No planet has an eccentricity of its own

The Astronomical Almanac gives the Earth’s orbital eccentricity as 0.0167. It is quoted to three figures, it appears in every textbook, and it is not a property of the Earth.

Across the four hundred thousand years either side of the present it runs between 0.0035 and 0.0436 — a factor of twelve — and it changes because the Earth’s orbit is not an isolated two-body problem but one voice in an eight-part chord. The chord, unlike any of the voices in it, holds its shape indefinitely.

The averaging that makes the problem linear #

The full planetary problem has no closed solution and, as the three-body problem establishes, no prospect of one. What makes secular theory possible is that most of the mutual perturbation between two planets averages to nothing.

Write the disturbing function — the extra potential each planet feels from the others — as a sum of cosines of combinations of the two orbital angles. Some of those combinations contain the mean longitudes, which run through a full circle once per orbit; over a few hundred revolutions their contribution averages to zero and stays there. What survives are the terms in which the fast angles cancel: the ones depending only on the apsidal longitudes and the nodes, which move on timescales of tens of thousands of years.

Averaging over the fast angles is not an approximation to the trajectory. It is a change of subject. The averaged system says nothing about where a planet is; it describes only the slow deformation of the orbit, and that is the object it turns out to be able to describe well.

The averaged equations, kept to first order in the planetary masses and second order in the eccentricities, are startling in their simplicity. Introduce the pair

$$h = e \sin\varpi, \qquad k = e \cos\varpi,$$

which are the eccentricity vector’s components — a point in a plane whose distance from the origin is the eccentricity and whose direction is the apse. Then

$$\dot h = +A\,k, \qquad \dot k = -A\,h,$$

with $A$ a matrix built from the planets’ masses and semi-major axes alone. The system is linear. Its coefficients are constants, its solution is a sum of sinusoids, and neither the eccentricities nor the apsidal longitudes appear anywhere in $A$.

What the matrix is made of #

The entries of $A$ are integrals over the relative geometry of two orbits, and they carry a name older than the theory:

$$b^{(j)}_{3/2}(\alpha) = \frac{1}{\pi}\int_0^{2\pi}\frac{\cos j\theta}{(1-2\alpha\cos\theta+\alpha^2)^{3/2}}\,\mathrm d\theta,$$

the Laplace coefficients, with $\alpha$ the ratio of the smaller semi-major axis to the larger. They are not obscure functions in disguise; they are what an inverse-cube force averaged around two circles amounts to. The diagonal entry $A_{jj}$ sums the effect of every other planet on planet $j$ and is positive, so an unperturbed apse precesses forwards. The off-diagonal entry $A_{jk}$ is negative and couples the two.

There is one condition on the matrix that is easy to state and load-bearing. Writing $\Lambda_j = m_j\sqrt{GMa_j}$ for each planet’s orbital angular momentum, the matrix must satisfy

$$\Lambda_j A_{jk} = \Lambda_k A_{kj}.$$

That relation makes $A$ similar to a symmetric matrix, which guarantees its eigenvalues are real. A complex eigenvalue would mean an exponentially growing eccentricity, and a solar system with one in it does not survive. The relation is a consequence of the disturbing function being a single scalar shared between two planets — it does not have to be imposed, and checking that it holds is the cheapest possible test of whether the matrix has been built correctly.

The eigenmodes #

Diagonalising $A$ turns eight coupled equations into eight independent ones. The eigenvalues $g_1,\dots,g_8$ are frequencies, and the eigenvectors say how much of each frequency each planet carries. The solution is

$$h_j = \sum_i E_{ji}\sin(g_i t + \beta_i), \qquad k_j = \sum_i E_{ji}\cos(g_i t + \beta_i),$$

so every planet’s eccentricity vector is a sum of eight circles, each turning at its own rate. The amplitudes $E_{ji}$ and phases $\beta_i$ are fixed once and for all by the present state; the frequencies belong to the system.

For the eight planets at their catalogued masses and semi-major axes, the linear theory gives frequencies of 0.63, 2.71, 3.73, 5.45, 7.33, 17.28, 17.96 and 22.51 arcseconds per year — periods from 57,000 years to two million. The two largest are Jupiter’s and Saturn’s; the two smallest belong to Uranus and Neptune, whose slow apsidal circulation sets the longest clock in the system.

The Earth’s case is worth spelling out because it has a consequence outside astronomy. The two modes with the largest amplitudes in the Earth’s eccentricity are the ones at 3.73 and 7.33 arcseconds per year — Jupiter’s and Venus’s. Their beat period is close to 400,000 years, and a second beat between neighbouring pairs gives about 100,000. Those two numbers are the eccentricity terms of the Milankovitch cycles, and they arrive in the ice record because the eccentricity modulates how much the seasonal insolation differs between the hemispheres.

