Secular perturbation

A secular perturbation survives averaging over fast orbital phases and drives slow precession and exchange of eccentricity or inclination.

For a quantity on a periodic Kepler orbit, its orbit average is $⟨x⟩=P^{-1}{\int }_0^{P}x(t)dt$. Equal increments of true anomaly are not equal increments of time. Eccentric anomaly gives the weight $dt=(1-e\cos E)dE/n$, which yields $⟨\cos f⟩=-e$. Slow orbital perturbations can be averaged using elements held fixed over a single orbital period.

The difference between the angular momentum of circular coplanar orbits at fixed semi-major axes and the system's actual angular momentum along the reference normal. With circular momenta $L_j=m_j\sqrt{G{M}_{\star }{a}_j}$ it is ${\sum }_j{L}_j(1-\sqrt{1-e_j^2}\cos I_j)$. For small planar orbital eccentricities its quadratic part is $\frac{1}{2}{\sum }_j{L}_j|z_j{|}^2$. Its conservation in $\dot{z}=iAz$ requires $L_j{A}_{jk}=L_k{A}_{kj}$.

The proper eccentricity is the magnitude of the homogeneous part of the complex eccentricity in linear secular perturbation theory. It remains constant in conservative theory, while its proper longitude of periapsis advances at the free precession frequency.

Laplace-Lagrange secular theory linearizes the orbit-averaged disturbing function to second order in eccentricities and inclinations. It represents planetary eccentricity evolution by a constant matrix.

For a test particle interior to a distant eccentric planet, write $\alpha =a/a_p\ll 1$. The leading Laplace coefficients are $b_{3/2}^1=3\alpha$ and $b_{3/2}^2=15{\alpha }^2/4$. Thus its forced eccentricity is $Z_f=5\alpha e_p/4$ and its secular frequency is $A=(3/4)n_p(m_p/M){\alpha }^{3/2}$. More distant inner particles have larger forced orbital eccentricities and faster secular precession, producing a growing gradient of complex eccentricity.

If both precession and forcing in $\dot{z}=ig(t)(z-z_f)$ share a common mass factor while $z_f$ stays fixed, then $z=z_f+(z(0)-z_f)\exp (i{\int }_0^{t}g(t\prime )dt\prime )$. The Argand diagram circle is unchanged; only its traversal speed varies. With $g(t)=g_{f}t/t_p$, the phase is $g_f{t}^2/(2t_p)$.

The eigenvalues of the Laplace-Lagrange secular matrix are apsidal precession frequencies, and its eigenvectors give the relative eccentricities and apsidal phases of the corresponding normal modes.

The complex eccentricity packages eccentricity and longitude of periapsis as $z=e{e}^{i\varpi }$, turning linear secular dynamics into a complex linear system.

With a fixed planetary pericentre, the linear secular equation $\dot{z}=iA(z-Z_f)$ has solution $z=Z_f+(z(0)-Z_f)e^{iAt}$. An initially circular orbit follows $z=Z_f(1-e^{iAt})$, a counterclockwise circle in the complex plane centered at the forced eccentricity $Z_f$, with free eccentricity amplitude $Z_f$. Its maximum orbital eccentricity is $2Z_f$ and its phase-averaged mean orbital eccentricity is $4Z_f/\pi$.

A secular eigenmode is a normal mode in which all complex eccentricities precess at one eigenfrequency with fixed amplitude ratios and relative apsidal phases.

For a real two-planet Laplace-Lagrange secular matrix with negative off-diagonal entries, the upper secular eigenmode has anti-aligned apsides and the lower has aligned apsides. Their positive planet-1 to planet-2 amplitude ratios obey $r_p{r}_m=A_{12}/A_{21}$. Under the conventional angular momentum deficit normalization this is $L_2/L_1$.

Complex inclination packages small orbital inclination $I$ and longitude of ascending node $\Omega$. It differs from complex eccentricity. Linear Laplace-Lagrange secular theory gives $\dot{\mathbf{y}}=iB\mathbf{y}$, whose matrix exponential solution decomposes into nodal secular eigenmodes.

The particular, forced part of a complex inclination solution. For a fixed, circular, inclined inner planet and an exterior test ring, linear Laplace-Lagrange secular theory gives $\dot{w}=-i\nu (w-w_p)$, so $w_f=w_p$ and $w=w_p+(w(0)-w_p)e^{-i\nu t}$. The ring describes a circle about the planetary tilt in an Argand diagram; relative to the planet, its mutual inclination stays constant while its ascending node precesses.

For circular planetary orbits with angular momenta $L_j=M_j\sqrt{G{M}_{\star }{a}_j}$, the inclination matrix has positive off-diagonal entries, zero row sums, and $L_j{B}_{jk}=L_k{B}_{kj}$. Positive diagonal symmetrization of a matrix with $W=\mathrm{diag}(L_j)$ makes $W^{1/2}B{W}^{-1/2}$ a symmetric matrix. Moreover,All eigenvalues are real and nonpositive. The zero eigenvalue is a common tilt of every orbital plane; nonzero modes describe relative nodal precession.

