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,









This definiteintegral definition of may also be found in:
 Lamb32: as Eq. (6) in §114 (p. 153); and as Eq. (5) in §373 (p. 700).
 T78: as Eq. (5) in §10.2 (p. 234), but note that there is an error in the denominator of the righthandside — appears instead of .
Evaluation of Coefficients
As is detailed below, 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.
Derivation of Expressions for A_{i}
Let's carry out the integrals that appear in the definition of the coefficients,



where,



Here, we are adopting the subscript notation to identify which semiaxis length is the (largest, mediumlength, smallest).
Evaluating A_{ℓ}
First, let's focus on the coefficient associated with the longest axis :



Changing the integration variable to , we obtain a definite integral expression that appears as equation (3.133.1) in I. W. Gradshteyn & I. M. Ryzhik (2007; 7^{th} Edition), Table of Integrals, Series, and Products — hereafter, GR7^{th} — namely,







… valid for 
GR7^{th}, p. 255, Eq. (3.133.1) 
where (see p. 254 of GR7^{th}),






and where, and are elliptic integrals of the first and second kind, respectively. (Note that in the notation convention adopted by GR7^{th}, the order of the argument list, , is flipped relative to the convention that we have adopted above and elsewhere throughout our online, MediaWikibased chapters.) Recognizing that,



we see that the expression for can be rewritten as,



This matches the expression that we have provided for , above in the context of triaxial configurations.
Evaluating A_{m}
Next, let's evaluate the coefficient associated with the axis of intermediate length :



This time, by changing the integration variable to , we obtain a definite integral expression that appears as equation (3.133.7) in GR7^{th}, namely,







… valid for 
GR7^{th}, p. 256, Eq. (3.133.7) 
(Here, the parameters, and , have the same definitions as in our above evaluation of .) This time it is useful to recognize that,



in which case,



So the coefficient, , may be rewritten as,












This matches the expression that we have provided for , above in the context of triaxial configurations.
Evaluating A_{s}
Finally, let's evaluate the coefficient associated with the shortest axis, :



By changing the integration variable to , this time we obtain a definite integral expression that appears as equation (3.133.13) in GR7^{th}, namely,







… valid for 
GR7^{th}, p. 256, Eq. (3.133.13) 
(And, again, the parameters, and , have the same definitions as in our above evaluation of .) Recognizing that,



the coefficient, , may be rewritten as,












This matches the expression that we have provided for , above in the context of triaxial configurations.
When a_{m} = a_{ℓ}
When the length of the intermediate axis is the same as the length of the longest axis — that is, when we are dealing with an oblate spheroid — the coefficient associated with the longest axis is,



Changing the integration variable to , we obtain an integral expression that appears as equation (2.228.1) in GR7^{th}, namely,









The remaining integral in this expression appears as equation (2.224.5) in GR7^{th}. Its resolution depends on the sign of the constant term in the denominator, . Given that this term is negative, the integration gives,















where, . Similarly, the coefficient associated with the shortest axis is,



This time, after changing the integration variable to , we obtain an integral expression that appears as equation (2.229.1) in GR7^{th}, namely,






As before, the remaining integral in this expression appears as equation (2.224.5) in GR7^{th}; and, as before, the sign of the constant term in the denominator, , is negative. Hence, the integration gives,












Because we are evaluating the case where , we alternatively should have been able to obtain the expression for immediately from our derived expression for via the known relation,



This approach gives,









which, indeed, matches our separately derived expression for .
When a_{m} = a_{s}
When the length of the intermediate axis is the same as the length of the shortest axis — that is, when we are dealing with a prolate spheroid — the coefficient associated with the longest axis is,



Changing the integration variable to , we obtain an integral expression that appears as equation (2.229.1) in GR7^{th}, namely,






The remaining integral in this expression appears as equation (2.224.5) in GR7^{th}. Its resolution depends on the sign of the constant term in the denominator, . Given that this term is positive, the integration gives,









where, as above, . Now, given that , in this case we appreciate that,












Inhomogeneous Ellipsoids Leading to Ferrers Potentials
Other references to Ferrers Potential:

Following §2.3.2 (beginning on p. 60) of BT87, let's examine inhomogeneous configurations whose isodensity surfaces (including the surface, itself) are defined by triaxial ellipsoids on which the Cartesian coordinates satisfy the condition that,



[ EFE, Chapter 3, §20, p. 50, Eq. (75) ] 
be constant. More specifically, let's consider the case (related to the socalled Ferrers potentials) in which the configuration's density distribution is given by the expression,







NOTE: In our accompanying discussion of compressible analogues of Riemann Stype ellipsoids, we have discovered that — at least in the context of infinitesimally thin, nonaxisymmetric disks — this heterogeneous density profile can be nicely paired with an analytically expressible stream function, at least for the case where the integer exponent is, n = 1. 
According to Theorem 13 of EFE — see his Chapter 3, §20 (p. 53) — the potential at any point inside a triaxial ellipsoid with this specific density distribution is given by the expression,



[ EFE, Chapter 3, §20, p. 53, Eq. (101) ] 
where, has the same definition as above, and,



For purposes of illustration, in what follows we will assume that, .
The Case Where n = 0
When , we have a uniformdensity configuration, and the "interior" potential will be given by the expression,












As a check, let's see if this scalar potential satisfies the differential form of the
Given that,



[ EFE, §21, Eq. (108) ] 
we find,



Q.E.D.
The Case Where n = 1
When , we have a specific heterogeneous density configuration, and the "interior" potential will be given by the expression,









The first definiteintegral expression inside the curly braces is, to within a leading factor of , identical to the entire expression for the normalized potential that was derived in the case where n = 0. That is, we can write,






Then, from §22, p. 56 of EFE, we see that,



[ EFE, Chapter 3, §22, p. 53, Eq. (125) ] 
Applying this result to each of the other three definite integrals gives us,












where,


and we have made use of the symmetry relation, . Again, as a check, let's see if this scalar potential satisfies the differential form of the
We find,












In addition to recognizing, as stated above, that , and making explicit use of the relation,



this last expression can be simplified to discover that,






This does indeed demonstrate that the derived gravitational potential is consistent with our selected mass distribution in the case where n = 1, namely,



Q.E.D.
Material that appears after this point in our presentation is under development and therefore
may contain incorrect mathematical equations and/or physical misinterpretations.
 Go Home 
Work In Progress
Derivation of Expression for Gravitational Potential
In §373 (p. 700) of his book titled, Hydrodynamics, Lamb32 states that, "The gravitationpotential, at internal points, of a uniform mass enclosed by the surface



[ Lamb32, §373, Eq. (1) ] 
… may be written



[ Lamb32, §373, Eq. (4) ] 
where, as in §114,"














[ Lamb32, §373, Eqs. (5) & (6) ] 
and,



[ Lamb32, §373, Eq. (3) ] 
Although different variable names have been used, it is easy to see the correspondence between these expressions and the defining integral expressions that we have drawn from the more recent publications of EFE and BT87 and presented above. Here, we are interested in demonstrating how Lamb32 derived his expression for the potential inside (and on the surface of) an homogeneous ellipsoid.
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
 Properties of Maclaurin Spheroids
 Excerpts from Maclaurin's (1742) A Treatise of Fluxions
 Inhomogeneous Ellipsoids Leading to Ferrers Potentials
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  2021 by Joel E. Tohline 