This page contains animated versions of some of the figures from:

*A multi-timescale analysis of phase transitions in precessing black-hole binaries*

Michael Kesden, Davide Gerosa, Ulrich Sperhake, Emanuele Berti, Richard O’Shaughnessy.

Phys.Rev.D 92 (2015) 064016. arXiv:1506.03492 [gr-qc]*Effective potentials and morphological transitions for binary black-hole spin precession.*

Davide Gerosa, Michael Kesden, Richard O’Shaughnessy, Emanuele Berti, Ulrich Sperhake.

Phys.Rev.Lett. 114 (2015) 081103. arXiv:1411.0674 [gr-qc]*Precessional instability in binary black holes with aligned spins.*

Davide Gerosa, Michael Kesden, Richard O’Shaughnessy, Antoine Klein, Emanuele Berti, Ulrich Sperhake, Daniele Trifirò.

Phys.Rev.Lett. 115 (2015) 141102. arXiv:1506.09116 [gr-qc]

Figures published in the papers are snapshots (at fixed binary separation) of these movies. References to other figures and equations in the captions correspond to those in the relevant paper.

All animations can be downloaded here in full quality (they play well on VLC ). You’re welcome to use them, but please make reference to our papers and to this webpage. Here is a Youtube playlist containing all movies.

Some older animated gif from 1302.4442 are also provided at the end of this page.

#### Turn on **HD** quality using the YouTube gear!

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**HD**Fig. 3 from 1506.03492Analytical solutions given by Eq. (20) for the evolution of the angles θ1 (top panel), θ2 (middle panel), and ∆Φ (bottom panel) during a precession cycle. The evolution of three binaries with ξ = 0.25 (blue), 0.3 (green) and 0.35 (red) is shown for q=0.8,χ1 =1,χ2 =0.8 and evolved on the radiation time using Eq.(38). In particular, J=1.29M^2 for all binaries at r=20M. The evolution of θ1 and θ2 is monotonic during each half of a precession cycle and is bounded by the dotted lines for which cos φ = ∓1 [these curves can be found by substituting ξ±(S) for ξ in Eq. (20)]. Three classes of solutions are possible and define the binary morphology: ∆Φ can oscillate about 0 (ξ = 0.25), circulate (ξ = 0.3) or oscillate about π (ξ = 0.35).

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**HD**Fig. 5 from 1506.03492The (J,ξ) parameter space for BBHs with different minimum allowed total angular momentum J_min. BBH spin morphology is shown with different colors, as indicated in the legend. The extrema ξ_min(J ) and ξ_max(J ) of the effective potentials constitute the edges of the allowed regions and are marked by solid blue (red) curves for ∆Φ = 0 (π). Dashed lines mark the boundaries between the different morphologies. The parameters q, χ1, χ2 and r are chosen as in Fig.4, whose panels can be thought of as vertical (constant J) “sections” of this figure (where we suppress the S dependence). The lowest allowed value of ξ occurs at J = |L − S1 − S2| in all three panels at all separations. Three phases are present for each vertical section with J > |L − S1 − S2|. This condition may either cover the entire parameter space or leave room for additional regions where vertical sections include two different phases in which ∆Φ oscillates about π and a circulating phase in between or only a single phase where the spins librate about ∆Φ = π.

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**HD**Fig. 6 from 1506.03492Time-dependent solutions for the total spin-magnitude S (top panel) and the orbital-angular-momentum phase ΦL (bottom panel). We set q = 0.7, χ1 = 0.7, χ2 = 0.9; binaries are evolved such that J = 1.48M^2 r=30M. We integrate Eq. (26) for three values of ξ corresponding to the three different spin morphologies at r=30M: ∆Φ oscillates about 0 (ξ = 0.17, blue), circulates (ξ = 0.25, green), and oscillates about π (ξ = 0.34, red). Initial conditions have been chosen such that S=S− and ΦL=0 at t=0. The oscillations in S induce small wiggles in ΦL on top of a mostly linear drift. Spin-orbit resonances (horizontal dashed lines, top panel) correspond to configurations for which S is constant and can be interpreted as zero-amplitude limits of generic oscillatory solutions. The projections of the effective potentials, i.e. parametric curves [τ(ξ)/2, S+(ξ)] and [τ(ξ),S−(ξ)], are shown with dotted lines.

