White Dwarf Binary Statistics in Globular ClusterSimulations by MOCCA
Dongming Jin, Matthew BenacquistaDept. of Physics, University of Texas at Brownsville
Contact: [email protected]
AbstractThe MOCCA code is one of the most advanced codes whichhas the capacity to simulate a realistic sized star cluster witha full dynamical history including star evolution using MonteCarlo methods for the cluster evolution and the Fewbody codefor scattering. The dynamical evolution of a cluster can resultin the formation of many binary system. Some of these bi-naries may be very close. Close white dwarf binaries may bepromising gravitational wave sources.
Our work uses MOCCA to simulate 90 globular clusters ofdifferent number of stars, binary fraction, metallicity and ini-tial mass function parameter. After ruling out models whichevaporate before 9 Gyrs and uninteresting models with a verylow number of white dwarf binaries, we do multiple runs ofthe remaining models for around a Hubble time in order to getstatistics on the overall white dwarf binary population of dif-ferent component types, orbital periods and cluster radii in thetime range from 8 Gyrs to 12 Gyrs. We consider that whitedwarf binaries which exist within a specified time range & La-grangian radius range and have orbit at periods less than a dayare observable. Thus we set up a map of potential white dwarfbinary detection rates for different types of globular clusters.
Free ParametersTo get a general estimate of the statistics for white dwarf binaries,simulations for globular clusters with a different total number ofstars N , initial binary fraction fb, metallicity Z and initial massfunction parameter α are performed.
Parameter ValueN 24000, 100000fb 0.1, 0.2, 0.3, 0.4, 0.5Z 0.02, 0.0002, 0.00001α 2, 2.35, 3
α is the factor for the power-law mass function [4]:
ξ(m) ∝ m−α
where m is the stellar mass in solar units and ξ(m)dm is the num-ber of stars in the mass interval m to m + dm.
White Dwarf Binary Population
0 2000 4000 6000 8000 10000 12000 140000
2
4
6
8
10
12
14
16Model 8, White Dwarf Binary Population:
Time: Myrs
Bin
ary
Popula
tion
HeHeWD Binary
COCOWD Binary
ONeONeWD Binary
HeCOWD Binary
HeONeWD Binary
COONeWD Binary
Total WD Binary
0 5000 10000 150000
100
200
300
400
500
600Model 85, White Dwarf Binary Population:
Time: Myrs
Bin
ary
Popula
tion
HeHeWD Binary
COCOWD Binary
ONeONeWD Binary
HeCOWD Binary
HeONeWD Binary
COONeWD Binary
Total WD Binary
0 5000 10000 150000
0.5
1
1.5
2
2.5
3Model 30, White Dwarf Binary Population:
Time: Myrs
Bin
ary
Popula
tion
HeHeWD Binary
COCOWD Binary
ONeONeWD Binary
HeCOWD Binary
HeONeWD Binary
COONeWD Binary
Total WD Binary
0 5000 10000 150000
50
100
150
200
250
300Model 82, White Dwarf Binary Population:
Time: Myrs
Bin
ary
Popula
tion
HeHeWD Binary
COCOWD Binary
ONeONeWD Binary
HeCOWD Binary
HeONeWD Binary
COONeWD Binary
Total WD Binary
The population of different types of WD-WD binaries during cluster evolution.
White Dwarf Binary Distribution
0 2000 4000 6000 8000 10000 12000 14000−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5Model 8, White Dwarf Binary Distribution
Time: Myrs
La
gra
ngia
n R
ad
ii: L
og
1%
10%
30%
50%
70%
90%
0 2000 4000 6000 8000 10000 12000 14000 16000−2
−1.5
−1
−0.5
0
0.5
1
1.5
2Model 85, White Dwarf Binary Distribution
Time: Myrs
La
gra
ngia
n R
ad
ii: L
og
1%
10%
30%
50%
70%
90%
0 5000 10000 15000−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5Model 30, White Dwarf Binary Distribution
Time: Myrs
La
gra
ngia
n R
ad
ii: L
og
1%
10%
30%
50%
70%
90%
0 2000 4000 6000 8000 10000 12000 14000 16000−1.5
−1
−0.5
0
0.5
1
1.5
2Model 82, White Dwarf Binary Distribution
Time: Myrs
La
gra
ngia
n R
ad
ii: L
og
1%
10%
30%
50%
70%
90%
Markers show the location when white dwarf binary is first formed, the size ofthe markers indicate the lifetime.
White Dwarf Binary Orbital PeriodThe trajectory begins with ‘×’ mark shows the evolution of a certain whitedwarf binary. Color of the trajectory indicates the orbital period range. TheNo. in title is the amount of certain white dwarf binary for the whole stellarhistory.
