Spin-Down, Braking Indices, and Alignment

A braking index is a local curvature of a pulsar’s rotational phase history, not a direct label on one radiation mechanism. A constant vacuum-dipole coefficient gives $n=3$, but plasma torque, magnetic evolution, changing inclination, variable moment of inertia, timing noise, and glitch recovery all enter the same second derivative. Measuring $n$ therefore requires a long, phase-connected timing likelihood and an explicit stochastic and glitch model. The ATNF catalogue is useful for defining and cross-referencing a sample, but its catalog rows are not substitutes for the underlying pulse arrival times.

Required background. Frames, Coframes, and Anholonomy supplies the local rotation and field components used in torque models.

Helpful background. Flat-Background Hilbert Stress-Energy from Metric Variation supplies energy and angular-momentum flux; The Einstein Equation supplies relativistic timing and moment-of-inertia corrections; Relativistic Matter and Compact Objects supplies internal coupling; Computational Relativity supplies statistical and reproducibility checks.

Torque exponent versus measured braking index

Write a general secular torque as

$$\dot\Omega=-K(t)\Omega^q.$$

Taking one derivative gives

$$n\equiv\frac{\Omega\ddot\Omega}{\dot\Omega^2} =q+\frac{\Omega}{\dot\Omega}\frac{\dot K}{K}.$$

Only constant $K$ makes the measured braking index equal to the torque exponent $q$. In a dipole-like model, $K$ contains the magnetic moment, inclination, moment of inertia, and magnetospheric current factor. A growing dipole, motion toward greater torque, or decoupling that reduces the participating $I$ can lower $n$ below $q$; the opposite evolution can raise it. Several effects can produce the same $n$, so alignment is not inferred from $n$ alone.

The characteristic age

$$\tau_c=-\frac{\Omega}{2\dot\Omega}=\frac{P}{2\dot P}$$

equals an actual age only under additional assumptions: $q=3$, constant $K$, and a birth spin much faster than the current spin. It remains a convenient catalog coordinate when labeled as characteristic rather than chronological age.

Integrating the constant-$K$ law rather than differentiating it gives, for $q\ne1$,

$$t=\frac{P}{(q-1)\dot P} \left[1-\left(\frac{P_0}{P}\right)^{q-1}\right],$$

where $P_0$ is the birth period and $P,\dot P$ are their present values. This exposes both assumptions behind $\tau_c$: $q=3$ and $P_0/P\ll1$. For example, $P_0=P/2$ gives $t=3\tau_c/4$ even with a perfectly constant dipole coefficient. A measured braking index that changes over time cannot simply be inserted as one constant $q$ for the star’s entire history.

Phase connection and nuisance dynamics

Pulse times of arrival are fitted with a phase model such as

$$\phi(t)=\phi_0+\nu(t-t_0) +\frac12\dot\nu(t-t_0)^2 +\frac16\ddot\nu(t-t_0)^3+\cdots .$$

Every rotation must be counted across the data span. A low-frequency red-noise process can covary strongly with $\ddot\nu$; a glitch contributes steps in $\nu$ and $\dot\nu$ plus recoveries that can mimic secular curvature. Pulse- profile evolution, dispersion changes, and mode switching can shift measured arrival times or the torque state.

The Crab record makes the distinction concrete: interglitch and long-term indices differ because glitches leave cumulative changes in spin-down. The data segmentation and recovery treatment are described in Lyne et al. 2015, §§2–5 and Figs. 2–7. More broadly, a Bayesian Parkes analysis modeled red timing noise and tested braking curvature against long glitch recoveries in Parthasarathy et al. 2020, §§2–5, Tables 1–3. These are likelihood-dependent measurements, not values recoverable from $P$ and $\dot P$ alone.

Named catalogue and selection boundary

Evidence cutoff: 2026-08-18. The named catalogue release is the official ATNF PSRCAT v2.8.1 archive, released 2026-06-18 with 4,393 entries according to the Australia Telescope National Facility 2026, v2.8.1 history row. The frozen tar archive, not the mutable web-query result, is the dataset version for a reproducible catalog selection. Its field definitions, derived-distance precedence, references, and query behavior are specified by Manchester et al. 2005, §§2–4 and Tables 1–2 and the release documentation.

