User:Tohline/2DStructure/ToroidalGreenFunction
Using Toroidal Coordinates to Determine the Gravitational Potential
NOTE: An earlier version of this chapter has been shifted to our "Ramblings" Appendix.
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Here we build upon our accompanying review of the types of numerical techniques that various astrophysics research groups have developed to solve for the Newtonian gravitational potential, <math>~\Phi(\vec{x})</math>, given a specified, three-dimensional mass distribution, <math>~\rho(\vec{x})</math>. Our focus is on the use of toroidal coordinates to solve the integral formulation of the Poisson equation, namely,
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<math>~ \Phi(\vec{x})</math> |
<math>~=</math> |
<math>~ -G \iiint \frac{\rho(\vec{x}^{~'})}{|\vec{x}^{~'} - \vec{x}|} d^3x^' \, .</math> |
For the most part, we will adopt the notation used by C.-Y. Wong (1973, Annals of Physics, 77, 279); in an accompanying discussion, we review additional results from this insightful 1973 paper, as well as a paper of his that was published the following year in The Astrophysical Journal, namely, Wong (1974).
In order to accomplish this task, we first present the expressions that define how toroidal coordinates, <math>~(\eta,\theta,\psi)</math>, map to and from Cartesian coordinates <math>~(x, y, z)</math>, and present the toroidal-coordinate expression for the differential volume element, <math>~d^3 x</math>.
Basic Elements of a Toroidal Coordinate System
Given the meridional-plane coordinate location of a toroidal-coordinate system's axisymmetric anchor ring, <math>~(\varpi,z) = (a,Z_0)</math>, the relationship between toroidal coordinates <math>~(\eta,\theta,\psi) </math>and Cartesian coordinates <math>~(x, y, z)</math> is,
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<math>~x</math> |
<math>~=</math> |
<math>~\frac{a \sinh\eta \cos\psi}{(\cosh\eta - \cos\theta)} \, ,</math> |
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<math>~y</math> |
<math>~=</math> |
<math>~\frac{a \sinh\eta \sin\psi}{(\cosh\eta - \cos\theta)} \, ,</math> |
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<math>~z - Z_0</math> |
<math>~=</math> |
<math>~\frac{a \sin\theta}{(\cosh\eta - \cos\theta)} \, .</math> |
This set of coordinate relations appears as equations 2.1 - 2.3 in Wong (1973). This set of relations may also be found, for example, on p. 1301 within eq. (10.3.75) of [MF53]; in §14.19 of NIST's Digital Library of Mathematical Functions; or even within Wikipedia. (In most cases the implicit assumption is that <math>~Z_0 = 0</math>.) It is clear, of course, that the cylindrical radial coordinate is,
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<math>~\varpi = (x^2 + y^2)^{1 / 2}</math> |
<math>~=</math> |
<math>~\frac{a \sinh\eta}{(\cosh\eta - \cos\theta)} \, .</math> |
Mapping the other direction [see equations 2.13 - 2.15 of Wong (1973) ], we have,
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<math>~\eta</math> |
<math>~=</math> |
<math>~\ln\biggl(\frac{r_1}{r_2} \biggr) \, ,</math> |
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<math>~\cos\theta</math> |
<math>~=</math> |
<math>~\frac{(r_1^2 + r_2^2 - 4a^2)}{2r_1 r_2} \, ,</math> |
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<math>~\tan\psi</math> |
<math>~=</math> |
<math>~\frac{y}{x} \, ,</math> |
where,
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<math>~r_1^2 </math> |
<math>~\equiv</math> |
<math>~[(x^2 + y^2)^{1 / 2} + a]^2 + (z-Z_0)^2 \, ,</math> |
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<math>~r_2^2 </math> |
<math>~\equiv</math> |
<math>~[(x^2 + y^2)^{1 / 2} - a]^2 + (z-Z_0)^2 \, ,</math> |
and <math>~\theta</math> has the same sign as <math>~(z-Z_0)</math>.
