User:Tohline/SSC/FreeEnergy/Equilibrium Sequence Instabilities

Instabilities Associated with Equilibrium Sequence Turning Points

Introduction

Spherically symmetric, self-gravitating, equilibrium configurations can be constructed from gases exhibiting a wide variety of degrees of compressibility. When examining how the internal structure of such configurations varies with compressibility, or when examining the relative stability of such structures, it can be instructive to construct models using a polytropic equation of state,

<math>~P = K_\mathrm{n} \rho^{1+1/n}</math>

because the degree of compressibility can be adjusted by simply changing the value of the polytropic index, <math>~n</math>, across the range, <math>~0 \le n \le \infty</math>. (Alternatively, one can vary the effective adiabatic exponent of the gas, <math>~\Gamma = 1 + 1/n</math>.) In particular, <math>~n = 0 ~~ (\Gamma = \infty)</math> represents a hard equation of state and describes an incompressible configuration, while <math>~n = \infty ~~(\Gamma = 1)</math> represents an isothermal and extremely soft equation of state.

As has been detailed in an accompanying discussion, the structural properties of spherical polytropes can be described entirely in terms of a dimensionless radial coordinate, <math>~\xi</math>, and by the radial dependence of the dimensionless enthalpy function, <math>~\theta_n(\xi)</math>, and its first radial derivative, <math>~\theta^'_n(\xi)</math>. At the center of each configuration <math>~(\xi=0)</math>, <math>~\theta_n = 1</math> and <math>~\theta^'_n = 0</math>. The surface of each isolated polytrope is identified by the radial coordinate, <math>~\xi_1</math>, at which <math>~\theta_n</math> first drops to zero. As a class, isolated polytropes exhibit three attributes that are especially key in the context of our present discussion:

1. The equilibrium structure is dynamically stable if <math>~n < 3</math>.
2. The equilibrium structure has a finite radius if <math>~n < 5</math>.
3. The equilibrium structure can be described in terms of closed-form analytic expressions for <math>~n = 0</math>, <math>~n = 1</math>, and <math>~n = 5</math>.

Isothermal spheres are discussed in a wide variety of astrophysical contexts because it is not uncommon for physical conditions to conspire to create an extended volume throughout which a configuration exhibits uniform temperature. But, as can be surmised from our list of three key polytrope attributes and recognition that equilibrium isothermal configurations are polytropes with index <math>~n=\infty</math>, mathematical models of isolated isothermal spheres are relatively cumbersome to analyze because they extend to infinity, they are dynamically unstable, and they are not describable in terms of analytic functions. In such astrophysical contexts, we have sometimes found it advantageous to employ an <math>~n=5</math> polytrope instead of an isothermal sphere. An isolated <math>~n=5</math> polytrope can serve as an effective surrogate for an isothermal sphere because it is both infinite in extent and dynamically unstable, but it is less cumbersome to analyze because its structure can be described by closed-form analytic expressions.

Bonnor-Ebert Sphere

In the mid-1950s, Ebert (1955) and Bonnor (1956) independently realized that an equilibrium isothermal gas cloud can be constructed with a finite radius by embedding it in a hot, tenuous external medium. As has been described in an accompanying chapter, the relevant mathematical model is constructed by chopping off the isolated isothermal sphere at some finite radius — call it, <math>~\xi_e</math> — and imposing an externally applied pressure, <math>~P_e</math>, that is equal to the pressure of the isothermal gas at the specified edge of the truncated sphere. By varying <math>~\xi_e</math>, a sequence of equilibrium models can be constructed, as illustrated in Figure 1.

Figures from (left) Ebert (1955) and (center) Bonnor (1956)		Our Construction

Figures from (left) Ebert (1955) and (right) Bonnor (1956)
Our Analytic Analysis of n = 5 Polytropic Sequence]

Figure from Stahler (1983)	From our Detailed Analysis of Pressure-Truncated Polytrope Sequences
	Order-of-Magnitude	Detailed Force-Balance

Schönberg-Chandrasekhar Mass

Figures from (left) Henrich & Chandraskhar (1941) and (center) Schönberg & Chandrasekhar (1942)		Bipolytropes with <math>~(n_c,n_e) = (5,1)</math>

