Burning and convective processes in compact objects

1. Burning and convective processes in compact objects Irene Parenti Department of Physics and INFN of Ferrara “XI Convegno sui problemi di fisica nucleare teorica”Cortona, 11-14 Ottobre 2006

2. Why to study this? Process of conversion of a neutron star into a quark or a hybrid star. Conversion time? Velocity and mode of conversion? Important for: • Supernovae explosion • Gamma Ray Burst • Kick NS

3. Outline Combustion theory Mode of combustion: detonation or deflagration? Hydrodynamical instabilities Can Convection develope? Astrophysical implications

4. Combustion theory We consider a front of transition from nuclear matter to quark matter. In the front frame: P2, e2, ρB2, w2=p2+e2 P1, e1, ρB1, w1=p1+e1 From the conservation of momentum- energy tensor and of the baryonic flux through the discontinuity surface, we have: flux Txx T0x

5. P v1>c1 v2<c2 detonation v1>c1 v2>c2 O fast detonation A v1<c1 v2<c2 A’ slow combustion v1<c1 v2>c2 1 unstable O’ X Combustion theory We define proper volume: Baryonic flux Detonation adiabatic:

6. Thermodynamics of relativistic system Corrections to the thermodynamics quantities in relativistic moving systems: [Tolman, R. “Relativity Thermodinamics and Cosmology” (1934)] Is the reaction esothermic? In the hadronic matter rest frame we can compare the energy for baryon of the two phases considering the corrections due to the relativistic effects.

7. internal energy variation of the system work done by the system Temperature • The hadron temperature is always TH=0. • Instead the quark temperature can be TQ≠0. • To evaluate TQ we considerthat all the relaised energy goes into heat (and than in temperature) except a small fraction that goes into kinetic energy. • First thermodynamics principle: • We can rewrite it in this form:

8. Equations of state Hadronic phase:Relativistic mean field theory of hadrons interacting via meson exch. [e.g. Glendenning, Moszkowsky, PRL 67(1991)] Quark phase 1: EoS based on the MIT bag model for hadrons. [Farhi, Jaffe, Phys. Rev. D46(1992)] Quark phase 2: Simple model of a CFL phase. [Alford, Reddy, Phys. Rev. D67(2003)] Mixed phase: Gibbs construction for a multicomponent system with two conserved “charges”. [Glendenning, Phys. Rev. D46 (1992)]

9. Betastability: yes or not? Implicit hypothesis: quark matter after deconfinament is in equilibrium. What happens if there is not time for β-processes? flavour conservation The EoS of quark phase is defined tohave the same quark’s fraction of the pure hadronic matter:

10. Detonation or not detonation? beta not beta

11. Combustion with hyperons β-stable phase The vertical line corrisponds to the central density of the most massive star.

12. With Temperature When temperature of the quark phase is taken into account B1/4=170 MeV not β-stable mixed phase Temperatures from 5 to 40 MeV.

13. B¼=155 MeV B¼=155 MeV B¼=165 MeV B¼=155 MeV CFL-phase Conversion from a phase of Normal Quark (NQ) to a phase of CFL. The two phases are both β-stable. We show only the results for B¼=155 MeV but changing B the behaviour is the same.

14. Hydrodynamical instabilities We always are in the case of unstable front. This means that the front doesn’t remain as a geometrical surface but hydrodynamical instabilities develop and wrinkles form. The dominant hydrodynamical instability is the Rayleigh-Taylor. The increase of the conversion velocity can be estimated using a fractal scheme: D is the fractal dimension where and D0=0.6 Typical values for  are 0.4 or smaller (for not β-stable) and 0.7 or smaller (for β-stable quark matter). The conversion velocity can increase by up to 2 orders of magnitude respect to vsc, but in general the process remains a deflagration. [Blinnikov, S. Iv. And Sasorov, P. V. Phys. Rev. E 53, 4827 (1995)]

15. Convection theory Mixing length theory The convective element travels, on the average, through a distance Λ, the Mixing Length. The characteristic dimension of this element is assumed to be equal to Λ. Quasi-ledoux convection Blob of fluid moves in pressure equilibrium and without heat transfer. The condition for a blob to became unstable is: This defines the dimension of the convective layer. It is possible to estimate the velocity of the blob from the relation between kinetic energy and the work done by the buoyancy forces. where:

16. Convection? Hadronic phase 2 ρQ < ρH PQ = PH ρQ ? ρH PQ = PH 1 ρQ > ρH PQ = PH ρQ < ρH PQ < PH Quark phase

17. Convection: results Cg C0 H LgH155 v = 18,5 Km/msec B0

18. Convection with hyperons v = 45,4 Km/msec LβHy155 LgHy155

19. Possible scenario • It is possible to have two transitions: • from hadronic matter to normal quark matter • (a subsonic process) • from normal quark matter to a quark condensate • (always a convective process, subsonic but • very rapid) Possible explanation of double bursts in GRBs (see talk of Pagliara)

20. Conclusions • The combustion is never a detonation • It’s always a subsonic process with an unstable front • Hydrodynamical instabilities develop • It is possible to have convection: - if hyperons are taken into account - in the transition to a quark condensate (B indipendent result)

21. Collaborators A. Drago, A. Lavagno and I. P. astro-ph/0512652 Alessandro Drago Physics Department and INFN of Ferrara Andrea Lavagno Politecnico of Torino Others:Ignazio Bombaci(Pisa) Isaac Vidaña(Barcelona) Giuseppe Pagliara(Ferrara)

22. Appendix

23. Combustion theory Relativistic case momentum-energy tensor In the combustion front system and in the unidimensional case: Quadrivelocity Momentum-energy tensor

24. Surface tension • We work in a model with surface tension ≠0. But what is its value? • σ» 30 MeV/fm2 • It is not possible to form • structure of finite • dimensiones. • Maxwell construction • (there is not mixed phase). • - σ« 30 MeV/fm2 • (very small value but not • vanishing). Gibbs construction. • σ < 30 MeV/fm2 • mixed phase shift respect to that obtained by Gibbs construction • (structures form to minimize the energy).

25. Thermal nucleation 1 In order to understand when a fluidodynamical description of the formation of mixed phase (MP) is realistic we have to estimate the dynamical time-scale of the formation of its structures. The thermal nucleation rate: Wc is the maximum of the free energy of the bubble of the new phase: Then the number of bubbles of new phase formed inside a volume V and in a time t is given by: If is the spacing between two drops in the MP and is the number of drops in a volume V than a fluidodynamical description of the formation of MP is realistic if the number of bubbles produced while the front moves over a distance is of the order of the number of bubbles that have to be present in the MP.|

26. Thermal nucleation 2 Then: Therefore the following constraint have to be satisfied:

27. Fluidodynamical description • The fluidodynamical description of the transition is allowed only for densities: • -σ» 30 MeV/fm2ρHyd > ρeq • - σ« 30 MeV/fm2ρHyd > ρ1G • σ < 30 MeV/fm2ρHyd > ρ • ρeq is the density for which if ρHyd > ρeq is energetically convenient to transform completely hadrons into quarks although the energy of the system can be further reduced forming mixed phase. ¯

28. Rayleigh-Taylor instability