According to p. 1301, eq. (10.3.75) of [MF53] — or, for example, as found in Wikipedia — in toroidal coordinates the differential volume element is,
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<math>~d^3x</math> |
<math>~=</math> |
<math>~h_\eta h_\theta h_\psi d\eta d\theta d\psi</math> |
<math>~=</math> |
<math>~\biggl[ \frac{a^3 \sinh\eta}{(\cosh\eta - \cos\theta)^3} \biggr] d\eta~ d\theta~ d\psi \, .</math> |
Green's Function Expression
As presented by Wong (1973)
Referencing [MF53], Wong (1973) states that, in toroidal coordinates, the Green's function is,
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<math>~\frac{1}{|~\vec{x} - {\vec{x}}^{~'} ~|} </math> |
<math>~=</math> |
<math>~ \frac{1}{\pi a} \biggl[ (\cosh\eta - \cos\theta)(\cosh \eta^' - \cos\theta^') \biggr]^{1 /2 } \sum\limits^\infty_{m,n=0} (-1)^m \epsilon_m \epsilon_n ~\frac{\Gamma(n-m+\tfrac{1}{2})}{\Gamma(n + m + \tfrac{1}{2})} </math> |
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<math>~ \times \cos[m(\psi - \psi^')]\cos[n(\theta - \theta^')] ~\begin{cases}P^m_{n-1 / 2}(\cosh\eta) ~Q^m_{n-1 / 2}(\cosh\eta^') ~~~\eta^' > \eta \\P^m_{n-1 / 2}(\cosh\eta^') ~Q^m_{n-1 / 2}(\cosh\eta)~~~\eta^' < \eta \end{cases}\, , </math> |
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Wong (1973), p. 293, Eq. (2.53) |
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where, <math>~P^m_{n-1 / 2}, Q^m_{n-1 / 2}</math> are Associated Legendre Functions of the first and second kind with order <math>~n - \tfrac{1}{2}</math> and degree <math>~m</math> (toroidal harmonics), and <math>~\epsilon_m</math> is the Neumann factor, that is, <math>~\epsilon_0 = 1</math> and <math>~\epsilon_m = 2</math> for all <math>~m \ge 1</math>. This Green's function expression can indeed be found as eq. (10.3.81) on p. 1304 of [MF53], but it should be noted that the MF53 expression differs from Wong's in two respects (see footnote 2 on p. 370 of Cohl et al. (2000) for a proposed explanation): First, the factor, <math>~(-1)^m</math>, appears as <math>~(-i)^m</math> in MF53; and, second, in the term that is composed of a ratio of gamma functions, the denominator appears in MF53 as <math>~\Gamma(n - m + \tfrac{1}{2})</math>, whereas it should be <math>~\Gamma(n + m + \tfrac{1}{2})</math>, as presented here.
Rearranging Terms and Incorporating Special-Function Relations
Let's focus on the situation when <math>~\eta^' > \eta</math>, and begin rearranging or substituting terms.
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<math>~\frac{1}{|~\vec{x} - {\vec{x}}^{~'} ~|} </math> |
<math>~=</math> |
<math>~ \frac{1}{\pi a} \biggl[ (\cosh\eta - \cos\theta)(\cosh \eta^' - \cos\theta^') \biggr]^{1 /2 } \sum\limits^\infty_{m=0} (-1)^m \epsilon_m \cos[m(\psi - \psi^')] </math> |
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<math>~ \times \sum\limits^\infty_{n=0} \epsilon_n \cos[n(\theta - \theta^')] ~\frac{\Gamma(n-m+\tfrac{1}{2})}{\Gamma(n + m + \tfrac{1}{2})} ~P^m_{n-1 / 2}(\cosh\eta) ~Q^m_{n-1 / 2}(\cosh\eta^') </math> |
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<math>~=</math> |
<math>~ \frac{ [ (\cosh\eta - \cos\theta)(\cosh \eta^' - \cos\theta^')]^{1 /2 } }{\pi a \sqrt{\sinh\eta^'} \sqrt{\sinh\eta} } \sum\limits^\infty_{m=0} (-1)^m \epsilon_m \cos[m(\psi - \psi^')] </math> |
