User:Tohline/Apps/Wong1973Potential

Wong's (1973) Analytic Potential

Drawing principally from the published works of C.-Y. Wong (1973, Annals of Physics, 77, 279) and H. S. Cohl & J. E. Tohline (1999, ApJ, 527, 86), in an accompanying discussion we have detailed how the gravitational potential of any finite mass distribution can be determined using toroidal coordinates, rather than, say, cartesian, spherical or cylindrical coordinates. Specifically for axisymmetric systems, the relevant expression derived by Wong (1973) — involving a pair of integrals over the meridional-plane coordinates, <math>~(\eta,\theta)</math>, and a summation over all polar angle modes — is,

<math>~\Phi(\eta,\theta)</math>	<math>~=</math>	<math>~ - 2Ga^2 (\cosh\eta - \cos\theta)^{1 / 2} \sum\limits^\infty_{n=0} \epsilon_n \int d\eta^' ~\sinh\eta^'~P^0_{n-1 / 2}(\cosh\eta_<) ~Q^0_{n-1 / 2}(\cosh\eta_>) \int d\theta^' \biggl\{ \frac{ ~\cos[n(\theta - \theta^')]}{(\cosh\eta^' - \cos\theta^')^{5/2}} \biggr\} \rho(\eta^',\theta^') \, , </math>
Wong (1973), p. 293, Eq. (2.55)

while the expression derived by Cohl & Tohline (1999) — involving different integrand expressions and no summation — is,

<math>~\Phi(\eta,\theta)</math>	<math>~=</math>	<math>~ - 2Ga^2 \biggl[ \frac{(\cosh\eta - \cos\theta)}{\sinh\eta } \biggr]^{1 / 2} \int d\eta^' \int d\theta^' \biggl[ \frac{\sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^5} \biggr]^{1 / 2} Q_{- 1 / 2}(\Chi) \rho(\eta^',\theta^') \, , </math>
Cohl & Tohline (1999), p. 88, Eqs. (31) & (32a)

where, <math>~a</math> is the radius of the toroidal-coordinate system's anchor ring, <math>~P^0_{n-\frac{1}{2}}</math> and <math>~Q^0_{n-\frac{1}{2}}</math> are zero-order half-integer-degree associated Legendre functions of, respectively, the first and second kind, and

<math>~\Chi</math>

<math>~\equiv</math>

<math>~ \frac{\cosh\eta \cdot \cosh\eta^' - \cos(\theta^' - \theta) }{ \sinh\eta \cdot \sinh\eta^'} \, . </math>

In the subsection of our accompanying discussion titled, "Rearranging Terms and Incorporating Special-Function Relations," we have explicitly demonstrated that these two expressions lead to identical evaluations of the gravitational potential.

Here we build upon this technical foundation and detail how Wong (1973) was able to complete both integrals to derive an analytic expression for the potential (inside as well as outside) of axisymmetric, uniform-density tori having an arbitrarily specified ratio of the major to minor (cross-sectional) radii, <math>~R/d</math>. This is an outstanding accomplishment that has received little attention in the astrophysics literature and, therefore, is underappreciated.

Uniform-Density Torus

From an accompanying review, we see that for a torus of major radius <math>~R</math> and minor radius <math>~d</math>, the most accommodating (cylindrical-coordinate-based) radial coordinate location of the toroidal coordinate system's anchor ring is defined such that,

<math>~a^2</math>	<math>~\equiv</math>	<math>~R^2 - d^2 \, .</math>
Wong (1973), Eq. (2.8)

Then the corresponding (toroidal-coordinate-based) radial coordinate location <math>~\eta_0</math> of the surface of the torus is,

<math>~\eta_0</math>	<math>~=</math>	<math>~\cosh^{-1}\biggl(\frac{R}{d}\biggr) \, .</math>
Wong (1973), Eq. (2.9)

Alternatively, given <math>~\eta_0</math> and the value of the parameter <math>~a</math>, we have,

<math>~R</math>	<math>~=</math>	<math>~a \coth\eta_0 \, ,</math>
<math>~d</math>	<math>~=</math>	<math>~\frac{a}{\sinh\eta_0} \, .</math>
Wong (1973), Eqs. (2.10) & (2.11)

If such an axisymmetric torus has a uniform density, <math>~\rho_0</math>, then according to the above-cited Cohl & Tohline (1999) expression, the gravitational potential is,

<math>~\Phi_\mathrm{CT}(\eta,\theta)</math>

<math>~=</math>

<math>~ - 2G\rho_0 a^2 \biggl[ \frac{(\cosh\eta - \cos\theta)}{\sinh\eta } \biggr]^{1 / 2} \int_{\eta_0}^\infty d\eta^' \int_{-\pi}^\pi d\theta^' \biggl[ \frac{\sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^5} \biggr]^{1 / 2} Q_{- 1 / 2}(\Chi) \, . </math>

In contrast, according to Wong's (1973) expression, the gravitational potential is,

<math>~\Phi_\mathrm{W}(\eta,\theta)</math>	<math>~=</math>	<math>~ - 2G \rho_0 a^2 (\cosh\eta - \cos\theta)^{1 / 2} \sum\limits^\infty_{n=0} \epsilon_n \int_{\eta_0}^\infty d\eta^' ~\sinh\eta^'~P_{n-1 / 2}(\cosh\eta_<) ~Q_{n-1 / 2}(\cosh\eta_>) \int_{-\pi}^\pi \frac{ \cos[n(\theta - \theta^')] d\theta^' }{(\cosh\eta^' - \cos\theta^')^{5/2}} \, . </math>
Wong (1973), p. 293, Eq. (2.55)

Which Is Easier to Evaluate?