The quantity that does not move #

Any linear system with a symmetric matrix conserves a quadratic form, and here it has a name and a meaning. The angular momentum deficit is

$$\mathrm{AMD} = \sum_j \Lambda_j\left(1 - \sqrt{1-e_j^2}\right) \approx \tfrac12\sum_j \Lambda_j e_j^2,$$

the shortfall between the system’s actual angular momentum and the angular momentum it would have if every orbit were circular in the same plane. The approximate form on the right is the conserved quadratic; the exact expression on the left is what a real system conserves, and the two agree to fourth order in the eccentricities.

That distinction is not pedantry, and the figures above measure it. The quadratic form holds to a part in $10^{15}$ across eight hundred thousand years, which is arithmetic. The true deficit drifts by seven parts in ten thousand over the same interval, which is the theory rather than the arithmetic: Mercury’s eccentricity reaches 0.21, and at that value $e^4$ is no longer negligible. The gap between the two numbers is a direct measurement of where the second-order expansion stops being true, and it is worth having, because it says the theory is good to a fraction of a per cent for the outer planets and to a fraction of a per cent less for Mercury.

The deficit also gives the system’s budget. Eccentricity and inclination can be traded between planets, and the modes are exactly the channels along which the trading happens, but the total is fixed. A planet cannot be handed a large eccentricity unless another gives one up, and there is only so much to give.

Proper elements, and a collision two billion years old #

The practical payoff of the eigenbasis is a change of coordinates that makes a family visible.

An asteroid’s osculating elements are what an orbit fit to a few weeks of astrometry produces. They contain the body’s own contribution plus the forced term — the part of its eccentricity that belongs to Jupiter, arriving through the same modes as everything else. Near the middle of the main belt that forced eccentricity is around 0.05, and it varies in phase from body to body according to each one’s semi-major axis.

Its proper elements are the amplitudes in the eigenbasis: the part left after the forced term is removed. Those are constant on secular timescales, which for the belt means hundreds of millions of years.

Hirayama noticed the clustering in 1918 using nothing but a plot of the osculating elements, which was possible because the families he found are tight enough to survive the smearing. Every family found since — and there are more than a hundred — has been found in proper elements, because the smearing is larger than the clustering for all but the youngest.

The ages follow from a second effect. Proper elements are constant under gravity alone, and the semi-major axis of a small body is not constant under sunlight: the Yarkovsky effect, a recoil from thermal re-emission, drifts small asteroids inwards or outwards depending on their spin. So a family that began as a compact cloud spreads in semi-major axis at a rate that depends on size, and the shape of the spread is a clock. The Karin family reads 5.8 million years; the Themis family, over two billion.

Where the linear theory stops #

Three limits are worth naming, and each is visible in the numbers.

The theory is second order in the eccentricities, so it degrades where they are large. Mercury at 0.21 is the worst case among the planets, and the drift in the true angular momentum deficit measures the damage at seven parts in ten thousand over eight hundred thousand years. For the asteroid belt, where eccentricities of 0.2 and inclinations of 20° are ordinary, the linear theory is not good enough and proper elements are computed by higher-order analytic theories or by direct numerical filtering of long integrations.

It is first order in the masses, and the neglected terms are of relative size $10^{-3}$. That shows up directly in the frequencies: the linear theory gives Jupiter’s mode at 3.73 arcseconds a year, and the value from a full numerical solution is 4.26 — a discrepancy of fourteen per cent, all of it in the terms this expansion drops. The eigenvector structure survives that correction; the numbers on the axis do not.

And it assumes no resonance. Where two frequencies are commensurable a small divisor appears and the averaging breaks down, and the solar system has several such places. The most important is the $g_1 - g_5$ secular resonance, which links Mercury’s apsidal frequency to Jupiter’s; the two are close enough that Mercury’s eccentricity can wander far beyond what the linear theory allows.

The other half of the system #

Everything above concerns the eccentricities and the apsidal longitudes. There is a second, entirely parallel system for the inclinations and the nodes, and it behaves the same way with one important difference.

Define $p = \sin i \sin\Omega$ and $q = \sin i \cos\Omega$ — the components of the orbit’s tilt, in the same way that $h$ and $k$ were the components of its eccentricity — and the averaged equations are again linear with a constant matrix. Diagonalising gives a second set of frequencies, conventionally written $f_1$ through $f_8$, and a second set of amplitudes.

