Apps/MaclaurinToroid

Maclaurin Toroid

Maclaurin Toroid MPT77

In a separate chapter, we focused on the pioneering work of 📚 F. W. Dyson (1893, Phil. Trans. Royal Soc. London. A., Vol. 184, pp. 43 - 95), 📚 F. W. Dyson (1893, Phil. Trans. Royal Soc. London. A., Vol. 184, pp. 1041 - 1106) and, more recently, 📚 C. -Y. Wong (1974, ApJ, Vol. 190, pp. 675 - 694), who determined the approximate equilibrium structure of axisymmetric, uniformly rotating, incompressible tori. We will refer to these uniformly rotating configurations as "Dyson-Wong tori."

Here, we summarize the work of 📚 P. S. Marcus, W. H. Press, & S. A. Teukolsky (1977, ApJ, Vol. 214, pp. 584 - 597) — hereafter, MPT77 — who constructed a sequence of toroidal-shaped, self-gravitating, incompressible configurations that are not uniformly rotating but, rather, have a distribution of angular momentum that is identical to the distribution found in a uniformly rotating, uniform-density sphere. As we have pointed out in our associated overview of "simple rotation curves", this chosen (cylindrical) radial distribution of specific angular momentum is given by the expression,

$\dot{\phi }{\varpi }^2$	$=$	$\frac{5J}{2M}\{1-[1-m(\varpi ){]}^{2/3}\},$
📚 Stoeckly (1965), §II.c, Eq. (12) 📚 Ostriker & Mark (1968), Eq. (45) 📚 Bodenheimer & Ostriker (1970), Eq. (12) 📚 Bodenheimer & Ostriker (1973), Eq. (3)

where, the mass fraction,

$m(\varpi )\equiv \frac{M_{\varpi }(\varpi )}{M},$

and $M_{\varpi }(\varpi )$ is the mass enclosed within a cylinder of radius, $\varpi$. Such equilibrium models are often referred to as $n^{\prime }=0$ configurations. Following the lead of MPT77, we will refer to each of their equilibrium configurations as a "Maclaurin Toroid."

Maclaurin Spheroid Reminder

As has been demonstrated in our accompanying discussion of the Maclaurin spheroid sequence, the (square of the) normalized angular momentum that is associated with a spheroid of eccentricity, $e\equiv (1-c^2/a^2{)}^{1/2}$, is,

$L_{*}^2\equiv \frac{L^2}{(G{M}^3\overline{a})}$	$=$	$\frac{6}{5^2}[(3-2e^2)(1-e^2{)}^{1/2}\cdot \frac{{\sin }^{-1}e}{e^3}-\frac{3(1-e^2)}{e^2}](1-e^2{)}^{-2/3}.$
📚 Marcus, Press, & Teukolsky (1977), §IVa, p. 591, Eq. (4.2)

In that same discussion, we have demonstrated that that the corresponding ratio of rotational to gravitational potential energy is given by the expression,

$\tau \equiv \frac{T_{\mathrm{r}\mathrm{o}\mathrm{t}}}{|W_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}|}$	$=$	$\frac{1}{2e^2{\sin }^{-1}e}[(3-2e^2){\sin }^{-1}e-3e(1-e^2{)}^{1/2}].$
📚 Marcus, Press, & Teukolsky (1977), §IVc, p. 594, Eq. (4.4)

Figure 4 from this accompanying discussion — reprinted here, but relabeled "Figure 1" — shows how $L_{*}$ varies with $\tau$. In an effort to conform to MPT77's presentation, our Figure 2 displays the same information as displayed in Figure 1, but the axes have been swapped and the maximum displayed value of $L_{*}$ has been extended from 1 to 3.

Figure 1            Figure 2 Maclaurin Spheroid Sequence n/a This solid black curve also appears in: Fig. 4.2 (p. 88) & Fig. 10.12 (p. 237) of [T78] This solid black curve also appears in: Fig. 5 (p. 594) of 📚 Marcus, Press, & Teukolsky (1977)

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