Astronomy & Celestial Mechanics
the sky, computed
Concept

Free-fall time — where it appears

The time a self-gravitating cloud takes to collapse with no pressure support, set by its mean density alone. It depends on the density alone and not on the size or the mass separately, which is why it is the same expression as a pulsation period and as a grazing orbit's period.

Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.

Forty-two minutes, from anywhere to anywhere. Left, the gravitational field inside the Earth: a straight line for a uniform sphere, because the enclosed mass grows as r³ and the field as r, and the PREM curve for the real one, which is nearly flat through the whole mantle at about 9.94 m/s² because the dense core is already all below. Right, what falls through it. A body dropped down a diametric tunnel through a uniform Earth executes simple harmonic motion with ω = √(g/R), reaching the far side in 42.2 minutes; a body dropped down a chord at 0.6 of the radius feels only the component along the tunnel, which is the same ω times a smaller distance, so it arrives in the same time from a shorter trip. The period does not contain the length of the tunnel, its direction, or where the body starts. And 2π/ω is also the period of a circular orbit grazing the surface, 84.3 minutes — the tunnel and the orbit are one ellipse, seen twice. The real Earth is not uniform and gets there in 38.2 minutes instead, which is the honest number and is 9% quicker.

The tunnel that takes the same time from anywhere

Inside a uniform sphere the field grows in proportion to the distance from the centre, which is Hooke's law. A body dropped down any straight tunnel arrives in the same time — and that time is the period of an orbit skimming the surface.

gravitation · Shell theorem The star-formation law, and the clock inside it. 46 model discs on the Kennicutt–Schmidt plane, built from Σ_SFR = 2.5e-4 Σ_gas^1.4 with 0.28 dex of scatter, with the exponent then fitted back off the drawn points at 1.42. The diagonal dashed lines are constant depletion times: gas divided by the rate consuming it. Every disc on this plane consumes its gas within a few gigayears, and a galaxy that has been forming stars for ten of them therefore cannot have been working from the gas it started with. The slope steeper than one is what makes the depletion time shorter where the gas is richer, which is the opposite of the intuition that a full tank lasts longer.

The gas runs out before the galaxy does

Divide a disc's gas by the rate it is turning that gas into stars and the answer is about two billion years — a seventh of the age of the disc. Every spiral still forming stars is therefore being fed from outside, and the star-formation law says the richer the disc, the sooner it starves.

galaxies · Star formation The critical mass falls as the cloud contracts, with a slope of −0.50. The Jeans mass — the least mass of molecular gas that its own pressure cannot support — against number density, for gas at 10, 20, 50 K, both axes logarithmic. The curves are straight because the mass goes as T^(3/2) ρ^(−1/2), and the slope measured off the drawn 10 K curve is −0.500 against the −0.5 that exponent requires. The consequence is the one that matters and it is a matter of sign: a cloud collapsing at constant temperature moves to the right along one of these lines, so the mass it takes to be unstable keeps falling, and sub-regions that were individually stable when the collapse began become individually unstable during it. A giant molecular cloud at 100 particles per cubic centimetre has a Jeans mass of 99 solar masses; a dense core at 10⁵ has one of 1.70. That is why a cloud of ten thousand solar masses makes a cluster rather than a star, and why the question about star formation is not what makes gas collapse but what stops the fragmenting.

The cloud that cannot hold itself up

A cold cloud collapses when gravity beats pressure, and the mass at which that happens falls as the square root of the density — so the collapse makes the condition for collapse easier, over and over, until the gas can no longer get rid of the heat. That is why a cloud of ten thousand solar masses makes a cluster and not a star.

galaxies · Molecular clouds A collapse that stops 29 astronomical units short. Equatorial rotation speed against radius for a collapsing cloud core of 1 solar mass, turning at 10⁻¹⁴ radians a second at a radius of 0.05 parsecs, compared with the Keplerian speed at the same radius. Both axes are logarithmic; both curves are computed from the same specific angular momentum, 2.4·10¹⁶ square metres a second, held fixed. The rotation speed rises as the reciprocal of the radius and the orbital speed only as its inverse square root, so they must cross, and the crossing is measured off the drawn curves at 4.27·10¹² metres — 29 astronomical units, or 6135 solar radii. Inside that radius the material is orbiting rather than falling, and no further collapse happens along the equator at all. To arrive at a star turning once in 25.38 days the core must dispose of a factor of 1.7·10⁴ in specific angular momentum, and nothing in the collapse itself removes any: it has to be handed to a magnetic field, to a disc, or to a companion. The figure assumes uniform rotation and a spherical core, both of which are simplifications a real core violates in the direction that makes the problem worse.

A cloud that cannot become a star

A molecular cloud core turning once every twenty million years sounds like a body at rest. Conserve its angular momentum through a collapse by a factor of ten thousand and it stops at the orbit of Neptune, spinning, having failed to make anything at all.

galaxies · Star formation The one number a cloud cannot change by squeezing. The mass-to-flux ratio in units of its critical value, against column density, for clouds threaded by fields of 3, 10, 30, 100 microgauss. λ below one is subcritical, and the field alone holds the cloud up, and no amount of compression changes that, because squeezing raises the magnetic and the gravitational energy at the same rate and leaves their ratio exactly where it was. λ above one is supercritical and the field is irrelevant to whether the cloud collapses. Each line has slope exactly one because λ is proportional to the column density at fixed field, and each crosses the boundary at 3.9·10²⁰ cm⁻² for 3 µG, 1.3·10²¹ cm⁻² for 10 µG, 3.9·10²¹ cm⁻² for 30 µG, 1.3·10²² cm⁻² for 100 µG. The Jeans mass is a threshold a cloud can cross by contracting; this one is a label it is born with, and the only way past it is to let the field leak out.

Support that cannot be squeezed away

A cloud held up by pressure can always be defeated by compressing it, because gravity gains faster than heat does. A cloud held up by a magnetic field cannot — squeezing raises both energies at exactly the same rate, so the ratio a cloud is born with is the one it keeps, and the only way out is to let the field leak.

galaxies · Molecular clouds Turbulence stops being supersonic at 0.04 parsecs. The velocity dispersion of molecular gas against the size of the region it is measured over, from σ = 1 (R/pc)^0.5 km/s, with the isothermal sound speed of 10 K gas drawn flat beneath it. The two cross at 0.035 parsecs, which is solved for here rather than quoted: below that scale the motions are subsonic and the gas is supported by its own pressure, above it they are supersonic and nothing thermal is relevant. At ten parsecs the Mach number is 17 and at a hundredth of a parsec it is 0.53. The crossing is the scale at which turbulent support runs out, and it is within a factor of two of the size of the dense cores that actually form stars — which is either the most important coincidence in the subject or the reason cores are the size they are.

The support and the seed are the same motions

A molecular cloud's lines are ten times wider than its temperature allows, and the width grows with the size of the region measured. The motions that widen them hold the cloud up as a whole and make the dense lumps inside it — so the same turbulence that delays star formation is what decides where it happens.

galaxies · Molecular clouds

Named alongside it

The objects these essays reach for when they reach for this one.

Jeans massMolecular cloudAmbipolar diffusionDense coreFragmentationInitial mass functionMass-to-flux ratioVirial parameterAngular momentum problemAngular momentum transportBinary fractionBonnor ebert sphere

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