1. Equation Of State and back bending phenomenon in rotating neutron stars 1 st Astro-PF Workshop – CAMK, 14 October 2004 Compact Stars: structure, dynamics, and gravitational waves M. Bejger E. Gourgoulhon P. Haensel L. Zdunik

2. Plan 1.Historical remarks Glendenning 1997 Spyrou, Stergioulas 2002 2.Back bending phenomenon for Neutron Stars (with hyperons) new approach M B – R eq dependence at fixed frequency J(f) at fixed baryon mass 3.Polytropic EOS and back bending Phase transition to quark phase through mixed phase 4.The role of instability

3. 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

4. Spin up Braking index -singularity Energy loss equation

5. 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

6. 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

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

8. Braking index – no singular behaviour

9. 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

10. M B – R eq at fixed frequency Analysis of the BB problem in the baryon mass M B – equatorial radius R eq plane: M B 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 ρ c ) no need to calculate the evolution of the star with fixed M B numerical procedure: input - (f, ρ c ), output – (M,M B,J,R) Discussion based on M B (ρ c ) f=const or M B (R) f=const

11. Softening of the EOS due to the core with hyperons

12. Signature of BB – minimum of M B at fixed frequency

13. Back-bending and M B (x) f=const x=R eq x= ρ c x= P c The softening of the EOS due to the hyperonization leads to the flattening of the M B (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). This condition does not depend on the choice of x.

14. The onset of back-bending

15. Interesting points minimum frequency for BB maximum frequency for deceleration after BB acceleration from Keplerian configuration

16. 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 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

17. Instability

18. ANGULAR MOMENTUM vs MOMENT OF INERTIA

19. Angular momentum vs rotational frequency

21. Importance of the accuracy The innermost zone boundary not adjusted to the surface of hyperon threshold except for f~920 Hz 2 domains in the interior of the star The boundary have to be adjusted to the point of the discontinuity of properties of EOS

22. 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

23. Mixed Phase – analytical EOS  <  1 nuclear matter - polytrope  1 <  <  2 mixed phase – polytrope  >  2 quark matter – linear EOS

24. Mixed Phase – analytical EOS

26. Mixed Stable

28. Mixed Unstable

30. Mixed Marginally Stable

32. 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) M B increases. If M B (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.

33. Onset of instability – test of the code Test of the code (GR effects) Test of the thermodynamic consistency of the equation of state Total angular momentum J Gravitational mass M Baryon mass M B The extrema of two of these quantities at third fixed at the same point Cusps in Figures First law of thermodynamics

34. Mixed unstable M(J) M B =const

35. Mixed unstable M(M B ) J =const