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and the corresponding values of the angular velocity are provided by the expression,
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and the corresponding values of the angular velocity (in units of [G &rho;]<sup>½</sup>) are provided by the expression,
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Revision as of 15:26, 5 November 2019


Contents

Riemann S-type Ellipsoids

Whitworth's (1981) Isothermal Free-Energy Surface
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General Coefficient Expressions

As has been detailed in an accompanying chapter, the gravitational potential anywhere inside or on the surface, ~(a_1,a_2,a_3) ~\leftrightarrow~(a,b,c), of an homogeneous ellipsoid may be given analytically in terms of the following three coefficient expressions:


~A_1


~=

~2\biggl(\frac{b}{a}\biggr)\biggl(\frac{c}{a}\biggr)
\biggl[  \frac{F(\theta,k) - E(\theta,k)}{k^2 \sin^3\theta} \biggr] \, ,


~A_3


~=


~2\biggl(\frac{b}{a}\biggr) \biggl[  \frac{(b/a) \sin\theta - (c/a)E(\theta,k)}{(1-k^2) \sin^3\theta} \biggr] \, ,


~A_2


~=

~2 - (A_1+A_3) \, ,

where, ~F(\theta,k) and ~E(\theta,k) are incomplete elliptic integrals of the first and second kind, respectively, with arguments,

~\theta = \cos^{-1} \biggl(\frac{c}{a} \biggr)

      and      

~k = \biggl[\frac{1 - (b/a)^2}{1 - (c/a)^2} \biggr]^{1/2} \, .

[ EFE, Chapter 3, §17, Eq. (32) ]


TEST (part 1)
Notation: Use ~\phi in place of ~\theta.
~\frac{b}{a} ~\frac{c}{a} ~\phi ~k Numerical Recipes ~A_1 ~A_2 ~A_3
(deg) (rad) (deg) (rad) ~F(\phi,k) ~E(\phi,k)
0.9 0.641 50.13357253 0.874995907 32.53852919 0.567904468 0.909025949 0.843118048 0.521450273 0.595131012 0.883418715


Equilibrium Conditions for Riemann S-type Ellipsoids

Pulling from Chapter 7 — specifically, §48 — of Chandrasekhar's EFE, we understand that the semi-axis ratios, ~(\tfrac{b}{a}, \tfrac{c}{a}) associated with Riemann S-type ellipsoids are given by the roots of the equation,

~
\biggl[  \frac{a^2 b^2}{a^2 + b^2} \biggr] f \biggl( \frac{\Omega^2}{\pi G \rho} \biggr)

~=

~a^2 b^2 A_{12} - c^2 A_3 \, ,

[ EFE, §48, Eq. (34) ]

and the associated value of the square of the equilibrium configuration's angular velocity is,

~\biggl[ 1 + \frac{a^2 b^2 \cdot f^2}{(a^2 + b^2)^2}  \biggr] \frac{\Omega^2}{\pi G \rho}

~=

~2B_{12} \, ,

[ EFE, §48, Eq. (33) ]

where,

~A_{12}

~\equiv

~-\frac{A_1-A_2}{(a^2 - b^2)} \, ,

[ EFE, §21, Eq. (107) ]

~B_{12}

~\equiv

~A_2 - a^2A_{12} \, .

[ EFE, §21, Eq. (105) ]

(Notice that if we set ~f \rightarrow 0, this pair of expressions simplifies to the pair we have provided in a separate discussion of the equilibrium conditions for Jacobi ellipsoids.) Following Chandrasekhar's lead and eliminating ~\Omega^2 between these two expressions, we obtain,

~0

~=

~
\biggl[ \frac{a^2 b^2}{(a^2 + b^2)^2} \biggr] f^2 
+ \biggl[ \frac{2a^2 b^2 B_{12}}{c^2 A_3 - a^2 b^2 A_{12}} \biggr]\frac{f}{a^2 + b^2} + 1 \, .

