About 25 per cent of hot Jupiters (extrasolar Jovian-mass planets with close-in orbits) are actually orbiting counter to the spin direction of the star 1 . Perturbations from a distant binary star companion 2, 3 can produce high inclinations, but cannot explain orbits that are retrograde with respect to the total angular momentum of the system. Such orbits in a stellar context can be produced through secular (that is, long term) perturbations in hierarchical triple-star systems. Here we report a similar application to planetary bodies, including both the key octupole-order effects and tidal friction, and find that it can produce hot Jupiters in orbits that are retrograde with respect to the total angular momentum. With distant stellar mass perturbers such an outcome is not possible 2, 3 . With planetary perturbers the inner orbit's angular momentum component parallel to the total angular momentum need not be constant 4 . In fact, as we show here, it can even change sign, leading to a retrograde orbit. A brief excursion to very high eccentricity during the chaotic evolution of the inner orbit can then lead to rapid capture, forming a retrograde hot Jupiter.Despite many attempts 2,3,[5][6][7][8][9][10][11] , there is no model that can account for all the properties of the known hot Jupiter (HJ) systems. One model suggests that HJs formed far away from the star and slowly spiraled in, losing angular momentum and orbital energy to the protoplanetary disk 12, 13 . This "migration" process should produce planets with low orbital inclinations and eccentricities. However, many HJs are observed to be on orbits with high eccentricities, and misaligned with the spin direction of the star (as measured through the Rossiter-McLaughlin effect 14 ) and some of these (8 out of 32) even appear to be orbiting counter to the spin of the star. In a second model, secular perturbations from a distant binary star companion can produce increases in the eccentricity and inclination of a planetary orbit 15 . During the evolution to high eccentricity, tidal dissipation near pericenter can force the planet's orbit to decay, potentially forming a misaligned HJ 2, 3 . Recently, secular chaos involving several planets has also been proposed as a way to form HJs on eccentric and misaligned orbits 11 . A different class of models to produce a tilted orbit is via planet-planet scattering 5 , possibly combined with other perturbers and tidal friction 7 . In such models the initial configuration is a densly-packed system of planets and the final tilted orbit is a result of dynamical scattering among the planets, in contrast to the secular interactions we study here.In our general treatment of secular interactions between two orbiting bodies we allow for the magnitude and orientation of both orbital angular momenta to change (see Figure 1). The outer body (here either a planet or a brown-dwarf) gravitationally perturbs the inner planet on time scales long compared to the orbital period (i.e., we consider the secular evolution of the 1 system). We de...

The secular approximation for the evolution of hierarchical triple configurations has proven to be very useful in many astrophysical contexts, from planetary to triple-star systems. In this approximation the orbits may change shape and orientation, on time scales longer than the orbital time scales, but the semimajor axes are constant. For example, for highly inclined triple systems, the Kozai-Lidov mechanism can produce large-amplitude oscillations of the eccentricities and inclinations. Here we revisit the secular dynamics of hierarchical triple systems. We derive the secular evolution equations to octupole order in Hamiltonian perturbation theory. Our derivation corrects an error in some previous treatments of the problem that implicitly assumed a conservation of the z-component of the angular momentum of the inner orbit (i.e., parallel to the total angular momentum of the system). Already to quadrupole order, our results show new behaviors including the possibility for a system to oscillate from prograde to retrograde orbits. At the octupole order, for an eccentric outer orbit, the inner orbit can reach extremely high eccentricities and undergo chaotic flips in its orientation. We discuss applications to a variety of astrophysical systems, from stellar triples to merging compact binaries and planetary systems. Our results agree with those of previous studies done to quadrupole order only in the limit in which one of the inner two bodies is a massless test particle and the outer orbit is circular; our results agree with previous studies at octupole order for the eccentricity evolution, but not for the inclination evolution.Secular perturbations in triple systems also play an important role in planetary system dynamics. Kozai (1962) studied the effects of Jupiter's gravitational perturbation on an inclined asteroid in our own solar system. In the assumed hierarchical conc 0000 RAS