Fallback, Direct Collapse, and Remnant Mass

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A collapsing stellar core can leave a neutron star, form a black hole after an initially weak explosion, or proceed to black-hole formation with little successful ejection. These are not three perfectly separated mechanisms. They are regions of a continuum set by the proto-compact-object mass, the energy and timing of the outgoing shock, the binding energy of the overlying star, neutrino losses, and late accretion. The useful result is a mass budget, not the slogan that a star either “explodes” or “directly collapses.”

Helpful background. Black-Hole Geometry and Mechanics fixes the mass and horizon conventions used for the final object; Relativistic Matter and Compact Objects supplies neutron-star binding energy and maximum-mass concepts; Black-Hole Astrophysics supplies the observational interpretation of a birth-mass distribution; and Black-Hole Dynamics and Collapse supplies the relativistic collapse problem whose outcome is summarized here by a remnant prescription.

The mass budget

Let $M_{\rm fin}$ be the stellar baryonic mass immediately before collapse, $M_{\rm proto}$ the baryonic mass assigned to the proto-compact object, and $M_{\rm fb}$ the mass that is initially outside it but ultimately falls back. A common schematic prescription writes

M_{\rm fb}=f_{\rm fb}\bigl(M_{\rm fin}-M_{\rm proto}\bigr), \qquad M_{\rm rem,bar}=M_{\rm proto}+M_{\rm fb},

with $0\leq f_{\rm fb}\leq1$ . The limiting cases are transparent: $f_{\rm fb}=0$ means no fallback in this bookkeeping, while $f_{\rm fb}=1$ assigns the entire remaining star to the remnant. “Direct collapse” is often implemented as, or approximated by, the latter limit. The label does not prove that every layer crossed the horizon promptly, nor that the event had no neutrino, electromagnetic, or gravitational-wave signal.

The baryonic mass is not the gravitational mass measured from far away. Energy carried by neutrinos, photons, ejecta, and gravitational radiation lowers the latter. Some rapid population prescriptions approximate a black-hole gravitational mass by $M_{\rm rem}\simeq0.9M_{\rm rem,bar}$ ; Fryer and collaborators state their proto-object, fallback, and baryonic-to-gravitational mappings explicitly in Fryer et al. 2012, §§ 3.2–3.5 and eqs. (10)–(20). That factor is a prescription, not a universal relativistic identity.

Why matter returns

Matter is unbound only if its asymptotic total energy is positive after pressure forces, nuclear recombination, and gravity have done their work. A shell launched outward can remain marginally bound and return on a long, eccentric trajectory. A useful Newtonian diagnostic is

e_{\rm tot}=\frac{v^2}{2}+\epsilon_{\rm int}+\Phi(r), \qquad \Phi(\infty)=0;

where $\epsilon_{\rm int}$ is specific internal energy and $\Phi$ is the full gravitational potential. The approximation $\Phi\simeq-GM_c/r$ applies when a fixed central mass $M_c$ dominates; replacing it by $-Gm(r)/r$ omits the potential of exterior shells in an extended star. Negative $e_{\rm tot}$ is an instantaneous boundness diagnostic only after the internal-energy convention and continuing pressure work, heating, and potential evolution are specified. It is not a conserved parcel energy in a general hydrodynamic flow. Early fallback can be driven by an underenergetic shock. Later fallback can arise when the outgoing shock decelerates and a reverse shock travels back through the ejecta at a composition or density interface.

Fallback can push a proto-neutron star above its supported mass and make a black hole after a delay. Conversely, a formally failed shock need not imply perfectly silent swallowing: the loss of gravitational mass to neutrinos changes the envelope’s potential, and weakly bound outer layers can respond. Rotation can support a disk outside the newly formed black hole, producing outflows that reduce the mass actually retained. These effects show why $f_{\rm fb}$ should be an output of a specified dynamical model, not merely a synonym for initial mass.

From collapse calculations to fast prescriptions

Detailed calculations evolve hydrodynamics, gravity, an equation of state, nuclear composition, and neutrino transport. Population studies cannot normally rerun that problem for every binary in a cosmic sample, so they replace it with a mapping from pre-collapse structure to remnant type, mass, kick, and sometimes spin. “Rapid” and “delayed” mappings in Fryer et al. are two calibrated examples, designed to produce different transition behavior in remnant mass Fryer et al. 2012, §§ 4.1–4.3.

