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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

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astronomy meets qcd
Astronomy meets QCD

arXiv: 0808.1279

hydrostatic equilibrium for a star

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
properties of polytropic stars
Properties of polytropic stars

Analytic solutions:



useful equations
Useful equations
  • White dwarfs
  • Non-relativistic electrons
  • γ=5/3, K=(32/3π4/3 /5) (ћ2/memu5/3μe5/3);
  • μe-mean molecular weight per one electron
  • K=1.0036 1013μe-5/3 (CGS)
  • 2. Relativistic electrons
  • γ=4/3, K=(31/3π2/3 /4) (ћc/mu4/3μe4/3);
  • K=1.2435 1015μe-4/3 (CGS)
  • Neutron stars
  • Non-relativistic neutrons
  • γ=5/3, K=(32/3π4/3 /5) (ћ2/mn8/3);
  • K=5.3802 109(CGS)
  • 2. Relativistic neutrons
  • γ=4/3, K=(31/3π2/3 /4) (ћc/mn4/3);
  • K=1.2293 1015 (CGS)

[Shapiro, Teukolsky]

neutron stars
Neutron stars

Superdense matter and superstrong magnetic fields

astrophysical point of view

The first four are related to the NS structure!

Astrophysical point of view
  • Astrophysical appearence of NSsis mainly determined by:
  • Spin
  • Magnetic field
  • Temperature
  • Velocity
  • Environment
equator and radius




Equator and radius


In flat space Φ(r) and λ(r) are equal to zero.

  • t=const, r= const, θ=π/2, 0<Φ<2π


  • t=const, θ=const, φ=const, 0<r<r0


gravitational redshift
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 λ(r)

outside of the star
Outside of the star


Bounding energy

Apparent radius

bounding energy
Bounding energy

If you drop a kilo on a NS, thenyou increase its mass for < kilo


tov equation
TOV equation

Tolman (1939)


Volkoff (1939)

structure and layers
Structure and layers

Plus an atmosphere...

neutron star interiors
Neutron star interiors

Radius: 10 km

Mass: 1-2 solar

Density: above the nuclear

Strong magnetic fields


NS mass vs.central density(Weber et al. arXiv: 0705.2708)

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)

mass radius

Mass-radius relations for CSs

with possible phase transition

to deconfined quark matter.

About hyperon stars see a recent review in 1002.1658.

About strange stars and some otherexotic options – 1002.1793


mass radius relation

Haensel, Zdunik


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.






Lattimer & Prakash (2004)


(Weber et al. ArXiv: 0705.2708 )

experimental results and comparison
Experimental results and comparison

1 Mev/fm3 = 1.6 1032 Pa

GSI-SIS and AGS data

(Danielewicz et al. nucl-th/0208016)

phase diagram25
Phase diagram

Phase diagram for isospin

symmetry using the most

favorable hybrid EoS studied

in astro-ph/0611595.


particle fractions
Particle fractions

Effective chiral model of

Hanauske et al. (2000)

Relativistic mean-field model

TM1 of Sugahara & Toki (1971)

superfluidity in nss
Superfluidity in NSs


Яковлев и др. УФН 1999

ns interiors resume
NS interiors: resume

(Weber et al. ArXiv: 0705.2708)

papers to read
Papers to read

1. astro-ph/0405262 Lattimer, Prakash "Physics of neutron stars"

2. 0705.2708 Weber et al. "Neutron stars interiors and

equation of state of superdense matter"

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 Nuclear Lab in the Universe ”

7. 1001.3294 Schmitt “Dense matter in compact stars - A pedagogical introduction ”

8. The book by Haensel, Yakovlev, Potekhin

lectures on the web
Lectures on the Web

Lectures can be found at my homepage: