ThreeDimensionalConfigurations/JacobiEllipsoids

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Jacobi Ellipsoids

Jacobi
Ellipsoids

As has been detailed in an accompanying chapter, the gravitational potential anywhere inside or on the surface, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~(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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~A_1 }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~= }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~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] \, , }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~A_3 }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~= }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~2\biggl(\frac{b}{a}\biggr) \biggl[ \frac{(b/a) \sin\theta - (c/a)E(\theta,k)}{(1-k^2) \sin^3\theta} \biggr] \, , }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~A_2 }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~= }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~2 - (A_1+A_3) \, ,}

where, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~F(\theta,k)} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~E(\theta,k)} are incomplete elliptic integrals of the first and second kind, respectively, with arguments,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~\theta = \cos^{-1} \biggl(\frac{c}{a} \biggr)}

      and      

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~k = \biggl[\frac{1 - (b/a)^2}{1 - (c/a)^2} \biggr]^{1/2} \, .}

[EFE], Chapter 3, §17, Eq. (32)

Equilibrium Conditions for Jacobi Ellipsoids

Pulling from Chapter 6 — specifically, §39 — of [EFE], we understand that the semi-axis ratios, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~(\tfrac{b}{a},\tfrac{c}{a})} associated with Jacobi ellipsoids are given by the roots of the equation,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~a^2 b^2 A_{12}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~=}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~c^2 A_3 \, ,}

[EFE], §39, Eq. (4)

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~\frac{\Omega^2}{\pi G \rho}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~=}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~2B_{12} \, ,}

[EFE], §39, Eq. (5)

where,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~A_{12}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~\equiv}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~-\frac{A_1-A_2}{(a^2 - b^2)} \, ,}

[EFE], §21, Eq. (107)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~B_{12}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~\equiv}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~A_2 - a^2A_{12} \, .}

[EFE], §21, Eq. (105)


Taken together, we see that, written in terms of the two primary coefficients, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~A_1} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~A_3} , the pair of defining relations for Jacobi ellipsoids is:


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~f_J}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~\equiv}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~\biggl(\frac{b}{a}\biggr)^2 \biggl[ \frac{2(1-A_1)-A_3}{1 - (b/a)^2} \biggr]-\biggl(\frac{c}{a}\biggr)^2 A_3 =0 }

and

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~\frac{\Omega^2}{\pi G \rho}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~=}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~2\biggl\{2 - (A_1+A_3) - \biggl[ \frac{2(1-A_1)-A_3}{1 - (b/a)^2} \biggr] \biggr\}}

Roots of the Governing Relation

Constraint on Axis-Ratio Relationship

To simplify notation, here we will set,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~\chi \equiv \frac{b}{a}}

      and      

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~\upsilon \equiv \frac{c}{a} \, ,}

in which case the governing relation is,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~f_J}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~=}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~\frac{\chi^2}{1-\chi^2} \biggl[ 2(1-A_1)-A_3\biggr]-\upsilon^2 A_3 =0 \, .}

Our plan is to employ the Newton Raphson method to find the root(s) of the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~f_J = 0} relation, typically holding Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~\upsilon} fixed and using the Newton-Raphson technique to identify the corresponding "root" value of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~\chi} . Using this approach, the Newton Raphson technique requires specification of, not only the function, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~f_J} , but also its first derivative,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~f_J^'}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~=}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~\frac{df_J}{d\chi} \, .}

Let's determine the requisite expression, using a prime superscript to indicate differentiation with respect to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~\chi} .

