We discuss the properties of orbits within the influence sphere of asupermassive black hole (BH), in the case that the surrounding star cluster isnonaxisymmetric. There are four major orbit families; one of these, the pyramidorbits, have the interesting property that they can approach arbitrarilyclosely to the BH. We derive the orbit-averaged equations of motion and showthat in the limit of weak triaxiality, the pyramid orbits are integrable: themotion consists of a two-dimensional libration of the major axis of the orbitabout the short axis of the triaxial figure, with eccentricity varying as afunction of the two orientation angles, and reaching unity at the corners.Because pyramid orbits occupy the lowest angular momentum regions of phasespace, they compete with collisional loss cone repopulation and with resonantrelaxation in supplying matter to BHs. General relativistic advance of theperiapse dominates the precession for sufficiently eccentric orbits, and weshow that relativity imposes an upper limit to the eccentricity: roughly thevalue at which the relativistic precession time is equal to the time fortorques to change the angular momentum. We argue that this upper limit to theeccentricity should apply also to evolution driven by resonant relaxation, withpotentially important consequences for the rate of extreme-mass-ratio inspiralsin low-luminosity galaxies. In giant galaxies, we show that capture of stars onpyramid orbits can dominate the feeding of BHs, at least until such a time asthe pyramid orbits are depleted; however this time can be of order a Hubbletime.
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