Periodic A Table for Black Hole Orbits Janna Levin
0802.0459 0809.3838 0811.3798 0811.3814 0811.3815
Dynamical Systems approach
Gabe Perez-Giz NSF Fellow
Becky Grossman NSF Fellow
Gravitational Waves are a fundamental prediction of General Relativity
Something changes in the universe -- a distant object moves -- and we dont know about it until a gravitational wave traveling at the speed of light makes it to us to communicate that change
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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
1. Stellar Mass BBH from a stellar binary 2. Stellar Mass BBH formed by tidal capture in globular clusters
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3. Intermediate Mass BBH
4. Supermassive Black Holes with a compact companion
Spectrum of orbits 2body problem is insoluble
Newtonian elliptical orbits
Relativistic precession
In the strong-field regime periodic table of BH orbits
Relativistic 2body with spins chaotic
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. Doesnt 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 Lies an infinite set of periodic orbits
All generic orbits are defined by this periodic set
And the energy of ionization -- the escape orbit (or merger)
Spinning Black Holes Kerr Spacetime We know orbits around a spinning black hole
C = K ab u a u b
Carter found the geodesics around a Kerr black hole
In the 30+ years since Carter found the solutions, the periodic spectrum was overlooked
Taxonomy of Periodic Orbits
Kerr, Equatorial
Every orbit is defined by a rational number
z=3
Look at one radial cycle
w=1 v=1
Taxonomy of Periodic Orbits
Kerr, Equatorial
Every orbit is defined by a rational number
z=3
And this is a unique 3 leaf orbit
w=1 v=2
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; verage 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)
q >1+1/3 Lowest Energy (stable circular orbit)
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 orbits zero
All 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
Perturb around closed orbit to get precession
(3,1, 1)
q=1+1/3 q=1+103/300
(300,1, 103)
We can apply the periodic taxonomy to any description of binary black holes
Comparable mass black hole pairs No known analytic solution to the metric or orbits.
Comparable mass black hole pairs
Not enough constants of motion Unless restrict to SO only
Fully three-dimensional motion
Perihelion precession + Spin precession
Understanding these requires some new machinery
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
Find closed orbits in orbital plane
Suggests a way to find I.C. for Numerical Relativity Test PN expansion Compute GWs
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
All eccentric orbits will show zooms and all high spin show whirls Data analysis (template of periodic orbits)
Computation of Inspiral
Inspiral through periodic orbits Fourier decomposition in 1 fr...