The Final States

Section 3 of 6

The TOV Limit

Part 3: The TOV Limit — The End of Neutron-Star Support

Another maximum mass

White dwarfs have a maximum mass (the Chandrasekhar limit, $M_{\rm Ch} \approx 1.4\,M_\odot$ ). Neutron stars also have one: the Tolman-Oppenheimer-Volkoff limit.

TOV limit

The Tolman-Oppenheimer-Volkoff limit: the maximum mass of a stable neutron star, set by general relativity and the equation of state of dense nuclear matter. In this course treat it as roughly $2$ – $3\,M_\odot$ — less precisely known than the Chandrasekhar limit.

Why the TOV limit is not a single clean number

The Chandrasekhar limit is relatively clean because it depends on electron degeneracy pressure. The TOV limit is harder because neutron-star interiors involve general relativity (gravity is strong), nuclear-density matter with an uncertain equation of state, strong-force physics at very small distances, and possible exotic phases such as hyperons or deconfined quarks.

Equation of state

A relationship between pressure, density, temperature, and composition. For neutron stars the high-density equation of state is uncertain, which is the main reason the TOV limit is imprecise.

For this course, use the order-of-magnitude statement:

When we need a simple dividing value, use $M_{\rm TOV} \approx 2.5\,M_\odot$ — a useful course-level boundary, not an exact universal constant.

Object	Main support	Approximate maximum mass	Above the limit?
White dwarf	electron degeneracy pressure	$1.4\,M_\odot$	collapse toward neutron-star densities
Neutron star	dense nuclear matter + neutron degeneracy	$\sim 2$ – $3\,M_\odot$	collapse to a black hole

The physical meaning

The TOV limit does not mean “neutrons suddenly disappear.” It means that for a sufficiently massive neutron star, adding mass increases gravity faster than pressure can respond — and in general relativity, pressure itself contributes to gravity, making the instability worse. So a remnant below $M_{\rm TOV}$ allows a stable neutron star, while one above $M_{\rm TOV}$ implies black-hole formation.

Two-panel plot of measured neutron star masses with horizontal error bars. Top panel shows neutron star-white dwarf binary systems; bottom panel shows double neutron star systems. Most masses cluster between 1.2 and 1.5 solar masses, with vertical dashed lines indicating the mean near 1.35 solar masses. A few outliers extend to about 2 solar masses. — Figure 4Measured neutron-star masses from binary pulsar systems cluster tightly around 1.3-1.5 solar masses, strikingly close to the Chandrasekhar limit of 1.4 solar masses. The clustering reflects core-collapse physics: the iron-core mass at collapse is set by electron degeneracy and the Chandrasekhar limit. A few neutron stars near ~2 solar masses push the upper limit of neutron-degeneracy support.

Quick check

An unseen compact object in a binary has a securely measured mass of $4\,M_\odot$ . (1) Why unlikely a white dwarf? (2) Why unlikely a neutron star? (3) What model is left? (4) What assumption about the TOV limit are you making?

Above the TOV limit

If the remnant exceeds the maximum stable neutron-star mass, the known pressure sources have all failed: thermal pressure cannot help permanently (the remnant cools), electron degeneracy already failed at the Chandrasekhar limit, and neutron-star matter cannot support it. The collapse forms an event horizon; in classical general relativity, continued collapse inside leads toward a singularity, and a full quantum theory of gravity would be needed to describe the innermost endpoint.