User:Tohline/SphericallySymmetricConfigurations/Virial
Virial Equilibrium of Spherically Symmetric Configurations
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Free Energy Expression
Review
As has been introduced elsewhere in a more general context, associated with any isolated, self-gravitating, gaseous configuration we can identify a total Gibbs-like free energy, <math>\mathfrak{G}</math>, given by the sum of the relevant contributions to the total energy of the configuration,
<math> \mathfrak{G} = W_\mathrm{grav} + \mathfrak{S}_\mathrm{therm} + T_\mathrm{kin} + P_e V + \cdots </math>
Here, we have explicitly included the gravitational potential energy, <math>~W_\mathrm{grav}</math>, the ordered kinetic energy, <math>~T_\mathrm{kin}</math>, a term that accounts for surface effects if the configuration of volume <math>~V</math> is embedded in an external medium of pressure <math>~P_e,</math> and <math>~\mathfrak{S}_\mathrm{therm}</math>, the reservoir of thermodynamic energy that is available to perform work as the system expands or contracts. A mathematical expression encapsulating the physical definition of each of these energy terms, in full three-dimensional generality, can be found in our introductory discussion of the scalar virial theorem and the free-energy function.
Expressions for Various Energy Terms
We begin, here, by deriving each of the terms in the Gibbs-like free-energy expression as appropriate for spherically symmetric systems. In deriving each term, we keep in mind two issues: First, for a given size system, <math>~R,</math> a determination of each term's total contribution to the free energy generally will involve integration through the entire volume of the configuration, effectively "summing up" the differential mass in each radial shell,
<math> dm = \rho(\vec{x}) d^3x = 4\pi \rho(r) r^2 dr \, , </math>
weighted by some specific energy expression. Second, each term must be formulated in such a way that it is clear how the energy contribution depends on the overall system size, <math>~R.</math>
Volume Integrals
We note, first, that the mass enclosed within each interior radius, <math>~r</math>, is
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<math>~M_r(r) = \int\limits_V dm</math> |
<math>~=</math> |
<math>~ \int_0^r 4\pi r^2 \rho dr \, ,</math> |
hence, the total mass is,
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<math>~M_\mathrm{tot}</math> |
<math>~=</math> |
<math>~ \int_0^R 4\pi r^2 \rho dr \, .</math> |
Confinement by External Pressure: For spherically symmetric configurations, the energy term due to confinement by an external pressure can be expressed, simply, in terms of the configuration's radius, <math>~R</math>, as,
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<math>~P_e V</math> |
<math>~=</math> |
<math>~\frac{4\pi}{3} P_e R^3 \, .</math> |
Gravitational Potential Energy: From our discussion of the scalar virial theorem — see, specifically, the reference to Equation (18), on p. 18 of EFE — the gravitational potential energy is given by the expression,
<math> W_\mathrm{grav} = - \int\limits_V \rho x_i \frac{\partial\Phi}{\partial x_i} d^3 x = - \int\limits_V \vec{r} \cdot \nabla\Phi dm = - \int_0^R \biggl( r \frac{d\Phi}{dr} \biggr) dm \, . </math>
For spherically symmetric systems, the
Poisson Equation
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<math>\nabla^2 \Phi = 4\pi G \rho</math> |
becomes,
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<math>~\frac{1}{r^2} \frac{d}{dr} \biggl( r^2 \frac{d\Phi}{dr} \biggr) </math> |
<math>~=</math> |
<math>~4\pi G \rho(r) \, , </math> |
which implies,
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<math>~r^2 \frac{d\Phi}{dr} </math> |
<math>~=</math> |
<math>~\int_0^r 4\pi G \rho(r) r^2 dr = GM_r(r) \, .</math> |
Hence — see, also, p. 64, Equation (12) of Chandrasekhar [C67] — the desired expression for the gravitational potential energy is,
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<math>~W_\mathrm{grav}</math> |
<math>~=</math> |
<math>~ - \int_0^R \biggl( \frac{GM_r}{r} \biggr) dm = - \int_0^R \frac{G}{r}\biggl[\int_0^r 4\pi r^2 \rho dr \biggr] 4\pi r^2 \rho dr \, .</math> |
