Revision as of 18:48, 22 June 2016

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,

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

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

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>,

<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>
<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>
<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>
<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>~\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>
		<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>
	<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>
		<math>~- k^2E(\theta,k) + E(\theta,k) - (1-k^2)F(\theta,k) - (a_3/a_2)k^2\sin\theta\biggr\}</math>
	<math>~=</math>	<math>~ \frac{1}{(1-k^2)\sin^2\theta} \biggl[ \frac{a_2}{a_3} - \frac{a_3}{a_2} \biggr]</math>
	<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,

<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>
<math> ~A_2 </math>	<math> ~= </math>	<math> ~A_1 ~~; </math>
<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>
<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>

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

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,

<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>
<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>
<math> ~A_3 </math>	<math> ~= </math>	<math> A_2 ~~; </math>
<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
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>
<math>~\frac{a_2}{a_1}</math>	<math>~\frac{a_3}{a_1}</math>	radians	degrees	<math>~k</math>	radians	degrees	<math>~F(\theta,k)</math>	<math>~E(\theta,k)</math>	<math>~A_1</math>	<math>~A_2</math>	<math>~A_3</math>
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

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

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,

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

where, as above,

<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,

<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>
	<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,

<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>
	<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>
	<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>
	<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,

<math>~\mathcal{Z}</math>	<math>~\equiv</math>	<math>~(1-\zeta) [ (2-e^2) - 2\zeta + e^2\zeta^2 ]^{-1/2} -1</math>
	<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,

<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>
	<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>
	<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>
	<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>
	<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>
	<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>
	<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.

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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