Astronomy & Celestial Mechanics
the sky, computed
Concept

Steady state — where it appears

A condition in which a population's gains and losses balance, so that its size distribution stays fixed while its members change. A collisional cascade reaches one, with its shape set by fragmentation and its level by the supply from its largest bodies.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

Mass at the top, area at the bottom, 10 decades apart. Two moments of a collisional cascade's size distribution, per logarithmic interval of diameter, over 10 decades from a ten-micron grain to a hundred-kilometre parent body. Both axes are logarithmic and the vertical scale is arbitrary; only the slopes carry the argument. A population in which every collision makes fragments that go on to collide reaches a steady state where the same mass flows through every size per unit time, and that fixes the differential number distribution at an index of 3.5. The two consequences pull opposite ways. Mass per decade goes as the diameter to the power 0.5, so it climbs and almost all the mass is in the largest few bodies. Cross-sectional area per decade goes as the diameter to the power -0.5, so it falls, and almost all the area — which is what scatters light, what is detected, and what anything passing through gets hit by — is in the smallest. Over the range drawn the small end carries 10⁵ times the area of the large end and 10⁻⁵ times its mass. A disc's brightness therefore measures a population whose mass it says nothing whatever about, and the two numbers are connected only through the index of this line.

Where the mass is and where the light is

A population that grinds itself up settles into a size distribution with a fixed slope, and that slope puts almost all the mass in the largest bodies and almost all the cross-section in the smallest. So a debris disc's brightness measures a population whose mass it says nothing about, and the two are connected only by the exponent.

orbits · Collisional cascade Debris discs of very different masses, fading to one line. The fractional luminosity of a debris belt at 50 AU — the share of its star's light the dust absorbs and re-radiates — against age, from a million to ten billion years, for belts that start with 1, 10, 100, 1000 Earth masses of planetesimals up to 60 km across. Each stays near its starting brightness until its largest bodies have had time to collide, 16.5 billion years, 1.7 billion years, 165 million years, 17 million years respectively, and then fades as one over the age. A heavier belt starts brighter and grinds itself down faster, and the two effects cancel: after their collisional times all belts lie on the same line, the dashed maximum, whatever their mass. At ten billion years the two heaviest differ by 1 per cent.

A ceiling that forgets how much there was

A belt of planetesimals around a star shows itself only through its dust, and the dust is ground from the belt's largest bodies by collisions. Give the belt more mass and it makes more dust — but it also collides more often, and grinds itself away faster. At late times the two effects cancel exactly, and every belt at a given distance and age lies on the same line of maximum brightness whatever it started with. A disc above that line cannot be a steady cascade, and it is telling of a recent catastrophe.

orbits · Collisional cascade The binaries in a fed core are spread evenly in energy. The steady distribution of binding energies among the hard binaries of a core where new binaries enter at x = 5 and leave by ejection from a cluster whose escape speed is 4 times its dispersion: the number per unit binding energy, against binding energy in units of mσ², from 20,000 simulated binaries weighted by the time each spends at each energy. Between twice the entry energy and a third of the typical ejection energy (about 120) it is flat to within 5 per cent, and it falls away smoothly near ejection because a single encounter can throw a binary out from anywhere above about 24. A flat distribution is what a population with a constant flux through energy and a constant drift must have: binaries pass every energy at the same rate, and spend the same time at each.

The binaries a core keeps are spread evenly in energy

A hard binary in a star cluster gains binding energy at the same average rate however hard it already is. So in a core that is fed new binaries and loses hardened ones, every binding energy is passed through at the same speed, and the steady population is flat in energy — which, counted per factor of energy, means most binaries are within a factor of ten of being thrown out. Where they were born hardly matters; where they leave decides everything.

gravitation · Binary heating

Named alongside it

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

Collisional cascadeDebris discAsteroid familyBinary heatingBinary period distributionBinding energyCatastrophic disruptionCollisional lifetimeCore collapseCross-sectionDustEscape velocity

All concepts