Quasiequilibrium Sequences of Binary Neutron Stars

8 October 2002

Quasiequilibrium Sequences ofBinary Neutron Stars

Keisuke Taniguchi

Department of Earth and Astronomy,Graduate School of Arts and Sciences,

University of Tokyo

§1. Introduction§2. Evolution of binary neutron stars§3. Formulation§4. Numerical method

§5. Analytical approach and tests§6. Numerical results§7. Summary

§1. Introduction

A common misconception outside of the gravitational-wave researchcommunity is that the primary purpose of observatories such as LIGO is todetect directly gravitational waves. Although the first unambiguous directdetection will certainly be a celebrated event, the real excitement will comewhen gravitational-wave detection can be used as an observational tool forastronomy.

(S. A. Hughes, et al., astro-ph/0110349)

Detection of gravitational waves

• Dynamics of strong gravitational field• Physics in high density regions• Black hole formations / Neutron star formations• Proof of general relativity in the strong gravitational field• Cosmology• etc.

Laser interferometers

Ground-based detectorsLIGO (USA) http://www.ligo.caltech.edu/

The data taking has just started.

Hanford, Washington, USA Livingston, Louisiana, USA4km, 2km 4km

VIRGO (France & Italy) http://www.pg.infn.it/virgo/

The operation has not started yet.

Pisa, Italy3km

GEO600 (England & Germany) http://www.geo600.uni-hannover.de/

The data taking has just started.

Hannover, Germany600m

TAMA300 (Japan) http://tamago.mtk.nao.ac.jp/

The data taking has started since summer in 1999.

Mitaka, Tokyo, Japan300km

Space-based detectors

LISA (ESA & NASA) http://www.lisa.uni-hannover.de/

It may launch in 2011.

5000000km

Targets for laser interferometers

For ground-based detectors

• Coalescing compact binaries(NS-NS, NS-BH, BH-BH)

• Oscillation of neutron stars• Rapidly rotating neutron stars• Stellar core collapse (supernovae)• Low-mass X-ray binaries• etc.

NS/NS Inspiral 300Mpc; NS/BH Inspiral 650Mpc

BH/BH Inspiral, z=0.4

10 20 50 100 200 500 1000

10-24

10-23

10-22

LIGO-I

NB LIGO-II

WB LIGO-II

10 Mo/10Mo BH/BH inspiral 100Mpc

50Mo/50MoBH/BH Merger, z=2

20Mo/20MoBH/BH Merger, z=1

Ω=10 -9Ω=10 -11

Ω=10 -7

Crab SpindownUpper Limit

Vela SpindownUpper Limit

Sco X-1

LMXBs

Known

Pulsa

r

ε=10

-6 , r=1

0kpc

ε=10

-7r=

10kp

c

frequency, Hz

BH/BH Inspiral 400Mpc

Merger

Merger

1/2

, Hz-1

/2h(

f) =

Sh

~

C. Cutler & K. S. Thorne, Proceeding of GR16

(2002), gr-qc/0204090

For space-based detector

• Short-period galactic binaries (WD-WD, WD-LMHS, Low-mass X-ray binaries)• Inspiral and merger of supermassive black hole binaries• Inspiral and capture of compact objects by supermassive black holes• Collapse of supermassive stars• etc.

0.0001 0.001 0.01 0.1

10− 21

10− 20

10− 19

10− 18

10− 17

merger waves

white dwarf binary noise

merger waves

106 Mo / 106 Mo BH Inspiral at 3Gpc



10 Mo BH into106 Mo BH@1Gpc

4U1820-30

maximal spin

no spin

brightest NS/NS binaries

Frequency, Hz

h n = (

f Sh

)1/

2 , H

z-1/2

SA

LISA noise

**

*RXJ1914+245

WD 0957-666

C. Cutler & K. S. Thorne, Proceeding of GR16 (2002), gr-qc/0204090

§2. Evolution of binary neutron stars

Inspiraling Phase

Numerical SimulationPost-NewtonianApproximation

Merging Phase

Inspiraling phaseMass and Spin of each NS, etc.

