Astronomy & Celestial Mechanics
the sky, computed
Generator

The closure-test generator

One initial condition, three force laws
One initial condition, three force laws. The same pericentre and the same angular momentum, integrated under an attractive central force proportional to r^−n at n = 1.7, 2, 2.3, over 3 radial periods each. The equation solved is u″ + u = k u^(n−2)/L² with u = 1/r and the independent variable the polar angle, so the drawn apsidal angle is measured from the path rather than imposed on it: n = 1.7 advances by 156.3°, n = 2 advances by 180.0°, n = 2.3 advances by 216.1°. Only the inverse square returns its pericentre to the same direction — every other exponent draws a rosette that never closes, and the two exponents that do close every bound orbit are n = 2 and the harmonic n = −1, which is Bertrand's theorem. The eccentricity is 0.45 in the n = 2 case; the others start from the same state and have no eccentricity of their own, because their orbits are not conics.

The same pericentre and the same angular momentum, integrated under an attractive central force proportional to r^−n at n = 1.7, 2, 2.3, over 3 radial periods each. The equation solved is u″ + u = k u^(n−2)/L² with u = 1/r and the independent variable the polar angle, so the drawn apsidal angle is measured from the path rather than imposed on it: n = 1.7 advances by 156.3°, n = 2 advances by 180.0°, n = 2.3 advances by 216.1°. Only the inverse square returns its pericentre to the same direction — every other exponent draws a rosette that never closes, and the two exponents that do close every bound orbit are n = 2 and the harmonic n = −1, which is Bertrand's theorem. The eccentricity is 0.45 in the n = 2 case; the others start from the same state and have no eccentricity of their own, because their orbits are not conics.

1 essay calls closure-test. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Every figure listed here is the same construction drawn at different numbers, so a correction to one is a correction to all of them.

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. Orbits

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.

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