As has already been alluded to, upon evaluation, we should expect the expressions for <math>~\Phi_\mathrm{CT}</math> and for <math>~\Phi_\mathrm{W}</math> to give identical values for the gravitational potential at all coordinate locations <math>~(\eta,\theta)</math> across the entire meridional plane. But which of the two expressions is easier to evaluate?

CT

If the desire is to perform the evaluation numerically, then the <math>~\Phi_\mathrm{CT}</math> expression is almost certainly easier to contend with. It only requires a double integration; and as has been detailed in an accompanying discussion, the only relatively unfamiliar special function that appears in the integrand, <math>~Q_{-\frac{1}{2}}(\Chi)</math>, can be re-expressed in terms of (the more familiar) complete elliptic integral of the first kind, namely,

<math>~Q_{-\frac{1}{2}}(\Chi)</math>

<math>~=</math>

<math>~ \mu K(\mu) \, , </math>

where,

<math>~\mu</math>

<math>~\equiv</math>

<math>~\biggl[ \frac{2}{\Chi + 1} \biggr]^{1 / 2}</math>

<math>~=</math>

<math>~\biggl[ \frac{2 \sinh\eta \cdot \sinh\eta^' }{ \cosh(\eta^' + \eta) - \cos(\theta^' - \theta) } \biggr]^{1 / 2} \, .</math>

The task is daunting, however, if the desire is to evaluate the expression for <math>~\Phi_\mathrm{CT}</math> analytically. To our knowledge, no one has yet been able to start from the expression as presented here for <math>~\Phi_\mathrm{CT}</math> and complete the integral over the angular coordinate, <math>~\theta^'</math>, analytically, let alone analytically evaluate the second, radial integral. This is principally because the integrand of the first (angular) integral contains a nontrivial function multiplied by a special function whose argument is, itself, a nontrivial function of both <math>~\theta^'</math> and <math>~\eta^'</math>.

W

At first glance, the expression for <math>~\Phi_\mathrm{W}</math> is much more complex than the expression for <math>~\Phi_\mathrm{CT}</math>. It not only requires a double integration, but also an infinite-series summation over the integer index, <math>~n</math>; and the radial integrand contains a product of two (relatively unfamiliar) associated Legendre functions of various half-integer orders (toroidal functions). Developing a numerical algorithm to evaluate <math>~\Phi_\mathrm{W}</math> to a certain accuracy — presumably set by the number of terms included in the summation over <math>~n</math> — would certainly be a more challenging task than developing a numerical algorithm to evaluate <math>~\Phi_\mathrm{CT}</math>.

Analytic Integration Over the Angular Coordinate

However, focusing only on the integration over the angular coordinate, we see that the integrand in the expression for <math>~\Phi_\mathrm{W}</math> is significantly less imposing than the one that appears in the expression for <math>~\Phi_\mathrm{CT}</math>. Wong (1973) was able to evaluate this definite integral in closed form, analytically. While Wong does not record the detailed steps that he used to evaluate this definite integral, he does indicate that he received guidance from Volume I of A. Erdélyi's (1953) Higher Transcendental Functions. We therefore presume that he adopted the line of reasoning that we have detailed in the Appendix, below, in deriving the expression labeled ③. Wong recognized, what we have explicitly demonstrated, namely,

<math>~\int_{-\pi}^\pi d\theta^' (\cosh\eta^' - \cos\theta^')^{- 5 / 2} \cos[n(\theta - \theta^')]</math>	<math>~=</math>	<math>~2\cos(n\theta) \int_0^\pi \frac{ \cos(n\theta^')~d\theta^' }{ (\cosh\eta^' - \cos \theta^')^{\frac{5}{2}} } </math>	<math>~=</math>	<math>~ \frac{8\sqrt{2}}{3} \biggl[ \frac{\cos(n\theta)}{\sinh^2\eta^'} \biggr] Q^2_{n- \frac{1}{2}} (\cosh\eta^') \,. </math>
Wong (1973), p. 293, Eq. (2.56)

CAUTION:  It is important to appreciate that, in this expression as well as in the expressions to follow, the term, <math>~Q^2_{n-\frac{1}{2}}(z)</math>, is not the square of the zero-order toroidal function, <math>~Q^0_{n - \frac{1}{2}}(z)</math>, but is instead the toroidal function of order two. In an accompanying discussion we present an analytic expression for <math>~Q^2_{-\frac{1}{2}}(z)</math> — and a separate analytic expression for <math>~Q^1_{-\frac{1}{2}}(z)</math> — in terms of complete elliptic integrals, as well as a recurrence relation that can be used to generate analytic expressions for all other order-two (and all other order-one) toroidal functions that have higher half-integer degrees, <math>~n-\tfrac{1}{2}</math> for <math>~n \ge 1</math>.

