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, 12 Ottobre 2006
Cortona, 11-14 Ottobre 2006
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Irene Parenti
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
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Outline
Combustion theory
Mode of combustion: detonation or deflagration?
Hydrodynamical instabilities
Can Convection develope?
Astrophysical implications
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Combustion theory
We consider a front of transition
from nuclear matter to quark
matter. In the front frame:
From the conservation of momentumenergy tensor and of the baryonic
flux through the discontinuity
surface, we have:
 B1u1   B 2u 2  j
p1  w u  p 2  w2u
2
1 1
flux
2
2
w1 1u1  w2 2u 2
Txx
T0x
P2, e2, ρB2,
w2=p2+e2
( p2  p1 )(e2  p1 )
v 
(e2  e1 )( e1  p2 )
2
1
v22 

Cortona, 12 Ottobre 2006
P1, e1, ρB1,
w1=p1+e1
2
B2
( p2  p1 )(e1  p2 )
(e2  e1 )( e2  p1 )

2
B1
(e2  p2 )(e2  p1 )
(e1  p2 )(e1  p1 )
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Irene Parenti
Combustion theory
We define
proper volume:
X
P
detonation
w
 B2
O
fast detonation
( p2  p1 )
j 
( X 2  X1)
Detonation
adiabatic:
v1>c1
v2>c2
A
Baryonic flux
2
v1>c1
v2<c2
1
A’
v1<c1
v2<c2
slow combustion
unstable
O’
v1<c1
v2>c2
X
X 2 w2  X1w1  ( X1  X 2 )( p2  p1 )
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Thermodynamics of relativistic system
Corrections to the thermodynamics quantities in relativistic
moving systems:
p  p0

v2 
E    E0  p0 V0 2 
c 

V0
V 
[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.
E

A
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 e0  p0v 2 / c 2

 B0




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Temperature
• The hadron temperature is always TH=0.
• Instead the quark temperature can be TQ≠0.
• To evaluate TQ we consider that 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:
internal energy variation
of the system
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work done by the system
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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)]
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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 to have the same
quark’s fraction of the
pure hadronic matter:
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Detonation or not detonation?
beta
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not beta
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Combustion with hyperons
β-stable phase
The vertical line
corrisponds to the
central density of the
most massive star.
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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.
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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.
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B¼=155 MeV
B¼=155 MeV
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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: [Blinnikov, S. Iv. And Sasorov, P. V. Phys. Rev. E 53, 4827 (1995)]
v eff
 lmax
 v sc 
 lmin



D2
D is the fractal dimension
D  2  D0  2
where
  1  e2 / e1 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.
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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:
 ( PD , S D , YeD )   ( P, S , Ye )
PD  P
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: g  
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1 2
 v   g  C
2
1 dP
 dR
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Convection?
Hadronic phase
2
ρQ < ρH
ρQ ? ρH
PQ = P H
PQ = P H
1
ρQ > ρH
ρQ <PQρ=
H PH
PQ < PH
Quark phase
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Convection: results
Cg
LgH155
H
C0
v = 18,5 Km/msec
B0
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Convection with hyperons
LβHy155
v = 45,4 Km/msec
LgHy155
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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)
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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)
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Collaborators
A. Drago, A. Lavagno and I. P. astro-ph/0512652
Alessandro Drago
Physics Department
and INFN of Ferrara
Andrea Lavagno
Others:
Politecnico of Torino
Ignazio Bombaci
(Pisa)
Isaac Vidaña
Giuseppe Pagliara
(Barcelona)
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(Ferrara)
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Appendix
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Combustion theory
Relativistic case
T  wu u   pg
momentum-energy tensor
In the combustion front system and in the
unidimensional case:
u
v
 v
1 v2
T0 x  w u
Txx  wu  p
2
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Quadrivelocity
Momentum-energy tensor
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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).
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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:
Ra   4 exp( Wc / T )
Wc is the maximum of the free energy of the bubble of the new
phase:
4 3
W ( R)  
R ( P2  P1 )   B 2 (  2  1 )  4  R 2
3
Then the number of bubbles of new phase formed
inside a volume V and in a time t is given by:
N  Ra V t
If  is the spacing between two drops in the MP and V / 3 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.|
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Thermal nucleation 2
Then:
Therefore
constraint
satisfied:
S
  V
Ra  V  t  Ra  S      3  2

v 
the following
have to be
  44   Wc 
Wc
  
 ln 

T
v
T
 max

 
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Fluidodynamical description
The fluidodynamical description of the transition is allowed only
for densities:
- σ » 30 MeV/fm2 ρHyd > ρeq
- σ « 30 MeV/fm2 ρHyd > ρ1G
- σ < 30 MeV/fm2 ρHyd > ¯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.
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Rayleigh-Taylor instability
lmax  10 Km
lmin 
4 e v
2
q sc
g e
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