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<math>~\sum_{\ell = 1}^3 A_\ell</math>
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<math>~2 \, ,</math>
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<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">&sect;21, Eq. (108)</font> ]</td></tr>
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Revision as of 15:11, 24 September 2020


Contents

Properties of Homogeneous Ellipsoids (1)

Whitworth's (1981) Isothermal Free-Energy Surface
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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 ~\rho(\vec{x}) can be derived from the gradient of a scalar potential ~\Phi(\vec{x}) defined as follows:


\Phi(\vec{x}) \equiv - \int \frac{G \rho(\vec{x}')}{|\vec{x}' - \vec{x}|} d^3 x' .

As has been explicitly demonstrated in Chapter 3 of EFE and summarized in Table 2-2 (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 semi-axes ~(x,y,z) = (a_1,a_2,a_3),


~\Phi(\vec{x}) = -\pi G \rho \biggl[ I_\mathrm{BT} a_1^2 - \biggl(A_1 x^2 + A_2 y^2 +A_3 z^2 \biggr) \biggr],

[ EFE, Chapter 3, Eq. (40)1,2 ]
[ BT87, Chapter 2, Table 2-2 ]

Comment by J. E. Tohline on 15 August 2020:  This integral definition of A_i also appears as Eq. (5) of §10.2 (p. 234) of T78, but it contains an error — in the denominator on the right-hand-side, a_1 appears instead of a_i.
where,


~A_i


~\equiv


~a_1 a_2 a_3 \int_0^\infty \frac{du}{\Delta (a_i^2 + u )} ,


~I_\mathrm{BT}


~\equiv


~\frac{a_2 a_3}{a_1} \int_0^\infty \frac{du}{\Delta} = A_1 + A_2\biggl(\frac{a_2}{a_1}\biggr)^2+ A_3\biggl(\frac{a_3}{a_1}\biggr)^2 ,


~\Delta


~\equiv


~\biggl[ (a_1^2 + u)(a_2^2 + u)(a_3^2 + u)  \biggr]^{1/2} .

[ EFE, Chapter 3, Eqs. (18), (15 & 22)1, & (8), respectively ]
[ BT87, Chapter 2, Table 2-2 ]

This definite-integral definition of ~A_i 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 right-hand-side — ~a_1 appears instead of ~a_i.

Evaluation of Coefficients

As is detailed below, the integrals defining ~A_i and ~I_\mathrm{BT} can be evaluated in terms of the incomplete elliptic integral of the first kind,


~F(\theta,k) \equiv \int_0^\theta \frac{d\theta '}{\sqrt{1 - k^2 \sin^2\theta '}} ~~ ,

and/or the incomplete elliptic integral of the second kind,


E(\theta,k) \equiv \int_0^\theta {\sqrt{1 - k^2 \sin^2\theta '}}~d\theta ' ~~ ,

where, for our particular problem,


~\theta \equiv \cos^{-1} \biggl(\frac{a_3}{a_1} \biggr) ,


~k \equiv \biggl[\frac{a_1^2 - a_2^2}{a_1^2 - a_3^2} \biggr]^{1/2} = \biggl[\frac{1 - (a_2/a_1)^2}{1 - (a_3/a_1)^2} \biggr]^{1/2},

[ EFE, Chapter 3, Eq. (32) ]

or the integrals can be evaluated in terms of more elementary functions if either ~a_2 = a_1 (oblate spheroids) or ~a_3 = a_2 (prolate spheroids).

 

Triaxial Configurations ~(a_1 > a_2 > a_3)

If the three principal axes of the configuration are unequal in length and related to one another such that ~a_1 > a_2 > a_3 ,


~A_1


~=


~\frac{2a_2 a_3}{a_1^2} \biggl[  \frac{F(\theta,k) - E(\theta,k)}{k^2 \sin^3\theta} \biggr] ~~;


~A_2


~=


~\frac{2a_2 a_3}{a_1^2} \biggl[ \frac{E(\theta,k) - (1-k^2)F(\theta,k) - (a_3/a_2)k^2\sin\theta}{k^2 (1-k^2) \sin^3\theta}\biggr]  ~~;


~A_3


~=


~\frac{2a_2 a_3}{a_1^2} \biggl[  \frac{(a_2/a_3) \sin\theta - E(\theta,k)}{(1-k^2) \sin^3\theta} \biggr] ~~;


~I_\mathrm{BT}


~=


~\frac{2a_2 a_3}{a_1^2} \biggl[ \frac{F(\theta,k)}{\sin\theta} \biggr] ~~.

[ EFE, Chapter 3, Eqs. (33), (34) & (35) ]

Notice that there is no need to specify the actual value of ~a_1 in any of these expressions, as they each can be written in terms of the pair of axis ratios, ~a_2/a_1 and ~a_3/a_1. 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,

~\sum_{\ell=1}^3 A_\ell = 2 \, .

~\frac{a_1^2}{2a_2 a_3} \biggl[A_1 + A_3 + A_2\biggr]

~=

~ \frac{F(\theta,k) - E(\theta,k)}{k^2 \sin^3\theta} 
+ \frac{(a_2/a_3) \sin\theta - E(\theta,k)}{(1-k^2) \sin^3\theta}

 

 

~+ \frac{E(\theta,k) - (1-k^2)F(\theta,k) - (a_3/a_2)k^2\sin\theta}{k^2 (1-k^2) \sin^3\theta}

 

~=

~ \frac{1}{k^2(1-k^2)\sin^3\theta} \biggl\{(1-k^2)F(\theta,k) - (1-k^2)E(\theta,k) 
+ k^2(a_2/a_3) \sin\theta

 

 

~- k^2E(\theta,k)  + E(\theta,k) - (1-k^2)F(\theta,k) - (a_3/a_2)k^2\sin\theta\biggr\}

 

~=

~ \frac{1}{(1-k^2)\sin^2\theta} \biggl[ \frac{a_2}{a_3} - \frac{a_3}{a_2} \biggr]

 

~=

~ \frac{a_1^2}{a_2 a_3} \, .

Q.E.D.

