Equation Of State and back bending phenomenon in rotating neutron stars

1st Astro-PF Workshop – CAMK, 14 October 2004Compact Stars: structure, dynamics, and gravitational waves

M. BejgerE. GourgoulhonP. HaenselL. Zdunik

Plan

1. Historical remarksGlendenning 1997 Spyrou, Stergioulas 2002

2. Back bending phenomenon for Neutron Stars (with hyperons)new approach MB – Req dependence at fixed frequency

J(f) at fixed baryon mass3. Polytropic EOS and back bending

Phase transition to quark phase through mixed phase4. The role of instability

• rapidly rotating pulsar spins down (looses angular momentum J)• central density increases with time• the density of the transition to the mixed quark-hadron phase is reached• the radius of the star and the moment of inertia significantly decreases • the increase of central density of the star is more important than decrease of angular momentum J=I Ω dJ=I dΩ+Ω dI dΩ = dJ /I - Ω dI /I > 0• the epoch of SPIN-UP BY ANGULAR MOMENTUM LOSS

Spin up

Braking index -singularity

Energy loss equation

Consequences of back-bending

• the braking index has very large value• the isolated pulsar may be observed to be spinning up

Signature of the transition to the mixed phase with quarks

Re-invistigate the deconfinement phase-transition of spinning-down PSR

• fully relativistic, rapidly rotating models (vs. Slow-rotation approximation)• analytic expression for quark phase (vs interpolation of tabulated EOS)• high accuracy of the code and EOS – extremely important

For normal pulsar the quark core appears without back-bending behaviour

Braking index – no singular behaviour

Back-bending for NS with hyperons

• 2-D multidomain LORENE code based on spectral methods• softening of the EOS due to the appereance of hypeons

MB – Req at fixed frequency

Analysis of the BB problem in the baryon mass MB – equatorial radius Req plane:• MB is constant during the evolution of solitary pulsars• at fixed frequency – the frequency are directly connected to the back-bending definition• numerical reasons: frequency is basic input parameter in the numerical calculations of rotating star (with central density ρc)• no need to calculate the evolution of the star with fixed MB

• numerical procedure: input - (f, ρc) , output – (M,MB ,J,R)

Discussion based on MB (ρc) f=const or MB (R) f=const

Softening of the EOS due to the core with hyperons

Signature of BB – minimum of MB at fixed frequency

Back-bending and MB (x) f=const

x=Req x= ρc x= Pc

The softening of the EOS due to the hyperonization leads to the flattening of the MB (x) f=const curves.

Back bending - between two frequencies defined by the existence of the point x of vanishing first and second derivative(point of inflexion).

002

2

dx

Md

dx

dM BB

This condition does not depend on the choice of x.

The onset of back-bending

02

2

constfc

B

rot

M

02

2

constfeq

B

rotR

M

0

constfc

B

rot

M

0

constfeq

B

rot

R

M

Interesting points

• minimum frequency for BB• maximum frequency for deceleration after BB • acceleration from Keplerian configuration

Importance of angular momentum

Why to use angular momentum J instead of moment of inertia I ?

• J is well defined quantity in GR describing the instantaneous state of rotating star• the evolution of rotating star can be easily calculated under some assumptions about the change of J

12/ ndt

dTJT magnetic braking n=3

GW emission n=5

• the moment of inertia defined as J/Ω does not describe the response of the star to the change of J or Ω (rather dJ/dΩ)• J enters the stability condition of rotating stars with respect to axially symmetric perturbations

Instability

0

constMcB

J

0

constJc

BM

0

constJc

M

ANGULAR MOMENTUM

vs

MOMENT OF INERTIA

Angular momentum vs rotational frequency

Angular momentum vs rotational frequency

Importance of the accuracy

The innermost zone boundary not adjusted to the surface of hyperonthreshold except for f~920 Hz

2 domains inthe interior ofthe star

The boundary have to be adjusted to the point of the discontinuity of properties of EOS

Conclusions for NS with hyperons

• the presence of hyperons neutron-star cores can strongly affect the spin evolution of solitary NS (isolated pulsar)• epochs with back-bending for normal rotating NS were found for two of four EOS • for these models pulsar looses half of its initial angular momentum without changing much its rotation period

Mixed Phase – analytical EOS

<1 nuclear matter - polytrope

1< <2 mixed phase – polytrope

>2 quark matter – linear EOS

nnP

nnnP 01

)()()(

4/300 )1()(

3

1)(

P

nnP

Mixed Phase – analytical EOS

Mixed Phase – analytical EOS

Mixed Stable

Mixed Stable

Mixed Unstable

Mixed Unstable

Mixed Marginally Stable

Mixed Marginally Stable

Rotation and stability

If nonrotating stars are stable (ie. softening of the EOS does not result in unstable branch) then for any value of total angular momentum J (fixed) MB increases.

If MB (x) J=0 has local maximum and minimum (unstable region) than for any value of total angular momentum J (fixed) such region exists.

In most cases rotation neither stabilizes nor destabilizes configurations with phase transitions.

Onset of instability – test of the code

• Test of the code (GR effects)• Test of the thermodynamic consistency of the equation of state

dn

ndn

dn

dP )/(2

• Total angular momentum J• Gravitational mass M• Baryon mass MB

The extrema of two of these quantities at third fixed at the same point

Cusps in Figures fixedJBfixedMBfixedM MMMJMJB )(,)(,)(

dNdJdM

First law of thermodynamics

Mixed unstable M(J)

MB =const

Mixed unstable M(MB)

J =const