Zoom–Whirl Orbit Taxonomy

• Zoom–whirl taxonomy is a classification scheme for strong-field black-hole orbits that encodes orbital geometry via rational frequency ratios and integer labels (z, w, v).
• It organizes orbital morphology into multi-leaf structures, capturing zoom and whirl phases to enable effective gravitational waveform analysis.
• The taxonomy extends across various gravitational models, including Kerr, deformed Schwarzschild, and scalar–tensor scenarios, aiding strong-field research.

Zoom–whirl taxonomy is a classification scheme for strong-field black-hole orbits in which the geometry of a bound trajectory is encoded by a rational relation between its radial and azimuthal frequencies. In the equatorial Kerr problem, a geodesic closes exactly after $z$ radial oscillations if and only if

$$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$

and the rational number is written in lowest terms as $q=w+v/z$, with $w,v,z\in\mathbb{N}$ and $\gcd(v,z)=1$. The three integers specify, respectively, the number of distinct radial leaves, the number of extra near-periastron windings, and the leaf-ordering rule; together they organize both periodic and aperiodic motion because every orbit is in or near the periodic set (0802.0459).

1. Rational-frequency correspondence

The core of the taxonomy is the correspondence between periodic orbits and rational numbers. In the original Kerr construction, every bound equatorial geodesic has two fundamental frequencies: the radial oscillation frequency $\Omega_r\equiv 2\pi/T_r$ and the azimuthal advance averaged over one radial cycle, $$\Omega_\phi\equiv \langle d\phi/dt\rangle_r=\Delta\phi_r/T_r$$. Periodicity is equivalent to commensurability of these frequencies, and the rational label is

$q=w+\frac{v}{z}.$

Here $z$ is the number of distinct radial leaves before closure, $w$ is the integer number of additional $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$0 windings near periastron between successive apastra, and $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$1 is the vertex-skip, with $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$2 and $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$3 (0802.0459).

The same three-integer structure recurs across later work on deformed Schwarzschild, Kerr–Sen, $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$4-metric, scalar–tensor, Euler–Heisenberg plus perfect fluid dark matter, asymptotically safe gravity, and quintessence backgrounds. In those settings the periodicity condition is again expressed through a rational frequency ratio or an equivalent apsidal-angle condition, with the orbit labeled by $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$5 (Hua et al., 2 Jan 2026, Liu et al., 2018, Zhang et al., 18 Nov 2025, Gogoi et al., 22 Jun 2026, Gogoi et al., 13 Apr 2026, Kumar et al., 8 May 2026, Ali, 22 Jun 2026).

The notation is not fully uniform across the literature. In the Kerr taxonomy, $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$6 (0802.0459). Some later summaries write $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$7 (0907.0671), while others introduce $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$8 and separately define $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$9 (Gogoi et al., 22 Jun 2026). This suggests that the invariant content of the taxonomy lies less in the symbol itself than in the rational decomposition of the apsidal advance.

2. Geodesic basis, turning points, and the separatrix

For equatorial Kerr geodesics in Boyer–Lindquist coordinates, the first-order equations may be written as

$q=w+v/z$0

with

$q=w+v/z$1

and

$q=w+v/z$2

The turning points $q=w+v/z$3 and $q=w+v/z$4 determine the azimuthal advance per radial oscillation,

$q=w+v/z$5

and therefore the rational label $q=w+v/z$6 (0802.0459).

A complementary Schwarzschild presentation emphasizes the effective potential

$q=w+v/z$7

so that

$q=w+v/z$8

Bound motion oscillates between the real roots $q=w+v/z$9 and $w,v,z\in\mathbb{N}$0 of $w,v,z\in\mathbb{N}$1. Circular orbits satisfy $w,v,z\in\mathbb{N}$2 and $w,v,z\in\mathbb{N}$3, while the separatrix is reached when the orbit asymptotes to the unstable circular orbit at the top of the potential barrier (Pakiela et al., 2023).