Hence, Wong was able to simplify the expression for <math>~\Phi_\mathrm{W}</math> to one that — albeit, in addition to an infinite summation over the index, <math>~n</math> — only requires integration over the radial coordinate, <math>~\eta^'</math>. Specifically, he obtained,

<math>~\Phi_\mathrm{W}(\eta,\theta)</math>	<math>~=</math>	<math>~ - 2G \rho_0 a^2 (\cosh\eta - \cos\theta)^{1 / 2} \sum\limits^\infty_{n=0} \epsilon_n \int_{\eta_0}^\infty d\eta^' ~\sinh\eta^'~P_{n-1 / 2}(\cosh\eta_<) ~Q_{n-1 / 2}(\cosh\eta_>)\biggl\{ \frac{8\sqrt{2}}{3} \biggl[ \frac{\cos(n\theta)}{\sinh^2\eta^'} \biggr] Q^2_{n- \frac{1}{2}} (\cosh\eta^') \biggr\} </math>
	<math>~=</math>	<math>~ - \biggl( \frac{16\sqrt{2}}{3} \biggr) G \rho_0 a^2 (\cosh\eta - \cos\theta)^{1 / 2} \sum\limits^\infty_{n=0} \epsilon_n \cos(n\theta) \int_{\eta_0}^\infty d\eta^' \Biggl[ \frac{Q^2_{n- \frac{1}{2}} (\cosh\eta^')}{\sinh\eta^'} \Biggr] ~P_{n-1 / 2}(\cosh\eta_<) ~Q_{n-1 / 2}(\cosh\eta_>) \, . </math>
Wong (1973), p. 294, Eq. (2.57)

Analytic Integration Over the Radial Coordinate

Now, in considering how to handle integration over the radial coordinate, <math>~\eta^'</math>, let's examine, first, the case where <math>~\eta^' \ge \eta_0 > \eta</math>, that is, the potential is being evaluated at a location that is entirely outside of the torus. In this case,

<math>~\Phi_\mathrm{W}(\eta,\theta)</math>

<math>~=</math>

<math>~ - \biggl( \frac{16\sqrt{2}}{3} \biggr) G \rho_0 a^2 (\cosh\eta - \cos\theta)^{1 / 2} \sum\limits^\infty_{n=0} \epsilon_n \cos(n\theta) ~P_{n-1 / 2}(\cosh\eta) \int_{\eta_0}^\infty d\eta^' \Biggl[ \frac{Q^2_{n- \frac{1}{2}} (\cosh\eta^')}{\sinh\eta^'} \Biggr] ~Q_{n-1 / 2}(\cosh\eta^') \, . </math>

So we are interested in carrying out the "radial" integral,

<math>~ \int_{\eta_0}^\infty d\eta^' \sinh\eta^' \Biggl[ \frac{Q^2_{n- \frac{1}{2}} (\cosh\eta^')}{\sinh^2\eta^'} \Biggr] ~Q_{n-1 / 2}(\cosh\eta^') </math>

<math>~=</math>

<math>~ \int_{\cosh(\eta_0)}^{\cosh(\infty)} dt \Biggl[ \frac{Q^2_{n- \frac{1}{2}} (t)~Q_{n-\frac{1}{2}}(t)}{t^2 - 1} \Biggr] \, , </math>

where we have made the association, <math>~\cosh\eta^' \rightarrow t</math>, in which case, <math>~dt = \sinh\eta^' d\eta^'</math>. Following the line of reasoning that we have detailed in the Appendix, below, in deriving the expression labeled ④, this integral can be evaluated in closed form to give,

<math>~ \int_a^b\biggl[ \frac{Q_{n - \frac{1}{2}}^2(t) ~Q_{n - \frac{1}{2}}(t) }{(t^2-1)}\biggr]~dt </math>	<math>~=</math>	<math>~ \frac{1}{4} \biggl[ (n - \tfrac{3}{2})Q^2_{n + \frac{1}{2}}(t) ~ Q_{n - \frac{1}{2}}(t) - (n+\tfrac{1}{2})Q_{n+\frac{1}{2}}(t) Q_{n - \frac{1}{2}}^2(t) \biggr]_a^b \, . </math>
Wong (1973), Eq. (2.58)

Hence, we have,

<math>~\Phi_\mathrm{W}(\eta,\theta)</math>	<math>~=</math>	<math>~ - \biggl( \frac{4\sqrt{2}}{3} \biggr) G \rho_0 a^2 (\cosh\eta - \cos\theta)^{1 / 2} \sum\limits^\infty_{n=0} \epsilon_n \cos(n\theta) </math>
		<math>~ \times P_{n-1 / 2}(\cosh\eta) \biggl[ (n - \tfrac{3}{2})Q^2_{n + \frac{1}{2}}(t) ~ Q_{n - \frac{1}{2}}(t) - (n+\tfrac{1}{2})Q_{n+\frac{1}{2}}(t) Q_{n - \frac{1}{2}}^2(t) \biggr]_{t = \cosh \eta_0}^{t = \cosh\infty} \, . </math>

Or, given that, in this uniform-density configuration, the density is exactly the mass divided by the torus volume, that is,

