A Periodic Table for Black Hole Orbits

Janna Levin

Dynamical Systems approach

Gabe Perez-Giz NSF Fellow

0802.0459

0809.3838

0811.3798

0811.3814

0811.3815

Becky Grossman NSF Fellow

Gravitational Waves are a fundamental prediction of

General Relativity

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Something changes in the universe -- a

distant object moves -- and we

don’t know about it until a gravitational wave traveling at the speed of light makes it to us to

communicate that change

Inspired by the imminent Advanced LIGO and the proposed space mission LISA

Gravitational Wave Detection Relies on a Knowledge Dynamics

Binary Black Holes (BBH) Many Varieties

3. Intermediate Mass BBH

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2. Stellar Mass BBH formed by tidal capture in globular clusters

1. Stellar Mass BBH from a stellar binary

4. Supermassive Black Holes with a compact companion

2body problem is insolubleSpectrum of orbits

Newtonian elliptical

orbits

Relativistic 2body with

spins - chaotic

Relativistic precession

In the strong-field regime periodic table of BH orbits

Might expect orbits of this kind

Circular orbit Nearby low-eccentricity precessing orbit

Highly-eccentric orbit

unbound

Instead, in strong-field zoom-whirl orbits prevalent

Multi-leaf clovers zooms whirls

Extreme form of perihelion precession. Doesn’t close.

Anatomy of zoom-whirl orbits

zooms

whirls

We build a periodic table of black hole orbits

Between the ground state -- the stable circular orbit

And the energy of ionization -- the escape orbit (or merger)

Lies an infinite

set of periodic

orbits

All generic orbits are defined by this

periodic set

Spinning Black HolesKerr Spacetime

In the 30+ years since Carter found the solutions, the periodic spectrum was overlooked

Carter found the geodesics around a

Kerr black hole

€

C = Kabuaub

We know orbits around a spinning black hole

Taxonomy of Periodic Orbits

Kerr, Equatorial

Every orbit is defined by a rational

number

z = 3

w = 1

v = 1

Look at one radial cycle

Taxonomy of Periodic Orbits

Kerr, Equatorial

Every orbit is defined by a rational

number

z = 3

w = 1

v = 2

And this is a unique 3 leaf orbit

Finding periodic orbits with rational numbers

Physically, the rational corresponds to the accumulated angle in one radial cycle

Finding periodic orbits with rational numbers

Physically, the rational corresponds to the accumulated angle in one radial cycle

Another derivation of q

Every equatorial orbit has two fundamental frequencies

Radial frequency; is the radial period

Angular frequency; average of

over

For periodic orbits, their ratio is rationally related

and are the same frequencies derived from an action-angle formulation of the dynamics

Are all periodic orbits allowed?

No, as illustrated in this energy-level diagram

Highest Energy (unstable circular orbit)

Lowest Energy (stable circular orbit) q >1+1/3

No Mercury-like orbits in the strong field

q > 1+1/3

there are no tightly precessing ellipses since q for these would be near zero

All orbits in the strong-field exhibit zoom and/or whirl behavior

A periodic table of

black hole orbits

Energy increases

up to down and left to

right

Columns are w-bands

Rows are v/z

All orbits, ALL, are defined by the periodic set

High z orbits as perturbations of low z orbits

(3,1, 1) q=1+1/3

(300,1, 103) q=1+103/300

Perturb around closed orbit to get precession

We can apply the periodic taxonomy to any description of binary black holes

No known analytic solution to the metric or orbits.

Comparable mass black hole pairs

Comparable mass black hole pairs

Not enough constants of motion Unless restrict to

SO only

Fully three-dimensional motion

Understanding these requires some new machinery

Perihelion precession + Spin precession

Fully 3D motion now since spin precession drags out of the plane

Fully three-dimensional motion as 2d orbital motion + precession of

plane

the orbit looks simple in the orbital plane

Suggests a way to

• find I.C. for Numerical Relativity

• Test PN expansion

• Compute GWs

Find closed orbits in orbital plane

A periodic Table in the Orbital Plane

These orbital plane periodics are generally not closed in 3D

We have not required a rational in

Not necessary to discuss fully closed

1. These are a subset of the set of periodics in the orbital plane.

Since the subset is dense, our set of periodics must be dense

2. Since q and are related for fixed L, an orbit that is near a given orbital periodic is

near it in the full 3D

Not necessary to discuss fully closed

Fully closed in three-dimensions

A fully closed orbit after one full orbit

A fully closed orbit after two full orbits

Utility of Rational Spectrum

Gravitational wave astronomy

Computation of Inspiral

Transition to chaos with higher-order spin

• All eccentric orbits will show zooms and all high spin show whirls

• Data analysis (template of periodic orbits)

• Inspiral through periodic orbits

• Fourier decomposition in 1 frequency instead of 3

The famed ISCO is a homoclinic orbit with zero e

An orbit with any residual e will transition to plunge through one of these separatrices

Separatrix from Inspiral to Plunge

The homoclinic orbit in phase space

The divide between bound and plunging, whirling and not-whirling, chaotic and

not chaotic

The periodic orbits

To pack a proliferating infinity of orbitsin a finite phase space form a

Fractal

Homoclinic tangle

Chaos develops around certain unstable periodic orbits

What don’t we Know

If the companion

spins - orbits are strongly altered

If there are two black holes of

comparable mass

There are no known exact solutions to the orbits of these BH binaries

Fractal Basin Boundaries

z

x

y

look at initial basins as the spin angles are variedfind they’re fractal

Post-Newtonian approximation to the 2body problem

€

θ =cos−1r S ⋅ ˆ z ( )

€

θ2

€

θ1

•Black merger

•White stable

•Grey escape

Fractal Basin Boundary

Chaotic scattering off an a fractal set of periodic orbits

€

m1

m2

= 3,S1 =1,S2 = 0,θ1 = 95o,r = 5M

In Brief

First test of GR the precession of the perhelion of mercury

Almost a century later, looking for precession of multi-leaf orbits

The precessions will shape the gravitational waves