lecture1. structure of neutron stars
DESCRIPTION
Lecture1. Structure of Neutron Stars. Sergei Popov (SAI MSU). Artistic view. 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 J/M ~ 0.25 (1 ms/P) (M/M )(R/10km) 2. - PowerPoint PPT PresentationTRANSCRIPT
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Lecture1.Structure of Neutron Stars
Sergei Popov (SAI MSU)
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Artistic view
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)( (4)
(3)
4 (2)
)( (1)
2
2
PP
Qdt
dS
rdr
dm
rmmr
Gm
dr
dP
{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
d
d
d
d
GKnaar
Kn
G
dr
dKn
dr
ddr
dKn
dr
dPKP
r
dr
d
dr
dPdr
d
r
Gmgg
r
Gm
dr
dPn
KKP
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:
|)('|33
4
|)('|44
1
13
12
132
0
M
R
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
RMn
RMn
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)
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Neutron stars
Superdense matter and superstrong magnetic fields
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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≠r0
0
r0
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Gravitational redshift
)( e
0
e ,e
r
dt
dNr
dt
dN
d
dNdtd
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
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Outside of the star
RrRR
MMMM
g
b
/1/
2.0~ sun
r
r
drdrr
rdtc
r
rds
c
GMr
r
r
rc
GM
MrmRr
gr
gg
gg
1
1 1
2 ,1
21e
(1) и (3) изconst )( При
222
1
222
222
Bounding energy
Apparent radius
redshift
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ikikik Tc
GRgR
4
8
2
1
)( )4(
1 1
(3)
4 )2(
21
41 1 (1)
1
22
2
1
22
3
22
PP
c
P
dr
dP
cdr
d
rdr
dm
rc
Gm
mc
Pr
c
P
r
mG
dr
dP
{ Tolman (1939)Oppenheimer-Volkoff (1939)
TOV equation
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Structure and layers
Plus an envelope and an atmosphere...
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Neutron star interiors
Radius: 10 kmMass: 1-2 solarDensity: above the nuclearStrong magnetic fields
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Configurations
Stable 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.
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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 radius
Spin-down can result in phase transition.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 Pa
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Experiments and soft EoS
(arViv: 0708.2810)
Sagert et al. claimthat at the momentexperiments, whichfavour soft EoSdo not contradictdirectly observationsas even for K<200 MeVit is possible to haveMmax > 2 Msolar
K-compressibility. It is smaller for softer EoS.
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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)
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Glitches
Unpinning of superfluid vortex lines results in a glitch.
Vortex density is about 104 cm-2 P-1
Flux lines density is 5 1018 B12 cm-2
Starquakes or vortex lines unpinning.
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NS interiors: resume
(Weber et al. ArXiv: 0705.2708)
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NS Masses Stellar masses are directly measured only in
binary systems Accurate NS mass determination for PSRs in
relativistic systems by measuring PK corrections
Gravitational redshift may provide M/R in NSs by detecting a known spectral line,
E∞ = E(1-2GM/Rc2)1/2
Fe and O lines in EXO 0748-676, M/R ~ 0.22 (Cottam et al 2002)
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White
dw
arfs
Neu
tro
n s
tars
Brown dwarfs,Giant planets
Maximum-massneutronstar
Minimum-massneutron star
Maximum-masswhite dwarf
km 250~
1.0~ Sun
R
MM
km 129~
)5.25.1(~ Sun
R
MM
c
Neutron stars and white dwarfs
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Minimal mass
In reality, minimal mass is determined by properties of protoNSs.Being hot, lepton rich they have much higher limit: about 0.7 solar mass.
Stellar evolution does not produce NSs with barion mass less thanabout 1.4 solar mass.
Fragmentation of a core due to rapid rotation potentially can lead to smallermasses, but not as small as the limit for cold NSs
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Page & Reddy (2006)
BHs ?
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Compact objects and progenitors.Solar metallicity.
(Woosley et al. 2002)
There can be a range of progenitormasses in which NSs are formed,however, for smaller and larger progenitors masses BHs appear.
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Mass spectrum of compact objects
(Timmes et al. 1996, astro-ph/9510136)
Results of calculations(depend on the assumed modelof explosion)
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Mass spectrum of compact objects
(Timmes et al. 1996, astro-ph/9510136)
Comparison of one ofthe model with observations.
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A NS from a massive progenitor
(astro-ph/0611589)
Anomalous X-ray pulsar in the associationWesterlund1 most probably has a very massive progenitor, >40 MO.
