Mass Measurements and Population Structure

A neutron-star mass is not read directly from a catalog: it is inferred from clocks, orbits, spectra, and a model of the observing selection. The distinction matters most at the upper end of the distribution, where one well-measured heavy star can rule out equations of state but a fitted population cutoff need not equal the maximum mass of a cold, nonrotating star. This page develops the measurement likelihood, embeds it in a population model, and identifies the checks needed before interpreting an apparent high-mass edge as dense-matter physics. Observational statements use an evidence cutoff of 2026-08-18.

Required background. Equivalence principles and gravitational measurement supplies the distinction between invariant source parameters and instrument- or model-dependent observables.

Helpful background. Stress–energy and conservation supplies the material variables constrained by a mass; the Einstein equation connects them to stellar equilibrium; relativistic matter and compact objects supplies the equation-of-state and stellar-response setting; gravitational radiation and waveforms supplies merger mass measurements; and computational relativity supplies numerical convergence and reproducibility practice.

Masses from relativistic clocks and binary dynamics

For a pulsar in a Keplerian binary, pulse arrival times determine the orbital period $P_b$ and projected pulsar semimajor axis $x=a_p\sin i$ (with $c=1$). They give the mass function

$$f_p =\frac{4\pi^2x^3}{G P_b^2} =\frac{(M_c\sin i)^3}{(M_p+M_c)^2},$$

where $M_p$, $M_c$, and $i$ are the pulsar mass, companion mass, and inclination. To see the origin of the combination, let $a=a_p+a_c$ be the relative semimajor axis. Center-of-mass balance gives $a_p=aM_c/(M_p+M_c)$, while Kepler’s law gives $4\pi^2a^3=G(M_p+M_c)P_b^2$. Substitution eliminates $a$ and leaves the measured projected orbit on the left. The mass function alone defines a family, not two masses.

Relativistic post-Keplerian effects can close the system: Shapiro delay constrains a range-like mass parameter and a shape related to $\sin i$; periastron advance, Einstein delay, and orbital decay constrain other combinations. A mass becomes a test of gravity as well as a stellar parameter when several such effects overdetermine the two-body model. The timing logic and its covariance are reviewed in Özel and Freire 2016, §§ 2.1–2.3.

Calibration here includes observatory clock corrections, dispersion and frequency-dependent profile evolution, Solar-System ephemerides, pulse-template choices, and kinematic terms such as the Shklovskii and Galactic-acceleration corrections. These are part of the timing model, not optional error bars appended afterward. The refined Shapiro-delay analysis of PSR J0740+6620 is a useful high-mass case because the model, observing cadence, nuisance parameters, and alternative statistical treatments are exposed in Fonseca et al. 2021, §§ 2–4 and Table 2.

Optical binaries use radial velocities, light curves, and a Roche-geometry or atmosphere model to infer inclination and mass ratio. In irradiated “black widow” systems, the measured lines preferentially sample the bright face of the companion, so the center-of-light velocity is not automatically the center-of-mass velocity. Heating pattern, limb darkening, spots, filling factor, distance, and atmosphere grids can move the inferred pulsar mass together. Radio Shapiro-delay masses and optical irradiation-model masses must therefore remain separate measurement classes even when both are reported in solar masses.

Population likelihood and the selection denominator

Let $d_i$ denote the data for system $i$, $m_i$ its latent mass, $\eta_i$ measurement nuisances, and $\lambda$ the population hyperparameters. For a sample whose detection process is described by $p_{\rm det}(m,\eta)$, the likelihood conditional on $N$ detections has the schematic form

$$\begin{aligned} \mathcal L(\lambda) \propto \frac{ \displaystyle\prod_{i=1}^{N} \int dm_i\,d\eta_i\, p(d_i\mid m_i,\eta_i)\, p(m_i,\eta_i\mid\lambda) }{\alpha(\lambda)^N},\\[0.4em] \alpha(\lambda)= \int dm\,d\eta\, p_{\rm det}(m,\eta)p(m,\eta\mid\lambda). \end{aligned}$$

The numerator is a population-prior average of each measurement likelihood. The denominator normalizes the distribution of detected systems. Here detection is a cut on the observed data, so it is already satisfied by each $d_i$; an additional $p_{\rm det}$ is not multiplied into that event numerator. An analysis of the total occurrence rate also needs a Poisson count model.

For a simple example, suppose two narrow mass groups contain equal intrinsic numbers, but their detection probabilities are 1 and $1/4$. The expected detected fractions are $0.5/(0.5+0.125)=0.8$ and 0.2. An observed four-to-one ratio would then be compatible with a one-to-one intrinsic ratio. Ignoring selection would mistake visibility for formation physics.

For targeted radio timing there is no single universal detection curve—survey discovery, binary acceleration, eclipses, timing precision, and the decision to conduct high-cadence follow-up are distinct selection stages. A transparent analysis either models those stages or limits its claim to the explicitly selected sample.

