAHYDRA


Foundation Document of ARI May 10, 1700Calendar Patent of Duke of Brandenburg

ARI Fields of Work today:

• Astrometry (Hipparcos, GAIA)

• Stellar Dynamics (Galaxies, Star Clusters)

• Calendar Data• Bibliography (Astronomy and Astroph. Abstracts, ceased 2001)

(ARI)


Introduction: HistoryIntroduction: History

S.v. Hoerner,Z.f.Astroph. 1960, 63

Siemens 2002N=4,8,12,16 (4 Trx)

N=16,25 (40 Trx)see also v. Hoerner 2001, in star2000-Proceedings...see also v. Hoerner 2001, in star2000-Proceedings...


Introduction: Star ClustersIntroduction: Star Clusters

Dynamical Time ScaleDynamical Time Scale

Relaxation Time ScaleRelaxation Time Scale

Age of UniverseAge of Universe

Laboratories for gravothermal N-Body Systems!Laboratories for gravothermal N-Body Systems!Note: Cosmological and Galactic N-Body Simulations need few crossing times, Note: Cosmological and Galactic N-Body Simulations need few crossing times,

and less than a relaxation time, while gravothermal systems need multiples of and less than a relaxation time, while gravothermal systems need multiples of N crossing times, several relaxation times! Complexity goes as NN crossing times, several relaxation times! Complexity goes as N3 3 !!

10106 6 yrsyrs

101088 yrs yrs

10101010 yrs yrs

← ← Virial Equilibrium Virial Equilibrium →→


Stellar Evolution TimeStellar Evolution Time

Stellar CollisionStellar Collision TimeTime

Readjustment of Collision ProductsReadjustment of Collision Products 　　Star Formation TimeStar Formation Time

Binary EvolutionBinary Evolution

Laboratories for dissipative orLaboratories for dissipative or

chemodynamical N-Body Systems!chemodynamical N-Body Systems!

10105-105-10 yrs yrs 10104-6 4-6 yrsyrs

101033 yrs yrs10106 6 yrsyrs

10104-104-10 yrs yrs

Introduction: Star ClustersIntroduction: Star Clusters


S.J.Aarseth, S. Mikkola (ca. 20.000 lines):•Hierarchical Block Time Steps•Ahmad-Cohen Neighbour Scheme•Kustaanheimo-Stiefel and Chain-Regular. for bound subsystems of N<6•4th order Hermite scheme (pred/corr)• Bulirsch-Stoer (for KS)



Some methods for studying the evolution of globular clusters (by D.C.Heggie)

Monte Carlo (Giersz 1998)

Hybrid (Giersz & Spurzem 2000)Hybrid (Giersz & Spurzem 2000)



Where it does work....(Spurzem & Aarseth 1996) (Giersz & Spurzem 1994)

N-Body / N-Body N-Body / Fokker-Planck

In spherical symmetry ...but...



N-body .vs. Continuum model w. rotationN-body .vs. Continuum model w. rotation

Kim, Lee,Kim, Lee,Spurzem,Spurzem,2003 (in prep.)2003 (in prep.)

Decay of rotationalDecay of rotationalenergy by relaxationenergy by relaxationand tidal fieldand tidal field


Physical and Numerical Methods: Modelling the DynamicsPhysical and Numerical Methods: Modelling the Dynamics

Full Fokker-Planck Equation, use Liouville‘s Theorem andFull Fokker-Planck Equation, use Liouville‘s Theorem andRosenbluth Potentials (local equation):Rosenbluth Potentials (local equation):



Rosenbluth PotentialsRosenbluth Potentials(Rosenbluth, McDonald(Rosenbluth, McDonald& Judd 1956)& Judd 1956)

Diffusion CoefficientsDiffusion Coefficients(Local)(Local)

Gas Models:Gas Models:up to l=2 for background f up to l=2 for background f (in principle higher possible)(in principle higher possible)

)()()(0

ll

l PvAvf



Orbit averagedOrbit averagedFokker-PlanckFokker-PlanckEquationEquation

(here in the 2D(here in the 2Dform for axisymm.form for axisymm.systems,systems,Einsel & SpurzemEinsel & Spurzem1999)1999)



