Download - Spin-induced Precession and its Modulation of Gravitational Waveforms from Merging Binaries
![Page 1: Spin-induced Precession and its Modulation of Gravitational Waveforms from Merging Binaries](https://reader033.vdocument.in/reader033/viewer/2022042615/56649d695503460f94a47335/html5/thumbnails/1.jpg)
Spin-induced Precession and its Modulation of Gravitational Waveforms from Merging
Binaries
![Page 2: Spin-induced Precession and its Modulation of Gravitational Waveforms from Merging Binaries](https://reader033.vdocument.in/reader033/viewer/2022042615/56649d695503460f94a47335/html5/thumbnails/2.jpg)
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
![Page 3: Spin-induced Precession and its Modulation of Gravitational Waveforms from Merging Binaries](https://reader033.vdocument.in/reader033/viewer/2022042615/56649d695503460f94a47335/html5/thumbnails/3.jpg)
Angular Momentum Evolution
( ) (( ) ( ) ) ( )
( ( ) ( ) )
( ( )
Lr
M M
MS
M M
MS L
rS L S S L S L
r
M
rL
Sr
M M
MM r L S S S S L L S
Sr
M M
MM r L
1 4 3
2
4 3
2
3
2
32
5
1 4 3
2
1
2
3
2
1 4 3
2
31 2
11
2 1
22 3 2 1 1 2
2 5
2
1 31 2
11 2 1 1 1
2 32 1
2
S S S S L L S2 1 2 2 2
1
2
3
2 ( ) )
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.
![Page 4: Spin-induced Precession and its Modulation of Gravitational Waveforms from Merging Binaries](https://reader033.vdocument.in/reader033/viewer/2022042615/56649d695503460f94a47335/html5/thumbnails/4.jpg)
Simplified Case
( )
( ) ( )
( ) ( )
S L S
d
d tS 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 S
d
d tL S L S L S
0
0
0
![Page 5: Spin-induced Precession and its Modulation of Gravitational Waveforms from Merging Binaries](https://reader033.vdocument.in/reader033/viewer/2022042615/56649d695503460f94a47335/html5/thumbnails/5.jpg)
Simplified Evolution Equations
( )
( )
( )| |
LM
M
J
rL J L
SM
M
J
rS J S
M
M
J
r
p
p
p
23
2
23
2
23
2
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
![Page 6: Spin-induced Precession and its Modulation of Gravitational Waveforms from Merging Binaries](https://reader033.vdocument.in/reader033/viewer/2022042615/56649d695503460f94a47335/html5/thumbnails/6.jpg)
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
dr
d tr
r f
dN
d t
dN
dr
dN
dtdr
d tL S
L
rr
N f
L S
S
rr
N f
p
p
p
p
p
p
p
p
3
23
32 5
1
33
23
,
.
![Page 7: Spin-induced Precession and its Modulation of Gravitational Waveforms from Merging Binaries](https://reader033.vdocument.in/reader033/viewer/2022042615/56649d695503460f94a47335/html5/thumbnails/7.jpg)
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
![Page 8: Spin-induced Precession and its Modulation of Gravitational Waveforms from Merging Binaries](https://reader033.vdocument.in/reader033/viewer/2022042615/56649d695503460f94a47335/html5/thumbnails/8.jpg)
Inspiral Waveformh t A t
A tM
rDL t N F L t N F
L t N F
L t N F
F
F
x
x
x
( ) ( ) co s( )
( ) ( ( ( ) ) ) ( ( ) )
tan (( ( ) )
( ( ( ) ) ))
( co s ( )) co s( ) co s( ) co s( ) s in ( ) s in ( )
( co s ( )) co s( ) s in ( )
2
21 4
2
1
1
21 2 2 2 2
1
21 2 2
2 2 2 2 2
12
2
2
cos( ) s in ( ) co s( )
( ) tan (( ) ( ( ) )( )
( ( ) ))
2 2
21tL t z L t N z N
N L t z
d t
C
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
![Page 9: Spin-induced Precession and its Modulation of Gravitational Waveforms from Merging Binaries](https://reader033.vdocument.in/reader033/viewer/2022042615/56649d695503460f94a47335/html5/thumbnails/9.jpg)
Amplitude Modulation
The 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.
![Page 10: Spin-induced Precession and its Modulation of Gravitational Waveforms from Merging Binaries](https://reader033.vdocument.in/reader033/viewer/2022042615/56649d695503460f94a47335/html5/thumbnails/10.jpg)
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
![Page 11: Spin-induced Precession and its Modulation of Gravitational Waveforms from Merging Binaries](https://reader033.vdocument.in/reader033/viewer/2022042615/56649d695503460f94a47335/html5/thumbnails/11.jpg)
Precession Phase Correction
cos( ( )) sin ( ( ))
( )
( ) cos( ( ))
( )( )
r t t L
r L r L r L
L N
L N
r t
L N
L NL N L
1
1
2
2
Note that the precession phase correction depends only on L and N, not on the detector orientation
![Page 12: Spin-induced Precession and its Modulation of Gravitational Waveforms from Merging Binaries](https://reader033.vdocument.in/reader033/viewer/2022042615/56649d695503460f94a47335/html5/thumbnails/12.jpg)
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)
![Page 13: Spin-induced Precession and its Modulation of Gravitational Waveforms from Merging Binaries](https://reader033.vdocument.in/reader033/viewer/2022042615/56649d695503460f94a47335/html5/thumbnails/13.jpg)
Other Cases: Numerical results
![Page 14: Spin-induced Precession and its Modulation of Gravitational Waveforms from Merging Binaries](https://reader033.vdocument.in/reader033/viewer/2022042615/56649d695503460f94a47335/html5/thumbnails/14.jpg)
Other Cases: Numerical results
![Page 15: Spin-induced Precession and its Modulation of Gravitational Waveforms from Merging Binaries](https://reader033.vdocument.in/reader033/viewer/2022042615/56649d695503460f94a47335/html5/thumbnails/15.jpg)
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
![Page 16: Spin-induced Precession and its Modulation of Gravitational Waveforms from Merging Binaries](https://reader033.vdocument.in/reader033/viewer/2022042615/56649d695503460f94a47335/html5/thumbnails/16.jpg)
Reference
• Apostolatos, Cutler, Sussman, and Thorne, Phys. Rev. D 49, p. 6274–6297 (1994)