Post-Keplerian Parameter Tests

A relativistic binary pulsar does not test gravity by measuring a mass and comparing it with an expected mass. It measures Keplerian and post-Keplerian (PK) timing parameters whose predicted values depend on the two component masses and on a theory of gravity. In a fixed theory, any two independent PK parameters can determine the masses; every additional parameter then supplies an overdetermined consistency test. This mass–mass construction is powerful precisely because the fitted timing coordinates remain distinct from the theory-specific map used to interpret them.

Required background. Probability and Statistical Inference supplies covariance propagation and joint inference for correlated timing parameters.

Helpful background. Weak-Field Gravity supplies the weak-field expansion; Relativistic Two-Body Dynamics supplies conservative binary motion; and Gravitational Radiation and Waveforms supplies the radiative flux underlying orbital damping.

Keplerian geometry and relativistic timing coordinates

Restore $c$ on this page so that measured timing quantities have seconds as their units. Let $P_b$ be the orbital period, $e$ the eccentricity, $x=a_p\sin i/c$ the pulsar’s projected semimajor axis expressed as a light-travel time, $i$ the inclination, and $n=2\pi/P_b$. The leading Newtonian mass function is

$$f(m_p,m_c,i)=\frac{(m_c\sin i)^3}{(m_p+m_c)^2}=\frac{c^3n^2x^3}{G},$$

where $m_p$ and $m_c$ are physical masses. To derive it, combine $n^2a^3=G(m_p+m_c)$ with center-of-mass geometry $a_p=a\,m_c/(m_p+m_c)$, then multiply by $\sin^3 i$. The measured combination $n^2x^3$ is a timing observable; interpreting it with the displayed masses assumes this gravitational coupling and Kepler law. A theory with body-dependent effective $G$ or higher-order corrections changes the mass map. By itself, $f$ is a surface in $(m_p,m_c,i)$, not a determination of both masses.

PK parameters encode effects not absorbed into the Keplerian ellipse: periastron advance $\dot\omega$, the combined gravitational-redshift and time-dilation amplitude $\gamma$, Shapiro-delay range and shape, orbital decay $\dot P_b$, and sometimes relativistic deformations or spin precession. They are initially phenomenological timing coefficients. General relativity maps them to masses. At leading relevant orders,

$$\begin{aligned} \dot\omega_{\rm GR}&=\frac{3n}{1-e^2}\left(\frac{GMn}{c^3}\right)^{2/3},\\ \gamma_{\rm GR}&=\frac{e}{n}\left(\frac{GMn}{c^3}\right)^{2/3}\frac{m_c(m_p+2m_c)}{M^2}. \end{aligned}$$

with $M=m_p+m_c$, while the conventional Shapiro parameters satisfy $r=Gm_c/c^3$ and $s=\sin i$. The symbol $\gamma$ here is the Einstein-delay amplitude, measured in seconds; it is unrelated to the dimensionless PPN parameter with the same letter. The leading quadrupole prediction can be written

$$\dot P_b^{\rm GR}=-\frac{192\pi}{5}\left(\frac{G\mathcal M n}{c^3}\right)^{5/3} \frac{1+\tfrac{73}{24}e^2+\tfrac{37}{96}e^4}{(1-e^2)^{7/2}},$$

where $\mathcal M=(m_pm_c)^{3/5}M^{-1/5}$ is the chirp mass. These expressions assume pointlike, slowly evolving compact bodies at the stated post-Newtonian order. At sufficiently high precision, higher-order conservative terms, spin–orbit coupling, aberration, mass loss, and propagation corrections must be included.

The timing coordinates and their GR mass maps are given with explicit conventions in Freire and Wex 2024, §3.4.2, Eqs. (28)–(34), and §4. The Double Pulsar analysis shows why higher-order pieces matter in practice: it combines multiple PK observables and includes second-post-Newtonian and spin-related contributions where they exceed the measurement precision Kramer et al. 2021, §V, Table IV, and Eqs. (39)–(48).

From two mass curves to a theory test

For each measured parameter $q_k$ with covariance, the equation

$$q_k^{\rm obs}=q_k^{\mathcal T}(m_p,m_c;\boldsymbol\lambda_k)$$

defines a band in the mass–mass plane for theory $\mathcal T$ and nuisance quantities $\boldsymbol\lambda_k$. Two nonparallel bands normally localize $(m_p,m_c)$. A third is not permission to choose another mass pair: it tests whether the same intersection predicts that observable. With $K$ measurements and two masses, the intuitive count is $K-2$ consistency directions, although correlations and shared nuisance parameters determine the actual rank.