Figures from (left) Henrich & Chandraskhar (1941) and (center) Schönberg & Chandrasekhar (1942)		Bipolytropes with <math>~(n_c,n_e) = (5,1)</math>

In the early 1940s, Chandrasekhar and his colleagues (see Henrich & Chandraskhar (1941) and Schönberg & Chandrasekhar (1942)) discovered that a star with an isothermal core will become unstable if the fractional mass of the core is above some limiting value. They discovered this by constructing models that are now commonly referred to as composite polytropes or bipolytropes, that is, models in which the star's core is described by a polytropic equation of state having one index — say, <math>~n_c</math> — and the star's envelope is described by a polytropic equation of state of a different index — say, <math>~n_e</math>. In an accompanying discussion we explain in detail how the two structural components with different polytropic indexes are pieced together mathematically to build equilibrium bipolytropes. For a given choice of the two indexes, <math>~n_c</math> and <math>~n_e</math>, a sequence of models can be generated by varying the radial location at which the interface between the core and envelope occurs. As the interface location is varied, the relative amount of mass enclosed inside the core, <math>~\nu \equiv M_\mathrm{core}/M_\mathrm{tot}</math>, quite naturally varies as well.

Henrich & Chandraskhar (1941) built structures of uniform composition having an isothermal core (<math>~n_c = \infty</math>) and an <math>~n_e = 3/2</math> polytropic envelope and found that equilibrium models exist only for values of <math>~\nu \le \nu_\mathrm{max} \approx 0.35</math>. Schönberg & Chandrasekhar (1942) extended this analysis to include structures in which the mean molecular weight of the gas changes discontinuously across the interface. Specifically, they used the same values of <math>~n_c</math> and <math>~n_e</math> as Henrich & Chandrasekhar, but they constructed models in which the ratio of the molecular weight in the core to the molecular weight in the envelope is <math>~\mu_c/\mu_e = 2</math>. This was done to more realistically represent stars as they evolve off the main sequence; they have inert, isothermal helium cores and envelopes that are rich in hydrogen. Note that introducing a discontinuous drop in the mean molecular weight at the core-envelope interface also introduces a discontinuous drop in the gas density across the interface. As the following excerpt from p. 168 of their article summarizes, in these models, Schönberg & Chandrasekhar (1942) found that <math>~\nu_\mathrm{max} \approx 0.101</math>. This is commonly referred to as the Schönberg-Chandrasekhar mass limit, although it was Henrich & Chandrasekhar who were the first to identify the instability.

Text excerpt from Schönberg & Chandrasekhar (1942)

In an effort to develop a more complete appreciation of the onset of the instability associated with the Schönberg-Chandrasekhar mass limit, Beech (1988) matched an analytically prescribable, <math>~n_e = 1</math> polytropic envelope to an isothermal core and, like Schönberg & Chandrasekhar, allowed for a discontinuous change in the molecular weight at the interface. [For an even more comprehensive generalization and discussion, see Ball, Tout, & Żytkow (2012, MNRAS, 421, 2713)]. Beech's results were not significantly different from those reported by Schönberg & Chandrasekhar (1942); in particular, the value of <math>~\nu_\mathrm{max}</math> was still only definable numerically because an isothermal core cannot be described in terms of analytic functions.

In an accompanying derivation [see, also, Eggleton, Faulkner, and Cannon (1998, MNRAS, 298, 831)] we have gone one step farther, matching an analytically prescribable, <math>~n_e = 1</math> polytropic envelope to an analytically prescribable, <math>~n_c = 5</math> polytropic core. For this bipolytrope, we show that there is a limiting mass-fraction, <math>~\nu_\mathrm{max}</math>, for any choice of the molecular weight ratio <math>~\mu_c/\mu_e > 3</math> and that the interface location, <math>~\xi_i</math>, associated with this critical configuration is given by the positive, real root of the following relation:

<math> \biggl(\frac{\pi}{2} + \tan^{-1} \Lambda_i\biggr) (1+\ell_i^2) [ 3 + (1-m_3)^2(2-\ell_i^2)\ell_i^2] - m_3 \ell_i [(1-m_3)\ell_i^4 - (m_3^2 - m_3 +2)\ell_i^2 - 3] = 0 \, , </math>

where,

<math> \ell_i \equiv \frac{\xi_i}{\sqrt{3}} \, ; </math>             <math> m_3 \equiv 3 \biggl( \frac{\mu_c}{\mu_e} \biggr)^{-1} \, ; </math>       and       <math> \Lambda_i \equiv \frac{1}{m_3\ell_i} [ 1 + (1-m_3)\ell_i^2] \, . </math>

