Burning and convective processes in compact objects

Cortona, 12 Ottobre Cortona, 12 Ottobre 20062006

11/21 Irene Parenti/21 Irene Parenti

Burning and convective Burning and convective processes in compact objectsprocesses in compact objects

““XI Convegno sui problemi di fisica nucleare XI Convegno sui problemi di fisica nucleare

teorica”teorica” Cortona, 11-14 Ottobre 2006Cortona, 11-14 Ottobre 2006

Irene ParentiIrene Parenti

Department of Physics and INFN of FerraraDepartment of Physics and INFN of Ferrara



Process of conversion of a neutron star into a quarkProcess of conversion of a neutron star into a quarkor a hybrid star.or a hybrid star.

Conversion time?Conversion time?

Important for:Important for:

●Supernovae explosionSupernovae explosion●Gamma Ray BurstGamma Ray Burst●Kick NSKick NS

Why to study this?Why to study this?

Velocity and mode of conversion?Velocity and mode of conversion?



OutlineOutline

Can Convection develope?Can Convection develope?

Astrophysical implicationsAstrophysical implications

Combustion theoryCombustion theory

Mode of combustion: detonation or deflagration?Mode of combustion: detonation or deflagration?

Hydrodynamical instabilitiesHydrodynamical instabilities




PP22, e, e22, , ρρB2B2,,ww22=p=p22+e+e22

PP11, e, e11, , ρρB1B1,,ww11=p=p11+e+e11

We consider a front of transitionWe consider a front of transitionfrom nuclear matter to quark from nuclear matter to quark matter. In the front frame:matter. In the front frame:

fluxflux TTxxxx

TT0x0x222111

2222

2111

2211

uwuw

uwpuwp

juu BB

From the conservation of momentum-From the conservation of momentum-energy tensor and of the baryonic energy tensor and of the baryonic flux through the discontinuity flux through the discontinuity surface, we have:surface, we have:

))((

))((

))((

))((

))((

))((

1121

122221

22

1212

211222

2112

121221

pepe

pepe

peee

peppv

peee

peppv

BB



PP

AA

A’A’

O’O’

XX

OO

detonationdetonation

slow combustionslow combustion

fast detonationfast detonation

unstableunstable

vv11>c>c11

vv22<c<c22

vv11<c<c11

vv22<c<c22

vv11>c>c11

vv22>c>c22

vv11<c<c11

vv22>c>c22

11


))(( 12211122 ppXXwXwX

)(

)(

12

122

XX

ppj

Baryonic fluxBaryonic flux

Detonation Detonation adiabatic:adiabatic:

2B

wX

We define We define proper volumeproper volume::



Thermodynamics of relativistic systemThermodynamics of relativistic system

Corrections to the thermodynamics quantities in Corrections to the thermodynamics quantities in relativistic moving systems:relativistic moving systems:

[Tolman, R. [Tolman, R. “Relativity “Relativity Thermodinamics Thermodinamics and Cosmology” and Cosmology” (1934)](1934)]

0

2

2

000

0

VV

c

vVpEE

pp

In the hadronic matter rest frame we can compare the In the hadronic matter rest frame we can compare the energy for baryon of the two phases considering the energy for baryon of the two phases considering the corrections due to the relativistic effects. corrections due to the relativistic effects.

Is the reaction esothermic?Is the reaction esothermic?

0

2200 /

B

cvpe

A

E



TemperatureTemperature• The hadron temperature is always TThe hadron temperature is always THH=0=0. . • Instead the quark temperature can be TInstead the quark temperature can be TQQ≠0≠0.. • To evaluate TTo evaluate TQQ we consider we consider that all the relaised energy that all the relaised energy goes into heat (and than in temperature) except a small goes into heat (and than in temperature) except a small fraction that goes into kinetic energyfraction that goes into kinetic energy..• First thermodynamics principle:First thermodynamics principle:

• We can rewrite it in this form:We can rewrite it in this form:

internal energy variationinternal energy variationof the system of the system

work done by the systemwork done by the system



Hadronic phaseHadronic phase:: Relativistic mean field theory of hadrons Relativistic mean field theory of hadrons interacting via meson exch. interacting via meson exch. [[e.g. Glendenning, Moszkowsky, PRL 67(1991)e.g. Glendenning, Moszkowsky, PRL 67(1991)]]

Quark phase 1Quark phase 1: : EoS based on the MIT bag model for hadronsEoS based on the MIT bag model for hadrons.. [[Farhi, Jaffe, Phys. Rev. D46(1992)]Farhi, Jaffe, Phys. Rev. D46(1992)]

Mixed phaseMixed phase: : Gibbs construction for a multicomponent system Gibbs construction for a multicomponent system with two conserved “charges”.with two conserved “charges”.[Glendenning, Phys. Rev. D46 (1992)][Glendenning, Phys. Rev. D46 (1992)]

Equations of stateEquations of state

Quark phase 2Quark phase 2: : Simple model of a CFL phase.Simple model of a CFL phase.[[Alford, Reddy, Phys. Rev. D67(2003)]Alford, Reddy, Phys. Rev. D67(2003)]



Betastability: yes or not?Betastability: yes or not?