Flux Conserving Form of Fokker-Planck EquationFlux Conserving Form of Fokker-Planck Equation(Einsel & Spurzem 1999), Chang-Cooper Scheme (1970)(Einsel & Spurzem 1999), Chang-Cooper Scheme (1970)


Rotation - Initial modelsRotation - Initial models

Rotating King modelRotating King model

* two parameter family * two parameter family

* W* W0 0 = 3,6 & = 3,6 & ωω0 0 = 0.0 ... 2.4= 0.0 ... 2.4

Mass functionMass function

* single mass system* single mass system

* two-component model* two-component model

* continuous mass spectrum (10 comp), * continuous mass spectrum (10 comp),

Boundary conditionBoundary condition

* tidally limited/isolated system* tidally limited/isolated system

),( 00 W


Rotation – single massRotation – single mass

(Einsel & Spurzem 1999) (Kim, Einsel, Lee, Spurzem & Lee 2002)

Rotation acceleratescore collapse significantly and takes away flattening


Rotation - Effect of mass functionRotation - Effect of mass function

Kim,Kim,Lee,Lee,Spurzem,Spurzem,2003 (in prep.)2003 (in prep.)


&

ic,

ic,

Kim, Lee,Kim, Lee,Spurzem,Spurzem,2003 (in prep.)2003 (in prep.)


Giersz & Spurzem (2000, 2003)

Fully self-consistent evolution of cluster and Binaries...3b and 4b integration of encounters using Regularisation techniques. No assumptions about any cross sections, but still point mass...remain challenges for theory AND modelling...

Gas Monte Carlo Hybrid Model and BinariesGas Monte Carlo Hybrid Model and Binaries

NNss =300.000, N =300.000, N bb=30.000=30.000


Gas Model and relativistic Gas Model and relativistic binaries?binaries?


Loss Cone TheoryLoss Cone Theory standard diffusive picture - fast out – slow in, Liouville Theorem ok,standard diffusive picture - fast out – slow in, Liouville Theorem ok, orbit averaged Fokker-Planck equationorbit averaged Fokker-Planck equation

lcDrxcross ttEq ,1/,1)( Non-standard cases (re Milosavljevic, Phd thesis 2002)1. Ejection (binary black hole) instead of tidal disruption θlc large!2. Binary black hole gets kicks3. External perturbations (fueling, bars, mergers)4. Orbit „diffusion“ in non-spherical potentials can be fast

(Malkov, Vilkoviski, Nuzhnova, Spurzem, 1993, in Russian) 5. Stars return in case of ejection tout=n tcross ; n > 1

→ time-dep. loss-cone required!

Gas Model and Black HoleGas Model and Black Hole


Milosavljevic 2002



Time-Dep. Loss-Cone Diffusion (Milos. & Merritt, 2003)Time-Dep. Loss-Cone Diffusion (Milos. & Merritt, 2003) in orbit averaged Fokker-Planck modelin orbit averaged Fokker-Planck model

Anisotropic gaseous model (Louis & Spurzem 1991, Spurzem 1994) Anisotropic gaseous model (Louis & Spurzem 1991, Spurzem 1994) plus local simplified diffusion equation (Amaro-Seoane, Freitag, plus local simplified diffusion equation (Amaro-Seoane, Freitag, Spurzem, in prep.)Spurzem, in prep.)Using gaseous model:Using gaseous model:

http://www.gaseous.model.dehttp://www.gaseous.model.de (S. Deiters) (S. Deiters)



Gas Model - Example of Black Hole GrowthGas Model - Example of Black Hole Growth

Amaro-Seoane,Amaro-Seoane,Spurzem, 2001Spurzem, 2001

Amaro-Seoane,Amaro-Seoane,Spurzem, 2003,Spurzem, 2003,in prep.in prep.

Black HoleBlack HoleGrowth Growth Self-RegulationSelf-RegulationGas ModelGas Modelsingle masssingle mass


Gas Model – Multi-Mass with Black HoleGas Model – Multi-Mass with Black Hole

Amaro-Seoane,Amaro-Seoane,Freitag,Freitag,Spurzem, 2003,Spurzem, 2003,in prep.in prep.

