Radius-to-Frequency Mapping in Neutron Stars

• RFM is the proposed relationship where lower radio frequencies originate at higher altitudes in neutron-star magnetospheres, defined by a power-law mapping.
• Observational techniques like aberration/retardation analyses in pulsars and FRBs test emission altitudes, often contradicting the simple nested-cone model.
• Recent studies suggest that propagation effects, fan-beam models, and multipolar magnetic fields may produce both RFM and anti-RFM behaviors, calling for revised theoretical frameworks.

Searching arXiv for the cited RFM papers to ground the article in the provided literature. Search query: radius-to-frequency mapping pulsar FRB aberration retardation fan beam multipole Radius-to-frequency mapping (RFM) is the postulated relation between radio frequency and emission altitude in neutron-star magnetospheres. In the classic pulsar picture, lower radio frequencies are thought to arise on field lines that extend further from the magnetic axis and therefore at larger magnetic colatitudes than high-frequency emission, so that the observed broadening of pulse profiles toward low frequency is interpreted geometrically (Rankin et al., 16 Nov 2025). In fast radio bursts (FRBs), an analogous construction links the observed frequency drift and burst-width evolution to a power-law relation between emission radius and frequency in a rotating neutron-star magnetosphere (Tong et al., 2021, Lyutikov, 2019). Recent work, however, has placed the simplest single-altitude, nested-cone form of RFM under sustained scrutiny, particularly through broadband aberration/retardation analyses, fan-beam models, and multipolar magnetic-field treatments (Rankin et al., 16 Nov 2025, Dyks et al., 2014, Jaroenjittichai et al., 15 Sep 2025, Qiu et al., 2023).

1. Classical formulation and geometric content

In the classic picture of pulsar emission, radiation at lower radio frequencies is thought to arise on field lines that extend further from the magnetic axis—and therefore at larger magnetic colatitudes—than high-frequency emission (Rankin et al., 16 Nov 2025). If the emission is tangential to dipolar field lines, a larger opening angle $\rho$ implies a higher altitude $h$ above the neutron-star surface, since $\rho \propto \sqrt{h/R_*}$, with $R_*$ the stellar radius (Rankin et al., 16 Nov 2025). Empirically, many pulsars show that the separation $W$ between outer conal components increases toward lower frequencies roughly as

$W(f) \simeq W_0 + A f^\eta ,$

with $\eta$ negative (Rankin et al., 16 Nov 2025).

A commonly adopted parametrization expresses RFM as a power-law relation between emission altitude and observing frequency,

$h(\nu) \propto \nu^{-k},$

with higher $\nu$ arising closer to the neutron star (Jaroenjittichai et al., 15 Sep 2025). In an equivalent FRB-oriented formulation,

$$\nu \propto r^{-\alpha} \quad \Rightarrow \quad r(\nu) \propto \nu^{-1/\alpha},$$

where $h$0 for curvature radiation and $h$1 for plasma-frequency-type emission under the assumptions summarized by Tong et al. (Tong et al., 2021). These formulations share the same central statement: higher frequencies come from smaller radii, while lower frequencies come from larger radii (Lyutikov, 2019).

In dipolar geometry, the half-opening angle of the emission cone obeys

$h$2

so that as $h$3, $h$4, $h$5, and the observed pulse width $h$6 narrows (Jaroenjittichai et al., 15 Sep 2025). In a pure-dipole curvature-radiation construction, the same idea yields

$h$7

recovering the standard narrower-at-higher-$h$8 version of RFM (Qiu et al., 2023).

This geometric interpretation has historically been influential because it converts observed profile-width evolution into an inferred altitude stratification. A plausible implication is that RFM operates as a bridge between phenomenological pulse morphology and magnetospheric structure, but the later literature shows that this bridge depends strongly on assumptions about field geometry, propagation, and how emission regions are sampled (Jaroenjittichai et al., 15 Sep 2025, Qiu et al., 2023).