Blanchet, Damour, Schafer, · · ·

Intermediate phaseInitial data for merging phase, Eq. of state of each NS, etc.

Bonazzola, Gourgoulhon, Marck & KTUryu, Eriguchi & UsuiWilson, Mathews, & Marronetti, · · ·

Merging phaseEq. of state of each NS, BH formation, etc.

ShibataOohara & NakamuraWashington University group, · · ·

§3. Formulation

§§3.1 Basic assumptions

Quasiequilibrium tGW

Porb' 1.1

(c2d

6GMtot

)5/2(Mtot

4µ

)

We assume that there exists a Killing vector:

l =∂

∂t+ Ω

∂

∂ϕ

Perfect fluidThe matter stress-energy tensor:

Tµν = (e + p)uµuν + pgµν

Synchronized and irrotational flowThe realistic rotation state will be irrotational one.

C. S. Kochaneck, ApJ. 398, 234 (1992).L. Bildsten & C. Cutler, ApJ. 400, 175 (1992).

Polytropic equation of statep = κργ

Conformally flat spatial metric

(Isenberg-Wilson-Mathews approximation)

A simplifying assumption for the metric:ds2 = −(N2 −BiB

i)dt2 − 2Bidtdxi + A2fijdxidxj

§§3.2 Basic equations

Fluid equations

First integral of fluid motion:

H + ν − ln Γ0 + lnΓ = constant

Differential eq. for the velocity potential of irrotational flow:

ζH∆Ψ0 +[(1− ζH)∇iH + ζH∇iβ

]∇iΨ0

= (W i −W i0)∇iH + ζH

[W i

0∇i(H − β) +W i

Γn∇iΓn

]

Gravitational field equations

The trace of the spatial part of the Einstein eq. combined

with the Hamiltonian constraint eq.:

∆ν = 4πA2(E + S) + A2KijKij − ∇iν∇iβ

∆β = 4πA2S +3

4A2KijKij − 1

2(∇iν∇iν + ∇iβ∇iβ)

Momentum constraint equation:

∆N i +1

3∇i(∇jNj) = −16πNA2(E + p)U i

+ 2NA2Kij∇j(3β − 4ν)

H := ln h

h : fluid specific enthalpy

ν := ln N

β := ln(AN)

Γ0 : Lorentz factor between the co-

orbiting observer and the Eulerian one

Γ : Lorentz factor between the fluid and

the co-orbiting observer

Γn : Lorentz factor between the fluid

and the Eulerian observer

Irrot. fluid 4-velocity: u =∇Ψ

hScalar potential: Ψ = Ψ0 + fijW i

0xj

E := Γ2n(e + p)− p

S := 3p + (E + p)UiUi

Ni = Bi + (Ω× r)i

Ui : fluid 3-velocity w.r.t. the Eulerian

observer

ζ =d ln H

d ln n

§4 Numerical methodSpectral methods lose much of their accuracy when

non-smooth functions are treated because of the

so-called Gibbs phenomenon.

For a C∞ function, the numerical error decreases

as exp(−N), where N is the number of coefficients

involved in the spectral expansion. On the other

hand, the coefficients of a function which is of class

Cp but not Cp+1 decrease as 1/Np only.

⇑

Multi-domain spectral method circumvents it.

§§4.1 Multi-domain spectral

methodS. Bonazzola, E.Gourgoulhon & J.-A. Marck,

PRD 58, 104020 (1998);

J. Comput. Appl. Math. 109, 433 (1999).

P. Grandclement, S. Bonazzola, E. Gourgoulhon

& J.-A. Marck, J. Comput .Phys. 170, 231 (2001).