The separatrix is the dynamical origin of large whirl counts. In the conservative description summarized for black-hole binaries, the homoclinic, infinite-whirl separatrix satisfies $w,v,z\in\mathbb{N}$4 and $w,v,z\in\mathbb{N}$5, and $w,v,z\in\mathbb{N}$6 there. Near the homoclinic orbit, $w,v,z\in\mathbb{N}$7 grows logarithmically as one approaches the separatrix energy (0907.0671). In the Kerr periodic-orbit taxonomy, for fixed spin $w,v,z\in\mathbb{N}$8 and angular momentum $w,v,z\in\mathbb{N}$9, $\gcd(v,z)=1$0 grows monotonically with orbital energy $\gcd(v,z)=1$1 or eccentricity. As $\gcd(v,z)=1$2 decreases toward the innermost bound circular orbit, one reaches a threshold $\gcd(v,z)=1$3 at which the minimum $\gcd(v,z)=1$4 exceeds unity. Below this critical $\gcd(v,z)=1$5, zoom–whirl motion with $\gcd(v,z)=1$6 is unavoidable; in Schwarzschild, $\gcd(v,z)=1$7 (0802.0459).

3. Morphology of orbit families

The integers $\gcd(v,z)=1$8 have an immediate geometric interpretation. The orbit closes after $\gcd(v,z)=1$9 radial oscillations, $\Omega_r\equiv 2\pi/T_r$0 counts extra full windings near periastron, and $\Omega_r\equiv 2\pi/T_r$1 specifies how the next apastron is connected to the previous one. When $\Omega_r\equiv 2\pi/T_r$2, the orbit is a pure multi-leaf precession; when $\Omega_r\equiv 2\pi/T_r$3, it exhibits true zoom–whirl structure (0802.0459).

A central implication of the strong-field taxonomy is that the simple precessing ellipse familiar from planetary orbits is not allowed in the strong-field regime. Instead, eccentric orbits trace precessions of multi-leaf clovers in the final stages of inspiral (0802.0459). Later papers retain the same geometric language: leaves, petals, lobes, vertices, zoom arcs, and periapsis loops (Gold et al., 2012, Liu et al., 2018, Gogoi et al., 22 Jun 2026).

$\Omega_r\equiv 2\pi/T_r$4	$\Omega_r\equiv 2\pi/T_r$5	Geometry
$\Omega_r\equiv 2\pi/T_r$6	$\Omega_r\equiv 2\pi/T_r$7	single-leaf, one-whirl orbit
$\Omega_r\equiv 2\pi/T_r$8	$\Omega_r\equiv 2\pi/T_r$9	two-leaf, one-whirl clover
$$\Omega_\phi\equiv \langle d\phi/dt\rangle_r=\Delta\phi_r/T_r$$0	$$\Omega_\phi\equiv \langle d\phi/dt\rangle_r=\Delta\phi_r/T_r$$1	three-leaf, zero-whirl clover
$$\Omega_\phi\equiv \langle d\phi/dt\rangle_r=\Delta\phi_r/T_r$$2	$$\Omega_\phi\equiv \langle d\phi/dt\rangle_r=\Delta\phi_r/T_r$$3	one whirl, triple zoom
$$\Omega_\phi\equiv \langle d\phi/dt\rangle_r=\Delta\phi_r/T_r$$4	$$\Omega_\phi\equiv \langle d\phi/dt\rangle_r=\Delta\phi_r/T_r$$5	two whirls, double zoom
$$\Omega_\phi\equiv \langle d\phi/dt\rangle_r=\Delta\phi_r/T_r$$6	$$\Omega_\phi\equiv \langle d\phi/dt\rangle_r=\Delta\phi_r/T_r$$7	four zoom lobes, one whirl, phase shift $$\Omega_\phi\equiv \langle d\phi/dt\rangle_r=\Delta\phi_r/T_r$$8

The low-order examples are the basic alphabet of the taxonomy. In Kerr, $$\Omega_\phi\equiv \langle d\phi/dt\rangle_r=\Delta\phi_r/T_r$$9 corresponds to a single-leaf orbit that reaches one apastron, whirls once near periastron, and returns; $q=w+\frac{v}{z}.$0 is a two-petal clover with one extra $q=w+\frac{v}{z}.$1 twist at each periastron; $q=w+\frac{v}{z}.$2 is a three-leaf zero-whirl clover (0802.0459). In scalar–tensor and other deformed backgrounds, the same families reappear as explicit sample morphologies, such as $q=w+\frac{v}{z}.$3, $q=w+\frac{v}{z}.$4, and $q=w+\frac{v}{z}.$5 (Gogoi et al., 22 Jun 2026).

