Abstract
In addition to excitation processes considered in Chaps. 9 and 10, we have another important excitation process of disk oscillations. This is a wave-wave resonant excitation in deformed disks. Disk deformation from axisymmetric state is widely expected in tidally deformed disks. Even in a single star, long-living (time-periodic) deformations may be present on disks (e.g., warps). Basics on a wave-wave resonant instability in deformed disks was studied by Kato (Publ Astron Soc Jpn 56:905, 2004; 60:111, 2008), and later examined from different ways by Ferreria and Ogilvie (Mon Not R Astron Soc 386:2297, 2008) and Oktariani et al. (Publ Astron Soc Jpn 62:709, 2010). Subsequently, Kato (Publ Astron Soc Jpn 65:75, 2013; 66:24, 2014, see also Kato et al. (Publ Astron Soc Jpn 63:363, 2011)) formulated the instability in a perspective way. In this chapter, we outline the formulation by Kato (Publ Astron Soc Jpn 65:75, 2013). Applications to time variations in dwarf novae and X-ray binaries are presented in Chap. 12
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Notes
- 1.
Adoption of equations (11.1) as resonant conditions means that we do not restrict ω’s and m’s to positive values.
- 2.
In expression for \(\boldsymbol{P}_{\mathrm{nonlinear}}\) given by equation (82) by Kato (2008) there are typographical errors.
- 3.
The cases where ω D = 0 can be included in our analyses.
- 4.
For oscillations to resonantly interact, additional conditions concerning wave forms in the vertical and radial directions are necessary, as mentioned in Sect. 11.1. These conditions are not considered here in detail. If these conditions are not satisfied, the value of the coupling term, W or W T, given by equation (11.48) or (11.49) vanishes.
- 5.
The formula
$$\displaystyle{\mathfrak{R}(A)\mathfrak{R}(B) = \frac{1} {2}\mathfrak{R}[AB + AB^{{\ast}}] = \frac{1} {2}\mathfrak{R}[AB + A^{{\ast}}B]}$$is used, where A and B are complex variables and B ∗ is the complex conjugate of B.
- 6.
The tidal instability in dwarf novae, which is discussed in Sect. 12.2, corresponds to this case.
References
Bernstein, I. B., Frieman, E. A., Kruskal, M. D., & Kulsrud, R. M. 1958, Proc. R. Soc. A-Math. Phys. Eng.Sci., A244, 17
Ferreria, B. T., & Ogilvie, G. I. 2008, Mon. Not. R. Astron. Soc., 386, 2297
Kato, S. 2004, Publ. Astron. Soc. Jpn., 56, 905
Kato, S. 2008, Publ. Astron. Soc. Jpn., 60, 111
Kato, S. 2013, Publ. Astron. Soc. Jpn., 65, 75
Kato, S. 2014, Publ. Astron. Soc. Jpn., 66, 24
Kato, S. & Unno, W. 1967, Publ. Astron. Soc. Jpn., 19, 1
Kato, S., Okazaki, A. T., & Oktariani, F. 2011, Publ. Astron. Soc. Jpn., 63, 363
Khalzov, I. V., Smolyakov, A. I., & Hgisonis, V. I. 2008, Phys. Plasmas, 15, 4501
Lynden-Bell, D., & Ostriker, J. P. 1967, Mon. Not. R. Astron. Soc., 136, 293
Oktariani, F., Okazaki, A. T., & Kato, S. 2010, Publ. Astron. Soc. Jpn., 62, 709
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Kato, S. (2016). Wave-Wave Resonant Instability in Deformed Disks. In: Oscillations of Disks. Astrophysics and Space Science Library, vol 437. Springer, Tokyo. https://doi.org/10.1007/978-4-431-56208-5_11
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DOI: https://doi.org/10.1007/978-4-431-56208-5_11
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