Physical domains and their mapping:

Nucleus

[0, 1]× [0, π]× [0, 2π[ → D0

(ξ, θ′, ϕ′) → (r, θ, ϕ)

r = R0(ξ, θ′, ϕ′), θ = θ′, ϕ = ϕ′

Intermediate domains (shell)

[−1, 1]× [0, π]× [0, 2π[ → Dl

(ξ, θ′, ϕ′) → (r, θ, ϕ)

r = Rl(ξ, θ′, ϕ′), θ = θ′, ϕ = ϕ′

Compactified infinite domain

[−1, 1]× [0, π]× [0, 2π[ → Dext

(ξ, θ′, ϕ′) → (r, θ, ϕ)

u := 1/r = U(ξ, θ′, ϕ′), θ = θ′, ϕ = ϕ′

Spectral expansion

f(ξ, θ, ϕ) =

Nϕ−1∑k=0

Nθ−1∑j=0

Nr−1∑i=0

fkjiXkji(ξ)Θkj(θ)Φk(ϕ) Nr, Nθ, Nϕ : numbers of degrees of freedom

in r, θ and ϕ

ϕ direction Φk(ϕ) = eikϕ (−Nϕ/2 ≤ k ≤ Nϕ/2)

Associated collocation points: ϕk =2πk

Nϕ(0 ≤ k ≤ Nϕ − 1)

θ direction Θkj(θ) = cos(jθ) (0 ≤ j ≤ Nθ − 1) for k : even

Θkj(θ) = sin(jθ) (1 ≤ j ≤ Nθ − 2) for k : odd

Associated collocation points: θj =πj

Nθ − 1(0 ≤ j ≤ Nθ − 1)

ξ direction NucleusXkji(ξ) = T2i(ξ) (0 ≤ i ≤ Nr − 1) for j : even

Xkji(ξ) = T2i+1(ξ) (0 ≤ i ≤ Nr − 2) for j : odd

Associated collocation points: ξi = sin(π

2

i

Nr − 1

)(0 ≤ i ≤ Nr − 1)

Intermediate and external domainsXkji(ξ) = Ti(ξ) (0 ≤ i ≤ Nr − 1)

Associated collocation points: ξi = − cos(π

i

Nr − 1

)(0 ≤ i ≤ Nr − 1)

Tn : nth degree Chebyshev polynomial

§§4.2 Improvement on the cases of stiff EOSIf a star has a stiff EOS (γ > 2), the density decreases

so rapidly that ∂n/∂r diverges at the surface.

=⇒ Gibbs phenomenon

⇑Regularization circumvents it.

Virial error for a spherical static star in Newtonian gravity

10 100Number of radial collocation points

10−14

10−12

10−10

10−8

10−6

10−4

Viri

al e

rror

no regularizationK = 1K = 2K = 3

γ=2.25(3)10−14

10−12

10−10

10−8

10−6

10−4

Viri

al e

rror


γ=3(1)

10 100Number of radial collocation points

no regularization

γ=2(4)


γ=2.5(2)

Virial error =|W + 3P ||W |

W : gravitational potential energy

P : volume integral of the fluid pressure

§§4.3 Iterative procedure

(1) We prepare two numerical solutions for

spherically symmetric, static, isolated

neutron stars, and

set the X coordinates of the two stellar

centers:

X<1> = − M<2>

M<1> + M<2>

d

and

X<2> =M<1>

M<1> + M<2>

d

(2) We demand that the enthalpy H be maxi-

mal at the center of each star:∂

∂XH

∣∣∣(X<a>,0,0)

= 0, a = 1, 2

=⇒ Ω and Xrot

(3) The X-axis is translated in order for the

rotation axis to coincide with the origin of

the coordinate system:

X<a>,new = X<a>,old −Xrot, a = 1, 2

Xrot,new = 0

(4) For irrotational configurations, the velocity

potential Ψ is obtained.

ZZ

YY

XX

(1)(1)

(1)zz

yy xx

zz

yyxx

(2)

(2)(2) XX

XX

XX(1)

(2)

00

ΩΩ

rot

(5) The new enthalpy field is calculated:

=⇒ We search for the location of H = 0

(stellar surface), and change the position

of the inner domain boundary.

(6) The new baryon density and so on are ob-

tained via the EOS.

(7) The gravitational field equations are

solved.

(8) The relative difference between the en-

thalpy fields of two successive steps is

checked:

δH :=

∑i|HJ (xi)−HJ−1(xi)|∑

i|HJ−1(xi)|

§5. Analytical approach and tests

In the irrotational case:

In the relativistic case

Numerical solutions :• S. Bonazzola, E. Gourgoulhon, & J.-A. Marck,

PRL82, 892 (1999).