A common misconception is that zoom–whirl structure requires large integer $q=w+\frac{v}{z}.$6. The taxonomy shows otherwise: $q=w+\frac{v}{z}.$7 already allows strongly relativistic multi-leaf precession, while $q=w+\frac{v}{z}.$8 or $q=w+\frac{v}{z}.$9 captures the onset of genuine whirl behavior. High-order whirls arise as the separatrix is approached, but low-order periodic families already encode much of the strong-field morphology (0802.0459).

4. Periodic skeleton and the “periodic table”

The taxonomy does not classify only exactly periodic motion. Generic bound orbits typically have irrational frequency ratio $z$0 and therefore never close, but the density of the rationals implies that any irrational can be approximated arbitrarily well by a rational. Since numerical and observational work is always performed at finite precision, any aperiodic Kerr orbit is effectively indistinguishable from some periodic skeleton orbit (0802.0459).

This observation motivates the “periodic table” viewpoint. For fixed $z$1 and black-hole spin $z$2, one plots $z$3 versus eccentricity or energy and marks rational values corresponding to low $z$4. The resulting grid organizes orbit shapes by increasing $z$5 or $z$6, while empty slots mark rationals that are dynamically forbidden because they lie outside the allowed $z$7 range (0802.0459).

The periodic skeleton has direct computational uses. In practice one expands waveforms and self-forces in Fourier series over the periodic-orbit basis; low-order leaves dominate the early inspiral, while high-order rationals fill in fine-grained structure. For extreme-mass-ratio inspirals, an inspiral can be viewed as an adiabatic drift of $z$8 that traverses the periodic-orbit skeleton. Closed-orbit basis functions yield sparse spectrograms with dominant lines at integer multiples of $z$9 and $w$0, and one may build a library of low-$w$1 templates and interpolate for intermediate parameters (0802.0459).

A pedagogical Schwarzschild treatment recasts the same structure in effective-potential language and uses the simplified family $w$2 to show how increasingly elaborate orbits cluster near the separatrix (Pakiela et al., 2023). This suggests that the taxonomy is simultaneously a topological classification, a dynamical approximation scheme, and a bookkeeping device for waveform construction.

5. Dissipation, comparable-mass binaries, and scattering

Zoom–whirl behavior is not confined to conservative test-particle motion. Full numerical relativity identifies it in comparable-mass black-hole binaries despite gravitational-wave dissipation. Larger mass ratios allow the pair to spend longer in orbit before merger and therefore display more zooms and whirls, while larger spins enhance zoom-whirliness. An important consequence is that eccentric binaries can merge during a whirl phase, before enough angular momentum has been lost to circularize. In the waveform, zooms appear as quiet phases and whirls as louder glitches (0907.0671).

A more detailed numerical-relativity taxonomy for non-spinning equal and unequal mass binaries parameterizes the initial data by total mass scale $w$3, symmetric mass ratio $w$4, initial separation $w$5, linear momentum $w$6 relative to the quasi-circular value $w$7, and shooting angle $w$8. In that framework, the zoom phase is the outward swing from periastron to apastron or infinity, while the whirl phase is the near-periastron segment of one or more near-circular revolutions at roughly constant small radius $w$9. Because of gravitational-wave damping, the whirl number is finite in numerical relativity, typically $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$00 for mass ratios up to $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$01 (Gold et al., 2012).

The same study distinguishes bound elliptic and unbound hyperbolic regimes. A necessary condition for binding is $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$02, with $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$03 determined by zero initial binding energy in the head-off limit; for $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$04 and $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$05, $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$06 and $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$07. For $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$08, a critical shooting angle $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$09 separates dynamical capture from fly-by. Zoom–whirl behavior appears for eccentricities as low as $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$10, and the resulting waveforms show a rich structure that effectively breaks degeneracies in parameter space and improves parameter estimation (Gold et al., 2012).