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~f_J^'}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~=}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~ \biggl[ 2(1-A_1)-A_3\biggr]\biggl[ \frac{2\chi}{(1-\chi^2)^2} \biggr] -\frac{\chi^2}{1-\chi^2} \biggl[ 2A_1^'+A_3^'\biggr] -\upsilon^2 A_3^' \, , }

where, given that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~\theta} does not depend on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~\chi} ,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~A_1^' }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~= }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~\frac{2\upsilon}{\sin^3\theta} \cdot \frac{d}{d\chi}\biggl\{ \frac{\chi}{k^2} \biggl[ F(\theta,k) - E(\theta,k) \biggr] \biggr\} }

 

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~= }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~\frac{2\upsilon}{k^3 \sin^3\theta} \cdot \biggl\{ [ F - E ] [k - 2\chi k^' ] +\chi k [ F^' - E^' ]\biggr\} \, , }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~A_3^' }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~= }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~\frac{2}{\sin^3\theta} \cdot \frac{d}{d\chi}\biggl\{ \frac{\chi}{(1-k^2)} \biggl[ \chi \sin\theta - \upsilon E(\theta,k)\biggr] \biggr\} }

 

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~= }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~\frac{2}{(1-k^2)^2\sin^3\theta} \biggl\{ \biggl[ \chi \sin\theta - \upsilon E\biggr]\biggl[ (1-k^2) +2\chi kk^' \biggr] + \chi(1-k^2) \biggl[ \sin\theta - \upsilon E^'\biggr] \biggr\}\, , }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~k^'}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~=}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~ \frac{d}{d\chi}\biggl[\frac{1 - \chi^2}{1 - \upsilon^2} \biggr]^{1/2} = \frac{-\chi}{(1 - \chi^2)^{1/2}(1 - \upsilon^2)^{1/2}} \, , }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~F^'}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~=}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~ \frac{\partial F(\theta,k)}{\partial k} \cdot k^' \, , }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~E^'}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~=}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~ \frac{\partial E(\theta,k)}{\partial k} \cdot k^' \, . }

Now, according to online WolframResearch documentation — see, in particular, the subsection titled, "Representations of Derivatives" —

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~\frac{\partial F(z|m)}{\partial m}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~=}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~ \frac{E(z|m)}{2(1-m)m} - \frac{F(z|m)}{2m} - \frac{\sin(2z)}{4(1-m)\sqrt{1-m\sin^2(z)}} \, , }

and,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~\frac{\partial E(z|m)}{\partial m}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~=}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~\frac{E(z|m) - F(z|m)}{2m} \, ,}

where, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~z~\leftrightarrow~\theta} , and,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~m \equiv k^2 ~~~~\Rightarrow~~~~\frac{dm}{dk} = 2k \ .}

Hence, we have,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~F^'}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~=}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~ \biggl[\frac{\partial F(z|m)}{\partial m} \cdot \frac{dm}{dk}\biggr] k^' }

 

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~=}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~ \biggl[ \frac{E(\theta,k)}{2(1-k^2)k^2} - \frac{F(\theta,k)}{2k^2} - \frac{\sin(2\theta)}{4(1-k^2)\sqrt{1-k^2\sin^2\theta}} \biggr] 2kk^' \, , }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~E^'}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~=}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~ \biggl[ \frac{\partial E(z|m)}{\partial m} \cdot \frac{dm}{dk}\biggr] k^' }

 

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~=}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~ \biggl[ E(\theta,k) - F(\theta,k) \biggr] \frac{k^'}{k} \, . }

This, then, gives us all of the expressions necessary to specify the derivative, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~f_J^'} analytically.


Table 1:  Double-Precision Evaluations

Related to Table IV in [EFE], Chapter 6, §39 (p. 103)

         b/a      c/a            omega2              angmom              5L/M                fJ              fJderiv