Implicit hypothesisImplicit hypothesis: quark matter after deconfinament : quark matter after deconfinament is in equilibrium.is in equilibrium.

What happens if there is not time for What happens if there is not time for ββ-processes?-processes?

flavour conservationflavour conservation

The EoS of quark phase is The EoS of quark phase is defined todefined to have the same have the same quark’s fraction of the quark’s fraction of the pure hadronic matter:pure hadronic matter:



Detonation or not detonation?Detonation or not detonation?

betabeta not betanot beta



Combustion with hyperonsCombustion with hyperons

ββ-stable phase-stable phase

The vertical line The vertical line corrisponds to the corrisponds to the central density of central density of the most massive the most massive

star.star.



With TemperatureWith Temperature

When temperature of When temperature of the quark phase is the quark phase is taken into accounttaken into account

BB1/41/4=170 MeV=170 MeV

not not ββ-stable mixed -stable mixed phasephase

Temperatures from 5 Temperatures from 5 to 40 MeVto 40 MeV..



CFL-phaseCFL-phase

Conversion Conversion fromfrom a phase a phase of Normal Quark (of Normal Quark (NQNQ) to a ) to a phase of CFL.phase of CFL.

The two phases are both The two phases are both ββ-stable.-stable.

We show only the results We show only the results for Bfor B¼¼=155 MeV but =155 MeV but changing B the behaviour changing B the behaviour is the same.is the same.

BB¼¼ =155 MeV

=155 MeV

BB¼¼ =165 MeV

=165 MeV

BB¼¼=155 MeV=155 MeV

BB¼¼=155 MeV=155 MeV



Hydrodynamical instabilitiesHydrodynamical instabilitiesWe always are in the case of unstable front. We always are in the case of unstable front. This means that the front doesn’t remain as a This means that the front doesn’t remain as a geometrical surface but hydrodynamical instabilities geometrical surface but hydrodynamical instabilities develop and wrinkles form.develop and wrinkles form.The dominant hydrodynamical instability is the The dominant hydrodynamical instability is the Rayleigh-Taylor.Rayleigh-Taylor.The increase of the conversion velocity can be The increase of the conversion velocity can be estimated using a fractal scheme:estimated using a fractal scheme:

D is the fractal dimensionD is the fractal dimension

where and Dwhere and D00=0.6=0.6

Typical values for Typical values for are 0.4 or smaller (for not are 0.4 or smaller (for not ββ-stable) -stable) and 0.7 or smaller (for and 0.7 or smaller (for ββ-stable quark matter).-stable quark matter).The conversion velocity can increase by up to 2 orders The conversion velocity can increase by up to 2 orders of magnitude respect to vof magnitude respect to vscsc, but in general the process , but in general the process remains a deflagration.remains a deflagration.

2

min

maxvv

D

sceff l

l 202 DD

12 /1 ee

[Blinnikov, S. Iv. And Sasorov, P. V. Phys. Rev. E 53, 4827 [Blinnikov, S. Iv. And Sasorov, P. V. Phys. Rev. E 53, 4827 (1995)](1995)]



Convection theoryConvection theory

Mixing length theoryMixing length theoryThe convective element travels, on the average, through The convective element travels, on the average, through a distance a distance ΛΛ, the , the Mixing LengthMixing Length. The characteristic . The characteristic dimension of this element is assumed to be equal to dimension of this element is assumed to be equal to ΛΛ..

Quasi-ledoux convectionQuasi-ledoux convectionBlob of fluid moves in pressure equilibrium and without Blob of fluid moves in pressure equilibrium and without heat transfer. The condition for a blob to became heat transfer. The condition for a blob to became unstable is:unstable is:

),,(),,( eDeDD YSPYSP PPD

This defines the dimension of the convective layer.This defines the dimension of the convective layer.

It is possible to estimate the velocity of It is possible to estimate the velocity of the blob from the relation between the blob from the relation between kinetic energy and the work done by the kinetic energy and the work done by the buoyancy forces.buoyancy forces.

where:where:

Cgv 2

2

1

dR

dPg

1



Convection?Convection?

Hadronic phaseHadronic phase

Quark phaseQuark phase

ρρQ Q < < ρρH H

PPQ Q < P< PHH

ρρQ Q ? ? ρρH H

PPQ Q = P= PHH

22ρρQ Q < < ρρH H

PPQ Q = P= PHH

11ρρQ Q > > ρρH H

PPQ Q = P= PHH



Convection: resultsConvection: results

CCgg

HHCC00

BB00

LLgHgH155155

v = 18,5 Km/msecv = 18,5 Km/msec



Convection with hyperonsConvection with hyperons

LLgHygHy155155

LLββHyHy155155

v = 45,4 Km/msecv = 45,4 Km/msec



Possible scenarioPossible scenario

It is possible to have two transitions:It is possible to have two transitions:

- from hadronic matter to normal quark matterfrom hadronic matter to normal quark matter (a subsonic process)(a subsonic process)

