spin alignment effects in stellar mass black hole binaries · ulrich sperhake, richard o ......

Contents Post-Newtonian Spin-Orbit Resonances Stellar Mass BH Binary Results Future Developments

Spin Alignment Effects inStellar Mass Black Hole Binaries

Davide GerosaCo-authors: Emanuele Berti, Michael Kesden,

Ulrich Sperhake, Richard O’Shaughnessy

University of MississippiOxford, MS

September 29, 2012

22nd Midwest Relativity MeetingChicago, IL

[email protected]

Davide Gerosa University of Mississippi

Spin Alignment Effects in Stellar Mass Black Hole Binaries


Contents

1. Post-Newtonian Spin-Orbit Resonances

2. Stellar-Mass Black Hole Binary Formation

3. Results

4. Future Developments




Post-Newtonian Evolution

Black hole binary inspiral:

• Large separation R > 1000M:interactions with the astrophysical environment.

• Post-Newtonian approximation:solve the Einstein field equations by a series in v/c .

• Small separation R < 10M:numerical relativity.

Can the PN evolution alter spin orientation?

If the evolution brings the spin parameter to be clustered in certainregions we could place more templates in these regions, increasing theefficiency of possible detections.

Evolution from R=1000M to R=10M: what happens to the spins?




PN Equations of MotionKidder 1995; Arun et al. 2011; Kesden, Sperhake, and Berti 2010a

Spin precessiondS1

dt= Ω1 × S1

dS2

dt= Ω2 × S2

Ω1 = v5 3

4+η

2−

3

4

δm

M

L +1

2

v6

M2

[S2 − 3

(L · S2

)L]−

3

2

v6

M2

m2

m1

(L · S1

)L

Angular momentum conservation and radiation reactiond L

dt= −

ηv

M2

S1 + S2

dv

dt=

32

5

η

Mv9

1 + v2 − 743

336−

11

4η

+ v34π − ∑

i=1,2χi (Si · L)

113

12

m2i

M2+

25

4η

+ v4

3410318144

+13661

2016η +

59

18η2+

+721

48ηχ1χ2(S1 · L)(S2 · L) −

247

48ηχ1χ2(S1 S2) +

∑i=1,2

5

2χ2i

mi

M

2 (3(Si · L)2 − 1

)+

+∑

i=1,2

1

96χ2i

mi

M

2 (7 − (Si · L)2

) + v5− 4159

672−

189

8η

π + v6

16447322263139708800

+16

3π2 −

1712

105

(γE + ln 4v

)+

+

− 56198689

217728+

451

48π2 η +

541

896η2 −

5603

2592η3 + v7

π

− 4415

4032+

358675

6048η +

91495

1512η2 + O(v8)

Results robust to the addition of further PN terms [Favata, in prep.]Davide Gerosa University of Mississippi



PN Resonances

θ1

S1

LN

ey

ex∆φ

θ2

S2

Schnittman 2004

Family of equilibrium solutions in which thetwo BH spins and the orbital angularmomentum are coplanar, precessing jointlyabout the total angular momentum

• Fixing L, i.e. at a particular point in spaceand time during the inspiral

• ∆Φ = 0, 180

• θ1 and θ2 solution of the PN equations




PN Resonances q = 9/11, χ1 = 1, χ2 = 1

From r = 1000M to r = 10M

Kesden, Sperhake, and Berti 2010a

Two Resonances

• θ1 < θ2 → ∆Φ = 0

• θ2 < θ1 → ∆Φ = 180

Gravitational radiation: emissionof energy. The system evolvestowards θ1 ' θ2 along the red curves

Locking: if the system gets close toa resonance, it can get trapped!

rlock =

(1 + q2

1− q2

)2

M




Isotropic sample q = 9/11, χ1 = 1, χ2 = 1

From left to right and from top to bottom: Initial conditions, r = 1000M,

r = 750M, r = 500M, r = 250M, r = 100M, r = 50M, r = 20M, r = 10M

cos θ1 vs. cos θ2 ∆Φ vs. cos θ12




Compact binary formation: looking for initial conditions

Standard population synthesis model by Belczynski et al. 2008

• Binary stars: spins are initially aligned (tidal interactions)

• Stellar evolution, tracking those that produce BH-BH.

