Pulsar Timing Models and Residuals

A pulsar timing experiment turns a sequence of telescope timestamps into a comparison between a phase-coherent clock model and the observed pulse train. The scientifically useful object is not a single arrival time but the structured residual left after propagating the signal through the Solar System, the interstellar medium, and—when present—a binary orbit. Different errors imprint different dependences on time, radio frequency, and orbital phase, so a gravity parameter is credible only when it cannot substitute for one of those signatures. The evidence discussed here is current through 2026-08-19.

Required background. Probability and Statistical Inference supplies normalized likelihoods, priors, covariance, and posterior-predictive checks used to distinguish correlated timing components.

Helpful background. Weak-Field Gravity supplies the weak-field propagation model; Relativistic Two-Body Dynamics supplies the orbital dynamics; and Gravitational Radiation and Waveforms supplies the radiative interpretation of secular orbital changes.

From pulse number to a timing residual

Choose a fiducial pulse profile for each receiver band and backend. Cross-correlation gives a topocentric time of arrival (TOA) $t_i^{\rm topo}$ with uncertainty $\sigma_i$, but that timestamp still belongs to an observatory clock and observing frequency. A timing pipeline converts it to a barycentric coordinate time, including clock realization, station position, Earth orientation, Solar-System ephemeris, and relativistic propagation. Schematically,

$$t_i^{\rm SSB}=t_i^{\rm topo}+\Delta_{\rm clk}+\Delta_{\rm R}+\Delta_{\rm E}+\Delta_{\rm S}-K\,{\rm DM}(t_i)\nu_i^{-2}+\cdots ,$$

where the Rømer, Einstein, and Shapiro terms are denoted by $\Delta_{\rm R}$, $\Delta_{\rm E}$, and $\Delta_{\rm S}$, while the dispersive term is chromatic. The signs depend on whether a delay is added to the observed or predicted time, so a software convention must be stated rather than inferred from this mnemonic. The phase model assigns an integer pulse count,

$$\phi(t)=\phi_0+\nu_0(t-t_0)+\frac12\dot\nu_0(t-t_0)^2+\phi_{\rm ast}(t)+\phi_{\rm bin}(t;\vartheta_{\rm K},\vartheta_{\rm PK}),$$

and the residual is $r_i=t_i^{\rm SSB}-t_i^{\rm model}$. A constant offset shifts phase; a slope shifts spin frequency; a parabola shifts spin-down; an annual sinusoid indicates position error; a six-month parallax pattern tests distance; a $\nu^{-2}$ pattern indicates dispersion measure; and orbital harmonics diagnose binary parameters. These templates are not perfectly orthogonal on a finite, irregular cadence.

The distinction is operationally demonstrated in the Double Pulsar analysis: its timing model includes telescope-dependent profile evolution, interstellar and Solar-System propagation, orbital effects, and relativistic parameters rather than interpreting a raw residual as gravity Kramer et al. 2021, §§III–V and Table IV. The general construction and post-Keplerian timing coordinates are developed in Freire and Wex 2024, §3.4.2 and Eqs. (28)–(34).

Why timing finds very small angles

Let $\hat{\mathbf n}$ point from the Solar-System barycenter toward the pulsar, and $\mathbf r_\oplus(t)$ point from that barycenter to the observatory. The leading barycentric correction is $\mathbf r_\oplus\cdot\hat{\mathbf n}/c$. A position error $\delta\hat{\mathbf n}$ therefore leaves an annual residual with amplitude at most $(1\ {\rm au}/c)\lvert\delta\hat{\mathbf n}\rvert$. Since one astronomical unit of light travel takes about 499 s and one milliarcsecond is $4.848\times10^{-9}$ radians, a one-milliarcsecond error can produce a $2.4\ \mu{\rm s}$ annual residual. The actual amplitude is reduced by sky projection. This elementary geometric effect can be larger than a desired relativistic correction.

The frequency symbol also has two distinct uses. The $\nu_i^{-2}$ propagation term uses the observing radio frequency; $\nu_0$ in the pulse-number polynomial is the pulsar’s much lower spin frequency. In that polynomial phase is counted in cycles. A small phase error $\delta\phi$ corresponds locally to a timing error of magnitude $\lvert\delta\phi\rvert/\nu_0$, with sign fixed by the residual convention.

