internal structure of neutron stars
DESCRIPTION
Internal structure of Neutron Stars. Artistic view . Astronomy meets QCD. arXiv: 0808.1279. For NSs we can take T=0 and neglect the third equation. Hydrostatic equilibrium for a star. For a NS effects of GR are also important. M/R ~ 0.15 (M/M ) (R/10 km ) -1 - PowerPoint PPT PresentationTRANSCRIPT
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Internal structure of Neutron Stars
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Artistic view
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Astronomy meets QCD
arXiv: 0808.1279
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)( (4)
(3)
4 (2)
)( (1)
2
2
PP
QdtdS
rdrdm
rmmrGm
drdP
{Hydrostatic equilibrium for a star
For NSs we can take T=0and neglect the third equation
For a NS effects of GR are also important.
M/R ~ 0.15 (M/M)(R/10 km)-1
J/M ~ 0.25 (1 ms/P) (M/M)(R/10km)2
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Lane-Emden equation. Polytrops.
n
nc
nn
c
nc
nnc
nnc
nc
dd
dd
GKnaar
KnG
drdKn
drd
drdKn
drdPKP
rdrd
drdP
drd
rGmgg
rGm
drdP
nKKP
1)4/()1( ,/
)1(4
)1(
)1( ,
0 при 1 ,
G4 ,
,
11 const,, ,
22
1/12
/11
/1
/111/11
22
0)(0)0(' ,1)0(
0)(
1
1
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12
1
1
2
3/1
1 5
sin 1
6 6
1 0
n
n
n
Analytic solutions:
|)('|334
|)('|44
1
13
12
132
0
MR
ardrM
cc
c
R
n 0 1 1.5 2 3
2.449 3.142 3.654 4.353 6.897
0.7789 0.3183 0.2033 0.1272 0.04243
1 3.290 5.991 11.41 54.04
1
|'| 1 /c
)1/()3(
)2/()1(
)2/()3(
~
~
~
nn
nnc
nnc
RM
R
M
3/1
3
3
~const 3
~~ 5.1
const ~ 1~ 0
c
c
c
RMn
RMn
RMnRMn
Properties of polytropic stars
γ=5/3 γ=4/3
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Useful equationsWhite dwarfs
1. Non-relativistic electrons γ=5/3, K=(32/3 π4/3 /5) (ћ2/memu
5/3μe5/3);
μe-mean molecular weight per one electron K=1.0036 1013 μe
-5/3 (CGS)
2. Relativistic electrons γ=4/3, K=(31/3 π2/3 /4) (ћc/mu
4/3μe4/3);
K=1.2435 1015 μe-4/3 (CGS)
Neutron stars
1. Non-relativistic neutrons γ=5/3, K=(32/3 π4/3 /5) (ћ2/mn
8/3); K=5.3802 109 (CGS)
2. Relativistic neutrons γ=4/3, K=(31/3 π2/3 /4) (ћc/mn
4/3); K=1.2293 1015 (CGS)
[Shapiro, Teukolsky]
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Neutron stars
Superdense matter and superstrong magnetic fields
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Proto-neutron stars
1202.5791
Mass fraction of nuclei in the nuclear chart for matter at T = 1 MeV, nB = 10−3 fm−3, and YP = 0.4. Different colors indicate mass fraction in Log10 scale.
NS EoS are also important for SN explosion calculation, see 1207.2184
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Astrophysical point of view
Astrophysical appearence of NSsis mainly determined by:
• Spin• Magnetic field• Temperature• Velocity• Environment The first four are related to the NS structure!
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Equator and radius
ds2=c2dt2e2Φ-e2λdr2-r2[dθ2+sin2θdφ2]
In flat space Φ(r) and λ(r) are equal to zero.