Close Binary Stars

Darwin's (1906) Two Illustrative Binary Sequences

From Pt. I, §1 (p. 164) of G. H. Darwin (1906) — verbatum text in green: It will be useful to make a rough preliminary investigation of the regions in which we shall have to look for cases of limiting stability in the two problems. For this purpose I consider [1] the case of two spheres as the analogue of [Darwin's] problem of the figure of equilibrium, and [2] the case of a sphere and a particle as the analogue of Roche's problem.

… let the mass of the whole system be <math>~M_\mathrm{tot} = \tfrac{4}{3}\pi \rho a^3</math>; let the masses of the two spheres be

<math>~M = M_\mathrm{tot} \biggl[\frac{\lambda}{1+\lambda}\biggr] </math>

and

<math>~M^' = M_\mathrm{tot} \biggl[\frac{1}{1+\lambda}\biggr] </math>

<math>~\Rightarrow</math>

<math>~\lambda = \frac{M}{M^'}</math>

or for Roche's problem let the latter <math>~(M^')</math> be the mass of the particle. Assuming that both spheres have the same characteristic density, <math>~\rho</math>, that has been used to specify the total mass, we furthermore know that the radius of the first sphere is,

<math>~a_M</math>

<math>~=</math>

<math>~\biggl(\frac{M}{\tfrac{4}{3}\pi \rho}\biggr)^{1 / 3} = a \biggl(\frac{\lambda}{1+\lambda}\biggr)^{1 / 3} \, ,</math>

and (for the Darwin problem) the radius of the second sphere is,

<math>~a_{M^'}</math>

<math>~=</math>

<math>~\biggl(\frac{M^'}{\tfrac{4}{3}\pi \rho}\biggr)^{1 / 3} = a\biggl(\frac{1 }{1+\lambda}\biggr)^{1 / 3} \, .</math>

Let <math>~r</math> be the distance from the centre of one sphere to that of the other, or to the particle, as the case may be; and <math>~\omega</math> the orbital angular velocity, where (setting the gravitational constant, <math>~G=1</math>),

<math>~\omega^2 r^3</math>

<math>~=</math>

<math>~M_\mathrm{tot} \, .</math>

The centre of inertia of the two masses is distant <math>~r/(1+\lambda)</math> and <math>~\lambda r/(1+\lambda)</math> from their respective centres, and we easily find the orbital momentum to be

<math>~L_\mathrm{orb} </math>	<math>~=</math>	<math>~M \biggl(\frac{r}{1+\lambda}\biggr)^2\omega + M^' \biggl(\frac{\lambda r}{1+\lambda}\biggr)^2\omega</math>
	<math>~=</math>	<math>~M_\mathrm{tot} \biggl[ \biggl(\frac{\lambda}{1+\lambda}\biggr)\biggl(\frac{1}{1+\lambda}\biggr)^2\omega + \biggl(\frac{1}{1+\lambda}\biggr) \biggl(\frac{\lambda }{1+\lambda}\biggr)^2 \biggr] r^2 \omega</math>
	<math>~=</math>	<math>~M_\mathrm{tot} \biggl[ \frac{\lambda}{(1+\lambda)^2} \biggr] r^2\omega \, .</math>

In both problems the rotational momentum of the first sphere is

<math>~L_\mathrm{M} = \frac{2}{5} Ma_M^2 \omega</math>

<math>~=</math>

<math>~ \frac{2}{5} M_\mathrm{tot}\biggl( \frac{\lambda}{1+\lambda} \biggr) \biggl(\frac{\lambda}{1+\lambda}\biggr)^{2 / 3}a^2 \omega =\frac{2}{5} M_\mathrm{tot} \biggl(\frac{\lambda}{1+\lambda}\biggr)^{5 / 3}a^2 \omega \, . </math>

In the [Darwin] problem the rotational momentum of the second sphere is

<math>~L_\mathrm{M^'} = \frac{2}{5} M^' a_{M^'}^2 \omega</math>

<math>~=</math>

<math>~ \frac{2}{5} M_\mathrm{tot}\biggl( \frac{1}{1+\lambda} \biggr) \biggl(\frac{1}{1+\lambda}\biggr)^{2 / 3}a^2 \omega =\frac{2}{5} M_\mathrm{tot} \biggl(\frac{1}{1+\lambda}\biggr)^{5 / 3}a^2 \omega \, , </math>

and in the [Roche] problem it is nil.