<math>~\rho_0 = \frac{M}{V}</math>

<math>~=</math>

<math>~\frac{M \sinh^3\eta_0}{2\pi^2 a^3 \cosh{\eta_0}} \, ,</math>

we have,

<math>~\Phi_\mathrm{W}(\eta,\theta)</math>	<math>~=</math>	<math>~ - \frac{2\sqrt{2}~ GM}{3\pi^2 a} \biggl[ \frac{\sinh^3\eta_0}{ \cosh{\eta_0}} \biggr] (\cosh\eta - \cos\theta)^{1 / 2} \sum\limits^\infty_{n=0} \epsilon_n \cos(n\theta) </math>
		<math>~ \times P_{n-1 / 2}(\cosh\eta) \biggl[ (n - \tfrac{3}{2})Q^2_{n + \frac{1}{2}}(t) ~ Q_{n - \frac{1}{2}}(t) - (n+\tfrac{1}{2})Q_{n+\frac{1}{2}}(t) Q_{n - \frac{1}{2}}^2(t) \biggr]_{t = \cosh \eta_0}^{t = \cosh\infty} \, . </math>

In a separate chapter, we have examined the asymptotic behavior of the toroidal functions, <math>~Q_{n-\tfrac{1}{2}}(t)</math>, which is,

<math>~\lim_{\chi\rightarrow \infty} Q_{n-\frac{1}{2}}(t)</math>

<math>~\propto</math>

<math>~ \frac{1}{t^{n+\frac{1}{2}} } \, . </math>

This means that, at the upper integration limit, these toroidal functions go to zero. As a result, we can write,

Exterior Solution:  <math>~\eta_0 \ge \eta</math>

<math>~\Phi_\mathrm{W}(\eta,\theta)</math> <math>~=</math> <math>~ - \frac{2\sqrt{2}~ GM}{3\pi^2 a} \biggl[ \frac{\sinh^3\eta_0}{ \cosh{\eta_0}} \biggr] (\cosh\eta - \cos\theta)^{1 / 2} \sum\limits^\infty_{n=0} \epsilon_n \cos(n\theta) </math>     <math>~ \times P_{n-1 / 2}(\cosh\eta) \biggl[ (n+\tfrac{1}{2})Q_{n+\frac{1}{2}}(\cosh \eta_0) Q_{n - \frac{1}{2}}^2(\cosh \eta_0) - (n - \tfrac{3}{2})Q^2_{n + \frac{1}{2}}(\cosh \eta_0) ~ Q_{n - \frac{1}{2}}(\cosh \eta_0) \biggr] \, . </math>

This expression matches the potential for the exterior region that is obtained by combining Wong's (1973) Eqs. (2.59), (2.61) and (2.63).

Now, let's examine the potential evaluated at a radial location, <math>~\eta</math>, that is positioned inside the surface of the torus. That is, <math>~\eta_0 < \eta \le \infty</math>. We need to add expressions that have two different sets of integration limits as follows. First, over the subregion, <math>~\eta \le \eta^' \le \infty</math>, we use the same expression for <math>~\Phi_\mathrm{W}</math> but employ the new limits, as indicated; and second, over the subregion <math>~\eta_0 \le \eta^' \le \eta</math>, we swap the roles of <math>~P_{n-\frac{1}{2}}</math> and <math>~Q_{n-\frac{1}{2}}</math>.

