Astronomy & Celestial Mechanics
the sky, computed
Series

Lagrange points — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. The five Lagrange points at mass fraction 0.12. The five points at which a small body can keep station with two larger ones. The three on the line of centres are roots of a quintic and are unstable; the two forming equilateral triangles are stable for a sufficiently lopsided mass ratio.

    Five places that keep station, in a problem with no solution

    Three bodies under gravity cannot be solved. Restrict the problem slightly and five exact answers fall out anyway — three of them roots of a quintic, two of them perfect equilateral triangles.

    part 1 · gravitation
  2. The surface a star stops at. The equipotential through L₁ — the Roche lobe — at mass fractions 0.50, 0.20, 0.05, in the frame that rotates with the pair. Each is a level set of exactly the same function the zero-velocity curves are level sets of, at exactly the critical value, so nothing here is a new construction: the Roche lobe is the last closed equipotential, and it is closed only because the two lobes touch at a single point. Material that reaches that point is no longer bound to the star it came from, and it leaves through an opening of zero area. The lobes are drawn in the orbital plane; in three dimensions each is a teardrop, and its volume-equivalent radius is what "the size of a Roche lobe" means. As the mass ratio becomes extreme the smaller star's lobe shrinks towards it, which is why a white dwarf accreting from a companion has a lobe smaller than the Sun.

    The surface a star stops at

    Around each star of a close pair there is a last closed equipotential, and the two touch at a single point. A star that swells to reach it hands its outer layers to its companion through an opening of zero area — and the transfer, once started, makes itself worse.

    part 2 · gravitation
  3. Two outcomes, and a boundary with no width. 26 trajectories launched from one point beyond L₂, all at the one speed the Jacobi constant C = 3.5124 permits there, differing only in the direction they set off in. The heavy curve is the zero-velocity boundary at that constant — the region no trajectory of this energy may enter — and it is open at L₂ by the neck the trajectories are aimed at. 11 of the 26 pass through into the secondary's realm and 15 turn back, and they are not interleaved — sweeping the launch direction through 65° finds one changeover and nothing in between. Bisecting the first of them pins it to 2.6e-12 radians, and the integrator runs out of digits before the boundary runs out of sharpness. That surface is the tube. It is the stable manifold of the periodic orbit about L₂, it separates transit from non-transit everywhere and not only in this fan, and a mission that wants to arrive for nothing has to be put inside it.

    The tube that leads out of a neck

    Below a certain energy the forbidden region opens at a Lagrange point, and a trajectory may pass. Which ones do is decided by a surface with no width at all — and two tubes that meet give a transfer that costs nothing at the join.

    part 3 · spaceflight
  4. A loop round the Earth that the Earth does not make. The path over one year, in the frame that turns with the Earth (Earth at the centre, the Sun far to the left, the Earth's motion upward), of bodies on heliocentric orbits with the Earth's period and eccentricities of 0.05, 0.1, 0.15, started at the Earth's mean longitude. Each traces a retrograde loop round the Earth, about 2ae long in the direction of motion and ae wide towards the Sun: for e = 0.1, 0.395 by 0.199 AU. The small circle is the Earth's Hill sphere, 0.010 AU in radius, inside which the Earth's gravity dominates the Sun's tide. Every loop lies far outside it. The loop is a difference between two ellipses with one period, and would be traced the same way if the Earth had no mass at all.

    A moon the Earth does not hold

    Seen from the Earth, the small asteroid Kamoʻoalewa circles the planet once a year, backwards, between forty and a hundred times as far away as the Moon. It is not a moon. It orbits the Sun, with the Earth's period and a slightly different eccentricity, and the loop is the difference between two ellipses — one the Earth would trace even with no mass at all. What the Earth's gravity does is subtler: it keeps the loop from drifting away.

    part 4 · gravitation

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