A statistically correct implementation fits all TOAs, or a faithful joint PK likelihood, rather than multiplying one-dimensional intervals. If $\widehat{\mathbf q}$ has covariance $\Sigma_q$, a useful local statistic is

$$\chi^2_{\mathcal T}(m_p,m_c)= [\widehat{\mathbf q}-\mathbf q^{\mathcal T}(m_p,m_c)]^{\mathsf T} \Sigma_q^{-1} [\widehat{\mathbf q}-\mathbf q^{\mathcal T}(m_p,m_c)].$$

Its calibration depends on nonlinearity, parameter boundaries, and whether $\Sigma_q$ is itself estimated. A posterior or confidence region in masses should therefore be obtained with the declared likelihood, priors or confidence procedure, and shared nuisance parameters. The mass ratio from two timed pulsars can add an especially different curve, but its theory independence holds only under the assumptions used to relate projected semimajor axes to inertial masses.

A synthetic mass–mass calculation

Take $P_b=0.3$ days, $e=0.2$, and source masses $(m_p,m_c)=(1.4,1.2)M_\odot$. These are invented teaching inputs, not measured pulsar data. With $T_\odot=GM_\odot/c^3\simeq4.92549\ \mu{\rm s}$, substitute $GM/c^3=T_\odot(2.6)$ into the formulas to obtain $\dot\omega\simeq2.91474$ degrees per Julian year, $r\simeq5.91059\ \mu{\rm s}$, and $\gamma\simeq1.18435$ ms. The Shapiro range fixes $m_c$ directly; periastron advance fixes $M$ through

$$\frac{M}{M_\odot} =\frac{1}{T_\odot n} \left[\frac{\dot\omega(1-e^2)}{3n}\right]^{3/2}.$$

The two measurements therefore determine one mass pair before the Einstein delay is used. That delay is a prediction at the pair, not a third freely adjustable mass. The synthetic mass–mass diagram illustrates the common intersection and what happens when an additional timing quantity disagrees. In an actual fit the uncertainty bands are correlated, $r$ can be poorly measured at low inclination, and $\gamma$ loses sensitivity as $e\to0$; merely counting parameters does not establish that the masses are identified.

Checks and boundaries

Three checks catch most logical mistakes. First, substitute the Newtonian limit: PK effects should vanish while the mass function remains. Second, verify dimensions—$\dot\omega$ has inverse-time units, $\gamma$ and $r$ have time units, and $\dot P_b$ is dimensionless. Third, generate synthetic PK values from one mass pair and ensure every GR curve intersects there. Injecting one displaced PK parameter should produce an inconsistency that cannot be removed by silently choosing a different pair for each curve.

A disagreement would reject the complete timing-plus-theory model used in the comparison, not automatically identify a new gravitational degree of freedom. An unmodeled kinematic acceleration can bias $\dot P_b$; plasma or profile evolution can bias propagation terms; spin coupling can bias $\dot\omega$; and a theory beyond GR may make stellar sensitivities and the equation of state additional parameters. Those cases require a new common map, not a relabeling of the GR curves.

Exercise

Three independent PK measurements yield narrow bands. The $\dot\omega$ and Shapiro-shape bands intersect at $(m_p,m_c)=(1.34,1.25)M_\odot$, but the $\gamma$ band misses that point. Explain why refitting masses with $\dot\omega$ and $\gamma$ does not make the inconsistency disappear.

Solution

Any two nonparallel PK curves can usually define a mass pair, so replacing the Shapiro band by the $\gamma$ band merely moves the chosen intersection. The theory predicts that one common pair must satisfy all three observables simultaneously. Therefore the relevant statistic uses the joint covariance of all three residuals after minimizing over only one mass pair and the declared nuisance parameters. If no common intersection exists within uncertainty, the timing-plus-theory model is inconsistent; pairwise refits conceal rather than resolve that overdetermination.

References

• 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.

Further reading

• Damour, T., and Taylor, J. H. (1992), “Strong-field tests of relativistic gravity and binary pulsars,” Physical Review D 45, 1840–1868. DOI.