[ EFE, §48, Eq. (35) ]

For a given ~f, this last expression determines the ratios of the axes of the ellipsoids that are compatible with equilibrium; and the value of ~\Omega^2, that is to be associated with a particular solution of this last expression, then follows from either one of the first two expressions. For convenience of evaluation and for greater clarity, let's rewrite this last (quadratic) equation in the form,

~0

~=

~
\alpha f^2 + \beta f + 1 \, ,

in which case the pair of solutions is,

~f

~=

~
\frac{1}{2\alpha}\biggr\{
- \beta \pm \biggl[ \beta^2 - 4\alpha \biggr]^{1 / 2}
\biggr\} \, ;

and the corresponding values of the angular velocity (in units of [G ρ]½) are provided by the expression,

~\omega \equiv \frac{\Omega}{\sqrt{G\rho}}

~=

~
\pm \biggl\{ 2 \pi B_{12} \biggl[ 1 + \alpha f^2 \biggr]^{-1} \biggr\}^{1 / 2}


TEST (part 2)
~\frac{b}{a} ~\frac{c}{a} ~a^2 A_{12} ~ B_{12} ~\alpha \equiv \frac{(b/a)^2}{[ 1 + (b/a)^2]^2}  ~\beta \equiv \biggl[ \frac{2  B_{12}}{(c/b)^2 A_3 - a^2 A_{12}} \biggr]\frac{1}{1 + (b/a)^2}  Direct Adjoint
~f ~\omega = \frac{\Omega}{\sqrt{G\rho}} ~f^\dagger ~\omega^\dagger = \frac{\Omega^\dagger}{\sqrt{G\rho}}
0.9 0.641 0.387793362 0.207337649 0.247245200 3.797483556 - 0.268008879 ± 1.131374734 -15.09117122 ± 0.150771618


Now, according to Ou (2006), at any coordinate position inside or on the surface of the ellipsoid, ~(x, y), the three components of the velocity as viewed from a frame of rotation that is spinning at the equilibrium configuration's frequency, ~\Omega, are,

~\vec{v}

~=

~\lambda \biggl( \frac{ay}{b} , - \frac{bx}{a} , 0 \biggr) \, ,

where, ~\lambda is an overall scale factor. But, according to §48 of EFE, we see that,

~\vec{u}

~=

~\biggl( Q_1 y , Q_2 x , 0 \biggr) \, ,

where,

~Q_1

~\equiv

~- \biggl[ \frac{a^2}{a^2 + b^2} \biggr]\zeta

      and,      

~Q_2

~\equiv

~+ \biggl[ \frac{b^2}{a^2 + b^2} \biggr]\zeta \, ,

and ~\zeta is the scalar magnitude of the vorticity vector, ~\vec\zeta. The transformation from EFE's notation to the one used by Ou is, then,

~\lambda \biggl( \frac{a}{b} \biggr)

~=

~- \biggl[ \frac{a^2}{a^2 + b^2} \biggr]\zeta

      and,      

~- \lambda \biggl( \frac{b}{a} \biggr)

~=

~+ \biggl[ \frac{b^2}{a^2 + b^2} \biggr]\zeta

~\Rightarrow ~~~ \lambda

~=

~- \biggl[ \frac{a b}{a^2 + b^2} \biggr]\zeta = - \biggl[ \frac{b}{a} + \frac{a}{b} \biggr]^{-1} \zeta\, ,

which, gratifyingly agrees with Ou's equation (17).