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<math>~ \times \sum\limits^\infty_{n=0} \epsilon_n \cos[n(\theta - \theta^')] \biggl\{ ~ \sqrt{ \frac{\pi}{2} }~\Gamma(n-m+\tfrac{1}{2}) \sqrt{\sinh\eta}~P^m_{n-1 / 2}(\cosh\eta) \biggl\}\biggr\{ ~\sqrt{ \frac{2}{\pi} }~\frac{\sqrt{\sinh\eta^'}}{\Gamma(n + m + \tfrac{1}{2})} Q^m_{n-1 / 2}(\cosh\eta^') \biggr\} </math> |
The term contained within the first set of curly braces on the right-hand side of this expression can be replaced by the derived expression labeled ① in the Appendix, below, and simultaneously the term contained within the second set of curly braces can be replaced by the derived expression labeled ② in the same Appendix. After making these substitutions, we have,
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<math>~\frac{1}{|~\vec{x} - {\vec{x}}^{~'} ~|} </math> |
<math>~=</math> |
<math>~ \frac{ [ (\cosh\eta - \cos\theta)(\cosh \eta^' - \cos\theta^')]^{1 /2 } }{\pi a \sqrt{\sinh\eta^'} \sqrt{\sinh\eta} } \sum\limits^\infty_{m=0} (-1)^m \epsilon_m \cos[m(\psi - \psi^')] </math> |
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<math>~ \times \sum\limits^\infty_{n=0} \epsilon_n \cos[n(\theta - \theta^')] \biggl\{ ~ (-1)^{-n}Q^n_{m-1 / 2}(\coth\eta) \biggl\}\biggr\{ ~(-1)^{-m} P^{-n}_{m-1 / 2}(\coth\eta^') \biggr\} </math> |
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<math>~=</math> |
<math>~ \frac{ [ (\cosh\eta - \cos\theta)(\cosh \eta^' - \cos\theta^')]^{1 /2 } }{\pi a \sqrt{\sinh\eta^'} \sqrt{\sinh\eta} } \sum\limits^\infty_{m=0} \epsilon_m \cos[m(\psi - \psi^')] </math> |
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<math>~ \times \sum\limits^\infty_{n=0} (-1)^{n} \epsilon_n \cos[n(\theta - \theta^')] Q^n_{m-1 / 2}(\coth\eta) P^{-n}_{m-1 / 2}(\coth\eta^') \, , </math> |
where, in writing this last expression we have acknowledged that, since <math>~n</math> is either zero or a positive integer, <math>~(-1)^{-n} = (-1)^n</math>. Next we draw upon the "Key Equation" relation,
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<math>~ Q_\nu[t t^' - (t^2-1)^{1 / 2} (t^{'2} - 1)^{1 / 2} \cos\psi] </math> |
<math>~=</math> |
<math>~ Q_\nu(t) P_\nu(t^') + 2\sum_{n=1}^\infty (-1)^n Q^n_\nu(t) P^{-n}_\nu(t^') \cos(n\psi) </math> |
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A. Erdélyi (1953): Volume I, §3.11, p. 169, eq. (4) |
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| Valid for: |
<math>~t, t^'</math> real |
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<math>~1 < t^' < t</math> |
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<math>~\nu \ne -1, -2, -3, </math> … |
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<math>~\psi</math> real |
which, after making the substitutions, <math>~\nu \rightarrow (m - \tfrac{1}{2})</math> and <math>~\psi \rightarrow (\theta - \theta^')</math>, and incorporating the Neumann factor, <math>~\epsilon_n</math>, becomes,
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<math>~ Q_{m - \frac{1}{2} }\ [t t^' - (t^2-1)^{1 / 2} (t^{'2} - 1)^{1 / 2} \cos(\theta- \theta^') ] </math> |
<math>~=</math> |
<math>~ \sum_{n=0}^\infty (-1)^n \epsilon_n Q^n_{m - \frac{1}{2} }(t) P^{-n}_{m - \frac{1}{2} }(t^') \cos[(n(\theta- \theta^')] \, . </math> |
Finally, after making the associations, <math>~t \rightarrow \coth\eta</math> and <math>~t^' \rightarrow \coth\eta^'</math>, this last expression allows us to rewrite Wong's (1973) Green's function in a significantly more compact form, namely,
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<math>~\frac{1}{|~\vec{x} - {\vec{x}}^{~'} ~|} </math> |
