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Revision as of 16:47, 26 June 2016

Whitworth's (1981) Isothermal Free-Energy Surface
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Jacobi Ellipsoids

General Coefficient Expressions

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

<math> ~A_1 </math>

<math> ~= </math>

<math>~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] \, , </math>

<math> ~A_3 </math>

<math> ~= </math>

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

<math> ~A_2 </math>

<math> ~= </math>

<math>~2 - (A_1+A_3) \, ,</math>

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

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


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

[ EFE, Chapter 3, §17, Eq. (32) ]

Equilibrium Conditions for Jacobi Ellipsoids

Pulling from Chapter 6 — specifically, §39 — of Chandrasekhar's EFE, we understand that the semi-axis ratios, <math>~(\tfrac{b}{a},\tfrac{c}{a})</math> associated with Jacobi ellipsoids are given by the roots of the equation,

<math>~a^2 b^2 A_{12}</math>


<math>~c^2 A_3 \, ,</math>

[ EFE, §39, Eq. (4) ]

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

<math>~\frac{\Omega^2}{\pi G \rho}</math>


<math>~2B_{12} \, ,</math>

[ EFE, §39, Eq. (5) ]




<math>~-\frac{A_1-A_2}{(a^2 - b^2)} \, ,</math>

[ EFE, §21, Eq. (107) ]



<math>~A_2 - a^2A_{12} \, .</math>

[ EFE, §21, Eq. (105) ]

Taken together, we see that, written in terms of the two primary coefficients, <math>~A_1</math> and <math>~A_3</math>, the pair of defining relations for Jacobi ellipsoids is:



<math>~\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 </math>


<math>~\frac{\Omega^2}{\pi G \rho}</math>


<math>~2\biggl\{2 - (A_1+A_3) - \biggl[ \frac{2(1-A_1)-A_3}{1 - (b/a)^2} \biggr] \biggr\}</math>

Roots of the Governing Relation

To simplify notation, here we will set,

<math>~x \equiv \frac{b}{a}</math>


<math>~y \equiv \frac{c}{a} \, ,</math>

in which case the governing relation is,



<math>~\frac{x^2}{1-x^2} \biggl[ 2(1-A_1)-A_3\biggr]-y^2 A_3 =0 \, .</math>

Our plan is to employ the Newton-Raphson method to find the root(s) of the <math>~f_J = 0</math> relation, typically holding <math>~y</math> fixed and using the Newton-Raphson technique to identify the corresponding "root" value of <math>~x</math>. Using this approach, the Newton-Raphson technique requires specification of, not only the function, <math>~f_J</math>, but also its first derivative,



<math>~\frac{df_J}{dx} \, .</math>

Let's determine the requisite expression, using a prime superscript to indicate differentiation with respect to <math>~x</math>.



<math>~ \biggl[ 2(1-A_1)-A_3\biggr]\biggl[ \frac{2x}{(1-x^2)^2} \biggr] -\frac{x^2}{1-x^2} \biggl[ 2A_1^'+A_3^'\biggr] -y^2 A_3^' \, , </math>

where, given that <math>~\theta</math> does not depend on <math>~x</math>,

<math> ~A_1^' </math>

<math> ~= </math>

<math>~\frac{2y}{\sin^3\theta} \cdot \frac{d}{dx}\biggl\{ \frac{x}{k^2} \biggl[ F(\theta,k) - E(\theta,k) \biggr] \biggr\} </math>


<math> ~= </math>

<math>~\frac{2y}{k^3 \sin^3\theta} \cdot \biggl\{ [ F - E ] [k - 2xk^' ] +xk [ F^' - E^' ]\biggr\} \, , </math>

<math> ~A_3^' </math>

<math> ~= </math>

<math> ~\frac{2}{\sin^3\theta} \cdot \frac{d}{dx}\biggl\{ \frac{x}{(1-k^2)} \biggl[ x \sin\theta - yE(\theta,k)\biggr] \biggr\} </math>


<math> ~= </math>

<math> ~\frac{2}{(1-k^2)^2\sin^3\theta} \biggl\{ \biggl[ x \sin\theta - yE\biggr]\biggl[ (1-k^2) +2xkk^' \biggr] + x(1-k^2) \biggl[ \sin\theta - yE^'\biggr] \biggr\}\, , </math>



<math>~ \frac{d}{dx}\biggl[\frac{1 - x^2}{1 - y^2} \biggr]^{1/2} = \frac{-x}{(1 - x^2)^{1/2}(1 - y^2)^{1/2}} \, , </math>



<math>~ \frac{\partial F(\theta,k)}{\partial k} \cdot k^' \, , </math>



<math>~ \frac{\partial E(\theta,k)}{\partial k} \cdot k^' \, . </math>

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

<math>~\frac{\partial F(z|m)}{\partial m}</math>


<math>~ \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)}} \, , </math>


<math>~\frac{\partial E(z|m)}{\partial m}</math>


<math>~\frac{E(z|m) - F(z|m)}{2m} \, ,</math>

where, <math>~z~\leftrightarrow~\theta</math>, and,

<math>~m \equiv k^2 ~~~~\Rightarrow~~~~\frac{dm}{dk} = 2k \ .</math>

Hence, we have,



<math>~ \biggl[\frac{\partial F(z|m)}{\partial m} \cdot \frac{dm}{dk}\biggr] k^' </math>