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The case of zero metallicity
(Woosley et al. 2002)
No intermediate mass rangefor NS formation.
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NS+NS binaries
Pulsar Pulsar mass Companion mass
B1913+16 1.44 1.39B2127+11C 1.35 1.36B1534+12 1.33 1.35J0737-3039 1.34 1.25J1756-2251 1.40 1.18
(PSR+companion)/2
J1518+4904 1.35J1811-1736 1.30J1829+2456 1.25
Secondary companion in double NS binaries can give a good estimateof the initial mass (at least, in this evolutionary channel).
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Binary pulsars
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Relativistic corrections and measurable parameters
For details seeTaylor, Weisberg 1989ApJ 345, 434
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Shapiro delay
PSR 1855+09 (Taylor, Nobel lecture)
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Mass measurementsPSR 1913+16
(Taylor)
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Double pulsar J0737-3039
(Lyne et al. astro-ph/0401086)
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Masses for PSR J0737-3039
(Kramer et al. astro-ph/0609417)
The most precise values.
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Neutron stars in binaries
Study of close binary systems gives an opportunity to obtain mass estimate forprogenitors of NSs (see for example, Ergma, van den Heuvel 1998 A&A 331, L29).For example, an interesting estimate was obtained for GX 301-2. The progenitor mass is >50 solar masses.On the other hand, for several other systems with both NSs and BHsprogenitor masses a smaller: from 20 up to 50. Finally, for the BH binary LMC X-3 the progenitor mass is estimated as >60 solar.So, the situation is tricky.Most probably, in some range of masses, at least in binary systems, stars canproduce both types of compact objects: NSs and BHs.
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Mass determination in binaries:mass function
mx, mv - masses of a compact object and of a normal star (in solar units), Kv – observed semi-amplitude of line of sight velocity of the normal star (in km/s),P – orbital period (in days), e – orbital eccentricity, i – orbital inclination (the angle between the prbital plane and line of sight).
One can see that the mass function is the lower limit for the mass of a compact star.
The mass of a compact object can be calculated as:
So, to derive the mass it is necessary to know (besides the line of sight velocity)independently two more parameters: mass ration q=mx/mv, and orbital inclination i.
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Recent mass estimates
ArXiv: 0707.2802
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Mass-radius diagram and constraints
(astro-ph/0608345, 0608360)
Unfortunately, there are nogood data on independentmeasurements of massesand radii of NSs.
Still, it is possible to putimportant constraints.Most of recent observationsfavour stiff EoS.
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Combination of different methods
(Ozel astro-ph/0605106)
EXO 0748-676
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Limits on the EoS from EXO 0748-676
(Ozel astro-ph/0605106)
Stiff EoS are better.Many EoS for strangematter are rejected.But no all! (see discussionin Nature).
X- hydrogene fractionin the accreted material
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Limits from RX J1856
(Trumper)
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PSR 0751+1807
(Trumper)
Massive NS: 2.1+/-0.3 solar masses
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Limits on the moment of inertia Spin-orbital interaction
PSR J0737-3039(see Lattimer, Schutzastro-ph/0411470)
The band refers to ahypothetical 10% error.This limit, hopefully,can be reached in several years of observ.
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Most rapidly rotating PSR716-Hz eclipsing binary radio pulsar in the globular cluster Terzan 5
(Jason W.T. Hessels et al. astro-ph/0601337)
Previous record (642-Hz pulsar B1937+21)survived for more than 20 years.
Interesting calculationsfor rotating NS have beenperformed recently by Krastev et al.arXiv: 0709.3621
Rotation starts to be important from periods ~3 msec.
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QPO and rapid rotationXTE J1739-285 1122 HzP. Kaaret et al.astro-ph/0611716
(Miller astro-ph/0312449)
1330 Hz – one of thehighest QPO frequency
The line corresponds tothe interpretation, thatthe frequency is thatof the last stable orbit,6GM/c2
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Rotation and composition
(Weber et al. arXiv: 0705.2708)
Computed for a particular model:density dependent relativistic Brueckner-Hartree-Fock (DD-RBHF)
(equatorial) (polar)
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Rotation and composition
(Weber et al. arXiv: 0705.2708) 1.4 solar mass NS (when non-rotating)
hyperonhyperon quark-hybridquark-hybrid quark-hybridquark-hybrid(quarks in CFL)(quarks in CFL)