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Also, as pointed out by Chandrasekhar [C67] — see p. 64, Equation (16) — it may sometimes prove advantageous to recognize that, if a spherically symmetric system is in hydrostatic balance, an alternate expression for the total gravitational potential energy is,
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Rotational Kinetic Energy: We will also consider a system that is rotating with a specified simple angular velocity profile, <math>~\dot\varphi(\varpi)</math>, in which case, from our discussion of the scalar virial theorem — see, specifically, the reference to Equation (8), on p. 16 of EFE — the (ordered) kinetic energy,
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<math>~T_\mathrm{kin}</math> |
<math>~=</math> |
<math>~ \frac{1}{2} \int\limits_V \rho |\vec{v} |^2 d^3x = \frac{1}{2} \int\limits_V |\vec{v} |^2 dm \, ,</math> |
is entirely rotational kinetic energy, specifically,
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<math>~T_\mathrm{kin} = T_\mathrm{rot}</math> |
<math>~=</math> |
<math>~ \frac{1}{2} \int\int\int \dot\varphi^2 \varpi^2 dm = \frac{1}{2} \int_0^R \dot\varphi^2 \varpi^2 \int_{-\sqrt{R^2 - \varpi^2}}^{\sqrt{R^2 - \varpi^2}} \rho(r(\varpi,z)) 2\pi \varpi d\varpi dz\, .</math> |
Reservoir of Thermodynamic Energy: As has been explained in our introductory discussion of the Gibbs-like free energy, formulation of an expression for the reservoir of thermodynamic energy, <math>~\mathfrak{S}_\mathrm{therm}</math>, depends on whether the system is expected to evolve adiabatically or isothermally. For isothermal systems,
<math> \mathfrak{S}_\mathrm{therm} ~~\rightarrow ~~\mathfrak{S}_I = + \int\limits_V c_s^2 \ln \biggl(\frac{\rho}{\rho_0}\biggr) dm = c_s^2 \int_0^R \ln \biggl(\frac{\rho}{\rho_0}\biggr) 4\pi r^2 \rho dr \, , </math>
where, <math>~c_s</math> is the isothermal sound speed and <math>~\rho_0</math> is a (as yet unspecified) reference mass density; while, for adiabatic systems,
<math> \mathfrak{S}_\mathrm{therm} ~~\rightarrow ~~ \mathfrak{S}_A = + \int\limits_V \frac{1}{({\gamma_g}-1)} \biggl( \frac{P}{\rho} \biggr) dm = \frac{1}{({\gamma_g}-1)} \int_0^R 4\pi r^2 P dr
\, ,</math>
where, <math>~P(r)</math> is the system's pressure distribution and <math>~\gamma_g</math> is the specified adiabatic index.
Idealized Configuration
In the idealized situation of a configuration that has uniform density, <math>~\rho_c</math>, has uniform pressure, <math>~P_c</math>, and is uniformly rotating with angular velocity, <math>~\dot\varphi_c</math>, the mass interior to each radius and the total mass are, respectively,
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<math>~M_r</math> |
<math>~=</math> |
<math>~4\pi \rho_c R^3 \int_0^x x^2 dx = \frac{4\pi}{3} \rho_c R^3 x^3 \, ,</math> |
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<math>~M_\mathrm{tot} </math> |
<math>~=</math> |
<math>~M_r (x=1) = \frac{4\pi}{3} \rho_c R^3 \, ,</math> |
and evaluation of the various energy integrals yields,
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<math>~W_\mathrm{grav}</math> |
<math>~=</math> |
<math>~ - \biggl( \frac{2^4 \pi^2}{3} \biggr) G\rho_c^2 R^5 \int_0^1 x^4 dx = - \frac{3}{5} \biggl( \frac{GM_\mathrm{tot}^2}{R} \biggr) \, ,</math> |
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<math>~T_\mathrm{kin}</math> |
<math>~=</math> |
<math>~ 2\pi R^5 \rho_c \dot\varphi^2_c \int_0^1 w^3 dw \int_0^{\sqrt{1 - w^2}} d\zeta = 2\pi R^5 \rho_c \dot\varphi^2_c \int_0^1 w^3 (1-w^2)^{1/2} dw </math> |
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<math>~=</math> |
<math>~ 2\pi R^5 \rho_c \dot\varphi^2_c \biggl[ -\frac{1}{15} (1-w^2)^{3/2} (3w^2 +2) \biggr]_0^1 = \frac{4\pi}{15} R^5 \rho_c \dot\varphi^2_c = \frac{1}{5} M_\mathrm{tot} R^2 \dot\varphi^2_c \, ,</math> |
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<math>~\mathfrak{S}_I</math> |
<math>~=</math> |
<math>~ c_s^2 \ln \biggl(\frac{\rho_c}{\rho_0}\biggr) 4\pi R^3 \rho_c \int_0^1 x^2 dx = c_s^2 M_\mathrm{tot} \ln \biggl(\frac{\rho_c}{\rho_0}\biggr) \, ,</math> |
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<math>~\mathfrak{S}_A</math> |
<math>~=</math> |
<math>~\frac{4\pi R^3 P_c}{({\gamma_g}-1)} \int_0^1 x^2 dx = \frac{M_\mathrm{tot}}{({\gamma_g}-1)} \frac{P_c}{\rho_c} \, ,</math> |
where the various dimensionless integration variables are, <math>~x \equiv (r/R)</math>, <math>~\zeta \equiv (z/R)</math>, and <math>~w \equiv (\varpi/R)</math>.
