Abstract
In Chaps. 6 and 7 effects of magnetic fields on oscillations are neglected. In real disks, however, global toroidal magnetic fields will present, because the magnetic fields generated by magneto-rotational instability (MRI) will be generally stretched in the azimuthal direction by the shear of differential rotation. In Sect. 8.1, we examine effects of toroidal magnetic fields on c-mode oscillations. The purpose is to know whether toroidal magnetic fields affect the interpretation that kHz QPOs comes from the c-mode oscillations. In Chaps. 6 and 7 we had neglected the effects of radial variations of disk thickness, disk density and others on oscillations. This was allowed as the first step, since we are mainly interested in trapped oscillations and their trapped regions are not wide. In some important cases, however, the propagation regions of such oscillations as one-armed precession modes and tilt modes are as wide as the whole disk size. Hence, it will be necessary to relax the above approximations to study, for example, precession of the V/R variations in Be stars (see Chap. 1). This issue is discussed in Sect. 8.2. The main parts in this chapter are special and technical. In particular, Sect. 8.2 treats a special issue. Thus, the readers who are not interested in mathematical procedures can skip this chapter, except for Sect. 8.1.3.
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Notes
- 1.
In nearly vertical oscillations, the vertical component of equation motion gives approximately u z ∼ h 1∕c s (e.g., see equation (8.6) in the limit of c A = 0). Furthermore, the radial component of equation of motion (equation (8.4)) gives u r ∼ (1∕Ω λ)h 1. Combining these two relations, we have u r ∼ (H∕λ)u z , i.e., u r is smaller than u z by a factor of H∕λ. Hence, the terms of the second brackets of equation (8.11) is smaller than the terms of the first brackets by a factor of (H∕λ)2.
- 2.
In Chap. 7, r out was used to denote r c. Both of them have the same meanings.
- 3.
In the lowest order of approximations, the equation of continuity gives \(i\tilde{\omega }h_{1} + (c_{\mathrm{s}}^{2}/H)(\partial /\partial \eta -\eta )u_{z} = 0\). If u z is taken to be proportional to \(\mathcal{H}_{n-1}(\eta )\), i.e., \(u_{z} = f_{z}(r)\mathcal{H}_{n-1}(\eta )\), the above continuity relation shows that h 1 has a component proportional to \(\mathcal{H}_{n}(\eta )\), i.e., \(h_{1} = f_{h}(r)\mathcal{H}_{n}(\eta )\) and z f(r) and f h (r) is related by \(i\tilde{\omega }f_{h} = (c_{\mathrm{s}}^{2}/H)f_{z}\).
- 4.
- 5.
- 6.
It is noted again that in the present section the subscript u to f is omitted in order to avoid complexity. f in the present section should not be confused with f when h 1(r, η) is separated as h 1(r, η) = g(η, r)f(r).
- 7.
- 8.
The coefficient a h, 0 (1) is not always zero, although a 0 (1) is taken to be zero as mentioned before.
- 9.
In Sect. 8.2.3, u r (r, η) was expanded as
$$\displaystyle{u_{r}(r,\eta ) = (g^{(0)} + g^{(2)}+\ldots )f(r) =\biggr [ \mathcal{H}_{ 0} + \frac{d\mathrm{ln}H} {d\mathrm{ln}r} (a_{2}^{(1)}\mathcal{H}_{ 2}) +\biggr ( \frac{d\mathrm{ln}H} {d\mathrm{ln}r} \biggr )^{2}(a_{ 2}^{(2)}\mathcal{H}_{ 2} + a_{4}^{(2)}\mathcal{H}_{ 4}) +\ldots \biggr ]\, f.}$$In the present subsection, u r (r, η) is expressed as (see equation (8.80))
$$\displaystyle{u_{r}(r,\eta ) = \mathcal{H}_{0}(\eta )f_{0}(r) + \mathcal{H}_{2}(\eta )f_{2}(r).}$$
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Kato, S. (2016). Two Examples of Further Studies on Trapped Oscillations and Application. In: Oscillations of Disks. Astrophysics and Space Science Library, vol 437. Springer, Tokyo. https://doi.org/10.1007/978-4-431-56208-5_8
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