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Register a new userA Simple Family of Models for Eccentric Keplerian Fluid Disks
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T. S. Statler
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In order to be in a long-lived configuration, the density in a fluid disk should be constant along streamlines to prevent compressional (PdV) work from being done cyclically around every orbit. In a pure Kepler potential, flow along aligned, elliptical streamlines of constant eccentricity will satisfy this condition. For most density profiles, differential precession driven by the pressure gradient will destroy the alignment; however, in the razor-thin approximation there is a family of simple equilibria in which the precession frequency is the same at all radii. These disks may therefore be long-lived at significant eccentricities. The density can be made axisymmetric as r goes to 0, while maintaining the precession rate, by relaxing the requirement of constancy along streamlines in an arbitrarily small transition region near the center. In the limit of small eccentricity, the models can be seen as acoustically perturbed axisymmetric disks, and the precession rate is shown to agree with linear theory. The perturbation is a traveling wave similar to an ocean wave, with the fluid rising and falling epicyclically in the gravitational field of the central mass. The expected emission line profiles from the eccentric disks are shown to be strongly asymmetric in general, and, in extreme cases, prone to misinterpretation as single narrow lines with significant velocity offsets.
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We study the development of finite eccentricity in accretion disks in close binary systems using a two-dimensional grid-based numerical scheme. We perform detailed parameter studies to explore the dependence on viscosity, disk aspect ratio, the inclusion of a mass-transfer stream and the role of the boundary conditions. We consider mass ratios 0.05<q<0.3 appropriate to superoutbursting cataclysmic binary systems. Instability to the formation of a precessing eccentric disk that attains a quasi-steady state with mean eccentricity in the range 0.3-0.5 occurs readily. The shortest growth times are ~15 binary orbits for the largest viscosities and the instability mechanism is for the most part consistent with the mode-coupling mechanism associated with the 3:1 resonance proposed by Lubow. However, the results are sensitive to the treatment of the inner boundary and to the incorporation of the mass-transfer stream. In the presence of a stream we found a critical viscosity below which the disk remains circular. Incorporation of a mass-transfer stream tends to impart stability for small enough viscosity (or, equivalently, mass-transfer rate through the disk) and does assist in obtaining a prograde precession rate that is in agreement with observations. For the larger q the location of the 3:1 resonance is pushed outwards towards the Roche lobe where higher-order mode couplings and nonlinearity occur. It is likely that three-dimensional simulations that properly resolve the disks vertical structure are required to make significant progress in this case.
It is usually thought that viscous torque works to align a circumbinary disk with the binarys orbital plane. However, recent numerical simulations suggest that the disk may evolve to a configuration perpendicular to the binary orbit (polar alignment) if the binary is eccentric and the initial disk-binary inclination is sufficiently large. We carry out a theoretical study on the long-term evolution of inclined disks around eccentric binaries, calculating the disk warp profile and dissipative torque acting on the disk. For disks with aspect ratio $H/r$ larger than the viscosity parameter $alpha$, bending wave propagation effectively makes the disk precess as a quasi-rigid body, while viscosity acts on the disk warp and twist to drive secular evolution of the disk-binary inclination. We derive a simple analytic criterion (in terms of the binary eccentricity and initial disk orientation) for the disk to evolve toward polar alignment with the eccentric binary. When the disk has a non-negligible angular momentum compared to the binary, the final polar alignment inclination angle is reduced from $90^circ$. For typical protoplanetary disk parameters, the timescale of the inclination evolution is shorter than the disk lifetime, suggesting that highly-inclined disks and planets may exist orbiting eccentric binaries.
We construct dynamical models of the ``double nucleus of M31 in which the nucleus consists of an eccentric disk of stars orbiting a central black hole. The principal approximation in these models is that the disk stars travel in a Kepler potential, i.e., we neglect the mass of the disk relative to the black hole. We consider both ``aligned models, in which the eccentric disk lies in the plane of the large-scale M31 disk, and ``non-aligned models, in which the orientation of the eccentric disk is fitted to the data. Both types of model can reproduce the double structure and overall morphology seen in Hubble Space Telescope photometry. In comparison with the best available ground-based spectroscopy, the models reproduce the asymmetric rotation curve, the peak height of the dispersion profile, and the qualitative behavior of the Gauss-Hermite coefficients h_3 and h_4. Aligned models fail to reproduce the observation that the surface brightness at P1 is higher than at P2 and yield significantly poorer fits to the kinematics; thus we favor non-aligned models. Eccentric-disk models fitted to ground-based spectroscopy are used to predict the kinematics observed at much higher resolution by the STIS instrument on the Hubble Space Telescope (Bender et al. 2003), and we find generally satisfactory agreement.
We present the derivation of distribution functions for the first four members of a family of disks, previously obtained in (MNRAS, 371, 1873, 2006), which represent a family of axially symmetric galaxy models with finite radius and well behaved surface mass density. In order to do this we employ several approaches that have been developed starting from the potential-density pair and, essentially using the method introduced by Kalnajs (Ap. J., 205, 751, 1976) we obtain some distribution functions that depend on the Jacobi integral. Now, as this method demands that the mass density can be properly expressed as a function of the gravitational potential, we can do this only for the first four discs of the family. We also find another kind of distribution functions by starting with the even part of the previous distribution functions and using the maximum entropy principle in order to find the odd part and so a new distribution function, as it was pointed out by Dejonghe (Phys. Rep., 133, 217, 1986). The result is a wide variety of equilibrium states corresponding to several self-consistent finite flat galaxy models.
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