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Recent observations of globular clusters imposed major revisions to the previous paradigm, in which they were considered to be isotropic in velocity space and non-rotating. However, the theory of collisionless spheroids with some kinematic richness has seldom been studied. We present here a first step in this direction, owing to new results regarding the linear stability of rotating Plummer spheres, with varying anisotropy in velocity space and total amount of angular momentum. We extend the well-known radial orbit instability to rotating systems, and discover a new regime of instability in fast rotating, tangentially anisotropic systems.

Although 17 alpha-hydroxyprogesterone caproate (17P) has been shown to reduce the rate of recurrent preterm birth in singleton gestations, 17P did not reduce the risk of delivery or fetal death before 35 weeks of gestation in this randomized, placebo-controlled trial involving women with twin gestations. These data do not support the use of 17P to reduce the risk of preterm birth in twin gestations. In this trial involving women with twin gestations, 17 alpha-hydroxyprogesterone did not reduce the risk of delivery or fetal death before 35 weeks of gestation. Preterm birth is responsible for a substantial portion of infant mortality and persistent disability. The problem of preterm birth has proved largely intractable. In 2004, 12.5% of all live-born infants in the United States were delivered preterm — that is, before 37 completed weeks of gestation. 1 In a study published in 2003, weekly injections of 17 alpha-hydroxyprogesterone caproate (17P) were shown to lower the risk of recurrent preterm birth by one third in women who had previously given birth to a preterm infant spontaneously. 2 Although this finding is encouraging, only a minority of women destined to deliver preterm would qualify . . .

We report results from long term numerical integrations and analytical studies of particular orbits in the circular restricted three-body problem. These are mostly high-inclination trajectories in 1 : 1 resonance starting at or near the triangular Lagrangian L sub(5) point. In some intervals of inclination these orbits have short Lyapunov times, from 100 to a few hundred periods, yet the osculating semi-major axis shows only relatively small fluctuations and there are no escapes from the 1 : 1 resonance. The eccentricity of these chaotic orbits varies in an erratic manner, some of the orbits becoming temporarily almost rectilinear. Similarly the inclination experiences large variations due to the conservation of the Jacobi constant. We studied such orbits for up to 10 super(9) periods in two cases and for 10 super(6) periods in all others, for inclinations varying from 0 to 180. Thus our integrations extend from thousands to 10 million Lyapunov times without escapes of the massless particle. Since there are no zero-velocity curves restricting the motion this opens questions concerning the reason for the persistence of the 1 : 1 resonant motion. In the theory sections we consider the mechanism responsible for the observed behavior. We derive the averaged 1 : 1 resonance disturbing function, to second order in eccentricity, to calculate a critical inclination found in the numerical experiment, and examine motion close to this inclination. The cause of the chaos observed in the numerical experiments appears to be the emergence of saddle points in the averaged disturbing potential. We determine the location of several such saddle points in the (h, w) plane, with h being the mean longitude difference and w the argument of pericentre. Some of the saddle points are illustrated with the aid of contour plots of the disturbing function. Motion close to these saddles is sensitive to initial conditions, thus causing chaos.