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Spherically Symmetric Configurations (Stability — Part II)
Suppose we now want to study the stability of one of the spherically symmetric, equilibrium structures that have been derived elsewhere. The identified set of simplified, time-dependent governing equations will tell us how the configuration will respond to an applied radial (i.e., spherically symmetric) perturbation that pushes the configuration slightly away from its initial equilibrium state.
Assembling the Key Relations
Governing Equations
After combining the Euler equation with the Poisson equation in essentially the manner outlined by the "structural solution strategy" we have called Technique 1, the relevant set of time-dependent governing equations is:
Equation of Continuity
<math>\frac{d\rho}{dt} = - \rho \biggl[\frac{1}{r^2}\frac{d(r^2 v_r)}{dr} \biggr]
= -\rho \biggl[ \frac{dv_r}{dr} + \frac{2v_r}{r} \biggr] </math>
Euler + Poisson Equations
<math>\frac{dv_r}{dt} = - \frac{1}{\rho}\frac{dP}{dr} - \frac{GM_r}{r^2} </math>
Adiabatic Form of the
First Law of Thermodynamics
<math>~\frac{d\epsilon}{dt} + P \frac{d}{dt} \biggl(\frac{1}{\rho}\biggr) = 0</math>
where,
<math>v_r \equiv \frac{dr}{dt}</math> ,
and, as before, the mass enclosed inside radius <math>r</math> is,
<math>M_r \equiv \int_0^r dm_r = \int_0^r 4\pi r^2 \rho dr</math> .
Consistent Lagrangian Formulation
The (Lagrangian) time derivatives in these equations define how a given physical parameter — for example, <math>~\rho</math>, <math>v_r</math>, or <math>~\epsilon</math> — should vary with time in a (Lagrangian) fluid element that is not fixed in space but, rather, is moving along with the flow. However, the radial derivatives describe the spatial variation of various physical parameters as measured at fixed locations in space; that is, as written, the radial derivatives do not track conditions as viewed by a (Lagrangian) fluid element that is moving along with the flow because the position <math>r</math> of each (Lagrangian) fluid element is itself changing with time. A proper Lagrangian representation of the spatial derivatives can be formulated in the case of one-dimensional, spherically symmetric flows by using <math>M_r</math> (or, equivalently, <math>m_r</math>) instead of <math>r</math> as the independent variable. Making the substitution,
<math>\frac{d}{dr} = \frac{dM_r}{dr}\frac{d}{dM_r} = 4\pi \rho r^2 \frac{d}{dM_r}</math>
in the first two equations above gives, respectively,
<math>\frac{d\rho}{dt} = - 4\pi \rho^2 r^2 \frac{dv_r}{dM_r} - \frac{2\rho v_r}{r} </math> ,
and,
<math>\frac{dv_r}{dt} = - 4\pi r^2 \frac{dP}{dM_r} - \frac{GM_r}{r^2} </math>
Supplemental Relations
As has been discussed elsewhere, in any analysis of time-dependent flows, the principal governing equations must be supplemented by adopting an equation of state for the gas and by specifying initial conditions. Here, initial conditions will be given by the structural properties — for example, <math>~\rho</math><math>(M_r)</math> and <math>~P</math><math>(M_r)</math> — of one of our derived, spherically symmetric equilibrium structures — for example, a uniform-density sphere or an n = 1 polytrope. We will adopt an ideal gas equation of state and, specifically, the relation,
<math>~P = (\gamma_\mathrm{g} - 1)\epsilon \rho </math>
As a result, the adiabatic form of the <math>1^\mathrm{st}</math> law of thermodynamics can be written as,
<math> \rho \frac{dP}{dt} - \gamma_\mathrm{g} P \frac{d\rho}{dt} = 0 . </math>
Summary
In summary, the following three one-dimensional ODEs define the physical relationship between the three dependent variables <math>~\rho</math>, <math>~P</math>, and <math>r</math>, each of which should be expressible as a function of the two independent (Lagrangian) variables, <math>~t</math> and <math>M_r</math>:
Equation of Continuity
<math>\frac{d\rho}{dt} = - 4\pi \rho^2 r^2 \frac{d}{dM_r}\biggl(\frac{dr}{dt}\biggr) - \frac{2\rho}{r} \biggl(\frac{dr}{dt}\biggr) </math>
,
Euler + Poisson Equations
<math>\frac{d^2 r}{dt^2} = - 4\pi r^2 \frac{dP}{dM_r} - \frac{GM_r}{r^2} </math>
Adiabatic Form of the
First Law of Thermodynamics
<math>
\rho \frac{dP}{dt} - \gamma_\mathrm{g} P \frac{d\rho}{dt} = 0 .
</math>
Linearizing the Key Relations
See Also
- Part I of Spherically Symmetric Configurations: Simplified Governing Equations
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© 2014 - 2021 by Joel E. Tohline |