2. Aberration/retardation as a direct emission-height test

The most direct physical test of RFM in pulsars uses aberration/retardation (A/R). Blaskiewicz, Cordes & Wasserman showed that aberration and retardation shift a conal pair’s midpoint toward earlier phase by an amount

$h$9

in radians, or equivalently

$\rho \propto \sqrt{h/R_*}$0

(Rankin et al., 16 Nov 2025). Dyks, Rudak & Harding corrected the original BCW factor of two, so that one measures the longitude offset $\rho \propto \sqrt{h/R_*}$1 between a core-marked fiducial and the conal midpoint, then uses

$\rho \propto \sqrt{h/R_*}$2

since $\rho \propto \sqrt{h/R_*}$3 for an early shift (Rankin et al., 16 Nov 2025). Equivalently,

$\rho \propto \sqrt{h/R_*}$4

The methodology depends on identifying a stable fiducial longitude. For PSR B1237+25, the $\rho \propto \sqrt{h/R_*}$5 core component marks the magnetic axis longitude and coincides both with the linear polarization angle inflection point and the zero-crossing of its antisymmetric circular signature, which makes it possible to estimate emission heights over a very broad band using A/R (Rankin et al., 16 Nov 2025). Rankin et al. confirm that in B1237+25 the core centre, defined through mode-separated PPA inflection and Stokes-$\rho \propto \sqrt{h/R_*}$6 zero, reliably marks the true fiducial longitude across 25 MHz–5 GHz (Rankin et al., 16 Nov 2025).

A broader observational program applies similar logic at population scale. In the Blaskiewicz–Cordes–Wasserman model, the pulse-profile midpoint lags the polarization-angle inflexion by $\rho \propto \sqrt{h/R_*}$7, giving

$\rho \propto \sqrt{h/R_*}$8

with $\rho \propto \sqrt{h/R_*}$9 defined as the phase offset between the RVM inflexion and the 50\% intensity midpoint (Jaroenjittichai et al., 15 Sep 2025). Such A/R estimates are attractive because they attempt to measure a physical emission altitude rather than an altitude inferred from an assumed last-open-field-line geometry.

This distinction is central to the modern debate. The longstanding last-open-field-line model of RFM rests on an unverified assumption that conal emission always sits on field lines bordering the open-zone rim, whereas A/R is presented as a directly physical height measure that does not track last-open-field-line heights (Rankin et al., 16 Nov 2025).

3. Broadband pulsar tests: PSR B1237+25 and the challenge to simple RFM

PSR B1237+25 is perhaps the canonical example of a pulsar with a core/double cone profile, and Rankin et al. assembled more than a dozen high-quality profiles from 30 MHz to 5 GHz from Arecibo, LOFAR, LWA, MWA, and NenuFAR for a broadband A/R analysis (Rankin et al., 16 Nov 2025). Each total profile was decomposed into eight Gaussians using Kramer’s bfit package: two Gaussians for the often-asymmetric core, two for the inner cone, two for the outer cone, and two to absorb residual weak features (Rankin et al., 16 Nov 2025). The peak positions and half-power widths of each fitted Gaussian pair yield $R_*$0 and $R_*$1, and the midpoint $R_*$2 is measured against the core centre; fit quality is $R_*$3 at high S/N, with residuals typically $R_*$4 of peak (Rankin et al., 16 Nov 2025).

The outer-cone measurements show uniformly negative A/R shifts over 30 MHz–5 GHz, with $R_*$5 typically between about $R_*$6 and $R_*$7, equivalent delays of roughly $R_*$8 to $R_*$9 ms, and emission heights between about 215 km and 497 km for the tabulated high-quality measurements (Rankin et al., 16 Nov 2025). Averaging over all bands gives $W$0 km for the outer cone and $W$1 km for the inner cone by the peak method, while using component widths yields $W$2 km and $W$3 km (Rankin et al., 16 Nov 2025). All four estimates agree to within their formal uncertainties of about 70–100 km (Rankin et al., 16 Nov 2025).

A concise summary of the outer-cone A/R measurements is given below.

