User:Tohline/ThreeDimensionalConfigurations/HomogeneousEllipsoids
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Properties of Homogeneous Ellipsoids (1)
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Gravitational Potential
The Defining Integral Expressions
As has been shown in a separate discussion titled, "Origin of the Poisson Equation," the acceleration due to the gravitational attraction of a distribution of mass can be derived from the gradient of a scalar potential defined as follows:
As has been explicitly demonstrated in Chapter 3 of EFE and summarized in Table 22 (p. 57) of BT87, for an homogeneous ellipsoid this volume integral can be evaluated analytically in closed form. Specifically, at an internal point or on the surface of an homogeneous ellipsoid with semiaxes ,
where,









Evaluation of Coefficients
The integrals defining and can be evaluated in terms of the incomplete elliptic integral of the first kind,
and/or the incomplete elliptic integral of the second kind,
where, for our particular problem,
or the integrals can be evaluated in terms of more elementary functions if either (oblate spheroids) or (prolate spheroids).
Triaxial Configurations
If the three principal axes of the configuration are unequal in length and related to one another such that ,












[ EFE, Chapter 3, Eqs. (33), (34) & (35) ]
Notice that there is no need to specify the actual value of in any of these expressions, as they each can be written in terms of the pair of axis ratios, and . As a sanity check, let's see if these three expressions can be related to one another in the manner described by equation (108) in §21 of EFE, namely,


















Q.E.D.
Oblate Spheroids
If the longest axis, , and the intermediate axis, , of the ellipsoid are equal to one another, then an equatorial crosssection of the object presents a circle of radius and the object is referred to as an oblate spheroid. For homogeneous oblate spheroids, evaluation of the integrals defining and gives,












where the eccentricity,
Prolate Spheroids
If the shortest axis and the intermediate axis of the ellipsoid are equal to one another, then a crosssection in the plane of the object presents a circle of radius and the object is referred to as a prolate spheroid. For homogeneous prolate spheroids, evaluation of the integrals defining and gives,












[ EFE, Chapter 3, Eq. (38) ]
where, again, the eccentricity,
Example Evaluations
Here we adopt the notation mapping, . In general, for a given pair of axis ratios, , a determination of the coefficients, , , and , requires evaluation of elliptic integrals. For practical applications, we have decided to evaluate these special functions using the fortran functions provided in association with the book, Numerical Recipes in Fortran; in order to obtain the results presented in our Table 2, below, we modified those default (singleprecision) routines to generate results with doubleprecision accuracy. Along the way (see results posted in our Table 1), we pulled cruder evaluations of both elliptic integrals, and , from the printed specialfunctions table found in a CRC handbook.
As we developed/debugged the numerical tool that would allow us to determine the values of these three coefficients for arbitrary choices of the pair of axis ratios, it was important that we compare the results of our calculations to those that have appeared in the published literature. As a primary point of comparison, we chose to use The properties of the Jacobi ellipsoids as tabulated in §39 (Chapter 6) of Chandrasekhar's EFE. In particular, for twentysix separate axisratio pairs, Chandrasekhar's Table IV lists the values of the square of the angular velocity, , and the total angular momentum, , of an equilibrium Jacobi ellipsoid that is associated with each axisratio pair. We should be able to duplicate — or, via doubleprecision arithmetic, improve — Chandrasekhar's tabulated results using the expressions for "omega2",
and, for "angmom",



[ EFE, §39, Eq. (16) ]  



Or, in connection with the freeenergy discussion found in D. M. Christodoulou, D. Kazanas, I. Shlosman, & J. E. Tohline (1995, ApJ, 446, 472),






Table 1: Example Evaluations  

Given  Determined using calculator and (crude) CRC tables of elliptic integrals  
radians  degrees  radians  degrees  
1.00  0.582724  0.94871973  54.3576  0.00000000  0.00000000  0.000000  0.94871973  0.94871973  0.51589042  0.51589042  0.96821916 
0.96  0.570801  0.96331527  55.1939  0.34101077  0.34799191  19.9385  0.975  0.946  +0.4937  +0.5319  +0.9744 
0.60  0.433781  1.12211141  64.292  0.88788426  1.09272580  62.609  1.3375  0.9547  0.3455  0.6741  0.9803 
With regard to our Table 1 (immediately above): To begin with, we picked three axisratio pairs from Table IV of EFE, and considered them to be "given." For each pair, we used a handheld calculator to calculate the corresponding values of the two arguments of the elliptic integrals, namely, and , as defined above. By default, each determined value of is in radians. Because the published CRC specialfunctions tables quantify both arguments of the special functions in angular degrees, we converted from radians to degrees (see column 4 of Table 1) and, similarly, we converted to degrees (see column 7 of Table 1). For the axisymmetric configuration — the first row of numbers in Table 1, for which — the coefficients, , , and , were determined to eight digits of precision using the appropriate expressions for oblate spheroids. Note that, in this axisymmetric case, , but these function values are irrelevant with respect to the determination of the coefficients.
Table 2: DoublePrecision Evaluations
Related to Table IV in EFE, Chapter 6, §39 (p. 103) 

