Illustrated Physics
physics, drawn
Concept

Apsidal angle — where it appears

The angle swept round the centre between an orbit's closest point and its farthest. It is exactly 180° for an inverse-square force, so ellipses close, and for a nearly circular orbit under a force falling as r to the power n it is 180° divided by the square root of n + 3.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

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.

The orbit that does not come back to itself

A bounded orbit under any central force oscillates between a smallest and a largest radius for ever. That it should also return to the same point is a further demand, and only two force laws in existence meet it — the inverse square, and the linear spring.

astrophysics · Orbit stability The orbit a stretched sheet draws. A ball released sideways on a stretched elastic sheet dimpled by a central weight, at three-quarters of the speed for a circular orbit, seen from above, integrated from the exact equation of motion for a frictionless ball on the surface. A sheet under a point load sags logarithmically, which gives a force falling as 1/r, not 1/r², and the orbit is a rosette whose closest approaches come 131° apart, near the 127° of a 1/r force, never an ellipse. The demonstration shows orbits; it does not show Kepler's.

The sheet that cannot keep Kepler's orbits

A stretched elastic sheet with a heavy ball in the middle is the most popular picture of gravity there is: marbles roll round the dip like planets round a star. But a sheet pulled down by a point load sags as the logarithm of the distance, which is the potential of gravity in two dimensions, not three. Its marbles feel a force falling as 1/r, trace rosettes whose closest approaches are 127° apart instead of ellipses, and take a period proportional to their radius. The surface that does draw Kepler's orbits is a funnel shaped as −1/r, and only where it is shallow.

astrophysics · Orbit stability

Named alongside it

The objects these essays reach for when they reach for this one.

Central forceAngular momentumBertrand theoremBertrands theoremClosed orbitConserved quantityDegeneracyEffective potentialGravitational potentialThe inverse-square lawKeplers third lawMembrane

All concepts