tidal interacons in compact binaries systems · post‐newtonian affine model the ns is an extended...
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Tidal interac,ons in compact binaries systems
Andrea Maselli
Supervisor: Prof. V. Ferrari
6 Giugno 2011
Summary
Introduc,on
Compact ObjectsCoalescing binaries
A model for Tidal interac,on
Post‐Newtonian Affine ModelTidal Interac,on
Results
Compact objects
A compact object is an astrophysical body born from the death of an ordinary star, when all nuclear reac,ons at its interior are exhausted
Compact objects
A compact object is an astrophysical body born from the death of an ordinary star, when all nuclear reac,ons at its interior are exhausted
Black Hole (BH)
Neutron star (NS)
White dwarf (WD)
Compact objects
A compact object is an astrophysical body born from the death of an ordinary star, when all nuclear reac,ons at its interior are exhausted
Black Hole (BH)
Neutron star (NS)
White dwarf (WD)
Compact objects
A compact object is an astrophysical body born from the death of an ordinary star, when all nuclear reac,ons at its interior are exhausted
Black Hole (BH)
Neutron star (NS)
White dwarf (WD)
Fully rela,vis,c treatment is required
Compact objects
Sources of poten,ally observable gravita,onal waves (GWs)
Gravita,onal luminosity
Adding the characteris,c internal velocity
Compact objects
Sources of poten,ally observable gravita,onal waves (GWs)
Gravita,onal luminosity
Adding the characteris,c internal velocity
Compact objects
Sources of poten,ally observable gravita,onal waves (GWs)
Gravita,onal luminosity
Adding the characteris,c internal velocity
Compact objects
Sources of poten,ally observable gravita,onal waves (GWs)
Gravita,onal luminosity
Adding the characteris,c internal velocity
extremely compact objects
rela,vis,c internal veloci,es
What we need:
Coalescing binaries
Possible progenitors of short gamma‐ray burst
NS can be ,dally disrupted only from a BH or another NS: we can derive useful informa,ons about equa,on of state
Most promising source of gravita,onal waves for terrestrial interferometers
V.Ferrari, L.Gual.eri, F.Pannarale, PRD 81, 0604026 (2010)
Luciano Rezzolla et a., Astrophys. J. LeD. 732, L6 (2011)
Coalescing binaries
Several approaches have been introduced to study differents phases of the coalescence
Inspiral: approximate methods
Merger: numerical rela,vity
Ringdown: perturba,on theory
Coalescing binaries
Several approaches have been introduced to study differents phases of the coalescence
Merger: numerical rela,vity
Ringdown: perturba,on theory
Inspiral: approximate methods
Coalescing binaries
Several approaches have been introduced to study differents phases of the coalescence
Merger: numerical rela,vity
Ringdown: perturba,on theory
Inspiral: approximate methods
Tidal interac,on
Coalescing binaries (BH‐NS)
Lack of simmetry make numerical simula,on an hard task
High number of parameter: spins, mass ra,o MBH/MNS , equa,on of state
The contribu,ons to gravita,onal signal are given by orbital mo,on and size effects
Perturba,ve methods treat bodies as point par,cles
Numerical rela,vity parameter space is poorly explored
We need approxima,ons to reduce computa,onal cost
Post‐Newtonian affine model
The NS is an extended body subjected to its internal pressure, self‐gravity and ,dal field
It is an ellipsoid and preserves this shape during the orbital evolu,on (affine hypotesis)
NS gravity is determined by a poten,al constructed using rela,vis,c stellar structure equa,ons
Possibility to use realis,c equa,ons of state (EoS)
BH isn’t affected by ,dal field due to its higher compactness
V.Ferrari, L.Gual.eri, F.Pannarale, CQG 26, 125004 (2009)
Post‐Newtonian affine model
Low‐velocity (v/c small) and weak‐field (M/R small) expansion of the Einstein equa,ons
g00 = !1 +2
c2V (t,x)! 2
c4V (t,x)2 +O(6)
g0i = ! 4
c3Vi(t,x) +O(5)
gij = !ij
!1 +
2
c2V (t,x)
"+O(4) .
ds2 = gµ!dxµdx!
= [!µ! + hµ! ] dxµdx!
Post‐Newtonian affine model
Low‐velocity (v/c small) and weak‐field (M/R small) expansion of the Einstein equa,ons
g00 = !1 +2
c2V (t,x)! 2
c4V (t,x)2 +O(6)
g0i = ! 4
c3Vi(t,x) +O(5)
gij = !ij
!1 +
2
c2V (t,x)
"+O(4) .
V =Gm1
r1!Gm1
r1c2
!(n1v1)2
2! 2v21 +
Gm2
r12
"r214r212
+5
4! r22
4r212
#$+ 1 " 2 +O(3)
ds2 = gµ!dxµdx!
= [!µ! + hµ! ] dxµdx!
Post‐Newtonian affine model
Low‐velocity (v/c small) and weak‐field (M/R small) expansion of the Einstein equa,ons
g00 = !1 +2
c2V (t,x)! 2
c4V (t,x)2 +O(6)
g0i = ! 4
c3Vi(t,x) +O(5)
gij = !ij
!1 +
2
c2V (t,x)
"+O(4) .