• E. Gourgoulhon, P. Grandclement, KT, J.-A.

Marck, & S. Bonazzola, PRD63, 064029 (2001).

• KT & E. Gourgoulhon, accepted to PRD, gr-

qc/0207098.

• K. Uryu & Y. Eriguchi, PRD61, 124023 (2000).

• K. Uryu, M. Shibata, & Y. Eriguchi, PRD62,

104015 (2000).

⇑ Check the validity of the code

(Semi-)Analytic solutions :

Up to now, there are no solutions !

In the Newtonian case

Numerical solutions :• K. Uryu & Y. Eriguchi, ApJS. 118, 563 (1998).

• KT, E. Gourgoulhon, & S. Bonazzola, PRD64,

064012 (2001).

• KT & E. Gourgoulhon, PRD65, 044027 (2002).

⇑ Check the validity of the code

Semi-analytic solutions :• D. Lai, F. A. Rasio & S. L. Shapiro, ApJ. 420,

811 (1994).

• KT & T. Nakamura, PRL84, 581 (2000);

PRD62, 044040 (2000).

§§5.1 Newtonian analytic solutionIn the irrotational case:KT & T. Nakamura, PRL84, 581 (2000); PRD62, 044040 (2000)

Background: two spherical stars

Expansion parameter: ε :=R0

d, d: orbital separation, R0: radius of a star for d →∞

The physical quantities are calculated upto O[(R0/d)6]:

Equal mass: M1 = M2 = M , Polytropic index: γ = 2

Total energyE =

GM2

R0

[−1− 1

2

(R0

d

)+2

(15

π2−1

)(R0

d

)6

+O

(R0

d

)8

+· · ·

]

Total angular momentumJ =

1

2Md

2Ω

[1 + O

(R0

d

)8

+ · · ·

]

Orbital angular velocityΩ

2=

2GM

d3

[1 + 6

(15

π2− 1

)(R0

d

)5

+ O

(R0

d

)7

+ · · ·

]

Relative change in central densityδρc =

ρc − ρc0

ρc= − 45

2π2

(R0

d

)6

+ O

(R0

d

)8

+ · · ·

§§5.2 Comparison with analytic solution

Global quantities

1 10Coordinate separation / Radius of a spherical star

10−8

10−7

10−6

10−5

10−4

10−3

10−2

Rel

ativ

e di

ffere

nce

Relative difference from analytic solutionγ=2

Difference in E (1.625Msol)Difference in J (1.625Msol)Difference in Ω (1.625Msol)Difference in δρc (1.625Msol)Difference in E (0.001Msol)Difference in J (0.001Msol)Difference in Ω (0.001Msol)Difference in δρc (0.001Msol)Line parallel to (d/R0)

−9

Line parallel to (d/R0)−7.5

1 10Coordinate separation / Radius of a spherical star

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

Rel

ativ

e di

ffere

nce

Relative difference from analytic solutionSynchronized case (γ=2)

Difference in EDifference in JDifference in ΩDifference in δρc

Line parallel to (d/R0)−9

Difference in E Difference in J

Enum − Eana

GM2/R0

Jnum − Jana

Md2ΩKep/2

Difference in Ω Difference in δρc

Ωnum − Ωana

ΩKep |δρc:num − δρc:ana|

where ΩKep =(

2GMd3

)1/2

Internal velocity field

The velocity field in the co-orbiting frame:

ux = Ω

[y +O(ε3)+O(ε4)+O(ε5)+ · · ·

]

uy = Ω

[−x+O(ε3)+O(ε4)+O(ε5)+ · · ·

]

uz = Ω

[O(ε3)+O(ε4)+O(ε5)+ · · ·

]

0 10 20Radial direction [km]

−3e−07

−2e−07

−1e−07

0

Vel

ocity

[c]

NumericalAnalytic

d=100km

0 10 20−1e−08

0

1e−08

2e−08

Vel

ocity

[c]

