Dynamics of Circumstellar Disks III: The case of GG Tau A

(abridged) We present 2-dimensional hydrodynamic simulations using the Smoothed Particle Hydrodynamic (SPH) code, VINE, to model a self-gravitating binary system similar to the GG Tau A system. We simulate systems configured with semi-major axes of either $a=62$~AU (`wide') or $a=32$~AU (`close'), and with eccentricity of either $e=0$ or $e=0.3$. Strong spiral structures are generated with large material streams extending inwards. A small fraction accretes onto the circumstellar disks, with most returning to the torus. Structures also propagate outwards, generating net outwards mass flow and eventually losing coherence at large distances. The torus becomes significantly eccentric in shape. Accretion onto the stars occurs at a rate of a few $\times10^{-8}$\msun/yr implying disk lifetimes shorter than $\sim10^4$~yr, without replenishment. Only wide configurations retain disks by virtue of robust accretion. In eccentric configurations, accretion is episodic, occurs preferentially onto the secondary at wrates peaked near binary periapse. We conclude that the \ggtaua\ torus is strongly self gravitating and that a major contribution to its thermal energy is shock dissipation. We interpret its observed features as manifestations of spiral structures and the low density material surrounding it as an excretion disk created by outward mass flux. We interpret GG Tau A as a coplanar system with an eccentric torus, and account for its supposed mutual inclination as due to degeneracy between the interpretation of inclination and eccentricity. Although the disks persist for long enough to permit planet formation, the environment remains unfavorable due to high temperatures. We conclude that the GG Tau A system is in an eccentric, $a\sim62$~AU orbit.Comment: Accepted for publication in the Astrophysical Journa

On the Early Evolution of Forming Jovian Planets I: Initial Conditions, Systematics and Qualitative Comparisons to Theory

(abridged) We analyze the formation and migration of a proto-Jovian companion in a circumstellar disk in 2d, during the period in which the companion makes its transition from `Type I' to `Type II' migration, using a PPM code. Spiral waves are generated by the gravitational torque of the planet on the disk. Their effects are to cause the planet to migrate inward and the disk to form a deep (low surface density) gap. Until a transition to slower Type II migration, the migration rate of the planet is of order 1 AU/10$^3$ yr, and varies by less than a factor of two with a factor twenty change in planet mass, but depends near linearly on the disk mass. Although the disk is stable to self gravitating perturbations (Toomre $Q>5$ everywhere), migration is faster by a factor of two or more when self gravity is suppressed. Migration is equally sensitive to the disk's mass distribution within 1--2 Hill radii of the planet, as demonstrated by our simulations' sensitivity to the planet's assumed gravitational softening parameter. Rapid migration can continue after gap formation. Gaps are typically several AU in width and display the \mplan$^{2/3}$ proportionality predicted by theory. Beginning from an initially unperturbed 0.05\msun disk, planets of mass $M_{\rm pl}> 0.3$\mj can open a gap deep and wide enough to complete the transition to slower \ttwo migration. Lower mass objects continue to migrate rapidly, eventually impacting the inner boundary of our grid. This transition mass is much larger than that predicted as the `Shiva mass' discussed in Ward and Hahn (2000), making the survival of forming planets even more precarious than they would predict.Comment: 39 pages incl 13 figures. High resolution color figures at http://www.maths.ed.ac.uk/~andy/publications.htm

Parsimoneous Modeling of Yield Curves for U.S. Treasury Bills

A new model is proposed for representinq the term to maturity structure of interest rates at a point in time.The model produces humped, monotonic and S-shaped yield curves using four parameters. Conditional on a time decay parameter, estimates of the other three are obtained by least squares. Yield curves for thirty-seven sets of U.S. Treasury bill yields with maturities up to one year are presented. The median standard deviation of fit is just over seven basis points and the corresponding median R-squared is .96. Study of residuals suggests the existence of specific maturity effects not previously identified. Using the models to predict the price of a long term bond provides a diagnostic check and suggests directions for further research.

The Divine Clockwork: Bohr's correspondence principle and Nelson's stochastic mechanics for the atomic elliptic state

We consider the Bohr correspondence limit of the Schrodinger wave function for an atomic elliptic state. We analyse this limit in the context of Nelson's stochastic mechanics, exposing an underlying deterministic dynamical system in which trajectories converge to Keplerian motion on an ellipse. This solves the long standing problem of obtaining Kepler's laws of planetary motion in a quantum mechanical setting. In this quantum mechanical setting, local mild instabilities occur in the Kelperian orbit for eccentricities greater than 1/\sqrt{2} which do not occur classically.Comment: 42 pages, 18 figures, with typos corrected, updated abstract and updated section 6.

Interaction of a giant planet in an inclined orbit with a circum-stellar disk

We investigate the dynamical evolution of a Jovian--mass planet injected into an orbit highly inclined with respect to its nesting gaseous disk. Planet--planet scattering induced by convergent planetary migration and mean motion resonances may push a planet into such an out of plane configuration with inclinations as large as $20^\circ-30^\circ$. In this scenario the tidal interaction of the planet with the disk is more complex and, in addition to the usual Lindblad and corotation resonances, it involves also inclination resonances responsible of bending waves. We have performed three--dimensional hydrodynamic simulations of the disk and of its interactions with the planet with a Smoothed Particle Hydrodynamics (SPH) code. A main result is that the initial large eccentricity and inclination of the planetary orbit are rapidly damped on a timescale of the order of $10^3$ yrs, almost independently of the initial semimajor axis and eccentricity of the planet. The disk is warped in response to the planet perturbations and it precesses. Inward migration occurs also when the planet is inclined and it has a drift rate which is intermediate between type I and type II migration. The planet is not able to open a gap until its inclination becomes lower than $\sim 10^\circ$ when it also begins to accrete a significant amount of mass from the disk.Comment: Accepted for publication on Astrophysical Journa