<math>~\Phi_\mathrm{W}(\eta,\theta)</math>	<math>~=</math>	<math>~ \Phi_\mathrm{W}|_\mathrm{subregion1} + \Phi_\mathrm{W}|_\mathrm{subregion2} </math>
	<math>~=</math>	<math>~ - \frac{2\sqrt{2}~ GM}{3\pi^2 a} \biggl[ \frac{\sinh^3\eta_0}{ \cosh{\eta_0}} \biggr] (\cosh\eta - \cos\theta)^{1 / 2} \sum\limits^\infty_{n=0} \epsilon_n \cos(n\theta) </math>
		<math>~ \times P_{n-1 / 2}(\cosh\eta) \biggl[ (n - \tfrac{3}{2})Q^2_{n + \frac{1}{2}}(t) ~ Q_{n - \frac{1}{2}}(t) - (n+\tfrac{1}{2})Q_{n+\frac{1}{2}}(t) Q_{n - \frac{1}{2}}^2(t) \biggr]_{t = \cosh \eta}^{t = \cosh\infty} </math>
		<math>~ - \frac{2\sqrt{2}~ GM}{3\pi^2 a} \biggl[ \frac{\sinh^3\eta_0}{ \cosh{\eta_0}} \biggr] (\cosh\eta - \cos\theta)^{1 / 2} \sum\limits^\infty_{n=0} \epsilon_n \cos(n\theta) </math>
		<math>~ \times Q_{n-1 / 2}(\cosh\eta) \biggl[ (n - \tfrac{3}{2})Q^2_{n + \frac{1}{2}}(t) ~ P_{n - \frac{1}{2}}(t) - (n+\tfrac{1}{2})P_{n+\frac{1}{2}}(t) Q_{n - \frac{1}{2}}^2(t) \biggr]_{t = \cosh \eta_0}^{t = \cosh\eta} </math>
	<math>~=</math>	<math>~ - \frac{2\sqrt{2}~ GM}{3\pi^2 a} \biggl[ \frac{\sinh^3\eta_0}{ \cosh{\eta_0}} \biggr] (\cosh\eta - \cos\theta)^{1 / 2} \sum\limits^\infty_{n=0} \epsilon_n \cos(n\theta) </math>
		<math>~ \times \biggl\{ P_{n-1 / 2}(\cosh\eta) \biggl[ (n+\tfrac{1}{2})Q_{n+\frac{1}{2}}(\cosh\eta) Q_{n - \frac{1}{2}}^2(\cosh\eta) - (n - \tfrac{3}{2})Q^2_{n + \frac{1}{2}}(\cosh\eta) ~ Q_{n - \frac{1}{2}}(\cosh\eta) \biggr] </math>
		<math>~ + Q_{n-1 / 2}(\cosh\eta) \biggl[ (n - \tfrac{3}{2})Q^2_{n + \frac{1}{2}}(\cosh\eta) ~ P_{n - \frac{1}{2}}(\cosh\eta) - (n+\tfrac{1}{2})P_{n+\frac{1}{2}}(\cosh\eta) Q_{n - \frac{1}{2}}^2(\cosh\eta) \biggr] </math>
		<math>~ - Q_{n-1 / 2}(\cosh\eta) \biggl[ (n - \tfrac{3}{2})Q^2_{n + \frac{1}{2}}(\cosh{\eta_0}) ~ P_{n - \frac{1}{2}}(\cosh{\eta_0}) - (n+\tfrac{1}{2})P_{n+\frac{1}{2}}(\cosh{\eta_0}) Q_{n - \frac{1}{2}}^2(\cosh{\eta_0}) \biggr] \biggr\} </math>
	<math>~=</math>	<math>~ - \frac{2\sqrt{2}~ GM}{3\pi^2 a} \biggl[ \frac{\sinh^3\eta_0}{ \cosh{\eta_0}} \biggr] (\cosh\eta - \cos\theta)^{1 / 2} \sum\limits^\infty_{n=0} \epsilon_n \cos(n\theta) </math>
		<math>~ \times \biggl\{ P_{n-1 / 2}(\cosh\eta) \biggl[ (n+\tfrac{1}{2})Q_{n+\frac{1}{2}}(\cosh\eta) Q_{n - \frac{1}{2}}^2(\cosh\eta) \biggr] - Q_{n-1 / 2}(\cosh\eta) \biggl[ (n+\tfrac{1}{2})P_{n+\frac{1}{2}}(\cosh\eta) Q_{n - \frac{1}{2}}^2(\cosh\eta) \biggr] </math>
		<math>~ - Q_{n-1 / 2}(\cosh\eta) \biggl[ (n - \tfrac{3}{2})Q^2_{n + \frac{1}{2}}(\cosh{\eta_0}) ~ P_{n - \frac{1}{2}}(\cosh{\eta_0}) - (n+\tfrac{1}{2})P_{n+\frac{1}{2}}(\cosh{\eta_0}) Q_{n - \frac{1}{2}}^2(\cosh{\eta_0}) \biggr] \biggr\} </math>
	<math>~=</math>	<math>~ - \frac{2\sqrt{2}~ GM}{3\pi^2 a} \biggl[ \frac{\sinh^3\eta_0}{ \cosh{\eta_0}} \biggr] (\cosh\eta - \cos\theta)^{1 / 2} \sum\limits^\infty_{n=0} \epsilon_n \cos(n\theta) </math>
		<math>~ \times \biggl\{ Q_{n-1 / 2}(\cosh\eta) \biggl[ (n+\tfrac{1}{2})P_{n+\frac{1}{2}}(\cosh{\eta_0}) Q_{n - \frac{1}{2}}^2(\cosh{\eta_0}) - (n - \tfrac{3}{2}) ~ P_{n - \frac{1}{2}}(\cosh{\eta_0})~Q^2_{n + \frac{1}{2}}(\cosh{\eta_0}) \biggr] </math>
		<math>~ - (n+\tfrac{1}{2})Q_{n - \frac{1}{2}}^2(\cosh\eta) \biggl[ Q_{n-1 / 2}(\cosh\eta) P_{n+\frac{1}{2}}(\cosh\eta) - Q_{n+\frac{1}{2}}(\cosh\eta) P_{n-1 / 2}(\cosh\eta) \biggr] \biggr\} </math>

Drawing on the identified "Key Equation" expression,

	<math>~(\xi - z)\sum_{m=0}^n (2m+1)P_m(z) Q_m(\xi)</math>	<math>~=</math>	<math>~ 1 - (\ell+1)[P_{\ell+1}(z) Q_\ell(\xi) - P_\ell(z)Q_{\ell+1}(\xi)] </math>
Abramowitz & Stegun (1995), p. 335, eq. (8.9.2)

and adopting the associations, <math>~z = \xi = \cosh\eta</math> and <math>~\ell \rightarrow (n-\tfrac{1}{2})</math>, we recognize that,

<math>~(n+\tfrac{1}{2})[P_{n+\frac{1}{2}}(\cosh\eta) Q_{n-\frac{1}{2}} (\cosh\eta) - P_{n-\frac{1}{2}}(\cosh\eta)Q_{n+\frac{1}{2}}(\cosh\eta)] </math>