 

Oblate Spheroids ~(a_1 = a_2 > a_3)

If the longest axis, ~a_1, and the intermediate axis, ~a_2, of the ellipsoid are equal to one another, then an equatorial cross-section of the object presents a circle of radius ~a_1 and the object is referred to as an oblate spheroid. For homogeneous oblate spheroids, evaluation of the integrals defining ~A_i and ~I_\mathrm{BT} gives,


~A_1


~=


~\frac{1}{e^2} \biggl[  \frac{\sin^{-1}e}{e} - (1-e^2)^{1/2} \biggr] (1-e^2)^{1/2} ~~;


~A_2


~=


~A_1  ~~;


~A_3


~=


~\frac{2}{e^2} \biggl[  (1-e^2)^{-1/2} - \frac{\sin^{-1}e}{e} \biggr] (1-e^2)^{1/2} ~~;


~I_\mathrm{BT}


~=


~2A_1 + A_3 (1-e^2) = 2 (1-e^2)^{1/2} \biggl[ \frac{\sin^{-1}e}{e} \biggr] ~~,

[ EFE, Chapter 3, Eq. (36) ]
[ T78, §4.5, Eqs. (48) & (49) ]

where the eccentricity,


~e \equiv \biggl[1 - \biggl(\frac{a_3}{a_1}\biggr)^2  \biggr]^{1/2} ~~.


 

Prolate Spheroids ~(a_1 > a_2 = a_3)

If the shortest axis ~(a_3) and the intermediate axis ~(a_2) of the ellipsoid are equal to one another, then a cross-section in the ~x-y plane of the object presents a circle of radius ~a_3 and the object is referred to as a prolate spheroid. For homogeneous prolate spheroids, evaluation of the integrals defining ~A_i and ~I_\mathrm{BT} gives,


~A_1


~=


\ln\biggl[ \frac{1+e}{1-e} \biggr] \frac{(1-e^2)}{e^3} - \frac{2(1-e^2)}{e^2} ~~;


~A_2


~=


\frac{1}{e^2} - \ln\biggl[ \frac{1+e}{1-e} \biggr]\frac{(1-e^2)}{2e^3}  ~~;


~A_3


~=


A_2 ~~;


~I_\mathrm{BT}


~=

~
A_1 + 2(1-e^2)A_2 = \ln\biggl[ \frac{1+e}{1-e} \biggr]\frac{(1-e^2)}{e} ~~,

[ EFE, Chapter 3, Eq. (38) ]

where, again, the eccentricity,


~e \equiv \biggl[1 - \biggl(\frac{a_3}{a_1}\biggr)^2  \biggr]^{1/2} ~~.

Example Evaluations

Here we adopt the notation mapping, ~(a_1, a_2, a_3) ~\leftrightarrow~ (a,b,c). In general, for a given pair of axis ratios, ~(\tfrac{b}{a}, \tfrac{c}{a}), a determination of the coefficients, ~A_1, ~A_2, and ~A_3, 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 (single-precision) routines to generate results with double-precision accuracy. Along the way (see results posted in our Table 1), we pulled cruder evaluations of both elliptic integrals, ~F(\theta,k) and ~E(\theta,k), from the printed special-functions 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 twenty-six separate axis-ratio pairs, Chandrasekhar's Table IV lists the values of the square of the angular velocity, ~\Omega^2, and the total angular momentum, ~L, of an equilibrium Jacobi ellipsoid that is associated with each axis-ratio pair. We should be able to duplicate — or, via double-precision arithmetic, improve — Chandrasekhar's tabulated results using the expressions for "omega2",

~\frac{\Omega^2}{\pi G\rho}

~=

~2B_{12}

[ EFE, §39, Eq. (5) ]

 

~=

~2\biggl[\frac{A_1 - (b/a)^2A_2}{1-(b/a)^2} \biggr] \, ,

[using, EFE, §21, Eqs. (105) & (107) ]

and, for "angmom",

~\frac{L}{(GM^3)^{1/2}(abc)^{1/6}}

~=

~\frac{\sqrt{3}}{10}\biggl[ \frac{a^2 + b^2}{(abc)^{2/3}} \biggr]\biggl(\frac{\Omega^2}{\pi G \rho}\biggr)^{1/2}

[ EFE, §39, Eq. (16) ]

 

~=

~\frac{\sqrt{3}}{10}\biggl[ \frac{1 + (b/a)^2}{(b/a)^{2/3}(c/a)^{2/3}} \biggr]\biggl(\frac{\Omega^2}{\pi G \rho}\biggr)^{1/2} \, .

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

~\frac{5L}{M}

~=

~a^2\biggl[ 1 + \biggl(\frac{b}{a}\biggr)^2 \biggr]\biggl[\frac{\Omega^2}{\pi G \rho}\biggr]^{1/2}

 

~=

~\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]\biggl[\frac{\Omega^2}{\pi G \rho}\biggr]^{1/2}

 

Table 1:  Example Evaluations
Given Determined using calculator and (crude) CRC tables of elliptic integrals
~\frac{a_2}{a_1} ~\frac{a_3}{a_1} ~\theta ~k ~\sin^{-1}k ~F(\theta,k) ~E(\theta,k) ~A_1 ~A_2 ~A_3
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 axis-ratio pairs from Table IV of EFE, and considered them to be "given." For each pair, we used a hand-held calculator to calculate the corresponding values of the two arguments of the elliptic integrals, namely, ~\theta and ~k, as defined above. By default, each determined value of ~\theta is in radians. Because the published CRC special-functions tables quantify both arguments of the special functions in angular degrees, we converted ~\theta from radians to degrees (see column 4 of Table 1) and, similarly, we converted ~\sin^{-1}k to degrees (see column 7 of Table 1). For the axisymmetric configuration — the first row of numbers in Table 1, for which ~a_2/a_1 = 1 — the coefficients, ~A_1, ~A_2, and ~A_3, were determined to eight digits of precision using the appropriate expressions for oblate spheroids. Note that, in this axisymmetric case, ~F(\theta,0) = E(\theta,0) = \theta, but these function values are irrelevant with respect to the determination of the ~A_\ell coefficients.

Table 2:  Double-Precision 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          -----               -----          5.158904180D-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

With regard to our Table 2 (immediately above): Next, given each pair of axis ratios, ~(\tfrac{b}{a},\tfrac{c}{a}) — 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 ~F(\theta,k) and ~E(\theta,k) (see columns 3 and 4 of our Table 2); we converted the routines to accommodate double-precision arithmetic. We subsequently evaluated the coefficients, ~A_1, ~A_2, and ~A_3, (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 Ai

Let's carry out the integrals that appear in the definition of the ~A_i coefficients,


~A_i


~\equiv


~a_\ell a_m a_s \int_0^\infty \frac{du}{\Delta (a_i^2 + u )} ,

where,


~\Delta


~\equiv


~\biggl[ (a_\ell^2 + u)(a_m^2 + u)(a_s^2 + u)  \biggr]^{1/2} \, .

Here, we are adopting the ~(\ell, m, s) subscript notation to identify which semi-axis length is the (largest, medium-length, smallest).