If, then, we write <math>~L_1</math> for the total angular momentum of the two spheres, and <math>~L_2</math> for that of the sphere and particle, we have

<math>~L_1 </math>	<math>~=</math>	<math>~ L_\mathrm{orb} + L_M + L_{M^'} </math>
	<math>~=</math>	<math>~ M_\mathrm{tot} \biggl[ \frac{\lambda}{(1+\lambda)^2} \biggr] r^2\omega + \frac{2}{5} M_\mathrm{tot} \biggl(\frac{\lambda}{1+\lambda}\biggr)^{5 / 3}a^2 \omega + \frac{2}{5} M_\mathrm{tot} \biggl(\frac{1}{1+\lambda}\biggr)^{5 / 3}a^2 \omega</math>
	<math>~=</math>	<math>~ M_\mathrm{tot} a^2 \omega \biggl[ \frac{\lambda r^2}{(1+\lambda)^2a^2} + \frac{2}{5} \frac{1+ \lambda^{5/3}}{(1+\lambda)^{5 / 3}} \biggr] \, , </math>
<math>~L_2 </math>	<math>~=</math>	<math>~ L_\mathrm{orb} + L_M </math>
	<math>~=</math>	<math>~ M_\mathrm{tot} a^2 \omega \biggl[ \frac{\lambda r^2}{(1+\lambda)^2a^2} + \frac{2}{5} \frac{\lambda^{5/3}}{(1+\lambda)^{5 / 3}} \biggr] \, . </math>

Note that the two binary components come into contact when, for the Darwin problem,

<math>~a_M + a_{M^'} = r</math>

<math>~\Rightarrow</math>

<math>~\frac{r}{a} = \frac{1 + \lambda^{1 / 3}}{(1+\lambda)^{1 / 3}} \, ; </math>

and, for the Roche problem,

<math>~a_M = r</math>

<math>~\Rightarrow</math>

<math>~\frac{r}{a} = \frac{\lambda^{1 / 3}}{(1+\lambda)^{1 / 3}} \, . </math>

Setup

Jeans (1919)

From § 50 (p. 46) of J. H. Jeans (1919) — verbatum text in green: Let the two bodies be spoken of as primary and secondary, and let their masses be <math>~M</math>, <math>~M^'</math> respectively; let the distance apart of their centres of gravity be <math>~R</math>, and let the angular velocity of rotation of the line joining them be <math>~\omega</math>. It will be sufficient to fix our attention on the conditions of equilibrium of one of the two masses, say the primary. Let its centre of gravity be taken as origin, let the line joining it to the centre of the secondary be axis of <math>~x</math>, and let the plane in which the rotation takes place be that of <math>~xy</math>. Then the equation of the axis of rotation is

<math>~x = \frac{M^'}{M + M^'} ~ R</math>

and

<math>~y = 0 \, .</math>

The problem may be reduced to a statical one (cf. § 31) by supposing the masses acted on by a field of force of [the centrifugal] potential

<math>~\frac{1}{2}\omega^2\biggl[ \biggl( x - \frac{M^' }{M + M^'} ~R \biggr)^2 + y^2\biggr] \, .</math>

Chandrasekhar (1969)

From pp. 189-190 of [EFE] — verbatum text in green: Let the masses of the primary and the secondary be <math>~M</math> and <math>~M^'</math>, respectively; let the distance between their centers of mass be <math>~R</math>; and let the constant angular velocity of rotation about their common center of mass be <math>~\Omega</math>. Choose a coordinate system in which the origin is at the center of mass of the primary, the <math>~x_1-</math>axis points to the center of mass of the secondary, and the <math>~x_3-</math>axis is parallel to the direction of <math>~\vec\Omega</math>. In this coordinate system, the equation of motion governing fluid elements of <math>~M</math> includes (see EFE's equation 1) a gradient of the centrifugal potential,