        1.00   0.582724     3.742297785D-01     3.037510987D-01     4.232965627D+00     0.000000000D+00     0.000000000D+00
        0.96   0.570801     3.739782202D-01     3.039551227D-01     4.235808832D+00     1.377942479D-06     1.636908401D-01
        0.92   0.558330     3.731876801D-01     3.046006837D-01     4.244805137D+00    -6.821687132D-07     1.676406830D-01
        0.88   0.545263     3.717835971D-01     3.057488283D-01     4.260805266D+00     8.533280272D-07     1.715558312D-01
        0.84   0.531574     3.696959199D-01     3.074667323D-01     4.284745355D+00    -4.622993727D-08     1.754024874D-01
        0.80   0.517216     3.668370069D-01     3.098368632D-01     4.317774645D+00     2.805300664D-08     1.791408327D-01
        0.76   0.502147     3.631138118D-01     3.129555079D-01     4.361234951D+00     3.221800126D-07     1.827219476D-01
        0.72   0.486322     3.584232032D-01     3.169377270D-01     4.416729718D+00     3.274773094D-08     1.860866255D-01
        0.68   0.469689     3.526490289D-01     3.219229588D-01     4.486202108D+00     1.202999164D-08     1.891636215D-01
        0.64   0.452194     3.456641138D-01     3.280805511D-01     4.572012092D+00     2.681560312D-07     1.918668912D-01
        0.60   0.433781     3.373298891D-01     3.356184007D-01     4.677056841D+00     1.037186290D-08     1.940927000D-01
        0.56   0.414386     3.274928085D-01     3.447962894D-01     4.804956583D+00     1.071021385D-07     1.957166395D-01
        0.52   0.393944     3.159887358D-01     3.559412795D-01     4.960269141D+00     8.098003093D-08     1.965890756D-01
        0.48   0.372384     3.026414267D-01     3.694732246D-01     5.148845443D+00     1.255768368D-07     1.965308751D-01
        0.44   0.349632     2.872670174D-01     3.859399647D-01     5.378319986D+00     1.329168636D-08     1.953277019D-01
        0.40   0.325609     2.696779847D-01     4.060726774D-01     5.658882201D+00    -9.783004411D-08     1.927241063D-01
        0.36   0.300232     2.496925963D-01     4.308722159D-01     6.004479614D+00     1.044268276D-07     1.884168286D-01
        0.32   0.273419     2.271530240D-01     4.617497270D-01     6.434777459D+00    -4.469279448D-08     1.820477545D-01
        0.28   0.245083     2.019461513D-01     5.007767426D-01     6.978643856D+00     7.996820889D-08     1.731984783D-01
        0.24   0.215143     1.740514751D-01     5.511400218D-01     7.680488329D+00     1.099319693D-07     1.613864645D-01
        0.20   0.183524     1.436093757D-01     6.180687545D-01     8.613182979D+00     5.068010978D-08     1.460685065D-01
        0.16   0.150166     1.110438660D-01     7.109267615D-01     9.907218635D+00    -2.170751250D-08     1.266576761D-01
        0.12   0.115038     7.728058393D-02     8.487699974D-01     1.182815219D+01     3.613784147D-09     1.025686850D-01
        0.08   0.078166     4.416740942D-02     1.079303624D+00     1.504078558D+01     3.319018649D-08     7.332782508D-02
        0.04   0.039688     1.541513490D-02     1.582762691D+00     2.205680933D+01    -6.674246644D-09     3.882477311D-02

With regard to our Table 1 (immediately above): Given each pair of axis ratios, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~(\tfrac{b}{a},\tfrac{c}{a})} — copied from Table IV of [EFE] (see columns 1 and 2 of our Table 1) — and the corresponding coefficient values, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~A_1} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~A_2} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~A_3} , as tabulated in Table 2 of our accompanying discussion, we calculated corresponding values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~\Omega^2} (column 3) and total angular momentum (column 4) in the units used in Table IV of [EFE], as well as (column 5) the total angular momentum in units used by Christodoulou, et al. (1995, ApJ, 446, 472) — see our related discussion of these physical quantities. We also have tabulated associated values of the function, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~f_J} , (column 6) and its first derivative, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~f_J^'} , (column 7) as defined immediately above. Notice that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~f_J} is very nearly zero in all cases, which indicates that each axis-ratio pair indeed identifies a configuration that lies along the Jacobi sequence.

