tidal interacons in compact binaries systems · post‐newtonian aﬃne model the ns is an extended...

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


Black Hole (BH)

Neutron star (NS)

White dwarf (WD)

Compact objects


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

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





Coalescing binaries





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)


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!



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!



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!



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

tidal interacons in compact binaries systems · post‐newtonian aﬃne model the ns is an extended...

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