Hyperbolic orbits — the series · Astronomy & Celestial Mechanics

1I/ʻOumuamua: an orbit at e = 1.201, and the angle it turned through. The open branch of a conic at eccentricity 1.201 and periapsis 0.2559 AU, with the Sun at the occupied focus. Two numbers fix everything else: |a| = q/(e−1) = 1.273 AU, an impact parameter b = |a|√(e²−1) = 0.847 AU, an asymptote at ν∞ = arccos(−1/e) = 146.37° from periapsis, and a deflection of δ = 2ν∞ − 180° = 112.74° — which is the same statement as sin(δ/2) = 1/e. The asymptotes cross at |a|e = 1.529 AU from the Sun on the periapsis side, where a bound orbit's centre would be on the other. The speed left over at infinity is √(μ/|a|) = 26.40 km/s. Schematic in one respect: the drawing runs at about 119 px to the AU, so the Sun's own disc would be far smaller than its marker.

The orbit that has no period

Above an eccentricity of 1 the conic is open, the energy is positive and the semi-major axis is negative — and the vis-viva relation survives the sign change without a single alteration. What replaces the period is a speed, and that speed is what says where a visitor came from.

part 1 · orbits

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.

part 2 · orbits

The spike, and where one passage through the planets sends it. The distribution of the reciprocal semi-major axis 1/a of long-period comets, in units of 10⁻⁴ AU⁻¹ and each curve scaled to its own peak. The narrow shaded peak is the Oort spike: comets arriving from the Oort cloud for the first time, with 1/a of about 4·10⁻⁵ AU⁻¹ and a spread of a few times 10⁻⁵. The broad curve is where those same comets go after one passage through the planetary region, whose perturbations change 1/a by a typical 5·10⁻⁴ AU⁻¹ — ten times the spike's width. 47 per cent end with negative 1/a, on hyperbolic orbits that leave the Solar System; only 4 per cent remain inside the spike's width. A comet in the spike is therefore on its first passage, almost by definition: the planets cannot put a comet back into the spike once it has been through them.

A spike only first-time visitors can be in

Long-period comets arriving from the Oort cloud have orbital energies packed into a spike a few hundred-thousandths of an inverse astronomical unit wide. One passage through the planets changes that energy by ten times as much, so no comet can be in the spike twice — and the spike's position, near thirty thousand astronomical units, is where the Galaxy's own tide is just strong enough to lift a comet's perihelion over Jupiter and Saturn in a single orbit.

part 3 · orbits

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.

part 4 · orbits

Jupiter keeps only the visitors already nearly bound

An interstellar object passing Jupiter can lose enough speed to stay, but one flyby can only turn the visitor's velocity relative to the planet, never shorten it, and the turn removes at most a few kilometres a second from its speed around the Sun. So capture is common for visitors that arrive with less than a kilometre a second to spare and almost impossible for the speeds the three known visitors had. A census of the slow tail of the interstellar population, and the orbits a capture leaves — more than a third of them backwards — follow from the same arithmetic.

part 5 · orbits

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.

part 6 · orbits