Z−axis component of internal velocity field

NumericalAnalytic

d=200km


−6e−06

−4e−06

−2e−06

0

NumericalAnalytic

d=70km

0 10 20−2e−08

0

2e−08

4e−08

(θ, ϕ) = (π/4, π/4)

NumericalAnalytic

d=140km


−1.5e−06

−1e−06

−5e−07

0

Vel

ocity

[c]

NumericalAnalytic

d=100km

0 10 20−4e−08

−3e−08

−2e−08

−1e−08

0

Vel

ocity

[c]


NumericalAnalytic

d=200km


−1.2e−05

−8e−06

−4e−06

0

NumericalAnalytic

d=70km

0 10 20−3e−07

−2e−07

−1e−07

0

(θ, ϕ) = (π/4, π/2)

NumericalAnalytic

d=140km


−1.5e−06

−1e−06

−5e−07

0

Vel

ocity

[c]

NumericalAnalytic

d=100km

0 10 20−5e−08

−4e−08

−3e−08

−2e−08

−1e−08

0

Vel

ocity

[c]


NumericalAnalytic

d=200km


−8e−06

−6e−06

−4e−06

−2e−06

0

NumericalAnalytic

d=70km

0 10 20−3e−07

−2e−07

−1e−07

0

(θ, ϕ) = (π/4, 3π/4)

NumericalAnalytic

d=140km

§§5.3 Relative error in the virial theorem

Newtonian identical mass case

2 4 6dG/R0

10−14

10−12

10−10

10−8

10−6

10−4

10−2

Viri

al e

rror

Virial errorSynchronized case

γ=3γ=2.5γ=2.25γ=2γ=1.8

2 4 6dG/R0

Irrotational case

γ=3γ=2.5γ=2.25γ=2γ=1.8

Virial error =|2T + W + 3P |

|W |

T : kinetic energy

W : gravitational potential energy

P : volume integral of the fluid pres-

sure

Relativistic identical mass case

2 3 4dG κ−1/2(γ−1)

10−6

10−5

10−4

Viri

al e

rror

Relative error in the virial theoremM/R=0.14vs.0.14, Synchronized, γ=2

Virial error (mass)Virial error (GB)Virial error (FUS)

2 3 4 5dG κ−1/2(γ−1)

M/R=0.14vs.0.14, Irrotational, γ=2

Virial error (mass)Virial error (GB)Virial error (FUS)

Virial error∣∣∣MADM −MKomar

MADM

∣∣∣∣∣∣V E(FUS)

MADM

∣∣∣∣∣∣V E(GB)

MADM

∣∣∣

ADM mass : MADM = − 1

2π

∮

∞∇i

A1/2

dSi

Komar mass : MKomar =1

4π

∮

∞∇i

NdSi

MADM −MKomar = 0

V E(FUS) =

∫ [2NA

3S +

3

8πNA

3K

ji

Kij +

1

4πNA(∇iβ∇i

β − ∇iν∇iν)

]= 0

J. L. Friedman, K. Uryu, and M. Shibata, PRD65, 064035 (2002).

V E(GB) =

∫ [2A

3S +

3

8πA

3K

ji

Kij +

1

4πA(∇iβ∇i

β − ∇iν∇iν − 2∇iβ∇i

ν)

]= 0

E. Gourgoulhon and S. Bonazzola, CQG.11, 443 (1994).

§6. Numerical results

Parameters

• Adiabatic index : γ = 2

• Rotation state of the binary system : synchronized and irrotational

• Compactness of the star :M/R = 0.12 vs. 0.12, 0.12 vs. 0.13, 0.12 vs. 0.14M/R = 0.14 vs. 0.14, 0.14 vs. 0.15, 0.14 vs. 0.16M/R = 0.16 vs. 0.16, 0.16 vs. 0.17, 0.16 vs. 0.18M/R = 0.18 vs. 0.18

Polytropic unites

Rpoly := κ1/2(γ−1)

d := dκ−1/2(γ−1) Coordinate separation

Ω := Ωκ1/2(γ−1) Orbital angular velocity

M := MADMκ−1/2(γ−1) ADM mass

J := Jκ−1/(γ−1) Angular momentum

§§6.1 Equilibrium configurations

Synchronized case M/R = 0.14 vs. 0.14 0.14 vs. 0.15 0.14 vs. 0.16

Irrotational case M/R = 0.14 vs. 0.14 0.14 vs. 0.15 0.14 vs. 0.16

§§6.2 ADM mass

Synchronized case

0.04 0.08 0.12Ω κ1/2(γ−1)