Frequency range	Measured behavior	Derived height range
4460–1177 MHz	$W$4 to $W$5	$W$6 km to $W$7 km
327–120 MHz	$W$8 to $W$9	$W(f) \simeq W_0 + A f^\eta ,$0 km to $W(f) \simeq W_0 + A f^\eta ,$1 km
79–30 MHz	$W(f) \simeq W_0 + A f^\eta ,$2 to $W(f) \simeq W_0 + A f^\eta ,$3	$W(f) \simeq W_0 + A f^\eta ,$4 km to $W(f) \simeq W_0 + A f^\eta ,$5 km

The crucial result is negative: the analysis finds no evidence whatsoever for an emission height increase with wavelength, the so-called radius-to-frequency mapping, and no significant difference in A/R effect between the outer and inner cones (Rankin et al., 16 Nov 2025). Were the last-open-field-line phenomenology correct, the inferred height would climb from about 200 km at 5 GHz to about 600 km at 50 MHz, a $W(f) \simeq W_0 + A f^\eta ,$6 km change that would be $W(f) \simeq W_0 + A f^\eta ,$7 relative to the A/R measurement error of about 100 km; no such trend is seen (Rankin et al., 16 Nov 2025). The measured A/R shifts are instead uniformly negative, with conal midpoints leading the core by approximately $W(f) \simeq W_0 + A f^\eta ,$8 or about 2 ms at all frequencies (Rankin et al., 16 Nov 2025).

Rankin et al. explicitly address possible systematics. Mode-segregated polarimetry shows the core centre reliable from 25 MHz to 1.4 GHz; scattering at 30 MHz is $W(f) \simeq W_0 + A f^\eta ,$9, too small to bias $\eta$0 by much more than $\eta$1; and the Gaussian fits are repeatable across six observatories (Rankin et al., 16 Nov 2025). Statistically, the absence of any trend in $\eta$2 at the $\eta$3 level is presented as robustly falsifying simple RFM in this star (Rankin et al., 16 Nov 2025).

This result has broader significance because B1237+25 had long served as a strong phenomenological case for RFM: the outer half-power separation doubles from about $\eta$4 at 5 GHz to about $\eta$5 at 50 MHz, suggesting a geometric RFM that would place low-frequency emission hundreds of kilometres above high-frequency emission (Rankin et al., 16 Nov 2025). The A/R analysis shows that profile-width evolution need not imply altitude evolution.

4. RFM in FRBs: drift rates, timescales, and width evolution

In FRB applications, RFM is formulated as a kinematic mapping in a rotating neutron-star magnetosphere rather than primarily as a pulse-width diagnostic. Tong et al. assume that the FRB engine is a rotating neutron-star magnetosphere with predominantly dipolar geometry, that a sudden flux tube is ignited and particles stream out along a narrow bundle of field lines, and that the emission mechanism sets an instantaneous local plasma frequency or curvature-radiation frequency (Tong et al., 2021). With

$\eta$6

they derive

$\eta$7

and hence

$\eta$8

(Tong et al., 2021). The drifting timescale

$\eta$9

then obeys

$h(\nu) \propto \nu^{-k},$0

For curvature radiation, $h(\nu) \propto \nu^{-k},$1 gives $h(\nu) \propto \nu^{-k},$2 and $h(\nu) \propto \nu^{-k},$3; for plasma-frequency emission, $h(\nu) \propto \nu^{-k},$4 gives $h(\nu) \propto \nu^{-k},$5 and $h(\nu) \propto \nu^{-k},$6 (Tong et al., 2021). In a narrow observing band with $h(\nu) \propto \nu^{-k},$7 and $h(\nu) \propto \nu^{-k},$8, one often writes approximately

$h(\nu) \propto \nu^{-k},$9

(Tong et al., 2021).

A related construction by Lyutikov and Lorimer assumes that an emission patch propagates along dipolar magnetic field lines, producing coherent emission with frequency, direction, and polarization defined by the local magnetic field (Lyutikov, 2019). In that formulation one writes

$\nu$0

with $\nu$1–3 for various plasma- or field-line scalings, and obtains generically

$\nu$2

matching both numerically and parametrically the rates observed in FRBs; more complicated behavior is also possible (Lyutikov, 2019).

The width-frequency prediction in the FRB version of RFM follows from pulsar-style beam geometry:

$\nu$3

For curvature radiation, this gives $\nu$4 (Tong et al., 2021). Tong et al. connect this to specific examples: FRB 121102 has typical widths 2–9 ms at 1.4 GHz versus $\nu$5 ms at 4.5 GHz, while FRB 20180916B has 40–160 ms at 150 MHz versus 2–3 ms at 1.7 GHz (Tong et al., 2021). They further argue that the longer intrinsic drifting-timescale $\nu$6 at low $\nu$7 broadens burst envelopes (Tong et al., 2021).