Table 2:  Jacobi Sequence

   b/a       c/a        A1        A2        A3      omega2       a       5L/M

  0.990699  0.580000  0.512818  0.518962  0.968220  0.374217  1.868761  4.233113
  0.901558  0.552381  0.481786  0.549836  0.968378  0.372621  1.960046  4.251259
  0.820783  0.524762  0.450993  0.580215  0.968792  0.368424  2.057217  4.299402
  0.747135  0.497143  0.420459  0.610088  0.969452  0.361716  2.161309  4.377683
  0.679613  0.469524  0.390210  0.639442  0.970348  0.352587  2.273548  4.486951
  0.617393  0.441905  0.360273  0.668258  0.971469  0.341129  2.395412  4.628802
  0.559798  0.414286  0.330684  0.696516  0.972800  0.327439  2.528716  4.805667
  0.506257  0.386667  0.301483  0.724187  0.974329  0.311620  2.675723  5.020964
  0.456291  0.359048  0.272719  0.751241  0.976040  0.293786  2.839307  5.279337
  0.409492  0.331429  0.244450  0.777636  0.977914  0.274062  3.023190  5.587020
  0.365507  0.303810  0.216744  0.803324  0.979931  0.252593  3.232298  5.952388
  0.324034  0.276190  0.189686  0.828246  0.982067  0.229546  3.473314  6.386811
  0.284807  0.248571  0.163376  0.852329  0.984295  0.205118  3.755577  6.906010
  0.247591  0.220952  0.137939  0.875480  0.986581  0.179549  4.092599  7.532311
  0.212179  0.193333  0.113527  0.897587  0.988885  0.153130  4.504785  8.298565
  0.178382  0.165714  0.090333  0.918505  0.991162  0.126229  5.024664  9.255452
  0.146026  0.138095  0.068601  0.938044  0.993355  0.099316  5.707871 10.486253
  0.114948  0.110476  0.048654  0.955953  0.995393  0.073010  6.659169 12.140357
  0.084989  0.082857  0.030927  0.971879  0.997194  0.048162  8.105501 14.522397
  0.055982  0.055238  0.016051  0.985298  0.998651  0.026008 10.663879 18.396951
  0.027738  0.027619  0.005032  0.995331  0.999637  0.008539 16.979084 26.660547

With regard to our Table 2 (immediately above): Here we specified twenty-one values of the axis ratio, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~\tfrac{c}{a}} , (column 2) and used our Newton-Raphson-based root finder to identify corresponding values of the companion axis ratio, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~\tfrac{b}{a}} , (column 1) that satisfies the governing relation, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~f_J = 0} .

Angular Momentum Constraint

Angular Momentum Determination

In the above tables, the square of the angular momentum, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L^2} , for each equilibrium Jacobi ellipsoid has been determined in the following manner:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L^2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I^2\Omega^2}
  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{M^2\Omega^2}{5^2}(a^2 + b^2)^2}

  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{M^2}{5^2}\biggl(\frac{\Omega^2}{\pi G \rho}\biggr)\biggl[\frac{3G M}{2^2abc} \biggr](a^2 + b^2)^2}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Rightarrow ~~~ \frac{L^2}{GM^3 \bar{a}}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{5^2}\biggl(\frac{\Omega^2}{\pi G \rho}\biggr)\biggl[\frac{3}{2^2abc} \biggr]\frac{(a^2 + b^2)^2}{\bar{a}}}

  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{3}{2^2\cdot 5^2}\biggl(\frac{\Omega^2}{\pi G \rho}\biggr)\frac{(a^2 + b^2)^2}{\bar{a}^4} \, .}

[EFE], §39, p. 103, Eq. (16)

Normalizing in a different manner, we have:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L^2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{M^2}{5^2}\biggl(\frac{\Omega^2}{4\pi G \rho}\biggr)\biggl[4\pi G\rho \biggr](a^2 + b^2)^2}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Rightarrow ~~~j^2 \equiv \frac{L^2}{4\pi G M^{10/3}\rho^{-1 / 3}}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{M^{-4/3}}{5^2}\biggl(\frac{\Omega^2}{4\pi G \rho}\biggr)\biggl[\rho \biggr]^{4 / 3}(a^2 + b^2)^2}

  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{M^{-4/3}}{5^2}\biggl(\frac{\Omega^2}{4\pi G \rho}\biggr)\biggl[\frac{3M}{4\pi abc} \biggr]^{4 / 3}(a^2 + b^2)^2}

  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{2^2\cdot 5^2}\biggl[\frac{3}{4\pi } \biggr]^{4 / 3}\biggl(\frac{\Omega^2}{\pi G \rho}\biggr)\frac{(a^2 + b^2)^2}{\bar{a}^4}}

