Keplerian disk

While a purely hydrodynamic source for turbulence has not yet been demonstrated, the situation is much different when MHD effects are considered in a shearing, Keplerian disk. In this case, the Rayleigh criterion for stability can be shown to be irrelevant provided only that the angular velocity of the disk decreases with radius, even an infinitesimal magnetic field will grow at the expense of the shear motions. 

Godon P. and Livio M. (1999) On the nonlinear hydrodynamic stability of thin Keplerian disks. Astrophys. J. 521, 319-327. 

Of those YSOs with CO emission for which high-resolution spectra have been obtained, the majority are consistent with the CO emission originating from the inner regions of a Keplerian disk, fri fact, these results provide some of the best kinematic evidence for disks around young stars. Even simple disk models for the emission, without reference to physical heating mechanisms, are able to place useful constraints on the properties of the gas in the disk, such as the column density, the radial temperature gradient, and physical sizes. 

The material falling from the parent molecular cloud directly on the forming star is rapidly dissociated and ionized and solids are vaporized, once the matter passes the accretion shock on the stellar surface. The material falling on the inner parts of the accretion disk suffers the same fate, since the matter has to pass through a standing shockwave on the surface of the disk. At this shock the infall of matter is stopped and the flow characteristics change from infall of envelope material to nearly Keplerian rotation of disk material. The location of this shock is shown in Fig. 2.10 as the heavy line marked by crosses. 

Because the radial pressure gradient is expected to be generally negative in a protoplanetary disk, the result is that the gas will orbit the star slower than the solids (Adachi el al. 1976 Weidenschilling 1977a). Large solids, therefore, will feel a headwind as they attempt to follow Keplerian orbits. This drag force will... 

Figure 3.5 Plotted are the inward drift velocities of particles of different sizes in a disk where the velocity differential between the gas and a Keplerian orbit is 70 ms-1. The kink at 10 cm is due to the change in the gas-drag law as the particles exceed the mean free path of the gas.

Low-density clumps orbit the star at roughly Keplerian speeds, while gas in the surrounding disk typically orbits at 50m s-1 more slowly. As a result, clumps are vulnerable to disruption by the ram pressure of the gas. However, numerical simulations suggest that the mutual gravitational attraction of solid particles within a clump is sufficient to keep them largely intact during collapse, provided the initial clump is > 103 km in diameter (Cuzzi et al. 2008 see Figure 10.3). 

While dust continuum emission can be used to estimate disk masses, line emission from optically thin molecules (e.g., CO) can be used to map the line-of-sight velocities in disks, using the Doppler shift of the moving gas. Evidence for Keplerian velocity profiles is typically found (e.g., Launhardt and Sargent, 2001), as is to be expected for gas in a stable orbit around a central protostar. These measurements only apply to the gas at considerable distances from the young star, however, typically at several hundred astronomical units or more. The situation inside these disks at planetary distances is not constrained by these observations, and even the outer disk measurements are subject to possible confusion with infalling gas in the envelopes and outflowing gas from stellar winds. 

Another strong case is in the nucleus of the spiral NGC 4258 where a number of H2O masers are observed. The radial (Doppler) velocities can be fit by rotation of a circumnuclear disk of pc scale with a Keplerian rotation velocity profile reaching up to 1080 km s-1. This means there is a point mass at the center to the tune of 4 x 107 M . Once again we would be hard put to squeeze this massive a cluster of brown dwarfs or collapsed objects inside the disk s inner radius at 0.13 pc [18]. It must thus be a BH. The Seyfert nuclei of NGC 1068 and the Scd NGC 4945 also have (more modest) SMBHs discovered in them by the maser method. 

Upon contact, the less massive white dwarf will have a larger radius (Sect. 13) and will fill its Roche lobe first. Upon transferring a small amount of mass, the white dwarf radius will increase at a rate faster than the binary orbit will widen, and at a rate much faster than matter can be accreted by the companion. This leads to a runaway (or dynamical) disruption of the less massive white dwarf which will form a hot disk in Keplerian orbit around the other more massive white dwarf. Simulations of the process show that the disruption of the white dwarf will take only a few orbits, PQrb 3 m at the time of contact [57,58,59]. 

Fig. 2.3 The HH 30 system ([27]). The background image, taken with the Hubble Space Telescope (HST [28]), shows an edge-tm disk traced by the dark bar, a jet perpendicular to the disk and scattered light from the embedded proto-star. The left panel presents the CO(/ = 2 — 1) emission at large positive and negative velocities relative to the dense core narrow emission. This high velocity CO emission follows the narrow jet. The middle panel presents the CO (7 = 2 — 1) emission in two velocity intervals indicated with blue and red contours the emission approaching us/ieceding from us. This velocity pattern is consistent with Keplerian rotation around a solar mass star. The right panel presents the continuum emission due to dust grains in the circumstellar disk. The spatial resolution of the millimetre observations is 1" [27]...

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