<math>~\frac{1}{2}\Omega^2\biggl[ \biggl( x_1 - \frac{M^' R}{M + M^'} \biggr)^2 + x_2^2\biggr] \, .</math>

Tassoul (1978)

From p. 449 of [T78] — verbatum text in green: Let the masses of the primary and the secondary be <math>~M</math> and <math>~M^'</math>, respectively; let the distance between their centers of mass be <math>~d</math>; and let the angular velocity of rotation about their common center of mass be <math>~\Omega</math>. Next choose a system of reference in which the origin is at the center of mass of the primary; for convenience, the <math>~x_1-</math>axis points toward the center of mass of the secondary, and the <math>~x_3-</math>axis is parallel to the direction of <math>~\vec\Omega</math>. Then, the equation of the rotation axis, which of course passes through the center of mass of the two bodies, is

<math>~x_1 = \frac{M^' }{M + M^'} ~d</math>

and

<math>~x_2 = 0 \, .</math>

Accordingly, the centrifugal force acting on the mass <math>~M</math> may be derived from the potential

<math>~-\frac{1}{2}\Omega^2\biggl[ \biggl( x_1 - \frac{M^'}{M + M^'} ~ d\biggr)^2 + x_2^2\biggr] \, .</math>

Roche Ellipsoids

Jeans (1919)

From § 51 (p. 47) of J. H. Jeans (1919) — verbatum text in green: The simplest problem occurs when the secondary may be treated as a rigid sphere; this is the special problem dealt with by Roche. As in § 47 the tide-generating potential acting on the primary may be supposed to be

<math>~\frac{M^'}{R} + \frac{M^'}{R^2} x + \frac{M^'}{R^3}(x^2 - \tfrac{1}{2}y^2 - \tfrac{1}{2}z^2) + \cdots </math>

We shall for the present be content to omit all terms beyond those written down. The correction required by the neglect of these terms will be discussed later, and will be found to be so small that the results now to be obtained are hardly affected.

On omitting these terms, and combining the two potentials … it appears that the primary may be supposed influenced by a statical field of potential

<math>~\frac{M^'}{R} x\biggr(1 - \frac{\omega^2 R^3}{M + M^'}\biggr) + \frac{M^'}{R^3}(x^2 - \tfrac{1}{2}y^2 - \tfrac{1}{2}z^2) + \tfrac{1}{2}\omega^2(x^2 + y^2) \, .</math>

The terms in <math>~x</math> may immediately be removed by supposing <math>~\omega</math> to have the appropriate value given by

<math>~\omega^2</math>

<math>~=</math>

<math>~\frac{M+M^'}{R^3}</math>

and the condition for equilibrium is now seen to be that we shall have, at every point of the surface,

<math>~V_b + \mu (x^2 - \tfrac{1}{2}y^2 - \tfrac{1}{2}z^2) + \tfrac{1}{2}\omega^2(x^2 + y^2) </math>

<math>~=</math>

constant

where <math>~\mu</math> … stands for <math>~M^'/R^3</math> .

Chandrasekhar (1969)

From p. 190 of [EFE] — verbatum text in green: In Roche's particular problem, the secondary is treated as a rigid sphere. Then, over the primary, the tide-generating potential, <math>~\mathfrak{B}^'</math> can be expanded in the form

<math>~\mathfrak{B}^'</math>

<math>~=</math>

<math>~ \frac{GM^'}{R} \biggl( 1 + \frac{x_1}{R} + \frac{x_1^2 - \tfrac{1}{2}x_2^2 - \tfrac{1}{2}x_3^2}{R^2} + \cdots \biggr) \, ;</math>

and the approximation which underlies this theory is to retain, in this expansion for <math>~\mathfrak{B}^'</math>, only the terms which have been explicitly written down and ignore all the terms which are of higher order. On this assumption, the equation of motion becomes

<math>~\frac{du_i}{dt} + \frac{1}{\rho} \frac{\partial p}{\partial x_i}</math>

<math>~=</math>

<math>~ \frac{\partial}{\partial x_i} \biggl[ \mathfrak{B} + \tfrac{1}{2}\Omega^2(x_1^2 + x_2^2) + \mu(x_1^2 - \tfrac{1}{2}x_2^2 - \tfrac{1}{2}x_3^2) + \biggl( \frac{GM^'}{R^2} - \frac{M^' R}{M+M^'} ~\Omega^2 \biggr)x_1 \biggr] + 2\Omega \epsilon_{i\ell 3} u_\ell </math>

where we have introduced the abbreviation

<math>~\mu = \frac{GM^'}{R^3} \, .</math>

So far, we have left <math>~\Omega^2</math> unspecified. If we now let <math>~\Omega^2</math> have the "Keplerian value"