Evaluating A

First, let's focus on the coefficient associated with the longest axis ~(i = \ell):

~\frac{A_\ell}{a_\ell a_m a_s}

~=


~\int_0^\infty \biggl[ (a_\ell^2 + u)^3(a_m^2 + u)(a_s^2 + u) \biggr]^{-1 / 2} du

Changing the integration variable to ~x \equiv -u, we obtain a definite integral expression that appears as equation (3.133.1) in I. W. Gradshteyn & I. M. Ryzhik (2007; 7th Edition), Table of Integrals, Series, and Products — hereafter, GR7th — namely,

~\frac{A_\ell}{a_\ell a_m a_s}

~=


~\int_{-\infty}^0 \biggl[ (a_\ell^2 - x)^3(a_m^2 - x)(a_s^2 - x) \biggr]^{-1 / 2} dx

 

 

~=


~\frac{2}{(a_\ell^2 - a_m^2) \sqrt{a_\ell^2 - a_s^2}} 
\biggl[ F(\alpha, p) - E(\alpha, p) \biggr]

      … valid for [a_\ell > a_m > a_s \ge 0]
GR7th, p. 255, Eq. (3.133.1)

where (see p. 254 of GR7th),

~\sin^2\alpha

~\equiv

~\frac{a_\ell^2 - a_s^2}{a_\ell^2 - 0} = 1 - \frac{a_s^2}{a_\ell^2} \, ,

 

~p

~\equiv

~\biggl[ \frac{a_\ell^2 - a_m^2}{a_\ell^2 - a_s^2} \biggr]^{1 / 2} \, ,

and where, ~E(\alpha, p) and ~F(\alpha, p) are elliptic integrals of the first and second kind, respectively. (Note that in the notation convention adopted by GR7th, the order of the argument list, ~(\alpha, p), is flipped relative to the convention that we have adopted above and elsewhere throughout our online, MediaWiki-based chapters.) Recognizing that,

~p^2 \sin^3\alpha

~=

~
\biggl[\frac{a_\ell^2 - a_s^2}{a_\ell^2 }\biggr]^{3 / 2}\biggl[ \frac{a_\ell^2 - a_m^2}{a_\ell^2 - a_s^2} \biggr]
=
\frac{(a_\ell^2 - a_s^2)^{1 / 2}}{a_\ell^3 } \biggl[ a_\ell^2 - a_m^2 \biggr] \, ,

we see that the expression for ~A_\ell can be rewritten as,

~\frac{A_\ell}{a_\ell a_m a_s}

~=


~\frac{2}{a_\ell^3 ~p^2 \sin^3\alpha} 
\biggl[ F(\alpha, p) - E(\alpha, p) \biggr] \, .

This matches the expression that we have provided for ~A_1, above in the context of triaxial configurations.


Evaluating Am

Next, let's evaluate the coefficient associated with the axis of intermediate length ~(i = m):

~\frac{A_m}{a_\ell a_m a_s}

~=


~\int_0^\infty \biggl[ (a_\ell^2 + u)(a_m^2 + u)^3(a_s^2 + u) \biggr]^{-1 / 2} du \, .

This time, by changing the integration variable to ~x \equiv -u, we obtain a definite integral expression that appears as equation (3.133.7) in GR7th, namely,

~\frac{A_m}{a_\ell a_m a_s}

~=


~\int_{-\infty}^0 \biggl[ (a_\ell^2 - x)(a_m^2 - x)^3(a_s^2 - x) \biggr]^{-1 / 2} dx

 

 

~=


~\frac{2\sqrt{a_\ell^2 - a_s^2}}{(a_\ell^2 - a_m^2) (a_m^2 - a_s^2)} E(\alpha, p) 
-
\frac{2}{(a_\ell^2 - a_m^2) \sqrt{a_\ell^2 - a_s^2}} F(\alpha, p)
-
\frac{2}{a_m^2 - a_s^2} \biggl[ \frac{a_s^2}{a_\ell^2 a_m^2} \biggr]^{1 / 2}

      … valid for [a_\ell > a_m > a_s \ge 0]
GR7th, p. 256, Eq. (3.133.7)

(Here, the parameters, ~\alpha and ~p, have the same definitions as in our above evaluation of ~A_\ell.) This time it is useful to recognize that,

~1 -p^2

~=

~
1 - \frac{a_\ell^2 - a_m^2}{a_\ell^2 - a_s^2} 
=
\frac{a_m^2 - a_s^2 }{a_\ell^2 - a_s^2}

in which case,

~p^2(1-p^2) \sin^3\alpha

~=

~
\frac{ ( a_\ell^2 - a_m^2 )( a_m^2 - a_s^2 )}{a_\ell^3 (a_\ell^2 - a_s^2)^{1 / 2}} \, .

So the coefficient, ~A_m, may be rewritten as,

~p^2 (1-p^2) \sin^3\alpha\biggl[ \frac{A_m}{a_\ell a_m a_s} \biggr]

~=

~
\frac{ ( a_\ell^2 - a_m^2 )( a_m^2 - a_s^2 )}{a_\ell^3 (a_\ell^2 - a_s^2)^{1 / 2}}
\biggl\{
\frac{2\sqrt{a_\ell^2 - a_s^2}}{(a_\ell^2 - a_m^2) (a_m^2 - a_s^2)} E(\alpha, p) 
~-~
\frac{2}{(a_\ell^2 - a_m^2) \sqrt{a_\ell^2 - a_s^2}} F(\alpha, p)
~-~\frac{2}{a_m^2 - a_s^2} \biggl[ \frac{a_s^2}{a_\ell^2 a_m^2} \biggr]^{1 / 2}
\biggr\}

 

~=

~
\frac{ 2}{a_\ell^3 }
\biggl\{
E(\alpha, p) 
\biggr\}
-~\frac{ 2( a_m^2 - a_s^2 )}{a_\ell^3 (a_\ell^2 - a_s^2)}
\biggl\{
F(\alpha, p)
\biggr\}
-~\frac{ 2( a_\ell^2 - a_m^2 )}{a_\ell^3 (a_\ell^2 - a_s^2)^{1 / 2}}
\biggl[ \frac{a_s}{a_\ell a_m} \biggr]

 

~=

~
\frac{ 2}{a_\ell^3 }
\biggl\{
E(\alpha, p) 
-~(1-p^2)
F(\alpha, p)
-~p^2\sin\alpha
\biggl[ \frac{a_s}{a_m} \biggr]
\biggr\}

~\Rightarrow ~~~ \frac{A_m}{a_\ell a_m a_s}

~=

~
\frac{ 2}{a_\ell^3 }
\biggl[ \frac{
E(\alpha, p) 
-~(1-p^2)
F(\alpha, p)
-~(a_s/a_m)p^2\sin\alpha}{p^2 (1-p^2)\sin^3\alpha}
\biggr] \, .

This matches the expression that we have provided for ~A_2, above in the context of triaxial configurations.


