Astronomy & Celestial Mechanics
the sky, computed
Series

Secular theory — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Earth's eccentricity is a sum of 8 sinusoids. The eccentricity of Earth over 800 thousand years, from the Laplace–Lagrange solution for all eight planets — the secular matrix built from the JPL masses and semi-major axes, symmetrised, and diagonalised by Jacobi rotations. It runs between 0.0035 and 0.0436, and it has no period, because it is a sum of 8 incommensurable frequencies. The two largest contributions to this planet are the modes at 3.73 and 7.33 arcseconds per year, drawn as the flat lines: those are constants, and everything moving in the figure is their beat. The check is the quadratic form ½ΣΛe², which a symmetric secular matrix conserves exactly and which drifts by 6.7e-16 across the whole interval — computed from the same curves the figure draws and from nothing the eigenvalues were fitted to. The true angular momentum deficit, Σ Λ(1 − √(1−e²)), drifts by 7.1e-4, and that difference is not an error either: the two agree only to fourth order in e, and Mercury at 0.206 supplies almost all of the gap.

    No planet has an eccentricity of its own

    Strip the short-period terms out of the planetary equations and what is left is a linear system. Its eigenvectors are modes of the whole solar system, and the number a catalogue quotes for a planet's eccentricity turns out to be a reading of a clock rather than a property of the planet.

    part 1 · orbits
  2. The divisor is 0.129″/yr and the theory's own error is 0.24. Six frequencies of the secular solar system on one logarithmic axis, in arcseconds per year. The top two are the pair whose near-equality is the whole story: the perihelia of Mercury and Jupiter separate at 1.333″ a year, the nodes of Mercury and Venus at 1.462, and the difference of those two differences is 0.129 — a resonant argument that turns once every 10.0 million years. A term with that argument in the disturbing function acts in one direction for five million years at a stretch, which is what pumps Mercury's eccentricity, and it is the reason the inner solar system's Lyapunov time is what it is. The bottom three bars are why this figure exists. The divisor is smaller than the corrections the theory that computes it leaves out. Relativity contributes 0.4298″ a year to g₁ alone — the same 43 arcseconds a century that broke Newtonian gravity — which is 3.3 times the divisor; the fourth-order terms in the eccentricity that Laplace–Lagrange truncates come to about 0.24″, which is 1.8 times it; and the second-order solution computed on this page gets 0.35″, missing the published value by more than the value itself. A theory cannot bound what it cannot resolve. Laplace's proof that the eccentricities stay bounded is a proof about a system whose frequencies are constants, and the frequency that decides the question is not one.

    The bound that holds only in the linear theory

    Laplace proved the planetary eccentricities bounded, and the proof is a proof about a linearised system with constant frequencies. One combination of those frequencies is nearly zero — and it is smaller than the terms the linearisation threw away, which is why the stability of the solar system is a probability rather than a theorem.

    part 2 · orbits
  3. The eccentricity the planets force on an asteroid, and where it diverges. The forced eccentricity of a body in the inner Solar System against its semi-major axis, from the linear secular theory with all eight planets — its two giant-planet frequencies set to the measured 4.257 and 28.245 arcseconds a year — on a logarithmic axis: the largest value it reaches over the planetary cycles (solid) and its present value (dashed). Across most of the asteroid belt it is a few hundredths — 0.048 at 2.8 AU — the part of every asteroid's eccentricity that belongs to Jupiter and Saturn rather than to the asteroid. Near 2.04 AU it diverges: there the asteroid's own apsidal precession rate equals the frequency of the planetary mode dominated by Saturn, the denominator A − g₆ vanishes, and the theory's forced eccentricity goes to infinity. That is the ν₆ secular resonance. Real asteroids near it do not reach infinite eccentricity; they reach eccentricities large enough to cross the orbits of Mars and the Earth, and are removed. The location is for a body of low eccentricity and inclination; the full theory curves the resonance to larger distances at higher inclination, which is why the belt's inner edge is not a single number.

    A denominator that emptied the inner belt

    The eccentricity Jupiter and Saturn force on an asteroid is a fraction, and its denominator is the difference between the asteroid's own precession rate and one of the planets'. At two astronomical units that difference is zero. Where the zero sits depends on where Saturn is — and the belt's quiet orbits say Saturn did not take long to get there.

    part 3 · orbits
  4. Which pairs of planets the angular momentum deficit can bring together. For each adjacent pair of planets, the system's angular momentum deficit divided by the critical deficit at which that pair's orbits could touch, on a logarithmic scale; below one the pair cannot collide under secular evolution, whatever happens. Blue: using only the deficit of the pair's own subsystem — the four terrestrial planets (7.8 × 10⁻⁹ in units of a solar mass times √AU) or the four giants (4.3 × 10⁻⁶). Orange: using the whole Solar System's deficit, which the giants hold almost entirely. On their own budget the inner pairs sit at 0.21, 0.12, 0.34 — safe. On the whole budget they sit at 112, 65, 187. The giants are safe either way, at 0.05, 0.18, 0.36. The terrestrial planets are stable because almost none of the giants' deficit reaches them, not because there is too little of it.

    A deficit the giants hold and Mercury could borrow

    The angular momentum the planets' orbits lack of circular motion is conserved by every secular interaction, to every order, so it is a budget no planet can overdraw. Compared with what it would take to bring a pair of orbits into contact, it proves without an integration that the terrestrial planets cannot collide among themselves — and shows that they are safe only because the giants, who hold five hundred times as much, lend them almost none. A loan of half a per cent of Jupiter's would be enough to unsettle Mercury.

    part 4 · orbits

All series