Laplace-Lagrange secular theory

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 $-(B_{12}+B_{21})$.