Astronomy & Celestial Mechanics
the sky, computed
Concept

Gravitational focusing — where it appears

The enlargement of a body's effective cross-section by its own gravity, which pulls in trajectories that would otherwise miss. The capture radius is R√(1 + v²ₑₛc/v²∞), so a slow approach sees a far larger target than a fast one and the effect dominates impact rates at low encounter speeds.

Named by 10 essays across 4 fields — each of them below, with the objects they name alongside it.

The aim point and the miss distance are the same number far out and nothing like it close in. Periapsis distance against aim point for a hyperbolic approach to Jupiter at v∞ = 5.6 km/s, both in planet radii. The diagonal is where the two would be equal — where gravity did nothing — and the curve falls below it everywhere, by more the closer in the aim is. A trajectory aimed at 10.68 radii grazes the surface, because gravitational focusing means the planet's effective size is √(1 + v_esc²/v∞²) times its radius. The slope of this curve is what a navigation team cares about: it is 0.208 at an aim of 12 radii and 0.936 at 150, so the same correction manoeuvre changes the periapsis distance by 4.5 times as much at one end of the range as at the other, while changing B by exactly the same amount at both; far outside this plot, where focusing has run out, it reaches 1.000 and the two numbers become the same one. That is why the aim point is the coordinate a manoeuvre is quoted in, why an error ellipse is published in the B-plane, and why the turn angle — 2 arctan(μ/Bv∞²), 156.0° at 12 radii and 77.8° at 70 — is thought of as a function of B rather than of anything the spacecraft does.

Aiming at a plane instead of at a planet

A spacecraft arriving at a planet is not aimed at a periapsis distance. It is aimed at a point in a plane perpendicular to its own incoming asymptote, because that is the one coordinate in which the miss distance responds linearly to a correction — and every navigation product ever published for a flyby is written in it.

orbits · Hyperbolic orbits After one revolution the error is 3π times longer than it is wide, and after 300 it is 2827. The two semi-axes of a fitted orbit's position uncertainty, against elapsed revolutions, for a solution whose semi-major axis is uncertain by 12 kilometres. The radial extent does not grow at all: a body on a slightly larger orbit is slightly further out and stays so. The along-track extent grows linearly, because δn/n = −(3/2)δa/a makes a semi-major-axis error into a mean-motion error and a mean-motion error into a phase that runs away — a·δM = 3πN·δa after N revolutions. The ratio is 3π ≈ 9.42 after a single revolution and 2827 after 300, which is why an asteroid recovered after one apparition is found within a few arcseconds of its predicted place along its own track and could be a long way from it in time. Every consequence of this in practice — that an impact probability is a one-dimensional integral rather than a volume, that a keyhole is an interval, that the next observation worth taking is the one across the track rather than the one that fits best — is a restatement of these two lines diverging.

An error that is nearly all in one direction

A fitted orbit's uncertainty is not a ball. Within a few revolutions it has collapsed onto a line along the track, because an error in the size of an orbit is an error in its period and an error in period is a phase that runs away — which is why an impact probability is an integral along a curve rather than over a volume.

orbits · Orbit determination A clock that is a straight line for three billion years and then is not. The lunar chronology function: craters of a kilometre or more per square kilometre against the age of the surface, with the count logarithmic and time not. The dashed line is the present impact rate extrapolated backwards, and it accounts for the whole curve up to 3.1 billion years — over that entire range, dating a surface is dividing a crater count by a constant. The rate itself, read off the slope of the drawn curve at the present day, is 8.4·10⁻⁴ craters per square kilometre per billion years, which over the whole Moon is about 32 new craters of a kilometre or more per million years. Past three and a half billion years the exponential term takes over and the curve turns almost vertical: ground that is 4.1 billion years old carries 26 times the crater density of ground 3.5 billion years old, for a difference in age of six hundred million years. Most of the craters on the Moon were made in a small fraction of its life, and nothing that happened after them is recorded anything like as densely. The six marked ages are laboratory measurements on returned rock, and they are what makes the curve a chronology rather than a shape — the Moon is the only body whose crater counts and whose radiometric ages have ever been measured on the same square kilometre.

