Astronomy & Celestial Mechanics
the sky, computed
Concept

Asteroid belt — where it appears

The region between Mars and Jupiter, from about 2.1 to 3.3 astronomical units, holding most of the Solar System's rocky minor bodies. Its gaps, families and edges record the resonances of the giant planets and the history of their orbits.

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

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 The eccentricity the planets force on an asteroid, and where it diverges. The forced eccentricity of a body in the inner Solar System against its semi-major axis, from the linear secular theory with all eight planets — its two giant-planet frequencies set to the measured 4.257 and 28.245 arcseconds a year — on a logarithmic axis: the largest value it reaches over the planetary cycles (solid) and its present value (dashed). Across most of the asteroid belt it is a few hundredths — 0.048 at 2.8 AU — the part of every asteroid's eccentricity that belongs to Jupiter and Saturn rather than to the asteroid. Near 2.04 AU it diverges: there the asteroid's own apsidal precession rate equals the frequency of the planetary mode dominated by Saturn, the denominator A − g₆ vanishes, and the theory's forced eccentricity goes to infinity. That is the ν₆ secular resonance. Real asteroids near it do not reach infinite eccentricity; they reach eccentricities large enough to cross the orbits of Mars and the Earth, and are removed. The location is for a body of low eccentricity and inclination; the full theory curves the resonance to larger distances at higher inclination, which is why the belt's inner edge is not a single number.

A denominator that emptied the inner belt

The eccentricity Jupiter and Saturn force on an asteroid is a fraction, and its denominator is the difference between the asteroid's own precession rate and one of the planets'. At two astronomical units that difference is zero. Where the zero sits depends on where Saturn is — and the belt's quiet orbits say Saturn did not take long to get there.

orbits · Secular theory Perihelia of a young family, run backwards until they meet. The longitude of perihelion of each of 26 members of a synthetic young asteroid family at the Karin family's position, relative to the family's mean at each moment, against time before the present. Today the angles are spread round the whole circle: members only a few thousandths of an astronomical unit apart precess at rates that differ by a fraction of an arcsecond a year, which over millions of years is many full turns. Run backwards — with each member's own drift in semi-major axis included — they converge, and the spread is smallest at 5.75 million years ago, where they were made. The nodes turn the other way, and in the real family, whose rates come from a full theory in which the two frequencies differ, they give a second, independent convergence at the same instant. An age read this way is a measurement of a time in the way a light curve's period is, with an error of a few per cent, rather than an inference from a drift rate.

An age read from angles that meet

A family of asteroids a few million years old has had its members' orbits turned round and round by the planets, each at a slightly different rate, until their orientations look random. Turn them back and they come together at one instant — the collision — with an uncertainty of a few tens of thousands of years, but only if each member's own slow drift in sunlight is put back too.

orbits · Asteroid families

Named alongside it

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

Proper elementsApsidal precessionAscending nodeAsteroid familyCollisional cascadeDebris discDisruption thresholdDohnanyi slopeForced eccentricityGravity regimeLaplace–Lagrange theoryLongitude of perihelion

All concepts