Free-fall time

The free-fall time is the characteristic timescale over which a self-gravitating body, such as a uniform-density gas cloud, collapses to a point under its own gravitational attraction in the absence of any supporting forces like thermal pressure, turbulence, or magnetic fields.^[1] This concept originates from classical Newtonian gravity and represents the duration of free radial infall from an initial configuration to the center of mass.^[2] In astrophysics, it provides a benchmark for dynamical processes driven by self-gravity, particularly in the formation and evolution of stars, protostellar cores, and larger structures like molecular clouds.^[3] The standard formula for the free-fall time of a pressureless, uniform-density sphere is t_\mathrm{ff} = \sqrt{\frac{3\pi}{32 G \rho}}, where G is the gravitational constant and \rho is the mean mass density of the body.^[4] This expression is obtained by applying conservation of energy to the motion of infalling shells, integrating the radial velocity from the initial radius to zero while accounting for the time-dependent gravitational potential during homologous collapse.^[1] The timescale scales inversely with the square root of density, meaning denser configurations collapse more rapidly; for example, at the average density of the Sun (\rho \approx 1.4 g/cm³), t_\mathrm{ff} \approx 30 minutes.^[1] Variations of this formula exist for non-spherical geometries, such as filaments or sheets, which are common in interstellar medium structures.^[5] Free-fall time plays a central role in modeling gravitational instabilities, such as the Jeans criterion for cloud fragmentation, where collapse occurs if the cloud mass exceeds a critical value relative to its thermal support.^[2] In star formation, it sets the pace for gas accretion onto protostars, with observations indicating that only a small fraction—typically \epsilon_\mathrm{ff} \approx 0.01 or 1%—of a molecular cloud's mass converts to stars per free-fall time in nearby galaxies, highlighting the regulatory effects of feedback mechanisms like radiation and outflows.^[3] For typical molecular clouds with densities \rho \sim 10^{-20} g/cm³, t_\mathrm{ff} ranges from 1 to 10 million years, influencing the overall efficiency and duration of star cluster formation.^[6] Beyond stellar contexts, the free-fall time informs larger-scale phenomena, including the dynamical relaxation of galaxy clusters and the infall phases in cosmological structure formation.^[4]

Fundamentals

Definition

The free-fall time is the characteristic timescale for gravitational collapse under idealized conditions, defined as the duration required for a uniform, pressure-free sphere to contract to a point solely under its self-gravity. This concept establishes the fundamental rate at which unbound material would coalesce in the absence of supporting forces, serving as a baseline for analyzing dynamical processes in self-gravitating systems.^[7] In precise terms, the free-fall time represents the interval for a test particle, initially at rest at a given radius within a gravitational potential, to fall radially inward to the center. It assumes no pressure gradients, angular momentum, or other resistive effects that could alter the motion, thereby isolating pure gravitational acceleration.^[2] The concept originates from early work in astrophysics on gravitational collapse and instability, such as analyses by James Jeans.^[8] Key assumptions underpinning this definition include purely radial infall trajectories, negligible environmental resistance akin to a vacuum, and an initial condition of rest at the starting radius from the center.^[2] Qualitatively, the free-fall time scales proportionally to the square root of the initial radius cubed divided by the gravitational constant times the enclosed mass, mirroring the dependence of orbital periods in Kepler's third law for bound systems.^[1]

Basic Formula

The free-fall time t_{\mathrm{ff}} provides a characteristic timescale for gravitational collapse under self-gravity, assuming no supporting forces such as pressure or rotation. For a uniform-density sphere of total mass M and initial radius R, the free-fall time is given by

t_{\mathrm{ff}} = \pi \sqrt{\frac{R^3}{8 G M}},

where G is the gravitational constant.https://iopscience.iop.org/article/10.1088/0004-637X/744/2/190 Equivalently, in terms of the uniform density \rho = 3M / (4\pi R^3), this expression becomes

t_\text{orb} = 2\pi \sqrt{\frac{R^3}{GM}}.

This formula arises from equating gravitational attraction to the [centripetal force](/page/Centripetal_force) required for [circular motion](/page/Circular_motion) and follows from Kepler's third law generalized to Newtonian [gravity](/page/Gravity).[](https://science.nasa.gov/solar-system/orbits-and-keplers-laws/) In contrast, the free-fall time $ t_\text{ff} $ governs purely radial motion, where a [test particle](/page/Test_particle) starts from rest at $ R $ and falls inward under [gravity](/page/Gravity) alone. For a point mass,

t_\text{ff} = \frac{\pi}{2\sqrt{2}} \sqrt{\frac{R^3}{GM}} \approx 0.18 , t_\text{orb}.

The shorter free-fall time reflects the absence of angular momentum support in radial trajectories, allowing faster collapse compared to orbital circulation. This ratio is derived from integrating the radial equation of motion using conservation of energy, where the velocity is $ v(r) = \sqrt{2GM \left( \frac{1}{r} - \frac{1}{R} \right)} $, yielding the time as the integral $ t_\text{ff} = \int_0^R \frac{dr}{v(r)} $.[](https://arxiv.org/pdf/1005.5279) The distinction between these timescales has key implications for gravitational dynamics, particularly in systems where radial infall competes with orbital stability. Since $ t_{\rm ff} < t_{\rm orb} $, material can [collapse](/page/Collapse) inward before completing even a fraction of an [orbit](/page/Orbit), leading to rapid accretion and potential instabilities in flows assuming near-Keplerian [rotation](/page/Rotation). In astrophysical accretion disks, this drives phenomena like the viscous instability or clumping, as infalling matter disrupts azimuthal balance on dynamical timescales comparable to but shorter than full orbital periods.[](https://arxiv.org/pdf/1005.5279) Examples illustrate this radial-angular contrast in specific contexts. During the final stages of binary mergers, such as [neutron star](/page/Neutron_star) or [black hole](/page/Black_hole) systems, the inspiral follows orbital dynamics governed by $ t_\text{orb} $, but the ultimate coalescence occurs on free-fall timescales, enabling prompt merger and [gravitational wave](/page/Gravitational_wave) emission as [angular momentum](/page/Angular_momentum) is shed.[](https://arxiv.org/pdf/2201.12218) Similarly, in warped accretion disks—where misalignment between inner and outer regions induces [precession](/page/Precession)—regions with $ t_\text{ff} < t_\text{orb} $ promote infall over sustained orbiting, contributing to disk breaking or material drainage toward the central object.[](https://arxiv.org/pdf/2303.14210) In [general relativity](/page/General_relativity), for compact objects like black holes, GR corrections modify the free-fall time, reducing it relative to the Newtonian value by factors incorporating the compactness $ GM/(c^2 R) $. In the weak-field limit, GR corrections to the free-fall [proper time](/page/Proper_time) approach the Newtonian value. Near the event horizon, relativistic effects significantly shorten the proper time relative to Newtonian expectations, while the [coordinate time](/page/Coordinate_time) for distant observers diverges logarithmically. This is particularly pronounced for marginally bound geodesics from [infinity](/page/Infinity), where proper time to the horizon remains finite.[](https://arxiv.org/pdf/1707.05187)

Free-fall time

Fundamentals

Definition

Basic Formula

References