The Be Star Newsletter, Volume 35 - April 2001

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1. Introduction Despite recent advances, many due to readers of this newsletter, the overall dynamics of Be disks remain an enigma. We don't fundamentally understand the processes that lead to either their (often recurrent) formation or destruction, nor do we understand the complex forms of variation they exhibit in between. A proper dynamical understanding requires knowledge of the role of forces associated with, e.g., gravity, viscosity, inertia, magnetic fields, and radiation. Here we focus on the last of these, specifically the radiative forces that arise from line scattering of a star's continuum radiation. Such line forces are well known to be important in driving the wind outflow in a broad range of luminous, early-type stars. But thus far, their consideration in Be stars has been mostly confined to modeling high-speed winds from higher latitudes (e.g., Bjorkman & Cassinelli 1993). Attempts have been made to include parametrized optically thin line forces (e.g. Chen and Marlborough 1994) in disk dynamics ( Okazaki 1997), but in winds, it is the marginally optically thick lines that dominate. We present arguments, in the main article summarized here, that the same should be expected in disks. When treated similarly to the CAK approach so successfully applied in winds, the signature of such forces is a unique sensitivity to the gas dynamics, accompanied by rich and surprising phenomena. At first glance, it may seem strange to apply CAK formalism to gas that is not radially expanding. This would indeed be inappropriate if the central source were a point star. But if one takes account of the nonradial radiation streams from a finite-size star, the Keplerian shear of a steady, orbiting disk can desaturate lines in much the same way as a radially accelerating outflow does. This leads to surprisingly effective optically thick line acceleration within a disk. We stress two potentially important consequences of such line forces. First, the self-consistent structure of the disk is no longer Keplerian. Although this may require only minor corrections in the densest portions of the disk, in the low-density surface layers it may lead to changes in the disk structure that alter or enhance the inexorable erosion by disk winds, and this may be a key factor in what appears to be the occasional complete disappearance of disks in many Be stars. Second, even in the denser regions, the perturbative effect of line forces can lead to a precession of elliptical orbits, and this may be an important factor for understanding long-term V/R variations and the precession of global disk oscillation modes. In both these cases, the actual computation of relevant timescales for disk destruction or mode precession may depend on various details and subtleties not yet accounted for. Nonetheless, our analysis shows how the basis for such further work can be built upon existing line-driven wind theory, with appropriate geometric corrections. 2. CAK-Type Line Driving for an Orbiting Disk Geometry One requirement for strong line driving is large line-of-sight velocity gradients. To understand how a Keplerian disk generates such gradients, consider the schematic in Figure 1. A central point is that the finite size of the star allows for nonradial radiation streams, including the tangential stream from the limb depicted in the figure. These streams encounter the disk at oblique angles that sample the Keplerian velocity gradient, even in the absence of any radial motions. Since Keplerian speeds are of the same order as wind speeds, and they vary over the same radial scale, the line-of-sight gradients can also be of the same order. Thus the justification for applying CAK-type line forces in winds is not as distinct from disks as has been assumed in the past. When standard CAK theory is applied, the radiative acceleration g is proportional to the flux-weighted average of the line-shadowing correction factor , integrated over direction . This factor in turn obeys and accounts for all optical depth effects within the line distribution, where is the line-of-sight velocity gradient. For such a power-law line list, is of order the cumulative Sobolev optical depth along summed over all lines ( Gayley 1995; see Cranmer & Owocki 1995 for a derivation of the detailed geometrical terms). The essential point of eq. (1) is that the radiative acceleration weakens as density is increased or as line-of-sight velocity gradient is decreased. Since figure 1 shows that the latter is comparable in winds and disks, it is really the former, the density, that represents the key factor distinguishing the forces in Be disks from those in winds of hot stars. Figure 1 - Comparison of the radial velocity gradient, dvr/dr, of a radial stellar wind outflow with the line-of-sight gradient, dvz/dz, along a ray direction z that is tangent to the stellar limb and passing through an orbiting Keplerian disk. The units of the gradient are arbitary, but the relative scaling is accurate for the case in which the terminal velocity of the wind is the same as the near-surface orbital