Astronomy & Celestial Mechanics
the sky, computed
Concept

Binary asteroid — where it appears

An asteroid with a satellite, or two of similar size orbiting each other. The satellite's period and distance give the pair's mass through Kepler's third law, and so the bulk density, which is how rubble piles were recognised.

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

A rubble pile that splits below a mass ratio of 0.204 can lose its piece; above it, the piece stays. The total energy of two spherical components of equal density in contact, spinning together at the rate at which their mutual gravity just holds them against the spin, against the mass ratio of the smaller to the larger, in units of G m₁²/R₁. The energy is the kinetic energy of the rotating pair minus their mutual gravitational binding. When the smaller piece is a small fraction of the whole, the spin carries more energy than the binding and the total is positive: a body spun to breakup that sheds a fragment of that size has enough energy for the fragment to escape entirely, becoming a separate asteroid on a nearly identical orbit. The total changes sign at q = 0.204. Above that ratio the pair cannot separate without an energy source; it stays as a binary, orbiting and eventually synchronising, or re-accretes. As q goes to zero the energy tends to 0.2 G m₁²/R₁, the rotational energy of the primary alone at its breakup rate. Nothing in the threshold depends on the size or the density of the body — it is a pure number from the geometry of two touching spheres — and asteroid pairs sharing an orbit have been found overwhelmingly with estimated mass ratios below it.

A split that decides whether the piece can leave

A rubble pile spun past its limit splits in two, and whether the smaller piece escapes or stays in orbit is not a matter of luck. Two touching spheres spinning at their shared limit have positive total energy only when the smaller is less than 0.204 of the larger's mass — a number with no size and no density in it. Below it the pieces can become a pair of asteroids on nearly identical orbits; above it, a binary. And the larger the piece that leaves, the slower the body left behind.

orbits · Rubble piles A moonlet's period measures its primary's density. The orbital period of a small satellite against the radius of its orbit in units of the primary's radius, both logarithmic, for primaries of bulk density 1, 2, 3.5 g/cm³. The size of the asteroid cancels: a moonlet at five radii from a body of 2 g/cm³ takes 26 hours whether the body is a kilometre or a hundred across. At one radius the curves give the spin period at which a strengthless rubble pile flies apart — 3.3, 2.3 and 1.8 hours. Didymos–Dimorphos, at 3.1 radii and 11.92 hours, gives 2.4 g/cm³; 1999 KW4, at 3.9 radii and 17.42 hours, gives 2.1 g/cm³.

A moonlet that weighs a pile of rubble

Put Kepler's third law round an asteroid and its size cancels: a moonlet's period depends only on how many of the asteroid's radii away it orbits and on the asteroid's density. So the first asteroid moon ever seen, a mile-wide speck beside Ida, turned a picture into a density — and the density came out too low for solid rock. A fifth to a half of the asteroids weighed this way is empty space.

orbits · Harmonic law

Named alongside it

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

Rubble pileSpin barrierAngular momentumAsteroidAsteroid pairBulk densityContact binaryFree energyKepler's third lawKinetic impactorMass ratioPorosity

All concepts