Structural Form Factors
Keeping in mind the expressions that arise in the case of an idealized spherical configuration, we generally will write the expression for the total mass as,
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<math>~M_\mathrm{tot}</math> |
<math>~=</math> |
<math>~ \frac{4\pi}{3} R^3 \rho_c \cdot \mathfrak{f}_M ~~~~~ \biggl(\Rightarrow ~ \mathfrak{f}_M = \frac{\bar\rho}{\rho_c} \biggr) \, ,</math> |
and we generally will write each energy term as follows:
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<math>~W_\mathrm{grav}</math> |
<math>~=</math> |
<math>~ - \frac{3}{5} \biggl( \frac{GM_\mathrm{tot}^2}{R} \biggr) \cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_M^2} \, ,</math> |
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<math>~T_\mathrm{kin}</math> |
<math>~=</math> |
<math>~ \frac{1}{2} \biggl( \frac{2}{5} M_\mathrm{tot} R^2 \biggr) \dot\varphi^2_c \cdot \frac{\mathfrak{f}_T }{\mathfrak{f}_M} \, ,</math> |
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<math>~\mathfrak{S}_I</math> |
<math>~=</math> |
<math>~ M_\mathrm{tot} c_s^2 \biggl[\ln\biggl( \frac{\rho_c}{\rho_0} \biggr) + \frac{\mathfrak{f}_I }{\mathfrak{f}_M} \biggl] \, ,</math> |
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<math>~\mathfrak{S}_A</math> |
<math>~=</math> |
<math>~ \frac{M_\mathrm{tot}}{({\gamma_g}-1)} \frac{P_c}{\rho_c} \cdot \frac{\mathfrak{f}_A}{\mathfrak{f}_M} \, ,</math> |
where the dimensionless form factors, <math>~\mathfrak{f}_i</math> — each usually of order unity — are,
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<math>~\mathfrak{f}_M = \frac{\bar\rho}{\rho_c}</math> |
<math>~=</math> |
<math>~ 3\int_0^1 \biggl[ \frac{\rho(x)}{\rho_c}\biggr] x^2 dx \, ,</math> |
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<math>~\mathfrak{f}_W</math> |
<math>~\equiv</math> |
<math>~ 3\cdot 5 \int_0^1 \biggl\{ \int_0^x \biggl[ \frac{\rho(x)}{\rho_c}\biggr] x^2 dx \biggr\} \biggl[ \frac{\rho(x)}{\rho_c}\biggr] x dx\, ,</math> |
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<math>~\mathfrak{f}_T</math> |
<math>~\equiv</math> |
<math>~ \frac{15}{2} \int_0^1 \biggl[ \frac{\dot\varphi(w)}{\dot\varphi_c} \biggr]^2 w^3 dw \int_0^{\sqrt{1 - w^2}} \biggl[ \frac{\rho(w,\zeta)}{\rho_c} \biggr] d\zeta\, ,</math> |
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<math>~\mathfrak{f}_I</math> |
<math>~\equiv</math> |
<math>~ 3\int_0^1 \biggl[ \frac{\rho(x)}{\rho_c}\biggr] \ln \biggl[ \frac{\rho(x)}{\rho_c}\biggr] x^2 dx \, ,</math> |
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<math>~\mathfrak{f}_A</math> |
<math>~\equiv</math> |
<math>~ 3\int_0^1 \biggl[ \frac{P(x)}{P_c}\biggr] x^2 dx \, .</math> |
In each case, the "idealized" energy expression is retrieved if/when the relevant form factor, <math>~\mathfrak{f}_i</math>, is set to unity.