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**HD**Fig. 8 from 1506.03492Precession-averaged BBH inspirals as described in Sec.IIIC (purple/darker) compared to numerical integration of the orbit-averaged PN equations [35,36] (orange/lighter). Marginalized distributions of the spin angles θ1, θ2, and |∆Φ| (rows) are shown at several separations along the inspirals. The three initial spin distributions are isotropic (top panels), one aligned BH (middle panels), and Gaussian spikes (bottom panels) as described in Sec.IIIC. The two approaches are in good agreement except for minor deviations in the distribution of ∆Φ at r∼10M. We take q=0.7, χ1=0.8 and χ2=0.4 for all BBHs.

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**HD**Fig. 10 from 1506.03492Precessional solutions ∆Φ(S) of Eqs. (20) as J and L evolve during inspirals according to Eq.(38). These solutions are colored according to the separation r/M as shown in the color bar on the right (orange/lighter for large separations and black/darker for small separations). Binaries in the left (right) panel transition from the circulating morphology to the morphology in which ∆Φ librates about 0 (π) at the transition radius r_tr ≃ 152M (18.9M).

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**HD**Fig. 13 from 1506.03492The fraction f of isotropic BBHs for which ∆Φ circulates (green, middle region), oscillates about 0 (blue, bottom region), or oscillates about π (red, top region) as functions of the mass ratio q. Dashed lines separate the different morphologies. Each panel corresponds to a different value of χ1 (columns) and χ2 (rows). The fraction of BBHs in librating morphologies increases as the mass asymmetry decreases (q→1). For nearly equal masses (q>0.9), asymmetry in the spin magnitudes increases the fraction of binaries in the circulating morphology as can be seen by comparing panels on and off of the diagonal.

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**HD**Fig. 14 from 1506.03492Spin morphologies at evolving separation r_f as functions of the asymptotic values of the spin angles θi∞. The mass ratio q and spin magnitudes χi for each panel are indicated in the legends. Evolving BBHs along the four lines cosθi=±1 at r_f out to r/M → ∞ using our new precession-averaged approach yields the dashed curves separating the different final morphologies: ∆Φ oscillates about 0 (blue), oscillates about π (red), circulates without ever having experienced a phase transition (plain green), or circulates after having experienced a phase transition to libration and then a second phase transition back to circulation (hatched green). The morphology within each region defined by the dashed boundaries is determined by which of the conditions cosθi=±1 these boundaries satisfy, as described in Sec. IV C. The points show the locations of binaries in the cosθ1 − cosθ2 plane at r_f and are colored by their morphology at that separation [∆Φ oscillates about 0 (blue circles), oscillates about π (green squares), or circulates (red trianges)]. Because morphology depends on ∆Φ in addition to θ1 and θ2 at finite separation, the projection onto the cosθ1 − cosθ2 plane can lead points of different morphologies to occur at the same positions, particularly for comparable-mass binaries q≃1 where the θi’s oscillate with greater amplitude.

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**HD**Fig. 2 from 1411.0674The three morphologies of BBH spin precession. The angular momenta J, L, and Si are all in the xz plane at S=S±. In all three panels, the BBHs have maximal spins and q=0.8. Binaries have J=0.85M^2 at r=10M (L=0.781M^2) as in Fig. 1. The left, middle, and right panels correspond to ξ=−0.025, 0.025, and 0.15, respectively. If the components of Si perpendicular to L are aligned with each other at both roots S, ΔΦ librates about 0°. If they are aligned at one root and antialigned at the other, ΔΦ circulates. If they are antialigned at both roots, ΔΦ librates about 180°.