A surface dated by counting holes in it

Every age quoted for a surface in the solar system outside the Earth — a Martian lava flow, a crater on Mercury, the ice of Europa — comes from counting craters and passing the count through one curve. That curve was calibrated on nine square kilometres of the Moon, and it is nearly a straight line for three billion years and then is not.

orbits · Surface chronology Hard below 8.3 astronomical units, soft above, and nothing settles at the line. The binding energy of a binary of two 0.7 solar-mass stars against its separation, both axes logarithmic, with the mean kinetic energy of a single cluster star at a velocity dispersion of 5 kilometres a second drawn as a level. Where the curve is above the level the binary is bound more tightly than a passing star's motion, and encounters on average take energy out of the field and put it into the pair; where it is below, they do the reverse. The crossing at 8.3 astronomical units is the hard–soft boundary, and it is the entire content of Heggie's law: hard binaries harden and soft binaries soften. The arrows are the direction each side moves, and they point away from the crossing in both directions rather than toward it. That is not a coincidence but a negative heat capacity, the same property that makes a star contract when it radiates: taking energy out of a bound pair moves it closer together and speeds it up, so a hard binary that gives energy to the cluster becomes harder still and gives more. A cluster with binaries in it therefore has a heat source that turns itself up, and the boundary drawn here is a watershed rather than an equilibrium.

A pair that heats what is trying to cool

A star cluster has a negative heat capacity, so it cannot reach equilibrium — its core contracts and gets hotter without limit. What stops it is a binary, and which way a binary exchanges energy with the stars around it is settled by one comparison of two energies.

gravitation · Binary heating Dynamical friction is strongest at 0.97 times the dispersion, and weaker either side. The dynamical friction on a body moving through a Maxwellian sea of stars, against its own speed in units of the square root of two times the velocity dispersion. The dashed curve is the fraction of the field moving slower than the body, which is the only part of the field that contributes: a star overtaking from behind pulls the body forward exactly as often as one being overtaken pulls it back, so the fast tail cancels out entirely. That fraction rises from nothing to one. The drag is that fraction divided by the square of the speed, and the quotient of a saturating numerator and a growing denominator has a single maximum, here at 0.97. The consequences run in both directions. A body moving much faster than the stars around it is barely slowed at all, which is why a galaxy passing through a cluster at a thousand kilometres a second does not sink; and a body already at rest with respect to the field feels nothing either, because there is no wake to be behind it. Sinking is therefore fastest in the middle of the process rather than at the start of it.

A drag that is strongest in the middle

Dynamical friction is not like air resistance. Only the stars moving slower than the body contribute at all, so the drag rises from nothing, peaks when the body has slowed to about the speed of everything around it, and falls away again — which is why a fast satellite is never captured and a slow one sinks in a hurry.

gravitation · Dynamical friction Accretion from a medium a body is moving through, against how fast it moves. The rate at which a gravitating body captures gas out of a medium it is ploughing through, divided by the rate it would capture at rest, against its Mach number, both axes logarithmic. The accretion radius is set by where the body's escape speed matches the speed of the gas relative to it, and that relative speed combines the sound speed with the motion in quadrature; the rate carries the square of that radius times the speed, which leaves one plus the square of the Mach number to the power minus three halves. Subsonic motion therefore costs almost nothing — the curve is flat below Mach one third — while supersonic motion costs the inverse cube of the speed, drawn here with a measured logarithmic slope of -2.99. The same focusing produces the wake and the drag: the body pulls a denser column behind it, that column pulls back, and the material closest to the axis is captured. Accretion and dynamical friction are not two processes but two accounts of one, which is why a body that grows by this mechanism is also being slowed by it.

The wake and the meal are one calculation

A body moving through gas gathers what passes inside the radius at which its escape speed matches the flow. That single radius sets both the drag it feels and the rate at which it grows, so accretion and friction are not two processes but two readings of one — and the rate falls as the inverse cube of the speed.

stars · Accretion Prograde and retrograde, the same encounter. The same encounter twice: a companion of 1 times the primary's mass passing at 1.5 disc radii, with the disc spinning the same way the companion orbits and then the other way. Nothing else differs. The prograde disc grows a bridge towards the companion and a tail away from it; the retrograde one is barely disturbed. The reason is a resonance rather than a force: in the prograde case the outer particles keep pace with the companion for a substantial part of the encounter and are pulled the whole time, and in the retrograde case they sweep past it and the impulses cancel.