A braking-index sample should state whether it includes isolated rotation-powered pulsars only, how binaries, accretors, magnetars, and cluster accelerations are handled, the minimum coherent baseline, required uncertainty on $\ddot\nu$, and the treatment of glitches and timing noise. Catalogue nulls are missing or unpublished information, not upper limits. Selecting only published significant $n$ values creates a reporting bias toward young, high-$\lvert\dot\nu\rvert$ objects with long monitoring.

PSRCAT is a literature compilation. Its period derivatives and references are useful discovery products, but a new braking likelihood requires the primary TOAs or folded archives. A quoted catalog uncertainty does not encode alternative red-noise, glitch, or clock models and cannot be treated as their joint posterior.

Calibration, likelihood, and priors

A reproducible radio timing analysis freezes the telescope and backend, frequency channels, pulse template, polarization and profile calibration, dispersion treatment, radio-frequency-interference selections, observatory clock corrections, terrestrial time realization, and Solar-system ephemeris. Backend delays and template changes require fitted jumps or a documented calibration. Kinematic corrections such as the Shklovskii effect and Galactic acceleration matter for intrinsic $\dot P$ and characteristic age, though ordinary rotational timing noise often dominates $\ddot\nu$ in young pulsars.

The forward model maps spin, astrometry, dispersion, glitches, and any binary parameters into barycentric arrival times. The likelihood should include white-noise scale terms and a stated red-noise covariance; priors on red-noise amplitude and slope, $\ddot\nu$, glitch epochs and recoveries, proper motion, and dispersion variations must be published. Model comparison between a secular $\ddot\nu$ and long exponential recovery is more informative than reporting whichever cubic coefficient a least-squares fit returns.

Systematic alternatives include changing the time span, excluding intervals around glitches, adding unresolved recoveries, changing the red-noise spectrum, using another ephemeris or terrestrial time standard, and allowing profile-domain timing. A robust result remains identifiable across reasonable alternatives or clearly reports their spread.

Reproduction boundary and physical interpretation

This page does not query PSRCAT, refit TOAs, or produce a new braking-index posterior. Reproduction requires the archived v2.8.1 tarball and checksum, exact query expression, source-reference list, TOA and template products, .par files, clock and ephemeris versions, TEMPO2 or PINT and sampler versions, noise and glitch model, priors, likelihood, posterior samples, and a script that regenerates residual and model-comparison diagnostics. The software environment and code configuration are therefore part of the inference product.

Even a well-measured $n$ constrains the total evolving torque. Vacuum-dipole, force-free, wind, inclination, magnetic-decay or growth, and changing coupled moment-of-inertia models must be propagated through $K(t)$ before comparison. Population claims add radio beaming, survey sensitivity, luminosity, sky coverage, and age distributions. A trend in a selected $P$–$\dot P$ sample is not automatically an evolutionary track for one star.

Check your understanding

Suppose the instantaneous torque exponent is $q=3$, the spin-down timescale is $-\Omega/\dot\Omega=1000\,\mathrm{yr}$, and the torque coefficient grows at $\dot K/K=10^{-4}\,\mathrm{yr}^{-1}$. What braking index is measured?

Solution

Because $\Omega/\dot\Omega=-1000\,\mathrm{yr}$,

$$n=3+(-1000\,\mathrm{yr}) \left(10^{-4}\,\mathrm{yr}^{-1}\right)=2.9.$$

The measured value is below three even though the instantaneous power of $\Omega$ remains three. The example shows why $n\ne3$ does not by itself identify a non-dipolar torque exponent.

References

• Australia Telescope National Facility. “The ATNF Pulsar Catalogue: History and Archived Releases.” PSRCAT v2.8.1, released 18 June 2026. Official version history.
• Lyne, Andrew G., Christine A. Jordan, Francis Graham-Smith, Cristóbal M. Espinoza, Benjamin W. Stappers, and Patrick Weltevrede. “45 Years of Rotation of the Crab Pulsar.” Monthly Notices of the Royal Astronomical Society 446 (2015): 857–864. DOI.
• Manchester, Richard N., George B. Hobbs, Alok Teoh, and Matthew Hobbs. “The Australia Telescope National Facility Pulsar Catalogue.” The Astronomical Journal 129 (2005): 1993–2006. DOI.
• Parthasarathy, Aditya, et al. “Timing of Young Radio Pulsars—II. Braking Indices and Their Interpretation.” Monthly Notices of the Royal Astronomical Society 494 (2020): 2012–2026. DOI.