Spin-induced Precession and its Modulation of Gravitational Waveforms from Merging

Binaries

Spin-induced Precession

• Two qualitatively different types of precession:– Simple Precession

• L moves in a tight, slowing growing spiral around a fixed direction

– Transitional Precession• Can only occur when L and S are ~

anti-aligned• L migrates from simple precession

about one direction to simple precession about another direction

Angular Momentum Evolution

( ) (( ) ( ) ) ( )

( ( ) ( ) )

( ( )

Lr

M MM

SM M

MS L

rS L S S L S L

rMr

L

Sr

M MM

M r L S S S S L L S

Sr

M MM

M r L

1 4 32

4 32

32

325

1 4 32

12

32

1 4 32

31 2

11

2 1

22 3 2 1 1 2

2 52

1 31 2

11 2 1 1 1

2 32 1

2

S S S S L L S2 1 2 2 2

12

32

( ) )

Time Evolution Equations for the Angular Momenta, Valid to 2PN order

The first term on each line is a spin-orbit interaction, and will dominate the other spin-spin interaction terms. Note the individual spins have constant magnitude, and the last term on the first line describes the loss of angular momentum magnitude to GW radiation.

Simplified Case

( )

( ) ( ) ( ) ( )

S L Sddt

S S S S S S

L S S S L S

L S S L S S

i i

1 2 1 2 1 2

1 2 1 2

1 2 2 1 0

If we ignore spin-spin effects, which we can do when S2 ~0, and/or M1~M2, and then S1S2 will be constant (thus total |S| is constant)

Also, the angle between L and S will be constant

( )

L S L L S

S L S L Sdd t

L S L S L S

0

0

0

Simplified Evolution Equations ( )

( )

( )| |

LMM

Jr

L J L

SMM

Jr

S J S

MM

Jr

p

p

p

232

232

232

2

13

2

13

2

13

Note that L and S precess around J with the same frequency, and since |L| is decreasing, J moves from L towards S as they spiral around it

Precession Rate• The precession frequency

is much slower than the orbital frequency

• But much faster than the inspiral (radial decrease) rate

• ~10 precessions during LIGO/VIRGO observation period, mostly at low frequencies (about 80-90%)

• Large and small S have a comparable number of precessions

drd t

r

r f

dNd t

dNdr

dNdt

drd t

L SLr

r

N f

L SSr

r

N f

pp

p

p

p

p

p

p

3

23

32 5

1

33

23

,

.

Transitional Precession• At large enough separation,

L>S and J~L• simple precession causes J

and L to spiral away from each other

• If L and S are anti-aligned, as |L| shrinks to |S|, J~0

• The system ‘tumbles’ when its total momentum is roughly 0

• As L continues to shrink, J->S• Simple precession begins

again, and J and S spiral towards each other

Inspiral Waveformh t A t

A tM

rDL t N F L t N F

L t N FL t N F

F

F

x

x

x

( ) ( ) co s( )

( ) ( ( ( ) ) ) ( ( ) )

tan (( ( ) )

( ( ( ) ) ))

( cos ( )) co s( ) co s( ) cos( ) sin ( ) sin ( )

( co s ( )) co s( ) sin ( )

22

1 4

21

121 2 2 2 2

121 2 2

2 2 2 2 2

12

2

2

co s( ) s in ( ) co s( )

( ) tan (( ) ( ( ) )( )

( ( ) ))

2 2

21tL t z L t N z N

N L t z

d tC

C

Precession modulates the waveform because L is not constant in time. Note that the modulation of the amplitude and polarization phase depends on the orientation of the detector through the antenna pattern functions

Amplitude ModulationThe modulation depends on the detector orientation. The +’ signal is when the principal + direction is || to the detector’s arm, the x’ signal is when the principal + direction is 45 degrees from the detector’s arm.

Two factors affect the observed amplitude: The orbital plane’s position relative to the detector arms, and the angle between N and L.

Polarization Phase• Same system as previous

slide• Modulation to Polarization

phase a small oscillation about zero for the +’ orientation

• Large secular increase/decrease for the x’ orientation

• Evolution determined by where the precession cone lies in the cell diagram in the lower right

Precession Phase Correction cos( ( )) sin ( ( ))

( )

( ) cos( ( ))

( )( )

r t t L

r L r L r L

L N

L N

r t

L NL N

L N L

1

1

2

2

Note that the precession phase correction depends only on L and N, not on the detector orientation

Other Cases: Numerical results

Fig. 11. Equal masses, One body maximally spinning, the other non-spinning. +’ detector orientation. Binary at 45 degrees above one arm of the detector

(Spin-Spin terms included)

Other Cases: Numerical results

Other Cases: Numerical results

Other Cases: Numerical resultsIn the second case, S2 can be treated as a perturbation of L, and it turns out that it precesses about L at a frequency much higher than the simple precession frequency, hence the epicycles

Reference

• Apostolatos, Cutler, Sussman, and Thorne, Phys. Rev. D 49, p. 6274–6297 (1994)