0.2698

0.27

0.2702

0.2704

0.2706

0.2708

MA

DM κ

−1/2

(γ−1

)

ADM mass (Synchronized case)M/R=0.14 vs. 0.14

0.04 0.08 0.12Ω κ1/2(γ−1)

0.2758

0.276

0.2762

0.2764

0.2766

0.2768M/R=0.14 vs. 0.15

0.04 0.08 0.12Ω κ1/2(γ−1)

0.2812

0.2814

0.2816

0.2818

0.282

0.2822M/R=0.14 vs. 0.16

Irrotational case

0.04 0.08 0.12Ω κ1/2(γ−1)

0.2692

0.2696

0.27

0.2704

0.2708

MA

DM κ

−1/2

(γ−1

)

ADM mass (Irrotational case)M/R=0.14 vs. 0.14

0.04 0.08 0.12Ω κ1/2(γ−1)

0.2752

0.2756

0.276

0.2764

0.2768M/R=0.14 vs. 0.15

0.04 0.08 0.12Ω κ1/2(γ−1)

0.2806

0.281

0.2814

0.2818

0.2822MR=0.14 vs. 0.16

§§6.3 End points of sequences

Identical mass binary systems

1 1.5 2 2.5d/(a1+a1’)

0

0.2

0.4

0.6

0.8

1

χ

Synchronized case, γ=2

M/R=0.12vs.0.12M/R=0.14vs.0.14M/R=0.16vs.0.16M/R=0.18vs.0.18

1 1.5 2 2.5d/(a1+a1’)

Irrotational case, γ=2

M/R=0.12vs.0.12M/R=0.14vs.0.14M/R=0.16vs.0.16M/R=0.18vs.0.18

Different mass binary systems

1 1.5 2 2.5d/(a1+a1’)

0

0.2

0.4

0.6

0.8

1

χ


M/R=0.14vs.0.14M/R=0.14vs.0.15: less massive starM/R=0.14vs.0.15: massive starM/R=0.14vs.0.16: less massive starM/R=0.14vs.0.16: massive star

1 1.5 2 2.5d/(a1+a1’)


M/R=0.14vs.0.14M/R=0.14vs.0.15: less massive starM/R=0.14vs.0.15: massive starM/R=0.14vs.0.16: less massive starM/R=0.14vs.0.16: massive star

Indicator for mass-shedding limit : χ :=(∂H/∂r)eq,comp

(∂H/∂r)pole

Indicator for contact limit :d

a1 + a′1

§§6.4 Central energy density

As a function of the orbital separation

2 3 4dG κ−1/2(γ−1)

−0.12

−0.1

−0.08

−0.06

−0.04

−0.02

(ec−

e c,oo

)/e c,

oo


M/R=0.14vs.0.14M/R=0.14vs.0.16: less massive starM/R=0.14vs.0.16: massive star

0

2 3 4dG κ−1/2(γ−1)

−0.0175

−0.015

−0.0125

−0.01

−0.0075

−0.005

−0.0025


M/R=0.14vs.0.14M/R=0.14vs.0.16: less massive starM/R=0.14vs.0.16: massive star

0

As a function of χ

0 0.2 0.4 0.6 0.8 1χ

−0.25

−0.2

−0.15

−0.1

−0.05

(ec−

e c,oo

)/e c,

oo


M/R=0.12vs.0.12M/R=0.12vs.0.14: less massive starM/R=0.12vs.0.14: massive starM/R=0.14vs.0.14M/R=0.14vs.0.16: less massive starM/R=0.14vs.0.16: massive starM/R=0.16vs.0.16M/R=0.16vs.0.18: less massive starM/R=0.16vs.0.18: massive starM/R=0.18vs.0.18

