Lecture1. Structure of Neutron Stars

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Lecture1.Structure of Neutron Stars

• Sergei Popov (SAI MSU)

I want ot thank D. Yakovlev and R. Turollafor a
kind permission to use their slides in these
lectures.
Phenomenology of NSs. Trento. August 2007
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Artistic view
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Hydrostatic equilibrium for a star
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Lane-Emden equation. Polytrops.
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Properties of polytropic stars
Analytic solutions
?5/3
?4/3
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
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Useful equations

• White dwarfs
• Non-relativistic electrons
• ?5/3, K(32/3 p4/3 /5) (?2/memu5/3µe5/3)
• µe-mean molecular weight per one electron
• K1.0036 1013 µe-5/3 (CGS)
• 2. Relativistic electrons
• ?4/3, K(31/3 p2/3 /4) (?c/mu4/3µe4/3)
• K1.2435 1015 µe-4/3 (CGS)

• Neutron stars
• Non-relativistic neutrons
• ?5/3, K(32/3 p4/3 /5) (?2/mn8/3)
• K5.3802 109 (CGS)
• 2. Relativistic neutrons
• ?4/3, K(31/3 p2/3 /4) (?c/mn4/3)
• K1.2293 1015 (CGS)

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Neutron stars
Radius 10 km Mass 1-2 solar Density above the
nuclear Strong magnetic fields
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Neutron stars - 2
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

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Equator and radius
ds2c2dt2e2F-e2?dr2-r2d?2sin2?df2
In flat space F(r) and ?(r) are equal to zero.

• tconst, r const, ?p/2, 0ltFlt2p

l2pr

• tconst, ?const, fconst, 0ltrltr0

dle?dr
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Gravitational redshift
Frequency emitted at r Frequency detected byan
observer at infinity This function
determinesgravitational redshift
It is useful to use m(r) gravitational mass
inside r instead of ?(r)
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TOV equation
Tolman (1939) Oppenheimer- Volkoff (1939)
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Outside of the star
redshift
Bounding energy
Apparent radius
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Structure and layers
Plus an envelope and an atmosphere...
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Configurations
Stable CS configurations for neutron stars and
hybrid stars.
(astro-ph/0611595)
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Mass-radius relation

• Main 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.

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Lattimer Prakash (2004)
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EoS
(Weber et al. ArXiv 0705.2708 )
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Experimental results and comparison
(Danielewicz et al. nucl-th/0208016)
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Experiments and soft EoS
Sagert et al. Claimthat at the
momentexperiments, whichfavour soft EoSdo not
contradictdirectly observationsas even for
Klt200 MeVit is possible to haveMmax gt 2 Msolar
(arViv 0708.2810)
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Phase diagram
Phase diagram for isospin symmetry using the
most favoravble hybrid EoS studied in
astro-ph/0611595.
(astro-ph/0611595)
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Mass-radius
Mass-radius relations for CSs with possible phase
transition to deconfined quark matter.
(astro-ph/0611595)
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Superfluidity in NSs
(Yakovlev)
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Glitches
Starquakes or vortex lines unpinning.
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
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NS Masses

• Stellar masses 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,
• E8 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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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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(No Transcript)
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Compact objects and progenitors.Solar
metallicity.
There can be a range of progenitormasses in
which NSs are formed,however, for smaller and
larger progenitors masses BHs appear.
(Woosley et al. 2002)
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Mass spectrum of compact objects
Results of calculations
(Timmes et al. 1996, astro-ph/9510136)
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Mass spectrum of compact objects
Comparison of one ofthe model with observations.
(Timmes et al. 1996, astro-ph/9510136)
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A NS from a massive progenitor
Anomalous X-ray pulsar in the associationWesterlu
nd1 most probably has a very massive progenitor,
gt40 MO.
(astro-ph/0611589)
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The case of zero metallicity
No intermediate mass rangefor NS formation.
(Woosley et al. 2002)
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NSNS binaries
Secondary companion in double NS binaries can
give a good estimateof the initial mass (at
least, in this evolutionary channel).
Pulsar Pulsar mass
Companion mass B191316 1.44
1.39 B212711C
1.35 1.36 B153412
1.33
1.35 J0737-3039 1.34
1.25 J1756-2251 1.40
1.18
(PSRcompanion)/2 J15184904
1.35 J1811-1736
1.30 J18292456
1.25
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Binary pulsars
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Relativistic corrections and measurable
parameters
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Shapiro delay
PSR 185509 (Taylor, Nobel lecture)
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Mass measurements
PSR 191316
(Taylor)
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Double pulsar J0737-3039
(Lyne et al. astro-ph/0401086)
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Masses for PSR J0737-3039
The most precise values.
(Kramer et al. astro-ph/0609417)
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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 AA 331, L29). For example,
an interesting estimate was obtained for GX
301-2. The progenitor mass is gt50 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 gt60 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 binariesmass 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 qmx/mv, and
orbital inclination i.
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Recent mass estimates
ArXiv 0707.2802
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Mass-radius diagram and constraints
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.
(astro-ph/0608345, 0608360)
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Combination of different methods
EXO 0748-676
(Ozel astro-ph/0605106)
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Limits on the EoS from EXO 0748-676
Stiff EoS are better. Many EoS for strangematter
are rejected.But no all! (see discussionin
Nature).
(Ozel astro-ph/0605106)
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Limits from RX J1856
(Trumper)
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PSR 07511807
Massive NS 2.1/-0.3 solar masses
(Trumper)
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Limits on the moment of inertia
PSR J0737-3039 (see Lattimer, Schutzastro-ph/0411
470)
(Trumper)
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Most rapidly rotating PSR
716-Hz eclipsing binary radio pulsar in the
globular cluster Terzan 5
Previous record (642-Hz pulsar
B193721)survived for more than 20 years.
(Jason W.T. Hessels et al. astro-ph/0601337)
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QPO and rapid rotation
XTE J1739-285 1122 Hz P. Kaaret et
al.astro-ph/0611716
1330 Hz one of thehighest QPO frequency
(Miller astro-ph/0312449)