  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{3}\biggl[\frac{3}{4\pi } \biggr]^{4 / 3} \biggl[\frac{L^2}{GM^{3}\bar{a}} \biggr] = 0.049365924 \biggl[\frac{L^2}{GM^{3}\bar{a}} \biggr] \, . }


Alternatively, let's choose a value for the system's total angular momentum, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~L > 4.23296} , and solve for the axis-ratio pair that identifies that configuration's location along the Jacobi sequence. We'll adopt the units used by Christodoulou et al (1995), that is, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~G = 1} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~\pi \rho = 1} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~M = 5} , hence,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~a^3}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~=}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~\frac{3Ma^2}{4\pi(bc)\rho} = \frac{15}{4}\biggl(\frac{b}{a}\biggr)^{-1} \biggl(\frac{c}{a}\biggr)^{-1}\, .}

Given that the relationship between Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~L} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~\Omega} in equilibrium Jacobi ellipsoids is,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~L}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~=}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~a^2\biggl[1 + \biggl(\frac{b}{a}\biggr)^2\biggr]\Omega }

 

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~=}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~\biggl[ \frac{15}{4}\biggl(\frac{b}{a}\biggr)^{-1} \biggl(\frac{c}{a}\biggr)^{-1} \biggr]^{2/3} \biggl[1 + \biggl(\frac{b}{a}\biggr)^2\biggr]\Omega }

the constraint on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~\Omega^2} given above implies that,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~L^2 \biggl[ \frac{4}{15}\biggl(\frac{b}{a}\biggr) \biggl(\frac{c}{a}\biggr) \biggr]^{4/3} \biggl[1 + \biggl(\frac{b}{a}\biggr)^2\biggr]^{-2}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~=}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~2\biggl\{2 - (A_1+A_3) - \biggl[ \frac{2(1-A_1)-A_3}{1 - (b/a)^2} \biggr] \biggr\} \, .}

Or, again adopting the shorthand notation,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~\chi \equiv \frac{b}{a}}

      and      

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~\upsilon \equiv \frac{c}{a} \, ,}

we seek roots of the function,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~f_L}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~\equiv}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~L^2 - \biggl[ \frac{3^4\cdot 5^4}{2^5} \biggr]^{1/3}\chi^{-4/3} \upsilon^{-4/3}(1 + \chi^2)^{2} \biggl\{[2 - (A_1+A_3)] - \biggl[ 2(1-A_1)-A_3\biggr](1-\chi^2)^{-1} \biggr\} = 0 \, .}

As above, we will hold Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~\upsilon} fixed and use the Newton-Raphson technique to identify the corresponding "root" value of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~\chi} . Hence, we need to specify, not only the function, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~f_L} , but also its first derivative,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~f_L^'}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~\equiv}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~\frac{\partial f_L}{\partial \chi} \, .}

 

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~=}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~- \biggl[ \frac{3^4\cdot 5^4}{2^5} \biggr]^{1/3}\upsilon^{-4/3} \frac{\partial}{\partial \chi} \biggl\{ \chi^{-4/3} (1 + \chi^2)^{2} [2 - (A_1+A_3)] - \chi^{-4/3} (1 + \chi^2)^{2}(1-\chi^2)^{-1}[ 2(1-A_1)-A_3] \biggr\}}

 

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~=}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~- \biggl[ \frac{3^4\cdot 5^4}{2^5} \biggr]^{1/3}\upsilon^{-4/3} \biggl\{ -\frac{4}{3}\chi^{-7/3} (1 + \chi^2)^{2}[2 - (A_1+A_3)] +4\chi^{-1/3} (1 + \chi^2)[2 - (A_1+A_3)] -\chi^{-4/3} (1 + \chi^2)^{2}(A_1^'+A_3^') }

 

 

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~ + \frac{4}{3} \chi^{-7/3} (1 + \chi^2)^{2}(1-\chi^2)^{-1}[ 2(1-A_1)-A_3] - 4\chi^{-1/3} (1 + \chi^2)(1-\chi^2)^{-1}[ 2(1-A_1)-A_3] }