Dependence on Size
Having completed carrying out the relevant volume integrals for a system of a particular size (radius), <math>~R,</math> we now establish how each term in the free-energy expression varies with size. For simplicity, we will assume that all dimensionless structural form factors remain constant during a phase of contraction or expansion, and will reference the system size to an, as yet unspecified, length scale, <math>~R_0.</math>
Mass Conservation: If the total mass of the system is held constant during a phase of contraction/expansion, the gravitational potential energy will scale as <math>~R^{-1},</math> specifically,
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<math>~W_\mathrm{grav}</math> |
<math>~=</math> |
<math>~ - \frac{3}{5} \biggl( \frac{GM_\mathrm{tot}^2}{R_0} \biggr) \biggl( \frac{R}{R_0} \biggr)^{-1} \cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_M^2} \, .</math> |
Angular Momentum Conservation: If, in addition, the system conserves its total angular momentum, <math>~J,</math> during a phase of contraction/expansion, we should write the expression for <math>~T_\mathrm{kin}</math> in terms of <math>~J.</math> For a system in uniform rotation, as is the case being considered here, we know that,
<math> T_\mathrm{rot} = \frac{1}{2}I \dot\varphi_c^2 ~~~~~ \mathrm{and} ~~~~~ J = I\dot\varphi_c \, , </math>
<math> \Rightarrow ~~~~ T_\mathrm{rot} = \frac{1}{2} \biggl( \frac{J^2}{I} \biggr) \, , </math>
where, from our above expression for <math>~T_\mathrm{kin}</math> we deduce that the scalar moment of inertia is
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<math>~I</math> |
<math>~=</math> |
<math>~ \biggl( \frac{2}{5} M_\mathrm{tot} R^2 \biggr) \frac{\mathfrak{f}_T }{\mathfrak{f}_M} \, .</math> |
[See also the formal definition of <math>~I</math> provided in our discussion of the scalar virial theorem; or see Eqs. (3) & (5) on p. 16 of EFE.] Hence, we can write,
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<math>~T_\mathrm{kin} = T_\mathrm{rot}</math> |
<math>~=</math> |
<math>~ \frac{5}{4} \frac{J^2}{MR_0^2} \biggl( \frac{R}{R_0} \biggr)^{-2} \cdot \frac{\mathfrak{f}_M}{\mathfrak{f}_T} \, .</math> |
Isothermal Contraction/Expansion: When a system expands or contracts isothermally, this means that the sound speed, <math>~c_s,</math> is held constant. Keeping in mind that mass conservation also implies,
<math> \frac{\rho_c}{\rho_0} = \biggl( \frac{R}{R_0} \biggr)^{-3} \, , </math>
we can write,
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<math>~\mathfrak{S}_I</math> |
<math>~=</math> |
<math>~ M_\mathrm{tot} c_s^2 \biggl[ \frac{\mathfrak{f}_I}{\mathfrak{f}_M} -3\ln\biggl( \frac{R}{R_0} \biggr) \biggl] \, .</math> |
Adiabatic Contraction/Expansion: If the system expands or contracts adiabatically, this means that <math>~K</math> — which specifies the material's specific entropy — is held constant. Therefore, the expression for <math>~\mathfrak{S}_A</math> must be written in terms of <math>~K</math>, rather than in terms of <math>~P_c</math>, in order to identify its proper variation with the system size. Appreciating that the relationship between <math>~K</math> and <math>~P_c</math> is governed by the (in this case, polytropic) equation of state,