0

0 0.2 0.4 0.6 0.8 1χ

−0.025

−0.02

−0.015

−0.01

−0.005


M/R=0.12vs.0.12M/R=0.12vs.0.14: less massive starM/R=0.12vs.0.14: massive starM/R=0.14vs.0.14M/R=0.14vs.0.16: less massive starM/R=0.14vs.0.16: massive starM/R=0.16vs.0.16M/R=0.16vs.0.18: less massive starM/R=0.16vs.0.18: massive starM/R=0.18vs.0.18

0

Relative change in central energy density : δe :=ec − ec,∞

ec,∞

§§6.5 Axial ratios

Identical mass binary systems

0 0.2 0.4 0.6 0.8χ

0.6

0.7

0.8

0.9

1

a 2/a1 o

r a3/a

1


M/R=0.12vs.0.12: a2/a1

M/R=0.12vs.0.12: a3/a1

M/R=0.16vs.0.16: a2/a1

M/R=0.16vs.0.16: a3/a1

0 0.2 0.4 0.6 0.8 1χ


M/R=0.12vs.0.12: a2/a1

M/R=0.12vs.0.12: a3/a1

M/R=0.16vs.0.16: a2/a1

M/R=0.16vs.0.16: a3/a1

Different mass binary systems

0 0.2 0.4 0.6 0.8χ

0.6

0.7

0.8

0.9

1

a 2/a1 o

r a3/a

1


M/R=0.14vs.0.14: a2/a1

M/R=0.14vs.0.14: a3/a1

M/R=0.14vs.0.15: a2/a1 (0.14)M/R=0.14vs.0.15: a3/a1 (0.14)M/R=0.14vs.0.15: a2/a1 (0.15)M/R=0.14vs.0.15: a3/a1 (0.15)M/R=0.14vs.0.16: a2/a1 (0.14)M/R=0.14vs.0.16: a3/a1 (0.14)M/R=0.14vs.0.16: a2/a1 (0.16)M/R=0.14vs.0.16: a3/a1 (0.16)

0 0.2 0.4 0.6 0.8 1χ


M/R=0.14vs.0.14: a2/a1

M/R=0.14vs.0.14: a3/a1

M/R=0.14vs.0.15: a2/a1 (0.14)M/R=0.14vs.0.15: a3/a1 (0.14)M/R=0.14vs.0.15: a2/a1 (0.15)M/R=0.14vs.0.15: a3/a1 (0.15)M/R=0.14vs.0.16: a2/a1 (0.14)M/R=0.14vs.0.16: a3/a1 (0.14)M/R=0.14vs.0.16: a2/a1 (0.16)M/R=0.14vs.0.16: a3/a1 (0.16)

§§6.6 Turning point

Synchronized case

0.12 0.14 0.16 0.18M/R

0.4

0.45

0.5

0.55

0.6

χ turn

ing

Turning pointSynchronized case, γ=2

§§6.7 Shift vector

Synchronized case (γ = 2)

M

R= 0.12 vs. 0.12

M

R= 0.12 vs. 0.14

Irrotational case (γ = 2)

M

R= 0.12 vs. 0.12

M

R= 0.12 vs. 0.14

§§6.8 Explanation of the difference in χ

We consider the equal mass binary in Newtonian gravity.

2 4 6 8d/a1

0

0.2

0.4

0.6

0.8

1

χ

γ=2

Synchronized caseIrrotational case

2 2.1 2.2 2.3 2.4 2.5d/a1

0

0.1

0.2

0.3

0.4

0.5

χ

Synchronized case

γ=3γ=2.5γ=2.25γ=2γ=1.8

2 2.1 2.2 2.3 2.4 2.5d/a1

Irrotational case

γ=3γ=2.5γ=2.25γ=2γ=1.8

χ :=(∂H/∂r)eq,comp

(∂H/∂r)pole

χsynch

=−( ∂ν

∂x1)eq,comp + Ω2(− d

2 + a1)

−( ∂ν∂z1

)pole

χirrot

=−( ∂ν

∂x1)eq,comp + Ω2(− d

2 + a1)− 12

∂∂x1

(~∇Ψ0 −Ws)2|eq,comp

−( ∂ν∂z1

)pole

where Ws := Ω(−y1, x1, 0)

Difference in χ along a sequence

2 3 4 5 6 7 8d/a1

−0.1

−0.05

0

0.05

δχ

Difference in χγ=2

δχflow = χirrotflow − χsynch

flow

δχpot = χirrotpot − χsynch

pot

At small separation, the difference in the grav-

itational force plus centrifugal force is smaller

than that in the force related to the internal

flow.