<math>~=</math> |
<math>~ \frac{ [ (\cosh\eta - \cos\theta)(\cosh \eta^' - \cos\theta^')]^{1 /2 } }{\pi a \sqrt{\sinh\eta^'} \sqrt{\sinh\eta} } \sum\limits^\infty_{m=0} \epsilon_m \cos[m(\psi - \psi^')] Q_{m - \frac{1}{2}}(\Chi) \, , </math> |
where the argument, <math>~\Chi</math>, of the toroidal function, <math>~Q_{m - \frac{1}{2}}</math>, is,
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<math>~\Chi</math> |
<math>~\equiv</math> |
<math>~ t t^' - (t^2-1)^{1 / 2} (t^{'2} - 1)^{1 / 2} \cos(\theta- \theta^') </math> |
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<math>~=</math> |
<math>~ \coth\eta \coth\eta^' - (\coth^2\eta-1)^{1 / 2} (\coth^2\eta'- 1)^{1 / 2} \cos(\theta- \theta^') </math> |
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<math>~=</math> |
<math>~ \frac{\cosh\eta \cosh\eta^'}{\sinh\eta \sinh\eta^'} - \biggl[ \frac{1}{\sinh^2\eta} \biggr]^{1 / 2} \biggl[ \frac{1}{\sinh^2\eta'}\biggr]^{1 / 2} \cos(\theta- \theta^') </math> |
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<math>~=</math> |
<math>~ \frac{\cosh\eta \cosh\eta^' - \cos(\theta- \theta^') }{\sinh\eta \sinh\eta^'} \, . </math> |
As Presented in Cohl & Tohline (1999)
This last, compact Green's function expression — which we have derived, here, from Wong's (1973) published Green's function by drawing strategically upon a variety of special function relations — precisely matches the "compact cylindrical Green's function expression" that has been derived independently by Cohl & Tohline (1999) via a less tortuous route, namely,
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<math>~ \frac{1}{|\vec{x} - \vec{x}^{~'}|}</math> |
<math>~=</math> |
<math>~ \frac{1}{\pi \sqrt{\varpi \varpi^'}} \sum_{m=0}^{\infty} \epsilon_m \cos[m(\psi - \psi^')] Q_{m- 1 / 2}(\Chi) </math> |
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Cohl & Tohline (1999), p. 88, Eq. (17) |
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<math>~=</math> |
<math>~ \frac{1}{a\pi} \biggl[ \frac{(\cosh\eta^' - \cos\theta^')}{\sinh\eta^' } \frac{(\cosh\eta - \cos\theta)}{\sinh\eta } \biggr]^{1 / 2} \sum_{m=0}^{\infty} \epsilon_m \cos[m(\psi - \psi^')] Q_{m- 1 / 2}(\Chi) \, , </math> |
where,
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<math>~\Chi</math> |
<math>~\equiv</math> |
<math>~ \frac{(\varpi^')^2 + \varpi^2 + (z^' - z)^2}{2\varpi^' \varpi} </math> |
<math>~=</math> |
<math>~ \frac{\cosh\eta \cdot \cosh\eta^' - \cos(\theta^' - \theta) }{ \sinh\eta \cdot \sinh\eta^'} \, . </math> |
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Cohl & Tohline (1999), p. 88, Eq. (16) |
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Note from J. E. Tohline (June, 2018): This is the first time that I have been able to formally demonstrate to myself that these two separately derived Green's function expressions are identical. See, however, the earlier identification of new addition theorems in association with equations (49) and (50) of Cohl et al. (2000).
Gravitational Potential
Quite generally, then, the gravitational potential can be obtained at any coordinate location, <math>~(\eta,\theta,\psi)</math> — both inside and outside of a specified mass distribution — by carrying out three nested spatial integrals over the product of: <math>~\rho(\vec{x}^{~'})</math>, the differential volume element, and the Green's function as specified either by Wong (1973) or by Cohl & Tohline (1999).
