Astronomy & Celestial Mechanics
the sky, computed
Series

Conic sections — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Every orbit one force allows. Circle, ellipse, parabola and hyperbola, all sharing a focus and a closest approach. The eccentricity alone decides which one a body is on, and whether it returns.

    Every orbit one force allows, and the number that picks between them

    Circle, ellipse, parabola, hyperbola. A single inverse-square force permits exactly these four, and one number decides which — including whether the body ever comes back.

    part 1 · orbits
  2. A vector that does not change, drawn five times. The Laplace–Runge–Lenz vector A = v × L − GM r̂, constructed at five points of one orbit at e = 0.45 under a force ∝ r^−2, each drawn from the body rather than from the focus so that its length and direction can be compared point by point. Under the inverse square every one of them is the same vector: identical to machine precision in length and in direction. Its length is 0.4500, which is the eccentricity, and it points at pericentre — so the orbit's orientation is a conserved quantity and not a constant of integration, which is why the ellipse does not turn.

    The third thing that is conserved

    Energy fixes an orbit's size and angular momentum fixes its shape. Neither fixes which way it points — so a curve that closes needs a third conserved quantity, and only two force laws in the universe supply one.

    part 2 · orbits
  3. A fall into the Sun from a standing start. The distance of a body released at rest 1 AU from the Sun, as a fraction of its starting distance, against time as a fraction of the fall's total duration, 64.6 days: the exact radial orbit (solid) and a fall at the constant acceleration felt at the start (dashed). The exact fall begins exactly like the constant one and finishes far faster, because the pull grows as the inverse square of the shrinking distance: it reaches the Sun at 79 per cent of the time the constant pull would take. The curve is a cycloid, and the total time is a fifth of the circular orbital period at the starting distance — exactly 1/(4√2) of it — because a straight fall is half of an ellipse whose semi-major axis is half the starting distance.

    A fall is half an orbit with no width

    Stop a planet dead and it falls into the Sun, in a time that is not a quarter of its year or a half but exactly one over four times the square root of two. The fall is an ellipse squeezed to a line, it spends four-fifths of its duration in the outer half of its path, and in a uniform cloud every shell of it arrives at the centre at the same moment — which is where the free-fall time of star formation comes from.

    part 3 · orbits
  4. One orbit seen from three angles. An orbit of eccentricity 0.5 with its periastron 40° from the line of nodes, seen face on and tilted by 60° and 75°, each panel centred on its apparent ellipse. Every projection is again an ellipse — of apparent eccentricity 0.88 and 0.97 — but only the face-on view has the star (red) at one of its foci (open circles). Tilted, the star sits 0.48 and 0.57 apparent semi-major axes from the nearer focus. The displacement is what measures the tilt.

    An orbit seen at an angle keeps everything but its focus

    No orbit outside the solar system is seen face on. It is seen as a shadow on the sky, and the shadow of an ellipse is an ellipse, swept at equal areas in equal times, indistinguishable in shape and timing from a real orbit. Only one thing gives the tilt away. The star the companion circles projects to where the true focus projects, and that is not where the apparent ellipse has its focus.

    part 4 · orbits

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