Evaluation of the force related to the internal flow

F := − 1

2

∂

∂x1(~∇Ψ0 −Ws)

2= −Ω

2[(f + y1)

∂f

∂x1+ (g − x1)

(∂g

∂x1− 1

)+ h

∂h

∂x1

]

where we assume the form for ~∇Ψ0:

~∇Ψ0 = Ω(f(d, x1, y1, z1), g(d, x1, y1, z1), h(d, x1, y1, z1))

Then the surface value at (x1, y1, z1) = (a1, 0, 0) becomes

Feq,comp = −Ω2a1

[f(a1)

a1

∂f

∂x1(a1) +

(g(a1)

a1− 1

)(∂g

∂x1(a1)− 1

)]' −Ω

2a1

(g(a1)

a1− 1

)(∂g

∂x1(a1)− 1

)

where we use the fact:

f ' Λ(d)y1

g ' Λ(d)x1 Λ = O

[(R0

d

)3]=

a21 − a2

2

a21 + a2

2

: ellipsoidal model

⇓(i)

g(a1)

a1< 1 and

∂g

∂x1(a1) ≤ 1 =⇒ χ : increase

(ii)g(a1)

a1< 1 and

∂g

∂x1(a1) > 1 =⇒ χ : decrease

∂g

∂x1at

d

R0= 2.851

0 0.2 0.4 0.6 0.8 1(center) Radius of a star x1/a1 (surface)

0

0.5

1

1.5

2

Der

ivat

ive

of v

eloc

ity

Y−axis component of (d(grad(Ψ0))/dx1)/ΩY−axis component of (dWs/dx1)/Ω

0

0.2

0.4

0.6

0.8

1

Nor

mal

ized

vel

ocity

Y−axis component of grad(Ψ0) toward (θ, ϕ)=(π/2, 0)Irrotational case (γ=2), d/R0 = 2.851

Y−axis component of grad(Ψ0)/Ωa1

Y−axis component of Ws/Ωa1

∂g

∂x1(a1, 0, 0) along a sequence

2 3 4 5 6 7 8d/a1

0

0.5

1

1.5

2

2.5

Der

ivat

ive

of v

eloc

ity

γ=3γ=2.5γ=2.25γ=2γ=1.8

Y−axis component of(d(grad(Ψ0))/dx1)/Ω

0

0.1

0.2

0.3

0.4

0.5

Nor

mal

ized

vel

ocity

Y−axis component of grad(Ψ0) along a sequenceIrrotational case, Values at (r, θ, ϕ) = (a1, π/2, 0)

γ=3γ=2.5γ=2.25γ=2γ=1.8

Y−axis component ofgrad(Ψ0)/Ωa1

§7. Summary

Constant baryon number sequences of binary neutron stars in quasiequilibrium have been computed in

both cases of synchronized and irrotational motion and in both cases of identical and different mass

systems.

We have performed a general relativistic treatment, with in the Isenberg-Wilson-Mathews

approximation (conformally flat spatial metric).

We have used a polytropic equation of state with an adiabatic index γ = 2.

(1) Turning point of the ADM mass

• for irrotational binary systems : no turning point

• for synchronized binary systems : one turning point

• is more difficult to be seen for different mass binaries

(2) End point of the sequences

• for synchronized identical star binaries : contact

• for the other types of sequences (synchronized different star binaries,

irrotational identical and different star binaries) : mass shedding point

(3) Decrease of the central energy density

• depends on the compactness of the star

• does not depend on that of its companion

(4) Deformation of the star

• is determined by the orbital separation and the mass ratio

• is not affected much by its compactness (in particular for synchronized cases)

Quasiequilibrium Sequences of Binary Neutron Stars

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