Whether it merges is a ratio of two times

Two galaxies passing each other either merge or do not, and what decides it is not how close they come. It is whether the encounter lasts long enough for the internal motions to respond — a slow, prograde passage transfers orbital energy into stellar orbits and the pair is bound; a fast one leaves both galaxies heated and still moving.

galaxies · Galaxy interactions One detection, and every density it could mean. The number density of interstellar objects implied by one detection of a given size in 8 years, against the distance from the Sun inside which a survey could have seen it, for objects arriving at 26.3 km/s, if the survey caught 100 per cent, 10 per cent, 1 per cent of those passing inside that distance. Each estimate is the count divided by the rate of passages per unit density, πq²v∞ with gravitational focusing, times the efficiency. If ʻOumuamua-sized bodies could be seen inside 0.5 AU at 10 per cent efficiency, there are about 0.05 per cubic AU; if inside 2 AU with every one caught, 7.9 × 10^−4. The inferred density is a statement about the survey as much as about the visitors.

A census of visitors is a census of the survey

Three bodies from other stars have passed through the inner solar system in eight years of looking. Turning that count into a number of such bodies per cubic astronomical unit needs the volume in which they could have been seen, and that volume is set by the Sun's gravity, by each body's size and by how many weeks a survey had to catch it — so the first number the census gives is a measurement of the survey.

orbits · Hyperbolic orbits A tilt that wanders up, one passing body at a time. The inclination of the Moon's orbit to the plane it started in, against the number of lunar-mass bodies that have passed the Earth–Moon pair at 5 km/s within 300 Earth radii, with the Moon at 30 Earth radii, for four different sequences of passages. Each passage tilts the orbit by a fraction of a degree in a random direction, so the inclination performs a random walk and grows as the square root of the number of passages: after 1,000 the four runs stand at 3.8, 1.8, 3.9, 0.2 degrees. The dashed line is the Moon's present inclination, 5.145 degrees. Whether a given history reaches it is a matter of chance; how far a typical one gets is set by the passages' number, mass and speed.

A tilt built from a thousand near misses

The Moon's orbit is tilted five degrees to the ecliptic, and slow tidal recession cannot have done it. One candidate is the leftover debris of planet formation: bodies the size of the Moon, passing the young Earth–Moon pair, each giving its orbit a kick of a fifth of a degree in a random direction. A random walk of a few thousand such kicks reaches five degrees — and the bodies that would supply them are the same population whose impacts left the Earth's mantle its gold.

orbits · Angular momentum Captures counted over the Sun's life, cluster and field together. The cumulative number of ʻOumuamua-sized bodies Jupiter has captured since the Sun formed, against time, for birth clusters of 300, 1000, 3000 stars per cubic parsec whose bodies were captured at 4.3 a year per unit density for 3·10⁷ years, followed by today's visitors at 7.2 × 10⁻⁴ a year (dotted). The cluster's 3·10⁷ years supply 99.5%, 99.9%, 100.0% of all the captures the Sun has made: 7.2·10⁸, 2.4·10⁹, 7.2·10⁹ bodies, against 3.3·10⁶ from four and a half billion years of fast strangers. Almost everything Jupiter has ever captured, it captured from its siblings.

What Jupiter kept of the Sun's siblings

The Sun was born among a few thousand other stars, and while their planets formed they flung out small bodies at a few kilometres a second — exactly the speeds at which Jupiter can capture a passing body. Counted honestly, the birth cluster supplied almost every capture Jupiter has ever made: billions of bodies in thirty million years, against a few million in the four and a half billion since. And billions is still a hundred-thousandth of the Sun's own comets. Whatever the siblings gave the Sun, Jupiter was not how it arrived.

orbits · Hyperbolic orbits

Named alongside it

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

Velocity dispersionDynamical frictionImpulse approximationAngular momentumGlobular clusterHyperbolic excess speedImpact parameterInterstellar objectRelaxation timeWakeAbsolute magnitudeAccretion

All concepts