<math>~=</math>

<math>~ 1 \, . </math>

Hence, we can write,

Interior Solution:  <math>~\eta \ge \eta_0 </math>

<math>~\Phi_\mathrm{W}(\eta,\theta)</math> <math>~=</math> <math>~ - \frac{2\sqrt{2}~ GM}{3\pi^2 a} \biggl[ \frac{\sinh^3\eta_0}{ \cosh{\eta_0}} \biggr] (\cosh\eta - \cos\theta)^{1 / 2} \sum\limits^\infty_{n=0} \epsilon_n \cos(n\theta) </math>     <math>~ \times \biggl\{ Q_{n-1 / 2}(\cosh\eta) \biggl[ (n+\tfrac{1}{2})P_{n+\frac{1}{2}}(\cosh{\eta_0}) Q_{n - \frac{1}{2}}^2(\cosh{\eta_0}) - (n - \tfrac{3}{2}) ~ P_{n - \frac{1}{2}}(\cosh{\eta_0})~Q^2_{n + \frac{1}{2}}(\cosh{\eta_0}) \biggr] - Q_{n - \frac{1}{2}}^2(\cosh\eta) \biggr\} \, . </math>

This expression matches the potential for the interior region that is obtained by combining Wong's (1973) Eqs. (2.59), (2.60) and (2.62). Finally, drawing on yet another identified "Key Equation," namely,

	<math>~ Q_{-\frac{1}{2}}^\mu(z) + 2\sum_{n=1}^\infty Q^\mu_{n-\frac{1}{2}}(z) \cos(n\nu) </math>	<math>~=</math>	<math>~ e^{i\mu\pi}~\biggl(\frac{\pi}{2} \biggr)^{1 / 2} \Gamma(\mu + \tfrac{1}{2})\biggl[ \frac{(z^2-1)^{\mu/2}}{(z - \cos\nu)^{\mu + \frac{1}{2}}} \biggr] </math>
A. Erdélyi (1953):  Volume I, §3.10, p. 166, eq. (3)

Valid for:

<math>~\mathrm{Re}~\mu > - \tfrac{1}{2}</math>

and adopting the associations, <math>~\mu \rightarrow 2</math>, <math>~\nu \rightarrow \theta</math>, and <math>~z \rightarrow \cosh\eta</math>, we see that,

<math>~ \sum_{n=0}^\infty \epsilon_n \cos(n\theta) ~Q^2_{n-\frac{1}{2}}(\cosh\eta) </math>	<math>~=</math>	<math>~ (-1)^2~\biggl(\frac{\pi}{2} \biggr)^{1 / 2} \Gamma(2 + \tfrac{1}{2})\biggl[ \frac{ \sinh^2\eta }{(\cosh\eta - \cos\theta)^{\frac{5}{2}}} \biggr] </math>
	<math>~=</math>	<math>~ \biggl(\frac{\pi}{2} \biggr)^{1 / 2} \biggl[ \frac{4! \sqrt{\pi}}{ 4^2 \cdot 2! } \biggr] \biggl[ \frac{ \sinh^2\eta }{(\cosh\eta - \cos\theta)^{\frac{5}{2}} } \biggr] </math>
	<math>~=</math>	<math>~ \biggl(\frac{1}{2} \biggr)^{1 / 2} \biggl[ \frac{ 3\pi}{ 2^2} \biggr] \biggl[ \frac{ \sinh^2\eta }{(\cosh\eta - \cos\theta)^{\frac{5}{2}} } \biggr] \, . </math>
Wong (1973), Eq. (2.64) except, see accompanying (pink, scroll-over balloon) COMMENT

Hence, the interior solution can be rewritten as,

<math>~\Phi_\mathrm{W}(\eta,\theta)\bigr|_\mathrm{interior}</math>	<math>~=</math>	<math>~ \frac{ GM}{2\pi a} \biggl[ \frac{\sinh^3\eta_0}{ \cosh{\eta_0}} \biggr] \biggl\{ \biggl[ \frac{ \sinh\eta }{(\cosh\eta - \cos\theta) } \biggr]^2 ~ - ~ \frac{2^2\sqrt{2}~ }{3\pi} (\cosh\eta - \cos\theta)^{1 / 2} \sum\limits^\infty_{n=0} \epsilon_n \cos(n\theta) \, . </math>
		<math>~ \times Q_{n-1 / 2}(\cosh\eta) \biggl[ (n+\tfrac{1}{2})P_{n+\frac{1}{2}}(\cosh{\eta_0}) Q_{n - \frac{1}{2}}^2(\cosh{\eta_0}) - (n - \tfrac{3}{2}) ~ P_{n - \frac{1}{2}}(\cosh{\eta_0})~Q^2_{n + \frac{1}{2}}(\cosh{\eta_0}) \biggr] \biggr\} \, . </math>
Wong (1973), Eq. (2.65) except, see accompanying (pink, scroll-over balloon) COMMENT

Evaluation

Appendix: Selected Toroidal Function Relationships

Here, we draw from the set of toroidal function relationships that have been identified as "Key Equations" in our accompanying Equations appendix. Two related appendices — numbered A.1 and A.2 — may be found in an accompanying discussion.