Likelihood, calibration, and selection

For a versioned TOA vector $\mathbf t$, design matrix $M$, deterministic parameter offset $\delta\boldsymbol\theta$, and stochastic covariance $C(\boldsymbol\lambda)$, a normalized Gaussian likelihood is

$$p(\mathbf r\mid\delta\boldsymbol\theta,\boldsymbol\lambda) =\frac{\exp\!\left[-\tfrac12(\mathbf r-M\delta\boldsymbol\theta)^{\mathsf T}C^{-1}(\mathbf r-M\delta\boldsymbol\theta)\right]} {(2\pi)^{N/2}\det(C)^{1/2}}.$$

White radiometer noise, pulse jitter, red spin noise, time-variable dispersion, and common clock or ephemeris components belong in $C$ or in explicit latent processes. Priors must be declared for noise amplitudes and spectral indices as well as for the gravity parameters; the determinant term cannot be discarded when comparing covariance models. Marginalizing over a nuisance component and projecting it out are not generally equivalent.

A reproducible use of data pins the TOA file, template, observing system and backend identifiers, clock-correction release, Solar-System ephemeris, dispersion model, binary timing model, and timing-software version. TEMPO2’s model and implementation are documented in Edwards, Hobbs, and Manchester 2006, §§2–4, but a software citation does not identify a particular reduction. Machine-readable inputs, their hashes, fit command, parameter file, and post-fit residuals are needed to reproduce one.

Selection occurs at several levels: telescopes target particular sources; observing schedules favor some orbital phases and frequencies; radio-frequency interference removes epochs or channels; low signal-to-noise profiles fail TOA cuts; and analysts may exclude glitches or profile-changing intervals. For a single known system this selection limits the times and frequency bands to which a residual check applies. For a population it also changes which systems enter the sample, so completeness and detection efficiency cannot be ignored.

Discriminating signatures and failure modes

A useful validation injects one perturbation at a time into realistic sampling. An astrometric error should return its annual signature, a dispersion perturbation should scale approximately as $\nu^{-2}$, and a clock error should be achromatic and common across pulsars. A binary perturbation should follow orbital phase. Recovery is then tested on withheld epochs and radio bands. If a proposed gravity term improves only the training data, or absorbs a chromatic feature, the result is a model-flexibility warning rather than evidence for new gravity.

The principal failure modes are covariance between annual and orbital signatures, profile evolution masquerading as frequency-dependent delay, backend jumps, unmodeled scattering, solar-wind dispersion, clock or ephemeris errors, red-noise leakage, glitches, and a binary model outside its eccentricity or inclination regime. Posterior-predictive residuals should be structureless not merely in time, but also versus radio frequency, orbital phase, telescope, backend, and sky position. A later calibration, ephemeris, timing release, or independently discovered residual pattern should trigger renewed analysis.

Exercise

Suppose post-fit residuals show a sinusoid with a one-year period whose amplitude is the same at every observing frequency. Which two model classes should be tested first, and why is dispersion measure a poor first explanation?

Solution

An error in sky position produces an annual barycentric-correction error, while a Solar-System ephemeris error can produce an achromatic annual or planet-linked common signal. Both should therefore be tested with astrometric and cross-pulsar diagnostics. Dispersion delay scales approximately as $\nu^{-2}$, so it predicts different amplitudes in different radio bands. Because the observed sinusoid is achromatic, substituting a dispersion term would fail the frequency-domain comparison even if it reduced a time-only residual statistic.

References

• Edwards, R. T., Hobbs, G. B., and Manchester, R. N. (2006), “TEMPO2, a new pulsar timing package. II: The timing model and precision estimates,” Monthly Notices of the Royal Astronomical Society 372, 1549–1574. DOI.
• Freire, P. C. C., and Wex, N. (2024), “Gravity Experiments with Radio Pulsars,” Living Reviews in Relativity 27, article 5. DOI.
• Kramer, M., et al. (2021), “Strong-Field Gravity Tests with the Double Pulsar,” Physical Review X 11, 041050. DOI.