<math> ~P_c = K \rho_c^{\gamma_g} \, , </math>
we can write,
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<math>~\mathfrak{S}_A</math> |
<math>~=</math> |
<math>~ \frac{M_\mathrm{tot}}{({\gamma_g}-1)} \biggl[ K\rho_c^{\gamma_g-1}\biggr] \cdot \frac{\mathfrak{f}_A}{\mathfrak{f}_M} = \frac{M_\mathrm{tot} K}{({\gamma_g}-1)} \biggl[ \frac{3M_\mathrm{tot}}{4\pi R^3} \cdot \frac{1}{\mathfrak{f}_M} \biggr]^{\gamma_g-1} \cdot \frac{\mathfrak{f}_A}{\mathfrak{f}_M} </math> |
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<math>~=</math> |
<math>\frac{M_\mathrm{tot} K}{({\gamma_g}-1)} \biggl[ \frac{3M_\mathrm{tot}}{4\pi R_0^3} \biggr]^{\gamma_g-1} \biggl( \frac{R}{R_0} \biggr)^{-3(\gamma_g-1)} \cdot \frac{\mathfrak{f}_A}{\mathfrak{f}_M^{\gamma_g}} \, .</math> |
Gathering it all Together
Gathering all of the terms together we find that, to within an additive constant, the expression for the free energy is,
<math> \mathfrak{G} = -3A\biggl( \frac{R}{R_0} \biggr)^{-1} -~ (1-\delta_{1\gamma_g})\frac{B}{(1-\gamma_g)}\biggl( \frac{R}{R_0} \biggr)^{3-3\gamma_g} -~ \delta_{1\gamma_g} 3B_I \ln \biggl( \frac{R}{R_0} \biggr) +~ C \biggl( \frac{R}{R_0} \biggr)^{-2} +~ D\biggl( \frac{R}{R_0} \biggr)^3 \, , </math>
where, <math>R_0</math> is an, as yet unspecified, scale length,
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<math>~A</math> |
<math>~\equiv</math> |
<math>\frac{1}{5} \frac{GM_\mathrm{tot} ^2}{R_0} \cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_M^2} \, ,</math> |
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<math>~B</math> |
<math>~\equiv</math> |
<math> K M_\mathrm{tot} \biggl( \frac{3M_\mathrm{tot} }{4\pi R_0^3} \biggr)^{\gamma_g - 1} \cdot \frac{\mathfrak{f}_A}{\mathfrak{f}_M^{\gamma_g}} = \bar{c_s}^2 M_\mathrm{tot} \cdot \frac{\mathfrak{f}_A}{\mathfrak{f}_M^{\gamma_g}} \, , </math> |
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<math>~B_I</math> |
<math>~\equiv</math> |
<math> c_s^2 M_\mathrm{tot} \, , </math> |
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<math>~C</math> |
<math>~\equiv</math> |
<math> \frac{5J^2}{4M_\mathrm{tot} R_0^2} \cdot \frac{\mathfrak{f}_M}{\mathfrak{f}_T} \, , </math> |
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<math>~D</math> |
<math>~\equiv</math> |
<math> \frac{4}{3} \pi R_0^3 P_e \, . </math> |
Once the pressure exerted by the external medium (<math>~P_e</math>), and the configuration's mass (<math>~M_\mathrm{tot}</math>), angular momentum (<math>~J</math>), and specific entropy (via <math>~K</math>) — or, in the isothermal case, sound speed (<math>~c_s</math>) — have been specified, the values of all of the coefficients are known and the above algebraic expression for <math>~\mathfrak{G}</math> describes how the free energy of the configuration will vary with the configuration's size (<math>~R</math>) for a given choice of <math>~\gamma_g</math>.