Models Examined by Ou (2006)

His Tabulated Model Parameters

Table 1 (see below) lists a subset of the Riemann S-type ellipsoids that were studied by Ou (2006); properties of various so-called Direct configurations can be found in Ou's Table 1, while properties of various Adjoint configurations can be found in his Table 5. Each row of our Table 1 was constructed as follows:

  • The pair of axis ratios ~(\tfrac{b}{a}, \tfrac{c}{a} ) associated with one of Ou's (2006) uniform-density, incompressible ~(n=0) ellipsoid models (columns 1 and 2 from Ou's Table 1) has been copied into columns 1 and 2 of our table.
  • Properties of Direct Configurations
    • The pair of parameter values ~(\omega_\mathrm{analytic}, \lambda_\mathrm{analytic}) that is required in order for this to be an equilibrium configuration — as specified by the above set of analytical expressions from EFE — is copied from, respectively, columns 11 and 13 of Ou's Table 1 into columns 3 and 4 of our table; in our table, the "analytic" subscript has been dropped from the column headings.
    • The value of the equilibrium configuration's vorticity, ~\zeta — see column 5 of our table — has been determined from the expression,
      ~\zeta = - \biggl[ \frac{1 + (b/a)^2}{b/a} \biggr] \lambda \, .
    • Column 6 of our table lists the value of the frequency ratio, ~f \equiv \zeta/\omega.
  • Properties of Adjoint Configurations [in order to distinguish from Direct configuration properties, a superscript † has been attached to each parameter name] …
    • As listed in column 7 of our Table, the "spin" angular velocity of the adjoint equilibrium configuration has been determined from the vorticity of the direct configuration via the relation,
      ~\omega^\dagger = \zeta \biggl[\frac{b/a}{1 + (b/a)^2}\biggr] \, .
    • As listed in column 10 of our Table, the ratio ~(f^\dagger) of the vorticity to the angular velocity in the adjoint equilibrium configuration has been determined from the same ratio ~(f) in the direct configuration via the relation,
      ~f^\dagger = \frac{1}{f} \biggl\{ \frac{[1 + (b/a)^2]^2}{(b/a)^2} \biggr\} \, .
    • As indicated, the value of the vorticity in the adjoint equilibrium configuration (column 9 of our table) has been determined from a product of ~\omega^\dagger and ~f^\dagger.
    • As listed in column 8 of our table, the value of the parameter, ~\lambda^\dagger, has been determined from the vorticity in the adjoint equilibrium configuration via the relation,
      ~\lambda^\dagger = -~ \zeta^\dagger \biggl[ \frac{b}{a} + \frac{a}{b}\biggr]^{-1} \, .

Table 1:   Example Riemann S-type Ellipsoids
[Cells with a pink background contain numbers copied directly from Table 1 of Ou (2006)]
[Cells with a yellow background contain numbers drawn from Table IV (p. 103) of EFE]

~\frac{b}{a} ~\frac{c}{a}

Properties of
Direct Configurations

Properties of
Adjoint Configurations

~\omega = \frac{\Omega}{\sqrt{G \rho}} ~\lambda ~\zeta ~f \equiv \frac{\zeta}{\omega} ~\omega^\dagger ~\lambda^\dagger ~\zeta^\dagger = \omega^\dagger f^\dagger ~f^\dagger
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
0.90 0.795 1.14704 0.43181 -0.86842 -0.75709 -0.43181 -1.14704 +2.30682 -5.3422
0.641 1.13137 0.15077 - 0.30322 - 0.26801 - 0.15077 -1.13137 2.27531 - 15.0913
0.590 1.10661 0.06406 -0.12883 -0.11642 -0.06406 -1.10661 +2.22552 -34.7411
0.564 1.09034 0.02033 -0.04089 -0.03750 -0.02033 -1.09034 +2.19279 -107.86
0.538 1.07148 - 0.02324 +0.04674 +0.04362 +0.02324 - 1.07148 +2.15487 +92.722
0.487 1.02639 - 0.10880 +0.21881 +0.21318 +0.10880 -1.02639 +2.06418 +18.972
0.333 0.79257 - 0.39224 +0.78884 +0.99529 +0.39224 -0.79257 +1.59395 +4.06370
0.28 0.256 0.80944 0.03668 -0.14127 -0.17453 -0.03668 -0.80944 +3.11750 -84.992
0.245083 0.796512a 0.0 0.0 0.0 0.0 ~\infty
0.231 0.77651 - 0.04714 +0.18156 +0.23381 +0.04714 -0.77651 +2.99067 +63.442
0.205 0.72853 - 0.13511 +0.52037 +0.71427 +0.13511 -0.72853 +2.80588 +20.7674