Population families commonly include one or more Gaussians, a broad component for recycled stars, and a truncated upper tail. Galactic double-neutron-star components should not automatically share a distribution with accretion-grown millisecond pulsars or merger components. Farrow, Zhu, and Thrane 2019, §§ 2–4 demonstrate how even a small double-neutron-star sample changes its preferred shape when recycled and nonrecycled components are distinguished. Alsing, Silva, and Berti 2018, §§ 2–5 show how a flexible mixture and an upper cutoff can be inferred jointly; their result is a model-conditioned population statement, not an identity between the cutoff and the Tolman–Oppenheimer–Volkoff limit.

If posterior samples rather than likelihoods are reused, each sample must be divided by the interim prior under which it was produced. Multiplying published posteriors directly double-counts those priors. The population prior, component labels, truncation rule, and normalization must all be stated before a Bayes factor or credible interval can be interpreted.

Versioned data and a reproducible analysis boundary

At the cutoff, the machine-readable discovery catalog used for sample bookkeeping is ATNF Pulsar Catalogue v2.8.1, released 2026-06-18 with 4,393 entries. Manchester et al. 2005, § 2 defines the catalog fields and construction; the ATNF v2.8.1 documentation, § 1 describes the software and citation practice, while the ATNF version history, v2.8.1 row records the release date and membership. ATNF is not a substitute for primary timing solutions: covariance matrices, timing ephemerides, mass-model assumptions, and data spans come from the cited analysis for each system. Measurement uncertainty and systematic model dependence must remain separate fields, because collecting them into one symmetric error bar would erase the alternatives that the robustness tests need to vary.

A reproducible population study should archive:

• the exact catalog release and query, plus an inclusion/exclusion table keyed to primary mass papers;
• the likelihood representation for every object, including asymmetric or non-Gaussian structure and the interim prior used to create posterior samples;
• clock, ephemeris, dispersion, kinematic, optical-atmosphere, and irradiation alternatives appropriate to each measurement class;
• the population family, label assignments, hyperpriors, selection model, sampler version, random seeds, and convergence diagnostics; and
• posterior-predictive draws, leave-one-system-out results, and injection–recovery tests for a simulated upper cutoff.

This page documents that route but does not claim a new catalog fit. Reproduction begins from the archived v2.8.1 query and primary likelihood products, not from rounded masses copied out of a review table.

What the high-mass tail can and cannot establish

Three scales must be kept distinct. The largest securely measured mass is a lower bound on the maximum mass admitted by the relevant stellar model. A population cutoff is a hyperparameter shaped by formation, accretion, survival, detection, and the assumed distribution. The cold, nonrotating maximum $M_{\rm TOV}$ is obtained from a stable equilibrium sequence for one equation of state. Rotation and temperature can support larger masses, while collapse and binary evolution can keep the observed distribution below $M_{\rm TOV}$.

The most informative robustness checks are therefore structural: replace a Gaussian mixture with a spline or nonparametric population; move systems between evolutionary subpopulations; include or omit optical high-mass candidates; vary the selection function; and remove each influential system in turn. A claimed cutoff should survive reasonable choices or be reported as prior- and model-sensitive. Posterior-predictive checks should compare the distribution of measurement precision and system type as well as the mass histogram, because a model can fit the latter for the wrong reason.

Check your understanding

A binary has a precisely measured mass function $f_p=0.20\,M_\odot$. Explain why this does not imply that either star has mass $0.20\,M_\odot$, and find the minimum companion mass if $M_p=1.40\,M_\odot$.

Solution

The mass function is a nonlinear combination of both masses and inclination. Its minimum companion mass occurs for an edge-on orbit, $\sin i=1$. One solves

$$0.20=\frac{M_c^3}{(1.40+M_c)^2}$$

with masses measured in $M_\odot$. Trial values give $M_c=1.00$: the right-hand side is $1/(2.40)^2\simeq0.174$, while $M_c=1.07$ gives $1.07^3/(2.47)^2\simeq0.201$. Thus $M_{c,\min}\simeq1.07\,M_\odot$. For any inclination below $90^\circ$, the companion must be heavier. Additional post-Keplerian or optical information is required to determine both masses.

References

• Alsing, Justin, Hector O. Silva, and Emanuele Berti. “Evidence for a Maximum Mass Cut-Off in the Neutron Star Mass Distribution and Constraints on the Equation of State.” Monthly Notices of the Royal Astronomical Society 478 (2018): 1377–1391. DOI.
• Farrow, Nicholas, Xing-Jiang Zhu, and Eric Thrane. “The Mass Distribution of Galactic Double Neutron Stars.” Astrophysical Journal 876 (2019): 18. DOI.
• Fonseca, Emmanuel, et al. “Refined Mass and Geometric Measurements of the High-Mass PSR J0740+6620.” Astrophysical Journal Letters 915 (2021): L12. DOI.
• Manchester, Richard N., George B. Hobbs, Andrew Teoh, and Matthew Hobbs. “The Australia Telescope National Facility Pulsar Catalogue.” Astronomical Journal 129 (2005): 1993–2006; database release v2.8.1 (2026). DOI; catalog documentation.
• Özel, Feryal, and Paulo Freire. “Masses, Radii, and the Equation of State of Neutron Stars.” Annual Review of Astronomy and Astrophysics 54 (2016): 401–440. DOI.