We investigate the dynamical evolution of coplanar hierarchical two-planet systems where the ratio of the orbital semimajor axes \alpha = a _ { 1 } / a _ { 2 } is small .
Hierarchical two-planet systems are likely to be ubiquitous among extrasolar planetary systems .
We show that the orbital parameters obtained from a multiple Kepler fit to the radial velocity variations of a host star are best interpreted as Jacobi coordinates and that Jacobi coordinates should be used in any analyses of hierarchical planetary systems .
An approximate theory that can be applied to coplanar hierarchical two-planet systems with a wide range of masses and orbital eccentricities is the octupole-level secular perturbation theory , which is based on an expansion to order \alpha ^ { 3 } and orbit-averaging .
It reduces the coplanar problem to one degree of freedom , with e _ { 1 } 
( or e _ { 2 } ) and { \varpi _ { 1 } - \varpi _ { 2 } } as the relevant phase-space variables ( where e _ { 1 , 2 } are the orbital eccentricities of the inner and outer orbits and \varpi _ { 1 , 2 } are the longitudes of periapse ) .
The octupole equations show that if the ratio of the maximum orbital angular momenta , \lambda = L _ { 1 } / L _ { 2 } \approx ( m _ { 1 } / m _ { 2 } ) \alpha ^ { 1 / 2 } , for given semimajor axes is approximately equal to a critical value \lambda _ { crit } , then libration of { \varpi _ { 1 } - \varpi _ { 2 } } about either 0 ^ { \circ } or 180 ^ { \circ } is almost certain , with possibly large amplitude variations of both eccentricities .
From a study of the HD 168443 and HD 12661 systems and their variants using both the octupole theory and direct numerical orbit integrations , we establish that the octupole theory is highly accurate for systems with \alpha \lesssim 0.1 and reasonably accurate even for systems with \alpha as large as 1 / 3 , provided that \alpha is not too close to a significant mean-motion commensurability or above the stability boundary .
The HD 168443 system is not in a secular resonance and its { \varpi _ { 1 } - \varpi _ { 2 } } 
circulates .
The HD 12661 system is the first extrasolar planetary system found to have { \varpi _ { 1 } - \varpi _ { 2 } } librating about 180 ^ { \circ } .
The secular resonance means that the lines of apsides of the two orbits are on average anti-aligned , although the amplitude of libration of { \varpi _ { 1 } - \varpi _ { 2 } } is large .
The libration of { \varpi _ { 1 } - \varpi _ { 2 } } and the large-amplitude variations of both eccentricities in the HD 12661 system are consistent with the analytic results on systems with \lambda \approx \lambda _ { crit } .
The evolution of the HD 12661 system with the best-fit orbital parameters and \sin i = 1 ( i is the inclination of the orbital plane from the plane of the sky ) is affected by the close proximity to the 11:2 mean-motion commensurability , but small changes in the orbital period of the outer planet within the uncertainty can result in configurations that are not affected by mean-motion commensurabilities .
The stability of the HD 12661 system requires \sin i > 0.3 .