 

 

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~ - 2 \chi^{-1/3} (1 + \chi^2)^{2}(1-\chi^2)^{-2}[ 2(1-A_1)-A_3] - \chi^{-4/3} (1 + \chi^2)^{2}(1-\chi^2)^{-1}[ -2A_1^'-A_3^'] \biggr\}}

 

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~=}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~- \biggl[ \frac{3\cdot 5^4}{2^5} \biggr]^{1/3}\upsilon^{-4/3} \chi^{-7/3} (1 + \chi^2)\biggl\{ [12\chi^2-4(1 + \chi^2)][2 - (A_1+A_3)] -3\chi (1 + \chi^2)(A_1^'+A_3^') + 3\chi (1 + \chi^2)(1-\chi^2)^{-1}[ 2A_1^' + A_3^'] }

 

 

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~ + (1-\chi^2)^{-2}\{ 4 (1 + \chi^2)(1-\chi^2)[ 2(1-A_1)-A_3] - 12\chi^{2} (1-\chi^2)[ 2(1-A_1)-A_3] - 6 \chi^{2} (1 + \chi^2)[ 2(1-A_1)-A_3] \} \biggr\} }

 

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~=}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~- \biggl[ \frac{3\cdot 5^4}{2^5} \biggr]^{1/3}\upsilon^{-4/3} \chi^{-7/3} (1 + \chi^2)\biggl\{ [8\chi^2-4]A_1 + 3\chi (1 + \chi^2)(1-\chi^2)^{-1} [ (1+\chi^2)A_1^' + \chi^2A_3^' ] }

 

 

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~ + 2(1-\chi^2)^{-2} [ 2-2A_1-A_3] [ (4\chi^2-2)(1-\chi^2)^{2} + 2 (1 + \chi^2)(1-\chi^2) - 6\chi^{2} (1-\chi^2) - 3 \chi^{2} (1 + \chi^2) ] \biggr\} }

 

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~=}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~- \biggl[ \frac{3\cdot 5^4}{2^5} \biggr]^{1/3}\upsilon^{-4/3} \chi^{-7/3} (1 + \chi^2)\biggl\{ 3\chi (1 + \chi^2)(1-\chi^2)^{-1} [ (1+\chi^2)A_1^' + \chi^2A_3^' ] }

 

 

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~+[8\chi^2-4]A_1 + 2(1-\chi^2)^{-2} [ 2-2A_1-A_3] [ - \chi^2 - 9\chi^4 + 4\chi^6 ] \biggr\} }


What values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~L} should we choose? In association with our discussion of warped free-energy surfaces, we'd like to specify the eccentricity, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~e} , of a Maclaurin spheroid and adopt the angular momentum of that configuration. According to our accompanying discussion of the properties of Maclaurin spheroids,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~L_\mathrm{Mac}^2}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~=}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~2^2a^4\Omega^2}

 

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~=}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~2^3a^4 [ A_1 -A_3(1-e^2)]_\mathrm{Mac} }

 

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~=}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~2^3 \biggl[\frac{3\cdot 5}{2^2}(1-e^2)^{-1/2} \biggr]^{4/3} [ A_1 -A_3(1-e^2)]_\mathrm{Mac} }

 

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~=}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~ [2\cdot 3^4\cdot 5^4]^{1/3} (1-e^2)^{-2/3} \biggl\{ \frac{1}{e^2} \biggl[\frac{\sin^{-1}e}{e} - (1-e^2)^{1/2} \biggr](1-e^2)^{1/2} -\frac{2}{e^2} \biggl[(1-e^2)^{-1/2} -\frac{\sin^{-1}e}{e} \biggr](1-e^2)^{3/2} \biggr\} }

 

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~=}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~ [2\cdot 3^4\cdot 5^4]^{1/3} (1-e^2)^{-2/3} \biggl\{ \frac{1}{e^2} \biggl[\frac{\sin^{-1}e}{e} \biggr](1-e^2)^{1/2} +\frac{2}{e^2} \biggl[\frac{\sin^{-1}e}{e} \biggr](1-e^2)^{3/2} -\frac{1}{e^2} \biggl[ (1-e^2)^{1/2} \biggr](1-e^2)^{1/2} -\frac{2}{e^2} \biggl[(1-e^2)^{-1/2} \biggr](1-e^2)^{3/2} \biggr\} }

