Astronomy & Celestial Mechanics
the sky, computed
Concept

Fragmentation — where it appears

The break-up of a collapsing cloud into smaller self-gravitating pieces. It happens because the critical mass falls as the square root of density in an isothermal collapse, so sub-regions that were stable at the start become individually unstable partway in.

Named by 4 essays across 3 fields — each of them below, with the objects they name alongside it.

A quadratic and a linear, crossing at 149 objects. The two rates that decide whether a shell at 900 km is stable, against how many objects are in it. Production goes as N² — every collision needs two objects, so the number of collisions is proportional to the square of the population, and each one is taken here to make 1600 trackable fragments. Removal goes as N, because drag acts on each object independently and takes 1,195 years to do it at this altitude. A quadratic and a linear cross exactly once, at 149 objects in this shell, and above that crossing the population grows with nothing launched. The shell presently holds about 2,280, which is 15 times the crossing. Every number on the production side is uncertain by a factor of a few — the fragment yield most of all, and the cross-section is calibrated against an observed collision rate rather than measured — so the position of the crossing carries that uncertainty with it. The shape does not, and the shape is the argument: a quadratic overtakes a linear once and never comes back, the crossing falls as the altitude rises because the lifetime is in the denominator, and what results is a threshold rather than a trend.

A collision rate that needs no collision

The flux through an orbital shell is a gas-kinetic calculation with no orbits in it. Production goes as the square of the population and removal goes as the first power, so a quadratic overtakes a linear once and never comes back — and which side of that a shell is on is decided by its altitude.

spaceflight · Orbital debris 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 A cloud of fragments that pinches where it was born. 240 fragments thrown from a satellite on a circular orbit 800 kilometres up, with kicks of about 100 metres a second in random directions in the orbital plane, each then following its own Kepler ellipse, drawn at 0.5, 1, 4.5, 20 orbital periods of the parent, with the Earth's centre marked and the break-up point at the top. Radial offsets from the parent's circle are exaggerated 2.5 times. Half a period after the break-up the fragments are spread 28 degrees along the orbit and 1374 kilometres in altitude, the kicks having carried some to higher apogees and others to lower perigees. After one period the along-track spread has grown to 53 degrees but the whole cloud is only 112 kilometres thick, and the 83 fragments within eight degrees of the break-up point span 35: every fragment's orbit passes through the break-up point, and the cloud pinches there once a period. By 4.5 periods it is 231 degrees long and 863 kilometres thick overall; by 20 it is 352 degrees long, still pinched to 11 kilometres where it began.

Every fragment comes back through the same point

When a satellite breaks up, its pieces fly off in every direction and every one of them returns, an orbit later, to the place where it broke. The cloud pinches there once a period, stretches along the orbit at a rate set by one component of each kick, and becomes a ring in hours.

orbits · Vis-viva

Named alongside it

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

Free-fall timeOrbital debrisAmbipolar diffusionAngular momentum problemAngular momentum transportBallistic coefficientBinary fractionBonnor ebert sphereCentrifugal barrierCollision cross-sectionConjunction analysisCritical density

All concepts