The same start, four force laws

At its defaults it draws the same start, four force laws. Orbits under 4 different central force laws, every one started at the same radius with the same fraction — 0.72 — of the local circular speed, and every one integrated for 6 radial oscillations. The paths are drawn to different scales because the excursions differ; what is comparable between the panels is whether the curve retraces itself. Under the inverse square it does: the orbit is a closed ellipse and the sixth circuit lies exactly on the first. Under the linear force it does too, and the ellipse is centred on the source rather than focused on it. Under anything in between the path is a rosette that never closes, because the angle between successive closest approaches is not a rational fraction of a full turn. Those angles are 180.0° at n = -2, 145.9° at n = -1.5, 126.4° at n = -1, 90.0° at n = 1. The closure is not a matter of degree — a rosette that nearly closes is not nearly a closed orbit, since after enough circuits it fills the annulus.

orbit-closure is one function in lib/figures/mechanics.js — motion, force, energy and rotation. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page. A figure here is the figure a reader meets in an essay, and if the generator changes, this page changes with it.

Orbits under 4 different central force laws, every one started at the same radius with the same fraction — 0.72 — of the local circular speed, and every one integrated for 6 radial oscillations. The paths are drawn to different scales because the excursions differ; what is comparable between the panels is whether the curve retraces itself. Under the inverse square it does: the orbit is a closed ellipse and the sixth circuit lies exactly on the first. Under the linear force it does too, and the ellipse is centred on the source rather than focused on it. Under anything in between the path is a rosette that never closes, because the angle between successive closest approaches is not a rational fraction of a full turn. Those angles are 180.0° at n = -2, 145.9° at n = -1.5, 126.4° at n = -1, 90.0° at n = 1. The closure is not a matter of degree — a rosette that nearly closes is not nearly a closed orbit, since after enough circuits it fills the annulus.

A Kepler orbit of eccentricity 0.6, integrated for two revolutions, with the Laplace–Runge– Lenz vector constructed from the position and velocity at five points along it. Every one of the five is the same arrow: its length varies by 3.7e-11 over the whole run and its direction by 2.1e-10 radians. It points at the perihelion and its length is 0.600000, which is the orbit's eccentricity measured independently from the closest and furthest radii as 0.600000. Energy and angular momentum fix the size and shape of an orbit and say nothing about which way it points; this vector is the missing statement, and only an inverse square has one.

The length of the Laplace–Runge–Lenz vector against the eccentricity measured from the orbit's own closest and furthest radii, for 5 orbits integrated separately. 0.100000 against 0.100000; 0.300000 against 0.300000; 0.500000 against 0.500000; 0.700000 against 0.700000; 0.850000 against 0.850000. The largest discrepancy anywhere is 8.8e-9, which is the integrator's error rather than a physical difference. So the vector carries the shape of the orbit in its length and the orientation in its direction, and between them they say everything about the orbit that the energy and the angular momentum do not.

The direction of the Laplace–Runge–Lenz vector against time, for an orbit of eccentricity 0.6 under an inverse square with a small extra inverse-cube term of strength 0, 0.004, 0.01. With no extra term the direction is a horizontal line. With one it turns steadily, and the rate is 11.4403° per orbit at β = 0.004; 32.9136° per orbit at β = 0.01 — each checked against the movement of the perihelion measured from the orbit itself, which is an independent quantity and agrees to better than two per cent. So the vector is not merely a bookkeeping device: it turns when the orbit turns, at the same rate, and its conservation is exactly the statement that the orbit closes.

The direction of the Laplace–Runge–Lenz vector against time, for an orbit of eccentricity 0.3 under an inverse square with a small extra inverse-cube term of strength 0, 0.002, 0.006. With no extra term the direction is a horizontal line. With one it turns steadily, and the rate is 2.6579° per orbit at β = 0.002; 8.2880° per orbit at β = 0.006 — each checked against the movement of the perihelion measured from the orbit itself, which is an independent quantity and agrees to better than two per cent. So the vector is not merely a bookkeeping device: it turns when the orbit turns, at the same rate, and its conservation is exactly the statement that the orbit closes.

The rate at which the Laplace–Runge–Lenz vector turns for 5 bodies orbiting the Sun, from the general-relativistic correction to the inverse square, in arcseconds per century. Mercury: 42.98 computed against 43 quoted; Icarus: 10.06 computed against 10 quoted; Venus: 8.62 computed against 8.6 quoted; Earth: 3.84 computed against 3.8 quoted; Mars: 1.35 computed against 1.35 quoted. The formula has no free parameters at all — it is six pi times the gravitational radius divided by the semi-latus rectum, per orbit — and it matches every body, not only the famous one. Mercury's forty-three arcseconds a century is what is left of its perihelion's motion once the pull of the other planets has been subtracted, and it is a rotation of a vector that ought not to rotate at all.

physicscheck asserts something about orbit-closure that could fail — it draws it and measures the result against a value reached some other way.

Across the library: 100 interrogated, 2 exercised only, 1 untouched, of 103. Read out of the gate's source by the gate's own two patterns — and the gate's last claim fails the build if that read disagrees with what it was handed while running.