A.3

Our objective is to evaluate the definite integral, <math>~\int_{-\pi}^{\pi} \frac{\cos[n(\theta - \theta^')] d\theta}{(\cosh\eta - \cos\theta)^{5 / 2}} </math> <math>~=</math> <math>~ \cos(n\theta^') \int_{-\pi}^{\pi} \frac{ \cos(n\theta) ~ d\theta}{(\cosh\eta - \cos\theta)^{5 / 2}} + \sin(n\theta^') \cancelto{0}{ \int_{-\pi}^{\pi} \frac{ \sin(n\theta) d\theta}{(\cosh\eta - \cos\theta)^{5 / 2}} } </math>   <math>~=</math> <math>~ 2 \cos(n\theta^') \int_{0}^{\pi} \frac{ \cos(n\theta)~ d\theta}{(\cosh\eta - \cos\theta)^{5 / 2}} \, . </math> Notice that, since the limits of the integration are <math>~-\pi</math> to <math>~+\pi</math>:   The second integral on the right-hand-side goes to zero because the numerator of its integrand — i.e., <math>~\sin(n\theta)</math> — is an odd function; and, with regard to the first integral on the right-hand-side, the lower integration limit can be set to zero and the result doubled because the numerator of its integrand — i.e., <math>~\cos(n\theta)</math> — is an even function. Drawing from Volume I of A. Erdélyi's (1953) Higher Transcendental Functions, we find the following "Key Equation": <math>~Q_\nu^\mu(z)</math> <math>~=</math> <math>~ e^{i \mu \pi} ~ (2\pi)^{-\frac{1}{2}} (z^2-1)^{\mu/2} ~\Gamma(\mu + \tfrac{1}{2})~\biggl\{ \int_0^\pi (z - \cos t)^{-\mu - \frac{1}{2}} \cos[(\nu + \tfrac{1}{2})t] ~dt -\cos(\nu\pi) \int_0^\infty (z + \cosh t)^{-\mu - \frac{1}{2}} e^{-(\nu + \frac{1}{2})t} ~dt \biggr\} </math> A. Erdélyi (1953):  Volume I, §3.7, p. 156, eq. (10) Valid for:     <math>~\mathrm{Re} ~\nu > -\tfrac{1}{2}</math>      and     <math>~\mathrm{Re} (\nu + \mu + 1) > 0 \, .</math> Next we adopt the associations, <math>~z \rightarrow \cosh\eta</math>, <math>~t \rightarrow \theta</math>, <math>~\mu \rightarrow 2</math>, and, <math>~\nu \rightarrow n - \tfrac{1}{2}</math>, where <math>~n</math> is zero or a positive integer. In this case we have, <math>~Q_{n - \frac{1}{2}}^2 (\cosh\eta)</math> <math>~=</math> <math>~ (2\pi)^{-\frac{1}{2}} (\cosh^2\eta-1) ~\Gamma(\tfrac{5}{2})~\biggl\{ \int_0^\pi (\cosh\eta - \cos \theta)^{-\frac{5}{2}} \cos(n\theta) ~d\theta - \cancelto{0}{\cos[(n-\tfrac{1}{2})\pi] }~~\int_0^\infty (\cosh\eta + \cosh \theta)^{- \frac{5}{2}} e^{-n\theta} ~d\theta \biggr\} \, , </math> where the prefactor of the second term — that is, <math>~\cos[(n-\tfrac{1}{2})\pi] </math> — goes to zero for all allowable values of the integer, <math>~n</math>. Hence, we conclude that, <math>~2\cos(n\theta^') \int_0^\pi \frac{ \cos(n\theta)~d\theta }{ (\cosh\eta - \cos \theta)^{\frac{5}{2}} } </math> <math>~=</math> <math>~\frac{ 2(2\pi)^{\frac{1}{2}} Q_{n - \frac{1}{2}}^2 (\cosh\eta) \cos(n\theta^')}{ (\cosh^2\eta-1) ~\Gamma(\tfrac{5}{2})~ } </math>   <math>~=</math> <math>~\biggl[ \frac{ 2^3 \sqrt{2} }{ 3 } \biggr] \frac{ Q_{n - \frac{1}{2}}^2(\cosh\eta) \cos(n\theta^')}{ \sinh^2\eta } \, . </math> ③ where we have set, <math>~ \Gamma(\tfrac{5}{2}) = \Gamma(\tfrac{1}{2} + 2) = \frac{ \sqrt{\pi} \cdot 4! }{4^2 \cdot 2!} = \frac{\sqrt{\pi} \cdot 2^3\cdot 3}{ 2^5 } = \frac{3 \sqrt{\pi}}{2^2} \, . </math>