velocity of the disk. 3. Radial Forces on Circular Orbits For strictly circular orbits, the expressions for the line force become quite simple, and the result is that both the Keplerian velocity shear and the orbital curvature serve to augment the velocity gradient for nonradial radiation streams. Relative to the inverse-square falloff of gravity, it then follows that the radiative acceleration varies as The overall scale of this ratio is order unity when the density is windlike, and the steepness of the reduction in a disk due to the density enhancement is controlled by the power-index. The appropriate value of this parameter in a disk is a matter for future study, but we note that wind models ( Puls, Springmann, & Lennon 2000) suggest a low value, in the vicinity of , may be relevant at B-star temperatures. If the radiation field is azimuthally symmetric, the force points radially outward. Since strictly Keplerian orbits do not include such an additional force, the overall results are not self consistent, and can only be used to test when the magnitude of the radiative force will represent a small perturbation. A key point is that this should not hold near the disk boundaries, and so future models of the seat of disk winds should include a self-consistent structure calculation that includes the effect of Keplerian shear on the line force. How this could alter our view of disk winds is not yet known, including the ramifications for accretion disks. 4. Precession of Slightly Elliptical Orbits Due to Radiative Forces Even when the radiative force is small, it can have an important cumulative effect on the long-term disk dynamics. For example, the apparent precession of global disk modes (e.g., Telting et al. 1994;  Hummel & Hanuschik 1997) requires of order 103 orbits, which implies a force perturbation that is roughly only 10-3 times gravity. Calculating radiative global modes is beyond our scope, so we simply attempt to guide expectations by considering slightly elliptical orbits embedded in such a mode, and show in the full paper that line driving does cause such orbits to precess. Indeed, there are two separate types of precession induced, and they often have similar magnitude but opposite sign. The first is the more obvious, and is caused by the force gradient given in eq. (2). If the eccentricity is not rapidly increasing, we argue that over a single elliptical orbit (not adjacent orbits) will fall off less rapidly than r-5/2, and so the radiative force falls off more rapidly than inverse-square. This induces precession in a way similar to the steeply falling quadrupole gravity term. The key difference is that the radiative force is outward, and so the induced precession is retrograde, not prograde. This creates difficulty when compared with observations that currently favor a prograde interpretation. There is, however, a second effect of radiative forces that is considerably more subtle, and may contribute to prograde precession. That is the radiative torque imposed by the line driving, which would peak when the radial and azimuthal speeds are of the same order. This torque was first postulated for winds by Gayley & Owocki (2000) under the assumption that the radial speed is much greater than the azimuthal speed. Here we consider the opposite limit, for which weak ellipticities yield weak torques that become apparent only after many orbits. Such torques are symmetric over an orbit, so changes in the overall angular momentum average out. However, the orbit ceases to be closed, and the resulting precession will be prograde whenever the torque is positive on the outbound leg. A self-consistent calculation of the impact of these effects on an actual one-arm mode remains to be carried out. Our results are applied only to gas parcels in elliptical orbits of constant eccentricity, and suggest in this case that the gradient effect dominates and the net precession will be retrograde. But considerable further work is needed before these effects can be confronted with observations. Self-consistent mode calculations are required, which must likely be followed into the nonlinear regime. Our current objectives are merely to call attention in the Be-star community to the potential importance of line forces for Be disks, to stimulate interest in including them in disk models, and to outline how familiar CAK concepts may be utilized with a minimum of additional effort. Of particular importance would be future calculations of the ionization conditions and the appropriate line lists, and inclusion of line driving into nonlinear hydrodynamic simulations of disk formation, destruction, and precession. REFERENCES Bjorkman, J. E. & Cassinelli, J. P. 1993, ApJ, 409, 429 Chen, H., & Marlborough, J. M. 1994, ApJ, 427, 1005 Cranmer, S. R. & Owocki, S. P. 1995, ApJ, 440, 308 Gayley, K. G. 1995, ApJ, 454, 410 Gayley, K. G., & Owocki, S. P. 2000, ApJ, 537, 461 Grinin, A. 1978, Sov. Astr., 14, 113 Hummel, W. & Hanuschik, R. W. 1997, A&A, 320, 852 Okazaki, A. T. 1997, A&A, 318, 548 Puls, J., Springmann, U., & Lennon, M. 2000, A&AS, 141, 23 Telting, J. H., Heemskerk, M. H. M., Henrichs, H. F., & Savonije, G. J. 1994, A&A, 288, 558