The collapse of a neutron star into a black hole is governed by its compactness, a measure of how much mass is squeezed into a given radius. The maximum compactness a non-rotating neutron star can achieve before gravitational forces overwhelm the internal pressure and cause it to collapse into a black hole is a fundamental limit in astrophysics, determined by the yet-to-be-fully-understood equation of state (EOS) for matter at nuclear and supra-nuclear densities.
The Compactness Limit and the Tolman-Oppenheimer-Volkoff (TOV) Limit
The Theoretical Compactness Ratio
In general relativity, the compactness of a spherical object is defined by the ratio $\mathcal{C} = \frac{GM}{Rc^2}$, where $G$ is the gravitational constant, $M$ is the mass, $R$ is the radius, and $c$ is the speed of light. For an object to be a black hole, its compactness must be $\mathcal{C} \ge 1/2$, corresponding to the Schwarzschild radius $R_s = 2GM/c^2$.
A neutron star must have a radius larger than its Schwarzschild radius to exist, so its compactness must be $\mathcal{C} < 1/2$.
Recent theoretical work, constrained by principles of general relativity, causality, and quantum chromodynamics (QCD), suggests a universal upper limit for the compactness of the most massive non-rotating neutron stars:
Where $M_{\text{max}}$ is the maximum stable mass (the TOV limit) and $R_{\text{min}}$ is the minimum possible radius for a stable neutron star of that mass. This finding suggests a lower limit for the radius of a neutron star: $R_{\text{min}} \gtrsim 3 \frac{GM}{c^2}$.
The Maximum Mass (TOV Limit)
The most critical factor in a neutron star's fate is its maximum stable mass, known as the Tolman-Oppenheimer-Volkoff (TOV) limit or maximum mass $M_{\text{max}}$. This mass is the turning point: any neutron star exceeding it must collapse to a black hole.
Theoretical Range: The TOV limit is not a fixed number but depends on the unknown EOS of ultra-dense matter. Theoretical models place it somewhere in the range of approximately $\mathbf{2.2}$ to $\mathbf{2.9}$ solar masses ($\mathbf{M_{\odot}}$).
Observational Constraints: Observations of the most massive known neutron stars, like PSR J0740+6620 ($\approx 2.07 M_{\odot}$), establish a firm lower limit on $M_{\text{max}}$. More stringent limits come from gravitational wave observations of binary neutron star mergers (e.g., GW170817), which suggest an upper bound on the maximum mass of around $\mathbf{2.1}$ to $\mathbf{2.3} \mathbf{M_{\odot}}$ for the non-rotating TOV limit, depending on assumptions about the merger remnant's immediate fate.
The Role of the Equation of State (EOS)
The maximum compactness, and therefore the maximum mass, is dictated by how strongly matter resists compression, described by the EOS—the relationship between pressure and density inside the star.
"Stiff" EOS: A "stiffer" EOS means the matter is more resistant to compression (pressure rises sharply with density). This allows a neutron star to support a larger mass before collapsing, resulting in a higher maximum mass and potentially a larger radius for a given mass.
"Soft" EOS: A "softer" EOS means the matter is less resistant. This leads to a lower maximum mass and a more compact star for a given mass. The softer the EOS, the closer the compactness $\mathcal{C}$ gets to the theoretical maximum $\approx 1/3$.
The challenge of determining the exact compactness limit is thus synonymous with the challenge of determining the EOS of matter at densities far exceeding that of atomic nuclei.
Observational Probes of Compactness
Astronomers use two primary methods to constrain the compactness and ultimately the black hole collapse limit:
1. Gravitational Wave Observations (Tidal Deformability)
During the inspiral phase of a binary neutron star merger (like GW170817), the stars' gravitational fields deform each other. This deformation is quantified by the tidal deformability parameter ($\Lambda$), which is highly sensitive to the star's internal structure, specifically its radius and compactness.
Tidal Deformability and Collapse: A larger radius (less compact, "stiffer" EOS) leads to a higher $\Lambda$. Measurements from GW170817 constrained $\Lambda$ and, by extension, the radius of a $1.4 M_{\odot}$ neutron star to be in the range of $\approx 11-13 \text{ km}$. This radius range places a strong indirect limit on the star's compactness and maximum mass.
Prompt Collapse: The total mass of a merging binary dictates the fate of the remnant. If the total mass is above a certain threshold, the remnant will promptly collapse to a black hole, producing a distinct gravitational wave signal. This prompt collapse mass provides another observational constraint on $M_{\text{max}}$.
2. X-ray and Radio Observations (Mass and Radius)
Precise measurements of both the mass ($M$) and radius ($R$) of individual neutron stars, often in binary systems, provide direct constraints on their compactness ($\mathcal{C} = GM/Rc^2$) and the EOS. Observational data for high-mass neutron stars are particularly crucial for placing a lower limit on the TOV mass.
Summary
The ultimate boundary for a neutron star's compactness before it collapses into a black hole is set by the Tolman-Oppenheimer-Volkoff (TOV) limit, the maximum mass a neutron star can support against gravity.
| Parameter | Limit/Value | Significance |
| Maximum Compactness ($\mathcal{C}_{\text{max}}$) | $\mathbf{\approx 1/3}$ | Theoretical upper bound for a stable star. |
| Maximum Mass ($M_{\text{max}}$ - TOV Limit) | $\mathbf{\approx 2.1}$ to $\mathbf{2.3} \mathbf{M_{\odot}}$ (observational estimate for non-rotating) | The critical mass above which collapse is inevitable. |
| Minimum Stable Radius ($R_{\text{min}}$) | $\mathbf{R_{\text{min}} \gtrsim 3 \frac{GM_{\text{max}}}{c^2}}$ | The radius corresponding to the maximum mass. |
The exact values of $M_{\text{max}}$ and the corresponding $R_{\text{min}}$ remain a subject of active research, driven by joint theoretical models and multi-messenger observations involving gravitational waves and electromagnetic radiation.