Visual Representation
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Figure 1: Free Energy Surface |
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This segment of the free energy "surface" shows how the free energy varies as the size of the configuration and the applied external pressure are varied, while all other relevant physical attributes are held fixed. The plotted function — derived from the above expression for <math>\mathfrak{G}</math>, with <math>\gamma_\mathrm{g} = 1</math> and <math>C=0</math> (see further discussion, below) — is, specifically, <math> \frac{\mathfrak{G}}{3Mc_s^2} = 3000\biggl[ - \frac{1}{\chi} - \ln\chi + \frac{\Pi}{3}\chi^3 + 0.9558 \biggr] \, . </math> As shown, the size of the configuration <math>(\chi)</math> increases to the right from <math>1.2</math> to <math>1.51</math>; the dimensionless external pressure <math>(\Pi)</math> increases into the screen from <math>0.103</math> to <math>0.104</math>; and the dimensionless free energy, <math>\mathfrak{G}/(3Mc_s^2)</math>, increases upward. |
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Energy Extrema
As is illustrated in Figure 1, the free energy surface generally will exhibit multiple local minima and local maxima, and may also possess one or more points of inflection. The locations along the energy surface where these special points arise identify equilibrium states, and the associated values of <math>(R/R_0)_\mathrm{eq}</math> give the radii of the equilibrium configurations.
For a given choice of the set of physical parameters <math>M</math>, <math>K</math>, <math>J</math>, <math>P_e</math>, and <math>\gamma_g</math>, extrema occur wherever,
<math> \frac{d\mathfrak{G}}{dR} = 0 \, . </math>
For the free energy function identified above,
<math> \frac{d\mathfrak{G}}{dR} = \frac{1}{R_0} \biggl[ 3A\chi^{-2} -~ (1-\delta_{1\gamma_g})~3 B\chi^{2 -3\gamma_g} -~ \delta_{1\gamma_g} 3B_I \chi^{-1} ~ -2C \chi^{-3} +~ 3D\chi^2 \biggr] \, . </math>
where,
<math>\chi \equiv \frac{R}{R_0} \, .</math>
So <math>\chi_\mathrm{eq} \equiv (R/R_0)_\mathrm{eq}</math> is obtained from the real root(s) of the equation,
<math> 3A \chi^{-2} -~ (1-\delta_{1\gamma_g})~3 B\chi^{2 -3\gamma_g} -~ \delta_{1\gamma_g} 3B_I \chi^{-1} ~ -2C \chi^{-3} +~ 3D\chi^2 = 0 \, , </math>
or, equivalently, from the roots of the equation,
<math> 2C \chi^{-2} + ~ (1-\delta_{1\gamma_g})~3 B\chi^{3 -3\gamma_g} +~ \delta_{1\gamma_g} 3B_I ~-~3A\chi^{-1} -~ 3D\chi^3 = 0 \, . </math>
Examples
- Polytropic Spheres
- Isothermal Spheres
Material that appears after this point in our presentation is under development and therefore
may contain incorrect mathematical equations and/or physical misinterpretations.
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BiPolytrope
[Following a discussion that Tohline had with Kundan Kadam on 3 July 2013, we have decided to carry out a virial equilibrium and stability analysis of nonrotating bipolytropes.]
We will adopt the following approach:
- Properties of the core <math>\cdots</math>
- Uniform density, <math>\rho_c</math>;
- Polytropic constant, <math>K_c</math>, and polytropic index, <math>n_c</math>;
- Surface of the core at <math>r_i</math>;
- Properties of the envelope <math>\cdots</math>
- Uniform density, <math>\rho_e</math>;
- Polytropic constant, <math>K_e</math>, and polytropic index, <math>n_e</math>;
- Base of the core at <math>r_i</math> and surface at <math>R</math>.
Use the dimensionless radius,
<math>\xi \equiv \frac{r}{r_i}</math>.
Then, <math>\xi_i = 1</math> and <math>\xi_s \equiv R/r_i</math>.
Expressions for Mass
Inside the core, the expression for the mass interior to any radius, <math>0 \le \xi \le 1</math>, is,
<math>M_\xi = \frac{4\pi}{3} \rho_c r_i^3 \xi^3</math> .
The expression for the mass interior to any position within the envelope, <math>1 \le \xi \le \xi_s</math>, is,
<math>M_\xi = \frac{4\pi}{3} r_i^3 \biggl[\rho_c + \rho_e(\xi^3 - 1) \biggr]</math> .
