User:Tohline/ThreeDimensionalConfigurations/HomogeneousEllipsoids
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Properties of Homogeneous Ellipsoids (1)
Gravitational Potential
The Defining Integral Expressions
As has been shown in a separate discussion (not yet typed!), the acceleration due to the gravitational attraction of a distribution of mass <math>~\rho</math><math>(\vec{x})</math> can be derived from the gradient of a scalar potential <math>~\Phi</math><math>(\vec{x})</math> defined as follows:
<math> \Phi(\vec{x}) \equiv - \int \frac{G \rho(\vec{x}')}{|\vec{x}' - \vec{x}|} d^3 x' . </math>
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 <math>~(x,y,z) = (a_1,a_2,a_3)</math>,
<math>
~\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],
</math>
[ EFE, Chapter 3, Eq. (40)1,2 ]
[ BT87, Chapter 2, Table 2-2 ]
where,
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<math> ~A_i </math> |
<math> ~\equiv </math> |
<math> ~a_1 a_2 a_3 \int_0^\infty \frac{du}{\Delta (a_i^2 + u )} , </math> |
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<math> ~I_\mathrm{BT} </math> |
<math> ~\equiv </math> |
<math> ~\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 , </math> |
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<math> ~\Delta </math> |
<math> ~\equiv </math> |
<math> ~\biggl[ (a_1^2 + u)(a_2^2 + u)(a_3^2 + u) \biggr]^{1/2} . </math> |
Evaluation of Coefficients
The integrals defining <math>~A_i</math> and <math>~I_\mathrm{BT}</math> can be evaluated in terms of the incomplete elliptic integral of the first kind,
<math> ~F(\theta,k) \equiv \int_0^\theta \frac{d\theta '}{\sqrt{1 - k^2 \sin^2\theta '}} ~~ , </math>
and/or the incomplete elliptic integral of the second kind,
<math> E(\theta,k) \equiv \int_0^\theta {\sqrt{1 - k^2 \sin^2\theta '}}~d\theta ' ~~ , </math>
where, for our particular problem,
<math>
~\theta \equiv \cos^{-1} \biggl(\frac{a_3}{a_1} \biggr) ,
</math>
<math>
~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},
</math>
[ EFE, Chapter 3, Eq. (32) ]
or the integrals can be evaluated in terms of more elementary functions if either <math>~a_2 = a_1</math> (oblate spheroids) or <math>~a_3 = a_2</math> (prolate spheroids).
Triaxial Configurations <math>~(a_1 > a_2 > a_3)</math>
If the three principal axes of the configuration are unequal in length and related to one another such that <math>~a_1 > a_2 > a_3 </math>,
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<math> ~A_1 </math> |
<math> ~= </math> |
<math> ~\frac{2a_2 a_3}{a_1^2} \biggl[ \frac{F(\theta,k) - E(\theta,k)}{k^2 \sin^3\theta} \biggr] ~~; </math> |
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<math> ~A_2 </math> |
<math> ~= </math> |
<math> ~\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] ~~; </math> |
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<math> ~A_3 </math> |
<math> ~= </math> |
<math> ~\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] ~~; </math> |
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<math> ~I_\mathrm{BT} </math> |
<math> ~= </math> |
<math> ~\frac{2a_2 a_3}{a_1^2} \biggl[ \frac{F(\theta,k)}{\sin\theta} \biggr] ~~. </math> |
[ EFE, Chapter 3, Eqs. (33), (34) & (35) ]
Notice that there is no need to specify the actual value of <math>~a_1</math> in any of these expressions, as they each can be written in terms of the pair of axis ratios, <math>~a_2/a_1</math> and <math>~a_3/a_1</math>. 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,
<math>~\sum_{\ell=1}^3 A_\ell = 2 \, .</math>
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<math>~\frac{a_1^2}{2a_2 a_3} \biggl[A_1 + A_3 + A_2\biggr]</math> |
<math>~=</math> |
<math>~ \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} </math> |
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<math>~+ \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}</math> |
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<math>~=</math> |
<math>~ \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 </math> |
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<math>~- k^2E(\theta,k) + E(\theta,k) - (1-k^2)F(\theta,k) - (a_3/a_2)k^2\sin\theta\biggr\}</math> |
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<math>~=</math> |
<math>~ \frac{1}{(1-k^2)\sin^2\theta} \biggl[ \frac{a_2}{a_3} - \frac{a_3}{a_2} \biggr]</math> |
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<math>~=</math> |
<math>~ \frac{a_1^2}{a_2 a_3} \, .</math> |
Q.E.D.
