Kerr Ringdown Frequency Calculator

What this calculator computes

After a black hole merger (or any strong perturbation), the remnant relaxes to a stationary Kerr black hole by emitting gravitational radiation in a superposition of quasinormal modes (QNMs). Each mode behaves like a damped oscillator with a complex angular frequency . The real part sets the oscillation frequency and the imaginary part sets how quickly the signal decays.

This tool estimates the dominant (“fundamental”) Kerr ringdown mode—commonly written —and reports:

• Ringdown frequency in Hz (the pitch of the oscillation),
• Quality factor (roughly how many cycles ring before the amplitude decays substantially),
• Damping time in seconds (the e-folding decay time of the amplitude).

Inputs

• Black hole mass in solar masses . The ringdown scales inversely with mass: heavier remnants ring at lower frequencies.
• Dimensionless spin (sometimes written ), constrained here to . It is defined by , where is the angular momentum. Higher spin generally increases the frequency and increases the quality factor (slower decay).

Equations used (semi-analytic fit)

The calculator uses a widely-cited approximate fit (originating with Echeverria and later refinements) for the fundamental mode. In terms of mass and dimensionless spin , the frequency is approximated by

The quality factor is approximated by

Given and , the damping time is computed as

Interpreting the results

• Frequency : This is the oscillation frequency of the dominant ringdown. As a rough observational guide, frequencies are in the band of ground-based detectors (LIGO/Virgo/KAGRA), while much lower frequencies are more relevant to space-based detectors (e.g., LISA). Actual detectability depends on source distance, detector noise, and signal amplitude.
• Quality factor : A higher means a narrower spectral line and a ringdown that persists for more cycles. For fixed mass, increasing spin increases .
• Damping time : This is the e-folding time of the amplitude decay (amplitude ). Larger implies a longer-lived ringdown.

Worked example (using the default inputs)

Suppose the remnant has mass and spin .

1. Compute the spin-dependent bracket term:
   , so .
2. The mass scaling factor for is about .
3. Frequency: .
4. Quality factor: .
5. Damping time:

Interpretation: a kHz ringdown is squarely in the ground-based band; the millisecond decay time indicates only a handful of visible cycles unless the signal is very loud.

How mass and spin change ringdown (quick comparison)

Assumptions & limitations (read before using)

• Single mode only: This calculator estimates only the fundamental QNM. Real ringdowns can contain overtones () and higher angular modes (), especially depending on mass ratio, inclination, and how soon after merger you start the ringdown model.
• Fit accuracy: The formulas are semi-analytic fits intended for quick estimates, not a substitute for full numerical relativity or modern ringdown fitting formulae spanning multiple modes and higher accuracy.
• Spin range: Inputs are restricted to . Very high spins can be sensitive to modeling choices; results near the upper end should be treated as indicative rather than definitive.
• Mass definition: The calculation uses the remnant’s gravitational mass as an input. In real analyses, the remnant mass differs from the total initial mass because energy is radiated away in gravitational waves.
• No cosmological redshift: The output is source-frame. Observed frequencies are redshifted by : and .
• Detectability not computed: Whether a detector can observe the ringdown depends on amplitude, distance, antenna pattern, noise curve, and analysis method; this tool reports only intrinsic mode scales ().

FAQ

Which ringdown mode is used?

The dominant fundamental Kerr mode , often the strongest component in binary black hole mergers.

Why does spin change the ringdown frequency and decay?

Spin changes the Kerr spacetime’s characteristic scales (e.g., the effective potential governing perturbations). Higher spin shifts the mode spectrum to higher real frequency and typically reduces damping (higher ).

Is this the same as the “chirp” frequency?

No. The chirp frequency describes the inspiral and merger evolution. Ringdown is the late-time, approximately exponentially damped oscillation of the remnant black hole.

How accurate are these numbers?

They are intended as first-order estimates. For precision work, use up-to-date QNM fitting formulae (or direct numerical relativity / perturbation solvers) and include multiple modes and redshift.

How do I compare this to detector bands?

Ground-based detectors are most sensitive from tens to a few thousand Hz; space-based detectors target much lower frequencies (milliHertz). Always compare against a specific detector noise curve and include redshift for distant sources.