Abstract
In Chap. 6 we have considered various instabilities which occur in the framework of adiabatic processes. Distinct from such instabilities, there are other kinds of many important instabilities which occur by dissipative processes. There are two ways for classifying these instabilities. One is phenomenological, and the other is physical. In the former phenomenological classification, instabilities are classified by whether their growth is monotonous in idealized situation (secular instability) or oscillatory (overstability). In the latter classification, instabilities are divided by physical processes involved (for example, thermal or viscous). Here we classify, for convenience, instabilities in the former way. That is, we classify them into secular instability and overstability. Instabilities which are former in the idealized situations (secular instability) are described in this chapter (Chap. 7) and the latter (overstability) in the next chapter (Chap. 8). It is noted that by this classification some instabilities whose cause is thermal are described in this chapter, and some others in the next chapter (Chap. 8).
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- 1.
If the thermometric conductivity due to radiative diffusion is written as cℓ, where c is the speed of light and ℓ is the mean free path of photons. The timescale of radiative heat diffusion is r 2∕cℓ, where r is a typical linear size in stars. On the other hand, the dynamical timescale is r∕c s. Hence, the thermal timescale is longer than the dynamical one by (r∕ℓ)(c s∕c), which is usually much larger than unity, because ℓ is much smaller than r due to large optical thickness inside stars.
- 2.
By using cylindrical coordinates (r, φ, z) and by expressing the flow in the corresponding directions by (v r, v φ, v z), we have the momentum equation in the φ-direction in the form
$$\displaystyle \begin{aligned}\rho\biggl(\frac{\partial }{\partial t}+v_r\frac{\partial v_\varphi}{\partial r} +v_z\frac{\partial v_\varphi}{\partial z}+ \frac{v_rv_\varphi}{r}\biggr)=\frac{1}{r^2}\frac{\partial}{\partial r}(r^2t_{r\varphi}), \end{aligned}$$where t rφ is the rφ-component of viscous stress tensor. This equation can be arranged to the form:
$$\displaystyle \begin{aligned}\rho \frac{\partial}{\partial t}(r v_\varphi) +\rho {\boldsymbol v}\cdot\nabla(rv_\varphi) =\frac{1}{r}\frac{\partial}{\partial r}(r^2t_{r\varphi}). \end{aligned}$$This is an equation representing angular momentum transport. It is noted that if this equation is combined with the continuity equation, we have an equation representing angular momentum conservation:
$$\displaystyle \begin{aligned}\frac{\partial (\rho rv_\varphi)}{\partial t}+\mathrm{div}\,({\boldsymbol v}\rho rv_\varphi) =\frac{1}{r}\frac{\partial}{\partial r}(r^2t_{r\varphi}). \end{aligned}$$ - 3.
The instability criterion has been studied for various boundary conditions. Furthermore, the gravothermal instability has been extended to rotating systems (Inagaki and Hachisu 1978).
- 4.
In the conventional method of Lagrange multipliers, Lagrange function L defined by L = S − λE, where λ is a Lagrange multiplier, is introduced and examine the state where L becomes stationary. This method of the Lagrange multipliers has been modified in the text in a slightly different form for convenience in the present study.
- 5.