Evaluating As

Finally, let's evaluate the coefficient associated with the shortest axis, ~(i = s):

~\frac{A_s}{a_\ell a_m a_s}

~=


~\int_0^\infty \biggl[ (a_\ell^2 + u)(a_m^2 + u)(a_s^2 + u)^3 \biggr]^{-1 / 2} du \, .

By changing the integration variable to ~x \equiv -u, this time we obtain a definite integral expression that appears as equation (3.133.13) in GR7th, namely,

~\frac{A_s}{a_\ell a_m a_s}

~=


~\int_{-\infty}^0 \biggl[ (a_\ell^2 - x)(a_m^2 - x)(a_s^2 - x)^3 \biggr]^{-1 / 2} dx

 

 

~=


~\frac{2}{(a_s^2 - a_m^2) \sqrt{a_\ell^2 - a_s^2}} E(\alpha, p) 
+
\frac{2}{a_m^2 - a_s^2} \biggl[ \frac{a_m^2}{a_\ell^2 a_s^2} \biggr]^{1 / 2}

      … valid for [a_\ell > a_m > a_s > 0]
GR7th, p. 256, Eq. (3.133.13)

(And, again, the parameters, ~\alpha and ~p, have the same definitions as in our above evaluation of ~A_\ell.) Recognizing that,

~(1-p^2) \sin^3\alpha

~=

~
\frac{ (a_\ell^2 - a_s^2)^{1 / 2}( a_m^2 - a_s^2 )}{a_\ell^3 } \, ,

the coefficient, ~A_s, may be rewritten as,

~(1-p^2) \sin^3\alpha\biggl[ \frac{A_s}{a_\ell a_m a_s} \biggr]

~=

~
\frac{ (a_\ell^2 - a_s^2)^{1 / 2}( a_m^2 - a_s^2 )}{a_\ell^3 }
\biggl\{
~\frac{2}{(a_s^2 - a_m^2) \sqrt{a_\ell^2 - a_s^2}} E(\alpha, p) 
+
\frac{2}{a_m^2 - a_s^2} \biggl[ \frac{a_m^2}{a_\ell^2 a_s^2} \biggr]^{1 / 2}
\biggr\}

 

~=

~
\frac{ 2}{a_\ell^3 }
\biggl\{
- E(\alpha, p) 
\biggr\}
~+~
\frac{ 2(a_\ell^2 - a_s^2)^{1 / 2}}{a_\ell^3 }
\biggl[ \frac{a_m}{a_\ell a_s} \biggr]

 

~=

~
\frac{ 2}{a_\ell^3 }
\biggl\{
\biggl(\frac{a_m}{a_s}\biggr) \sin\alpha
- E(\alpha, p) 
\biggr\}

~\Rightarrow ~~~ \frac{A_s}{a_\ell a_m a_s}

~=

~
\frac{ 2}{a_\ell^3 }
\biggl[\frac{
(a_m/a_s) \sin\alpha
- E(\alpha, p)}{ (1-p^2) \sin^3\alpha } 
\biggr] \, .

This matches the expression that we have provided for ~A_3, above in the context of triaxial configurations.

When am = a

When the length of the intermediate axis is the same as the length of the longest axis — that is, when we are dealing with a prolate spheroid — the coefficient associated with the longest axis is,

~\frac{A_\ell}{a_\ell^2 a_s}

~=


~\int_0^\infty \frac{du}{ (a_\ell^2 + u)^2(a_s^2 + u)^{1 / 2} } \, .

Changing the integration variable to ~x \equiv (a_\ell^2 + u), we obtain an integral expression that appears as equation (2.228.1) in GR7th, namely,

~\frac{A_\ell}{a_\ell^2 a_s}

~=


~\int_{a_\ell^2}^\infty \frac{dx}{ x^2(a_s^2 - a_\ell^2 + x)^{1 / 2} }

 

~=


- \biggl[ \frac{\sqrt{a_s^2 - a_\ell^2 + x}}{(a_s^2 - a_\ell^2) x}\biggr]_{a_\ell^2}^\infty - \frac{1}{2(a_s^2 - a_\ell^2)}
~\int_{a_\ell^2}^\infty \frac{dx}{ x(a_s^2 - a_\ell^2 + x)^{1 / 2} }

 

~=

-
\frac{a_s}{(a_\ell^2 - a_s^2 ) a_\ell^2} + \frac{1}{2(a_\ell^2 - a_s^2)}
~\int_{a_\ell^2}^\infty \frac{dx}{ x(a_s^2 - a_\ell^2 + x)^{1 / 2} } \, .

The remaining integral in this expression appears as equation (2.224.5) in GR7th. Its resolution depends on the sign of the constant term in the denominator, ~(a_s^2 - a_\ell^2). Given that this term is negative, the integration gives,

~\frac{A_\ell}{a_\ell^2 a_s}

~=

~-
\frac{a_s}{(a_\ell^2 - a_s^2) a_\ell^2} + \frac{1}{2(a_\ell^2 - a_s^2)}
~\biggl\{
\frac{2}{ (a_\ell^2 - a_s^2)^{1 / 2} } \tan^{-1}\bigg[ \frac{(a_s^2 - a_\ell^2 + x)^{1 / 2} }{ (a_\ell^2 - a_s^2)^{1 / 2}} \biggr]
\biggr\}_{a_\ell^2}^\infty

~\Rightarrow~~~A_\ell

~=

~-
\frac{a_s^2}{(a_\ell^2 - a_s^2) } + 
~\frac{a_\ell^2 a_s}{ (a_\ell^2 - a_s^2)^{3 / 2} } 
\biggl\{\frac{\pi}{2} -
\tan^{-1}\bigg[ \frac{a_s }{ (a_\ell^2 - a_s^2)^{1 / 2}} \biggr]
\biggr\}

 

~=

~-
\frac{(1-e^2)}{e^2} + 
~\frac{(1-e^2)^{1 / 2}}{ e^3 } 
\biggl\{\frac{\pi}{2} -
\tan^{-1}\bigg[ \frac{ (1-e^2)^{1 / 2}}{ e} \biggr]
\biggr\}

 

~=

~-
\frac{(1-e^2)}{e^2} + 
~\frac{(1-e^2)^{1 / 2}}{ e^3 } \biggl\{\frac{\pi}{2} -
\cos^{-1}e
\biggr\}

 

~=

~-
\frac{(1-e^2)}{e^2} + 
~\frac{(1-e^2)^{1 / 2}}{ e^3 } \biggl\{
\sin^{-1}e
\biggr\} \, ,

where, ~e \equiv (1 - a_s^2/a_\ell^2)^{1 / 2}. Similarly, the coefficient associated with the shortest axis is,

~\frac{A_s}{a_\ell^2 a_s}

~=


~\int_0^\infty \frac{du}{ (a_\ell^2 + u)(a_s^2 + u)^{3 / 2} } \, .