Hence, the mass of the core, the mass of the envelope, and the total mass are, respectively,
<math>M_\mathrm{core} = \frac{4\pi}{3} \rho_c r_i^3 = M_0 \biggl[ \frac{\rho_c}{\rho_0} \biggl( \frac{r_i}{R_0}\biggr)^3 \biggr]</math> <math>\Rightarrow</math> <math>\frac{\rho_c}{\rho_0} = \frac{M_\mathrm{core}}{M_0} \biggl( \frac{r_i}{R_0}\biggr)^{-3}</math> ;
<math>M_\mathrm{env} = \frac{4\pi}{3} r_i^3 \biggl[\rho_e (\xi_s^3 - 1) \biggr] = M_0 (\xi_s^3 - 1) \biggl[ \frac{\rho_e}{\rho_0} \biggl( \frac{r_i}{R_0}\biggr)^3 \biggr]</math> <math>\Rightarrow</math> <math>\frac{\rho_e}{\rho_0} = \frac{M_\mathrm{env}}{M_0} \biggl( \frac{r_i}{R_0}\biggr)^{-3} (\xi_s^3 - 1)^{-1}</math> ;
<math>M_\mathrm{tot} = \frac{4\pi}{3} r_i^3 \biggl[\rho_c + \rho_e(\xi_s^3 - 1) \biggr] = M_0 \biggl( \frac{\rho_c}{\rho_0} \biggr) \biggl( \frac{r_i}{R_0}\biggr)^3 \biggl[ 1 + \frac{\rho_e}{\rho_c} (\xi_s^3 - 1) \biggr] </math> ;
where, <math>M_0 \equiv 4\pi \rho_0 R_0^3/3</math>. Letting <math>\nu \equiv M_\mathrm{core}/M_\mathrm{tot}</math> — which also means, <math>M_\mathrm{env}/M_\mathrm{tot} = (1-\nu) </math> — we can write,
<math>\frac{\rho_e}{\rho_c} = \frac{M_\mathrm{env}}{M_\mathrm{core}} (\xi_s^3 - 1)^{-1} = \frac{(1-\nu)}{\nu (\xi_s^3 - 1)} </math> ,
and,
<math>\nu (\xi_s^3 - 1) \biggl( \frac{\rho_e}{\rho_c} \biggr) = (1-\nu) </math> <math>\Rightarrow</math> <math>\nu = \biggl[ 1 + \biggl( \frac{\rho_e}{\rho_c} \biggr) (\xi_s^3 - 1) \biggr]^{-1}</math> .
Following the work of Schönberg & Chandrasekhar (1942) — see our accompanying discussion — we are seeking equilibrium configurations in the <math>\nu - q</math> plane where,
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<math>\nu</math> |
<math>\equiv</math> |
<math>\frac{M_\mathrm{core}}{M_\mathrm{tot}} </math>, (as also defined here) |
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<math>q</math> |
<math>\equiv</math> |
<math>\frac{r_i}{R} = \frac{1}{\xi_s}</math> . |
We also will be examining the stability of configurations, looking for extrema in the free energy plane where <math>\nu</math> is allowed to vary while holding <math>q</math> fixed. According to the above relations, this can be accomplished by varying the ratio, <math>\rho_e/\rho_c</math>. Notice that this can also be viewed as follows: Changing the ratio of densities, <math>\rho_e/\rho_c</math>, while holding <math>q</math> fixed will cause <math>\nu</math> to vary. If we want to impose this perturbation on a configuration of constant total mass, then an additional contraint on the choice of <math>\rho_e</math> and <math>\rho_c</math> is,
<math>\rho_c|_0 + \rho_e|_0 (\xi_s^3 - 1) = \rho_c|_\mathrm{new} + \rho_e|_\mathrm{new}(\xi_s^3 - 1) </math>
<math>\Rightarrow</math> <math>\rho_c|_0 - \rho_c|_\mathrm{new} = \rho_e|_\mathrm{new}(\xi_s^3 - 1) - \rho_e|_0 (\xi_s^3 - 1) </math>
<math>\Rightarrow</math> <math>\rho_c|_0 ( 1 - f_c ) = \rho_e|_0 (\xi_s^3 - 1) \biggl[ f_e - 1 \biggr]</math>
<math>\Rightarrow</math> <math>f_e = 1 + \biggl[ \frac{( 1 - f_c )}{ (\xi_s^3 - 1) } \biggr] \biggl( \frac{\rho_c}{\rho_e} \biggr)_0 \, ,</math>
where, <math>f_c \equiv \rho_c|_\mathrm{new}/ \rho_c|_0</math> and <math>f_e \equiv \rho_e|_\mathrm{new}/ \rho_e|_0</math>. This also means that, for a given initial choice of the density ratio <math>(\rho_e/\rho_c)</math> and the factor, <math>f_c</math>, by which you want the core density to increase or decrease, the new density ratio will be,
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<math> \biggl( \frac{\rho_e}{\rho_c} \biggr)_\mathrm{new} </math> |
<math>=</math> |
<math> \frac{f_e}{f_c} \biggl( \frac{\rho_e}{\rho_c} \biggr)_0 </math> |
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<math>=</math> |
<math> \frac{1}{f_c} \biggl\{ \biggl( \frac{\rho_e} {\rho_c}\biggr)_0 + \biggl[ \frac{( 1 - f_c )}{ (\xi_s^3 - 1) } \biggr] \biggr\} \, . </math> |
A variation in <math>\nu</math> will imply that the mass contained in the core will vary, since <math>\nu = M_\mathrm{core}/M_\mathrm{tot}</math>. The particular algebraic relation makes sense because, a decrease in the ratio <math>\rho_e/\rho_c</math> will mean that <math>\nu</math> increases, which also means that a relatively larger fraction of the mass is in the core.
