Astronomy & Celestial Mechanics
the sky, computed
Concept

Bulk density — where it appears

A body's mass divided by its volume, without reference to how either is distributed. It is the easiest interior quantity to obtain and the weakest: a mixed body and a strongly layered one of the same composition have the same value, and for a small body it is well below the density of the meteorites believed to come from it.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

Mass against radius, and what lies between the curves. Planetary radius against mass on logarithmic axes, in Earth units, with composition curves computed from interior models rather than drawn through the points. The solid-planet curves are R ∝ M^(1/3.7) — flatter than a constant density because a heavier planet compresses itself — and the hydrogen curve turns over near three Jupiter masses, where degeneracy pressure takes over and adding mass makes the planet smaller. A mass alone or a radius alone places a planet on a line; only both together place it between two curves, and that is the whole argument for measuring a planet twice.

Two methods, and one density

A transit gives a radius. A wobble gives a mass. Neither says what a planet is made of, and the two together say it in one number — which is the only reason both are worth doing on the same object.

exoplanets · Planet composition The 2.3-hour spin barrier, and the small bodies that are allowed through it. Rotation period against diameter for a synthetic asteroid population, both axes logarithmic, with the horizontal lines marking where a body held together by nothing but its own gravity would fly apart. That limit is P = √(3π/Gρ) and it contains only the density: 3.30 hours at 1 gram per cubic centimetre, 2.33 hours at 2 gram per cubic centimetre, 1.91 hours at 3 gram per cubic centimetre. Size does not appear in it, which is what makes the figure's shape informative rather than obvious — a barrier that depended on size would be drawn as a slope, and a horizontal line crossing five decades of diameter is a much stronger statement. The observed population respects it. Above about two hundred metres nothing rotates faster than the 2.3-hour line, and the crowding just below that line is real: bodies pile up against a limit they cannot cross. Below two hundred metres the wall stops applying and the fast rotators appear, some of them turning in minutes. Nothing about gravity changes at that size. What changes is that a body small enough is a single coherent rock with tensile strength, while a body large enough is a pile of fragments with almost none, and the barrier is a measurement of which is which. The picture is a synthetic population drawn at random from a fixed seed rather than a catalogue, so the individual points are not asteroids; what is real is the barrier, its value, and the fact that only the smallest bodies are found beyond it.

A wall with no size in it

Spin a body held together by nothing but its own gravity, and past a certain rate it comes apart. The rate depends on density alone — 2.3 hours for the stuff asteroids are made of — and the observed population respects the limit exactly, for every body larger than a couple of hundred metres.

orbits · Rubble piles One mass and one radius, and every composition that gives them. A planet of 5 Earth masses and 1.6 Earth radii, and the compositions consistent with it. The horizontal axis is the fraction of the planet's mass in an iron core and the vertical axis the fraction in a water layer outside the rock; the heavy curve is every pair that reproduces the measurement exactly, and the band around it is what the 0.05 Earth-radius uncertainty allows. The answer is a curve, not a point, and that is not a failure of precision. Two numbers cannot determine three components: a planet can be made denser by adding iron or lighter by adding water, and along this locus the two changes cancel exactly. The ends of it are not small variations on one planet. At the left is a body with no iron at all and 0 per cent of its mass in water; at the right, one with an iron core like Mercury's and 23 per cent water. Those have different formation histories, different interiors, different everything, and the same mass and radius to the precision anybody can measure them. Breaking the degeneracy needs an observation that is neither a mass nor a radius. The usual one is a transmission spectrum, which measures the atmosphere's scale height and so its mean molecular weight — a hydrogen envelope and a steam envelope differ by a factor of nine in that, and the corresponding factor in the size of the spectral features. What the picture assumes is that the planet is differentiated into clean layers, which is the standard assumption and is false in detail: water dissolves into silicate melt at these pressures, and a mixed interior sits at neither end of this curve.

One density, and every planet that has it

A mass and a radius are two numbers, and a differentiated planet has at least three components. The set of compositions matching a measurement is therefore a curve rather than a point — and its two ends are a body with no iron and half its mass in water, and a body with a Mercury-like core.

exoplanets · Planet composition A wall at 2.33 hours that bends into a slope below 598 metres. The fastest rotation period a body of bulk density 2 g/cm³ can hold against its own spin, against its diameter, on logarithmic axes, for cohesive strengths of 0, 10, 100, 1000 pascals. With no cohesion the limit is the density-only barrier √(3π/Gρ) = 2.33 hours at every size. A cohesion C adds a stress that does not depend on size to a gravitational stress that goes as the square of the radius, so small bodies are held mostly by cohesion and can spin faster in proportion to their smallness: below the corner the limiting period goes as the diameter. The corner is where the two stresses are equal, at a diameter of 189 m for 10 Pa, 598 m for 100 Pa, 1.9 km for 1000 Pa. The scaling is the strength-regime form with a single coefficient of order one; detailed limits depend on the angle of friction and the shape. What the figure makes plain is why the observed spin barrier is sharp for kilometre-sized asteroids and fades below a few hundred metres, and why a handful of fast rotators a few hundred metres across can be rubble piles with a few tens of pascals of cohesion — a strength far below that of any rock — rather than monoliths.

A spin barrier with a corner in it

A rubble pile cannot spin faster than a period set by its density alone — 2.3 hours for most asteroids — at any size. Give it a cohesion of a few tens of pascals, weaker than any rock, and the barrier bends into a slope below a corner at a few hundred metres. The thermal torque that drives bodies to the barrier doubles their spin in a time that grows as the square of their size, so the bodies it pushes hardest are the ones that can go past.

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.

PorosityRubble pileSpin barrierCohesionCore mass fractionMass radius relationSub-neptuneSuper-EarthYORP effectAngular momentumAsteroidAsteroid family

All concepts