What hundreds of thousands of orbits can reveal about a black hole

Sandbox Physics · 2026-11-12

Picture a black hole ten times the mass of the Sun circling one a million times heavier. It is too small to disturb its partner, so it follows the giant’s spacetime almost exactly. It also falls inward so slowly that it completes hundreds of thousands of laps first. Every lap is a reading of the same spacetime, and the gravitational waves it sends out carry those readings to us.

A slow fall into a giant

Astronomers call such a pair an extreme mass ratio inspiral, or EMRI. Galaxy centres hold black holes of about a million solar masses, surrounded by dense clusters of stars and stellar remnants. Now and then a remnant is scattered onto a tight orbit around the central hole.

The small body loses orbital energy to gravitational waves, but only a tiny share each lap, roughly in proportion to its mass divided by the giant’s. With a mass ratio of one in a hundred thousand, the last year or so before it plunges still holds about a hundred thousand wave cycles.

Around a million-solar-mass hole these waves oscillate a few thousandths of a time per second, a few millihertz. Ground detectors cannot hear that low. The planned space interferometer LISA is designed for this band.

Reading: Babak et al. (2017) · Barack & Pound (2019)

The orbit never closes

A planet’s ellipse comes back almost to where it started. Mercury’s closest point to the Sun advances by less than a hundredth of a degree per orbit. Near a black hole the same advance is huge. In the Lab’s default orbit, the closest point swings forward by about 203 degrees every time the small body goes in and out.

Push the orbit closer and the advance grows past a full turn. The body dives in, whirls around the hole several times near its closest point, then zooms back out. Physicists call these zoom-whirl orbits.

Closer still, there is a boundary. Inside it no stable orbit exists, and the body plunges. For a circular orbit around a non-spinning hole that boundary is at six times the hole’s mass, in units where the gravitational constant and the speed of light are one.

Reading: Glampedakis & Kennefick (2002) · Abramowicz & Kluźniak (2004)

Spin changes every rate

A spinning black hole drags the space around it. An orbit that goes the same way as the spin can sit much closer before it becomes unstable: at a spin of 0.9 the last stable circular orbit moves in from six to about 2.3 mass units. An orbit going against the spin is pushed out to about 8.7.

Spin also changes how fast the closest point advances. At a fixed orbit size, the Lab shows a smaller advance with the spin and a larger one against it. The orbit’s shape, the boundary of stability and the advance rate all respond to the same two numbers, the hole’s mass and spin.

Reading: Abramowicz & Kluźniak (2004) · Glampedakis & Kennefick (2002)

Every lap leaves a mark in the wave

The orbit has two rhythms: the in-and-out motion and the turning around the hole. Without the advance they would match. Near the hole they differ, and the wave’s frequencies are built from both. Its spectrum is a comb of lines that sits slightly off the evenly spaced comb a closed ellipse would give.

As the body spirals in, both rhythms speed up along a path set by the spacetime. A detector compares the arriving wave with predicted waves cycle by cycle. A wrong guess for the spin drifts out of step after enough cycles. Hundreds of thousands of cycles make even a small mismatch visible.

Reading: Babak et al. (2017)

What LISA could measure

A 2017 study for the LISA mission simulated many possible populations of these sources. It forecast at least a few detectable EMRIs per year under every population model it tried, and up to a few thousand per year under the most optimistic assumptions.

For a typical detected source, the same study forecast fractional errors of about one in a million to one in ten thousand on the black hole masses, the spin and the orbital eccentricity. The distance would be known to about ten percent and the sky position to a few square degrees. A test of whether the hole’s shape matches the prediction for a spinning black hole could reach percent level or better.

These numbers are forecasts from models, not measurements. They rest on simplified waveforms and on assumptions about how many such pairs exist.

Reading: Babak et al. (2017)

What still has to be computed

To follow hundreds of thousands of cycles, a waveform model has to keep the phase right to a fraction of a cycle. Energy balance alone, the method behind the Lab’s cycle count, gets the leading part. The small body’s own gravity adds a correction of order one radian, and computing it is the job of self-force theory.

Orbits tilted out of the equator add another complication. When two of the orbit’s rhythms fall into a simple ratio, such as one to two, the inspiral briefly speeds up or slows down. For mass ratios near one in a million, this can shift the phase by a few tens of cycles.

Reading: Barack & Pound (2019) · Flanagan & Hinderer (2012)

Try it in the Lab

Watch an orbit swing forward and change its spin

Leave the default orbit running and follow the orange dots marking each closest approach.
Choose Zoom and whirl, then Far out, nearly Kepler, and compare how far the dots move per lap.
Drag the spin slider from positive to negative and watch the pink ring, the last stable circular orbit, move outward.

The Lab draws an exact equatorial orbit around a spinning black hole. Its wave spectrum and cycle count use simple approximations, described under the stage.

Primary sources and what we read in each

Babak et al. (2017), Science with the space-based interferometer LISA. V. Extreme mass-ratio inspirals, Phys. Rev. D 95, 103012

arXiv version 2, stamped 11 April 2018. Abstract; Table I, the twelve population models and their rates; Section IV on the AKS and AKK kludge waveforms; Fig. 5, the detection horizon for a prograde circular equatorial inspiral into a hole of spin 0.99; Figs. 11–13, Fisher-matrix error distributions for intrinsic parameters, distance and sky position, and the quadrupole deviation; the conclusions.

Flanagan & Hinderer (2012), Transient resonances in the inspirals of point particles into black holes, Phys. Rev. Lett. 109, 071102

arXiv version 3, dated 11 July 2012. Abstract, introduction and the order-of-magnitude estimates with eqs. (1)–(2): resonances occur when the ratio of polar and radial frequencies passes a low-order rational number, only for orbits that are both eccentric and inclined around a spinning hole; the phase shift scales as the inverse square root of the mass ratio, a few tens of cycles near one in a million.

Barack & Pound (2019), Self-force and radiation reaction in general relativity, Rep. Prog. Phys., doi:10.1088/1361-6633/aae552

arXiv version 2, dated 19 November 2018. Section 5.2 and eqs. (72)–(74): the two-timescale phase expansion and what the adiabatic and post-adiabatic orders need; Section 5.3 on transient resonances; Section 6.1 and eqs. (77)–(78), the orbit-averaged balance laws, including negative horizon fluxes.

Glampedakis & Kennefick (2002), Zoom and whirl: eccentric equatorial orbits around spinning black holes and their evolution under gravitational radiation reaction, Phys. Rev. D 66, 044002

arXiv version 1. Section II, eqs. 1–24: equatorial geodesic constants in terms of p and e, the quartic for L − aE, and the separatrix condition; Appendix A. We did not use the printed integrands for the periods; ours were derived separately and checked numerically.

Abramowicz & Kluźniak (2004), Epicyclic frequencies derived from the effective potential: simple and practical formulae, arXiv:astro-ph/0411709, Astrophys. Space Sci. doi:10.1007/s10509-005-1173-z

arXiv version 1. Eqs. 20–34: circular-orbit angular velocity, energy and angular momentum in a stationary axisymmetric metric, and the radial and vertical epicyclic frequencies from the effective potential.