This time, after changing the integration variable to ~x \equiv (a_\ell^2 + u), we obtain an integral expression that appears as equation (2.229.1) in GR7th, namely,

~\frac{A_s}{a_\ell^2 a_s}

~=


~\int_{a_\ell^2}^\infty \frac{dx}{ x(a_s^2 - a_\ell^2 + x)^{3 / 2} }

 

~=


\biggl[ \frac{2}{(a_s^2 - a_\ell^2) \sqrt{a_s^2 - a_\ell^2 + x}}\biggr]_{a_\ell^2}^\infty 
+ \frac{1}{(a_s^2 - a_\ell^2)}
~\int_{a_\ell^2}^\infty \frac{dx}{ x(a_s^2 - a_\ell^2 + x)^{1 / 2} } \, .

As before, the remaining integral in this expression appears as equation (2.224.5) in GR7th; and, as before, the sign of the constant term in the denominator, ~(a_s^2 - a_\ell^2), is negative. Hence, the integration gives,

~\frac{A_s}{a_\ell^2 a_s}

~=


\frac{2}{(a_\ell^2 - a_s^2) a_s} 
- \frac{1}{(a_\ell^2 - a_s^2)}
~\biggl\{
\frac{2}{ (a_\ell^2 - a_s^2)^{1 / 2} } \tan^{-1}\bigg[ \frac{(a_s^2 - a_\ell^2 + x)^{1 / 2} }{ (a_\ell^2 - a_s^2)^{1 / 2}} \biggr]
\biggr\}_{a_\ell^2}^\infty

~\Rightarrow ~~~ A_s

~=


\frac{2a_\ell^2 a_s}{(a_\ell^2 - a_s^2) a_s} 
- \frac{2a_\ell^2 a_s}{(a_\ell^2 - a_s^2)^{3 / 2}}
\biggl\{\frac{\pi}{2} -
\tan^{-1}\bigg[ \frac{a_s }{ (a_\ell^2 - a_s^2)^{1 / 2}} \biggr]
\biggr\}

 

~=


\frac{2}{e^2} 
- \frac{2(1-e^2)^{1 / 2}}{e^3}
\biggl\{\frac{\pi}{2} -
\tan^{-1}\bigg[ \frac{ (1-e^2)^{1 / 2}}{ e} \biggr]
\biggr\}

 

~=


\frac{2}{e^2} 
- \frac{2(1-e^2)^{1 / 2}}{e^3}
\biggl\{\sin^{-1}e \biggr\} \, .

Because we are evaluating the case where ~A_m = A_\ell, we alternatively should have been able to obtain the expression for ~A_s immediately from our derived expression for ~A_\ell via the known relation,

~2

~=

~A_\ell + A_m + A_s = 2A_\ell + A_s \, .

This approach gives,

~A_s

~=

~2 - 2A_\ell

 

~=

~2 + 2\biggl\{
\frac{(1-e^2)}{e^2} - 
~\frac{(1-e^2)^{1 / 2}}{ e^3 } \biggl[
\sin^{-1}e
\biggr] 
\biggr\}

 

~=

~
\frac{2}{e^2} - 
~\frac{2(1-e^2)^{1 / 2}}{ e^3 } \biggl[
\sin^{-1}e
\biggr] \, ,

which, indeed, matches our separately derived expression for ~A_s.

When am = as

When the length of the intermediate axis is the same as the length of the shortest axis — that is, when we are dealing with an oblate spheroid — the coefficient associated with the longest axis is,

~\frac{A_\ell}{a_\ell a_s^2}

~=


~\int_0^\infty \frac{du}{ (a_s^2 + u)(a_\ell^2 + u)^{3 / 2}  } \, .

Changing the integration variable to ~x \equiv (a_s^2 + u), we obtain an integral expression that appears as equation (2.229.1) in GR7th, namely,

~\frac{A_\ell}{a_\ell a_s^2}

~=


~\int_{a_s^2}^\infty \frac{dx}{ x^2(a_\ell^2 - a_s^2 + x)^{1 / 2} }

~

~=

~
\biggl[ \frac{2}{(a_\ell^2 - a_s^2) (a_\ell^2 - a_s^2 + x)^{1 / 2}} \biggr]_{a_s^2}^\infty
+
\frac{1}{(a_\ell^2 - a_s^2)} \int_{a_s^2}^\infty \frac{dx}{ x (a_\ell^2 - a_s^2 + x)^{1 / 2} } \, .

The remaining integral in this expression appears as equation (2.224.5) in GR7th. Its resolution depends on the sign of the constant term in the denominator, ~(a_\ell^2 - a_s^2). Given that this term is positive, the integration gives,

~\frac{A_\ell}{a_\ell a_s^2}

~=

~
- \frac{2}{(a_\ell^2 - a_s^2) a_\ell} 
+
\frac{1}{(a_\ell^2 - a_s^2)} 
\biggl\{
\frac{1}{\sqrt{(a_\ell^2 - a_s^2)}} \ln \biggl[ \frac{ (a_\ell^2 - a_s^2 + x)^{1 / 2} - \sqrt{(a_\ell^2 - a_s^2)} }{(a_\ell^2 - a_s^2 + x)^{1 / 2} + \sqrt{(a_\ell^2 - a_s^2)}  } \biggr]
\biggr\}_{a_s^2}^\infty

~\Rightarrow ~~~ A_\ell

~=

~
- \frac{2a_\ell a_s^2}{(a_\ell^2 - a_s^2) a_\ell} 
-
\frac{a_\ell a_s^2}{(a_\ell^2 - a_s^2)^{3 / 2}} 
\biggl\{
\ln \biggl[ \frac{ 1-e }{1+e}   \biggr]
\biggr\}

 

~=

~
\frac{ (1-e^2)}{e^3} 
\biggl\{
\ln \biggl[ \frac{1+e}{ 1-e }   \biggr]
\biggr\}
- \frac{2(1-e^2)}{e^2 } 
\, ,

where, as above, ~e \equiv (1 - a_s^2/a_\ell^2)^{1 / 2}. Now, given that ~A_m = A_s, in this case we appreciate that,

~2

~=

~A_\ell + A_m + A_s = A_\ell + 2A_s

~\Rightarrow ~~~ A_s

~=

~1 - \frac{A_\ell}{2}

 

~=

~1 - \frac{1}{2}\biggl[ \frac{ (1-e^2)}{e^3} 
\cdot
\ln \biggl( \frac{1+e}{ 1-e }   \biggr)
- \frac{2(1-e^2)}{e^2 } 
 \biggr]

 

~=

~
\frac{1}{e^2 } 
- \frac{ (1-e^2)}{2e^3} 
\cdot
\ln \biggl( \frac{1+e}{ 1-e }   \biggr) \, .