aAccording to Table IV (p. 103) of EFE, the square of the angular velocity of this Jacobi ellipsoid is, ~\Omega^2/(\pi G\rho) = 0.201946; from this value, we find that, ~\omega = \sqrt{\pi} \cdot \sqrt{0.201946} = 0.796512.


Our Parameter Determinations

The parameter values that have been posted above in our Table 1 are typically given with five digits of precision. This is because, as explained, the values were determined from the analytically determined values, ~\omega_\mathrm{analytic} and ~\lambda_\mathrm{analytic}, that were provided by Ou (2006) with only five digit accuracy. Our Table 2 (shown immediately below) provides values of this same set of model parameters to better than eleven digits accuracy. We calculated these parameter values by following the steps detailed in earlier subsections of this chapter and, as a foundation, using double-precision versions of Numerical Recipes algorithms to evaluate the special functions, ~F(\phi,k) and ~E(\phi,k). As an example, the above pair of brief tables titled, TEST (part 1) and TEST (part 2) detail all of the intermediate steps that were used in order to determine the high-precision parameter values specifically for the model having the axis-ratio pair ~(0.9,0.641). This table of higher precision parameter values was primarily generated in order to convince ourselves that we understood from first principles how to accurately determine the properties of Riemann S-type ellipsoids; the lower-precision parameter values that we derived from Ou's work provided a handy means of cross-checking these "first principles" determinations.

In generating our Table 2, we wondered what the approriate signs were of the various model parameters — especially when part of our objective is to distinguish between direct and adjunct configurations. We took the following approach: First we decided that the spin frequency of every direct configuration should be positive. (Evidently, Ou made this same choice.)

Table 2:   Example Riemann S-type Ellipsoids (double-precision evaluation)

~\frac{b}{a} ~\frac{c}{a}

Properties of
Direct Configurations

Properties of
Adjoint Configurations

~\omega = \frac{\Omega}{\sqrt{G \rho}} ~\lambda ~\zeta ~f \equiv \frac{\zeta}{\Omega} ~\omega^\dagger ~\lambda^\dagger ~\zeta^\dagger = \omega^\dagger f^\dagger ~f^\dagger
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
0.90 0.795 +1.147036091720 +0.431809451699 -0.868416786194 -0.757096320116 -0.431809460593 -1.147036104571 +2.306817054749 -5.342210487323
0.641 +1.131374738327 +0.150771621841 -0.303218483925 -0.268008886644 -0.150771621877 -1.131374730590 +2.275320291519 -15.091170863305
0.590 +1.106612583610 +0.064060198174 -0.128832176328 -0.116420305902 -0.064060197762 -1.106612576964 +2.225520849228 -34.741086358509
0.564 +1.090339840378 +0.020334563779 -0.040895067155 -0.037506716440 -0.020334563809 -1.090339837153 +2.192794561386 -107.8358300897
0.538 +1.071485625744 -0.023236834336 +0.046731855720 +0.043614077664 +0.023236835120 -1.071485656401 +2.154876708984 +92.735376233270
0.487 +1.026387311947 -0.108799837242 +0.218808561563 +0.213183225210 +0.108799835209 -1.026387320039 +2.064178943634 +18.972261524065
0.333 +0.792566980901 -0.392440787995 +0.789242029190 +0.995804846843 +0.392440793882 -0.792566979129 +1.593940258026 +4.061606964516
0.41 0.333 +0.929630138695 +0.003311666790 -0.009435019456 -0.010149218281 -0.003311666699 -0.929630099681 +2.648538827896 -799.7601146950
0.28 0.256 +0.809436834686 +0.036676037913 -0.141255140305 -0.174510396110 -0.036676038521 -0.809436833116 +3.117488145828 -85.000678306244
0.245083 0.796512a 0.0 0.0 0.0 0.0 ~\infty
0.231 +0.776514825339 -0.047142035397 +0.181564182043 +0.233819345828 +0.047142037070 -0.776514835457 +2.990691423416 +63.440011724689
0.205 +0.728526018042 -0.135108121071 +0.520359277725 +0.714263156392 +0.135108125079 -0.728526039364 +2.805866003036 +20.767558718483