In what follows we will make an effort to elucidate the pros and cons of adopting one Green's function expression over the other. In each case we begin by writing the expression for the potential in such a way that variations in the azimuthal coordinate, <math>~\psi</math>, are described by Fourier components, <math>~\Phi_m^{(1)}(\eta,\theta)</math> and <math>~\Phi_m^{(2)}(\eta,\theta)</math>, of the potential, such that,
<math>~\Phi(\vec{x}) = \tfrac{1}{2}\Phi_0^{(1)}(\eta,\theta) + \sum_{m=1}^\infty \cos (m\psi) \Phi_m^{(1)}(\eta,\theta) + \sum_{m=1}^\infty \sin (m\psi) \Phi_m^{(2)}(\eta,\theta) \, .</math>
Each Fourier component of the potential depends explicitly on the corresponding Fourier component of the density distribution, defined such that,
<math>~\rho(\vec{x}) = \tfrac{1}{2}\rho_0^{(1)}(\eta,\theta) + \sum_{m=1}^\infty \cos (m\psi) \rho_m^{(1)}(\eta,\theta) + \sum_{m=1}^\infty \sin (m\psi) \rho_m^{(2)}(\eta,\theta) \, .</math>
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LaTeX mathematical expressions cut-and-pasted directly from
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As an additional primary point of reference, note that according to §1.8(i) of NIST's Digital Library of Mathematical Functions, a Fourier Series is defined as follows:
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Notice, therefore, that,
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<math>~\rho_m^{(1)}(\eta,\theta)</math> |
<math>~=</math> |
<math>~\frac{1}{\pi}\int^{\pi}_{-\pi}\rho(\eta,\theta,\psi)\cos\bigl(m\psi\bigr)\mathrm{d}\psi,</math> |
and, |
<math>~\rho_m^{(2)}(\eta,\theta)</math> |
<math>~=</math> |
<math>~\frac{1}{\pi}\int^{\pi}_{-\pi}\rho(\eta,\theta,\psi)\sin\bigl(m\psi\bigr)\mathrm{d}\psi \, .</math> |
The CT99 Expression for the Potential
In Three-Dimensional Generality
Employing the Green's function expression derived by Cohl & Tohline (1999), the gravitational potential for any three-dimensional matter distribution is,
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<math>~ \Phi(\eta,\theta,\psi)</math> |
<math>~=</math> |
<math>~ -G \iiint \rho(\eta^',\theta^',\psi^') \biggl\{ \frac{1}{|\vec{x}^{~'} - \vec{x}|} \biggr\} \biggl[ \frac{a^3 \sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^3} \biggr] d\eta^'~ d\theta^'~ d\psi^' </math> |
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<math>~=</math> |
<math>~ -\frac{Ga^2}{\pi} \int d\eta^' \int d\theta^' \int d\psi^' \iiint \rho(\eta^',\theta^',\psi^') \biggl[ \frac{\sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^3} \biggr] </math> |
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<math>~ \times \biggl[ \frac{(\cosh\eta^' - \cos\theta^')}{\sinh\eta^' } \frac{(\cosh\eta - \cos\theta)}{\sinh\eta } \biggr]^{1 / 2} \sum_{m=0}^{\infty} \epsilon_m \cos[m(\psi - \psi^')] Q_{m- 1 / 2}(\Chi) </math> |
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<math>~=</math> |
<math>~ -\frac{Ga^2}{\pi} \biggl[ \frac{(\cosh\eta - \cos\theta)}{\sinh\eta } \biggr]^{1 / 2} \sum_{m=0}^{\infty} \epsilon_m \int d\eta^' \int d\theta^' \biggl[ \frac{\sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^5} \biggr]^{1 / 2} Q_{m- 1 / 2}(\Chi) \int d\psi^' \rho(\eta^',\theta^',\psi^') </math> |
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<math>~ \times [\cos(m\psi)\cos(m\psi^') + \sin(m\psi)\sin(m\psi^') ] </math> |
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<math>~=</math> |
<math>~ -\frac{Ga^2}{\pi} \biggl[ \frac{(\cosh\eta - \cos\theta)}{\sinh\eta } \biggr]^{1 / 2} \biggl\{ \int d\eta^' \int d\theta^' \biggl[ \frac{\sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^5} \biggr]^{1 / 2} Q_{- \frac{1}{2}}(\Chi) \int_{-\pi}^{\pi} d\psi^' \rho(\eta^',\theta^',\psi^') </math> |
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<math>~ + \sum_{m=1}^{\infty} 2\cos(m\psi) \int d\eta^' \int d\theta^' \biggl[ \frac{\sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^5} \biggr]^{1 / 2} Q_{m- 1 / 2}(\Chi) \int_{-\pi}^\pi d\psi^' \rho(\eta^',\theta^',\psi^') \cos(m\psi^') </math> |
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<math>~ + \sum_{m=1}^{\infty} 2\sin(m\psi) \int d\eta^' \int d\theta^' \biggl[ \frac{\sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^5} \biggr]^{1 / 2} Q_{m- 1 / 2}(\Chi) \int_{-\pi}^\pi d\psi^' \rho(\eta^',\theta^',\psi^') \sin(m\psi^') \biggr\} </math> |
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<math>~=</math> |
<math>~ - Ga^2 \biggl[ \frac{(\cosh\eta - \cos\theta)}{\sinh\eta } \biggr]^{1 / 2} \biggl\{ \int d\eta^' \int d\theta^' \biggl[ \frac{\sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^5} \biggr]^{1 / 2} Q_{- \frac{1}{2}}(\Chi) \rho_0^{(1)}(\eta^',\theta^') </math> |