V =Gm1
r1!Gm1
r1c2
!(n1v1)2
2! 2v21 +
Gm2
r12
"r214r212
+5
4! r22
4r212
#$+ 1 " 2 +O(3)
Post‐Newtonian order
ds2 = gµ!dxµdx!
= [!µ! + hµ! ] dxµdx!
Post‐Newtonian affine model
Low‐velocity (v/c small) and weak‐field (M/R small) expansion of the Einstein equa,ons
g00 = !1 +2
c2V (t,x)! 2
c4V (t,x)2 +O(6)
g0i = ! 4
c3Vi(t,x) +O(5)
gij = !ij
!1 +
2
c2V (t,x)
"+O(4) .
Equa,ons of mo,on including spin‐spin and spin‐orbit couplings
Tidal field
We derive from the metric
ds2 = gµ!dxµdx!
= [!µ! + hµ! ] dxµdx!
Tidal interac,on
We study the devia,on of two geodesics due to the BH‐NS gravitata,onal interac,on
d2n!
d!2= !R!
"#$u"u#n$ d2xi
d!2= !Ci
jxj
Cij = R!"#$e!(0)e
"(i)e
µ(0)e
%(j)
Using the tidal tensor we can calculate the NS deformation during the inspiral
We estimate the tidal tensor up to the 1/c5 order
Tidal interac,on
es,mate the deriva,ves of scalar and vectorial poten,als
compute the ,dal tensor at the source loca,on
express all quan,,es in the center of mass frame
switch to the principal frame
Cxx = ! 1
c2!xxV
(0) ! 1
c4
!4!xtV
(0)x ! 4vy!xxV
(0)y + 4vy!xyV
(0)x + (!yV
(0))2 + !xxV(2)
!!!tt + v2y(!yy + 2!xx) + 2vy!yt ! (2Qxy + vxvy) !xy
"V (0)
+2!xxV(0)V (0) + 2(!xV
(0))2!
! 4
c5
"(!xt + vy!xy)V
(1)x ! vy!xxV
(1)y
#
To do:
[Cij ]NS
Tidal interac,on
es,mate the deriva,ves of scalar and vectorial poten,als
compute the ,dal tensor at the source loca,on
express all quan,,es in the center of mass frame
switch to the principal frame
Cxx = ! 1
c2!xxV
(0) ! 1
c4
!4!xtV
(0)x ! 4vy!xxV
(0)y + 4vy!xyV
(0)x + (!yV
(0))2 + !xxV(2)
!!!tt + v2y(!yy + 2!xx) + 2vy!yt ! (2Qxy + vxvy) !xy
"V (0)
+2!xxV(0)V (0) + 2(!xV
(0))2!
! 4
c5
"(!xt + vy!xy)V
(1)x ! vy!xxV
(1)y
#
To do:
[Cij ]NS
spin
Orbital evolu,on
H = Horb +HT +HI
We use an hamiltonian approach to describe the BH‐NS orbital evolu,on
HT = !LT =1
2CijIij
We place a spherical star in equilibrium at r ! RNS from the black hole
Horb =p2
2µ! Gmµ
r+O(2)
rshedWe evolve numerically the equa,ons of mo,on un,l the system reachs the distance at which the mass flow from the NS starts
Tidal disrutp,on
We define iden,fying the Roche lobe surface embeddingthe neutron star and the Innermost Circular Orbit (ICO)
RNS > Rroche
rorb > rICOTidal disrup,on
NS and Roche lobe radii
BH-NS orbital separation and ICO
We have to compare
rshed
Results
We tuned our model with the numerical results for a BH‐NS dynamic simula,on
Z. E.enne, Y. Liu, S. Shapiro and T. Baumgarte, PRD 79, 044024 (2009)
Polytropic EoS
aMNS = 1.35M! and
Black Hole spin values aBH = {0, 0.75}MBH
q = MBH/MNS = 3
Results
0
0.05
0.1
0.15
0.2
0.25
0.3
0 100 200 300 400 500
!
t/m
aBH
/MBH
= 0
Numerical simulation
Post Newtonian affine model
Results
0
0.05
0.1
0.15
0.2
0.25
0.3
0 100 200 300 400 500 600 700
!
t/m
aBH
/MBH
= 0.75
Numerical simulation
Post Newtonian affine model
Conclusions
We improved a semi-analytic model to describe the last phase of the inspiral in BH-NS binary systems
BH-NS dynamic evolution by means of PN equations of motion including gravitational waves dissipation and orbital corrections due to the tidal field
Relativistic tidal tensor up to the 1/c5 order, including spin terms
We defined a unique Post-Newtonian framework to study both orbital evolution and tidal interactions
We can use realistic equations of state to describe the NS internal structure (numerical simulations use only polytropic EoS)
Future Works
Generalize the model in order to describe NS-NS tidal interactions
Derive initial data for numerical relativity simulations:
Wide range of mass ratio values can be considered
Differents spin configurations for both bodies
Study tidal deformations using different EoS
Sample the parameter space
Characterize the features of the BH accretion disk for differents EoS
Investigate NS crust stress
EoS !MNS ! q ! aBH
THE END