The Dead Star That Breaks Physics' Bathroom Scale

Right now, somewhere out in the galaxy, a ball of crushed atomic nuclei the size of a city is spinning hundreds of times every second. It weighs more than the Sun. You could tuck the whole thing inside a midsize town. Now toss a little extra mass onto it, and at one exact instant, gravity stops losing. The whole object caves in and becomes a black hole.

Here's the strange part. We have a century of nuclear physics behind us. We have a fleet of gravitational-wave detectors listening to the universe. And nobody can tell you where that tipping point is. The heaviest a neutron star can possibly get is one of the most solid-sounding numbers in all of astrophysics, and science still can't pin it down.

What We Know For Sure

Start with the corpse. A neutron star is what's left when a massive star blows itself apart as a supernova and its core falls inward. The crush is unimaginable: protons and electrons get squeezed together into neutrons, and roughly a Sun's worth of matter ends up packed into a sphere about 12 miles wide. What keeps it from collapsing further is neutron degeneracy pressure, plus the shoving match between nuclear particles jammed shoulder to shoulder. That outward push holds gravity off. For a while.

But every standoff has a snapping point. In 1939, physicists J. Robert Oppenheimer and George Volkoff, building on earlier work by Richard Tolman, did the math on it. They found an upper mass limit: pile on more than this, and a neutron star simply can't hold itself up. It has to become a black hole. That number is the Tolman–Oppenheimer–Volkoff limit, the TOV limit, and it's the neutron-star cousin of the Chandrasekhar limit for white dwarfs (Wikipedia: TOV limit). Their first guess came out near 0.7 solar masses. But they had left something out on purpose: the strong nuclear force between neutrons. That one simplification made their early number way too small.

The sky proves it. Astronomers have now weighed neutron stars heavier than two Suns. Take PSR J0740+6620, spotted in 2019 with the Green Bank Telescope. It tips the scale at about 2.08 solar masses, measured by the Shapiro delay, a tiny relativistic lag in its pulses as light from the star threads through the warped space-time around its white-dwarf companion (PSR J0740+6620, Wikipedia). And then there's the current champion: PSR J0952–0607, a "black widow" pulsar that has eaten its own companion star and spun itself up to a dizzying 707 rotations per second. Using the Keck I telescope in Hawai'i, Romani and colleagues clocked it at 2.35 ± 0.17 solar masses, the heaviest well-measured neutron star we know of (Romani et al. 2022, ApJ Letters).

So one thing is locked down. Whatever the true ceiling is, it sits at least around 2.35 solar masses, because we've actually seen one that heavy and it didn't collapse. The real question is: how much higher does the ceiling go?

The Question Nobody Can Answer

The whole puzzle fits in one sentence: we don't know the equation of state of matter at the insane densities inside a neutron star's core, so we can't calculate the exact maximum mass.

What's an "equation of state"? It's just the rulebook linking pressure to density. For water or steel here on Earth, we know that rulebook cold. But inside a neutron star, the density blows past what you'd find inside an atomic nucleus, and we have zero laboratories that can recreate those conditions. Theorists aren't even sure what matter is down there. At the very deepest, densest core, weird ingredients might show up: hyperons, meson condensates, even deconfined quark matter, where individual neutrons melt into a free-flowing soup of quarks (review of dense-matter equations of state, A&A 2024). Those exotic ingredients tend to "soften" the equation of state, which drags the maximum mass down. Do they actually exist inside real neutron stars? Open question. Nobody has the smoking-gun observation.

That ignorance is why the published guesses for the TOV limit are all over the map. Different models put it anywhere from roughly 2.2 to 2.9 solar masses for a star that isn't spinning (Wikipedia: TOV limit). That's a spread of nearly a whole Sun. And it's not sloppy measurement. It's a flat admission that we don't understand the underlying physics yet.