• t=const, r= const, θ=π/2, 0<Φ<2π l=2πr
• t=const, θ=const, φ=const, 0<r<r0 dl=eλdr l=∫eλdr≠r00
r0
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Gravitational redshift
)( e
0
e ,e
rdtdNr
dtdN
ddNdtd
r
r
rcGm
2
2
21
1e
It is useful to use m(r) – gravitational mass inside r –instead of λ(r)
Frequency emitted at r
Frequency detected byan observer at infinity
This function determinesgravitational redshift
<1
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Outside of the star
RrRR
MMMM
g
b
/1/
2.0~ sun
rr
drdrrr
dtcrr
ds
cGMr
rr
rcGM
MrmRr
gr
gg
gg
1
1 1
2 ,121e
(1) и (3) изconst )( При
2221
222
222
Bounding energy
Apparent radius
redshift
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Bounding energy
1102.2912
If you drop a kilo on a NS, thenyou increase its mass for < kilo
Macc is shown with color
Macc=ΔMG+ΔBE/c2=ΔMB
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ikikik TcGRgR 4
821
)( )4(
1 1 (3)
4 )2(
21 41 1 (1)
1
22
2
1
22
3
22
PPcP
drdP
cdrd
rdrdm
rcGm
mcPr
cP
rmG
drdP
{ Tolman (1939)Oppenheimer-Volkoff (1939)
TOV equation
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Structure and layers
Plus an atmosphere...
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Neutron star interiors
Radius: 10 kmMass: 1-2 solarDensity: above the nuclearStrong magnetic fields
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Neutron star crust
Nuclear pasta. Watanabe and Maruyama. 1109.3511
Many contributions to the book are available in the arXiv.
Mechanical properties of crusts are continuosly discussed, see 1208.3258
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Accreted crust
1104.0385
It is interesting that the crust formed by accreted matter differs from the crustformed from catalyzed matter. The former is thicker.
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ConfigurationsStable configurations for neutron stars and hybrid stars(astro-ph/0611595).
A RNS code is developedand made available to the publicby Sterligioulas and FriedmanApJ 444, 306 (1995)http://www.gravity.phys.uwm.edu/rns/
NS mass vs.central density(Weber et al. arXiv: 0705.2708)
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Mass-radius
(astro-ph/0611595)
Mass-radius relations for CSswith possible phase transitionto deconfined quark matter.
About hyperon stars see a review in 1002.1658.About strange stars and some otherexotic options – 1002.1793
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Mass-radius relationMain features
• Max. mass• Diff. branches (quark and normal)• Stiff and soft EoS• Small differences for realistic parameters• Softening of an EoS with growing mass
Rotation is neglected here. Obviously, rotation results in:• larger max. mass• larger equatorial radiusSpin-down can result in phase transition,as well as spin-up (due to accreted mass),see 1109.1179
Haensel, Zdunikastro-ph/0610549
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R=2GM/c2
P=ρR~3GM/c2
ω=ωKR∞=R(1-2GM/Rc2)-1/2
Lattimer & Prakash (2004)
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EoS
(Weber et al. ArXiv: 0705.2708 )
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Au-Au collisions
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Experimental results and comparison
Danielewicz et al. nucl-th/0208016
1 Mev/fm3 = 1.6 1032 PaGSI-SIS and AGS data
Also laboratory measurements of lead nuclei radius can be important, see 1202.5701
New heavy-ion data and discussion: 1211.0427
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Phase diagram
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Phase diagram
(astro-ph/0611595)
Phase diagram for isospin symmetry using the most favorable hybrid EoS studiedin astro-ph/0611595.
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Effective chiral model ofHanauske et al. (2000)
Relativistic mean-field modelTM1 of Sugahara & Toki (1971)
Particle fractions
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Superfluidity in NSs
(Yakovlev)
Яковлев и др. УФН 1999
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Quark stars
1210.1910
See also 1112.6430
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Formation of quark stars
1109.0539
Turbulent deflagration,as in SNIa.
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NS interiors: resume
(Weber et al. ArXiv: 0705.2708)
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Papers to read1. astro-ph/0405262 Lattimer, Prakash "Physics of neutron stars"
2. 0705.2708 Weber et al. "Neutron stars interiors and equation of state …”
3. physics/0503245 Baym, Lamb "Neutron stars"
4. 0901.4475 Piekarewicz “Nuclear physics of neutron stars” (first part)
5. 0904.0435 Paerels et al. “The Behavior of Matter Under Extreme Conditions”
6. 1001.1272 Pizzochero “Neutron Stars, the Most Exotic Lab in the Universe ”
7. 1001.3294 Schmitt “Dense matter in compact stars - A pedagogical introduction ”
8. The book by Haensel, Yakovlev, Potekhin
9. 1210.1910 Weber et al. Structure of quark star
10. 1302.1928 Stone “High density matter ”
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Lectures on the WebLectures can be found at my homepage:
http://xray.sai.msu.ru/~polar/html/presentations.html