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<math>~ + \sum_{m=1}^{\infty} 2\cos(m\psi) \int d\eta^' \int d\theta^' \biggl[ \frac{\sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^5} \biggr]^{1 / 2} Q_{m- 1 / 2}(\Chi) \rho_m^{(1)}(\eta^',\theta^') </math> |
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<math>~ + \sum_{m=1}^{\infty} 2\sin(m\psi) \int d\eta^' \int d\theta^' \biggl[ \frac{\sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^5} \biggr]^{1 / 2} Q_{m- 1 / 2}(\Chi) \rho_m^{(2)}(\eta^',\theta^') \biggr\} \, . </math> |
We conclude, therefore, that each one of the Fourier components of the gravitational potential is given by the expression,
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<math>~\Phi_m^{(1),(2)}(\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_{m- 1 / 2}(\Chi) \rho_m^{(1),(2)}(\eta^',\theta^') \, , </math> |
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Cohl & Tohline (1999), p. 88, Eq. (20) |
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where, as above,
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<math>~\Chi</math> |
<math>~\equiv</math> |
<math>~ \frac{\cosh\eta \cdot \cosh\eta^' - \cos(\theta^' - \theta) }{ \sinh\eta \cdot \sinh\eta^'} \, . </math> |
For Axisymmetric Systems
For axisymmetric systems, the density distribution has no dependence on the azimuthal coordinate, <math>~\psi</math>. Hence, for all <math>~m > 0</math>, the Fourier components of the density, <math>~\rho_m^{(1),(2)}</math>, are zero. The only nonzero component is, <math>~\rho_0^{(1)}(\eta,\theta) = 2\rho(\eta,\theta)</math>. For axisymmetric systems, then, the gravitational potential is,
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<math>~\Phi(\eta,\theta)</math> |
<math>~=</math> |
<math>~\tfrac{1}{2}\Phi_0^{(1)}(\eta,\theta)</math> |
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<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> |
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Cohl & Tohline (1999), p. 88, Eqs. (31) & (32a) |
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Wong's Expression for the Potential
Fully Three-Dimensional Case
Employing Wong's (1973) Green's function expression, the gravitational potential for any three-dimensional matter distribution is,
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<math>~ \Phi(\vec{x})</math> |
<math>~=</math> |
<math>~ -G \iiint \rho(\eta^',\theta^',\psi^') \biggl\{ \frac{1}{|\vec{x}^{~'} - \vec{x}|} \biggr\} \biggl[ \frac{a^3 \sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^3} \biggr] d\eta^'~ d\theta^'~ d\psi^' </math> |
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<math>~=</math> |
<math>~ - \frac{a^2G}{\pi} \int d\eta^' \int d\theta^' \int d\psi^' \biggl[ \frac{\rho(\eta^',\theta^',\psi^') ~\sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^3} \biggr] \biggl[ (\cosh\eta - \cos\theta)(\cosh \eta^' - \cos\theta^') \biggr]^{1 /2 } \sum\limits^\infty_{n=0} \sum\limits^\infty_{m=0}(-1)^m \epsilon_m \epsilon_n ~\frac{\Gamma(n-m+\tfrac{1}{2})}{\Gamma(n + m + \tfrac{1}{2})} </math> |
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<math>~ \times \cos[m(\psi - \psi^')]\cos[n(\theta - \theta^')] ~P^m_{n-1 / 2}(\cosh\eta_<) ~Q^m_{n-1 / 2}(\cosh\eta_>) </math> |
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<math>~=</math> |
<math>~ - \frac{a^2G}{\pi} (\cosh\eta - \cos\theta)^{1 / 2} \sum\limits^\infty_{n=0} \epsilon_n \sum\limits^\infty_{m=0}(-1)^m \epsilon_m ~\frac{\Gamma(n-m+\tfrac{1}{2})}{\Gamma(n + m + \tfrac{1}{2})} \int d\eta^' ~P^m_{n-1 / 2}(\cosh\eta_<) ~Q^m_{n-1 / 2}(\cosh\eta_>) </math> |
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<math>~ \times~ \int d\theta^' \biggl[ \frac{ ~\sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^{5/2}} \biggr]\cos[n(\theta - \theta^')] \int d\psi^' \rho(\eta^',\theta^',\psi^') [\cos(m\psi)\cos(m\psi^') + \sin(m\psi) \sin(m\psi^')] </math> |
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<math>~=</math> |
<math>~ - \frac{a^2G}{\pi} (\cosh\eta - \cos\theta)^{1 / 2} \sum\limits^\infty_{n=0} \epsilon_n \biggl\{ \int d\eta^' ~P_{n-1 / 2}(\cosh\eta_<) ~Q_{n-1 / 2}(\cosh\eta_>) \int d\theta^' \biggl[ \frac{ ~\sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^{5/2}} \biggr]\cos[n(\theta - \theta^')] \int_{-\pi}^\pi d\psi^' \rho(\eta^',\theta^',\psi^') </math> |
