Astronomy & Celestial Mechanics
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Hyperbolic orbits — the series

2 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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
  2. 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.

    part 2 · orbits

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