 

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~=}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~ [2\cdot 3^4\cdot 5^4]^{1/3} (1-e^2)^{-2/3} \biggl\{ \frac{1}{e^2} \biggl[\frac{\sin^{-1}e}{e} \biggr](1-e^2)^{1/2} \biggl[3-2e^2\biggr] -\frac{3(1-e^2)}{e^2} \biggr\} }

 

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~=}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~ [2\cdot 3^4\cdot 5^4]^{1/3} \frac{(1-e^2)^{1/3} }{e^2} \biggl\{ \biggl[\frac{\sin^{-1}e}{e} \biggr](1-e^2)^{-1/2} \biggl[3-2e^2\biggr] - 3 \biggr\} \, . }

Note, for example, that if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~e = 0.85} , the square-root of this expression gives, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~L_\mathrm{Mac} = 4.7148806} , which matches the angular momentum that was used by Christodoulou et al (1995) to generate their Figure 3.

Sequence Plots

EFE Diagram

Jacobi Sequence: (blue) Points defined by data in Table IV of [EFE], Chapter 6, §39 (p. 103); (red) points generated here from above-defined roots of the governing relation. Figure 2 extracted from p. 902 of S. Chandrasekhar (1965)

"The Equilibrium and the Stability of the Riemann Ellipsoids. I"

ApJ, vol. 142, pp. 890-921 © American Astronomical Society

Jacobi Sequence

Chandrasekhar Figure2

  Original figure has been annotated (maroon-colored text and arrow added) for clarification.

Jacobi Sequence

Chandrasekhar Figure2

Other Sequence Depictions

Jacobi Omega2 versus J2 Jacobi Omega2 versus Tau Jacobi Tau versus J2

Note that,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \equiv}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \biggl( \frac{3}{2^8 \pi^4} \biggr)^{1/6} \frac{L}{(GM^3\bar{a})^{1 / 2}} \, . }

Bifurcation from Maclaurin to Jacobi Sequence

According to Table IV of [EFE], Chapter 6, §39 (p. 103), the Jacobi Sequence bifurcates from the Maclaurin sequence at …

  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (b/a, c/a) = (1.000, 0.582724)}
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega^2/(\pi G \rho) = 0.374230 ~~~~ \Rightarrow ~~~ \omega_0^2/(4\pi G \rho) = 0.093558}
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L/(G M^3 \bar{a})^{1 / 2} = 0.303751 ~~~~ \Rightarrow ~~~ j^2 = \biggl( \frac{3}{2^8 \pi^4} \biggr)^{1/3} \frac{L^2}{(GM^3\bar{a})} = 0.049365924 (0.303751)^2 = 0.004554731}

According to Tables D.3 and D.4 (pp. 485 & 486) of [T87], the Jacobi Sequence bifurcates from the Maclaurin sequence at …

  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a/\bar{a}, b/\bar{a}, c/\bar{a}) = (1.197234, 1.197234, 0.697657) ~~~~ \Rightarrow ~~~ (b/a, c/a) = (1.000000, 0.582724)}
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega^2/(2\pi G \rho) = 0.1871148 ~~~~ \Rightarrow ~~~ \omega_0^2/(4\pi G \rho) = 0.0935574}
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J/(G M^3 \bar{a})^{1 / 2} = 0.303751 ~~~~\Rightarrow ~~~~ j^2 = 0.004555}
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tau \equiv T/|W| = 0.1375 }

In the paragraph on p. 467 of { {Hachisu86bfull }} that immediately follows Eq. (47), we find …

  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega^2/(4\pi G\rho) = 0.09356}
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j^2 = 0.004555}

Bifurcation of Poincaré's Sequence of Pear-Shaped Configurations from the Jacobian Sequence

According to Eq. (28) of [EFE], Chapter 6, §40 (p. 106), a pear-shaped configuration bifurcates from the Jacobi Sequence at …