High-energy scattering of like-charged black holes introduces an impact-parameter taxonomy. At fixed initial Lorentz factor $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$11, the outcomes are classified by the immediate-merger threshold $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$12 and the scattering threshold $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$13: immediate merger for $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$14, zoom–whirl behavior for $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$15, and scattering for $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$16. In ADM-mass units, both thresholds decrease with charge-to-mass ratio $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$17, but when normalized by the sum of irreducible masses they become approximately universal: $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$18 Within the zoom–whirl window $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$19, where $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$20, the whirl count increases from a few whirls near the lower edge to $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$21 logarithmically as $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$22 (Smith et al., 2024).

6. Extensions to deformed spacetimes and waveform diagnostics

Subsequent work generalizes the zoom–whirl taxonomy to a wide range of non-Kerr backgrounds while preserving the same rational classification. In a deformed Schwarzschild spacetime, increasing the deformation parameter $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$23 lowers $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$24 and raises $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$25 at fixed $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$26, so $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$27 grows and typically both $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$28 and the fractional part $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$29 increase. Above a critical $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$30, a new inner region of bound motion appears, characterized by large $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$31 and large $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$32; circular orbits can disappear when the deformation is large enough (Hua et al., 2 Jan 2026).

Around a Kerr–Sen black hole, bound periodic zoom–whirl orbits occur for $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$33, and increasing the dilaton–axion charge parameter $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$34 lowers both $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$35 and $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$36. For a fixed $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$37, the required energy drops as $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$38 grows, so the same multi-leaf clover can be supported at lower $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$39 (Liu et al., 2018). In the $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$40-metric, numerical solutions show monotonic growth of $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$41, $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$42, and the corresponding critical angular momenta as $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$43 increases from $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$44 to $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$45; deviations from $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$46 induce phase shifts and amplitude modulations correlated with changes in zoom–whirl structure (Zhang et al., 18 Nov 2025).

Scalar and environmental couplings alter the taxonomy through systematic shifts of the separatrix and the turning-point structure. In Freund–Nambu scalar–tensor gravity, the total precession per radial cycle satisfies $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$47 and $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$48 at fixed $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$49, so stronger geometric coupling or stronger scalar–particle coupling enhances relativistic periapsis precession. Increasing $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$50 pushes the separatrix to larger $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$51 for the same $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$52, while positive $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$53 pulls $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$54 to smaller values; as $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$55, the precession parameter diverges (Gogoi et al., 22 Jun 2026). In Euler–Heisenberg black holes surrounded by perfect fluid dark matter, increasing the PFDM parameter $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$56 lowers the depth of the potential well and the height of its barrier, moves $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$57 and $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$58 outward, and decreases $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$59 and $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$60; PFDM suppresses the waveform amplitude, while QED corrections enhance high-frequency structure generated near the horizon (Gogoi et al., 13 Apr 2026).

Other extensions make similar use of the same classification. In regular black holes in asymptotically safe gravity, increasing the quantum parameter $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$61 decreases $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$62, $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$63, $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$64, $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$65, and $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$66, allowing orbits to penetrate deeper and boosting $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$67 for fixed $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$68 or $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$69; peak strain increases monotonically with $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$70, and the spectra peak in the millihertz band relevant for LISA, Taiji, and TianQin (Kumar et al., 8 May 2026). For a magnetically charged black hole surrounded by quintessence, increasing the quintessence parameter $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$71 increases $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$72, $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$73, and the corresponding critical angular momenta, while at fixed $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$74 the energies of selected periodic orbits decrease and at fixed $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$75 the required angular momenta increase; the topological class is preserved even as zoom lobes and whirl radii shift (Ali, 22 Jun 2026).

Across these deformed and matter-coupled spacetimes, the recurring result is that the integers $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$76 remain the topological labels, while deformation parameters move the loci of periodic families in the $$\frac{\Omega_\phi}{\Omega_r}=\frac{\Delta\phi_r}{2\pi}=1+q,\qquad q\in\mathbb{Q},$$77 plane and imprint phase evolution, burst timing, harmonic content, or amplitude modulation on the associated gravitational radiation (Hua et al., 2 Jan 2026, Zhang et al., 18 Nov 2025, Gogoi et al., 13 Apr 2026, Ali, 22 Jun 2026). A plausible implication is that zoom–whirl taxonomy functions as a transportable language for comparing strong-field orbital structure across otherwise disparate black-hole models.