User:Tohline/ThreeDimensionalConfigurations/JacobiEllipsoids
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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:
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<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> |
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<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> |
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<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,
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<math>~\theta = \cos^{-1} \biggl(\frac{c}{a} \biggr)</math> |
and |
<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,
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<math>~a^2 b^2 A_{12}</math> |
<math>~=</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,
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<math>~\frac{\Omega^2}{\pi G \rho}</math> |
<math>~=</math> |
<math>~2B_{12} \, ,</math> |
| [ EFE, §39, Eq. (5) ] | ||
where,
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<math>~A_{12}</math> |
<math>~\equiv</math> |
<math>~-\frac{A_1-A_2}{(a^2 - b^2)} \, ,</math> |
| [ EFE, §21, Eq. (107) ] | ||
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<math>~B_{12}</math> |
<math>~\equiv</math> |
<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:
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Roots of the Governing Relation
To simplify notation, here we will set,
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<math>~x \equiv \frac{b}{a}</math> |
and |
<math>~y \equiv \frac{c}{a} \, ,</math> |
in which case the governing relation is,
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<math>~f_J</math> |
<math>~=</math> |
<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,
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<math>~f_J^'</math> |
<math>~=</math> |
<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>.
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<math>~f_J^'</math> |
<math>~=</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>,
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<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> |
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<math> ~= </math> |
<math>~\frac{2y}{k^3 \sin^3\theta} \cdot \biggl\{ [ F - E ] [k - 2xk^' ] +xk [ F^' - E^' ]\biggr\} \, , </math> |
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<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> |
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<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> |
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<math>~k^'</math> |
<math>~=</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> |
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<math>~F^'</math> |
<math>~=</math> |
<math>~ \frac{\partial F(\theta,k)}{\partial k} \cdot k^' \, , </math> |
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<math>~E^'</math> |
<math>~=</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" —
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<math>~\frac{\partial F(z|m)}{\partial m}</math> |
<math>~=</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> |
and,
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<math>~\frac{\partial E(z|m)}{\partial m}</math> |
<math>~=</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,
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<math>~F^'</math> |
<math>~=</math> |
<math>~ \biggl[\frac{\partial F(z|m)}{\partial m} \cdot \frac{dm}{dk}\biggr] k^' </math> |
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<math>~=</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> |
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<math>~E^'</math> |
<math>~=</math> |
<math>~ \biggl[ \frac{\partial E(z|m)}{\partial m} \cdot \frac{dm}{dk}\biggr] k^' </math> |
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<math>~=</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.
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Table 1: Double-Precision Evaluations
Related to Table IV in EFE, Chapter 6, §39 (p. 103) |
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precision
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
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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
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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 without modification 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 |
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See Also
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© 2014 - 2021 by Joel E. Tohline |