A.4

Beginning with the identified "Key Equation", <math>~ \int_a^b\biggl[(\nu - \sigma)(\nu + \sigma + 1) + (\rho^2 - \mu^2)(1 - z^2)^{-1} \biggr] w_\nu^\mu ~w_\sigma^\rho ~dz </math> <math>~=</math> <math>~ \biggl[ z(\nu-\sigma) w_\nu^\mu ~w_\sigma^\rho + (\sigma+\rho) w_\nu^\mu ~ w_{\sigma-1}^\rho - (\nu + \mu) w_{\nu - 1}^\mu ~w_\sigma^\rho \biggr]_a^b </math> A. Erdélyi (1953):  Volume I, §3.12, p. 169, eq. (1) where, <math>~w_\nu^\mu(z)</math> and <math>~w_\sigma^\rho(z)</math> denote any solutions of Legendre's differential equation we will adopt the associations:   <math>~z \rightarrow t</math>, <math>~\mu \rightarrow 2</math>, <math>~\nu \rightarrow (n - \tfrac{1}{2})</math>, <math>~\rho \rightarrow 0</math>, and <math>~\sigma \rightarrow ( n - \tfrac{1}{2})</math>. As a result, Erdélyi's (1953) expression becomes, <math>~ \int_a^b\biggl[ -4(1 - t^2)^{-1} \biggr] Q_{n - \frac{1}{2}}^2(t) ~X_{n - \frac{1}{2}}(t) ~dt </math> <math>~=</math> <math>~ \biggl[ (n - \tfrac{1}{2} ) Q_{n - \frac{1}{2}}^2(t) ~ X_{n - \frac{3}{2}}(t) - (n + \tfrac{3}{2}) Q_{n - \frac{3}{2}}^2(t) ~X_{n - \frac{1}{2}}(t) \biggr]_a^b \, . </math> Drawing upon the recurrence "Key Equation," <math>~(\nu - \mu + 1)P^\mu_{\nu + 1} (z)</math> <math>~=</math> <math>~ (2\nu + 1)z P_\nu^\mu(z) - (\nu + \mu)P^\mu_{\nu-1}(z) </math> Abramowitz & Stegun (1995), p. 334, eq. (8.5.3) NOTE: <math>~Q_\nu^\mu</math>, as well as <math>~P_\nu^\mu</math>, satisfies this same recurrence relation. which means, after making the associations, <math>~z \rightarrow t</math>, <math>~\mu \rightarrow 0</math> and <math>~\nu \rightarrow (n-\tfrac{1}{2})</math>, that, <math>~(n+\tfrac{1}{2})X_{n+\frac{1}{2}}(t)</math> <math>~=</math> <math>~2n t X_{n-\frac{1}{2}}(t) - (n-\tfrac{1}{2})X_{n - \frac{3}{2}}(t)</math> <math>~\Rightarrow ~~~(n-\tfrac{1}{2})X_{n - \frac{3}{2}}(t) </math> <math>~=</math> <math>~2n t X_{n-\frac{1}{2}}(t) - (n+\tfrac{1}{2})X_{n+\frac{1}{2}}(t)</math> the integral can be rewritten as, <math>~ \int_a^b\biggl[ \frac{Q_{n - \frac{1}{2}}^2(t) ~X_{n - \frac{1}{2}}(t) }{(t^2-1)}\biggr]~dt </math> <math>~=</math> <math>~ \frac{1}{4} \biggl\{ \biggl[ 2n t X_{n-\frac{1}{2}}(t) - (n+\tfrac{1}{2})X_{n+\frac{1}{2}}(t) \biggr] Q_{n - \frac{1}{2}}^2(t) - (n + \tfrac{3}{2}) Q_{n - \frac{3}{2}}^2(t) ~X_{n - \frac{1}{2}}(t) \biggr\}_a^b </math>   <math>~=</math> <math>~ \frac{1}{4} \biggl\{ \biggl[ 2n t Q_{n - \frac{1}{2}}^2(t) - (n + \tfrac{3}{2}) Q_{n - \frac{3}{2}}^2(t) \biggr] X_{n - \frac{1}{2}}(t) - (n+\tfrac{1}{2})X_{n+\frac{1}{2}}(t) Q_{n - \frac{1}{2}}^2(t) \biggr\}_a^b \, . </math> Returning to the same recurrence "Key Equation," but this time adopting the associations, <math>~z \rightarrow t</math>, <math>~\mu \rightarrow 2</math> and <math>~\nu \rightarrow (n-\tfrac{1}{2})</math>, we can write, <math>~(n - \tfrac{3}{2})Q^2_{n + \frac{1}{2}}(t)</math> <math>~=</math> <math>~ 2n t Q^2_{n - \frac{1}{2}}(t) - (n + \tfrac{3}{2}) Q^2_{n - \frac{3}{2}} (t) \, , </math> in which case the integral becomes, <math>~ \int_a^b\biggl[ \frac{Q_{n - \frac{1}{2}}^2(t) ~X_{n - \frac{1}{2}}(t) }{(t^2-1)}\biggr]~dt </math> <math>~=</math> <math>~ \frac{1}{4} \biggl\{ (n - \tfrac{3}{2})Q^2_{n + \frac{1}{2}}(t) ~ X_{n - \frac{1}{2}}(t) - (n+\tfrac{1}{2})X_{n+\frac{1}{2}}(t) Q_{n - \frac{1}{2}}^2(t) \biggr\}_a^b \, . </math> ④

See Also

© 2014 - 2021 by Joel E. Tohline |   H_Book Home   |   YouTube   | Appendices: | Equations | Variables | References | Ramblings | Images | myphys.lsu | ADS | Recommended citation:   Tohline, Joel E. (2021), The Structure, Stability, & Dynamics of Self-Gravitating Fluids, a (MediaWiki-based) Vistrails.org publication, https://www.vistrails.org/index.php/User:Tohline/citation