The difference is that one of those frequencies is exactly zero, and it has to be. The system’s total angular momentum vector is conserved, so the plane perpendicular to it — the invariable plane — does not move. That fixed direction appears in the solution as a mode with zero frequency, and its eigenvector is the invariable plane’s own orientation.

The consequence for any individual planet is that its inclination is measured relative to a plane that is itself one of the modes. An inclination quoted with respect to the ecliptic is a quantity that mixes the Earth’s own inclination mode with the planet’s, which is why the invariable plane is the reference a dynamicist uses and the ecliptic is the reference an observer uses.

The inclination modes matter outside astronomy for the same reason the eccentricity modes do. The Earth’s obliquity — the tilt of its spin axis relative to its orbit — depends on the orbit’s orientation as well as on the spin, so the inclination modes drive a variation in obliquity with a period near forty-one thousand years. That is the second of the Milankovitch cycles, and it is the one that controls how strongly the seasons differ rather than how they are distributed between hemispheres.

Two linear systems, eight frequencies each, and between them they set the pacing of the ice ages — from a calculation whose only inputs are eight masses and eight semi-major axes.

There is a symmetry between the two systems worth noticing before leaving them. The eccentricity modes conserve the angular momentum deficit and the inclination modes conserve the direction of the total angular momentum, and both conservation laws come from the same place — the disturbing function being a single scalar shared between every pair of planets. A secular theory that violated either would be a secular theory built from a matrix that is not similar to a symmetric one, which is the same failure the reality of the eigenvalues depends on.

The edge of the belt, cut by a frequency #

The most visible thing secular theory has done to a population is not to the planets but to the asteroids, and it is a boundary rather than a gap.

An asteroid’s own apsidal precession rate depends on where it sits: closer in, the planets’ combined effect precesses it faster. At one particular semi-major axis that rate equals one of the planetary frequencies — the one dominated by Saturn — and there the perturbation stops averaging away and starts accumulating.

The consequence is that a body near that location has its eccentricity pumped, over a few million years, to values large enough that its perihelion falls inside the terrestrial planets. It is then removed, by a close encounter or by falling into the Sun.

The boundary that results is the inner edge of the main asteroid belt, at about two astronomical units, and it is sharp. It is not a gap in the sense the Kirkwood gaps are: it is a region depopulated by a secular resonance rather than by a mean-motion one, and it separates the belt from the empty region inside it.

Its practical importance is that it is a delivery route. A body drifting inward under the thermal recoil described in this collection’s essay on orbits moved by heat reaches this resonance eventually, is pumped, and becomes a near-Earth object within a few million years. A substantial fraction of the objects that cross the Earth’s orbit arrived by that route, and so did a substantial fraction of the meteorites in collections.

A frequency computed from an eight-by-eight matrix decides where the asteroid belt stops and how the meteorites get here, and the calculation has no free parameters in it at all.

What the change of basis actually bought #

It is worth being precise about what was gained, because the same manoeuvre recurs across the subject and it is not always recognised as the same one.

Nothing was solved. The planetary problem is as unsolvable after the averaging as before it. What happened is that a quantity that varies was replaced by a quantity that does not, at the cost of describing less: the eight amplitudes and their eight phase angles say nothing whatever about where any planet is, and they say everything about what the orbits will do over the next million years. The same trade is what makes an asteroid family recognisable, what makes the Milankovitch periods computable, and what allows a stability argument to be made about a system whose trajectories cannot be computed at all. It is also what makes the eccentricity in an almanac a slightly misleading number: it is correct, it is measured, and it belongs to this century rather than to the Earth.

Both of the applications above use the theory in the same way: not to predict where anything will be, but to identify a quantity that does not move.

Where this ladder goes next #

This rung establishes the linear theory and the elements that are constant in it. The rungs above it go three ways.

One follows the expansion upward: what changes when the second-order truncation in eccentricity is dropped, how the analytic theories that compute real proper elements handle inclinations of thirty degrees, and where the resulting series stop converging.

One follows the resonances, which is where the averaging fails by construction. A secular resonance is the place two of the frequencies above become commensurable, and the small divisor that follows is what drives Mercury’s long-term instability and what carves the $\nu_6$ boundary at the inner edge of the asteroid belt.

And one follows the frequencies themselves out of the solar system. The same eigenvalue problem, with two planets instead of eight, sets whether a warm Jupiter’s companion can maintain an eccentricity — and the same conserved deficit says how much eccentricity a compact system of planets is permitted to have at all, which is a real constraint on the architectures that survive migration.