Such a mapping must declare at least four things: the structural input used to decide explodability; the assigned proto-object mass; the fallback rule; and the conversion from baryonic to gravitational mass. It should also state what happens near a neutron-star maximum mass and whether winds or binary stripping have already changed $M_{\rm fin}$ . Sukhbold and collaborators demonstrate that calibrated one-dimensional engines can yield nonmonotonic islands of explosion and failure rather than a single initial-mass boundary Sukhbold et al. 2016, § 5.2 and Figs. 19–20.

Results and limitations through 2026-08-18

The existence of fallback and black-hole-forming collapse is durable. The event-by-event division among prompt collapse, delayed collapse, weak ejection, and disk-assisted outflow remains unresolved, and competing calculations depend on progenitor structure and transport choices.

Position or evidence	Result	Scope and limitation
Energy and mass conservation	Marginally bound ejecta can return, increasing the remnant mass	Robust mechanism; the returned fraction depends on multidimensional flow and continued heating
Calibrated one-dimensional engines	Produce conditional remnant maps and nonmonotonic success/failure patterns	Efficient for grids; calibration and transport closure are model dependent
Multidimensional collapse and disk calculations	Reveal anisotropic accretion and outflows that a scalar $f_{\rm fb}$ cannot represent	Expensive and presently sparse across the full progenitor parameter regime
Failed-supernova searches and compact-object populations	Constrain, but do not uniquely identify, black-hole-forming channels	Selection effects, dust, binary interaction, and the source population confound inversion

The interpretation could change after a public, systematically sampled multidimensional progenitor grid changes the retained-mass mapping, after a statistically decisive failed-supernova census constrains the silent-collapse fraction, or after revised neutrino-loss or disk-outflow calculations alter the baryonic-to-gravitational conversion used in population models.

What the remnant mass can and cannot tell us

A measured black-hole mass constrains the combined effect of the pre-collapse core, envelope removal, explosion, fallback, and later accretion. It does not by itself distinguish a massive single star from a stripped binary, or weak fallback from nearly direct collapse followed by a disk wind. Natal kicks and binary survival provide complementary evidence, but they too require assumptions about mass loss and orbital phase.

The boundary of this treatment is deliberate. It does not derive neutrino-radiation hydrodynamics, determine the nuclear equation of state, or perform a gravitational-wave population likelihood. It supplies the remnant-mass interface those calculations consume. Within that scope, the best practice is to propagate multiple physically motivated prescriptions rather than attach false precision to one curve.

Common pitfalls

Equating direct collapse with zero signal. Little successful envelope ejection does not rule out neutrinos, gravitational radiation, an envelope response to neutrino mass loss, or disk-powered emission.

Mixing baryonic and gravitational mass. The difference is an energy budget. A fixed ten-percent correction is convenient for some population recipes but cannot replace a declared relativistic or calibrated mapping.

Exercise

A prescription assigns $M_{\rm proto}=1.30M_\odot$ , $M_{\rm fin}=12.0M_\odot$ , and $f_{\rm fb}=0.60$ . It then uses $M_{\rm rem}=0.9M_{\rm rem,bar}$ . Find both remnant masses and identify the main interpretive limitation.

Solution

The fallback mass is

M_{\rm fb}=0.60(12.0-1.30)M_\odot=6.42M_\odot.

Therefore $M_{\rm rem,bar}=1.30M_\odot+6.42M_\odot=7.72M_\odot$ , and the adopted conversion gives $M_{\rm rem}=6.95M_\odot$ to three significant figures. The arithmetic is exact within the stated recipe, but the answer is not a first-principles prediction: changing the explosion, fallback, maximum-neutron-star-mass, neutrino-loss, or disk-outflow prescription can change both steps.

References

Fryer, C. L., Belczynski, K., Wiktorowicz, G., Dominik, M., Kalogera, V., and Holz, D. E. (2012). “Compact Remnant Mass Function: Dependence on the Explosion Mechanism and Metallicity.” The Astrophysical Journal 749, 91. DOI.
Sukhbold, T., Ertl, T., Woosley, S. E., Brown, J. M., and Janka, H.-T. (2016). “Core-collapse Supernovae from 9 to 120 Solar Masses Based on Neutrino-powered Explosions.” The Astrophysical Journal 821, 38. DOI; model archive.