Color Green Magenta CyanOrbital period less than 10−2 day 10−2 to 1 day larger than 1 day
8 9 10 11 12
−1
0
1
Time: Gyrs
La
gra
ngia
n R
ad
ii: L
og
Model 8, He−He WDB No.:2
8 9 10 11 12
−1
0
1
Time: Gyrs
La
gra
ngia
n R
ad
ii: L
og
Model 8, CO−CO WDB No.:58
8 9 10 11 12
−1
0
1
Time: Gyrs
Lagra
ng
ian R
adii:
Log
Model 8, ONe−ONe WDB No.:4
8 9 10 11 12
−1
0
1
Time: Gyrs
Lagra
ng
ian R
adii:
Log
Model 8, He−CO WDB No.:9
8 9 10 11 12
−1
0
1
Time: Gyrs
Lag
ran
gia
n R
adii:
Lo
g
Model 8, He−ONe WDB No.:1
8 9 10 11 12
−1
0
1
Time: Gyrs
Lag
ran
gia
n R
adii:
Lo
g
Model 8, CO−ONe WDB No.:24
8 10 12 14
−1
0
1
Time: Gyrs
Lag
ran
gia
n R
ad
ii: L
og
Model 85, He−He WDB No.:264
8 10 12 14
−1
0
1
Time: Gyrs
Lag
ran
gia
n R
ad
ii: L
og
Model 85, CO−CO WDB No.:490
8 10 12 14
−1
0
1
Time: Gyrs
Lagra
ng
ian
Rad
ii: L
og
Model 85, ONe−ONe WDB No.:54
8 10 12 14
−1
0
1
Time: Gyrs
Lagra
ng
ian
Rad
ii: L
og
Model 85, He−CO WDB No.:466
8 10 12 14
−1
0
1
Time: Gyrs
Lag
ran
gia
n R
adii:
Log
Model 85, He−ONe WDB No.:53
8 10 12 14
−1
0
1
Time: Gyrs
Lag
ran
gia
n R
adii:
Log
Model 85, CO−ONe WDB No.:215
8 10 12 14
−1
0
1
Time: Gyrs
La
gra
ng
ian
Radii:
Lo
g
Model 30, He−He WDB No.:1
8 10 12 14
−1
0
1
Time: Gyrs
La
gra
ng
ian
Radii:
Lo
g
Model 30, CO−CO WDB No.:5
8 10 12 14
−1
0
1
Time: Gyrs
Lagra
ngia
n R
adii:
Log
Model 30, ONe−ONe WDB No.:0
8 10 12 14
−1
0
1
Time: Gyrs
Lagra
ngia
n R
adii:
Log
Model 30, He−CO WDB No.:2
8 10 12 14
−1
0
1
Time: Gyrs
Lagra
ngia
n R
adii:
Log
Model 30, He−ONe WDB No.:0
8 10 12 14
−1
0
1
Time: Gyrs
Lagra
ngia
n R
adii:
Log
Model 30, CO−ONe WDB No.:0
8 10 12 14
−1
0
1
Time: Gyrs
Lag
ran
gia
n R
adii:
Log
Model 82, He−He WDB No.:82
8 10 12 14
−1
0
1
Time: Gyrs
Lag
ran
gia
n R
adii:
Log
Model 82, CO−CO WDB No.:208
8 10 12 14
−1
0
1
Time: Gyrs
Lagra
ngia
n R
adii:
Log
Model 82, ONe−ONe WDB No.:17
8 10 12 14
−1
0
1
Time: Gyrs
Lagra
ngia
n R
adii:
Log
Model 82, He−CO WDB No.:356
8 10 12 14
−1
0
1
Time: Gyrs
Lag
rang
ian R
adii:
Log
Model 82, He−ONe WDB No.:60
8 10 12 14
−1
0
1
Time: Gyrs
Lag
rang
ian R
adii:
Log
Model 82, CO−ONe WDB No.:89
Potential Detectable Rate Mapping
0.1 0.2 0.3 0.4 0.5
2
2.5
3
N=24000, Z=0.02
Binary Fraction
IMF
pa
ram
ete
r Model 30
0.1 0.2 0.3 0.4 0.5
2
2.5
3
N=24000, Z=0.001
Binary Fraction
IMF
pa
ram
ete
r
0.1 0.2 0.3 0.4 0.5
2
2.5
3
N=24000, Z=0.0002
Binary Fraction
IMF
pa
ram
ete
r
Model 8
0.1 0.2 0.3 0.4 0.5
2
2.5
3
N=100000, Z=0.02
Binary Fraction
IMF
pa
ram
ete
r
Model 82
0.1 0.2 0.3 0.4 0.5
2
2.5
3
N=100000, Z=0.001
Binary Fraction
IMF
para
mete
r
Model 85
0.1 0.2 0.3 0.4 0.5
2
2.5
3
N=100000, Z=0.0002
Binary Fraction
IMF
para
mete
r
Figure 1: Size indicates the number of WD-WD binaries under following con-ditions of a certain model:
1. Exists during 8 Gyr to 12 Gyr,
2. has a orbital period less than 1 day,
3. locates between 30% to 70% Lagrangian radii.
The marked models are plotted for more detail information.
Acknowledgements: MJB and DJ acknowledge the supportof NASA grant NNX08AB74G and the Center for GravitationalWave Astronomy, supported by NSF Cooperative Agreements HRD-0734800 and HRD-1242090.
References
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[4] P. Kroupa, C. A. Tout, and G. Gilmore. The distribution oflow-mass stars in the Galactic disc. , 262:545–587, June 1993.
