Spin-induced Precession and its Modulation of Gravitational Waveforms from Merging Binaries
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Spin-induced Precession and its Modulation of Gravitational Waveforms from Merging
Binaries
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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
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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.
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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
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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
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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
,
.
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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
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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
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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.
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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
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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
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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)
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Other Cases: Numerical results
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Other Cases: Numerical results
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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
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Reference
• Apostolatos, Cutler, Sussman, and Thorne, Phys. Rev. D 49, p. 6274–6297 (1994)