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<math>~ +~ \sum\limits^\infty_{m=1} 2\cos(m\psi)(-1)^m ~\frac{\Gamma(n-m+\tfrac{1}{2})}{\Gamma(n + m + \tfrac{1}{2})} \int d\eta^' ~P^m_{n-1 / 2}(\cosh\eta_<) ~Q^m_{n-1 / 2}(\cosh\eta_>) \int d\theta^' \biggl[ \frac{ ~\sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^{5/2}} \biggr]\cos[n(\theta - \theta^')] \int_{-\pi}^\pi d\psi^' \rho(\eta^',\theta^',\psi^') \cos(m\psi^') </math> |
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<math>~ +~ \sum\limits^\infty_{m=1} 2 \sin(m\psi)(-1)^m ~\frac{\Gamma(n-m+\tfrac{1}{2})}{\Gamma(n + m + \tfrac{1}{2})} \int d\eta^' ~P^m_{n-1 / 2}(\cosh\eta_<) ~Q^m_{n-1 / 2}(\cosh\eta_>) \int d\theta^' \biggl[ \frac{ ~\sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^{5/2}} \biggr]\cos[n(\theta - \theta^')] \int_{-\pi}^\pi d\psi^' \rho(\eta^',\theta^',\psi^') \sin(m\psi^') \biggr\} </math> |
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<math>~=</math> |
<math>~ - a^2G (\cosh\eta - \cos\theta)^{1 / 2} \sum\limits^\infty_{n=0} \epsilon_n \biggl\{ \int d\eta^' ~P_{n-1 / 2}(\cosh\eta_<) ~Q_{n-1 / 2}(\cosh\eta_>) \int d\theta^' \biggl[ \frac{ ~\sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^{5/2}} \biggr]\cos[n(\theta - \theta^')] \rho_0^{(1)}(\eta^',\theta^') </math> |
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<math>~ +~ \sum\limits^\infty_{m=1} 2\cos(m\psi)(-1)^m ~\frac{\Gamma(n-m+\tfrac{1}{2})}{\Gamma(n + m + \tfrac{1}{2})} \int d\eta^' ~P^m_{n-1 / 2}(\cosh\eta_<) ~Q^m_{n-1 / 2}(\cosh\eta_>) \int d\theta^' \biggl[ \frac{ ~\sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^{5/2}} \biggr]\cos[n(\theta - \theta^')] \rho_m^{(1)}(\eta^',\theta^') </math> |
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<math>~ +~ \sum\limits^\infty_{m=1} 2 \sin(m\psi)(-1)^m ~\frac{\Gamma(n-m+\tfrac{1}{2})}{\Gamma(n + m + \tfrac{1}{2})} \int d\eta^' ~P^m_{n-1 / 2}(\cosh\eta_<) ~Q^m_{n-1 / 2}(\cosh\eta_>) \int d\theta^' \biggl[ \frac{ ~\sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^{5/2}} \biggr]\cos[n(\theta - \theta^')] \rho_m^{(2)} (\eta^',\theta^') \biggr\} \, . </math> |
We conclude, therefore, that each one of the Fourier components of the gravitational potential is given by the expression,
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<math>~\Phi_m^{(1),(2)} (\eta,\theta)</math> |
<math>~=</math> |
<math>~ - 2Ga^2 (\cosh\eta - \cos\theta)^{1 / 2} \sum\limits^\infty_{n=0} \epsilon_n (-1)^m ~\frac{\Gamma(n-m+\tfrac{1}{2})}{\Gamma(n + m + \tfrac{1}{2})} </math> |
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<math>~\times \int d\eta^' ~\sinh\eta^' ~ P^m_{n-1 / 2}(\cosh\eta_<) ~ Q^m_{n-1 / 2}(\cosh\eta_>) \int d\theta^' \biggl\{ \frac{ \cos[n(\theta - \theta^')]}{(\cosh\eta^' - \cos\theta^')^{5/2}} \biggr\} \rho_m^{(1),(2)}(\eta^',\theta^') \, . </math> |
Axisymmetric Systems
For axisymmetric systems, the density distribution has no dependence on the azimuthal coordinate, <math>~\psi</math>. Hence, for all <math>~m > 0</math>, the Fourier components of the density, <math>~\rho_m^{(1),(2)}</math>, are zero. The only nonzero component is, <math>~\rho_0^{(1)}(\eta,\theta) = 2\rho(\eta,\theta)</math>. For axisymmetric systems, then, the gravitational potential is,
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<math>~\Phi(\eta,\theta)</math> |
<math>~=</math> |
<math>~\tfrac{1}{2}\Phi_0^{(1)}(\eta,\theta)</math> |
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<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> |
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Wong (1973), p. 293, Eq. (2.55) |
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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, <math>~a</math>, is defined such that,
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<math>~a^2</math> |
<math>~\equiv</math> |
<math>~R^2 - d^2 \, .</math> |
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Wong (1973), Eq. (2.8) |
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Then the corresponding (toroidal-coordinate-based) radial coordinate location <math>~\eta_0</math> of the surface of the torus is,
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<math>~\eta_0</math> |
<math>~=</math> |
<math>~\cosh^{-1}\biggl(\frac{R}{d}\biggr) \, .</math> |
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Wong (1973), Eq. (2.9) |
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Alternatively, given <math>~\eta_0</math> and the value of the parameter <math>~a</math>, we have,
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<math>~R</math> |
<math>~=</math> |
<math>~a \coth\eta_0 \, ,</math> |
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<math>~d</math> |
<math>~=</math> |
<math>~\frac{a}{\sinh\eta_0} \, .</math> |
