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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 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**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**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**Simplified Evolution Equations**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**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 Waveform**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 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.**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**Note that the precession phase correction depends only on L and N, not on the detector orientation**Other Cases: Numerical results**(Spin-Spin terms included) Fig. 11. Equal masses, One body maximally spinning, the other non-spinning. +’ detector orientation. Binary at 45 degrees above one arm of the detector**Other Cases: Numerical results**In 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)