Free-Fall Time Calculator

Leave a cloud of gas with nothing holding it up — no pressure, no rotation, no magnetic field, nothing but its own gravity — and it collapses to a point in a finite, calculable time. That number, the free-fall time, is one of the most-used timescales in star formation: it’s the natural clock against which every slowing-down mechanism (turbulence, magnetic support, rotation, radiative feedback) gets measured, and it sets the baseline question “is this cloud collapsing about as fast as gravity alone would allow, or is something holding it back?”

The formula

$\rho$ is the sphere’s (uniform) mass density and $G$ is the gravitational constant. Only density enters — not size, not mass on its own — because for a uniform sphere, size and mass cancel out of the collapse time exactly: a denser cloud always collapses faster than a puffier one, regardless of how big or how massive either actually is.

Two ways to specify density

Molecular clouds are almost always quoted as a number density of particles per cubic centimeter, not a mass density, so this calculator accepts either:

• Number density — enter $n$ (particles/cm³) and a mean molecular weight $\mu$ (in units of the hydrogen mass); the tool converts to mass density via $\rho = n\mu m_H$ . $\mu \approx 2.3$ is the standard default for molecular gas (H₂ plus a standard helium fraction); $\mu \approx 1.27$ is more appropriate for atomic (HI) gas.
• Mass density — enter $\rho$ directly, in kg/m³ or g/cm³ — useful for comparing against a planet’s or star’s mean density, or any density quoted directly in the literature.

Reading the visuals

• The collapse curve plots the exact trajectory of a uniform sphere’s outer edge as it falls — not a schematic, the real analytic solution to the pressure-free collapse problem, normalized to the sphere’s own initial radius $r_0$ and its own $t_{\rm ff}$ . Every uniform sphere collapses along this same normalized shape regardless of density: slow at first, then a sudden plunge right at the end. By the time half of $t_{\rm ff}$ has elapsed, the radius has only shrunk to about 84% of its starting value — most of the actual shrinking happens in the second half of the time, not the first.
• The comparison scale places this density’s free-fall time on a log scale next to the same five presets available as buttons above — spanning everything from a rocky planet’s mean density (tens of minutes) to a diffuse molecular cloud’s (millions of years), a spread of upward of ten orders of magnitude.

What this does and doesn’t capture

A real molecular cloud isn’t uniform, isn’t spherical, and isn’t pressure-free — it has internal structure, turbulence, magnetic fields, and (usually) at least some thermal or non-thermal support against gravity. A cloud with a nonuniform density collapses hierarchically — denser clumps inside it reach their own, shorter, local free-fall time and collapse first, well before the cloud’s overall average density would suggest — so $t_{\rm ff}$ computed from an average density is best read as an order-of-magnitude reference clock, not a precise prediction for any one particular clump. Comparing a cloud’s actual star-formation timescale to this reference clock is exactly how astronomers quantify how much something besides gravity must be holding star formation back.