Aberration and twist can complicate the observed drift. In a rapidly rotating magnetosphere, co-rotation bends the emission beam forward by

$\nu$8

with an arrival-time shift

$\nu$9

and this may reverse the sign of $$\nu \propto r^{-\alpha} \quad \Rightarrow \quad r(\nu) \propto \nu^{-1/\alpha},$$0 so that one might observe upward drifting if aberration dominates over retardation (Tong et al., 2021). In magnetars, a field-line twist $$\nu \propto r^{-\alpha} \quad \Rightarrow \quad r(\nu) \propto \nu^{-1/\alpha},$$1 introduces an additional time shift

$$\nu \propto r^{-\alpha} \quad \Rightarrow \quad r(\nu) \propto \nu^{-1/\alpha},$$2

which can be positive or negative, again allowing both downward and upward $$\nu \propto r^{-\alpha} \quad \Rightarrow \quad r(\nu) \propto \nu^{-1/\alpha},$$3–$$\nu \propto r^{-\alpha} \quad \Rightarrow \quad r(\nu) \propto \nu^{-1/\alpha},$$4 drifts (Tong et al., 2021).

An observationally compact estimate comes from the characteristic scale

$$\nu \propto r^{-\alpha} \quad \Rightarrow \quad r(\nu) \propto \nu^{-1/\alpha},$$5

Using drift rates of $$\nu \propto r^{-\alpha} \quad \Rightarrow \quad r(\nu) \propto \nu^{-1/\alpha},$$6 MHz ms$$\nu \propto r^{-\alpha} \quad \Rightarrow \quad r(\nu) \propto \nu^{-1/\alpha},$$7 at frequencies around GHz yields a physical size of a few $$\nu \propto r^{-\alpha} \quad \Rightarrow \quad r(\nu) \propto \nu^{-1/\alpha},$$8 cm, consistent with the hypothesis of FRB origin in neutron-star magnetospheres (Lyutikov, 2019). This places FRB RFM in a regime analogous in geometry, but not in environment, to both pulsar RFM and Solar type-III bursts (Lyutikov, 2019).

5. Alternatives to simple nested-cone RFM

Several recent studies argue that pulse-profile evolution commonly attributed to RFM can arise without a monotonic single-height mapping.

Dyks & Rudak develop a stream-based model in which emission is produced by a small number of narrow plasma streams confined to limited azimuthal intervals, each projecting on the sky as a fan-shaped wedge of emission (Dyks et al., 2014). In this framework one assumes a monotonically changing local spectrum along each stream and introduces an angular spectral gradient

$$\nu \propto r^{-\alpha} \quad \Rightarrow \quad r(\nu) \propto \nu^{-1/\alpha},$$9

In the flat-pulsar approximation, the pulse longitude of a component is

$h$00

so that

$h$01

For two symmetric streams at $h$02, the half-separation is

$h$03

implying

$h$04

This geometry naturally predicts weaker apparent RFM for inner pairs because inner streams have smaller $h$05, so $h$06 is smaller (Dyks et al., 2014).

The same model is presented as explaining why millisecond pulsars, despite more strongly flaring magnetic field lines, do not exhibit as strong RFM as normal pulsars (Dyks et al., 2014). In angular terms, the angular spectral gradient $h$07 is small in millisecond pulsars, so $h$08 is suppressed and the fan-beam contours for different $h$09 overlap heavily (Dyks et al., 2014). Dyks & Rudak therefore argue that the apparent RFM, including its reduced strength for the inner pair of components, can be explained with no reference to the flaring boundary of the polar magnetic flux tube (Dyks et al., 2014).

A wider observational study using 704–4032 MHz Murriyang observations of over 100 pulsars likewise finds results difficult to reconcile with the classic single-altitude RFM picture (Jaroenjittichai et al., 15 Sep 2025). From 128 pulsars, the distribution of component separations has median

$h$10

indicating almost no net frequency evolution of component centroids, while the individual Gaussian widths yield

$h$11

and the 20\%-edge separation gives

$h$12

(Jaroenjittichai et al., 15 Sep 2025). The Gaussian decomposition shows that the peak locations of the components vary little with frequency, whereas the component widths do, in general, narrow with increasing frequency (Jaroenjittichai et al., 15 Sep 2025). The authors therefore argue that propagation effects are responsible for the width evolution of the profiles rather than emission height (Jaroenjittichai et al., 15 Sep 2025).

