Maximum Neutron-Star Mass

The maximum neutron-star mass is not one universal number. For a specified cold EOS, the nonrotating maximum $M_{\rm TOV}$ is the endpoint of the stable Tolman–Oppenheimer–Volkoff sequence. Uniform rotation raises the supported mass, while differential rotation and thermal pressure can raise it further for a transient remnant. Observations constrain these theoretical ceilings from below when a reliably weighed neutron star exists; turning that lower bound into an EOS statement requires the measurement likelihood, not merely a quoted central value.

Required background. The Einstein Equation supplies the relativistic stellar-structure equations that define the nonrotating mass limit.

Helpful background. Flat-Background Hilbert Stress-Energy from Metric Variation supplies pressure and energy-density conventions; Relativistic Matter and Compact Objects supplies the EOS setting; Computational Relativity supplies sequence, interpolation, and convergence checks.

The theoretical ceiling

Take a causal, thermodynamically stable, barotropic cold EOS and integrate the TOV equations through a one-parameter sequence labeled by central pressure. On the branch connected continuously to low-density stars, the first maximum of gravitational mass normally coincides with the onset of the fundamental radial instability. If a strong phase transition creates a disconnected stable branch, its modes must be counted separately; $M_{\rm TOV}$ denotes the largest stable nonrotating mass over all admissible branches, not automatically the first plotted maximum. The stable nonrotating configurations therefore obey $M\leq M_{\rm TOV}$ for that EOS. The value is sensitive to pressure at several times nuclear saturation density, so it cannot be predicted from the saturation point alone. The EOS dependence and causal upper-bound logic are reviewed in Lattimer and Prakash 2016, §2.2 and §4.2.

Uniform rotation supplies centrifugal support and produces a two-dimensional equilibrium sequence. A supramassive star has baryon mass above the largest stable nonrotating baryon mass, but below the corresponding ceiling for uniform rotation. It can be stationary while spinning rapidly; on a constant-baryon-mass spin-down track it cannot reach a stable nonrotating endpoint. Comparing gravitational mass with $M_{\rm TOV}$ alone does not define this track, because rotational energy contributes to gravitational mass and is lost during spin-down. A differentially rotating, hot merger remnant may be hypermassive. Its support depends on an evolving rotation law, entropy, and composition, so its initial mass is not a direct measurement of $M_{\rm TOV}$. Confusing these three limits is a common source of overstrong claims.

What a massive pulsar establishes

If a cold neutron star has a mass likelihood $p(d\mid M)$ and known spin frequency $\nu$, a viable EOS must support stable configurations at that spin over a nonnegligible part of the likelihood. The relevant ceiling is $M_{\max}(\nu;\mathrm{EOS})$. Only in the nonrotating approximation does it reduce to $M_{\rm TOV}$. A spin correction may be small for a particular pulsar, but must be checked before asserting a bound on $M_{\rm TOV}$. PSR J0740+6620 supplies a particularly clean mass measurement: radio timing resolves Shapiro delay and yields a mass near two solar masses. The joint timing analysis, noise treatment, and mass–inclination posterior are given in Fonseca et al. 2021, §§3–5 and Table 2. This is a probabilistic lower bound, not permission to replace the likelihood by a hard cut at its median.

Optical modeling can reach apparently larger values but introduces additional physics. For PSR J0952−0607, the companion’s irradiated surface brightness, spectral-line velocities, inclination, and center-of-light correction enter the mass inference. The result and alternative heating models are set out in Romani et al. 2022, §§2–4 and Table 2. It is credible evidence, but its model dependence differs from Shapiro-delay timing and should be represented as such rather than silently pooled as an equivalent datum.

Dataset, likelihood, and reproduction boundary

Evidence cutoff: 2026-08-18. A reproducible radio-timing baseline is the official NANOGrav 12.5-year narrowband data release, whose pulsar timing files, noise products, observing systems, and release construction are documented in Alam et al. 2021, §§2–4 and Tables 1–3. The specialized J0740+6620 mass analysis includes observations and modeling choices beyond treating that release as a single catalog row. The public timing data product, the derived mass posterior, and a reusable likelihood are therefore three distinct objects.