Black HoleBlack HoleGrowth Growth Self-RegulationSelf-RegulationGas Model Gas Model multi-massmulti-mass


Exotic Exotic Processes... Processes...

Star-Gas Star-Gas InteractionsInteractions

Just (2003)Just (2003)

Gas Model and Star-Gas DragGas Model and Star-Gas Drag


Ongoing Project: High-Resolution 1D Chemo+Stellar DynamicsOngoing Project: High-Resolution 1D Chemo+Stellar Dynamics Stellar Dynamical EquationsStellar Dynamical Equations Gas Dynamics, 1D Radiative Transfer (Yorke, 1980)Gas Dynamics, 1D Radiative Transfer (Yorke, 1980) Stellar Collisions, Star-Gas InteractionStellar Collisions, Star-Gas Interaction

Gas Model and Radiative TransferGas Model and Radiative Transfer


Status of Continuum ModelsStatus of Continuum Models

st.dyn.st.dyn.

st. evol.st. evol.

FP 1DFP 1Df(E) sphf(E) sph

FP 2D FP 2D f(E,J) sphf(E,J) sph

FP 2DFP 2Df(E,Jf(E,Jzz) rot) rot

GasGasf(E,J) sph. f(E,J) sph.

M-CarloM-Carlof(E,J) sph.f(E,J) sph.

multi-massmulti-mass

post-collapsepost-collapse ☼☼ ☼☼ ☼☼ ☼☼ ☼☼

stell. evolutionstell. evolution ☼☼ ☼☼ ☼☼ ☼☼tidal cut/tidal cut/

mass lossmass loss ☼☼ ☼☼ ☼☼ ☼☼ ☼☼

tidal shock tidal shock ☼☼ many hardmany hard

binariesbinaries((☼☼))approx.approx.

☼ ☼ eq.m.eq.m.

(hybr. MC)(hybr. MC)

☼☼

binarybinary

stell. evolutionstell. evolution

☼☼ yellow sun: accomplished yellow sun: accomplished green key: in progress or straightforwardgreen key: in progress or straightforward

☼☼


Status of Continuum ModelsStatus of Continuum Models

moremore FP 1DFP 1Df(E) sphf(E) sph

FP 2D FP 2D f(E,J) sphf(E,J) sph

FP 2DFP 2Df(E,Jf(E,Jzz) rot) rot

GasGasf(E,J) sph. f(E,J) sph.

M-CarloM-Carlof(E,J) sph.f(E,J) sph.

cent. bl. hole cent. bl. hole

star accretion star accretion ☼☼ ☼☼ ☼☼ ☼☼

stell. collisionsstell. collisions ((☼☼)) ☼☼gas dynamics gas dynamics

(single comp.)(single comp.) ((☼☼))hyd.-stat.hyd.-stat.

star-gas star-gas interactionsinteractions

☼☼ star formationstar formation full chemo-full chemo-

dynamicsdynamics

☼☼ yellow sun: accomplished yellow sun: accomplished green key: in progress or straightforwardgreen key: in progress or straightforward


Credit: R. Blandford, in Active Galactic Nuclei 1991, Saas Fee Lecture 20

Future Modelling Required for Galactic NucleiFuture Modelling Required for Galactic Nuclei


Yu & Lu 2001

Can we detect them?

Spin-Orbit Couplings Black Holes / Stellar Orbits?Spin-Orbit Couplings Black Holes / Stellar Orbits?


ConclusionConclusion

Continuum Models still very fast even in Continuum Models still very fast even in times of GRAPE-6,8,...,NBODY6++,...times of GRAPE-6,8,...,NBODY6++,...

Continuum Models help understanding Continuum Models help understanding physicsphysics

Continuum Models are far from being Continuum Models are far from being exploited to their limits – much to do still!exploited to their limits – much to do still!

Comparisons with NBODY and Monte-Comparisons with NBODY and Monte-Carlo should be continued (KyotoII)Carlo should be continued (KyotoII)

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Continuum Models of Globular Cluster Dynamics