• SN explosions: each new compact object receives a kick

• Kick’s magnitudes are taken from the observed pulsar propermotions distribution and the direction is assumed isotropic

• Asymmetric mass ejection: fallback material

• Kicks change the orbital plane, not the direction of the spins θ1 ∼ θ2Is there any realignment due to mass transfers and tidal interactionsbetween the two SN?




Simulation Setup

• Mass ratio q = 9/11 ' 0.81:• It is the value used in all the previous works (comparison)• Median and average of the catalog by Dominik et al. 2012 are ' 0.83

• Maximally spinning BHs: χ1 = χ2 = 1

• Misalignment of 10 with a dispersion of 3

• ∆Φ free (initial phase of the precession)

Phenomenological scenarios

• 10/10: Both the BHs are tiltedθ1 ∈ [10 − 3, 10 + 3] θ2 ∈ [10 − 3, 10 + 3]

• 10/0: Secondary BH realigned*θ1 ∈ [10 − 3, 10 + 3] θ2 ∈ [0, 3]

• 0/10: Primary BH realignedθ1 ∈ [0, 3] θ2 ∈ [10 − 3, 10 + 3]




10/10 sample q = 9/11, χ1 = 1, χ2 = 1


r = 750M, r = 500M, r = 250M, r = 100M, r = 50M, r = 20M, r = 10M





10/0 sample 0/10 sample


r = 750M, r = 500M, r = 250M, r = 100M, r = 50M, r = 20M, r = 10M

∆Φ vs. cos θ12 ∆Φ vs. cos θ12




Towards an Astrophysical Model

Results

• Locking strongly depends on initial conditions

• Tidal interactions and mass transfer events are crucial

Outlook:

• Analytical Model of the kick, using conservation laws

• Generalization of Kalogera 2000 to elliptic orbits and double kicks

• Priors: progenitor masses, initial separation, eccentricity

• Tidal interactions and mass transfers

⇒ Initial tilt angle distributions

• PN evolution and resonant effects




Acknowledgements

E. Berti,M. Kesden,U. Sperhake,R. O’Shaughnessy,M. Favata,G. Lodato.

Thanks

Davide [email protected]




General Picture

BH - BH Binary main parameters:

• Source position: right ascension, declination, distance

• Orbital plane: two angles (orientation and polarization angle)

• Phase at coalescence, time of coalescence

⇒ extrinsic parameters: affect the amplitude of the signal, but not its form

• Masses: (m1,m2) or (M, q)

⇒ 9 parameters for non-spinning binaries (circular orbit)

• Spin vectors: components, or magnitude and two angles for each BH

⇒ 15 parameters for spinning binaries (circular orbit)

⇒ intrinsic parameters: affect the evolution of amplitude and phase in time

• Orbit: eccentricity and orientation of semi-major axis (two angles)

⇒ 18 parameters for spinning binaries on a general orbit




Black Hole Spins

Do we need all of these parameters?

The orbits circularize on a timescale shorter than the inspiral timescale(Peters 1964; Peters and Mathews 1963)

Are astrophysical black holes spinning?

For Kerr black holes, the spin angular momentum is given by

J = χM2G

c

with 0 ≤ χ ≤ 1

We cannot avoid the use of the 6 spins parameters but we can (maybe)find out something about spin alignment.




Black Hole Spins

Measurements indicate that many BHs could be rapidly spinning

• Fe K-α line profile

• Continuos X-Ray emission of the accretion disc

0.5 1 1.5

Line profile

GraviGeneral relativity

Tran

Beami

Special relativity

Newtonian

r t tr t

❳

❳

❯

❳

❳

rrtr r qt P t t♦❱ ❲rtt r rt r t ❨ ❩❱

Fabian et al. 2000

McClintock et al. 2011

We cannot avoid the use of the 6 spins parameters but we can (maybe)find out something about spin alignment.




