Three dimensional, axisymmetric cusps without chaos

We construct three dimensional axisymmetric, cuspy density distributions, whose potentials are of St\"ackel form in parabolic coordinates. As in Sridhar and Touma (1997), a black hole of arbitrary mass may be added at the centre, without destroying the St\"ackel form of the potentials. The construction uses a classic method, originally due to Kuzmin (1956), which is here extended to parabolic coordinates. The models are highly oblate, and the cusps are "weak", with the density, $\rho \propto 1/r^k$, where $0<k<1$.Comment: 5 pages, 2 figures, submitted to MNRA

Secular Instabilities of Keplerian Stellar Discs

We present idealized models of a razor-thin, axisymmetric, Keplerian stellar disc around a massive black hole, and study non-axisymmetric secular instabilities in the absence of either counter-rotation or loss cones. These discs are prograde mono-energetic waterbags, whose phase space distribution functions are constant for orbits within a range of eccentricities (e) and zero outside this range. The linear normal modes of waterbags are composed of sinusoidal disturbances of the edges of distribution function in phase space. Waterbags which include circular orbits (polarcaps) have one stable linear normal mode for each azimuthal wavenumber m. The m = 1 mode always has positive pattern speed and, for polarcaps consisting of orbits with e < 0.9428, only the m = 1 mode has positive pattern speed. Waterbags excluding circular orbits (bands) have two linear normal modes for each m, which can be stable or unstable. We derive analytical expressions for the instability condition, pattern speeds, growth rates and normal mode structure. Narrow bands are unstable to modes with a wide range in m. Numerical simulations confirm linear theory and follow the non-linear evolution of instabilities. Long-time integration suggests that instabilities of different m grow, interact non-linearly and relax collisionlessly to a coarse-grained equilibrium with a wide range of eccentricities.Comment: Manuscript accepted for publication in MNRA

Stellar Dynamics around Black Holes in Galactic Nuclei

We classify orbits of stars that are bound to central black holes in galactic nuclei. The stars move under the combined gravitational influences of the black hole and the central star cluster. Within the sphere of influence of the black hole, the orbital periods of the stars are much shorter than the periods of precession. We average over the orbital motion and end up with a simpler problem and an extra integral of motion: the product of the black hole mass and the semimajor axis of the orbit. Thus the black hole enforces some degree of regularity in its neighborhood. Well within the sphere of influence, (i) planar, as well as three dimensional, axisymmetric configurations-both of which could be lopsided-are integrable, (ii) fully three dimensional clusters with no spatial symmetry whatsover must have semi-regular dynamics with two integrals of motion. Similar considerations apply to stellar orbits when the black hole grows adiabatically. We introduce a family of planar, non-axisymmetric potential perturbations, and study the orbital structure for the harmonic case in some detail. In the centered potentials there are essentially two main families of orbits: the familiar loops and lenses, which were discussed in Sridhar and Touma (1997, MNRAS, 287, L1-L4). We study the effect of lopsidedness, and identify a family of loop orbits, whose orientation reinforces the lopsidedness, an encouraging sign for the construction of self-consistent models of eccentric, discs around black holes, such as in M31 and NGC 4486B.Comment: to appear in MNRAS, 10 pages, latex, 20 POstScript figure

