Astronomy & Celestial Mechanics
the sky, computed
Concept

Size distribution — where it appears

How the number of bodies in a population varies with their size, usually a power law. A collisional cascade among bodies of size-independent strength settles at an exponent of 3.5, so a measured slope is a statement about how thoroughly a population has been processed.

Named by 5 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 Below 0.46 microns a grain is not in orbit at all. The ratio of the radiation force to the gravitational force on a dust grain, against the grain's radius, for three densities. Every line has slope exactly −1 because gravity acts on the mass and radiation on the cross-section, and the ratio of a volume to an area is a length. Two horizontal lines matter and they are different statements. At β = 1 the star does not attract the grain at all. At β = 1/2 a grain released at rest from a circular orbit is already unbound, because it keeps the speed appropriate to the full stellar mass while feeling only half of it — and since dust is made by breaking up larger bodies that were on circular orbits, the lower line is the one that applies. For rock at 2500 kilograms a cubic metre that is 0.46 microns; for ice it is 1.15, and for iron 0.15. A collisional cascade that grinds material finer runs into this floor and stops, and the material that would have been finer leaves the system on a hyperbola.

The drag that sorts a disc by size

Starlight does three different things to a dust grain depending on how big it is — blows it out of the system, drags it inward over millennia, or ignores it entirely. Which one happens is decided by a single length, and whether it happens at all is decided by how crowded the disc is.

orbits · Collisional cascade Below 1.3 km a shadow stops getting smaller. The relative rate at which a star is occulted, against the smallest body a survey can detect, for size distributions with slopes 3.5, 4, 4.5. The cross-section of a body is not its own diameter: diffraction gives every shadow a minimum width of about the Fresnel scale, √(λD/2), which at 40 AU and 550 nm is 1.28 km. Above that the cross-section grows with the body, so lowering the limit gains events as the limit to the power -1.5 for the shallowest distribution drawn; below it the cross-section stops shrinking and only the number of bodies keeps rising, which is a shallower gain by one power. Pushing the limit from 40 km to 0.2 multiplies the rate by 3106196 at the steepest slope and 14131 at the shallowest — so the rate a survey measures is a measurement of the size distribution, which is the quantity a collisional history predicts and nothing else can reach at these sizes.

A population counted by shadows that never repeat

A body a kilometre across at forty astronomical units is a hundred million times too faint to image and casts a shadow just as dark as a large one. Monitoring enough stars fast enough catches those shadows — each one a single unrepeatable event of a fraction of a second, and the measurement is not any event but the rate.

sky · Occultations The weakest bodies are a few hundred metres across. The energy per unit mass needed to shatter a body and disperse half of it, against the body's diameter, for basalt and water ice, from fits to impact simulations calibrated on laboratory fragmentation. Small bodies are held together by material strength, which falls with size because a larger body contains larger flaws; large bodies are held together by their own gravity, which rises with size because the fragments have to be thrown apart as well as broken. The two regimes meet in a minimum at 235 m for basalt and 204 m for water ice, where a body needs the least energy per kilogram to destroy of any size — 127 J/kg and 57 J/kg, the kinetic energy per kilogram of motion at only 16 and 11 metres a second. A collisional population is therefore not self-similar: it has a preferred size, and its size distribution records it.

The size at which a body is easiest to break

A pebble is held together by the strength of its rock and a moon by its own gravity, and somewhere between them is a size that has too little of either. Bodies a few hundred metres across are the weakest in the solar system — sixteen metres a second of impact energy per kilogram is enough to scatter one — and a population grinding itself down records that weakness as a knee in its size distribution.

orbits · Collisional cascade Spun apart, or run out of ice: two clocks against size. Numbers of orbits against nucleus radius, both axes logarithmic, for a comet on 41P's orbit losing mass at 41P's rate per unit area. Solid: the orbits until the nucleus has lost all its mass to sublimation, growing in proportion to the radius. Dashed: the orbits until outgassing torques, changing sense at random from one orbit to the next, spin it past the rate at which a strengthless body of density 500 kg/m³ comes apart (a period of 4.7 hours) — growing as the fourth power of the radius, because each orbit's change of spin goes as the inverse square and a random walk needs the square of the number of steps. With a torque efficiency of 0.01, about what other comets' measured spin changes suggest, the two clocks cross at a radius of 1.2 km; with 41P's own 0.044, at 3.3 km. Smaller nuclei are spun apart before they run out of ice; larger ones run out first. The crossing moves as the two-thirds power of the efficiency.

The jet that pushes a comet also turns it

The gas a comet sheds near the Sun pushes its orbit, and because the vents are not arranged symmetrically about its spin axis, it also twists the nucleus. In 2017 one small comet's day more than doubled in two months. The twist per orbit grows as the inverse square of the nucleus's radius while its supply of ice grows only as the radius, so below about a kilometre a comet is spun apart before it runs out of ice — and the shortage of small comet nuclei may be the result.

orbits · Non-gravitational forces

Named alongside it

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

Collisional cascadeDebris discPower lawPoynting robertson dragRadiation pressureAsteroid beltAsteroid familyBeta meteoroidBlowout sizeCollision timescaleComet nucleusCross-section

All concepts