From Eq. (7.62) we have
$$\displaystyle \begin{aligned}\delta\phi=-G\!\int\!\frac{dm'}{\vert({\boldsymbol r}+\delta{\boldsymbol r}) -({\boldsymbol r}'+\delta{\boldsymbol r}')\vert}+G\!\int\!\frac{dm'}{\vert{\boldsymbol r}-{\boldsymbol r}'\vert} = G\!\int\!\frac{\delta r_i(r_i-r_i^{\prime})} {\vert{\boldsymbol r}-{\boldsymbol r}'\vert^{3/2}}dm' -G\!\int\!\frac{\delta r_i^{\prime}(r_i-r_i^{\prime})} {\vert{\boldsymbol r}-{\boldsymbol r}'\vert^{3/2}}dm' \end{aligned}$$and
$$\displaystyle \begin{aligned}\int\delta\phi dm =G\int\int\frac{(\delta r_i-\delta r_i^{\prime})(r_i-r_i^{\prime})} {\vert{\boldsymbol r}-{\boldsymbol r}'\vert^{3/2}}dmdm'. \end{aligned}$$On the other hand,
$$\displaystyle \begin{aligned}\frac{\partial \phi}{\partial r_i}=G\int \frac{r_i-r_i^{\prime}} {\vert{\boldsymbol r}-{\boldsymbol r}'\vert^{3/2}}dm', \end{aligned}$$and
$$\displaystyle \begin{aligned} \begin{array}{rcl} \int\frac{\partial\phi}{\partial r_i}\delta r_idm & &\displaystyle =G\int\int\frac{\delta r_i(r_i-r_i^{\prime})} {\vert{\boldsymbol r}-{\boldsymbol r}'\vert^{3/2}}dmdm' = G\int\int\frac{\delta r_i^{\prime}(r_i^{\prime}-r_i)} {\vert{\boldsymbol r}-{\boldsymbol r}'\vert^{3/2}}dmdm' \\ & &\displaystyle =\frac{1}{2} G\int\int\frac{(\delta r_i-\delta r_i^{\prime})(r_i-r_i^{\prime})} {\vert{\boldsymbol r}-{\boldsymbol r}'\vert^{3/2}}dmdm'. \end{array} \end{aligned} $$ - 6.
From Eq. (7.75) we have
$$\displaystyle \begin{aligned}\varDelta\phi=-\frac{G}{r+\varDelta r}\int_0^{r+\varDelta r}dm' -G\int_{r+\varDelta r}^\infty\frac{1}{r'}dm'-\phi, \end{aligned}$$which gives for small Lagrangian variations
$$\displaystyle \begin{aligned} \frac{d}{dr}\varDelta\phi=\biggl(-2\frac{\varDelta r}{r}+\frac{d\varDelta r}{dr}\biggr) \frac{d\phi}{dr}. \end{aligned}$$The above equation is written as Eq. (7.85) by use of the mass conservation given by 2Δr∕r + Δρ∕ρ + dΔr∕dr = 0, which comes from 4π(r + Δr)2 ρ(r + Δr)d(r + Δr) = 4πr 2 ρ(r)dr.
- 7.
As will be noticed in the Virial theorem (Sect. 15.1) this characteristic of negative heat capacity is demonstrated as follows. The Virial theorem tells us that in dynamically equilibrium state of self-gravitating system 2T + Ω = 0 holds, where T and Ω are, respectively, thermal and gravitational energies of the system. Since the total energy of the system, E, is given by E = T + Ω, we have
$$\displaystyle \begin{aligned} T = - E. \end{aligned}$$
References
Antonov, V.A.: Vestn. Leningr. Gos. Univ. 7, 135 (1962)
Caleo, A., Balbus, S.A., Tognelli, E.: Mon. Not. R. Astron. Soc. 460, 338 (2016)
Field, G.: Astrophys. J. 142, 531 (1965)
Fricke, K.: Z. Astrophys. 68, 317 (1968)
Goldreich, P., Schubert, G.: Astrophys. J. 150, 571 (1967)
Inagaki, S., Hachisu, I.: Publ. Astron. Soc. Jpn. 30, 39 (1978)
Kato, S., Fukue, J., Mineshige, S.: Black-Hole Accretion Disks – Towards a New Paradigm. Kyoto University Press, Kyoto (2008)
Lynden-Bell, D., Wood, R.: Mon. Not. R. Astron. Soc. 138, 495 (1968)
Pringle, J.E.: Mon. Not. R. Astron. Soc. 177, 65 (1976)
Schwarzschild, M., Härm, R.: Astrophys. J. 142, 855 (1965)
Shakura, N.I., Sunyaev R.A.: Astron. Astrophys. 24, 337 (1973)
Shapiro, S.L., Lightman, A.P., Eardley, D.H.: Astrophys. J. 204, 187 (1976)
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Kato, S., Fukue, J. (2020). Instabilities Due to Dissipative Processes I (Secular Instability). In: Fundamentals of Astrophysical Fluid Dynamics. Astronomy and Astrophysics Library. Springer, Singapore. https://doi.org/10.1007/978-981-15-4174-2_7
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