PN Equations of MotionKidder 1995; Arun et al. 2009; Arun et al. 2011

Spin PrecessiondS1

dt= Ω1 × S1

dS2

dt= Ω2 × S2

Ω1 = v5(

3

4+η

2−

3

4

δm

M

)L Ω2 = v5

(3

4+η

2+

3

4

δm

M

)L

Angular Momentum Conservation and Radiation Reaction

d L

dt= −

ηv

M2

S1 + S2

dv

dt=

32

5

η

Mv9

1 + v2

[−

743

336−

11

4η

]+ v3

[4π −

∑i=1,2

χi (Si · L)

(113

12

m2i

M2+

25

4η

)]+ O(v4)

where M = m1 + m2, η =m1m2M2 , δm = m1 − m2.




PN Equations of MotionKesden, Sperhake, and Berti 2010a; Kesden, Sperhake, and Berti 2010b; Berti, Kesden, and Sperhake 2012

Spin PrecessiondS1

dt= Ω1 × S1

dS2

dt= Ω2 × S2

Ω1 = v5 3

4+η

2−

3

4

δm

M

L +1

2

v6

M2

[S2 − 3

(L · S2

)L]−

3

2

v6

M2

m2

m1

(L · S1

)L

Angular Momentum Conservation and Radiation Reaction

d L

dt= −

ηv

M2

S1 + S2

dv

dt=

32

5

η

Mv9

1 + v2 − 743

336−

11

4η

+ v34π − ∑

i=1,2χi (Si · L)

113

12

m2i

M2+

25

4η

+ v4

3410318144

+13661

2016η +

59

18η2+

+721

48ηχ1χ2(S1 · L)(S2 · L) −

247

48ηχ1χ2(S1 S2) +

∑i=1,2

5

2χ2i

mi

M

2 (3(Si · L)2 − 1

)

+∑

i=1,2

1

96χ2i

mi

M

2 (7 − (Si · L)2

) + v5− 4159

672−

189

8η

π + v6

16447322263139708800

+16

3π2 −

1712

105

(γE + ln 4v

)+

+

− 56198689

217728+

451

48π2 η +

541

896η2 −

5603

2592η3 + v7

π

− 4415

4032+

358675

6048η +

91495

1512η2 + O(v8)




PN Equations of MotionNew PN terms computed by M.Favata

Spin PrecessiondS1

dt= Ω1 × S1

dS2

dt= Ω2 × S2

Ω1 = v5 3

4+η

2−

3

4

δm

M

L +1

2

v6

M2

[S2 − 3

(L · S2

)L]−

3

2

v6

M2

m2

m1

(L · S1

)L + v7

9

16+

5η

4−η2

24+δm

M

− 9

16+

5η

8

L

Angular Momentum Conservation and Radiation Reactiond L

dt= −

ηv

M2

1 + v2 3

2+η

6

−1

S1 + S2 −2η

Mv7

η +

m2

M

21 +

3

16

m2

m1

(L × S1

)−

(1 ↔ 2

)dv

dt=

32

5

η

Mv9

1 + v2 − 743

336−

11

4η

+ v34π − ∑

i=1,2χi (Si · L)

113

12

m2i

M2+

25

4η

+ v4

3410318144

+13661

2016η +

59

18η2+

+721

48ηχ1χ2(S1 · L)(S2 · L) −

247

48ηχ1χ2(S1 S2) +

∑i=1,2

5

2χ2i

mi

M

2 (3(Si · L)2 − 1

)+

∑i=1,2

1

96χ2i

mi

M

2 (7 − (Si · L)2

) +

+ v5− 4159

672−

189

8η

π +∑

i=1,2χi (Si · L)

−21219

1008

m2i

M2+

1159

24η

m2i

M2−

809

84η +

281

8η

+

−1

4

∑i=1,2

mi

M

3 χi (Si · L)(1 + 3χ2i )