Population-level A/R trends in that study are similarly non-universal. Of 157 pulsars, 83 show $h$13 increasing with $h$14 and 75 decreasing, implying no universal sign for the height index

$h$15

while the weighted density peak of $h$16 lies at about 100 km with no clear $h$17-dependence (Jaroenjittichai et al., 15 Sep 2025). The standard RFM picture—single altitude, nested cones—therefore cannot account for the diversity of width and height behaviors in that sample (Jaroenjittichai et al., 15 Sep 2025).

These results do not eliminate every form of frequency-dependent magnetospheric geometry. Rather, they shift attention toward broadband, multi-altitude, plasma-guided streams with strong propagation effects, including refraction and birefringence (Jaroenjittichai et al., 15 Sep 2025). This suggests that what is often labeled RFM may in many cases be an observational composite of geometry and propagation rather than a direct map from frequency to a single radius.

6. Multipoles, anti-RFM, and broader theoretical implications

The presence of multipole magnetic structure modifies both the beam geometry and the interpretation of frequency evolution. For an axisymmetric potential multipole of order $h$18, the small-angle beam-opening relation becomes

$h$19

so higher multipoles produce wider beams at the same emission polar angle than a pure dipole does (Qiu et al., 2023). Qiu et al. argue that this may account for the increasing pulse width at higher frequency of pulsars, described as anti-radius-to-frequency mapping (Qiu et al., 2023).

In a composite dipole plus quadrupole field with aligned axes, the beam opening becomes

$h$20

where $h$21 is the surface-field strength ratio of quadrupole to dipole (Qiu et al., 2023). Introducing a dimensionless frequency variable yields

$h$22

so over some intermediate range the beam can broaden with increasing frequency, producing anti-RFM (Qiu et al., 2023). If the quadrupole is tilted relative to the dipole, then

$h$23

and depending on the sign of the phase-dependent term the width can either decrease or increase with frequency, allowing both RFM and anti-RFM in different pulse components or at different phases (Qiu et al., 2023).

The same paper emphasizes that the classical rotating-vector-model expression for polarization position angle remains unchanged in an axisymmetric potential multipole. Only the fitted values of $h$24 and $h$25 change, while $h$26 and $h$27 remain fixed by the spin-and-sight geometry (Qiu et al., 2023). In magnetars, sudden changes in the slope or sign of the PA swing can then be interpreted as the episodic appearance or disappearance of a multipole component (Qiu et al., 2023). The authors explicitly extend this line of reasoning to repeating FRBs, where burst-to-burst changes in PA swings are described as analogous to magnetar multipole episodes (Qiu et al., 2023).

The implications for RFM theory are substantial. If emission originates from a nearly constant altitude of 200–400 km over three decades of frequency, as found for B1237+25, then profile-width evolution must arise from other effects, such as frequency-dependent magnetospheric refraction of the ordinary mode or systematic spectral variations across the polar fluxtube (Rankin et al., 16 Nov 2025). More generally, analyses that assume $h$28—including beam-radius statistics and population synthesis—are called into question by direct A/R tests that do not recover a monotonic height increase toward low frequency (Rankin et al., 16 Nov 2025).

Across pulsars and FRBs, RFM therefore occupies a dual role. It remains a useful zeroth-order guide in models where frequency links to altitude and the magnetosphere is approximately dipolar (Jaroenjittichai et al., 15 Sep 2025, Tong et al., 2021). At the same time, the simplest version of the paradigm—single altitude, nested cones, monotonic decrease of height with increasing observing frequency—fails to capture a wide range of observed behaviors, including constant A/R heights across broad bands, fixed component centroids, pulse broadening at higher frequency, and multipole-induced anti-RFM (Rankin et al., 16 Nov 2025, Jaroenjittichai et al., 15 Sep 2025, Qiu et al., 2023). The current literature therefore frames RFM less as a settled law than as a family of frequency-geometry hypotheses whose validity is source-dependent and strongly conditioned by plasma propagation, beam topology, and magnetic-field structure.