- from normal quark matter to a quark condensate from normal quark matter to a quark condensate (always a convective process, subsonic but (always a convective process, subsonic but very rapid)very rapid)

Possible explanation of double bursts in GRBsPossible explanation of double bursts in GRBs

(see talk of Pagliara)(see talk of Pagliara)



ConclusionsConclusions

The combustion is never a detonationThe combustion is never a detonation

It’s always a subsonic process with an unstable It’s always a subsonic process with an unstable frontfront

It is possible to have convectionIt is possible to have convection::- - if hyperons are taken into accountif hyperons are taken into account

- - in the transition to a quark condensate in the transition to a quark condensate (B indipendent result)(B indipendent result)

Hydrodynamical instabilities developHydrodynamical instabilities develop



CollaboratorsCollaborators

Alessandro DragoAlessandro Drago

Andrea LavagnoAndrea Lavagno

Physics DepartmentPhysics Departmentand INFN of Ferraraand INFN of Ferrara

Politecnico of TorinoPolitecnico of Torino

OthersOthers:: Ignazio BombaciIgnazio Bombaci (Pisa)(Pisa)

Isaac VidaIsaac Vidaññaa (Barcelona)

Giuseppe PagliaraGiuseppe Pagliara (Ferrara)

A. Drago, A. Lavagno and I. P. astro-ph/0512652A. Drago, A. Lavagno and I. P. astro-ph/0512652



AppendixAppendix




Relativistic caseRelativistic case

pguwuT momentum-energy tensormomentum-energy tensor

vv

vu

21

In the combustion front system and in the In the combustion front system and in the unidimensional case:unidimensional case:

QuadrivelocityQuadrivelocity

pwuT

uwT

xx

x

2

0 Momentum-energy tensorMomentum-energy tensor



We work in a model with surface tension ≠0. But what is its value?We work in a model with surface tension ≠0. But what is its value?

-σσ » 30 MeV/fm» 30 MeV/fm22 It is not possible to form It is not possible to form structure of finite structure of finite dimensiones. dimensiones. Maxwell construction Maxwell construction (there is not mixed phase).(there is not mixed phase).

- - σσ « 30 MeV/fm« 30 MeV/fm22 (very small value but not (very small value but not vanishing). Gibbs construction.vanishing). Gibbs construction.

-σσ < < 30 MeV/fm 30 MeV/fm22 mixed phase shift respect to that obtained by Gibbs construction mixed phase shift respect to that obtained by Gibbs construction (structures form to minimize the energy).(structures form to minimize the energy).

Surface tensionSurface tension



Thermal nucleation 1Thermal nucleation 1In order to understand when a fluidodynamical In order to understand when a fluidodynamical description of the formation of mixed phase (MP) is description of the formation of mixed phase (MP) is realistic we have to estimate the dynamical time-scale of realistic we have to estimate the dynamical time-scale of the formation of its structures.the formation of its structures.

The thermal nucleation rate:The thermal nucleation rate:

WWcc is the maximum of the free energy of the bubble of is the maximum of the free energy of the bubble of

the new phase:the new phase:

Then the number of bubbles of new phase formed Then the number of bubbles of new phase formed inside a volume V and in a time t is given by:inside a volume V and in a time t is given by:

If is the spacing between two drops in the MP and If is the spacing between two drops in the MP and is the number of drops in a volume V than a is the number of drops in a volume V than a fluidodynamical description of the formation of MP is fluidodynamical description of the formation of MP is realistic if the number of bubbles produced while the realistic if the number of bubbles produced while the front moves over a distance is of the order of the front moves over a distance is of the order of the number of bubbles that have to be present in the MP.number of bubbles that have to be present in the MP.||

)/exp(4 TWRa c

212212

3 4)()(3

4)( RPPRRW B

tVRaN

3/V



Thermal nucleation 2Thermal nucleation 2

Then: Then:

Therefore the Therefore the following constraint following constraint have to be satisfied:have to be satisfied:

23 SV

vSRatVRa

max

44

ln

T

W

vT

W cc



The fluidodynamical description of the transition is The fluidodynamical description of the transition is allowed only for densities:allowed only for densities:

-- σσ » 30 MeV/fm» 30 MeV/fm22 ρρHydHyd > > ρρeqeq

- - σσ « 30 MeV/fm« 30 MeV/fm22 ρρHydHyd > > ρρ11GG

- σσ < < 30 MeV/fm 30 MeV/fm22 ρρHydHyd > > ρρ

ρρeqeq is the density for which if is the density for which if ρρHydHyd > > ρρeqeq is energetically is energetically convenient to transform completely hadrons into convenient to transform completely hadrons into quarks although the energy of the system can be quarks although the energy of the system can be further reduced forming mixed phase.further reduced forming mixed phase.

Fluidodynamical descriptionFluidodynamical description

¯̄



Rayleigh-Taylor instabilityRayleigh-Taylor instability

eg

vel

l

scq

2

min

max

4

Km10

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Burning and convective processes in compact objects