EMRI Orbits Around a Spinning Black Hole

What can hundreds of thousands of orbits reveal about the black hole underneath? Close to a massive black hole, an orbit does not close: its nearest point swings forward every lap, and the black hole’s spin changes how fast. Those rates set the frequencies of the gravitational waves. Because a small companion completes hundreds of thousands of laps before it falls in, the accumulated wave phase could measure the hole’s mass, spin and shape.

Drag an orbit around a spinning black hole and watch it precess. See the wave harmonics it emits and how many cycles remain before the last stable orbit.

Periapsis advance per radial lap $203^{\circ}$

Strongest wave harmonic $4.59\,\mathrm{mHz}$

Wave cycles left, circular reference $78\,600$

Time left, circular reference $0.559\,\mathrm{yr}$

Shape the orbit

Black hole spin

0.7

a/M

Orbit size, semi-latus rectum

7\,M

p

Eccentricity

0.4

e

Mass ratio

1\times10^{-5}

q=\mu/M

Drag the orange or blue handle to move the nearest or farthest point. Negative spin means the orbit runs against the rotation.

Black hole mass and orbit constants

Large black hole mass

1\times10^{6}\,M_\odot

M

Mass rescales every frequency and time; the orbit in units of $M$ does not change.

$E/\mu=0.9446$

$L/(\mu M)=3.11$

$\Omega_\phi/\Omega_r=1.564$

$r_{\rm ISCO}=3.39\,M,\quad p_{\rm s}=3.9\,M$

$\mathcal F_{\rm kludge}/\mathcal F_{\rm PM}=0.834$

$r_0=7\,M$

The flux ratio compares this orbit’s quadrupole-kludge energy flux with the Peters–Mathews flux of a Newtonian ellipse with the same $p$ and $e$ ; it tends to one far from the hole. The inspiral reference starts at $r_0$ .

Model assumptions

Units set the gravitational constant, the speed of light and the large black hole mass to one, in Boyer–Lindquist coordinates. The orbit is an exact bound equatorial Kerr geodesic. Spin is signed: a negative value means the orbit runs against the hole’s rotation. Orbit constants follow Glampedakis and Kennefick (2002); the radial and azimuthal periods come from quadrature over the radial phase and agree with direct Hamiltonian integration. The wave spectrum is a quadrupole kludge: the geodesic is placed in flat space and the flat-space quadrupole formula is applied. It is not a Teukolsky waveform. For circular orbits its energy flux is half to one and a fifth of the Fujita–Tagoshi Teukolsky value at six gravitational radii, and within fifteen percent from ten gravitational radii outward. The cycle count and time left follow a separate circular adiabatic inspiral that starts at the semi-latus rectum, uses that quadrupole flux and the exact circular-orbit energy, and stops just outside the innermost stable circular orbit. It omits horizon absorption, higher multipoles, eccentricity decay, inclination, resonances and every self-force correction beyond adiabatic order. It is a reference curve, not a LISA template. The canvas draws radius and azimuth as flat polar coordinates.

Harmonics of this orbit

Simulated, quadrupole kludge

2\Omega_\phi+n\Omega_r

n\Omega_r

Whole multiples

Frequency rise before the last stable orbit

Circular adiabatic reference

f_{\rm GW}=\Omega_\phi/\pi

How the charts are made, and how far to trust them

Left: the orbit’s mass quadrupole, taken in flat space, is split into lines at $2\Omega_\phi+n\Omega_r$ and $n\Omega_r$ . Grey lines are whole multiples of the radial frequency, where the lines would sit without precession. Right: a circular orbit starting at the same $p$ loses energy at the quadrupole rate until just outside the last stable orbit.