Inhomogeneous Ellipsoids Leading to Ferrers Potentials

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 ~(x_1, x_2, x_3) satisfy the condition that,

~m^2

~\equiv

~a_1^2 \sum_{i=1}^{3} \frac{x_i^2}{a_i^2} \, ,

[ EFE, Chapter 3, §20, p. 50, Eq. (75) ]
[ BT87, §2.3.2, p. 61, Eq. (2-97) ]

be constant. More specifically, let's consider the case (related to the so-called Ferrers potentials) in which the configuration's density distribution is given by the expression,

~\rho(m^2)

~=

~\rho_c \biggl[1 - \frac{m^2}{a_1^2}\biggr]^n

~=

~
\rho_c \biggl[1 - 
\sum_{i=1}^{3} \frac{x_i^2}{a_i^2} \biggr]^n

~=

~
\rho_c \biggl[1 - \biggl( \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2}\biggr) \biggr]^n 
\, .

NOTE:     In our accompanying discussion of compressible analogues of Riemann S-type 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,

~\Phi_\mathrm{grav}(\bold{x})

~=

~
- \frac{\pi G \rho_c a_1 a_2 a_3}{(n+1)}  \int_0^\infty \frac{ du}{\Delta } Q^{n+1}  \, ,

[ EFE, Chapter 3, §20, p. 53, Eq. (101) ]

where, ~\Delta has the same definition as above, and,

~Q

~\equiv

~
1 - \sum_{\ell = 1}^3 \frac{x_\ell^2}{ a_\ell^2 + u }  \, .

For purposes of illustration, in what follows we will assume that, ~a_1 > a_2 > a_3.

The Case Where n = 0

When ~n = 0, we have a uniform-density configuration, and the "interior" potential will be given by the expression,

~\Phi_\mathrm{grav}(\bold{x})

~=

~
- \pi G \rho_c a_1 a_2 a_3  \int_0^\infty \frac{ du}{\Delta } \biggl[ 1 - \sum_{\ell = 1}^3 \frac{x_\ell^2}{ a_\ell^2 + u }  \biggr]

 

~=

~
- \pi G \rho_c a_1 a_2 a_3 \biggl\{  
\int_0^\infty \frac{ du}{\Delta } 
- \int_0^\infty \frac{ du}{\Delta } \biggl( \frac{x^2}{ a_1^2 + u } \biggr)
- \int_0^\infty \frac{ du}{\Delta } \biggl( \frac{y^2}{ a_2^2 + u } \biggr) 
- \int_0^\infty \frac{ du}{\Delta } \biggl( \frac{z^2}{ a_3^2 + u }  \biggr)
\biggr\}

 

~=

~
- \pi G \rho_c a_1 a_2 a_3 \biggl\{  
\int_0^\infty \frac{ du}{\Delta } 
~ - ~x^2 \int_0^\infty \frac{ du}{\Delta (a_1^2 + u) } 
~ - ~y^2 \int_0^\infty \frac{ du}{\Delta (a_2^2 + u) }  
~ - ~ \int_0^\infty \frac{ du}{\Delta (a_3^2 + u) } 
\biggr\}

 

~=

~
-\pi G \rho_c \biggl[ I_\mathrm{BT} a_1^2 - \biggl(A_1 x^2 + A_2 y^2 +A_3 z^2 \biggr) \biggr] \, .

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

Poisson Equation

\nabla^2 \Phi = 4\pi G \rho

Given that,

~\sum_{\ell = 1}^3 A_\ell

~=

~2 \, ,

[ EFE, §21, Eq. (108) ]

we find,

~\nabla^2\Phi_\mathrm{grav} = \biggl[\frac{\partial^2}{\partial x^2}  + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}\biggr]\Phi_\mathrm{grav}

~=

~
+ 2\pi G \rho_c (A_1 + A_2 + A_3) = 4\pi G\rho_c \, .

Q.E.D.

The Case Where n = 1

When ~n = 1, we have a specific heterogeneous density configuration, and the "interior" potential will be given by the expression,

~\frac{ \Phi_\mathrm{grav}(\bold{x})}{(-\pi G\rho_c)}

~=

~
\frac{1}{2} a_1 a_2 a_3  \int_0^\infty \frac{ du}{\Delta } \biggl[ 1 - \sum_{\ell = 1}^3 \frac{x_\ell^2}{ a_\ell^2 + u }  \biggr]^2

 

~=

~
\frac{1}{2} a_1 a_2 a_3  \biggl\{  
\int_0^\infty \frac{ du}{\Delta } \biggl[ 1 - \sum_{\ell = 1}^3 \frac{x_\ell^2}{ a_\ell^2 + u }  \biggr]
~- ~ x^2 \int_0^\infty \frac{ du}{\Delta (a_1^2 + u)} \biggl[ 1 - \sum_{\ell = 1}^3 \frac{x_\ell^2}{ a_\ell^2 + u }  \biggr]

 

 

~
~- ~y^2 \int_0^\infty \frac{ du}{\Delta (a_2^2 + u)}  \biggl[ 1 - \sum_{\ell = 1}^3 \frac{x_\ell^2}{ a_\ell^2 + u }  \biggr]
~ - ~z^2 \int_0^\infty \frac{ du}{\Delta (a_3^2 + u)} \biggl[ 1 - \sum_{\ell = 1}^3 \frac{x_\ell^2}{ a_\ell^2 + u }  \biggr]
\biggr\} \, .

The first definite-integral expression inside the curly braces is, to within a leading factor of ~\tfrac{1}{2}, identical to the entire expression for the normalized potential that was derived in the case where n = 0. That is, we can write,

~\frac{ \Phi_\mathrm{grav}(\bold{x})}{(-\pi G\rho_c)}

~=

~
\frac{1}{2} \biggl[ I_\mathrm{BT} a_1^2 - \biggl(A_1 x^2 + A_2 y^2 +A_3 z^2 \biggr) \biggr]
~- \frac{1}{2} a_1 a_2 a_3  \biggl\{ ~ x^2 \int_0^\infty \frac{ du}{\Delta (a_1^2 + u)} \biggl[ 1 - \sum_{\ell = 1}^3 \frac{x_\ell^2}{ a_\ell^2 + u }  \biggr]

 

 

~
~+~y^2 \int_0^\infty \frac{ du}{\Delta (a_2^2 + u)}  \biggl[ 1 - \sum_{\ell = 1}^3 \frac{x_\ell^2}{ a_\ell^2 + u }  \biggr]
~ + ~z^2 \int_0^\infty \frac{ du}{\Delta (a_3^2 + u)} \biggl[ 1 - \sum_{\ell = 1}^3 \frac{x_\ell^2}{ a_\ell^2 + u }  \biggr]
\biggr\} \, .