  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (b/a, c/a) = (0.432232, 0.345069)}
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega^2/(\pi G \rho) = 0.284030 ~~~~ \Rightarrow ~~~ \omega_0^2/(4\pi G \rho) = 0.0710075}
  • From the above expression, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{L}{(GM^3\bar{a})^{1 / 2}} = 0.389536} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j^2 = \biggl( \frac{3}{2^8 \pi^4} \biggr)^{1/3} \frac{L^2}{(GM^3\bar{a})} = 0.0074907 }

According to the first row of properties in Table I of 📚 Eriguchi, Hachisu, & Sugimoto (1982)

  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega^2/(4\pi G \rho) = 0.07101 }
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j^2 = \biggl( \frac{3}{2^8 \pi^4} \biggr)^{1/3} \frac{L^2}{(GM^3\bar{a})} = 0.007821 }
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tau \equiv T/|W| = 0.1628 }

Bifurcation of Dumbbell sequence from Jacobian Sequence

According the last pair of equations on p. 128 of [EFE], Chapter 6, §45, a dumbbell-shaped configuration (neutral point belonging to the fourth harmonic distortion) bifurcates from the Jacobi Sequence at …

  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (b/a) = (0.2972)} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos^{-1}(c/a) = 75.081~\mathrm{degrees}~~~~\Rightarrow ~~~~ (c/a) = 0.2575} .

Chronologically, this result for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (b/a, c/a)} appears first in Eq. (93) on p. 635 of 📚 S. Chandrasekhar (1967b, ApJ, Vol. 148, pp. 621 - 644) — referred to in [EFE] as Publication XXXII. Then, in Eq. (66) on p. 302 of 📚 S. Chandrasekhar (1968, ApJ, Vol. 152, pp. 293 - 304) — referred to in [EFE] as Publication XXXV — we find Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos^{-1}(c/a) = 75.068~\mathrm{degrees}} , along with a footnote [5] which states, "The value Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos^{-1}(c/a) = 75.081~\mathrm{degrees}} found earlier differs slightly; but the difference is not outside the limits of accuracy of the numerical evaluation."

According to the first row of properties in Table II of 📚 Eriguchi, Hachisu, & Sugimoto (1982)

  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega^2/(4\pi G \rho) = 0.0532 }
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j^2 = \biggl( \frac{3}{2^8 \pi^4} \biggr)^{1/3} \frac{L^2}{(GM^3\bar{a})} = 0.01157 }
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tau \equiv T/|W| = 0.1863 }

I have not (yet) found the corresponding value of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega^2} in any of Chandrasekhar's publications, but if we combine the value of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega^2} obtained from 📚 Eriguchi, Hachisu, & Sugimoto (1982) with the values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (b/a, c/a)} obtained from [EFE], we find …

  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega^2/(\pi G \rho) = 0.2128}
  • From the above expression, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{L}{(GM^3\bar{a})^{1 / 2}} = 0.48242}   and   Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j^2 = \biggl( \frac{3}{2^8 \pi^4} \biggr)^{1/3} \frac{L^2}{(GM^3\bar{a})} = 0.01149}

This value of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j^2} is very close to the value obtained by 📚 Eriguchi, Hachisu, & Sugimoto (1982).

In the paragraph at the top of the right-hand column of p. 467 of 📚 I. Hachisu (1986b, ApJS, Vol. 62, pp. 461 - 499), we find …

  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega^2/(4\pi G\rho) = 0.0535}
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j^2 = 0.01157}

📚 D. M. Christodoulou, D. Kazanas, I. Shlosman, & J. E. Tohline (1995b, ApJ, Vol. 446, pp. 485 - 499) grab parameter values from a variety of sources. In subsection "B" (Jacobi Ellipsoid to Binary) of their Table 1 (p. 494) and in the first paragraph of their §3.2 (p. 492), they state …

  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (b/a, c/a) = (0.29720, 0.25746)}
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega^2/(4\pi G \rho) = 0.0532790}
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j^2 = 0.0115082}

See Also


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