If two nearby inner planets couple much more strongly to each other than to a distant third planet, their slow secular eigenmode has a common complex inclination. With $L_{12}=L_1+L_2$, the effective inner-to-outer coupling isAngular momentum balance makes the reverse coupling $c{L}_{12}/L_3$. The reduced matrix has eigenvalues $0$ and $-c(1+L_{12}/L_3)$. If $L_3\gg L_{12}$, the nonzero frequency is approximately $-c$, while the fast inner differential mode has frequency $-(B_{12}+B_{21})$.

The forced eccentricity is the particular secular response of a small body to planetary eigenmodes, as distinct from its freely precessing homogeneous eccentricity.

A secular resonance occurs when a small body's free precession frequency equals a planetary secular eigenfrequency. Undamped linear theory then predicts resonant eccentricity growth.

Changing planetary masses or other system parameters move the frequencies of secular eigenmodes. A particle can cross a secular resonance when its free precession frequency meets a forcing frequency. Passage can change its free eccentricity and mix modes; a single scalar phase replacement generally cannot describe the response.

Linear eccentricity damping contributes $\dot{z}=-z/\tau$ to the complex-eccentricity equation, exponentially suppressing the free mode and regularizing a secular resonance.

Linear eccentricity damping on one planet changes $\dot{z}=iAz$ to $\dot{z}=i[A+i\mathrm{diag}({\tau }^{-1},0)]z$. For a nondegenerate, weakly damped two-planet matrix, let $D=\sqrt{(A_{11}-A_{22}{)}^2+4A_{12}{A}_{21}}$. The modal damping rates are $g_{p,m}=[1\pm (A_{11}-A_{22})/D]/(2\tau )>0$. Both secular eigenmodes damp even though only one planet is directly affected; the coupling shares the dissipation.

A secular torque is the orbit-averaged gravitational torque that changes orbital orientation or shape after fast orbital phases have been averaged out.

In a hierarchical orbital system, the quadrupole approximation keeps the first nontrivial term beyond the monopole in the orbit-averaged expansion in the ratio of inner to outer separation.

A binary star of total mass $M$, reduced mass $\mu$ and separation vector $\mathbf{a}$ has far-field potential functionwhere $\theta$ is the angle between $\mathbf{x}$ and $\mathbf{a}$. The dipole term vanishes in centre of mass coordinates. Averaging a circular coplanar binary over its fast phase gives $\overline{\Phi }=-GM/x-G\mu a^2/(4x^3)$.

For a distant coplanar circular orbit about a binary star, the averaged quadrupole potential of a circular binary givesAt fixed outer radius $x$, the quadrupole shortens the period relative to the monopole result. If the inner binary evolves slowly, the outer specific angular momentum stays constant in the axisymmetric averaged potential, but the outer energy can change. The outer radius must then be allowed to adjust when computing a period derivative.

Nodal precession is the gradual rotation of an orbit's line of nodes, usually driven by a nonspherical or external gravitational torque.

Differential nodal precession occurs when neighboring orbits precess at different rates, winding a flat collection of rings into a warp or a phase-mixed thick structure.

An inclined inner planet causes exterior debris disk rings to undergo nodal precession at different rates. In the quadrupole approximation, $\nu \simeq (3/4)n(M_p/M_{\star })(a_p/a{)}^2\propto a^{-7/2}$. Rings that have precessed appreciably tilt away from their original orbital plane, whereas distant rings remain near it, making an evolving warp. Conservative secular perturbations alone do not damp every ring into the planet's plane; unresolved phase mixing can instead produce a thick inner disk whose mean normal is near the forced plane.

The Laplace plane is the local equilibrium orbital plane about which a satellite or disk annulus precesses under competing secular torques.

The Laplace radius is the satellite distance where the small-tilt nodal precession contributions from a planet's axisymmetric quadrupole and the stellar tide have equal magnitudes. In the circular hierarchical limit those rates are $3G{M}_p{J}_2{R}_p^2/(2n{a}^5)$ and $3G{M}_{\star }/(4n{a}_p^3)$. Equating them proves the displayed fifth-root expression. Inside this radius the planetary quadrupole coefficient controls the preferred plane; outside it the double-averaged circular tidal potential dominates. When equator and planetary orbit are inclined to one another, their balance determines the local Laplace plane.

Apsidal precession is the gradual rotation of an orbit's line of apsides, equivalently a change in its longitude of periapsis.

The Kozai–Lidov mechanism exchanges eccentricity and inclination in a hierarchical gravitational system while preserving a component of angular momentum in the quadrupole limit.