Generator

The hyperbolic-path generator

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.

10 essays call hyperbolic-path. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Every figure listed here is the same construction drawn at different numbers, so a correction to one is a correction to all of them.

Orbits

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.

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.

One admissible root, 0.01% from the truth. Gauss's reduction of three directions to a distance, drawn as the two relations whose intersection it is. Three observations of Ceres on days 0, 20, 40 of an arc, generated from its elements and used only as sight directions — no range, no radial velocity. The rising curve is geometry: the heliocentric distance a candidate at geocentric distance ρ₂ would have, r₂² = ρ₂² + 2ρ₂(R₂·L̂₂) + R₂², which contains no dynamics at all. The falling curve is dynamics: ρ₂ = A + µB/r₂³, with A and B built from the three sight vectors, the three observer positions and the three times, and containing no orbit. Eliminating ρ₂ between them gives r₂⁸ + a r₂⁶ + b r₂³ + c = 0 — an eighth-degree equation, from a problem with exactly as many equations as unknowns. Here they cross once at a positive ρ₂, at r₂ = 2.5893 AU against the true 2.5890. The other 2 real roots are rejected not by fitting but by sign: the ρ₂ each implies is negative, and an object behind the observer was not the thing observed.

Orbits

Three observations and no orbit at all

Three directions in space give six numbers for the six elements of an orbit, which sounds like a solved problem. The algebra that solves it is of the eighth degree, and for a near-Earth asteroid three perfect observations can be consistent with three different orbits.

Along a contour is free; across one costs, and 6 resonances sit on this one. Perihelion against aphelion, in units of Jupiter's orbit, with contours of constant v∞ — which is the Tisserand parameter through v∞² = (3 − T)v_p². Every contour crosses the line r_p = r_a = 1, because a spacecraft has to be at the planet's orbit to have an encounter there at all, and every point on one contour is reachable from every other point on it with no propellant: a flyby rotates the v∞ vector without changing its length, which moves the pump angle and slides the spacecraft along the curve. A burn is the only thing that moves it between curves, and that is the whole economy of a gravity-assist tour. The diagonals are resonant orbits — 1:1, 3:2, 2:1, 5:2, 3:1, 4:1 with the planet — and each is a straight line of slope −1 because a resonance fixes the semi-major axis and r_p + r_a = 2a. They matter because a spacecraft on one comes back to the same place at the same time as the planet, which is what makes a second encounter possible without waiting for a chance alignment. The marked points are where the v∞ = 0.3 contour meets each: a tour is a walk along the highlighted curve from one to the next, and the arithmetic that decides whether it can be walked is the turn one flyby delivers. At 1.35 Jupiter radii and 3.9 km s⁻¹ that turn is 163°, so the longest step drawn here needs 0.1 encounters — which is why a real tour has dozens of flybys and why the ones with the largest steps are the ones that need a deep-space manoeuvre in between.

Spaceflight

A map of the transfers that are free

Drawn as contours of perihelion against aphelion, the invariant that survives an encounter becomes a map. A flyby slides a spacecraft along its own contour and costs nothing; a burn is the only thing that moves it between contours — so tour design is reading a graph.

Orbits

An invariant that is only almost one

The Tisserand parameter survives a close encounter with Jupiter exactly, and comet families are separated by boundaries in it drawn to two decimal places. The exactness holds for a Jupiter on a circle. Jupiter's eccentricity is 0.0489, and integrating the same encounter twice shows what that costs.

What is left at infinity, and what it costs from 200 km. The velocity change needed to leave Earth from a circular orbit 200 kilometres up, against the speed the vehicle still has when the pull has run out. Escaping with nothing left over costs 3.22 km/s. Arriving at infinity with three kilometres a second in hand costs 3.63 — an extra 0.40 km/s to buy three, a discount of 87 per cent, because the two speeds do not add and their squares do — the departure speed is √(2μ/r + v∞²) — and adding a small square to a large one barely moves the root. The curve is therefore almost flat where the planetary missions are and straightens to slope 0.79 at the right-hand edge, where v∞ has overtaken escape speed and each metre a second bought costs a metre a second spent. The practical reading is that the expensive part of an interplanetary departure is getting to escape, and the destination is a surcharge on it.

Spaceflight

The speed that is left over

No mission is designed to sit on the escape threshold. What an interplanetary departure is specified by is the speed still remaining when the planet's pull has run out — and because energies add while speeds do not, the first three kilometres a second of it cost four hundred metres a second to buy.

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.

Orbits

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.

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.

Orbits

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

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.

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.

Orbits

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.

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