Oblate Spheroids <math>~(a_1 = a_2 > a_3)</math>
If the longest axis, <math>~a_1</math>, and the intermediate axis, <math>~a_2</math>, of the ellipsoid are equal to one another, then an equatorial cross-section of the object presents a circle of radius <math>~a_1</math> and the object is referred to as an oblate spheroid. For homogeneous oblate spheroids, evaluation of the integrals defining <math>~A_i</math> and <math>~I_\mathrm{BT}</math> gives,
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<math> ~A_1 </math> |
<math> ~= </math> |
<math> ~\frac{1}{e^2} \biggl[ \frac{\sin^{-1}e}{e} - (1-e^2)^{1/2} \biggr] (1-e^2)^{1/2} ~~; </math> |
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<math> ~A_2 </math> |
<math> ~= </math> |
<math> ~A_1 ~~; </math> |
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<math> ~A_3 </math> |
<math> ~= </math> |
<math> ~\frac{2}{e^2} \biggl[ (1-e^2)^{-1/2} - \frac{\sin^{-1}e}{e} \biggr] (1-e^2)^{1/2} ~~; </math> |
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<math> ~I_\mathrm{BT} </math> |
<math> ~= </math> |
<math> ~2A_1 + A_3 (1-e^2) = 2 (1-e^2)^{1/2} \biggl[ \frac{\sin^{-1}e}{e} \biggr] ~~, </math> |
where the eccentricity,
<math> ~e \equiv \biggl[1 - \biggl(\frac{a_3}{a_1}\biggr)^2 \biggr]^{1/2} ~~. </math>
Prolate Spheroids <math>~(a_1 > a_2 = a_3)</math>
If the shortest axis <math>~(a_3)</math> and the intermediate axis <math>~(a_2)</math> of the ellipsoid are equal to one another, then a cross-section in the <math>~x-y</math> plane of the object presents a circle of radius <math>~a_3</math> and the object is referred to as a prolate spheroid. For homogeneous prolate spheroids, evaluation of the integrals defining <math>~A_i</math> and <math>~I_\mathrm{BT}</math> gives,
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<math> ~A_1 </math> |
<math> ~= </math> |
<math> \ln\biggl[ \frac{1+e}{1-e} \biggr] \frac{(1-e^2)}{e^3} - \frac{2(1-e^2)}{e^2} ~~; </math> |
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<math> ~A_2 </math> |
<math> ~= </math> |
<math> \frac{1}{e^2} - \ln\biggl[ \frac{1+e}{1-e} \biggr]\frac{(1-e^2)}{2e^3} ~~; </math> |
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<math> ~A_3 </math> |
<math> ~= </math> |
<math> A_2 ~~; </math> |
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<math> ~I_\mathrm{BT} </math> |
<math> ~= </math> |
<math>~ A_1 + 2(1-e^2)A_2 = \ln\biggl[ \frac{1+e}{1-e} \biggr]\frac{(1-e^2)}{e} ~~, </math> |
[ EFE, Chapter 3, Eq. (38) ]
where, again, the eccentricity,
<math> ~e \equiv \biggl[1 - \biggl(\frac{a_3}{a_1}\biggr)^2 \biggr]^{1/2} ~~. </math>
Example Evaluations
| Table 1: Example Evaluations | |||||||||||
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| Given | Determined using calculator and (crude) CRC tables of elliptic integrals | ||||||||||
| <math>~\frac{a_2}{a_1}</math> | <math>~\frac{a_3}{a_1}</math> | <math>~\theta</math> | <math>~k</math> | <math>~\sin^{-1}k</math> | <math>~F(\theta,k)</math> | <math>~E(\theta,k)</math> | <math>~A_1</math> | <math>~A_2</math> | <math>~A_3</math> | ||
| 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 |
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Table 2: 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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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 <math>~x</math>-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 <math>~z</math>-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,
<math>~\frac{\varpi^2}{a_3^2} + \frac{z^2}{a_1^2} = 1 ~~~~\Rightarrow ~~~~ \varpi = a_3\sqrt{1-z^2/a_1^2} \, ,</math>
and the gravitational potential will be given by the expression,
<math> ~\Phi(\vec{x}) = -\pi G \rho \biggl[ I_\mathrm{BT} a_1^2 - \biggl(A_1 z^2 + A_3 \varpi^2 \biggr) \biggr]. </math>
The magnitude of the gravitational acceleration at the pole <math>~(\varpi, z) = (0, a_1)</math> of this prolate spheroid can be obtained from the gravitational potential via the expression,