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

~a_1 a_2 a_3 \int_0^\infty \frac{ du}{\Delta (a_i^2 + u)}  \biggl[ 1 - \sum_{\ell = 1}^3 \frac{x_\ell^2}{ a_\ell^2 + u }  \biggr]

~=

~
\biggl( A_i - \sum_{\ell=1}^3 A_{i\ell} x_\ell^2 \biggr) \, .

[ EFE, Chapter 3, §22, p. 53, Eq. (125) ]

Applying this result to each of the other three definite integrals gives us,

~\frac{ \Phi_\mathrm{grav}(\bold{x})}{(-\pi G\rho_c)}

~=

~
\frac{1}{2} \biggl[ I_\mathrm{BT} a_1^2 - \biggl(A_1 x^2 + A_2 y^2 +A_3 z^2 \biggr) \biggr]
~- \frac{x^2}{2}  \biggl( A_1 - \sum_{\ell=1}^3 A_{1\ell} x_\ell^2 \biggr)
~- \frac{y^2}{2}  \biggl( A_2 - \sum_{\ell=1}^3 A_{2\ell} x_\ell^2 \biggr)
~- \frac{z^2}{2}  \biggl( A_3 - \sum_{\ell=1}^3 A_{3\ell} x_\ell^2 \biggr) \, .

 

~=

~
\frac{1}{2} \biggl[ I_\mathrm{BT} a_1^2 - \biggl(A_1 x^2 + A_2 y^2 +A_3 z^2 \biggr) \biggr]
~- \frac{x^2}{2}  \biggl[ A_1 - \biggl( A_{11}x^2 + A_{12}y^2 + A_{13}z^2 \biggr) \biggr]

 

 

~
~- \frac{y^2}{2}  \biggl[ A_2 - \biggl( A_{21}x^2 + A_{22}y^2 + A_{23}z^2 \biggr) \biggr]
~- \frac{z^2}{2}  \biggl[ A_3 - \biggl( A_{31}x^2 + A_{32}y^2 + A_{33}z^2 \biggr) \biggr]

 

~=

~
\frac{1}{2} I_\mathrm{BT} a_1^2 
- \biggl(A_1 x^2 + A_2 y^2 +A_3 z^2 \biggr) 
~+ \biggl( A_{12} x^2y^2 + A_{13} x^2z^2 + A_{23} y^2z^2\biggr)
~+ \frac{1}{2}  \biggl(A_{11}x^4 +  A_{22}y^4 + A_{33}z^4  \biggr) 
\, ,

where,

~A_{ij}

~\equiv

~-\frac{A_i-A_j}{(a_i^2 - a_j^2)} \, ,

[ EFE, §21, Eq. (107) ]

and we have made use of the symmetry relation, ~A_{ij} = A_{ji}. Again, as a check, let's see if this scalar potential satisfies the differential form of the

Poisson Equation

\nabla^2 \Phi = 4\pi G \rho

We find,

~\nabla^2 \biggl[ \frac{\Phi_\mathrm{grav}}{-2\pi G \rho_c} \biggr]

~=

~
\frac{1}{2}\biggl[\frac{\partial^2}{\partial x^2}  + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}\biggr] 
\biggl[- \biggl(A_1 x^2 + A_2 y^2 +A_3 z^2 \biggr) 
~+ \biggl( A_{12} x^2y^2 + A_{13} x^2z^2 + A_{23} y^2z^2\biggr)
~+ \frac{1}{2}  \biggl(A_{11}x^4 +  A_{22}y^4 + A_{33}z^4  \biggr) 
\biggr]

 

~=

~
\frac{\partial}{\partial x}   \biggl[- A_1 x  ~+ A_{12} x y^2 + A_{13} x z^2 ~+ A_{11}x^3  \biggr]
+\frac{\partial}{\partial y}  \biggl[- A_2 y   ~+  A_{12} x^2y  + A_{23} y z^2~+ A_{22}y^3  \biggr]
+ \frac{\partial}{\partial z}\biggl[- A_3 z  ~+ A_{13} x^2z + A_{23} y^2z~+ A_{33}z^3  \biggr]

 

~=

~
\biggl[- A_1  + A_{12} y^2 + A_{13} z^2 ~+ 3A_{11}x^2  \biggr]
+ \biggl[- A_2 +  A_{12} x^2  + A_{23} z^2~+ 3A_{22}y^2  \biggr]
+ \biggl[- A_3  + A_{13} x^2 + A_{23} y^2~+ 3A_{33}z^2  \biggr]

 

~=

~ - (A_1 + A_2 + A_3) + x^2(3A_{11} + A_{12} + A_{13}) + y^2( 3A_{22} + A_{12} + A_{23}) + z^2( 3A_{33} + A_{13} + A_{23})

~\sum_{\ell = 1}^3 A_\ell

~=

~2 \, ,

[ EFE, §21, Eq. (108) ]



Material that appears after this point in our presentation is under development and therefore
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Derivation of Expression for Gravitational Potential

In §373 (p. 700) of his book titled, Hydrodynamics, Lamb32 states that, "The gravitation-potential, at internal points, of a uniform mass enclosed by the surface

~
\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2}

~=

~1

[ Lamb32, §373, Eq. (1) ]

… may be written

~\frac{\Phi(\vec{x})}{G}

~=

~
\pi \rho (\alpha_0 x^2 + \beta_0 y^2 + \gamma_0 z^2 - \chi_0) \, ,

[ Lamb32, §373, Eq. (4) ]

where, as in §114,"

~\frac{\alpha_0}{abc}

~=

~\int_0^\infty \frac{d\lambda}{(a^2 + \lambda)\Delta} \, ,

     

~\frac{\beta_0}{abc}

~=

~\int_0^\infty \frac{d\lambda}{(b^2 + \lambda)\Delta} \, ,

~\frac{\gamma_0}{abc}

~=

~\int_0^\infty \frac{d\lambda}{(c^2 + \lambda)\Delta} \, ,

     

~\frac{\chi_0}{abc}

~=

~\int_0^\infty \frac{d\lambda}{\Delta} \, ,

[ Lamb32, §373, Eqs. (5) & (6) ]

and,

~\Delta

~=

~[(a^2 + \lambda)(b^2 + \lambda)(c^2 + \lambda)]^{1 / 2} \, .