1 −15

8χ2i

1 − (Si · L)2

1 + 3χ2i

+ v6

16447322263139708800

+16

3π2 −

1712

105

(γE + ln 4v

)+

+

− 56198689

217728+

451

48π2 η +

541

896η2 −

5603

2592η3 + v7

π

− 4415

4032+

358675

6048η +

91495

1512η2 + O(v8)




Correlation between θ12 and ∆Φ

• θ12: angle between the two spins

• ∆Φ: angle between their projections on the orbital plane

cos θ12 = sin θ1 sin θ2 cos ∆Φ + cos θ1 cos θ2

Expected correlation: integrate over θ1 and θ2 providing initialdistributions f1 = f1(cos θ1) and f2 = f2(cos θ2)

< cos θ12 >=

∫sin θ1 f1 d cos θ1∫

f1 d cos θ1

∫sin θ2 f2 d cos θ2∫

f2 d cos θ2cos ∆Φ+

+

∫cos θ1 f1 d cos θ1∫

f1 d cos θ1

∫cos θ2 f2 d cos θ2∫

f2 d cos θ2




Constant of Motion

Effective One-Body problem (Damour 2001)

S0 = (1 + q)S1 + (1 + q−1)S2

At 2PN order (v/c)4

S0 · L

is a constant of motion (Racine 2008)

• (cos θ1, cos θ2) plane: propagation with slopes − 1q




10/0 sample q = 9/11, χ1 = 1, χ2 = 1


r = 750M, r = 500M, r = 250M, r = 100M, r = 50M, r = 20M, r = 10M





0/10 sample q = 9/11, χ1 = 1, χ2 = 1


r = 750M, r = 500M, r = 250M, r = 100M, r = 50M, r = 20M, r = 10M





Average < |∆Φ| >Is

otr

op

ic

10

/1

0

0/

10

10

/0




Average < cos θ12 >

Iso

tro

pic

10

/1

0

0/

10

10

/0




SN explosion on Eccentric Orbits

Elliptic orbit M0, a0, e0.

r0 =a0(1− e20 )

1 + e0 cosψ0

The explosion happens with true anomaly ψ0.Kick: mangnitude vk , direction θ, φ.

Conservation laws −→ new orbit parameters M1, a1, e1

α =a1a0

β =M1

M0uk =

vk√GM0/a0





Conservation of energy: semi-major axis

a1a0

= β

[2 (β − 1)

(1 + e0 cosψ0

1− e20

)+1−u2

k−2uk

(2

1 + e0 cosψ0

1− e20− 1

)1/2

cos θ

]−1

Conservation of angular momentum: eccentricity

(1− e21

)=

1

αβ

[(√1− e20 +

1− e201 + e0 cosψ0

uk cos θ

)2

+

+

(1− e20

1 + e0 cosψ0uk sin θ cosφ

)2].





Tilt Angle (single kick)

cos γ =L0 · L1

|L0||L1|=

=

(√1− e20 +

1−e201+e0 cosψ0

uk cos θ)

[(√1− e20 +

1−e201+e0 cosψ0

uk cos θ)2

+(

1−e201+e0 cosψ0

uk sin θ cosφ)2 ]1/2 .

• Generalization of the expression in Kalogera 2000 for e0 6= 0

• Double kick: using the same expressions, updating the system ofreference




Preliminary: nothing happens between the two SN

• No mass loss: ms1 = m1 = 11M, ms2 = m2 = 9M• a0 = 10R, e0 = 0

0 50 100 1500

0.005

0.01

0.015

tilt angle

0 50 100 1500

0.2

0.4

0.6

0.8

1

tilt angle

Tilt angle distributions (double kick)





r = 750M, r = 500M, r = 250M, r = 100M, r = 50M, r = 20M, r = 10M




spin alignment effects in stellar mass black hole binaries · ulrich sperhake, richard o ......

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