precision b/a c/a F E A1 A2 A3 [2(A1+A2+A3)]/2 1.00 0.582724   5.158904180D01 5.158904180D01 9.682191640D01 0.0D+00 0.96 0.570801 9.782631357D01 9.487496699D01 5.024584655D01 5.292952683D01 9.682462661D01 4.4D16 0.92 0.558330 1.009516282D+00 9.489290273D01 4.884500698D01 5.432292722D01 9.683206580D01 0.0D+00 0.88 0.545263 1.042655826D+00 9.492826127D01 4.738278227D01 5.577100115D01 9.684621658D01 2.2D16 0.84 0.531574 1.077849658D+00 9.498068890D01 4.585648648D01 5.727687434D01 9.686663918D01 2.2D16 0.80 0.517216 1.115314984D+00 9.505192815D01 4.426242197D01 5.884274351D01 9.689483451D01 4.4D16 0.76 0.502147 1.155290552D+00 9.514282210D01 4.259717080D01 6.047127268D01 9.693155652D01 2.2D16 0.72 0.486322 1.198053140D+00 9.525420558D01 4.085724682D01 6.216515450D01 9.697759868D01 4.4D16 0.68 0.469689 1.243931393D+00 9.538724717D01 3.903895871D01 6.392680107D01 9.703424022D01 2.2D16 0.64 0.452194 1.293310292D+00 9.554288569D01 3.713872890D01 6.575860416D01 9.710266694D01 4.4D16 0.60 0.433781 1.346645618D+00 9.572180643D01 3.515319835D01 6.766289416D01 9.718390749D01 3.3D16 0.56 0.414386 1.404492405D+00 9.592491501D01 3.307908374D01 6.964136019D01 9.727955606D01 6.7D16 0.52 0.393944 1.467522473D+00 9.615263122D01 3.091371405D01 7.169543256D01 9.739085339D01 4.4D16 0.48 0.372384 1.536570313D+00 9.640523748D01 2.865506903D01 7.382563770D01 9.751929327D01 2.2D16 0.44 0.349632 1.612684395D+00 9.668252052D01 2.630231082D01 7.603153245D01 9.766615673D01 8.9D16 0.40 0.325609 1.697213059D+00 9.698379297D01 2.385623719D01 7.831101146D01 9.783275135D01 0.0D+00 0.36 0.300232 1.791930117D+00 9.730763540D01 2.132011181D01 8.065964525D01 9.802024294D01 2.2D15 0.32 0.273419 1.899227853D+00 9.765135895D01 1.870102340D01 8.307027033D01 9.822870627D01 1.3D15 0.28 0.245083 2.022466812D+00 9.801112910D01 1.601127311D01 8.553054155D01 9.845818534D01 2.4D15 0.24 0.215143 2.166555572D+00 9.838093161D01 1.327137129D01 8.802197538D01 9.870665333D01 1.4D14 0.20 0.183524 2.339102805D+00 9.875217566D01 1.051389104D01 9.051602520D01 9.897008376D01 1.6D14 0.16 0.150166 2.552849055D+00 9.911267582D01 7.790060179D02 9.296886827D01 9.924107155D01 3.4D14 0.12 0.115038 2.831664019D+00 9.944537935D01 5.180880535D02 9.531203882D01 9.950708065D01 1.4D13 0.08 0.078166 3.229072310D+00 9.972669475D01 2.817821170D02 9.743504218D01 9.974713665D01 3.9D13 0.04 0.039688 3.915557866D+00 9.992484565D01 9.281550546D03 9.914470033D01 9.992714461D01 9.8D13 
With regard to our Table 2 (immediately above): Next, given each pair of axis ratios, — copied from Table IV of EFE (see columns 1 and 2 of our Table 2) — we used some fortran routines from Numerical Recipes to calculate and (see columns 3 and 4 of our Table 2); we converted the routines to accommodate doubleprecision arithmetic. We subsequently evaluated the coefficients, , , and , (columns 5, 6, & 7 of Table 2) using the expressions given above, then demonstrated that, in each case, the three coefficients sum to 2.0 to better than twelve digits accuracy.
Material that appears after this point in our presentation is under development and therefore
may contain incorrect mathematical equations and/or physical misinterpretations.
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Acceleration at the Pole
Prolate Spheroids
In our above review, for consistency, we assumed that the longest axis of the ellipsoid was aligned with the axis in all cases — for prolate spheroids as well as for oblate spheroids and for the more generic, triaxial ellipsoids. In this discussion, in order to better align with the operational features of a standard cylindrical coordinate system, we will orient the prolatespheroidal configuration such that its major axis and, hence, its axis of symmetry aligns with the axis while the center of the spheroid remains at the center of the (cylindrical) coordinate grid. In this case, the surface will be defined by the ellipse,
and the gravitational potential will be given by the expression,
The magnitude of the gravitational acceleration at the pole of this prolate spheroid can be obtained from the gravitational potential via the expression,



where, as above,



We should also be able to derive this expression for by integrating the component of the differential acceleration over the mass distribution, that is,






where the distance, , has been measured from the pole, that is,
Performing the integral over gives,












where, . For later reference, we will identify the expression inside the curly braces as the function, ; specifically,






where, in an effort to line up with notation found in integral tables, in this last expression we have used the notation, and, in our case,
and
We find that,





















Hence, we have,
which exactly matches the result obtained, above, by taking the derivative of the potential.
See Also
Footnotes
 In EFE this equation is written in terms of a variable instead of as defined here. The two variables are related to one another straightforwardly through the expression, .
 Throughout EFE, Chandrasekhar adopts a sign convention for the scalar gravitational potential that is opposite to the sign convention being used here.
© 2014  2020 by Joel E. Tohline 