[ 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 ~x-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 prolate-spheroidal configuration such that its major axis and, hence, its axis of symmetry aligns with the ~z-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,

~\frac{\varpi^2}{a_3^2} + \frac{z^2}{a_1^2} = 1 ~~~~\Rightarrow ~~~~ \varpi = a_3\sqrt{1-z^2/a_1^2} \, ,

and the gravitational potential will be given by the expression,


~\Phi(\vec{x}) = -\pi G \rho \biggl[ I_\mathrm{BT} a_1^2 - \biggl(A_1 z^2 + A_3 \varpi^2 \biggr) \biggr].

The magnitude of the gravitational acceleration at the pole ~(\varpi, z) = (0, a_1) of this prolate spheroid can be obtained from the gravitational potential via the expression,

~\mathcal{A} \equiv \biggl|- \frac{\partial \Phi}{\partial z}\biggr|_{a_1}

~=

~2\pi G \rho A_1 a_1 \, ,

where, as above,


~A_1


~=


\ln\biggl[ \frac{1+e}{1-e} \biggr] \frac{(1-e^2)}{e^3} - \frac{2(1-e^2)}{e^2} \, .

We should also be able to derive this expression for ~\mathcal{A} by integrating the ~z-component of the differential acceleration over the mass distribution, that is,

~\mathcal{A}

~=

~\int \biggl[ \frac{G }{r^2} \cdot \frac{(a_1-z)}{r} \biggr] dm = \int \biggl[ \frac{(a_1-z)G }{r^3} \biggr] 2\pi \varpi d\varpi dz

 

~=

~2\pi G\rho \int^{a_1}_{-a_1} (a_1-z)dz \int_0^{a_3\sqrt{1-z^2/a_1^2}} [\varpi^2+(z-a_1)^2]^{-3/2}\varpi d\varpi \, ,

where the distance, ~r, has been measured from the pole, that is,

~r^2 = \varpi^2 + (z-a_1)^2 \, .

Performing the integral over ~\varpi gives,

~\mathcal{A}

~=

~2\pi G\rho \int^{a_1}_{-a_1} (a_1-z)dz \biggl\{ -[\varpi^2+(z-a_1)^2]^{-1/2} \biggr\}_0^{a_3\sqrt{1-z^2/a_1^2}}

 

~=

~2\pi G\rho \int^{a_1}_{-a_1} (a_1-z)dz \biggl\{ \frac{1}{z - a_1} -\biggl[ a_3^2 \biggl(1-\frac{z^2}{a_1^2} \biggr)
+ a_1^2\biggl(1-\frac{z}{a_1}\biggr)^2 \biggr]^{-1/2} 
\biggr\}

 

~=

~ - 2\pi G\rho a_1 \int^{1}_{-1}  d\zeta \biggl\{ \frac{1-\zeta}{1-\zeta } - (1-\zeta)\biggl[ \biggl(\frac{a_3}{a_1}\biggr)^2 \biggl(1-\zeta^2 \biggr)
+ \biggl(1-\zeta\biggr)^2 \biggr]^{-1/2} 
\biggr\}

 

~=

~ 2\pi G\rho a_1 \int^{1}_{-1}  d\zeta \biggl\{ 
(1-\zeta) [ (2-e^2) - 2\zeta + e^2\zeta^2 ]^{-1/2} -1
\biggr\} \, ,

where, ~\zeta\equiv z/a_1. For later reference, we will identify the expression inside the curly braces as the function, ~\mathcal{Z}; specifically,

~\mathcal{Z}

~\equiv

~(1-\zeta) [ (2-e^2) - 2\zeta + e^2\zeta^2 ]^{-1/2} -1

 

~=

~- 1 - \frac{\zeta}{\sqrt{X}} + \frac{1}{\sqrt{X}}  \, ,

where, in an effort to line up with notation found in integral tables, in this last expression we have used the notation, ~X \equiv a + b\zeta + c\zeta^2 and, in our case,

a \equiv (2-e^2)\, ,       b \equiv -2\, ,       and       c \equiv e^2\, .

We find that,

~\int_{-1}^1 \mathcal{Z} d\zeta

~=

~- \zeta\biggr|_{-1}^{1} - \biggl\{ \frac{\sqrt{X}}{c} \biggr\}_{-1}^1
+\biggl[1 + \frac{b}{2c} \biggr]\int_{-1}^1 \frac{d\zeta}{\sqrt{X}}

 

~=

~- 2 - \biggl\{ \frac{\sqrt{(2-e^2) -2\zeta + e^2\zeta^2}}{e^2} \biggr\}_{-1}^1
+\biggl[1 - \frac{1}{e^2} \biggr] \biggl\{ \frac{1}{\sqrt{c}} \ln \biggl[2\sqrt{cX} + 2c\zeta + b  \biggr] \biggr\}_{-1}^1

 

~=

~- 2 - \biggl\{ \frac{\sqrt{(2-e^2) -2 + e^2}}{e^2} \biggr\} + \biggl\{ \frac{\sqrt{(2-e^2) +2 + e^2}}{e^2} \biggr\}
+ \biggl[1 - \frac{1}{e^2} \biggr] \biggl\{ \frac{1}{e} \ln \biggl[2\sqrt{e^2[(2-e^2) -2\zeta + e^2\zeta^2]} + 2e^2\zeta - 2  \biggr] \biggr\}_{-1}^1

 

~=

~- 2 + \frac{2}{e^2} 
+\biggl[\frac{e^2-1}{e^3} \biggr] \biggl\{ \ln \biggl[2e^2 - 2  \biggr] 
- \ln \biggl[4e - 2e^2 - 2  \biggr] \biggr\}

 

~=

~- 2\biggl[\frac{e^2 - 1}{e^2}\biggr]  
+\biggl[\frac{e^2-1}{e^3} \biggr] \biggl\{ \ln \biggl[-2(1-e^2)  \biggr] 
- \ln \biggl[-2(1-e)^2\biggr] \biggr\}

 

~=

~\biggl[\frac{1-e^2}{e^3} \biggr] \ln \biggl[\frac{1+e}{1-e} \biggr] -2\biggl[\frac{1-e^2 }{e^2}\biggr]

 

~=

~A_1 \, .

Hence, we have,

~\mathcal{A} = 2\pi G\rho a_1  \biggl[ \int_{-1}^1 \mathcal{Z} d\zeta\biggr]= 2\pi G \rho A_1 a_1 \, ,

which exactly matches the result obtained, above, by taking the derivative of the potential.

See Also

Footnotes

  1. In EFE this equation is written in terms of a variable ~I instead of ~I_\mathrm{BT} as defined here. The two variables are related to one another straightforwardly through the expression, ~I = I_\mathrm{BT} a_1^2.
  2. Throughout EFE, Chandrasekhar adopts a sign convention for the scalar gravitational potential that is opposite to the sign convention being used here.


Whitworth's (1981) Isothermal Free-Energy Surface

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