Circular-orbit energy flux of the quadrupole kludge divided by the Teukolsky flux to infinity tabulated by Fujita and Tagoshi (2004), multipoles through six, horizon flux not included:

$r/M$	$a=-0.9$	$a=0$	$a=0.9$
$6$	$0.506$	$0.879$	$1.19$
$10$	$0.889$	$1.04$	$1.15$
$20$	$1.03$	$1.07$	$1.1$
$50$	$1.03$	$1.04$	$1.05$

Near the hole the cycle counts are good to tens of percent; far out they are much better. A LISA analysis needs Teukolsky fluxes, eccentric and inclined orbits, and first-order self-force corrections.

Focus question: Which features of the orbit and its wave depend on the spin?
One-sentence intuition: A bound equatorial geodesic has two frequencies, $\Omega_r$ for the in-and-out motion and $\Omega_\phi$ for the turning. Every wave line sits at a combination of the two, and spin moves both.

Core mathematical model

Circular orbits and the last stable one

\Omega_\phi=\frac{1}{r^{3/2}+a},\qquad \Omega_r^2=\Omega_\phi^2\left(1-\frac{6}{r}+\frac{8a}{r^{3/2}}-\frac{3a^2}{r^2}\right)

In units with G, c and M equal to one. The radial frequency reaches zero at the innermost stable circular orbit; prograde spin moves that point inward.

Two frequencies set every harmonic

\omega_{n}=2\Omega_\phi+n\,\Omega_r,\qquad \Delta\phi_{\rm peri}=2\pi\left(\frac{\Omega_\phi}{\Omega_r}-1\right)

The rotating quadrupole radiates at these frequencies. Without precession the two frequencies are equal and the lines fall on whole multiples of the radial frequency.

Energy loss drives a slow inspiral

\frac{dr}{dt}=-\frac{\mathcal F}{dE/dr},\qquad N_{\rm GW}=\int\frac{\Omega_\phi}{\pi}\,dt\propto\frac{1}{q}

The flux and the energy per unit mass are fixed by the orbit, so the cycle count scales as one over the mass ratio. This is the adiabatic order; the first self-force correction adds a phase of order one radian, still large enough to matter for a measurement.

Common difficulties

Precession is not a slow drift here

Typical misconceptionPeriapsis precession is always a small correction, as for Mercury.

Better mental modelNear the separatrix the advance per lap can exceed a full turn, giving zoom-whirl orbits. Compare the zoom-whirl preset with the far-out one.

The wave lines are not orbital multiples

Typical misconceptionAn eccentric orbit radiates at whole multiples of one orbital frequency.

Better mental modelWith precession the lines shift away from the grey whole-multiple ticks by twice the difference of the two frequencies. That shift is a spacetime measurement.

The cycle count belongs to a circular reference

Typical misconceptionThe cycles-left readout follows the eccentric orbit on screen.

Better mental modelIt is computed for a circular orbit starting at the same semi-latus rectum, with a quadrupole flux. Eccentricity decay, horizon absorption and self-force terms are left out.

Run the experiment

01
Watch the rosette
Leave the default running and follow the orange periapsis dots.
What to observe: Each radial lap ends further round; the readout gives the advance in degrees.
02
Flip the spin
Keep the default orbit and drag the spin slider from 0.7 down to zero, then below.
What to observe: The precession per lap grows and the pink last-stable-orbit ring moves outward. Once the spin turns negative this orbit is no longer stable, and the closest stable one is shown.
03
Find the zoom-whirl edge
Drag the orange periapsis handle inward until the notice appears.
What to observe: Close to the separatrix the body lingers near periapsis and whirls before zooming out again.
04
Count the cycles
Change the mass ratio by a factor of ten.
What to observe: The cycles left and the time left change by the same factor of ten; the orbit and the spectrum do not.

EMRI Orbits Around a Spinning Black Hole

Harmonics of this orbit

Frequency rise before the last stable orbit

Read a black hole from the orbit around it

Core mathematical model

Circular orbits and the last stable one

Two frequencies set every harmonic

Energy loss drives a slow inspiral

Common difficulties

Precession is not a slow drift here

The wave lines are not orbital multiples

The cycle count belongs to a circular reference

Run the experiment

Watch the rosette

Flip the spin

Find the zoom-whirl edge

Count the cycles

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