<math>~\Omega^2 = \frac{G(M+ M^')}{R^3} = \mu \biggl(1 + \frac{M}{M^'} \biggr) \, ,</math>

the "unwanted" term in <math>~x_1</math>, on the right-hand side of [this equation,] vanishes and we are left with

<math>~\frac{du_i}{dt} + \frac{1}{\rho} \frac{\partial p}{\partial x_i}</math>

<math>~=</math>

<math>~ \frac{\partial}{\partial x_i} \biggl[ \mathfrak{B} + \tfrac{1}{2}\Omega^2(x_1^2 + x_2^2) + \mu(x_1^2 - \tfrac{1}{2}x_2^2 - \tfrac{1}{2}x_3^2) \biggr] + 2\Omega \epsilon_{i\ell 3} u_\ell \, . </math>

This is the basic equation of this theory; and Roche's problem is concerned with the equilibrium and the stability of homogeneous masses governed by [this relation].

Tassoul (1978)

From pp. 449-450 of [T78] — verbatum text in green: In Roche's particular problem, the secondary is treated as a rigid sphere; hence, over the primary, the tide-generating potential can be expanded in the form

<math>~ -\frac{GM^'}{d} \biggl( 1 + \frac{x_1}{d} + \frac{x_1^2 - \tfrac{1}{2}x_2^2 - \tfrac{1}{2}x_3^2}{d^2} + \cdots \biggr) \, .</math>

The approximation that underlies the theory is to omit all terms beyond those written down. On this assumption, we find that, apart from its own gravitation, the primary may be supposed to be acted upon by a total field of force derived from the potential

<math>~-\tfrac{1}{2}\Omega^2(x_1^2 + x_x^2) - \mu(x_1^2 - \tfrac{1}{2}x_2^2 - \tfrac{1}{2}x_3^2) - \biggl(\mu - \frac{M^'}{M+M^'} \Omega^2\biggr) dx_1 \, ,</math>

where

<math>~\mu</math>

<math>~=</math>

<math>~\frac{GM^'}{d^3} \, .</math>

Further letting <math>~\Omega^2</math> have its "Keplerian value"

<math>~\Omega^2</math>

<math>~=</math>

<math>~\frac{G(M+M^')}{d^3} \, ,</math>

we can thus write the conditions of relative equilibrium for the primary in the form

<math>~\frac{1}{\rho} \nabla p</math>

<math>~=</math>

<math>~ -\nabla [ V -\tfrac{1}{2}\Omega^2(x_1^2 + x_x^2) - \mu(x_1^2 - \tfrac{1}{2}x_2^2 - \tfrac{1}{2}x_3^2) ] \, , </math>

where <math>~V</math> is the self-gravitating potential of the primary.

References

• G. H. Darwin (1906) On the Figure and Stability of a Liquid Satellite, Philosophical Transactions of the Royal Society A, 206, 161.
• J. H. Jeans (1919) Problems of Cosmogony and Stellar Dynamics -- see especially beginning in Ch. III (p. 46)
• Z. Kopal (1959), Close Binary Systems -- see especially, Ch. III (p. 125); but not particularly useful.
• Chapter 8 (pp. 189-241) of [EFE]
• Chapter 16 (pp. 448-469) of [T78]
• Lai, Rasio, & Shapiro (1993a, ApJL, 406, L63)
• Lai, Rasio, & Shapiro (1993b, ApJS, 88, 205)

Figures from (left) Lai, Rasio, & Shapiro (1993) and (right) New & Tohline (1997)

See Also

The discussion presented here is supported by detailed reviews and new derivations presented in the following associated chapters:

• A free-energy synopsis.
• Material associated with an overarching PowerPoint presentation.
• Limiting Masses
• An accompanying chapter on the Bonnor-Ebert sphere.

Overlapping discussions of this topic may also be found in the following key references:

• §6.8 of [ST83].

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