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**HD**Fig. 1 from 1506.09116Effective-potential loops ξ±(S) for binary BHs with mass ratio q=0.9, dimensionless spins χ1=1, χ2=0.14, and total angular momentum J=|L+S1-S2|, corresponding to the up-down configuration. For binary separations r>rud+~337M, the up-down configuration at Smin marked by a red circle is also a minimum (marked by the lower triangle). At intermediate separations rud+>r>rud−~17M misaligned binaries with the same value of the conserved ξ exist along the dashed red line. Perturbations δJ, δξ will cause S to oscillate between the points S± where this line intersects the loop, making the up-down configuration unstable. For r<rud−, the up-down configuration is again a stable

extremum, now a maximum (marked by the upper triangle).

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**HD**Fig. 2 from 1506.09116The angles cos(θi) for spin-orbit resonances [extrema of ξ±(S)] for BHs with q=0.95, χ1=0.3, and

χ2=1. The solid (dashed) curves indicate the ∆Φ=0 (π) family. The up-down configuration (bottom right corner) belongs to the ∆Φ=0 family for r>rud+~ 2149M, to the ∆Φ=π family for r<rud-~13M, and is unstable for intermediate values rud−<r<rud+.

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**HD**Fig. 3 from 1506.09116Precession-averaged radiation reaction dJ/dL as a function of J and ξ for binaries with q=0.8, χ1=χ2=1. Spin-orbit resonances including the up-up, down-down, and down-up configurations are extrema of ξ±(S) and constitute the boundary of the allowed region. All four aligned configurations are maxima where dJ/dL = 1, but the unstable up-down configuration (shown in the inset) is a cusp.

### Older animated gif from 1302.4442

**tides-isotropic.gif **

**Fig. 5 from 1302.4442 **: Scatter plots of the PN inspiral of maximally spinning BH binaries with mass ratio q = 0.8 from an initial separation a_PNi just above 1000M to a final separation aPNf = 10M. The left panel shows this evolution in the (θ1, θ2) plane and the right panel shows the evolution in the (∆Φ, θ12) plane. Darker (red) and lighter (green) dots refer to the SMR and RMR scenarios, respectively. The initial distribution for these Monte Carlo simulations was constructed from an astrophysical model with efficient tides and isotropic kicks.

**tides-polar.gif **

**Fig. 6 from 1302.4442 **: Scatter plots of the PN inspiral of maximally spinning BH binaries with mass ratio q = 0.8 from an initial separation a_PNi just above 1000M to a final separation aPNf = 10M. The left panel shows this evolution in the (θ1, θ2) plane and the right panel shows the evolution in the (∆Φ, θ12) plane. Darker (red) and lighter (green) dots refer to the SMR and RMR scenarios, respectively. The initial distribution for these Monte Carlo simulations was constructed from an astrophysical model with efficient tides and polar kicks.

**notides-isotropic.gif **

**Fig. 6 from 1302.4442 **: Scatter plots of the PN inspiral of maximally spinning BH binaries with mass ratio q = 0.8 from an initial separation a_PNi just above 1000M to a final separation aPNf = 10M. The left panel shows this evolution in the (θ1, θ2) plane and the right panel shows the evolution in the (∆Φ, θ12) plane. Darker (red) and lighter (green) dots refer to the SMR and RMR scenarios, respectively. The initial distribution for these Monte Carlo simulations was constructed from an astrophysical model with inefficient tides and isotropic kicks.

**notides-polar.gif **

**Fig. 6 from 1506.09116**: Scatter plots of the PN inspiral of maximally spinning BH binaries with mass ratio q = 0.8 from an initial separation a_PNi just above 1000M to a final separation aPNf = 10M. The left panel shows this evolution in the (θ1, θ2) plane and the right panel shows the evolution in the (∆Φ, θ12) plane. Darker (red) and lighter (green) dots refer to the SMR and RMR scenarios, respectively. The initial distribution for these Monte Carlo simulations was constructed from an astrophysical model with inefficient tides and polar kicks.

**Picture**: wave inspiral in Lanzarote, Canary Islands