<math>~ \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^' \, , </math>



<math>~ \biggl[ \frac{\partial E(z|m)}{\partial m} \cdot \frac{dm}{dk}\biggr] k^' </math>



<math>~ \biggl[ E(\theta,k) - F(\theta,k) \biggr] \frac{k^'}{k} \, . </math>

This, then, gives us all of the expressions necessary to specify the derivative, <math>~f_J^'</math> analytically.

Table 1:  Double-Precision Evaluations

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

         b/a      c/a              F                   E                  A1                  A2                  A3          [2-(A1+A2+A3)]/2

        1.00   0.582724          -----               -----          5.158904180D-01     5.158904180D-01     9.682191640D-01        0.0D+00
        0.96   0.570801     9.782631357D-01     9.487496699D-01     5.024584655D-01     5.292952683D-01     9.682462661D-01        4.4D-16
        0.92   0.558330     1.009516282D+00     9.489290273D-01     4.884500698D-01     5.432292722D-01     9.683206580D-01        0.0D+00
        0.88   0.545263     1.042655826D+00     9.492826127D-01     4.738278227D-01     5.577100115D-01     9.684621658D-01        2.2D-16
        0.84   0.531574     1.077849658D+00     9.498068890D-01     4.585648648D-01     5.727687434D-01     9.686663918D-01        2.2D-16

        0.80   0.517216     1.115314984D+00     9.505192815D-01     4.426242197D-01     5.884274351D-01     9.689483451D-01       -4.4D-16
        0.76   0.502147     1.155290552D+00     9.514282210D-01     4.259717080D-01     6.047127268D-01     9.693155652D-01        2.2D-16
        0.72   0.486322     1.198053140D+00     9.525420558D-01     4.085724682D-01     6.216515450D-01     9.697759868D-01       -4.4D-16
        0.68   0.469689     1.243931393D+00     9.538724717D-01     3.903895871D-01     6.392680107D-01     9.703424022D-01        2.2D-16
        0.64   0.452194     1.293310292D+00     9.554288569D-01     3.713872890D-01     6.575860416D-01     9.710266694D-01        4.4D-16

        0.60   0.433781     1.346645618D+00     9.572180643D-01     3.515319835D-01     6.766289416D-01     9.718390749D-01       -3.3D-16
        0.56   0.414386     1.404492405D+00     9.592491501D-01     3.307908374D-01     6.964136019D-01     9.727955606D-01       -6.7D-16
        0.52   0.393944     1.467522473D+00     9.615263122D-01     3.091371405D-01     7.169543256D-01     9.739085339D-01        4.4D-16
        0.48   0.372384     1.536570313D+00     9.640523748D-01     2.865506903D-01     7.382563770D-01     9.751929327D-01       -2.2D-16
        0.44   0.349632     1.612684395D+00     9.668252052D-01     2.630231082D-01     7.603153245D-01     9.766615673D-01        8.9D-16

        0.40   0.325609     1.697213059D+00     9.698379297D-01     2.385623719D-01     7.831101146D-01     9.783275135D-01        0.0D+00
        0.36   0.300232     1.791930117D+00     9.730763540D-01     2.132011181D-01     8.065964525D-01     9.802024294D-01        2.2D-15
        0.32   0.273419     1.899227853D+00     9.765135895D-01     1.870102340D-01     8.307027033D-01     9.822870627D-01       -1.3D-15
        0.28   0.245083     2.022466812D+00     9.801112910D-01     1.601127311D-01     8.553054155D-01     9.845818534D-01       -2.4D-15
        0.24   0.215143     2.166555572D+00     9.838093161D-01     1.327137129D-01     8.802197538D-01     9.870665333D-01        1.4D-14

        0.20   0.183524     2.339102805D+00     9.875217566D-01     1.051389104D-01     9.051602520D-01     9.897008376D-01       -1.6D-14
        0.16   0.150166     2.552849055D+00     9.911267582D-01     7.790060179D-02     9.296886827D-01     9.924107155D-01       -3.4D-14
        0.12   0.115038     2.831664019D+00     9.944537935D-01     5.180880535D-02     9.531203882D-01     9.950708065D-01        1.4D-13
        0.08   0.078166     3.229072310D+00     9.972669475D-01     2.817821170D-02     9.743504218D-01     9.974713665D-01        3.9D-13
        0.04   0.039688     3.915557866D+00     9.992484565D-01     9.281550546D-03     9.914470033D-01     9.992714461D-01        9.8D-13
         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

Sequence Plots

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.

See Also

Whitworth's (1981) Isothermal Free-Energy Surface

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