Dynamical evolution of globularclusters using N-body simulation
Naoto Yoshinari
The University of Tokyo

Introduction
Globular cluster• Star : 105 - 106

NGC 6093• Size : 1 - 10 pc• Age : 10 Gyr• Shape : Nearly spherical• 150 globular clusters in our Galaxy
They are very closely related to the formationof galaxies, but their formation process is still
unclear.
3

Introduction
Gravothermal dynamics of star clusters• Core collapse increases the core density• Massive stars concentrate on the cluster core due to theenergy equipartition• Binaries are formed because of three-body encounter• Formation of hard binary halts the collapse
Binary star systems can provide an important source of energy for
globular clusters
These proceed on the relaxation timescale'/)𝑁𝑟#𝑡"# = +/) +/)∼ 2×107 year
𝑀 𝐺 ln(0.1𝑁)

𝑟#1pc'/)

𝑀10>M⊙+/)

𝑚M⊙

B+4

Introduction
Dynamical evolution of BH in GC
3body encounter
BH
ExchangeMS

BBH

HaloCore
BH

Hard binary star becomes
harder due to encounters
BH

Binary star systems escapeoutside the cluster
5

Introduction
Recent N-body simulation
Dragon simulation (Wang et al. 2016)
NBODY6++GPU• Star : 106• Age : 12 Gyr• King model• Direct integration• (KS) Regularization
BUT, the cluster density isrelatively low
BBHs were not formed much

Red: BBH8

Problems
Our targets• Simulation of binary stars in the globular cluster• Investigation of the formation rate of BBHs (Detection of GW)

Difficulties• When the distance between two stars becomes
extremely close, the calculation breaks down.
• Direct calculation cost 𝑂(𝑁 ) )Compute-intensive!
• Binary stars have short orbital period (days), but globular cluster
life time is more than 10Gyr.
It is difficult to simulate the evolution of globular clusters.
9PENTACLE( Parallelized Particle-Particle Particle-Tree code for Planet formation)

P3T scheme

Oshino et al.(2011), Iwasawa et al.(2015)• Open-source library designed for full automaticparallelization• New hybrid algorithm for the time integration• For Collisional N-body system
Short-range : Direct integration + Hermite method(Individual timestep)Long-range : Tree method + Leap-Frog integrator(Block timestep)
Calculation cost : 𝑂(𝑁log𝑁)
103Method of P TAlgorithm
The cutoff function 𝐾(𝑟JN ) splits thegravitational force between particles tocontribution of close encounters andothers
O

𝑭#G"H,J = − LJMNO𝑭RSTU,J = − L
JMN

𝐺𝑚J 𝑚N𝑟JN'𝐺𝑚J 𝑚N
𝑟JN'

𝐾 𝑟JN 𝒙JN
1 − 𝐾 𝑟JN 𝒙JN

𝑟JN𝑦−𝛾
𝑟[\𝑥=, 𝑦=, 𝛾=1−𝛾
𝑟YZU
𝑟YZU

𝑟[\ : inner cutoﬀ radius𝑟YZU : outer cutoﬀ radius
11

GORILLAGORILLA ( Tanikawa & Fukushige 2009 )Ø N-body simulation code for star clusters
Ø Forth-order Hermite scheme with individual timestep
Calculation cost : 𝑂(𝑁 ))Ø Tight Binaries are approximated as Kepler motions.
𝑚binary𝑚𝑚𝑚2𝑚𝛼𝑟"^_𝛼𝑟"^_
Regarded as
Isolated star𝑟"^_ : separation of binary
α : constant

14

Initial condition• Unit : N-body unit
(G = 1, M = 1, E = -1/4)
𝑦• Distribution : King model• Particle : N = 4096
(equal-mass particles)
Follow the evolution of the system up tothe moment of the core collapse𝑥15

ComparisonCore radius𝑟Y =9𝑣Y)
4𝜋𝐺𝜌Y𝑣Y : central velocity dispersion
𝜌Y : central density

PENTACLE breaks down when the cluster
core contracts enough.

𝑟YCore radius becomes smaller gradually.

Collapse time𝑡YY ∼ 20𝑡"#∼ 1400 (𝑁 = 4096)

Calculation time
PENTACLE : 470m

GORILLA : 2000m
16

Comparison
Show the time evolution of therelative energy error until T = 1500.
Growth of energy error is nearly
constant during an early stage.
The time that the energy error starts
to increase rapidly coincides with thecore collapse time 𝑡YY .

| 𝐸 − 𝐸i /𝐸i |Long term integration

Hard binaries are formed at thecollapse time.
17Summary of PENTACLE and GORILLAResults• PENTACLE cannot calculate star cluster evolution aftercore collapse• Energy conserves until core collapse• Both of two codes collapsed at the expected time
• Calculation speed of PENTACLE is about 5 times as fastas GORILLA (N=4096)• Hard binary affects the evolution of whole system

18New code
We are developing the new simulation code (PENTACLE + GORILLA)

New code

Expected calculation cost
∼ 𝑂(𝑁log𝑁)

Calculate the orbit of binarywithout regularization
Present situation

Kick

PENTACLE(Soft)GORILLA(Hard)

DriftKick𝑡

𝛥𝑡RSTU /2𝑡
𝛥𝑡J• The time that the energy error starts to increase rapidly before
core collapse time.
• New code cannot deal with three-body encounter
19

Summary & Future plans
Summary• P3T scheme can reduce the calculation cost of gravitational force• GORILLA code can calculate star cluster evolution with binaries• If new hybrid code is developed, the orbits of binaries can becalculated exactly with short time

Future plans• Implementation of new simulation code
• Application to BBH in globular cluster
20