Then gravitational waves handed us a fresh clue. In 2017, two neutron stars smashed together (the event called GW170817) and lit up both the gravitational-wave detectors and the regular sky with an electromagnetic afterglow. By modeling how fast the wreckage collapsed into a black hole, several teams worked backward and figured the maximum mass for a non-spinning neutron star is probably no more than about 2.17 solar masses (Margalit & Metzger 2017; see also constraints from Shibata et al.). Now look at that number. It's tighter and lower than a lot of theory allows, and it sits uncomfortably close to the 2.35-solar-mass pulsar we already found spinning out there. Two real numbers that don't want to fit together. Squaring them is one of the hottest fights in the field.

One more twist before we go: spin. A neutron star whirling fast can carry more mass than a still one, because the centrifugal fling helps hold it up against gravity. Theory says rapid rigid rotation can push the limit up by roughly 18 to 20 percent (Wikipedia: TOV limit). But these "supramassive" stars are living on borrowed time. As they slow down, they can drop dead in an instant, collapsing the moment the spin can no longer save them.

The Suspects and the Arguments

Here's a case that keeps astronomers up at night: GW190814, caught by LIGO and Virgo in 2019. A 23-solar-mass black hole collided with something weighing about 2.6 solar masses (Abbott et al. 2020, ApJ Letters). And that "something" landed dead center in the so-called "lower mass gap," the no-man's-land between the heaviest neutron stars we know and the lightest black holes we know. The detection showed no measurable tidal stretching and no light at all, so its true identity is a genuine mystery. Researchers have pitched it both ways: the lightest black hole ever seen, or the heaviest neutron star ever seen. The discovery team leaned toward black hole. But if it really were a neutron star, it would force the equation of state to be shockingly "stiff," shoving the maximum mass higher than a lot of models can stomach (arXiv:2007.03799).

That word, stiff, is what splits the theorists into camps. "Stiffer" equations of state fight back hard against compression and allow heavier neutron stars, climbing toward 2.5 to 2.9 solar masses. "Softer" equations of state, the ones that lean on quark matter or hyperons in the core, slam the cap down nearer 2.2 solar masses. Every new precise mass measurement tugs the whole field toward one camp or the other, but none has landed the knockout punch yet.

And here's the part that makes this mystery so satisfying. It's solvable. We're not chasing ghosts; we're waiting for data. The next generation of gravitational-wave observatories, and X-ray missions built to measure neutron-star radii, should eventually squeeze the equation of state tight enough to fix the maximum mass once and for all. Until that day, the heaviest a dead star can get before it winks out into a black hole stays exactly what it is: a real, beautifully fenced-in mystery, somewhere between roughly 2.2 and 2.9 solar masses, with the final answer still out there, still spinning, still waiting for someone to read it.

Sources and Further Reading

• Tolman–Oppenheimer–Volkoff limit, Wikipedia
• PSR J0740+6620, Wikipedia
• Romani et al. 2022, "PSR J0952–0607: The Fastest and Heaviest Known Galactic Neutron Star," ApJ Letters
• Margalit & Metzger 2017, joint constraint on the neutron-star equation of state, ApJ Letters
• Abbott et al. 2020, "GW190814," ApJ Letters
• Unified neutron-star equations of state, Astronomy & Astrophysics 2024

Sources & further reading

• https://en.wikipedia.org/wiki/Tolman%E2%80%93Oppenheimer%E2%80%93Volkoff_limit
• https://en.wikipedia.org/wiki/PSR_J0740%2B6620
• https://iopscience.iop.org/article/10.3847/2041-8213/ac8007
• https://arxiv.org/abs/2207.05124
• https://iopscience.iop.org/article/10.3847/2041-8213/aaa402
• https://www2.yukawa.kyoto-u.ac.jp/~masaru.shibata/PhysRevD.100.023015.pdf
• https://iopscience.iop.org/article/10.3847/2041-8213/ab960f
• https://arxiv.org/abs/2007.03799
• https://www.aanda.org/articles/aa/full_html/2024/09/aa49292-24/aa49292-24.html