Migration into a Companion's Trap: Disruption of Multiplanet Systems in Binaries

Most exoplanetary systems in binary stars are of S--type, and consist of one or more planets orbiting a primary star with a wide binary stellar companion. Gravitational forcing of a single planet by a sufficiently inclined binary orbit can induce large amplitude oscillations of the planet's eccentricity and inclination through the Kozai-Lidov (KL) instability. KL cycling was invoked to explain: the large eccentricities of planetary orbits; the family of close--in hot Jupiters; and the retrograde planetary orbits in eccentric binary systems. However, several kinds of perturbations can quench the KL instability, by inducing fast periapse precessions which stabilize circular orbits of all inclinations: these could be a Jupiter--mass planet, a massive remnant disc or general relativistic precession. Indeed, mutual gravitational perturbations in multiplanet S--type systems can be strong enough to lend a certain dynamical rigidity to their orbital planes. Here we present a new and faster process that is driven by this very agent inhibiting KL cycling. Planetary perturbations enable secular oscillations of planetary eccentricities and inclinations, also called Laplace--Lagrange (LL) eigenmodes. Interactions with a remnant disc of planetesimals can make planets migrate, causing a drift of LL mode periods which can bring one or more LL modes into resonance with binary orbital motion. The results can be dramatic, ranging from excitation of large eccentricities and mutual inclinations to total disruption. Not requiring special physical or initial conditions, binary resonant driving is generic and could have profoundly altered the architecture of many S--type multiplanet systems. It can also weaken the multiplanet occurrence rate in wide binaries, and affect planet formation in close binaries.Comment: The published version of the paper in compliance with Nature's embargo policy is available at http://nature.com/articles/doi:10.1038/nature1487

Modeling human trophoblast, the placental epithelium at the maternal fetal interface.

Appropriate human trophoblast lineage specification and differentiation is crucial for the establishment of normal placentation and maintenance of pregnancy. However, due to the lack of proper modeling systems, the molecular mechanisms of these processes are still largely unknown. Much of the early studies in this area have been based on animal models and tumor-derived trophoblast cell lines, both of which are suboptimal for modeling this unique human organ. Recent advances in regenerative and stem cell biology methods have led to development of novel in vitro model systems for studying human trophoblast. These include derivation of human embryonic and induced pluripotent stem cells and establishment of methods for the differentiation of these cells into trophoblast, as well as the more recent derivation of human trophoblast stem cells. In addition, advances in culture conditions, from traditional two-dimensional monolayer culture to 3D culturing systems, have led to development of trophoblast organoid and placenta-on-a-chip model, enabling us to study human trophoblast function in context of more physiologically accurate environment. In this review, we will discuss these various model systems, with a focus on human trophoblast, and their ability to help elucidate the key mechanisms underlying placental development and function. This review focuses on model systems of human trophoblast differentiation, including advantages and limitations of stem cell-based culture, trophoblast organoid, and organ-on-a-chip methods and their applications in understanding placental development and disease

Stellar Dynamics around a Massive Black Hole III: Resonant Relaxation of Axisymmetric Discs

We study the Resonant Relaxation (RR) of an axisymmetric low mass (or Keplerian) stellar disc orbiting a more massive black hole (MBH). Our recent work on the general kinetic theory of RR is simplified in the standard manner by ignoring the effects of `gravitational polarization', and applied to a zero-thickness, flat, axisymmetric disc. The wake of a stellar orbit is expressed in terms of the angular momenta exchanged with other orbits, and used to derive a kinetic equation for RR under the combined actions of self-gravity, 1 PN and 1.5 PN relativistic effects of the MBH and an arbitrary external axisymmetric potential. This is a Fokker-Planck equation for the stellar distribution function (DF), wherein the diffusion coefficients are given self-consistently in terms of contributions from apsidal resonances between pairs of stellar orbits. The physical kinetics is studied for the two main cases of interest. (1) `Lossless' discs in which the MBH is not a sink of stars, and disc mass, angular momentum and energy are conserved: we prove that general H-functions can increase or decrease during RR, but the Boltzmann entropy is (essentially) unique in being a non-decreasing function of time. Therefore secular thermal equilibria are maximum entropy states, with DFs of the Boltzmann form; the two-Ring correlation function at equilibrium is computed. (2) Discs that lose stars to the MBH through an `empty loss-cone': we derive expressions for the MBH feeding rates of mass, angular momentum and energy in terms of the diffusive flux at the loss-cone boundary.Comment: Submitted to MNRAS; 28 preprint pages, 3 figure