Energy Expressions
The gravitational potential energy of the bipolytropic configuration is obtained by integrating over the following differential energy contribution,
<math>dW = - \biggl( \frac{GM_r}{r} \biggr) dm</math> .
Hence,
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<math>W = W_\mathrm{core} + W_\mathrm{env}</math> |
<math> = - G \biggl\{ \int_0^{r_i} \biggl( \frac{M_r}{r} \biggr) 4\pi r^2 \rho_c dr + \int^R_{r_i} \biggl( \frac{M_r}{r} \biggr) 4\pi r^2 \rho_e dr \biggr\} </math> |
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<math> = - G \biggl\{ \int_0^1 \biggl( \frac{4\pi }{3} \rho_c r_i^3 \xi^3 \biggr) 4\pi r_i^2 \rho_c \xi d\xi + \int_1^{\xi_s} \frac{4\pi}{3} \rho_c r_i^3 \biggl[ 1 + \frac{\rho_e}{\rho_c}(\xi^3 - 1) \biggr] 4\pi r_i^2 \rho_e \xi d\xi \biggr\} </math> |
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<math> = - \frac{3GM^2_\mathrm{core}}{r_i} \biggl\{ \int_0^1 \xi^4 d\xi + \int_1^{\xi_s} \biggl[ 1 + \frac{\rho_e}{\rho_c}(\xi^3 - 1) \biggr] \biggl( \frac{\rho_e}{\rho_c} \biggr) \xi d\xi \biggr\} </math> |
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<math> = - \frac{3GM^2_\mathrm{core}}{r_i} \biggl\{ \frac{1}{5} + \biggl( \frac{\rho_e}{\rho_c} \biggr) \int_1^{\xi_s} \xi d\xi + \biggl( \frac{\rho_e}{\rho_c} \biggr)^2 \int_1^{\xi_s} (\xi^3 - 1) \xi d\xi \biggr\} </math> |
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<math> = - \biggl( \frac{GM^2_\mathrm{tot}}{R} \biggr) 3\nu^2 \xi_s \biggl\{ \frac{1}{5} + \frac{1}{2} \biggl( \frac{\rho_e}{\rho_c} \biggr) (\xi_s^2 - 1) + \biggl( \frac{\rho_e}{\rho_c} \biggr)^2 \biggl[ \frac{1}{5}(\xi_s^5 - 1) - \frac{1}{2}(\xi_s^2-1) \biggr] \biggr\} </math> |
I like the form of this expression. The leading term, which scales as <math>R^{-1}</math>, encapsulates the behavior of the gravitational potential energy for a given choice of the internal structure, namely, a given choice of <math>\xi_s</math>, <math>\nu</math>, and density ratio <math>(\rho_e/\rho_c)</math>. Actually, only two internal structural parameters need to be specified — <math>\xi_s</math> and <math>f_c</math>; from these two, the expressions shown above allow the determination of both <math>(\rho_e/\rho_c)</math> and <math>\nu</math>.
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© 2014 - 2021 by Joel E. Tohline |