Abstract

Models of extended decretion disks of critically rotating stars

Models of extended decretion disks of critically rotating stars

Petr Kurfürst, Achim Feldmeier, and Jiří Krtička

Ústav teoretické fyziky a astrofyziky, Masarykova univerzita, Brno, Czech Republic

Institut für Physik und Astronomie, Universität Potsdam, Potsdam-Golm, Germany

Abstract

Massive stars may during their evolution reach the phase of critical rotation when the further increase in rotational speed is no longer possible. The ejection of matter in the equatorial region forms the gaseous outflowing disk, which allows the star to remove the excess of its angular momentum. The outer part of the disk can extend up to large distance from the parent star. We study the evolution of density, radial and azimuthal velocity and angular momentum loss rate of equatorial decretion disk up to quite distant outer regions. We calculate the evolution of density, radial and azimuthal velocity from the initial Keplerian state until the disk approaches the final stationary state.

1 Viscous decretion disks of critically rotating stars

The basic scenario follows the model of the viscous decretion disk proposed by Lee et al. (1991), which naturally leads to the formation of Keplerian disk, assuming the outward transport of matter from star’s near-equatorial surface occurs through the gradual drifting and feeding of the disk due to the viscous torque. The disk mass loss rate of critically rotating star is determined by the angular momentum loss rate needed to keep the star at critical rotation (Krtička et al., 2011). The main uncertainties are the viscous coupling and the radial temperature distribution. The modelling of the viscosity is typically constrained to regions close to the star (Penna et al., 2013), consequently we assume a power low decrease of the viscous coupling. In this model we maintain the temperature distribution as a free parameter. Following the stationary models (Kurfürst & Krtička, 2012) we introduce the results of calculations of the time-evolving converging solutions for governing physical quantities in the disk up to quite distant supersonic regions from the parent star.

2 Radial thin disk structure

Basic hydrodynamic equations (mass, momentum and energy conservation equations) determine the disk behaviour. We assume axial symmetry in cylindrical coordinate system and fully integrated vertical structure with the vertically integrated density and with constant temperature and hydrostatic equilibrium in -direction (Lee et al., 1991; Krtička et al., 2011; Kurfürst & Krtička, 2012). Disk models (Carciofi & Bjorkman, 2008) found nearly isothermal () temperature distribution in the inner disk region. For calculations of the structure of outer part of the disk it is reasonable to consider the power law temperature decline , where is the cylindrical radial distance and is a free parameter. In our models we employ the second order Navier-Stokes viscosity. Penna et al. (2013) find the nonrelativistic viscosity coefficient being constant close to the star. We examine also disk behaviour with the introduced power law viscosity dependence , where is the viscosity parameter (Shakura & Sunyaev, 1973), is a free parameter and denotes viscosity near the stellar surface.

3 Results of numerical solution

For the purpose of modelling we developed the time-depedent hydrodynamic grid code. As a central object we selected the B0 type star with the following parameters (Harmanec, 1988): The results of our calculations are given in Figs. 12.

Figure 1: The dependence of the relative disk column density (), the relative radial () and azimuthal () disk velocities (in the units of sound speed and Keplerian velocity ) and the relative angular momentum loss rate () on radius in case of isothermal disk (left graph) and of the power law temperature decrease with parameter (right graph) in final stationary models. Viscosity profiles are represented by parameter in the graph, inner boundary viscosity is considered. The unphysical drop of the angular momentum loss rate at large radii can be avoided with decreasing viscosity coefficient . In this case the disks become angular momentum conserving at large radii.
Figure 2: Left panel: The same as Fig. 1, however with decreasing viscosity for different temperature profiles parameterized by . Right panel: Snaphsots of the density and velocity profiles plotted each 5.1 years of the disk evolution. The graph shows the density wave propagation on the viscous time scale , which can be considered as as the timescale for a disk annulus to move a radial distance (Pringle, 1981).

Acknowledgments

This work was supported by the grant GA ČR 13-10589S.

References

  1. Carciofi, A. C., & Bjorkman, J. E. 2008, ApJ, 684, 1374
  2. Harmanec, P. 1988, BAICz, 39, 329
  3. Krtička, J., et al. 2011, A&A, 527, 84
  4. Kurfürst, P., & Krtička, J. 2012, ASPC, 464, 223
  5. Lee, U., et al. 1991, MNRAS, 250, 432
  6. Penna, R. F., et al. 2013, MNRAS, 428, 2255
  7. Pringle, J. E. 1981, ARA&A, 19, 137
  8. Shakura, N. I., & Sunyaev, R. A. 1973, A&A, 24, 337
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