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<math>~\mathcal{A} \equiv \biggl|- \frac{\partial \Phi}{\partial z}\biggr|_{a_1}</math> |
<math>~=</math> |
<math>~2\pi G \rho A_1 a_1 \, ,</math> |
where, as above,
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<math> ~A_1 </math> |
<math> ~= </math> |
<math> \ln\biggl[ \frac{1+e}{1-e} \biggr] \frac{(1-e^2)}{e^3} - \frac{2(1-e^2)}{e^2} \, . </math> |
We should also be able to derive this expression for <math>~\mathcal{A}</math> by integrating the <math>~z</math>-component of the differential acceleration over the mass distribution, that is,
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<math>~\mathcal{A}</math> |
<math>~=</math> |
<math>~\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</math> |
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<math>~=</math> |
<math>~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 \, ,</math> |
where the distance, <math>~r</math>, has been measured from the pole, that is,
<math>~r^2 = \varpi^2 + (z-a_1)^2 \, .</math>
Performing the integral over <math>~\varpi</math> gives,
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<math>~\mathcal{A}</math> |
<math>~=</math> |
<math>~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}} </math> |
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<math>~=</math> |
<math>~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\} </math> |
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<math>~=</math> |
<math>~ - 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\} </math> |
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<math>~=</math> |
<math>~ 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\} \, , </math> |
where, <math>~\zeta\equiv z/a_1</math>. For later reference, we will identify the expression inside the curly braces as the function, <math>~\mathcal{Z}</math>; specifically,
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<math>~\mathcal{Z}</math> |
<math>~\equiv</math> |
<math>~(1-\zeta) [ (2-e^2) - 2\zeta + e^2\zeta^2 ]^{-1/2} -1</math> |
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<math>~=</math> |
<math>~- 1 - \frac{\zeta}{\sqrt{X}} + \frac{1}{\sqrt{X}} \, ,</math> |
where, in an effort to line up with notation found in integral tables, in this last expression we have used the notation, <math>~X \equiv a + b\zeta + c\zeta^2</math> and, in our case,
<math>a \equiv (2-e^2)\, ,</math> <math>b \equiv -2\, ,</math> and <math>c \equiv e^2\, .</math>
We find that,
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<math>~\int_{-1}^1 \mathcal{Z} d\zeta</math> |
<math>~=</math> |
<math>~- \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}} </math> |
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<math>~=</math> |
<math>~- 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 </math> |
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<math>~=</math> |
<math>~- 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 </math> |
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<math>~=</math> |
<math>~- 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\} </math> |
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<math>~=</math> |
<math>~- 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\} </math> |
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<math>~=</math> |
<math>~\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] </math> |
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<math>~=</math> |
<math>~A_1 \, . </math> |
Hence, we have,
<math>~\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 \, ,</math>
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 <math>I</math> instead of <math>I_\mathrm{BT}</math> as defined here. The two variables are related to one another straightforwardly through the expression, <math>I = I_\mathrm{BT} a_1^2</math>.
- Throughout EFE, Chandrasekhar adopts a sign convention for the scalar gravitational potential that is opposite to the sign convention being used here.
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© 2014 - 2021 by Joel E. Tohline |