Quasiequilibrium Sequences of Binary Neutron Stars
Transcript of 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
105 Mo / 105 Mo BH Inspiral at 3Gpc
104 Mo / 104 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
no regularizationK = 1K = 2K = 3
γ=3(1)
10 100Number of radial collocation points
no regularization
γ=2(4)
no regularizationK = 1K = 2K = 3
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
0 10 20Radial direction [km]
−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
0 10 20Radial direction [km]
−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]
Z−axis component of internal velocity field
NumericalAnalytic
d=200km
0 10 20Radial direction [km]
−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
0 10 20Radial direction [km]
−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]
Z−axis component of internal velocity field
NumericalAnalytic
d=200km
0 10 20Radial direction [km]
−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
Synchronized case, γ=2
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’)
Irrotational case, γ=2
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
Synchronized case, γ=2
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
Irrotational case, γ=2
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
Synchronized case, γ=2
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
Irrotational case, γ=2
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
Synchronized case, γ=2
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
Irrotational case, γ=2
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
Synchronized case, γ=2
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
Irrotational case, γ=2
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)