A clean reanalysis would freeze the .par and .tim data version, observatory clock corrections, terrestrial time standard, Solar-system ephemeris, dispersion treatment, backend offsets, and software version for TEMPO, TEMPO2, or PINT. Its selection rule would be declared in advance—for example, include only masses whose Shapiro-delay parameters are resolved, while placing irradiated-companion fits in a separately modeled class. The forward model maps timing parameters to pulse arrival times. A residual likelihood must include radiometer error, epoch-correlated noise, dispersion variations, and red timing noise; priors on inclination and companion mass must be reported because they shape a weak Shapiro signal.

For optical binaries, the corresponding record includes spectra and photometry, wavelength and flux calibration, passbands, orbital selections, irradiated-atmosphere model, velocity extraction, distance or extinction priors, and the likelihood for light curves and radial velocities. Important systematics are surface-temperature maps, asymmetric heating, star spots, line formation, contamination, and center-of-light corrections. This page does not rerun either timing or optical code and does not create a combined posterior; it identifies what must accompany one. Reproduction requires the released data, exact software environment, machine-readable priors and noise model, and posterior samples or a likelihood implementation.

From a lower bound to EOS pressure

A massive star disfavors those EOS models whose stable sequence at its measured spin ends below the relevant part of the measurement likelihood. It does not by itself give a unique pressure at any density. Radius or tidal data may further constrain the curve, but then prior transport and common EOS parameterization matter. Selection also matters at the population level: discovering a $2\,M_\odot$ star is strong existence evidence, whereas failure to discover a heavier star is not an upper bound without a survey selection model and an assumed mass population.

The inference should also separate physical alternatives. Rapid uniform rotation changes the allowed mass at measured spin; thermal pressure is irrelevant for an old radio pulsar but essential for a newborn remnant; and a strong first-order phase transition can create a disconnected stable branch. Causality, monotonic pressure, and radial stability must be checked on every candidate sequence. These are physical filters on the EOS ensemble, not extra astronomical observations.

Check your understanding

A hypothetical slowly rotating pulsar has negligible rotational mass correction for this exercise. Its timing analysis reports a Gaussian mass likelihood $M=(2.08\pm0.07)M_\odot$, with the uncertainty interpreted as one standard deviation. An EOS has $M_{\rm TOV}=2.00M_\odot$. By how many standard deviations does the measured mean exceed its ceiling, and is the EOS excluded with mathematical certainty?

Solution

The standardized separation is

$$z=\frac{2.08-2.00}{0.07}\simeq1.14.$$

The model is disfavored by this datum, but not excluded with certainty: a Gaussian likelihood retains probability below $2.00M_\odot$, and additional systematic alternatives may broaden it. A valid EOS comparison integrates the likelihood over its prediction and any nuisance parameters rather than using the median as a hard inequality.

References

• Alam, Md. F., et al. “The NANOGrav 12.5 yr Data Set: Observations and Narrowband Timing of 47 Millisecond Pulsars.” The Astrophysical Journal Supplement Series 252 (2021): 4. DOI.
• Fonseca, Emmanuel, et al. “Refined Mass and Geometric Measurements of the High-Mass PSR J0740+6620.” The Astrophysical Journal Letters 915 (2021): L12. DOI.
• Lattimer, James M., and Madappa Prakash. “The Equation of State of Hot, Dense Matter and Neutron Stars.” Physics Reports 621 (2016): 127–164. DOI.
• Romani, Roger W., D. Kandel, Alexei V. Filippenko, Thomas G. Brink, and WeiKang Zheng. “PSR J0952−0607: The Fastest and Heaviest Known Galactic Neutron Star.” The Astrophysical Journal Letters 934 (2022): L17. DOI.