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Wong (1973), Eqs. (2.10) & (2.11) |
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If such an axisymmetric torus has a uniform density, <math>~\rho_0</math>, then according to the Cohl & Tohline (1999) expression, the gravitational potential is,
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<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,
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<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> |
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Wong (1973), p. 293, Eq. (2.55) |
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Which Is Easier to Evaluate?
Upon evaluation, the expressions for <math>~\Phi_\mathrm{CT}</math> and for <math>~\Phi_\mathrm{W}</math> should 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,
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<math>~Q_{-\frac{1}{2}}(\Chi)</math> |
<math>~=</math> |
<math>~ \mu K(\mu) \, , </math> |
where,
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<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 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
On the surface, 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,
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<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> |
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Wong (1973), p. 293, Eq. (2.56) |
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Hence, he 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>. He obtained, specifically,
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<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> |
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<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> |
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Wong (1973), p. 294, Eq. (2.57) |
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Analytic Integration Over the Radial Coordinate
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,
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<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,
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<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>. According to our accompanying discussion, this integral can be evaluated in closed form to give,
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<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) ~ 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> |
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Wong (1973), Eq. (2.58) |
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Hence, we have,
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<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> |
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<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> |
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.
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Beginning with the identified "Key Equation",
we'll identify <math>~x</math> with <math>~\cosh\eta</math> — in which case we have <math>~\lambda = \coth\eta</math> — and switch the index notations, <math>~n \leftrightarrow m</math>. This gives,
Drawing upon the Euler reflection formula for gamma functions, namely,
where it is understood that <math>~m</math> and <math>~n</math> are each either zero or a positive integer, this toroidal-function relation becomes,
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Again, beginning with the identified "Key Equation",
this time, without switching index notations, we'll identify <math>~x</math> with <math>~\coth\eta</math> — in which case we have <math>~\lambda = \cosh\eta</math>. This gives,
Drawing upon the same Euler reflection formula for gamma functions, as quoted above, this toroidal function relation can be rewritten as,
Finally, calling upon the "Key Equation" relation,
making the index notation substitution, <math>~\nu \rightarrow (m-\tfrac{1}{2})</math>, and associating <math>~z</math> with <math>~ \coth\eta</math> gives,
As a result, we can write,
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Here, our objective is to evaluate the definite integral,
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":
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,
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,
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> |
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See Also
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© 2014 - 2021 by Joel E. Tohline |