aAccording to Table IV (p. 103) of EFE, the square of the angular velocity of this Jacobi ellipsoid is, ~\Omega^2/(\pi G\rho) = 0.201946; from this value, we find that, ~\omega = \sqrt{\pi} \cdot \sqrt{0.201946} = 0.796512.

Feeding a 3D Animation

Let's examine the elliptical trajectory of a Lagrangian particle that is moving in the equatorial plane of a Riemann S-Type ellipsoid. As viewed in a frame that is spinning about the Z-axis at angular frequency, ~\Omega, the trajectory is defined by,

r2


~=

~
\biggl(\frac{x}{a} \biggr)^2 + \biggl(\frac{y}{b}\biggr)^2 \, ,

where ~0 < r \le 1. (The surface of the relevant ellipsoid is associated with the value, ~r=1.)

Let's choose a pair of axis ratios — for example, ~b/a = 0.28 and ~c/a = 0.231 — then, from Table 1 of our above discussion, draw the associated value of either ~\lambda or ~\zeta that corresponds to the Jacobi-like equilibrium configuration — in this example, ~\lambda = -0.04714 and ~\zeta = +0.18156. Then, for any point ~(x,y) inside of the ellipsoid, the fluid's velocity components (as viewed from the rotating frame of reference) are,


v_x  = \frac{dx}{dt}  = \lambda \biggl( \frac{ay}{b} \biggr) = -0.16836 ~y

      and,      

~
v_y  = \frac{dy}{dt}  = - \lambda \biggl( \frac{bx}{a} \biggr) = + 0.01320~x \, .

Alternatively, we have,


u_x = \frac{dx}{dt} = Q_1  y = - \biggl[ 1 + \frac{b^2}{a^2} \biggr]^{-1}\zeta ~y = -0.16836 ~y

      and,      

~
u_y  = \frac{dy}{dt}= Q_2 x  = + \biggl[ 1 + \frac{a^2}{b^2} \biggr]^{-1}\zeta ~x = + 0.01320~x \, .

Now, each Lagrangian fluid element's motion is oscillatory in both the ~x and ~y coordinate directions. So let's see how this plays out. Suppose,


x = x_\mathrm{max} \cos(\varphi t)

      and,      

~
y = y_\mathrm{max} \sin(\varphi t) \, .

Then,


\frac{dx}{dt} = -  x_\mathrm{max}\varphi \sin(\varphi t) = -  \biggl( \frac{x_\mathrm{max}}{y_\mathrm{max}}\biggr) \varphi y = - \varphi \biggl(\frac{ay}{b}\biggr)

      and,      

~
\frac{dy}{dt} = y_\mathrm{max} \varphi \cos(\varphi t) = + \biggl( \frac{y_\mathrm{max}}{x_\mathrm{max}}\biggr) \varphi x = + \varphi \biggl(\frac{bx}{a}\biggr) \, .

Hence our functional representation of the time-dependent behavior of both ~x and ~y works perfectly if, for each orbit inside of or on the surface of the configuration, we set ~\varphi = - \lambda and if the ratio ~y_\mathrm{max}/x_\mathrm{max} = (b/a). Hooray!

See Also


Whitworth's (1981) Isothermal Free-Energy Surface

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