User:Tohline/Appendix/Ramblings/Dyson1893Part1

Dyson (1893a) Part I: Some Details

This chapter provides some derivation details relevant to our accompanying discussion of Dyson's analysis of the gravitational potential exterior to an anchor ring.

Overview

In his pioneering work, F. W. Dyson (1893a, Philosophical Transactions of the Royal Society of London. A., 184, 43 - 95) and (1893b, Philosophical Transactions of the Royal Society of London. A., 184, 1041 - 1106) used analytic techniques to determine the approximate equilibrium structure of axisymmetric, uniformly rotating, incompressible tori. C.-Y. Wong (1974, ApJ, 190, 675 - 694) extended Dyson's work, using numerical techniques to obtain more accurate — but still approximate — equilibrium structures for incompressible tori having solid body rotation. Since then, Y. Eriguchi & D. Sugimoto (1981, Progress of Theoretical Physics, 65, 1870 - 1875) and I. Hachisu, J. E. Tohline & Y. Eriguchi (1987, ApJ, 323, 592 - 613) have mapped out the full sequence of Dyson-Wong tori, beginning from a bifurcation point on the Maclaurin spheroid sequence.

External Potential

On p. 59, at the end of §6 of Dyson (1893a), we find the following expression for the potential at point "P", anywhere exterior to an anchor ring:

<math>~\frac{\pi V(r,\theta)}{M}</math>	<math>~=</math>	<math>~ \mathfrak{I}(r,\theta,c) ~+~ \frac{1}{2^3}\biggl(\frac{a^2}{c}\biggr) \frac{d}{dc} \biggl[ \mathfrak{I}(r,\theta,c)\biggr] ~-~ \frac{1}{2^6\cdot 3}\biggl(\frac{a^2}{c}\biggr)^2 \frac{d^2}{dc^2} \biggl[ \mathfrak{I}(r,\theta,c)\biggr] ~+~\cdots </math>
		<math>~ ~+~(-1)^{n+1} \frac{2}{2n+2} \biggl[ \frac{1\cdot 3\cdot 5 \cdots (2n-3)}{2^2\cdot 4^2\cdot 6^2\cdots(2n)^2} \biggr] \biggl(\frac{a^2}{c}\biggr)^n \frac{d^n}{dc^n} \biggl[ \mathfrak{I}(r,\theta,c)\biggr] ~+~ \cdots </math>

where (see beginning of §8 on p. 61),

<math>~\mathfrak{I}(r,\theta,c)</math>	<math>~\equiv</math>	<math>~ \int_0^\pi d\phi \biggl[r^2 - 2cr\sin\theta \cos\phi +c^2\biggr]^{-1 / 2} </math>
	<math>~\equiv</math>	<math>~ 2\int_0^{\pi/2} d\phi \biggl[ R_1^2 - (R_1^2-R^2)\sin^2\phi \biggr]^{-1 / 2} </math>
	<math>~\equiv</math>	<math>~ \frac{2}{R_1}\int_0^{\pi/2} d\phi \biggl[ 1 - \biggl( \frac{R_1^2-R^2}{R_1^2}\biggr) \sin^2\phi \biggr]^{-1 / 2} </math>
	<math>~\equiv</math>	<math>~ \frac{4K(\mu)}{R+R_1} \, , </math>

and, where furthermore,

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

<math>~=</math>

<math>~ \int_0^{\pi/2} d\phi \biggl[1 - \mu^2\sin^2\phi \bigg]^{-1 / 2} </math>

and

<math>~\mu</math>

<math>~\equiv</math>

<math>~ \frac{R_1-R}{R_1+R} \, . </math>

 

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